THE NATURE OF SOLAR PROMINENCES

ASTROPHYSICS AND
SPACE SCIENCE LIBRARY
VOLUME 199

Executive Committee
W. B. BURTON, Sterrewacht, Leiden, The Netherlands
C. DE JAGER, Foundation Space Research, Utrecht, The Netherlands
E. P. J. VAN DEN HEUVEL, Astronomical Institute, University ofAmsterdam,
The Netherlands
H. VAN DER LAAN, Astronomical Institute, University of Utrecht,
The Netherlands

Editorial Board
I. APPENZELLER, Landessternwarte Heidelberg-Konigstuhl, Germany
J. N. BAHCALL, The Institute for Advanced Study, Princeton, U.S.A.
F. BERTOLA, Universita di Padova, Italy
W. B. BURTON, Sterrewacht, Leiden, The Netherlands
J. P. CASSINELLI, University of Wisconsin, Madison, U.S.A.
C. J. CESARSKY, Centre d'Etudes de Saclay, Gif-sur-Yvette Cedex, France
C. DE JAGER, Foundation Space Research, Utrecht, The Netherlands
R. McCRAY, University of Colorado, lIlA, Boulder, U.S.A.
P. G. MURDIN, Royal Greenwich Observatory, Cambridge, U.K.
F. PACINI, Istituto Astronomia Arcetri, Firenze, Italy
V. RADHAKRISHNAN, Raman Research Institute, Bangalore, India
F. H. SHU, University of California, Berkeley, U.S.A.
B. V. SOMOV, Astronomical Institute, Moscow State University, Russia
S. TREMAINE, CITA, University of Toronto, Canada
Y. TANAKA, Institute of Space & Astronautical Science, Kanagawa, Japan
E. P. J. VAN DEN HEUVEL, Astronomical Institute, University ofAmsterdam,
The Netherlands
H. VAN DER LAAN, Astronomical Institute, University of Utrecht,
The Netherlands
N. O. WEISS, University of Cambridge, U.K.

THE NATURE OF
SOLAR PROMINENCES
by

EINAR TANDBERG-HANSSEN
NASA,
George C. Marshall Space Flight Center,
Huntsville, Alabama, U.S.A.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-4526-3
ISBN 978-94-017-3396-0 (eBook)
DOI 10.1007/978-94-017-3396-0

cover picture: Surge prominence erupting out of active region
Courtesy: Sacramento Peak Observatory,
Air Force Cambridge Research Laboratory (AFCRL)
Printed on acid-free paper
No copyright is asserted in the United States under Title 17, U.S. Code.
The U.S. Government has a royalty-free license to exercise all rights
under the copyright claimed herein for governmental purposes.
All other rights are reserved by the copyright owner.

© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.

THE NATURE OF SOLAR PROMINENCES

Les protuberances se presentent sous des aspects si bizarres et si capricieux qu'il est
absolument impossible de les decrire avec quelque exactitude.
SECCHI, Le Soleil, 1877.

TABLE OF CONTENTS

PREFACE
LIST OF SYMBOLS AND CONSTANTS
CHAPTER 1. INTRODUCTION
1.1
1.2
1.3
1.4

Outline of Book
What is a Prominence?
Historical Background
Morphological Classifications

CHAPTER 2. INTERPRETATION OF OBSERVATIONAL DATA
2.1 Spectroscopy-Atomic Physics
2.1.1 Basic Notations
2.1.2 Radiative Transfer, General Formalism
2.1.3 Radiative Transfer in Prominences
2.1.4 Continuous Radiation in Prominences
2.1.5 Line Profiles
2.1.6 Quantum Numbers and Selection Rules
2.2 Magnetohydrodynamics-Plasma Physics
2.2.1 Magnetic Field Observations
2.2.2 Elements of Magnetohydrodynamics
2.3 Waves in a Plasma
2.3.1 Acoustic Waves
2.3.2 Magnetic Waves
2.3.3 Magnetoacoustic Waves
2.3.4 Gravity Waves
2.3.5 Plasma Oscillations
2.4 Modeling-Computer Simulation

CHAPTER 3. PHYSICAL PARAMETERS OF THE PROMINENCE
PLASMA
3.1
3.2
3.3
3.4

Temperature
Spectroscopic Classifications
Density
Degree oflonization

XI
XIII
1
1
2
11
14
19

19
19
22
25
30
35
41
44
45
54
69
70
71
71
72
74
76

81
81

85

88
91

VIII

THE NATURE OF SOLAR PROMINENCES

3.5 Magnetic Field
3.6 Motions
3.6.1 Quiescent Prominences
3.6.2 Active Prominences
3.7 The Hvar Reference Model

CHAPTER 4. FORMATION OF PROMINENCES
4.1 Filament Channels and Magnetic Arcades
4.2 Photospheric Motions and Filament Formation
4.3 Condensations
4.3.1 Condensation as a Thermal Instability
4.3.2 Condensation of Prominences
4.4 Injections
4.4.1 Siphon-Type Injections
4.4.2 Diamagnetic Effects
4.4.3 Surges, Spicules, and Fibrils
4.4.4 Particle Acceleration

CHAPTER 5. PROMINENCE MODELS
5.1 The Location of Prominences
5.2 Radio Waves and Prominences
5.3 Fine Structure of the Prominence Plasma
5.3.1 Quiescent Prominences
5.3.2 Active Prominences
5.4 Early Models, Historical Notes
5.4.1 Some General Comments on Magnetohydrostatic Models
5.4.2 Field Configurations Capable of Supporting Prominences
5.5 Global Magnetohydrostatic Equilibrium
5.5.1 Support in Normal Polarity Fields
5.5.2 Support in Inverse Polarity Fields
5.5.3 The Importance of Dips in the Field Lines
5.6 Dynamic Support
5.7 Prominence Feet
5.7.1 Observations, Empirical Models
5.7.2 Physical Models
5.8 Internal Equilibrium
5.8.1 Magnetohydrostatic Equilibrium
5.8.2 Thermal Equilibrium
5.9 Fine Structure Revisited

91
95
95
105
110
113
114
121
122
123
128
142
143
148
151
158
167
167
170
172
173
182
183
184
185
191
192
198
200
210
213
214
216
219
220
223
233

TABLE OF CONTENTS

IX

CHAPTER 6. THE DEATH OF PROMINENCES

237

6.1 Comments on Active Prominences
6.2 Thermal Equilibrium Breakdown
6.2.1 Thermal Disparitions Brusques
6.3 M~gnetohydrostatic Equilibrium Breakdown
6.3.1 Prominence Stability
6.3.2 Destabilization of Prominences
6.3.3 Dynamic Disparitions Brusques
6.4 Coronal Mass Ejections
6.5 Global Magnetic Field Restructuring
6.5.1 From Local to Global Destabilization
6.5.2 Low's Model
6.5.3 The Disparition Brusque Revisited

237
238
240
242
244
249
258
264
267
267
269
269

REFERENCES

275

AUTHOR INDEX

297

SUBJECT INDEX

305

PREFACE

"He is beautiful and radiant
with great splendor..."
St. Francis,
from Cantico del sole
Two decades have elapsed since the publication of Solar Prominences, 20 years
that have seen a nearly phenomenal increase in the interest, as well as the information, concerning these fascinating and beautiful manifestations of solar activity. During this period many meetings have been held, and several books and
proceedings have been published, all dealing with specific aspects of solar
prominences. However, no unifying and comprehensive accord has appeared.
Recently some of my colleagues suggested that the time was ripe for a new
addition of Solar Prominences, and Kluwer Academic Publishers wanted to publish such a book. I, therefore, venture to present this monograph in the hope of
kindling the interest of some graduate students in the study of this-probably
the most spectacular and often the most beautiful of solar activity manifestation.
However, since it is the physical processes behind these events that will particularly interest us, I also hope the book may be of help to some of my colleagues.
In a rapidly developing field of science it is difficult, if not impossible, to
present an overview that is up to date in every respect. I have made nearly every
effort to include the latest contributions in the broad area of prominence
research, but I am sure I have overlooked some important investigations. For
these oversights, I apologize.
I have included in the discussions many older contributions and historical
notes, both to make us remember that ours is a science with deep roots in the
past, and to realize that many of the hot ideas of today had been conceived and
discussed by our older colleagues.
It is a pleasure to acknowledge the valuable help of S. Burrer and S. Morris in
typing the manuscript and of the personel in MSFC's Graphics Branch for their
competent art work. Special thanks are due to T. Moorehead for her careful
checking of the manuscript and planning its layout. Her professional touch has
been indispensable.
I have benefited greatly from interactions with many of my colleagues, either
through discussions or by being provided with preprints, illustrations, or good
advice, and I offer them my grateful thanks: M. Bruner, C.-C. Cheng, F. Chiuderi
Drago, M. Dryer, O. Engvold, T. G. Forbes, V. Gaizauskas, P. Heinzel, T.
Hirayama, A. Hundhausen, J.-L. Leroy, B. C. Low, J. M. Malherbe, S. F. Martin, P.
Mein, Z. Mouradian, J.-C. Noens, B. Rompolt, D. Rust, S. T. Wu, and J. B. Zirker.
In particular, I should like to express my appreciation to B. Schmieder who read
early versions of Chapters 2 and 3 and offered many valuable comments, and to
R. Moore for reading the entire manuscript and suggesting numerous improvements.
Huntsville, Alabama, August 1994

LIST OF SYMBOLS AND CONSTANTS

Symbols
A
b

B
c
C

d

D
e
E

f

F

g

G
h
H
i
I
j

J
k

K

I
L

m
M
n
N
p

P
q

Q

r

R
s
S
t

Einstein coefficient; area; amplitude
parameter expressing deviation from local thermodynamic equilibrium
Einstein coefficient; magnetic field; Planck function
velocity of light; specific heat
collisional rate coefficient
distance
distance; prominence thickness
base of natural logarithm; electronic charge
electric field; energy
oscillator strength; distribution function; filling factor
flux function
gravitational acceleration; statistical weight; Gaunt factor; Lande'sg-factor
constant of gravitation; energy gain function
Planck's constant; height
scale height
imaginary unit (...t=i)
Stokes parameter; specific intensity; electric current
current density; emission coefficient; total angular momentum quantum
number
mean intensity; action variable
Boltzmann's constant; absorption coefficient; wave number
degree absolute; thermal conductivity; force
characteristic length; angular momentum quantum number
characteristic length; orbital angular momentum quantum number; energy
loss function
mass
mass; magnetic quantum number
principal quantum number; number density of particles; vector normal
number of particles in column of cross section 1 cm2
pressure; momentum
rate coefficient (P =R + C); probability; power; period =2ft/ CO; degree of
polarization
heat flow
Stokes parameter; energy; cross section
cylindrical polar coordinate; distance (radial); ratio of continuum absorption
coefficient to line absorption coefficient
gas constant; radiative rate coefficient; Reynolds number
entropy; coordinate along field line; spin quantum number
spin quantum number; source function; Poynting number
time

XIV

THE NATURE OF SOLAR PROMINENCES

T
u

temperature
velocity
U partition function; heat energy; Stokes parameter
v
flow velocity (macroscopic)
V gross velocity; wave velocity; volume; Stokes parameter
w particle velocity (microscopic)
W energy
x cartesian coordinate
X prominence thickness
y
cartesian coordinate
z
cartesian and cylindrical polar coordinate
Z atomic number

a

filamentary structure ratio; absorption coefficient per atom; angle
ratio between gas pressure and magnetic pressure
y
ratio of specific heats; damping constant; angle; filamentary degree; Euler's
constant
l)
delta function; angle
1M.. halfwidth of spectral line
£
emissivity; dielectric constant; energy density
£'
complex dielectric constant (£' =£ - i(4 1tCJ/
11 coefficient of viscosity
e angle
1C
continuum quantum number
A wavelength; mean free path
AB Larmor radius
f.1
magnetic moment; mean molecular weight
v frequency
~
microturbulent velocity
p density; net radiative bracket
(J
Stefan's constant; electrical conductivity
t
optical depth; diffusion time
cp cylindrical polar coordinate; latitude; phase; absorption profile; scalar
potential; angle variable
ell angle
X electric potential
00
circular frequency; angular velocity
.a solid angle
~

(0»

LIST OF SYMBOLS

xv

Physical Constants
Boltzmann's constant
Electron rest mass
Elementary charge
Gravitational constant
Planck's constant
Proton rest mass
Stefan-Boltzmann constant
Velocity of light

k

me
e
G
h
mp
CJ

c

=1.381 X 10-16 erg deg-1
=9.109 x 10-28 g
=4.803 x 10-10 esu

=1.602 x 10-19 coulomb
=6.673 x 10-8 dyn em2 g-2
=6.626 X 10-27 erg s
=1.673 x 10-24 g
=5.669 x 10-5 erg em-2 s-l deg-4

=2.998 X 1010 cm s-l =1.8 x 1012 furlongs
fortnighr-l

Astronomical Constants
Astronomical unit
Solar mass
Solar radius
Solar luminosity
Gravity at solar surface
Solar effective temperature

=

AU 1.496 x 1013 cm
Mo =1.989 x 1033 g
Ro =6.960 X 1010 em
LO = 3.826 x 1033 erg s-l
go = 2.74 x 104 cm s-2
Teff =5800K

Commonly Used Acronyms
HAO
ISAS
NASA
NAOJ
SPO

High Altitude Observatory
Institute of Space and Astronautical Sciences
National Aeronautics and Space Administration
National Astronomical Observatory Japan
Sacramento Peak Observatory

CHAPTER 1

INTRODUCTION

1.1. Outline of Book
While the aim of this book is both to describe the different aspects of prominences and to understand the physics behind the various manifestations, the
emphasis will be on the latter. Consequently, we shall devote considerable
space to the mathematical and physical background necessary to interpret the
often confusing observational information gathered by a great variety of instruments.
In this first chapter we shall first try to define what we mean by a prominence. While there is little doubt that most solar scientists will agree that a
certain observation refers to a quiescent prominence, the identification becomes
more tricky when we look at pictures of sunspot-related filaments or loops and
arches in the corona (Section 1.2). Several attempts have been made to classify
prominences according to their morphology, and since this approach furnishes a
convenient framework for further discussion, we shall present morphological
classifications in Section 1.4.
Our understanding of the nature of prominences has changed considerably,
even dramatically in some aspects, over the last 20 years or so. Nevertheless, it
is both illuminating and useful to consider the historical development of this
branch of solar physics, and in Section 1.3 we shall touch on certain highlights.
The mathematical and physical background, necessary to interpret the
great variety of available observations, is presented in Chapter 2. Much of the
observational data pertains to spectra of prominences, and the bas.ics of spectroscopy can be found in Section 2.1. Similarly, much of the theoretical work on
prominences relies on our understanding of the interaction of the prominence
plasma with the magnetic fields present, and in Section 2.2 we treat magnetic
field observations and the magnetohydromagnetic background.
Finally, the last 20 years have seen a phenomenal growth in the use of
computer simulations to try to understand different prominence models. While
it is not possible to cover all aspects of these computer simulations, we shall
benefit from many of them, and Section 2.3 will introduce this fairly recent
addition to the arsenal of prominence-research tools.

2

CHAPTER 1

Chapter 3 summarizes our present knowledge regarding the values of the
relevant parameters that best describe the prominence plasma, including
temperatures, densities, magnetic fields, etc.
Equipped with a basic knowledge of what we mean by prominences and
with an adequate background in mathematics and physics, we enter in Chapter
4 the real realm of prominence physics: how do they form? The two main processes thought active in this phase of a prominence's life, i.e., condensation and
injection, are treated in Sections 4.3 and 4.4.
Once formed, prominences take on either a stable or an unstable nature.
Chapter 5 develops the concepts of magnetohydrostatic and thermal equilibria, and presents models for quiescent and active prominences using the information on equilibrium conditions.
The story of prominences ends with their disappearance, treated in Chapter 6. Of special interest is the disparition brusque phenomenon and the connection with coronal disturbances.
While many of the basic ideas concerning the nature of prominences were
formulated in the 1950's and 1960's, it took the more sophisticated treatments
of the last 20 years to develop these ideas and put them on a reasonably solid
mathematical foundation. A good example is furnished by the study of the role
of magnetic fields in prominences. One knew in the 1950' s that the field was a
necessary component of a prominence and the first one-dimensional models were
presented. However, one had to await the 1980's before treatments of more
realistic three-dimensional field configurations became available.
Similarly, the study of prominence spectra in the post-World War II years
made rough temperature and density determinations possible, but it took the
extensive treatment of many-level atoms and new insight into the processes involved in radiative transfer to explain many line profiles and deduce the more
accurate thermodynamic properties of the prominence plasma.
Hand in hand with these theoretical improvements went the development
of vastly improved high resolution observations of prominences. It is imperative that such observations be used as guidelines, as well as constraints, in
building and evaluating theoretical models.
When we discuss the models that describe the formation and stability of
prominences, we shall see how recent, more sophisticated models generally
build on previous investigations, thereby giving a sense of the historical development of this field of research.
1.2. What is a Prominence?
The term prominence is used to describe a variety of objects, ranging from relatively stable structures with lifetimes of many months, to transient phenomena
that last but hours, or less. It is not easy to give a short, concise definition of

INTRODUCTION

3

prominences that encompasses the necessary and sufficient criteria, one difficulty being to distinguish them from some flares. When seen projected against
the solar disk, nearly all prominences show up in absorption, looking dark
against the bright disk, while flares show in emission, being brighter than the
normal, quiescent disk. However, there are active, short-lived prominences
that show up in emission, at least during certain stages of their development.
When observed above the solar limb, prominences invariably show up bright
against the dark sky background, as do flares. An often-used definition of a
prominence is an object in the chromosphere or corona that is denser and cooler
than its surroundings.
We shall refer to the long-lived (days to months), only slowly changing
prominences seen, e.g., in Ha away from active regions with sunspots, as
quiescent prominences (see Figure 1.1). When these are seen in absorption
against the disk, they are often referred to as filaments. Quiescent prominences
are long, sheet-like structures, nearly vertical to the solar surface, with typical
dimensions: length 60,000-600,000 km, height 15,000-100,000 km, and thickness
5000-15,000 km. They consist of a series of arches with feet anchored in the
photosphere (see Figure 1.2). In and around active regions different kinds of
more short-lived, rapidly changing prominences occur. We refer to these as
active region prominences, or simply active prominences. On the disk, their
appearance in Ha is not unlike quiescent filaments, but they are generally
smaller and curve around in the active region (see Figure 1.1). There are
intermediate types, prominences that do not fall easily into either the
definitely quiescent or the definitely active class. We shall return to these
below.
With the above discussion we have come up with a simple, crude definition
for most prominences, and they fall into one of two major classes: quiescent and
active. That has turned out to be a convenient operational definition or description, even though we shall soon find out that we need to further subdivide, particularly the latter class. Also, it is interesting to note that we really have not
made much progress since already Secchi (1875) described prominences as quiescent or active objects (he called the latter eruptive).
A useful subdivision of active region prominences leads us to distinguish between:
a. Plage filaments, relatively stable prominences which are found above
the inversion line of the magnetic field polarity in active regions or at the
border of active regions. They are predecessors of quiescent prominences and
usually have phases of activity;
b. Active prominences like surges, sprays, and loops which exhibit fast
changes and often violent motions. Some are related to sunspots or associated
with flares-and may even be confused with flares. However, we need at this

4

CHAPTER 1

Fig. 1.1. Spectroheliogram in Ha taken 16 March 1990 showing three large quiescent prominences on
western hemisphere and several active region prominences (courtesy Observatoire de Paris-Meudon
and P. Mein).

stage to keep in mind that so far as the different types of prominences are concerned, there is no universally adopted definition, or even description, for many
of them. It is easy to run into one of two difficulties: either too coarse a classification is given, so that important variations between different types of prominences cannot be utilized, or one devises so fine a classification that the physics
is drowned in insignificant observational details. To read the many attempted
defining descriptions of different prominence manifestations is a walk through
a morass of details that often seems irrelevant to the underlying physical
cause. We shall walk some of this path to get familiar with commonly-used
nomenclature, keeping in mind that many of the descriptions have mostly

INTRODUCTION

5

Fig. 1.2. Quiescent prominence seen in Ha above the solar limb, showing predominantly vertical fine
structure and feet connecting to the chromosphere (courtesy SPO, AFCRL).

historical interest. First, however, let us define some commonly used terms from
the vocabulary of solar activity.
Photospheric faculae are areas seen on white-light photographs of the Sun
that are brighter than the surrounding photosphere.
Chromospheric faculae are areas on photographs taken in the nearly
monochromatic light of a strong spectral line, like Ho. or the Ca II, K-line, that
are brighter than the surrounding chromosphere. The French notation is plage
faculaire, and the term plage will be adopted as an alternative name for a
chromospheric facula. There is a continuous transition as one goes out in the
atmosphere from the deep-lying photospheric faculae to the plages. The two
types resemble each other in occurrence, shape, and their relation to sunspots.
Sunspots nearly always occur in pre-existing faculae.

6

CHAPTER 1

1737 U.T.

1752 U.T.

1807 U.T.

1823 U.T.

1837 U.T.

1847 U.T.

Fig. 1.3. Development of surge prominence, 12 June 1946, Climax Station of the HAO, Boulder,
Colorado.

INTRODUCTION

7

Fig. 1.4. Surge prominence erupting out of active region (courtesy SPO, AFCRL).

Active regions are areas of the Sun's atmosphere where excess magnetic flux
is found. The magnetic field causes a local heating of the atmosphere, and this
is observed as a plage. Above the optical plage other signs of activity are often
observed. The corona is hotter and dense!:, (coronal enhancements and condensations), leading to excess emission of forbidden coronal lines, radio waves, Xrays, and particles.
Surges are prominences that seem to be shot out of active regions as long
straight or curved columns, and return along the same trajectory (see Figures 1.3
and 1.4). They may reach to great heights (several hundreds of thousands of
km) and their velocities may exceed several hundred km s·l. Some active
regions produce nearly identical surges during part of their lives (homologous
surges).
Sprays are shot out from flare regions at velocities often exceeding the
velocity of escape. The ejection is so violent that the matter is not contained, as
in surges, but may fly out in fragments.

8

CHAPTER 1

Coronal clouds are irregular objects of cool material suspended in the corona
with matter streaming out of them into nearby active regions along curved
trajectories. The coronal clouds last for a day or more at heights of several tens
of thousands of km (see Figure 1.5).
Loops, as the name suggests, are prominences that have a loop structure and
their feet are anchored in or near sunspots. They occur as the result of a major
flare. Material is generally seen to stream down the two legs of the loop. A loop
system is the manifestation of the highest degree of activity observed
optically in the solar atmosphere. At the tops of such loops the corona is very
hot and condensed into a coronal condensation. Loop arcades are seen when
several loops occur in a configuration of more or less parallel loops (see Figure
1.6).

Fig. 1.5. Coronal cloud observed in Ha, 22 February 1989 (courtesy Observatoire de Pic-du-Midi and

J.-C. Noens).

9

INTRODUCTION

Coronal rain is closely related to loops, but the complete loop structure is
absent, giving the phenomenon its descriptive name.
Of the more commonly used descriptive names for prominences, based on
their shapes, we mention:
Fibrils are long, thin dark threads visible in Ha on the disk at the edge of
plages.

0820 UT

1991.06.15

Fig. 1.6. Composite Ha picture showing loop prominence system in the corona above the solar limb
(at 1213 un and the flares near the loops' feet in active region with sunspots (at 0820 un, 15 June
1991 (courtesy Astronomical Observatory, Wroclaw, Poland and B. Rompolt).

10

CHAPTER 1

Fig. 1.7. Development of active prominences observed in Ha on 30 July 1990: (a) at 1400 UT, (b) at
1644 UT. Notice eruption of prominence on southeast limb and appearance of surge at east limb

(courtesy Observatoire du Pic-du-Midi and J.-C. Noens).

INTRODUCTION

11

Caps are seen in emission above the limb as bright, low-lying objects near
active regions. Surges frequently are ejected from the edges of caps. They may be
the limb manifestation of fibrils and plage filaments.
Hedgerow, Tree, and Tree-Trunk are self-explanatory terms describing the
main types of quiescent prominences.
Eruptive prominences are quiescent prominences or active filaments that
become unstable, erupt, and disappear. The French term for this phenomenon is
disparition brusque, and we shall refer to this phase in the life of a quiescent
prominence by its French name. Generally, the prominence reforms in the same
place.
The display of prominence activity can be quite dramatic. In Figure 1.7, we
see the development over 2-3 hours of different kinds of activity. Note the
surge appearance on the east limb and the ascending part of the large
prominence in the southeast.
1.3. Historical Background
Prominences have been seen more or less accidentally for hundreds of years, and
systematic observations go back about 125 years. The first recordings of this
strange phenomenon took place during total solar eclipses, and explanations in
terms of "lunar clouds" or "hole in the moon" were invoked. In 1239 Muratori (see
Secchi, 1875) observed the corona during a total eclipse and reported "a burning
hole" in it. This burning hole in all probability was a prominence, and
Muratori's report is one of the earliest we have of this sign of solar activity.
Medieval Russian chronicles (see Vyssotsky, 1949) also mention prominences,
but the first semi-scientific deSCription of them came after the eclipse of May 2,
1733. During this event Vassenius (1733) observed three or four prominences
from Gothenburg, Sweden. He called them "red flames," and believed them to
be clouds in the lunar atmosphere. Celsius (1735) edited a report of the Swedish
observations which shows that other observers agreed with. Vassenius'
description (see also Grant, 1852). Ulloa (1779) observed what probably was an
active prominence during the eclipse of 1778, and attributed it to a hole in the
Moon.

These early observations of prominences were subsequently forgotten, and
Bailey, Airy, Struve and Schidlofscky, Arago, and others were all taken by
surprise when they rediscovered the phenomenon during the eclipse of July 8,
1842, in France and Italy. They were so baffled and amazed that hardly any
reliable account is available of what they saw. Hence, their descriptions of the
shape of prominences were so vague that they could not prevent later observers
from believing that prominences were mountains on the Sun (see Grant, 1852).
Not until the eclipse in 1851 observed in Norway and Sweden did a proper
solar interpretation emerge (see Secchi, 1875), but it was with the introduction

12

CHAPTER 1

of photography at the eclipse in Spain in 1860 and spectrography at the eclipse
in India and Malacca in 1868 that one realized that prominences were masses of
glowing gas (Secchi, 1875).
We may for convenience consider three epochs in the history of prominence
research, viz.
a. The speculative period, before 1860
b. The spectrographic period, 1860-1960
c. The polarimetric period, after 1960
The dates are approximate, and probably the only merit in presenting these
epochs lies in emphasiZing the importance of being able to analyze spectral
lines and, later, to discover the all-important electric currents and associated
magnetic fields.
In his truly remarkable book Le Solei! Father Secchi described the speculative period and led us into the next. The use of spectroscopy was such a potent
tool, that in 1875 Secchi could start his chapter on prominences with the statement ''The phenomenon of prominences is now so well known by everybody that
it may seem unnecessary to retrace the history of their discovery." The statement makes one marvel at the alleged educational level of "everybody" at the
time. It also makes one humble to realize that 120 years later this "well-known
phenomenon" still poses some of the most puzzling questions in solar activity.
Nevertheless, considerable progress has been made in our understanding of
the nature of solar prominences. Already at the eclipse in 1868 Janssen (1868)
and Lockyer (1868) realized that many of the observed emission lines were so
bright that they should be visible even without an eclipse, and since then regular observations of prominences have been carried out in that manner. By
opening the spectrograph slit one obtains a series of monochromatic images,
corresponding to the emission lines observed with a normal slit. By using this
method, first tried by Huggins (1869), one can better study the complex morphology of prominences. Figure 1.8 shows an early photograph obtained by
Janssen, the year he died, at the Observatoire de Meudon (Paris), which he
had founded.
The list of the many observations made through the first third of this
century would be too long to present here, but those observations provided the
foundation for our knowledge of the different types of prominences and their
spectra. We only mention a few. Schwarzschild (1906) published the results of
the first systematic photometric measurements of spectral lines in prominences,
using the data from the August 30, 1905, eclipse.
In the 1890's spectroheliographs became available and prominences could
be studied on the disk as absorption features (Hale and Ellerman, 1903;
Deslandres, 1910).

INTRODUCTION

13

Fig. 1.8. Photograph of prominence obtained by Janssen, 8 September 1908 (Observatoire de Meudon,
courtesy P. Mein; see text).

With Lyot's invention of the corona graph it became possible to observe
limb prominences at any time nearly as thoroughly as during eclipses (Lyot,
1936).
The actual magnetograph measurements of the longitudinal magnetic field
via the Zeeman effect in spectral lines from prominences ushered in our third
epoch, the polarimetry period (Zirin and Sevemy, 1961; Ioshpa, 1962). By then
it had already become clear that magnetic fields were likely to playa role in
the physics of prominences. Babcock and Babcock (1955) had pointed out that
quiescent prominences were to be formed along the neutral line between the two
opposite polarities in bi-polar magnetic regions. A little later Hyder (1964)
used the theory of resonance polarization and impact polarization to estimate
the longitudinal magnetic field in prominences from studies of the observed
linear polarization of prominence emission lines.
The progress that followed is due to the analysis of several sets of data
from coronagraph-magnetograph combinations especially built for prominence
research. At the High Altitude Observatory (HAO) a new corona graphmagnetograph (Lee et a1., 1965) enabled observations, via the Zeeman effect, of
limb prominences with greatly reduced scattered light and with accurate calibration, mostly in the Ha line, but also in NaI, D3 (Rust, 1966; Harvey and
Tandberg-Hanssen, 1968; Malville, 1968; Harvey, 1969; Tandberg-Hanssen,
1970).
One of the most important data sets then came from the use of the Hanle
effect and was due to the pioneering work at Pic-du-Midi Observatory, with its

14

CHAPTER 1

new instrumentation (Ratier, 1975), and at Meudon Observatory where the theoretical foundation and the innovative data analysis took place (SahalBrechot et a1., 1977; Leroy, 1979; Bommier and Sahal-Brechot, 1979; SahalBrechot, 1981; Leroy et al., 1983, 1984).
In the meantime Zeeman-effect observations were obtained at Sacramento
Peak Observatory (SPO) in collaboration with HAO using new "Stokesmeters"
(Baur et al., 1980, 1981) and interpreted theoretically (Landi Degl'Innocenti,
1982). Other Zeeman measurements were performed with a spectrally scanned
magnetograph at the Kislovodsk coronagraph (Nikolsky et al., 1982) and with
the magnetograph of Sibizmiran (Bashkirtsev and Mashnich, 1980).
The advent of the space era brought us the series of Orbiting Solar Observatories (05O's), Skylab, Solar Maximum Mission (SMM), Spacelab missions, and
the YOHKOH satellite. Instrumentation onboard several of these spacecrafts
made it possible to push prominence observations into the UV, EUV, and even
X-ray domains. We now have observations of prominences in many of the important emission lines and continua that populate these parts of the spectrum.
While the space research efforts pushed observations toward shorter
wavelengths, the radio astronomers provided us with information on the
prominence plasma from studies in the mm to dm wavelength region; e.g.,
Chiuderi Drago (1990). Disk observations of prominences in their filament
channels can be used to assess the physical conditions in the chromosphere
around these objects (Gary, 1986; Hiei et al., 1986; Kundu, 1986).
1.4. Morphological Classifications
From the preceding sections it follows that prominences may take on very different forms, their lifetimes may range from minutes to many months, and the
degree of dynamic activity varies greatly from one prominence to another. It
would seem natural to classify prominences using some of these characteristics,
and, as we have seen, Secchi (1875) already divided them into quiescent and
active prominences. Secchi further subdivided the quiescent and active prominences into subclasses (clouds, filaments, stems, plumes, horns, cyclones, flames,
jets, sheafs, spikes-see also Young, 1896). It is difficult to maintain the distinction between these subgroups, but the main two-class arrangement is of
lasting value, and we still use it as the basic classification.
There are also significant differences in the spectra of quiescent and active
prominences. How this fact can be used to furnish classification criteria will be
treated in the next chapter.
The last 30 years have given us essential, new information regarding the
magnetic field in prominences, and this has led to an important subdivision of
quiescent objects (Leroy et al., 1984). We return to this in Chapter 2.2.

INTRODUCTION

15

For many years the most widely known and used classification was that due
to Pettit (1925, 1932). He divided prominences into five classes, as shown in
Table 1.1.
Classes 1 and 2 are closely related according to Pettit; a given prominence
may for instance pass from the active to the eruptive state. Class 3 contains
some of the most dynamically active objects. The tornado prominences of class 4
seem to be very rare objects, only occasionally discussed in the literature
(Harvey, 1969). Pettit's class 5 are the classical quiescent prominences that
take the form of enormous sheets standing vertically in the solar atmosphere.
Characteristically, the sheets touch the solar surface only at certain fairly
regularly spaced intervals, which we may call the feet of the prominence (see
Figure 1.1). The dimensions of these, the largest of all prominences, are about
200,000 km in length and 50,000 km in height, but only 5000 to 10,000 km in
thickness. No prominence is really quiescent, but this type shows slower and
less pronounced changes than any other class.

Class
1

Name
Active

2

Eruptive

3

Sunspot

4

Tornado

5

Quiescent

TABLE 1.1
Pettit's classification
Description
Material seems to be streaming into nearby
active center Uike sunspots).
The whole prominence ascends with uniform
velocity (of several hundred km s-l often). The
velocity may at times suddenly increase.
These are found near sunspots and take the
shape of "water in a fountain" or loops.
A vertical spiral structure gives these
prominences the appearance of a closely wound
rope or whirling column.
Large prominence masses which show only minor
changes over periods of hours or days.

Both Hale (Hale and Ellerman, 1903) and Deslandres (1910) realized that
the dark filaments seen in absorption on the disk in their spectroheliograms
were nothing more than prominences seen against the bright photosphere.
However, further progress along these lines was hampered by two drawbacks of
the spectroheliograph, viz. (1) it takes considerable time to obtain a spectroheliogram and get ready for the next, and (2) for prominences with motions in
the line-of-sight, the Doppler effect will throw the spectral line outside the
second slit of the instrument, and the observation will be lost. When Hale
(1929) constructed the spectrohelioscope, these drawbacks were overcome. The
observations were now made visually and the line kept on the second slit by a

16

CHAPTER 1

line-shifter whose position directly gave the radial velocity. Newton (1934)
used this technique to study filaments and offered the following classifications:
Class I - Prominences that avoid the neighborhood of sunspots (but not the
whole sunspot zone). Long filaments, lasting several days.
Class II - Prominences that are associated with sunspots or plage areas.
Smaller objects, lasting minutes or hours.
Newton's class I corresponds to Pettit's quiescent prominences. All Pettit's
classes (1 through 4) are comprised in Newton's class II (active prominences).
While this classification today mostly has historical interest, Newton alluded to the importance of considering the influence of active regions, which
today we associate with magnetic field effects.
Another important consideration was emphasized by Menzel and Evans
(1953), whose classification distinguished between:
Class A - Prominences originating from above (i.e., the corona) and
Class B - Prominences originating from below.
In addition, Menzel and Evans subdivided class A and B into two subclasses,
depending on whether the prominences were associated with sunspots or not,
i.e., similar to Newton's classification. In this, they defined a class that originates from below and is not associated with sunspots: namely spicules. Whether
these objects are to be considered prominences is open to debate. We shall see
later that the distinction between elass A and elass B is fundamental in the discussion of prominence formation.
Two classifications from the 1950's emphasized the importance of motions.
In one, due to Severny (1950) and Severny and Khoklova (1953), there are three
elasses, viz.
Class I: Eruptive - Quiescent prominences becoming eruptive. Velocities
several hundred km s-1.
Class II: Electromagnetic - Prominence motions along definite curved trajectories. Velocities a few tens to a few hundred km s-1.
Class III: Irregular - Prominences with irregular random motions of individualknots.
In the other classification, due to de Jager (1959), there are two main groups:
quiescent and moving prominences. A finer subdivision followed fairly closely
Pettit's classification. de Jager distinguishes between normal quiescent prominences and high-latitude quiescent prominences. The latter ultimately form the

INTRODUCTION

17

important polar crowns. Furthermore, de Jager classifies specifically the
important types of surges and spicules, as did Menzel and Evans:
Class I - Quiescent Prominences
• Normal (low to medium latitudes)
• Polar (high latitudes)
Class II - Moving Prominences
• Active
• Eruptive
• Spot
• Surges
• Spicules
Last, we mention a classification due to Zirin (1966) who divided prominences into two sharply defined classes depending on the relation to solar
activity-in particular whether they are related to flares or not. Those that
are related to flares, class I, exhibit violent motions and are short lived.
Class I - Short Lived (associated with flares, active regions)
• Sprays
• Surges
• Loops, Coronal Rain
Class II - Long Lived, Quiescent
• Polar Cap Filaments
• Sunspot Zone Filaments
Class III - Intermediate
• Ascending Prominences
• Sunspot Filaments
We notice that Zirin subdivides quiescent prominences in the same way as
de Jager did. In addition, Zirin defines an intermediate class comprising quiescent prominences during their disparition brusque phase as well as the important dark filaments seen in active regions.
While today much of the painstaking classification efforts discussed above
mostly have historical interest, the classifications have emphasized different, important aspects that will become clues in our quest for a deeper physical
understanding of the nature of solar prominences.

CHAPTER 2

INTERPRETATION OF OBSERVATIONAL DATA

In Section 1.4 we have seen how the shapes and motions of prominences have
been used to define different classes of these objects. Many of these observations
can be obtained with fairly unsophisticated instrumentation, and their interpretation is straightforward. To go beyond this morphological information, i.e., the
shapes and overall, gross motions, requires knowledge of the prominence's spectrum, and nearly all other information on the physics of prominences comes from
more or less complex spectrographs and demands often considerable theoretical
background to be properly interpreted.
In Section 2.1 we shall look at the interpretation of spectroscopic data that
can give information on the temperature, density, and velocity of the prominence
plasma. Similarly, Section 2.2 will treat the question of how magnetic fields in
prominences are determined and how these fields interact with the prominence
plasma. We shall see that spectroscopy, with its underpinning of atomic physics,
and magnetohydrodynamics, with its discussion of transport equations, will be
the necessary tools for correct interpretation of the data.
2.1. Spectroscopy-Atomic Physics
Historically, most of our information on prominence parameters like temperature
and density comes from analyses of spectra obtained in their visible or ultraviolet
part. In spectra of prominences seen above the solar limb the lines show up in
emission. With very few exceptions lines from prominences on the disk are in absorption.
2.1.1. BASIC NOTATIONS
As radiation of frequency v travels a distance ds through the prominence plasma,
its specific intensity Iv (erg cm·2 s·l Hz·l srl) changes due to interaction with the
particles
(2.1)

CHAPTER 2

20

where £vand lev are the emission and absorption coefficients, respectively.
One commonly introduces the optical depth tv along a direction r making
angle e with ds
.
(2.2)

and writes the equation for the radiation transport in the form
(2.3)

where Il =cos e.
Under conditions of local thermodynamic equilibrium (LTE) the ratio £ vIle v'
which is called the source function Sw depends only on temperature (Kirchhoff's
law), and the source function is simply given by the Planck function Bv (1'), i.e.,

3[

2hv exp(hV)
Sv=Bv(T)=---;rkT -1

]-1 ,

(2.4)

where h is Planck's constant, k is Boltzmann's constant, e the velocity of light,
and T is the kinetic temperature.
If the intensity is expressed in wavelength, A. = elv, instead of frequency, i.e.,
h (erg cm-2 s-I sr-I), Planck's function takes the form

72hc2 [ exp(he)
AkT -1 J-

BA, (T) =

1

(2.5)

The classical Rayleigh-Jeans law

(2.6)
applies to radio waves where hv« kT and is a particularly simple form of the
Planck function.
For a discussion of the transport of continuum radiation, Equations (2.1) and
(2.4) provide the framework.
Also when we deal with weak spectral lines, the conditions may not depart
much from LTE. In the language of quantum mechanics, we can write for the
emergent intensity of an emission line

(2.7)

INTERPRETATION OF OBSERVATIONAL DATA

21

Here nj denotes the population of the upper atomic level from which the electrons jump to the lower level i affecting the emission of quanta (photons) of
energy hVji. Aji is Einstein's coefficient for spontaneous emission and V the
volume of plasma radiating the spectral line Vji. The crucial task is to evaluate the
population nj' which depends on four factors, viz. (1) the ratio of nj to the total
number of atoms in that same state of ionization, njon; (2) the ionization equilibrium njon/nEL, where nEL is the abundance of the element relative to hydrogen;
(3) the abundance of the element relative to hydrogen nEUnH, and the hydrogen
number density, nH. Consequently, for njwe can write
(2.8)

To proceed we recall that the different energy states in atoms and ions are
populated according to Boltzmann's law

(X" ),

n· g. exp __JI_
-L=-L
nj

(2.9)

kTex

gj

where 8j and 8i are the statistical weights of the two levels in question, Xji the excitation potential, and Tex the excitation temperature parameter. Furthermore, the
distribution of ions on the different ionization stages is given by Saha's equation
(Saha, 1920)
nion+l
- ne
nion

Uion+
=(21dcT)312
-;:r-1 exp(Zion)
--- ,
h

Uion

kTion

(2.10)

where ne is the electron density, lion the ionization potential for ionization from
stage ion to the next stage ion + 1, and Tion is a temperature parameter associated
with this process. The partition functions U for the two stages are given by expression of the form
Uj

=Igi
exp
j

(_-XL).
kTion

(2.11)

If the prominence plasma is in LTE, and only then, we shall have

Tex

=Tion= T,

(2.12)

22

CHAPTER 2

and Equations (2.9) and (2.10) will provide the value of njin Equation (2.8), and
the intensity of the line, Equation (2.7), can be evaluated.
2.1.2. RADIATIVE TRANSFER, GENERAL FORMAUSM
When we deal with strong spectral lines that often provide the best information
on the prominence plasma, the source function may not simply be given by the
Planck function, but may depart significantly from it. Under such departures
from LTE, Equation (2.12) does not hold, and the temperature concept becomes
very complicated. The source function will more closely resemble the intensity
averaged over directions (Chandrasekhar, 1960), i.e.,
J v =_1 fI dO,

41r

(2.13)

v

where dO shows integration of the solid angle.
In these cases, we replace the condition of LTE with the assumption that
there is a statistically steady state in the population of each atomic energy level.
We shall include both radiative and collisionally induced transitions between the
states, a treatment that is a generalization of the principle of reversibility, already
discussed by Rosseland (1926) and developed by Thomas (1948a,b) and others. If
we let 1C denote the continuum, Thomas (1957) showed that a general form of the
source function could be written as
_ 2hvjj 3(njgj
sJI.. -c2- -n .g.- 1)-1 -'" v
J

I

(2.14)

lit.
'
." v

where cIIv and 'l'v are the profiles of the absorption and spontaneous emission coefficients, respectively. To evaluate the source function we must determine the
ratio of the populations nJnj. This is done via the equation for statistical equilibrium:
dn.

_J

dt

=.~.'
~ (n.
Ir

I=I,I,tJ

p ..
IJ

-noJJI
P.. )=O

j

=1,2...

1(.

(2.15)

Pij is the total rate for transitions from level i to level j and consists of two terms:

(2.16)

23

INTERPRETATION OF OBSERVATIONAL DATA

where Rij is the radiative transition rate and Cij refers to the collisionally induced
transitions. For j > i, Rji is the Einstein coefficient for spontaneous emission, Aji,
plus the transition rate due to induced (stimulated) emission, Bji I JvWvd", with
Bji being the Einstein coefficient for induced emission, and J", the mean intensity,
being given by Equation (2.13). It is convenient to introduce a mean intensity, J,
weighted by the emission coefficient, i.e.,

whence
(2.17)
The Einstein coefficients satisfy the equations
(2.18)
Notice that in most discussion the profiles, which are normalized to unity, i.e.,

I 'l'vdv = I iPvdv = 1,
are assumed equal, Wv =cjlv·
To take care of transitions to and from the continuum; i.e., photo-ionization
and photo-recombination, we write for the rates to the continuum, indicated by
the subscript 1C
(2.19)
where O'vis the photo-ionization cross section. The rates from the continuum, the
photo-recombination rates Rid, are calculated by balancing ionization and recombination under LTE conditions which are assumed to hold in the continuum.
Indicating LTE conditions with an asterisk, we can write
(2.20)
where B(Te) is the Planckian radiation field at the electron temperature, Te; see
Equation (2.4).
We now need to consider the other term in Equation (2.16), the collisionally
induced transitions that help populate and depopulate both bound levels i and j

24

CHAPTER 2

as well as the continuum K. Let Qij be the cross section for collisional excitation,
we then can write
(2.21)
where Va is the threshold velocity of the electron with a velocity distributionj(v).
For collisional ionization we similarly have a cross section QilC' Collisional deexcitation rates are formed by balancing them against excitation rate, assuming
LTE conditions, i.e.,
(2.22)
and the ratio of the populations is determined by Boltzmann's law.
With all the appropriate rates in Equation (2.16) calculated, we can now go
back to Equation (2.15) and from its algebraic solution in the form

!!L = pi~.,
n.
J

pJI

(2.23)

where pij is the co-factor of the element Pij in the matrix of the coefficients of
Equation (2.16) (Rosseland, 1936; White, 1961), we can find the population ratio
that goes into Equation (2.14) to determine the source function Sji' However,
several of the radiative transition rates depend on the radiation field ly, whence
also the ratio ni/nj will depend on Iv. Furthermore, the mean intensity Iv is
coupled to the source function (see Equation (2.3», which means that the equations involved are strongly coupled. The problem has been solved by using a
form for the transfer Equation (2.3) appropriate for lines. By integrating Equation
(2.3) over all directions J.l. we obtain
_1 dF v =2(J -S),
2n d-r v
v
v

where

is the radiation flux (erg cm-2 s-l Hz-I). With the approximation

INTERPRETATION OF OBSERVATIONAL DATA

25

and having again integrated Equation (2.3) over all directions after multiplication
we find

by~.

When we eliminate Fv between the two equations, we obtain the transfer equation in the Eddington approximation
(2.24)
In this equation 'tv is now the total optical depth, i.e., line plus continuum, and
the source function depends on both the line source function Sji and the continuum source function SIC' the latter being taken equal to the Planck function,
Equation (2.4). We define the total source function by the expression
E;:.,."
ji _+_
Sv =_Et.:...

1(ji + 1("

It is convenient to introduce the ratio rv = lelC /leji and to write for the source func-

tion

(2.25)

Jefferies and Thomas (1958) showed that one can solve Equations (2.24) and
(2.25) together, and obtain an analytic solution of the resulting integrodifferential
equation for J v in terms of a Gaussian quadrature over frequency.
2.1.3. RADIATIVE TRANSFER IN PROMINENCES
When the populations of atomic levels are accomplished by collisional processes,
the assumption of LTE reigning in the plasma is good. This condition may not be
applicable to prominences which are irradiated from the hot corona as well as
from the transition region, the chromosphere, and the photosphere. Under such

26

CHAPTER 2

conditions, the ratio of radiatively induced transitions to collisionally induced
transitions is large, and the plasma may be far removed from LTE. The source
function is then given by Equation (2.14).
One of the goals of prominence models is to predict the intensity and shape
of spectral lines, which is done by adjusting the different parameters to fit the
predicted line to the one observed in a prominence. This procedure implies the
use of a model of the atom or ion in question. Early work often used to treat the
atom as having two energy levels plus a continuum. As sophisticated computer
programs became available, and more and more transitions rates were calculated, model atoms with many levels could then predict the strength and shape,
many not only of a resonance line, but of additional lines associated with higher
levels in the atom; see, e.g., Gouttebroze et al. (1993).
Intimately linked to the evaluation of spectral lines in prominences is the
question of radiative losses from prominences. As we shall see in Chapter 4, one
of the ways prominences may form is by condensation from the corona. This
mechanism requires the plasma to cool down to prominence temperatures which
may be done by radiating away energy. The radiative losses in prominences are
dominated by hydrogen (Zhang and Fang, 1987), and using a sufficiently sophisticated hydrogen model, one can evaluate the losses and compute the relevant
line profiles. We note that the loss even in relatively strong lines from Ca II is
negligible in comparison with the loss in hydrogen lines (Fang et al., 1990).
In very tenuous plasmas the radiative losses in spectral lines may be evaluated by considering the lines to be optically thin; i.e., t « 1 (Equation (2.2».
However, in most prominences, the stronger hydrogen lines, as well as helium
lines, become optically thick at temperatures below 40,000 K, and the optically
thin approximation overestimates the losses (Kuin and Poland, 1991). Optical
depth effects, therefore, must be included in all diagnostic work on prominence
spectra.
In addition to difficulties arising from large opacity in prominences, a
number of other phenomena complicate radiative transfer in this part of the solar
atmosphere. We mention first the effect that bulk mass motion of the prominence
can have. The brightness of a spectral line may depend on the velocity of the
plasma relative to the solar surface. These brightness changes are induced by
velocity-dependent variations of the populations of atomic levels and are
referred to as Doppler brightening and Doppler dimming. The Doppler brightening
of a certain line occurs when the radiation from the photosphere and chromosphere, incident on the prominence, in the wavelength range of the line's
absorption profile (<<Pv in Equation (2.14» increases due to the relative motion between the scattering prominence and the solar surface. The radiative excitation
rate for the line transition will then increase. On the other hand, Doppler
dimming takes place when the intensity of radiation over the absorption profile
decreases due to prominence motion. A study of line profiles affected by Doppler

INTERPRETATION OF OBSERVATIONAL DATA

27

brightening or Doppler dimming may consequently be used as a diagnostic for
prominence velocity (Heinzel and Rompolt, 1987).
Another effect of importance for the interpretation of spectral lines from
prominences is related to the way the atoms scatter incoming radiation. In LTE
an absorption at frequency v is balanced by a re-emission at the same frequency
v, and the absorption coefficient cpvand the emission coefficient 'IIvare equal.
Also, if there is complete reshuffling of atoms in their excited level such that
there is no correlation between frequencies of incoming and scattered photons,
we shall again have CPv = 'IIv- This situation is referred to as complete (frequency)
redistribution (CRD), or complete noncoherence; see, e.g., Athay (1972) or Mihalas
(1978) for indepth discussions. Complications arise if one abandons the concept
of CRD and treats prominence lines in the more realistic case of partial redistribution (PRO). Line profiles of strong hydrogen lines like Lyex calculated using PRO
differ markedly from lines calculated using CRD (Heinzel et aI., 1987; Paletou et
aI., 1992); see Figure 2.1a. The PDR case more closely matches observed profiles
(see Figure 2.1b) that refer to observations of the Lyex radiation from active region
events seen above the solar limb (Fontenla et aI., 1989). In a further development
Paletou et al. (1993) used improved computational codes to treat PRO scattering
and studied the formation of resonance lines, not only of hydrogen, but of the
strong Mg II and Ca II lines, for which they found good agreement with
observations.
A third effect that profoundly influences the radiative transfer in prominences stems from the fact that prominences are made up of fine-structure
elements (FSE); they have a certain degree of porosity. Early work on radiative
transfer approximated the prominence with a one-dimensional slab model. This
fairly standard but crude one-dimensional slab geometry nevertheless provided
interesting insight into many aspects relating to large quiescent prominences.
However, it fails to account for a number of important characteristics that are
vital to a proper interpretation of prominence spectra. In fact, it is not possible to
reproduce observed profiles of, e.g., resonance lines of hydrogen without taking
into account the prominence porosity (e.g., Heinzel, 1989). Such a filamentary
structure allows radiation from outside to penetrate deeper into the prominence
plasma than if the prominence merely consisted of a slab of material.
The formation of hydrogen lines in filamentary prominences has been
studied by Morozhenko (1978), Fontenla and Rovira (1983, 1985), Zharkova
(1984, 1989), Heinzel (1989), Vial et al. (1989), and others, while Orrall and
Schmahl (1980) considered the transfer of Lyman continuum radiation in inhomogeneous prominence plasmas. Both Morozhenko and Zharkova were able
to account for the mutual radiative interaction between the individual finestructure elements (FSEs), but they could not account for the penetration of the
external and diffusive radiation between the FSEs in the body of the prominence.
On the other hand, Fontenla and Rovira studied the non-LTE problem associated
with one representative slab to account for the FSE interface using an energy

28

CHAPTER 2

equation where radiative losses balance thermal conduction. They then added
several FSEs and integrated the transfer equation through all slabs to obtain line
profiles of the emergent radiation. However, Fontenla and Rovira could not treat
the mutual interaction between the different elements.

4.20,..-----------------------,
PRD

3.60

CRD

3.00

"C-

~

2.40

:::::!.

1.80

,,

1.20
0.60

-1.00

-0.75

-0.50

-0.25

0.00

0.25

,

'-

0.50

....

--- -- --.
0.75

1.00

Wavelength (A)

Fig. 2.1a. Lya profile computed with complete frequency distribution (CRD) and with partial
redistribution (PRO) (after Heinzel et aI., 1987).

The porosity of a prominence was defined by Morozhenko (1978) in terms of
a filamentary degree, "I, given by "I =i'/(i + i'), where i is the thickness of a filament (i.e., the FSE, often referred to as a thread) and i' is the distance between
filaments. Note that the filamentary degree is the fraction of the volume of the
prominence free of filaments, and should not be confused with the filling factor
which is more commonly used to describe the porosity and which gives the
fraction of the volume filled with fine-structure elements.
Heinzel (1989) built on the work of Fontenla and Rovira (1985), used partial
frequency redistribution-instead of complete redistribution, which also was applied by Zharkova-and devised an iterative boundary condition (ISC) method
to obtain the line profiles from a prominence irradiated by a field with incident
intensity of the form

29

INTERPRETATION OF OBSERVATIONAL DATA

where Ie is the mean intensity of radiation emerging from an individual FSE, and
Is is the mean intensity of the diluted solar radiation. Figure 2.2 shows line profiles obtained by Heinzel for different values of the prominence porosity, expressed by a filling factor a.. The case a. -+ 0 means a very sparse prominence so
that the few threads are irradiated mainly by Is. If a. = 1 we have a homogeneous
prominence (y= 0).
400000

r----.--,.---r-....---.--~~--.-__,-....,._-~_,

300000

>-

·enc:

Q)

200000

c:

100000

oL-~~~

-1.2

-0.8

__L-~-L~__L-~~~~
-0.4

0.0

0.4

0.8

1.2

Delta Wavelength
Fig. 2.1b. Lya profile observed above active solar region (after Fontenla et al., 1989).

Since prominences are made up of fine-structure elements, one expects
strong temperature gradients to exist, as one expects similar gradients associated
with the chromosphere-corona transition region. Under such conditions ambipolar
diffusion may be important. This process results in neutral atoms penetrating into
hot regions, while ions conversely invade the cooler parts of the plasma. As a
consequence, the ionization of hydrogen is changed significantly in the lower
parts of the chromosphere-corona transition region (Fontenla et al., 1990). Vial et
al. (1989) applied the effect of ambipolar diffusion to the fine structure of
prominences and compared the density and temperature structures of the
threads with and without ambipolar diffusion; see Figure 2.3. The presence of the
diffusion leads to less steep gradients and therefore to an increase in hydrogen at

30

CHAPTER 2

2x10-8 , - - - - - - - - - - - - - - - - ,

1.5x10-8

c

~

1x10-8

-

0.4

0.8

1.2

Wavelength (A)
Fig. 2.2. Computed Lya profiles for models with different values of the prominence porosity a. Curve
1: a 0; Curve 2: a O.S (after Heinzel, 1989).

=

=

lower temperatures (=2 x 104 K), which again means increased intensities in the
hydrogen lines. Vial et al. (1989) further solved the radiative transfer equations
for different thread models, including partial frequency redistribution and
ambipolar diffusion and reasonably predicted Lya and Ly~ lines, provided a
large number of threads (several hundred) were present. This number is much
higher than values reported by Zirker and Koutchmy (1989) and Mein et al.
(1989a) from Ha studies of prominences. We shall return to models of prominence fine structure in Chapter 5.
2.1.4.

CONTINUOUS RADIATION IN PROMINENCES

The emission lines observed in spectra of prominences above the limb are often
superimposed on an observabl~ continuous emission. This emission is made up
of contributions from hydrogen continua, i.e., free-bound transition or photorecombination and, in particular, from photospheric radiation scattered by the
electrons in the prominence plasma. Similarly, it is this electron-scattered photospheric radiation that produces the K-corona, the observation of which leads to a
determination of the electron density of the coronal plasma. In addition to the

31

INTERPRETATION OF OBSERVATIONAL DATA

scattered radiation and the free-bound hydrogen continua, optically thin radiation from the prominence plasma itself, i.e., a free-free transition contribution,
may be observably present. This Bremsstrahlung is often conspicuous in flares, but
probably is only of academic interest in all but the most active prominences like
loops and some surges.
120,000 r - - - r - - - - - , ; - - - - . - - - . - - - . , - - - - ,

100,000

T

80,000

",.,

60,000

40,000
20,000

,,

,
,,
,
,,

,
,,

,, . '

,

. , ..

,

.. " ..

.,'

,
,,

OL---~--~--~--~--~--~

o

20

40

60

80

100

120

km
Fig. 2.3. Computed temperature profile in fine-structure element. Solid curve with ambipolar
diffusion, dotted curve without ambipolar diffusion (after Vial et al., 1989).

In the relatively low density of a prominence plasma the scattered radiation
is directly proportional to the scattering electrons. The electric field E' in the radiation generated at large distances r from the accelerated scattering electron is
(Billings, 1966)
(2.26)

where a is the angle that the direction of acceleration, dv/ds, makes with the line
of sight to the electron. Let E denote the electric field that accelerates the electron;
i.e.,
dv
e2 Esina
eE=me - . then Es =
2
ds
me rc

32

CHAPTER 2

As E oscillates in the wave, so will E' and the expression (E s / E)2 will give the
ratio of the power in the plane-polarized scattered radiation to the power in the
incident wave. The differential cross section der for scattering into the solid angle
dO is

d(1) -_r2(E8)2 -_e4sin2 24 a •
(dfJ
E
pol

(2.27)

mec

We now integrate Equation (2.27) over the sphere and find
(2.28)

The cross section ere is called the Thomson cross section, and determines the important electron scattering in many parts of the solar atmosphere, including
prominences. Since Equation (2.28) is derived classically, it does not hold for very
high frequencies, i.e., when the photon energy hv approaches mc in which case
quantum effects must be incorporated.
While Equation (2.27) pertains to polarized radiation, we find the equivalent
expression for unpolarized radiation by regarding the latter as a superposition of
two orthogonal, linearly polarized components, and derive

2,

dC1
(
- )
dD

unpol

1 e4
. 2
=-24'(l+sm
2 me C

a).

(2.29)

Instead of using the angle (X between the direction of acceleration and the line
of sight, we choose to introduce the angle 9 between the incident and scattered
radiations, and find

d(1)
(-dD

unpol

2)

e4 ( 1+ cos O.
= -1 24'
2

(2.30)

me C

Let J(r) be the scattered radiation per unit solid angle and unit volume from a
point in the corona observed above the limb at a distance r from the center of the
Sun, and denote by ro the perpendicular distance from the center of the Sun to
the line of sight, s, through the point in the corona. The observed, projected
brightness at that point is given by the equation

JJ(r)ds JJ<;>rd:
'
r -ro

+00

I(ro) =

00

=2

-00

0

(2.31)

INTERPRETATION OF OBSERVATIONAL DATA

33

assuming spherical symmetry. This integral equation was first solved by Abel
(see e.g., discussions by Rosseland, 1936; van de Hulst, 1950) and yields through
inversion

=- f
00

J(r)

1

n:

K(ro)

r

rodro
(r6-r2)

112 '

(2.32)

.

(2.33)

where
K(ro) =

_2. d/(ro)
ro

dro

If one also assumes isotropic scattering, one can show that J is related to I
through the expression
(2.34)

Free-free emission can be described by classical theory in the non-relativistic
case, which holds for prominences. When an electron is accelerated during an
encounter with a proton or other ion, it will emit radiation, and Kramers (1923)
derived an expression for the rate of radiation in this free-free transition. We can
use Equation (2.26), remembering equipartition of the energy densities in the
radiation,
E2

B2

E2

8n:

8n:

41t"

_s +_s =_s ,

and find the rate, or power P, by integrating the energy flux density
a sphere with radius r

2(dV)2 ergs-1 .

2 -e- p=
3 c3 dt

(2.35)

cE; 147tover
(2.36)

To find the free-free emission per unit volume, elf, one needs to evaluate the
acceleration dv/ds in the classical case. Oster (1961a~b) has considered both the
classical and the quantum-mechanical treatment, and concludes that for temperatures below about 5 x 105 K the classical expressions should be used. At million
degree temperatures, when X-ray Bremsstrahlung is emitted from flares, a

CHAPTER 2

34

quantum-mechanical approach is essential. When dv/dt is evaluated for classical
orbits of the electron, one finds the expression
(2.37)
for the emission per unit volume per frequency interval dv, where nj is the number density of ions of charge Ze and dne is the number of electrons with velocity
in the range dv. The correction factor g, the Gaunt factor, is according to Oster
(1961a,b)

41rme (2kT
_ 3 1 [~
g--n
- - )3/2] ,
1C
Ze v ymc

(2.38)

where y is Enler's constant =1.78 ...
When we assume a Maxwellian velocity distribution, and integrate Equation
(2.37) over this distribution, we end up with the following expression for Eft
Elf

=

64 1C

r;;

2Z2 e6njne

3v3 (21CkT)

112

312 3 g=

me c

6
-38 2
-112
-3 -1
.8xlO Z njne T
g em Hz.

(

2.39)

Prominence spectra at times show emission due to a free-bound continuum.
The free-bound emission, Efb, radiated as an electron with velocity v is captured
onto an energy level n in the ion, can be evaluated in a simple Bohr quantum
picture using the Balmer formula for the emitted frequency. For details concerning the derivation of the expression, see, e.g., Billings (1966). We obtain for E{b the
equation
(2.40)

INTERPRETATION OF OBSERVATIONAL DATA

35

2.1.5. LINE PROFILES
With a known source function, we can write for the intensity of an emission line
instead of Equation (2.7)
(2.41)

where the optical depth is 'tv = CJ.vN, and CJ. v is the absorption coefficient per
atom. N is the number of atoms in the line of sight (cm-2).
We are, therefore, led to look at the important question of the shape of the
absorption coefficient which determines the shape of the observed speetralline.
To find the shape of the absorption coefficient we need to know what the dominant line-broadening mechanism is.
Going back to Equation (2.7) we recall that the frequency Vji emitted as electrons jump from level j to level i is given by

(2.42)
where Ej and Ejare the energies of the levels j and i, respectively. While a jump
from j to i gives an emission line, the transition i to j produces the corresponding
absorption line. According to Heisenberg's uncertainty principle the levels Ej and
Ei are not infinitely sharp; hence, the emitted line Vjj is not infinitely narrow, or,
stated equivalently, the lifetime of the electrons in that energy level is not infinite.
This can be expressed quantum-mechanically by Weisskopf and Wigner's (1930 )
probability distribution law: The energy of a level with mean energy Ej is distributed according to the formula
A·
dE
P(Ej)dE=:.:L
2
'
h .!A~+~(E_E.)2
4:J
h.L
J

(2.43)

where Aj is the total probability of transition from level j.
The probability function P(Ej) has a sharp maximum for E =Ej' so that most
of the atoms have energies near the mean energy Ej. When an electron jumps
from level j to level i, the intensity of the ensuing line will depend on the widths
of both levels and is then given by

CHAPTER 2

36

Iji

=hVjiAji

A·+A;
J 2
.

'!(A.
+A.)2 +~(E-(E'
_E.))2
4 J
h2
J
I

,

I

where Aji is the probability for the transition j --+ i. This equation can be written
(2.44)

where 'Y R = Aj + Ai, the radiation damping constant, is the sum of the decay
constants for the two energy levels involved in the electron jump. The curve
gives the so-called natural half width of the line, which is a fundamentally lower
limit to its width. Curves of the shape given by Equation (2.44) are referred to as
damping profiles; see curve D in Figure 2.4.
1.0~----------------------~-----------------------,

~
1/1
c:

.s

..5 0.5

~

Q)

a:

o
o~--------~--------------------------~--------~

Wave length, A. Fig. 2.4. Line profiles. G = Gaussian shape; 0 = damping profile.

In the classical treatment of radiation where the atom is viewed with an
oscillating electron executing forced vibrations, vo, under the influence of the
electric vector of the radiation field, one finds an expression for the intensity
formally equal to Equation (2.44) and with the damping constant given by

INTERPRETATION OF OBSERVATIONAL DATA

37

Expressing "leI in Angstroms, we find "leI = 1.8 x 10-4 which means that radiation
damping always constitutes an insignificant fraction of the total broadening of
any solar line. The quantity "IR also gives the half width of the absorption coefficient, another concept of great importance in spectroscopy. The line absorption coefficient per atom measures the absorption of incident radiation of frequency v in a given direction. The total absorption is then found by integrating
over all directions. We define the probability that a photon of frequency v will be
absorbed by an atom as (4n/hv)(Xv and find
TCe 2

rR

av = - - - - 2

f

(rR)

me 4TC ( v-va )2 + 4TC

2 '

(2.45)

wherefis the oscillator strength (or Landenbergf-value) and gives the number of
equivalent classical oscillators, or the effective number of electrons in the atom.
The integral over the absorption profile yields

In most work on prominences S v has been assumed independent of frequency and constant with optical depth in the line-forming region. Only under
these simplifying conditions can we integrate Equation (2.41) to give the line profile
(2.46)
where an index zero refers to the line center. The absorption coefficient (Xv that,
via the optical depth 'tv! determines the shape of the profile (2.46), will in the
idealized case be given by (2.45). However, in practice all solar lines will have
other, far more important, broadening mechanisms at work. The most important
source of line broadening in prominences is the Doppler effect, due to the fact
that the photons we observe are not radiated from atoms at rest with respect to
the observer. At any finite temperature, the radiating atoms move with a range of
velocities which depends on the value of the temperature. Let the velocity component of the radiating atom along the line of sight be v and the frequency of
radiated photons (i.e., in the atom's frame) be va. The observed frequency will be

Vo

=Vo[l-(v/e)]

38

CHAPTER 2

and the expression for the absorption coefficient (2.45) takes the form

rre 2
a~=­

f

me

(

)2 (r )2.
V-Vo+--:- + 4~

(2.47)

VV

We find the average absorption coefficient by integrating over the velocity distribution f(v), assumed Maxwellian, and find

av

exp[_(L1V2 )]d(L1V I L1vD)
oo
Tre2
r roo
L1vo
=r a~f(v) dv = ( )112 4R2 f -(--=---'----'-=)'-:::2,--(--)2::-.
mnc

-00

Tr

-00

VVo

V-Vo+C

rR

+-

(2.48)

4Tr

Equation (2.48) is known as the Voigt profile, and dAD is the Doppler width. This
wavelength shift is caused by the velocity Vo of the atom in the line of sight; i.e.,

dADIAO

=volc.

Substituting for Vo =
Doppler width

..j 2 kT1m , we find the following expression for the
(2.49)

T is the kinetic temperature of the plasma and m is the mass of the atom.

The integral in Equation (2.48) cannot be evaluated in closed form, but numerical integration tables are available (Harris, 1948). Two limiting cases are
easily recognized if we recast Equation (2.48) in the form

av

=const

f exp(-X2)dX
+Z

+00

(X-Y)

2

2'

(2.48')

-00

where
X= L1v =

L1vo

v

..j2kT 1m'

Y= Vo-v ~ Z= rRcl4Tr
vo vo'
vvO

First, for large Y, i.e., in the extreme wing of the line, we neglect X in the
denominator, and retrieve Equation (2.47), meaning that the thermal motion has
no influence on the radiation. In the other limiting case both Y and Z are small,

INTERPRETATION OF OBSERVATIONAL DATA

39

and the absorption coefficient shows the influence of the Maxwellian distribution
of the radiating atoms, displayed as curve G in Figure 2.4.
In general, the atoms possess a non-thermal velocity component !;t (the
astronomical "micro-turbulence"), whose influence on the line profile mimics
that of the thermal velocity, giving rise to a total Doppler width
(2.50)

In principle, we can separate the thermal and non-thermal velocity components
and solve for the kinetic temperature and the turbulent velocity !;f, provided we
can determine the widths of two lines from different elements, i.e., different
masses, mi =J1mH, i =1,2, where J1is the atomic weight and mHthe mass of anH
atom. This procedure is only permissible if the two lines are excited in the same
volume element of the prominence plasma. From Equation (2.50) we then find

and
(2.52)
The indiscriminate use of Equations (2.51) and (2.52), when there was no obvious
reason to expect the two lines to come from the same region of the prominence
plasma, has at times led to unintelligible results.
Even though the Doppler effect in general is the most important linebroadening mechanism in prominences, we find that for many H lines collisional
broadening may at times contribute significantly to the line shape. This broadening owes its existence to the fact that when the pressure in the plasma gets high
enough, collisions of the radiating atoms and ions will occur with surrounding
particles. The effects of collisions are complicated, and two limiting cases, collisional damping and statistical broadening, are generally considered. For details
on these approximations, which also are called the impact and the quasi-static
approximations, see, for instance, Jefferies (1968). The collisional damping, which
mainly pertains to the core of the line, can formally be treated like the natural
broadening (radiation damping) of the line, and the absorption coefficient per
atom is

40

CHAPTER 2

where the damping constant Y=Yrati + Ycoll is the reciprocal of the mean lifetime of
the upper level of the transition giving rise to the line in question. The resulting
line shape is described by a symmetrical dispersion profile. We have a Lorentz
broadened line
~c

Il

To

DC

(A _ Ao)2 + (~c)2 '

(2.53)

where the damping constant ~c is a collisional halfwidth; see curve 0 in Figure
2.4.
In the other approach, the statistical theory, which goes back to Stark (1916)
and Holtsmark (1919), one assesses the value at a given atom of the (microscopic)
electric field produced by the surrounding particles. Each field will cause a
frequency shift of the radiation from the given atom, and by assigning statistical
weights to the shifts, one can find the line profile by integrating over all shifts.
This method is mainly used for studies of the wings of the lines. The shift in
frequency, ~v, may be expressed in terms of the distance r between the two
interacting particles, i.e.,
C

DV=-,

,.s

where s is an integer. The constant C can often be calculated quantum mechanically, or measured in the laboratory. For s =2 the statistical broadening is symmetrical, but s = 4 and s = 6 lead to asymmetrical broadening, so that the line is
both broadened and shifted. The case s =6 pertains to the van der Waals forces.
Stark broadening is probably the best known example of statistical broadening. In this case the electrostatic field F is due to the charged particles in the
plasma. The line is split into several components, and in some cases, e.g., for
hydrogen, there is direct proportionality between F and the shift aVi of the ith
line component:
3h
Sir me

aVi =C;F=-2- qjF.

(2.54)

This is called the linear Stark effect. The number qj can be calculated and for the
hydrogen Balmer lines it is given by qj S; n(n -1) + 1, where n is the number of the
Balmer line. In other atoms the shift aVi is proportional to the square of the field

F,

(2.55)

and pertains to the quadratic Stark effect.

INTERPRETATION OF OBSERVATIONAL DATA

41

As the density of the plasma increases, the Sta~k broadened wings of the
hydrogen lines become more and more pronounced.Svestka(1976) and Canfield
et al. (1984) have used this fact to analyze solar lines as a diagnostic tool for
density determination. Also, as we go to higher and higher Balmer lines, i.e., to
greater and greater n-numbers, the lines lie closer and closer together in the
spectrum, and their wings will progressively overlap. In other words, the higher
the density, the more difficult it becomes to resolve the lines. Inglis and Teller
(1939) and Kurochka and Maslennikova (1970) used this situation to derive a
simple formula that gives the electron density ne in terms of the quantum
number n17UlX of the highest resolvable Balmer line
log ne ='22.7-7.0 log nmax.

(2.56)

Because of instrumental difficulties, estimates of ne by this formula are always
upper limits of the true electron density.
In all the studies mentioned above, it is assumed that Stark broadening by
macroscopic electric fields may be neglected, so that pressure broadening is responsible; whence the observed dependence of line width on Balmer number can
be used as a diagnostic of plasma density. However, Foukal et al. (1983) observed
hydrogen lines in post-flare loops and interpreted the line broadening as due to
macroscopic fields. The reason is that if pressure broadening were assumed, an
ion density of about 2 x 1012 cm-3 would be required, while mean electron densities measured directly from the Thomson scattering (Section 2.1.4) were found to
be in the range 3-7 x 1010 cm-3•
2.1.6. QUANTUM NUMBERS AND SELECI'ION RULES
The energy levels in an atom or ion, which define where an electron can be
found, are characterized by their quantum numbers: n (principal), 5 (spin), I
(angular momentum), and j (total angular momentum, i.e., coupling of spin and
orbital angular momentum).
The principal quantum number n gives the ordering of the levels with increasing orbital radii (the simple Bohr picture) or with increasing energy, En. A
photon is emitted by the atom when the electron jumps from one energy level (n,
5,1, j) to another (n', 5', 1', j'), previously called level j and level i in Equation (2.7).
If there is more than one electron in an atom, one uses the lower case symbols to
denote the angular momenta for a certain electron and capital letters L, 5, and J
to symbolize the momenta of the entire atom, measured in units h/21t and satisfying the inequality J ~ I L+5 I . With more than one electron, they are arranged in
levels (or "shells"), obeying Pauli's exclusion principle which states that two
electrons may not have the same quantum numbers. The levels (n, 5, L, J) are
designated, for historical reasons, by a letter (5, P, D, F, G, etc.) according to the

CHAPTER 2

42

value of L (i.e., L = 0 is called 5, L = 1 is P, etc.), by a superscript giving the
multiplicity, 25 + I, which indicates how many I values are available, and by a
subscript giving the I value. The value of n is placed in front of the letter. For
example, the ground level of the helium atom is designated 1150.
When an electron jumps it follows certain selection rules for how the
quantum numbers can change; viz. a change from level (n, 5, L, J) to level (n', 5',
L', J') is permitted provided

s: =S (spin

is preserved); i. e., 4S =0 }

L=LorL±l
I'=IorI±l

aL=O,±l
Al=O, ±1

(2.57)

Furthermore the transition I = 0 to I' = 0 is forbidden, and parity-the sum of
the I-values of all the electrons-must change by 1. No restriction applies to n.
These selection rules pertain to atoms where the so-called Russell-Saunders
coupling holds. In this coupling scheme there is strong coupling between the individual angular momenta I and the individual spins s so that they combine into
quantized resultants Land 5 in the manner considered above, while the coupling
between Land 5 is assumed to be slight.
A further concept of importance for our study of spectral lines is the
multiplet, by which we understand the collection of all possible spectral lines
between all levels with the same n, 5, and L quantum numbers, but with different
values of 1. A collection of such levels is called a term.
Lines that obey the selection rules (2.57) are referred to as allowed lines. We
call forbidden lines those that result from quantum jumps in violation of one of the
rules, and such lines are formed in the tenuous solar atmosphere, albeit generally
with low intensity. Certain atoms and ions may exist in systems with different
values of the quantum number,S, i.e., different multiplicity. Since a change as ¢O
is not allowed for radiation processes, we do not expect to observe an inter-system
line. However, they do exist, and are due to collisional excitation.
Neutral helium, He I, offers an example of an atom that exists in two forms,
orthohelium with multiplicity 3, that gives rise to lines in the so-called triplet
series, and parahelium with multiplicity I, that produces the singlet series; see
the energy level diagram in Figure 2.5. Notice that the ground level is very low, 20
volts below the next lowest. The metastable level 2 35 acts as a sort of ground level
for the triplet series and can be populated by collisions from the ground level
(Massey and Moiseiwitsch, 1954).
We have seen that a transition between two levels results in a spectral line
subject to selection rules (2.57). If one applies a magnetic field to the radiating
atom, quantum theory shows that the total angular momentum I may be pointed
only in certain "quantized" directions with respect to the magnetic field, such
that its projection along the field takes one of the following values: I, I - I, ..., -I,

INTERPRETATION OF OBSERVATIONAL DATA

43

in a1l2J + 1 values. This projection, M or Mit furnishes a fifth quantum number
and indicates that a level is composed of 2J + 1 sublevels that often are indistinguishable. The level is said to be degenerate and the degree of degeneracy is the
number 2J + 1 which is also called the statistical weight of the level; i.e., g =21 + 1;
see Equation (2.9). It is basically the number of electrons that can occupy a given
level without violating Pauli's exclusion principle. The sublevels, called states,
into which a level is split, are designated by the quantum numbers, n, 5, L, J, and
M. Of great importance for our study of magnetic fields in prominences is the
fact that the stronger the applied magnetic field, the larger the separation of the
states. Not all states can communicate with each other, since also for M do we
have a selection rule, viz.

M' =M or M±I; i.e., aM= O,±1.

(2.57')

The presence of a magnetic field in a plasma will remove the degeneracy of
an atomic level. Then instead of observing a single line with wavelength AO, we
see several line components, each occurring as the result of a transition from one
of the split states of the upper level. In the simplest case this emission Zeeman
effect will exhibit three lines, a n-component, which is not shifted, at Ao, and two
a-components, shifted in wavelength a distance ±aAB from AO, one to the red, the
other to the blue. The wavelength shift Jl')..B is given by

ILUBI=

e
2
-13 2
2A.ogB=4.67xlO A.o gB,
4nmec

(2.58)

where Ao is in Angstrom and the magnetic field in Gauss. In frequency units the
line shift is given by aVB = eBg/4n"'ec. The Lande g-factor determines the splitting
of the line through the magnetic moment J.l.B of the atom in the direction of the
magnetic field

eh

J.lB=--gM,

4nmec

where M, the magnetic quantum number, is subject to the selection rule (2.57');
aM = 0 leads to the unshifted n-component. Assuming that Russell-Saunders
coupling holds, the Lande g-factor is given by (Condon and Shortley, 1953)
g=

1 J(J+l)-L(L+l)+S(S+I)
+
21(J+l)
.

(2.59)

44

CHAPTER 2

PARA HELIUM
Ip

IS

24

s

~

...

21

*'

,<1)0

10

-5p

5d 5s

-40

4d \

~

22

ORTHOHELIUM

q)

&' ~

>

5d
,

~\!IJ,l

2p

~'Oo

<0

O....

2s <'t.o

,;
0 /
",0'0"3 /
~
/
/

"/ 2p

_?

V)

~
o

-5p
-4.

~ U'~

~~;

'\

\

20

/

19
/

/
~

/

/

/

/

/

/j
/

/

/

n~:---i

Oe/

Fig. 2.5. Energy level diagram of He I.

2.2. Magnetohydrodynamics-Plasma Physics
The following quote occurs in Solar Prominences: ''The single, physically most
important, parameter to study in prominences may be the magnetic field." The

INTERPRETATION OF OBSERVATIONAL DATA

45

research of the intervening 20 years fully supports this assessment, whence magnetohydrodynamics, MHO, has proven such an important tool in the study of the
nature of prominences. Before we lay the groundwork for using this tool, we
shall discuss how magnetic fields are observed.
2.2.1. MAGNETIC FIELD OBSERVATIONS
The most important development in the area of observations of magnetic fields in
prominences over the last two decades has been the use of the Hanle effect. Of
the two methods to spectroscopically measure the magnetic field, i.e., the
Zeeman effect and the Hanle effect, the former has a long history in solar physics
(see Section 1.4). We owe it mainly to the French school that the application of the
Hanle effect was successfully developed in prominence research.

2.2.1.1. The Zeeman effect
For strong magnetic fields the two a-components of a spectral line are observed
to be well separated in the spectrum, and by measuring /lAB (Equation (2.58»,
one can determine the magnetic field. However, in solar physics this only applies
to sunspots where the magnetic field is several thousand Gauss. In prominences
the field is too weak to make this procedure possible. Instead one takes
advantage of the fact that the line components are not only shifted in
wavelength, but they are differently polarized, linearly or right- or left-circularly
polarized. It is these polarization properties that make it possible to determine
magnetic fields in prominences with the help of the Zeeman effect.
For observations outside the limb of the Sun, i.e., in prominences and in the
corona, we see emission lines, and the presence of magnetic fields leads to the
Zeeman effect just described. When we observe the disk of the Sun, we see the
spectral liens in absorption (Fraunhofer lines), and the presence of a magnetic
field gives rise to the so-called inverse Zeeman effect. One must then consider the
more complicated case of a gas which both emits and absorbs radiation, and one
faces a fairly complicated radiative transfer problem. The absorption properties
may be described by introducing three selective absorption coefficients, Kit for
plane polarized light in the direction of the magnetic field lea, I for left-circularly
polarized light, and K a , r for right-circularly polarized light (Unno, 1956). Similarly, selective emission coefficients, EIt , Ea,l, and Ea,r may be used to account
for the emission properties.
There are several ways by which one may describe the state of polarization of
a light beam (see, for instance, Shurdiff, 1962; Harvey, 1969). We may represent
the beam by a pair of orthogonal plane waves

CHAPTER 2

46

(2.60)

where Ex~y(t) are the instantaneous amplitudes of the electric field, co is the
frequency, and ax, yare the phase factors. From Equation (2.60) we can find the
expression for the polarization ellipse, which is a familiar way of describing a
beam of elliptically polarized light. Instead, we may describe it by the equivalent
four Stokes parameters (Chandrasekhar, 1960)
1- E02
x +E02
y'

Q =E0x 2 _E0y2'

U=2E2E~sino,

(2.61)

V = 2E2E~ sin 0,

ay -

where a=
ax.
The parameter I is the total intensity of the radiation and Q, U, and V
describe the state of polarization. Furthermore, we see that J2 = Q2 + U2 + V2. In
other words, the Stokes parameters are the observables of the polarization
ellipse, or of the light beam. If the beam is incompletely polarized, then the intensity referred to above represents the polarized part, Ip, only, so that IplI gives
the degree of polarization,

We may write the Stokes parameters as four elements of a single column
matrix {I, Q, U, V} and the intensities of the Zeeman components are then
I tr

=Etr {sin 2 r,sin 2 r,o,o},

r),-t sin2 r, 0, cos r},
1(1,1 =E(I,/ {t(1+cos 2 r),-t sin2 r,o,-cosr},

1(I,r = E(I,r{ t( 1 + cos 2

(2.62)

where y is the angle between the line of sight and the direction of the magnetic
field, and the emission coefficients have the form

INTERPRETATION OF OBSERVATIONAL DATA

47

etc =e(vo)(i.e.,identical to the coefficient in the absence ofa field),
eq,r = e(vo +AvB) = e(v)+AvB :
eq,l =e(vO-AvB)=e(v)-AvB

+... (Taylor series for small B),

(2.63)

de

dv +...

In the case of absorption lines (the inverse Zeeman effect) analogous expressions
are used for the absorption coefficients K,r, 1\o,r, and 1\0,/.
In the case of the simple Zeeman triplet we find the following relationship
between the direction of the magnetic field and the observed polarization of the
line components: When we observe parallel to the field, the longitudinal case, Y =
0, we see only the two c.T-components, circularly polarized in opposite directions,
c.TR and c.Tv; see Figure 2.6. If the line is in emission, and the field is pointing
toward the observer, c.TR will be right-circularly and c.Tv left-circularly polarized.
For an absorption line, and with the same orientation of the field, the c.TR component will be left-circularly and c.Tv right-circularly polarized. In the other
limiting case, the transverse case when the line of sight is perpendicular to the
magnetic field, y = 90·, and the 1t-component is twice as strong as a c.T-component.
It is linearly polarized, parallel to the magnetic field in emission Zeeman effect,
and perpendicular to the field in the inverse Zeeman effect. In this case, also the
c.T-components are plane polarized. For all intermediate cases, i.e., in nearly all
practical applications, we see elliptically polarized c.T-components, while the 1tcomponent remains linearly polarized.
It is difficult to find suitable solar spectral lines that show the simple Zeeman
splitting. Most lines actually used in magnetograph work show the anomalous
Zeeman effect; i.e., they split up into an often large number of c.T- and 1tcomponents, and the overlapping of these differently polarized components
causes serious difficulties in interpreting the observed polarization in terms of a
magnetic field. One should, therefore, divide the problem of measuring solar
magnetic fields into its two parts, viz. (1) determination of the state of polarization in certain selected spectral lines, and (2) interpretation of the observed polarization in terms of a magnetic field. While the first part of the problem is mainly
a question of technique and is well understood (albeit being difficult), the second
part relies on our understanding of the formation of spectral lines in the presence
of a magnetic field, and a general theory is not available. The problem has been
solved for special cases (Unno, 1956; Stepanov, 1958, 1960; Michard, 1961;
Rachovsky, 1961, 1963, 1967; Obridko, 1965, 1968).
The basic principle of the longitudinal polarimeter (magnetograph) consists
in alternately blocking out one of the two c.T-components with appropriate polarizing optics, and measuring the difference in signal from the two components by
using two detectors, one fed by light from an entrance slit in the violet wing of

48

CHAPTER 2

Intensity

L---------~~~~--~~--~r---------~x

Slit V

26X B Slit R

Wavelength

Fig. 2.6. Observation of the longitudiinal magnetic field via the Zeeman components
simple Zeeman triplet.

Gv

and GR of a

the O'v-components, slit V, the other light transmitted through slit R in the red
wing of the O'R-component (see Figure 2.6) (Beckers, 1968; Harvey, 1969; Leroy,
1989).
A better solution is to use a multichannel detector (Baur et al., 1981) to observe the line profile in one exposure. To find how the signal from the detectors
is related to the magnetic field one needs a theory for the formation of the line. In
the simple case of a uniform magnetic field and an optically thin line, it can be
shown that the resulting intensities of the Zeeman components (Equation (2.62»
lead to the following Stokes parameters
I=E1t'sin 2 r+~(EO',r+EG,r)(1+COs2 r),

.2

1(

).2 r,

Q =E1t'sm r-'2 Ea,l + Ea,r sm

(2.64)

U=O,
V

=(Ea "r -

EO' t)COS r

"" 2B dEv cos r·
dv

In the case of the longitudinal polarimeter this light is transmitted through a
circular polarizer, i.e., through a y/4 plate followed by a linear polarizer oriented
with its transmission axis 45° with respect to the fast axis of the y/4 plate. The
transmitted intensity is then

INTERPRETATION OF OBSERVATIONAL DATA

49

Using Equation (2.63) for the emission coefficients e and retaining only the first
term of the Taylor expansion, we find
de
dv

de
dv

v =ev±const Bcosr-v.
It(lI) = ev ±.1vBcosrWith the help of the two slits of the polarimeter the detectors measure the intensity at a point in the line profile with alternately the CJR and the CJv component
blocked out. By taking the difference one obtains a signal S( II) = It•R( II) -It. v( II),
or

r

S( II) = const Bcos de v ;
dv

(2.65)

i.e., the resultant signal varies as B cos y. We see that for small magnetic fields the
5(11) signal is just the Stokes parameter V of Equation (2.64). To derive B cos y
from V, one must determine devldv. This is done with the help of the Stokes
parameter I, measuring the slope of the line profile as well as its intensity.
The problems involved in measuring polarization are vastly increased when
we want to know the complete state of polarization, which is needed to find the
vector magnetic field. The transverse component of the field is associated with
the Stokes parameter giving linear polarization, i.e., with the x-component of the
Zeeman-affected spectral line. In this transverse case the light beam of Equation
(2.64) is transmitted through a linear polarizer and the transmission axis is
switched between a direction parallel to and a direction perpendicular to the
magnetic field. One can then show that the transmitted intensity is given by

Again, using Equations (2.63) with the two terms of the Taylor expansion retained, we derive the following expression for the signal due to the intensity difference It,R{.l) -It, v{.l) after transmission through the alternating linear polarizer
(2.66)
For small magnetic fields of the strength one finds in prominences, it is inherently more difficult to measure the transverse component, which depends on

50

CHAPTER 2

B2 (Equation (2.66», than the longitudinal component, which is proportional to B
(Equation (2.65». An additional difficulty is encountered in measuring the transverse field in that there is a 180· ambiguity in the deduced direction of the component. This limitation of the method has its root in the nature of the photonatom interaction and is fundamental to the Zeeman effect (Sahal-Brechot, 1981).

2.2.1.2. The Hanle effect
The greatest improvement in measuring magnetic fields in prominences during
the last two decades is due to the use of the Hanle effect, pioneered by the French
groups at Pic-du-Midi and Meudon observatories. For reviews see Sahal-Bnkhot
(1981) and Leroy (1985, 1989).
If one excites an upper level of an atom by a polarized photon flux, the subsequently emitted resonance line will be linearly polarized. This resonance polarization will be affected by the presence of a magnetic field in the emitting plasma,
an effect studied in the laboratory by Hanle (1924, 1925) and referred to as the
Hanle effect. The magnetic field can, therefore, be deduced by studying the details of the polarized resonance line in question. The diagnostic possibilities of
the Hanle effect was recognized already by Hyder (1965), but the correct theory
is due to House (1971). In simple atomic structures the effect can be qualitatively
understood by classical theory, while generally a quantum-mechanical treatment
is required.
To use classical theory we consider the scattering atom a classical oscillator.
Let the direction of the vibration of the electric vector in the exciting radiation be
parallel to the X-axis and let there be a magnetic field (25-100 Gauss) in the same
direction. The polarization of the resonance-scattered light will then be in the
same direction as in the field-free case; i.e., we see that a high degree of incident
polarization along the X-axis produces scattered light that is also highly
polarized in the X-direction. However, if the (weak) magnetic field is parallel to
the Y-axis, we observe a strong resonance-scattered line which is highly polarized in the X-direction. If the field strength increases, one observes that the percentage polarization decreases only slightly from that of the field-free case, but as
the field increases-and the polarization decreases-the plane of polarization rotates through large angles.
This dependence of the observed polarization on the orientation of the magnetic field can be explained as due to the precession of the atom about the magnetic vector, and the sensitivity of polarization to the strength of the field is explained by assuming that the atom is a damped oscillator. Due to the precession,
the direction of vibration will deviate from the original direction and cause the
plane of polarization of the emitted light to rotate. For observations along the
magnetic field the oscillator describes a rosette (see Figure 2.7), and the shape of
the rosette-and, therefore, the nature of polarization-will depend on the ratio

INTERPRETATION OF OBSERVATIONAL DATA

51

between the angular velocity of precession, co, and the radiative damping constant, 'Y1V of the oscillator. If co »'YR, or
CiJT»

1,

(2.67)

where't =11 'Y R = mean radiative lifetime, the rosette will be axially symmetric,
since the atom will have time to precess many times before it is damped out.
Consequently, there is no polarization observed along the magnetic field if the
field is strong enough to make inequality (2.67) hold. If co '" 'Y R, or cot '" 1, the
oscillator describes an asymmetrical rosette which means that the degree of
polarization is reduced relative to the value in the field-free case, and the
direction of polarization is rotated an angle Q with respect to the direction of
polarization of the exciting beam. Finally, if cot« 1, the oscillator hardly has
time to precess before it is damped out, which explains the weak field results.
The percentage polarization, P, is given (Breit, 1925) by
(2.68)
for the polarization observed along the field. Here Po is the percentage polarization for the field-free case and COB =eBlmec is the Larmor precession velocity for
an electron of mass me and charge e (see Equation (2.74». The angle of rotation is
given by
(2.69)
In Equations (2.68) and (2.69), g is a correction factor necessary to obtain the observed results from classical theory, and Breit (1933) showed that classical and
quantum mechanical results agree if this correction factor is the Lande g-factor
for the upper level of the transition in question; see Equation (2.59).
Hanle (1923) went beyond the purely classical theory and considered an elementary quantum-mechanical treatment of the simplest resonance line, viz. the
transition from a 3Ptlevel to the ground leveltSOi see Figure 2.8. If the 3Ptlevel is
excited by light polarized parallel to the X-axis and if there is a strong magnetic
field in the same direction, only the 1t-component (which, according to the classical Zeeman effect, is polarized parallel to the field) can be absorbed. Hence, we
populate only the M = 0 state of the 3Pt intra level. Then, if no level transition
takes place, the emitted radiation will be a 1t-component whose electric vector is
parallel to the field. On the other hand, if the magnetic field is parallel to the Zaxis, the circularly polarized components will be absorbed. The circularly polarized components will subsequently be re-emitted, but along the line of sight, the

52

CHAPTER 2

Precession

Damping

/1\
\
\

Fig. 2.7. Path described by an oscillator when the angular velocity of precession,
the radiative damping constant 1/ t.

0>,

is of the order of

Y-axis, which is perpendicular to the field, the radiation will appear linearly
polarized parallel to the X-axis.
When there is no magnetic field present the upper level is degenerate, and
one cannot a priori predict whether the 1t- or the a-components will be absorbed.
This uncertainty is overcome by applying Heisenberg's (1926) principle of
spectroscopic stability which states that the absorption of radiation by the atom
when there is no magnetic field present is the same as when there is a field parallel to the direction of vibration of the incident polarized radiation; in other
words, the absorption takes place in 1t-components.
To go beyond this simple case of resonance radiation we need to consider
scattering that takes place between levels that are only weakly removed from de-

53

INTERPRETATION OF OBSERVATIONAL DATA

generacy by a magnetic field, so that the different M states nearly overlap. Under
such conditions mutual interactions take place between these close atomic states;
i.e., coherence effects occur. Since the different M states under these conditions
are more or less indistinguishable, one must sum over those states that could be
excited by the incident radiation. House (1971) pointed out that the summation
over the excited states should be carried out before one squares the amplitudes to
obtain the scattered intensity, and from a mathematical point of view one sees
that the physics behind the coherence nature of the Hanle effect is expressed in
the cross-terms that result from this procedure.

M
1

o
-1

(J

1t

1 So

(J

o

Fig. 2.8. (J and 7t components due to transitions tso _3Pt.

The importance of the coherence was realized already by Warwick and
Hyder (1965), and described phenomenologically by Obridko (1968). House
(1970a,b, 1971) gave the general discussion mentioned above. A complete quan-

54

CHAPTER 2

tum mechanical explanation of the Hanle effect has now become available
(Bommier, 1977, 1987). Leroy et al. (1977), Sahal-Brechot et al. (1977), Landi
Degl'Innocenti (1982), and others have applied it to the physics of solar
prominences.
A serious problem is encountered in magnetic field observations of prominences using both the transverse Zeeman effect and the Haole effect; namely, the
inability to distinguish two magnetic vectors symmetrical about the line of sight.
This degeneracy is fundamental and results from the nature of the photon-atom
interaction (see, for example, Sahal-Brechot, 1981). With the Zeeman effect there
is no obvious way to remedy this situation, generally referred to as "the 180·
ambiguity." Remember also that the Stokes parameters Qand U that describe the
linear polarization are much smaller than the circular polarization parameter V.
In the Hanle effect the situation is quite different. First the U and Q Stokes
parameters are an order of magnitude greater than V due to the fact that resonance scattering leads to linearly polarized radiation, so that the Haole effect first
and foremost concerns such radiation. Second, the observation of one spectral
line gives two necessary quantities, viz. the decrease of the degree of polarization
P, relative to the theoretically determined maximum degree of polarization in the
field-free case, and the rotation n of the direction of polarization. These two
quantities are not sufficient to determine the full vector nature of the magnetic
field, B, but if one simultaneously observes another spectral line, one ends up
with two more quantities, and the vector B can, in principle, be recovered, even
with one quantity to spare (Bommier et al., 1981; Landi Degl'Innocenti, 1982). In
practice, this procedure is difficult, but if one of the observed lines is optically
thick, Landi Degl'Innocenti et al. (1987) have shown that the field B can be completely determined. For a lucid review of the use of the Hanle effect, see Landi
Degl'Innocenti (1990).
2.2.2. ELEMENTS OF MAGNETOHYDRODYNAMICS
The solar atmosphere, including the prominence plasma, is in constant motion,
and magnetohydrodynamics (MHO) is often the tool by which the interaction of
these motions with electric and magnetic fields can be examined. This is appropriate when a macroscopic "fluid" point of view may be used in the mathematical treatment of the phenomenon. However, under more extreme conditions,
found in flares and maybe in very active prominences, a microscopic point of
view is necessary, and we then consider the interactions of the individual particles involved. Roughly speaking, this latter case is referred to as plasma
physics.

INTERPRETAnON OF OBSERVATIONAL DATA

55

2.2.2.1. Plasma parameters
We have used the term plasma somewhat loosely to mean a gas made up of
roughly equal numbers of positive and negative electric charges, mainly protons
and electrons, but with important minor constituents like helium ions, ionized
iron atoms, etc.
A useful concept, taken over from the theory of electrolytes, is the Debye
length, AD, or Oebye shielding distance. It indicates the distance to which a point
charge can make its influence felt. The electric field of the charge falls off with
distance r as
Epot

-.!.
e -rf)"D ,
r

(2.70)

and for distances r > AD the electric field is shielded by charges of opposite sign,
and AD is a measure of the distance over which strict neutrality (npos =nneg) may
not be obeyed. A rough estimate of the Oebye length may be obtained by equating the energy due to the electric potential (Equation (2.70» and the thermal
energy, whence
(2.71)
In the plasma, the gas, which remains quasi-neutral, is controlled by the
electromagnetic forces, which implies that the volume given by r < AD, will contain a large number of particles nD; i.e.,
nD

4
3
=-nnAD
»1,
3

and to be able to define a plasma, its characteristic length L must be much larger
than AD. Under such conditions the gas is controlled by the long-range Coulomb
forces. A particle's motion is then given not only by its interactions with the
nearest neighbors in binary collisions, but also with the nD particles within the
Oebye length, so-called collective interactions. This argument leads to a more
formal definition of a plasma as a quasi-neutral gas of ionized particles subject to
collective behavior.
When a magnetic field is present in the plasma, the ionized particles will gyrate around the field lines. With both an electric field E and a magnetic field B
present, the force on a particle of charge e is given by the Lorentz equation

56

CHAPTER 2

(2.72)
When B = 0, the particle experiences a uniform acceleration dvldt =eE/m. The
acceleration is perpendicular to both v and B in the case of E =0 and corresponds
to a centrifugal acceleration, viz.

mv2

evB

AB

C

--=-,
where AB is known as the gyro radius
1

_

mvc

"'B--·
eB

(2.73)

The orbital frequency-the gyro frequencyvIe

OJB=-=--B,
AB c m

(2.74)

is proportional to the charge-to-mass ratio of the particle and to the magnetic
field. This gyrating particle will, since it experiences an acceleration, emit radiation in accordance with the laws of electrodynamics. We consider here the case of
electrons. In a prominence plasma the electron velocity v is small compared to
the velocity of light, and we can express the electric intensity in the emitted, 80called gyro radiation by Equation (2.26), or better, by its vector form

e2 1
me r

E=-3 '3 rx [rx(vxB)].

(2.75)

The electric vector in Equation (2.75) gives an elliptically polarized radiation that
at times can be quite strong in the solar atmosphere. The power radiated per electron can be found from Equation (2.36) by substituting the expression for the acceleration, namely, the Lorentz acceleration, in Equation (2.72), viz. dvldt =
(e I mec)( v x B), giving for the power
(2.76)

INTERPRETATION OF OBSERVATIONAL DATA

57

2.2.2.2. Moments of the Boltzmann equation
With these more plasma physical concepts in mind, we now return to the fluid,
magnetohydrodynamic treatment of the prominence gas. First we recall the
framework for the gas dynamic treatment as given by the Boltzmann equation
for the velocity distribution function I( r, w, t), i.e.,

The function I gives the density of particles in phase space as a function of position r and microscopic velocity w, and Boltzmann's equation describes how I
changes as a result of collisions between the particles. This partial differential
equation may be written

~ +w. Vrl+!....
Vw/=(~) coli .
m

(2.77)

There will be one such equation for each species of particle present in the plasma,
but we shall here drop the suffix. The term (all at)coll is the collision integral that
makes the complete solution of Boltzmann's equation very complicated (see, e.g.,
Chapman and Cowling, 1952). The symbol Vr indicates differentiation with
similarly indicates difrespect to the coordinates r = (x,y,z) = (Xl, X2, X3) and
ferentiation with respect to the velocity components w = (w x, wy, wz) = (WI, w2,
W3)' F is the external force experienced by the particles of mass m. With no collision (allat)coll=O and Equation (2.77) reduces to the collisionless Boltzmann
equation-the Vlasov equation-that approximately holds in very tenuous
plasmas. However, it cannot be used to study transport phenomena since they
owe their existence to the collisions.
We can now continue to treat the individual particles in a microscopic sense
by using statistical theory to get information on the distribution function. This
procedure, the "kinetic" approach, should be used when we study, e.g., highenergy particle beams as they interact with the solar atmosphere during flares.
The other alternative, the macroscopic "fluid" point of view, is used to develop
the gas or hydrodynamic equations. This continuous plasma dynamics viewpoint
is adequate in most prominence research. We then consider large-scale phenomena where the particle mean free path A. is short compared to the characteristic
length L of the phenomenon under study. In this macroscopic picture we are
interested in the flow velocity v, rather than in the individual particle velocities
w. We consider a plasma with total mass M, volume V, and consisting of electrons with particle density ne and ions with particle density nj. The gas pressure p

Vw

58

CHAPTER 2

is Pe + Pj, which for a perfect gas is P =k (n e+ nj) T =knT where the kinetic temperature is given by
(2.78)
The flow velocity can be written
(2.79)
where p is the mass density, and
1
v'=-Iw"
I
n.V
I

(2.80)

I

We now introduce the important concept of the electric current density, j,
generated by the motion of the electric charges on ne and nj, viz.
(2.81)
Z is the number of elementary charges e on the ions nj • It is sometimes useful to
introduce the diffusion velocity u, which for the electrons is defined as Ue =Ve - v,
with a similar expression for the ions. We write then for the current density
(2.82)
The first term on the right-hand side of Equation (2.82) is referred to as the conduction current. The second term is the convection current and constitutes the
bulk flow of a charged fluid which vanishes when the plasma is neutral.
The final step of going from the microscopic Boltzmann equation to the
macroscopic conservation equations used in hydrodynamics is accomplished by
multiplying Equation (2.77) by appropriate functions of the microscopic velocity
and integrating over velocity space, dw x, dw y, dwz. If the multiplying function is
unity, we obtain the zeroth moment of the Boltzmann equation and find the
equation of continuity of matter, viz.

an + V • (nv) =0,
-at

(2.83)

where the integral over (ajlat)coll has vanished since collisions do not change
the number density of particles.

INTERPRETATION OF OBSERVATIONAL DATA

59

With P = nm, Equation (2.83) can be written

ap V • (pv) = o.
a;+

(2.83')

The particle density is given by

fff
+00

n(r,t) =

f(r, w,t)dwxdwydwZ

(2.84)

and the macroscopic velocity is

-l-f f f
+00

v(r,t) =

n(r,t)

wf(r, w,t)dwxdwydwz ·

(2.85)

We notice that there is an equation for the continuity of electric charge Pe similar
to Equation (2.83') for the continuity of matter, viz.
(2.86)
To obtain the first moment of the Boltzmann equation we let the multiplying
function be mw and integrate as before. This gives us the equation for the momentum transfer, i.e., the equation of motion in the form

The right-hand side of Equation (2.87) is the momentum transfer by collisions;
we denote this vector by P. The left-hand side can be simplified by introducing
the stress tensor P =nmww and rearranging terms using Equation (2.83') leading
to the equation of motion
dv
nm-=nF-V.P +P.
dt

(2.88)

Here the mobile operator dvldt = olot + v. V. The stress tensor reduces to the
scalar pressure p for an isotropic distribution of the random particle velocities.

60

CHAPTER 2

When both an electric field, E, and a magnetic field Do are present in the plasma,
the force F takes the form

F=e( E+;VXDo )+mg,

(2.89)

where g is the acceleration due to gravity.
Both the electrons and the ions obey an equation of the form (2.89). For the
electrons we write, neglecting convective derivative terms,

A similar equation holds for the ions. We add these two equations to find the
bulk fluid behavior of the plasma and obtain

where we have assumed neutrality, i.e., Znj = ne, and where Pie and Pei have canceled by Newton's third law. Furthermore we assume a scalar pressure, and note
that the terms involving E cancel in the neutral plasma. Using Equations (2.79)
and (2.81) we can write Equation (2.91) in the form
iJvI
P-at=;jXBo-Vp+pg.

(2.92)

This is the equation generally used to study the behavior of prominence
plasmas and it is worth noticing the restrictions and assumptions underlying it.
The Lorentz force, (11 c)j x Bo, provides the all-important coupling between the
mass motions and the magnetic field that determines the behavior of the plasma.
One could now find the second moment of the Boltzmann equation to obtain
the energy transfer equation. This would be accomplished by using w2 as the
multiplying function and integrating. However, we would then also introduce
the third moments of the distribution function, namely, the components of the
heat flow tensor, and we would not have a closed system of equation. Consequently, one seeks an independent expression for the heating effect, q, to obtain
closure. In ordinary gas dynamics, the heat flow is often small, and, if it can be
ignored, the energy conservation equation reduces to the equation for the conservation of entropy
(2.93)

INTERPRETATION OF OBSERVATIONAL DATA

61

In solar plasmas the heating density, q, may be due to, e.g., heat conduction,
radiation, and Joule heating, and the general expression for q becomes very
cumbersome. We often use
·2

q =V .(KVT)- V .F rad +L,

a

(2.94)

where K is the thermal conductivity, Frad is the net radiative flux, and (J the
electrical conductivity. Let U denote the internal heat energy (per unit mass),
then the desired energy equation becomes
dU pdp
p-=--+q=-pV.v+q.
dt
p dt

(2.95)

2.2.2.3. Ohm's law

It has become common practice among solar physicists to picture the magnetic
field with the help of field lines, and to construct mental pictures of magnetic flux
ropes or tubes bounded by field lines connecting-through the solar atmospheric
plasma-one polarity with the opposite. Furthermore, in the highly electrically
conductive plasma one involves the notion of frozen-in magnetic fields, and derives a magnetodynamic picture of the evolution of magnetic structures, including prominences.
This procedure may often be very useful and can help us in visualizing conditions for which we normally have only a rudimentary intuitive feeling regarding the relevant physical processes involved. However, a word of caution may be
in order. First, one needs to bear in mind that the magnetic line of force is a
manmade invention. In the solar plasma we encounter electrically charged particles that in spiraling motions constitute electric currents. These currents furnish
the fundamental description of the physics involved. At times the ionization of
the plasma is sufficient, and the electrical conductivity is high enough that we
may use the magnetodynamic picture, but one should bear in mind that it is the
electric current that is the primary parameter, and Ohm's law is the fundamental
equation.
On the other hand, since the magnetic field itself is real enough, most discussions in solar physics can be cast in either of the equivalent frameworks of
electric currents or magnetic fields, and we shall often discuss the magnetic field
of prominences in terms of flux tubes and field lines. One only needs to bear in
mind the conditions under which these terms are valid, and that in many
instances the frozen-in concept may fail.

62

CHAPTER 2

In free space only electromagnetic waves are possible, and Maxwell's equations govern the situation. On the other hand, in a plasma there is a coupling
between the radiation and the particles. We have looked at this from the radiation point of view in Section 2.1 Here we consider the coupling by using equations of motion for the particles-and the concept of electrical conductivity is
essential.
In the fundamental equation (2.91) for the electrons we define the electrical
conductivity in terms of the momentum coupling between electrons and ions:
(2.96)

Inserting Equation (2.96) into (2.91) and assuming a scalar pressure, we find
(2.97)
and a similar equation for the ions. We can combine these two equations, assume
charge neutrality, Znj = ne, and use Equations (2.79) and (2.81) to obtain the generalized Ohm's law, viz.

(2.98)

This is the fundamental equation describing the physics of the system and exposing the decisive role played by the electric current. The left-hand side gives the
change in an electric current when an electric field is turned on, and may be important in flare research at the injection of an electron beam into the flare plasma.
However, in prominences we expect a steady state to have been reached and
aj lat =o. If further there is force balance for the electrons, VPe =(11 c)( v x Do),
and if Zme «mi, we obtain Ohm's law for a steady state:
(2.99)
When no external magnetic field is present, we retrieve Ohm's law in a form familiar from elementary physics:

j=oE.

(2.100)

INTERPRETATION OF OBSERVATIONAL DATA

63

It becomes clear that only under extremely simplified conditions is Equation
(2.100) the correct form of Ohm's law. Instead of Equation (2.99), we may write
(2.101)
where we have introduced the plasma resistivity 11. In the general case neither
the electrical conductivity C1 nor the resistivity 11 is a scalar quality; both are tensors. This circumstance is due to the fact that resistance to motion parallel and
perpendicular to the magnetic field Bo will not be equal; currents flow more
easily along Bo than perpendicular to it. The simple scalar form for a can be
found from Equation (2.96) by setting P ei ne me v ve, i.e., equal to the rate of loss
of momentum, where ve is the collision frequency, and letting j = nee ve. The
classical result is

=

(2.102)
We have seen (Equation 2.74) that when a magnetic field is present in the
plasma, the charged particles will gyrate in the field. The frequency is referred to
as the gyro, cyclotron, or Larmor frequency, roB =eB/mc. The conductivity is then
a tensor, and the electric current has three components. Let us decompose the
electric field into two components, one along B, called E II' and one perpendicular
to B, called E1.. Along the field B the conductivity is unaltered from the simple
scalar case, i.e.,
(2.103)
Equation (2.103) shows that the magnetic field has no influence on the parallel
conductivity. On the other hand, for currents flowing perpendicularly to the
magnetic field inhibiting effects take place. In the case of the current flowing
along E1. one finds an electrical conductivity, a1., reduced relative to ao by the
factor [1 + (roB/Ve)2]-1; i.e.,
(2.104)

Finally, for the current flowing at right angles to both Band E, the so-called Hall

current, we have a conductivity aHgiven by

(2.105)

CHAPTER 2

64

In addition to describing the currents flowing in plasmas, the electrical conductivity also gives the energy dissipated by the currents as heat, WJ, referred to
as Joule heating. We have
·2

J -~E2
WJ -J'
- • E---u.
CJ

(2.106)

2.2.2.4. Maxwell's equations
The fundamental force vectors in space, free from matter, are the electric field
intensity E and the magnetic density B (Stratton, 1941), both already introduced
in our discussions. We now introduce the electric displacement D and the
magnetic field intensity H in anticipation of the presence of a plasma, and write
the Maxwell equations in the general form
4n. 1 an
VxH=-J+--

c

c dt '

IdB
c dt '

and

(2.107)

VxE=---

(2.108)

V.D = 4nPe'

(2.109)

V.B=O,

(2.110)

where Pe is the electric charge density; see Equation (2.86). The derived vectors H
and D are associated with the presence of matter, and D is assumed to be related
to E through the dielectric constant E,
(2.111)

D=eE.

In the simple case when also Equation (2.100) holds, one combines 0' and E into a
complex dielectric permittivity E'

e' =e-{

where
form

CO

4:CJ),

(2.112)

is the frequency of the time-dependent behavior of the variables of the
E - exp[i(k. r-

cot)]'

There is no distinction between Band H in a non-magnetic medium, and one will
often see H used in solar physics discussion instead of B-as employed in this

INTERPRETATION OF OBSERVATIONAL DATA

65

book. We can now combine Equations (2.107) and (2.108) and use (2.111) to find
the basic equation for electromagnetic waves
2

4 nO)
c

0)2

•

V E- VV oE+2 £E+--::r J =0

c

or

(2.113)

2

(2.114)
c
A similar equation can be derived for B. Different approximations for the
equation of motion lead to different expressions for the quantities e and (J that
enter the equation for E (and the similar equation for B).
Equation (2.114) is extremely complicated. When we consider a wave propagating normally to a plane parallel atmosphere, say in the Z-direction, Equation
(2.114) reduces to
V2E-VV oE+;'E'E=O.

(2.114')

Consider now Equation (2.113) for propagation in the Z-direction. We obtain

(2.115)

and

In Equation (2.115) we now insert the simplified form of (2.92) and of Ohm's law
and neglect the Lorentz force, j x B; i.e.,

p

dol.

i}t

jyBo
.
=c- =-IO)PV

l.

and
E y -- vzBo •
c

The resulting waves are then governed by the equation
(2.116)

66

CHAPTER 2

and represent a wave propagating through the plasma whose dielectric constant
Eisgiven by
41rpc2

e=1+-2- ·
Bo

These are called Alfuen waves and even though they are often treated as a special
form of oscillation, the above derivation points to the fact, already emphasized
by Spitzer (1962), that they are bona fide electromagnetic waves, deducible from
the basic Equation (2.114). What set the Alfven wave somewhat apart is the very
high value of E when regarded as a normal electromagnetic wave. Since E » I,
the wave speed, the Alfven speed, is
(2.117)
Before we leave this general discussion of the basic equations, let us return to
the fundamental question of the coupling between the electromagnetic field and
the material motion. For many practical purposes, we can write Ohm's law in the
form
(2.118)
where the electric field Ev is the field measured in a system moving with the velocity v of the plasma. It may contain both an externally applied component E as
well as the one induced by the motion of the plasma, vIc x Bo. On the other hand,
the magnetic field is independent of the coordinate system in the nonrelativistic
case and is made up both of any externally applied field and the induced field.
When the electrical conductivity is very large, Ev must be very small for j to
remain finite, and hence we have

v
E+-xBo =0;
c

(2.118')

that is, the electric field is approximately equal to the induced electric field v x Bo.
If we do not consider very rapid fluctuations, Maxwell's displacement current
will be negligible compared to the curl of the magnetic field, and the appropriate
Maxwell equation is
41r.
V x B =-J.

c

INTERPRETATION OF OBSERVATIONAL DATA

67

Next, if we approximate Ohm's law in the form
I. I
c
I
E=-J--vxB =--VxB--vxB,
eJ
c
4neJ
c

we can, with the two last equations, express all the electromagneti!- variables in
terms of the magnetic field B. Therefore, in the important case of very large electrical conductivity, all but the very rapid oscillations can be treated in terms of
the interaction of the magnetic field with the gas dynamic variables.
We will often find that the equations of hydromagnetics can be written in an
approximation sufficient in many plasmas in a form slightly simpler than the one
given so far. By neglecting the displacement current and the dielectric properties
of the plasma (assuming no polarization of the ions) and considering a nonmagnetic medium, we find that
n

v

4n.
x B =-J,
c

laB
c at '

VxE=---

v .j= 0,

V.B=O.

Using Maxwell's equations in this form together with Equation (2.118) for the
current density, we can deduce-provided that the electrical conductivity is uniform in space-an equation that describes the changes in the magnetic field, the
so-called magnetic diffusion equation:
(2.119)
If the plasma is at rest, we have v = 0, and Equation (2.119) reduces to

(2.120)
an equation that shows how the magnetic field diffuses through the plasma. Here
llm == c2 / 47t0" is defined as the magnetic diffusivity, and from dimensional arguments one concludes that the time of decay of the field due to this resistive diffusion, or Joule diffusion, is
(2.121)
where L is a characteristic length of the plasma in which the currents flow.
If the plasma has infinite electrical conductivity, Equation (2.119) reduces to

aB/at=Vx(vxB).

(2.122)

CHAPTER 2

68

We interpret Equation (2.122) to mean that in a plasma of infinite conductivity,
the magnetic lines of force move with the fluid, or in the nomenclature of Alfven,
the lines of force are "frozen" into the material. Motion along the lines of force is
not affected, but when material motion perpendicular to the lines of force occurs,
the lines are carried with the material.
From dimensional analysis we find again that the transport expressed by
Equation (2.122) dominates over the diffusion as given by Equation (2.120) when
LV/TIm »1. The ratio
(2.123)
is defined as the magnetic Reynold's number, in analogy with the equation defining
the Reynold's number of fluid dynamics R = LV/TI.
It is important to realize the restrictions that apply to the use of the "frozenin" concept, a notion that can easily lead the researcher astray. In prominences,
even though the electrical conductivity is large, it is not infinite, and electric
currents will be generated by the motion of the charged particles. It is the
generalized Ohm's law that governs this situation, and one must realize that
drastic simplifications have been invoked to deduce the magnetic field-line
picture.
Last, let us introduce another useful, but again often misused, concept: the
force-free magnetic field. In the solar plasma energy can be built up in the
magnetic field by stressing the field configuration, resulting in a non-potential
field. If the buildup occurs due to mass motions with velocities small compared
to the Alfven speed, one can at each instance approximate the magnetic field by
a static configuration. Hence, the equation of motion (Equation (2.92» becomes
1

-jxB =Vp-pg.
c

Using Amp~re's law (Equation (2.107», we find
1

-(VxB)xB=Vp-pg,
4n

and, after some rearranging of terms:

(B2)

1
-(B.V)B=V
-+p -pg.
4n
Sn

(2.124)

INTERPRETATION OF OBSERVATIONAL DATA

69

Probably everywhere in the solar atmosphere p and pg are small compared to
B2/8n; hence, to a good approximation, Equation (2.124) can be replaced by the
equation for a force-free field, i.e., vanishing Lorentz force j x B:
(VxB)xB=O.

Apart from the trivial case B
for a potential field, and for

= 0, Equation (2.125) is satisfied for
VxB= aB,

(:2.125)
V x B =0, i.e.,
(2.126)

where a = a(r) is a scalar function. Equation (2.126) represents a stressed, nonpotential field where the current j = V x B is everywhere parallel B. For a general
function a(r), Equation (2.126) is complicated. Solutions have been discussed for
an a everywhere constant in space; however, this may be an overly simplified
situation, since there is a priori no reason why a should be constant everywhere
in the domain of interest.
2.3. Waves in a Plasma
There are a bewildering number of wave modes possible in a plasma, and we
shall not go into details regarding their generation and propagation here. We
shall give, however, a brief account of the more important modes that may come
into play in the realm of prominence physics. We recall that in empty space only
electromagnetic waves are possible, their behavior being governed by Maxwell's
equations, Equations (2.107)-(2.110). If we, e.g., consider plane waves propagating in the Z-direction, these equations simplify, and with
B = ,uoH, D = foE,

(2.127)

where Ilo and £0 are the permeability and permittivity of free space, we obtain instead of Equation (2.116) the following equation for this wave propagation:
(2.128)
These waves, therefore, travel through space at a velocity c = 1 / ~£o~o = velocity
of light in vacuum =2.998 x 1010 em s-l.
In Section 2.2.2.4 we have already derived the equation for Alfven waves
(Equation (2.116», waves whose treatment really belong in the present section,

70

CHAPTER 2

but we presented them earlier to show their special electromagnetic nature and
the special role they play in discussions of solar physics.
In general we find that there are typically four modes of wave motion possible in a plasma like the solar atmosphere. The modes may be discussed in terms
of the restoring forces present. The tension and pressure of a magnetic field alone
drive magnetic waves, the Alfven waves. Acoustic (sound) waves owe their existence
to the plasma pressure, while gravity may excite gravity waves. However, when
these three forces act together, we get mixed wave forms, viz. two magnetoacoustic
gravity waves. With no gravity we refer to these modes as magnetoacoustic
waves, and similarly with no magnetic field they become acoustic gravity waves.
2.3.1.

ACOUSTICWAVES

These waves owe their existence to the compressibility of the plasma. The perturbation velocity, v', is in the direction of propagation, k. From elementary
physics, we find their dispersion relation
(2.129)

ro=kVs,

where the frequency ro and wavenumber k define a plane-wave solution of the
form
v'(r, t) =v' exp[i(k • r - co t).

(2.130)

These waves propagate in all direction at a phase speed Vph == rolk =Vs , and
with a group velocity Vgr =dro I dk =Vs.
When the pressure obeys the adiabatic law, p/pY =const, the sound speed in
given by

V; r.!!..= rkT,
5i

p

m

(2.131)

where y is the ratio of specific heats, cp/cv, cpfor constant pressure, Cv for constant
volume. With y = 5/3 and m = 0.5 m (proton), Vs '" 1.67 x 1CJ4.JT em s-l. This expression gives for the sound speed in the corona a value about 200 km s-l, and
about 10 km s-l for a quiescent prominence.

INTERPRETATION OF OBSERVATIONAL DATA

71

2.3.2. MAGNETIC WAVES

In this case the restoring force is the Lorentz force, j x B, see Equation (2.92). This
force acts perpendicularly to B, and consists of a magnetic tension force B2/41t, a
magnetic pressure force B2/81t,
j xB

~ (V

x B) x B

~ (B. V)B - V(B2/8n).

The magnetic tensio~produces a transverse wave along the magnetic field with
velocity VA = BI(41tp) ,i.e., the Alfven speed (Equation (2.117».
The magnetic pressure generates in its turn longitudinal magnetic waves that
propagate across B, again at the Alfven speed. We, therefore, have two types of
Alfven waves, given by the equation
(2.132)

See, e.g., Priest (1982) for a good discussion of waves in a plasma, and derivation
of dispersion relations.
H the plasma is incompressible, i.e., V .v' = 0, the product k.v' = O. One can
then show that
(2.133)

These Alfven waves, called shear Alfven waves, have a phase velocity VA cos eB,
and for propagation along the magnetic field, cos eB = I, their velocity is the
Alfven speed; Equation (2.117). On the other hand shear Alfven waves do not
propagate perpendicularly to the field, where eB = 90°; see Figure 2.9 for the
variation of the phase speed with eB.
Equation (2.132) also has the solution
(2.134)

which pertains to compressional Alfven waves. Their phase speed is always VA (see
Figure 2.9), and they have a group velocity Vgr = k VA, indicating that energy is
propagated isotropically by these Alfven waves. For comparison with velocities
of other wave forms treated in this section, we notice that the Alfven velocity, VA
is typically of the order of 300 km s-l under coronal conditions with B =10 Gauss.
For a prominence with B = 200 Gauss and ne = 1010 cm-3 we find VA'" 70 km s-l.
2.3.3. MAGNETOACOUSTIC WAVES

Magnetoacoustic, or magnetosonic, waves obey a dispersion relation of the form

72

CHAPTER 2

(2.135)
where V A is again the Alfven speed (Equation (2.117» and aB is the angle between the direction of propagation k and the equilibrium magnetic field vector B.
The dispersion relation (2.135) gives two solutions
(2.135')
viz. the fast and the slow magnetoacoustic wave. Since
(2.136)
one sometimes refers to the Alfven wave as the intermediate mode.
2.3.4.

GRAVITYWAVES

We have mentioned that gravity may excite a special oscillatory motion in a gas,
a mode called gravity waves. A blob of material that rises a distance ~z from
equilibrium in the gas, while remaining in pressure equilibrium with its surround,...-

I
I

\
\

I

\

I

\

/

"""

,,

",,'"

..... ..... .....

---- -... ...... ..........

.....

,,

\

............

_-_ ....... ""

;'

"" "

"

/

I

\

I

\

\

I

I

Fig. 2.9. Polar diagram of Alfven waves (solid curve) and compressional Alfven waves (dashed
curve). The phase speed, ro/k, in a direction given by the angle OB relative to the magnetic field is
given by the length of the corresponding radius vector.

INTERPRETATION OF OBSERVATIONAL DATA

73

ing and undergoing adiabatic density changes, will experience a buoyancy force
g(8po - 8p), where Po is the density at the original height z, and 8po the increase in
density outside the blob at z =8 z. Inside the blob the changes in density and
pressure at height z + 8z are 8p =8po =-pg8 z. The blob rises abiabatically, i.e.,
plpY= constant, and 8p = V~8p (see Equation (2.131», which gives for 8p the
expression

The buoyancy force may be written g(8Po - 8p) =-ro~vPo8z, where
WBV

=[_g(_1 dPo) + ...L]t
Po dz

V§

(2.137)

is the Brunt-Viiis~ilii frequency, when the expression (2.137) is real. The equation
of motion in this case is
(2.138)
and the blob will execute harmonic motion with frequency roBvif
(2.139)
Equation (2.137) can also be written in the form

wiv =..!...[dTo
_(dT)
],
To dz
dz ad

(2.137')

and we see that condition (2.139) holds when the temperature decreases more
slowly with height than adiabatic. This is known as Schwarzschild's criterion for
convective stability. If the temperature should increase too rapidly with height to
violate (2.139), the solution to the equation of motion (Equation (2.138» grows
exponentially, and convective instability sets in.

74
2.3.5.

CHAPTER 2

PLASMA OSCILLATIONS

We consider a quasi-neutral plasma as discussed in Section 2.2.2.1. If the plasma
is separated into positively and negatively charged parts, the parts will attract
each other and provide restoring force in the opposite direction to the original
separation. The result is simple harmonic oscillations about the equilibrium
(neutrality) configuration.
Consider the equations of continuity and momentum for the electrons in the
plasma (the ions, being much more massive, are less likely to contribute to the
oscillation), i.e., using Equations (2.83) and (2.90) with Do = g = Pei = Pe = 0,

~; + V • (ne v e) = 0,

and Gauss' law in the form

We shall consider motions small enough that these equations can be linearized,
and we seek solutions of the form given by Equation (2.130). After some rearrangements we find for the frequency the following expressions
(2.140)
where a subscript zero indicates equilibrium condition.
This is the electron plasma frequency, denoted Olpe. It depends only on the equii
librium density no and has the numerical value ffipe(s-l) =5.64 x 104 [no (cm-3)] 2
A cold plasma will oscillate with this frequency; however, since the group velocity dml dk = 0, waves do not propagate by this mechanism.
If the plasma is warm, however, thermal motion of the electrons will carry
information from one part of the plasma to another. The plasma oscillations now
have a finite group velocity Vgr = dml dk and represent a traveling wave. The dispersion relation for this wave is formed by adding a -VPe term to the momentum
equation. With an isentropic energy equation Pe oc nero The final relation for m is
given by

INTERPRETATION OF OBSERVATIONAL DATA

75
(2.141)

where ve =(kTe1me)t is the electron thermal speed and AD is the Debye length
(Equation (2.71». The group velocity for such waves is finite;

rv;

dw
2 k
V,r =-= rVe - = - - ,
dk

Vph

W

where Vpz is the phase velocity of the wave. The Rhase velocity can be arbitrarily
large, but the group velocity is always less than ytve'
Next consider wave motions associated with oscillations of the ions. Since
ions are relatively massive, they do not have wave amplitudes large enough to
actually collide with other ions, and they use the intervening electron gas as a
means of transmitting impulses. The ions have large inertia compared to electrons, and any ion motion is immediately accompanied by a corresponding electron motion-a process known as ambipolar diffusion. Therefore, although oscillating electrons can cause a change in E, the relatively slow oscillations associated
with ions do not.
We end up with an expression for the frequency given by
(2.142)

where"ie and"ii are the adiabatic indices for electrons and protons, respectively.
Both the phase velocity and group velocity are seen to be equal to the sound
speed,
(2.143)

In this case, even for vanishing ion temperature, the thermal motion of the electrons will carry the wave, but with the inertia term provided by the ions. Because
the thermal electrons resp~nsible for carrying the wave travel a relatively large
distance equal to (mj /me ) wavelengths in one wave period, the wave can be
considered isothermal, with "ie =1. Thus, in the limit Te» Tj,

V=(kTe)t =V·
9

the ion-acoustic speed.

mj

1,9

(2.144)

76

CHAPTER 2

2.4. Modeling-Computer Simulation

In the previous sections of this chapter, we have discussed some tools necessary
to properly interpret solar observational data. These tools are first and foremost
spectroscopy, based on atomic physics, and magnetohydrodynamics, incorporating transport phenomena. During the two last decades, a somewhat different
type of tool has been developed, namely, model building and simulation theory,
based on numerical computations. The incorporation of this tool in solar physics
research has led to new and exciting insight also in the case of prominences. We
shall consequently briefly consider some aspects of this technique, but the field is
much too broad to cover it completely in this text.
Perhaps no area of applied mathematical physics has seen a more impressive
growth in the last two decades than the model building by computer simulations
based on massive numerical calculations. The reason for this evolution is the
rapidly increasing availability of large (vector) computers, and the necessity for
having this new tool is the complexity of the systems of non-linear equations that
govern the different areas of the physics involved. Solar scientists are increasingly taking advantage of having this tool, and many of the prominence
models owe their sophistication to it. However, this powerful tool with all its
possible applications may also pose a problem. Our intuition can only give us
limited assurance that an obtained numerical solution really applies to the physical problem we are investigating. Therefore, great care should be exercised when
interpreting the simulations in terms of actual physical conditions. One has to
realize that the "back-of-the-envelope" calculations may help steer one away
from accepting unreasonable solutions. However, when that is said, one also has
to realize that further insight into the complex interplay of the many physical
forces as encountered in prominence research is possible to a large extent only by
employing numerical calculations to simulate the actual phenomenon.
In considering numerical solutions, the question arises whether these solutions are legitimate solutions, i.e., whether they are self-consistent. To answer
this question one needs to consider the boundary conditions for the equations
describing the physical situation one is investigating. In a laboratory, one can
often assign specific values for the physical parameters at fixed boundaries. In
the solar atmospheric plasma, we have free boundaries and most dynamic processes of interest give initial-value problems. The boundary conditions evolve as
the dynamic process evolves, and we are faced with a free boundary value problem. Mathematically, this means that there is a coupling between the solution
and the boundary conditions in this initial-value problem.
A number of authors have discussed the solar applications of this type of
initial-value problem, and, even though it is beyond the scope of this text to go
into much detail concerning the solution of initial-value problems, we shall highlight a few points of interest regarding applications to solar physics.

INTERPRETAnON OF OBSERVATIONAL DATA

77

It can be shown (Wu and Wang, 1987) that the correct set of boundary
conditions should be derived from the method of characteristics, the basic idea of
which is as follows.
Consider a partial differential equation for a function f(x,t) of two independent variables x and t

(2.145)
and write it as a total differential equation of the form
df =g.
ds

(2.146)

This transformation is possible since the directional derivative in the xlt-plane
along the curve s(x,t) is given by
df a.r at a.r ax
-=--+-ds at as ax as ,

(2.147)

and the direction of the curve s(x,t) is

-dx
=-( as I at) I (as I ax) =u.
dt

(2.148)

This means that the solution of Equation (2.145) may be obtained by integrating Equation (2.146) along the curve s(x,t). This applies even for the nonlinear case where u and g are general functions of x and t in the form

J
s

f(s) = f(O)+

gds,

(2.149)

o

where f(O) is the value of f(x,t) at s = O. The method allows solutions of nonlinear hyperbolic partial differential equations, and the curves along which total
differential equations can be derived are called the characteristics (e.g., Akhiezer
et al., 1975). The resultant equations are the compatibility equations (Sauerwein,
1966).
In general one constructs solutions for the initial-value boundary problem by
using the characteristics directly. However, with more than one spatial
dimension, this direct application of the method of characteristics becomes an
exceedingly complicated numerical procedure. To remedy this situation,

78

CHAPTER 2

Nakagawa (1981a,b) developed the method of projected characteristics which was
studied further by Hu and Wu (1984). This method is designed to reduce the
complexity by writing the compatibility equations projected onto the component
planes: (x,t), (y,t), and (z,t)-whence its name. It was first tried by Shih and Kot
(1978) for a two-dimensional hydrodynamic problem, and used in the threedimensional case to compute plasma wave and mass motions (Wu et al., 1983,
1986) and energy buildup in solar flares (WU et al., 1984). Nakagawa et al. (1987a)
studied the evolution of magnetic arches in the solar corona when their footpoints are subjected to shear motions, a situation that may be applicable to
prominence structures; see Figure 2.10.
The numerical simulation methods developed over the last two decades are
very powerful and can provide new insight into the physical processes taking
place under conditions found in solar plasmas. Nevertheless, one needs to bear
in mind that even with supercomputers, the simulations cannot match realistically the physical time- and spatial scales in many active region solar events. This
is due to limitations in memory capacity and computation speed for desired resolution and accuracy. Also, the large number of computational operations will introduce inherent truncation errors that will affect the accuracy of the computation and prevent a realistic simulation of the physical phenomenon under study.
One possible way out of this dilemma is to construct a simulation model on the
basis of the best available resolution and accuracy and then scale the physical
phenomenon with the simulation model and its physical parameters using classical similitude principles (Kalikhman, 1967). This is the method used, e.g., in airplane design where small-scale models are tested to predict the characteristics of
the full-scale model (corresponding to the solar phenomenon in our case). For
this procedure to be valid, certain similitude rules must be followed, and one develops scaling laws and iimensionless similitude parameters. We have already
encountered one such parameter, viz. the magnetic Reynold's number (Equation
(2.123». For an example of the application of this procedure to a solar
atmospheric phenomenon, see, e.g., Wu et al. (1988).

INTERPRETATION OF OBSERVATIONAL DATA

103 km

79

Z

8
7

6

(a)

5

4

3
2
1

X
1 2

103 km Z

3 4

T

8

5

6

7

8

103 km

= 2,000 s

7

6
(b)

5

4

3
2
1

X

00

103 km

1 2

Z

3 4
T

8

=

5

6

7

8

103 km

2,500 s

7

6

(c)

5
4

3
2

X
1 2

3 4

5

6

7

8

103 km

Fig. 2.10. Computed evolution of magnetic arches due to shearing motion of their footpoints. T refers
to time, in seconds, for a maximum shear velocity of 2 km S·l (after Nakagawa et ai., 1987a).

CHAPTER 3

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

With the help of the framework laid down in Chapter 2 we are now in a position to interpret pertinent observations in order to derive values for the crucial
parameters that describe the prominence plasma, e.g., temperature, density,
degree of ionization, magnetic field, and velocity.
3.1. Temperature

The temperature is one of the better-known parameters of the prominence
plasma, at least for some types of prominences. In quiescent objects where there
is reason to believe that optically thin spectral lines from different elements,
e.g., hydrogen and metals, or even hydrogen and helium, come from the same
volume elements of the plasma, one may use Equations (2.51) and (2.52) to deduce both the electron temperature and the temperature-mimicking microturbulence. One measures the Doppler widths aAD of the lines and inserts in Equations (2.51) and (2.52). Typical values for the solution set {T,;t} are

T =5000-8000 K
~t = 5-8 km s-l

(3.1)

(Hirayama, 1985; Engvold and Brynildsen, 1986; Zhang et al., 1987; Mein and
Mein, 1991). Temperatures as low as 4300 K have been reported by Hirayama et
al. (1979), who also found non-thermal velocities of 3.4 km s-l, from studies of
Balmer lines, He I lines, and certain metal lines (from Ti, Fe, AI, and Mg).
There is a tendency for both T and ;, to increase somewhat from the central
parts of the prominence toward the outer edges adjacent to the corona
(Hirayama, 1971), where the temperature may reach values of 1()4 to 2 x 104 K,
and the non-thermal velocity attains values of 10 to 20 km s-l. However, there
is still controversy on this last point, in that some observers do not report,:lY
increased temperature or velocity in the outer parts of quiescent prominences;
Zhang et al. (1987) even found the values to decrease away from the central
regions. The situation is complicated by the fact that prominences are made up

82

CHAPTER 3

of fine-structure elements, as well as being imbedded in a hot corona via a

prominence-corona transition region. It is, therefore, not obvious how the tem-

perature or the microturbulence should behave as we extend the observations
across a prominence. We shall return to the implications concerning both the
fine structure and the transition region later when we discuss prominence models.
With the advent of space-borne instrumentation it is possible to observe
prominences in lines from the UV part of the spectrum; e.g., the C 1,1665 A line,
the H I, 1215 A line, the He II, 1640 A line, the C N, 1548 A line, the 0 IV, 1402
A line, and several others. These lines also show up in quiescent-looking prominences, albeit sometimes with low intensity (Poland and Tandberg-Hanssen,
1983). To ionize some of the atoms in question and excite the observed lines requires temperatures far in excess of the values quoted for quiescent prominences.
While a picture in the C I line is almost identical to the Ha picture, indicating
a temperature of about 5000 K, the details of a picture of a prominence in the C
IV or 0 N line differ from the Ha picture. It seems that even semi-quiescent
prominences contain some hot plasma associated with the fine-structure
elements of these objects (see Section 5.3.1).
Considerable work has been done also on the thermal structure of active
region prominences. The spectral lines in these objects are markedly broader
than the low-temperature lines in quiescent prominences. If we write Equation
(2.50) in the form
A (';2
.!lAD =-V ~6,
(3.2)
c

i.e., where we have introduced the concept of a line-broadening velocity, we
can describe the widths of the observed lines by this velocity parameter. In the
outer edges of some quiescent prominences this line-broadening velocity may
reach 10 km s-1 and there exist quasi-quiescent prominences where ~o is typically 12-15 km s-1. We might call them spectroscopically semi-active, even
though they show no sign of activity in a dynamical sense of the word. Things
are quite different when we look at the hot regions somehow embedded in quiescent prominences. The line widths in these hot fine-structure elements indicate values of ~o of 30 km s-1 or more. Similar conditions are found when we
come to bona fide active prominences, i.e., active region filaments associated
with sunspots, surges, and, in particular, loop prominences. Often the lines in
these objects retain a nearly Gaussian shape so that we can talk about their
broadening velocity, which may reach values in excess of 50 km s-1. In loops the
line half widths, or the ~o, are generally not constant with height, being
greater in the highest part of the loop in the corona from whence they seem to
originate.
On the other hand, in surges that are shot up from below the opposite effect
takes place: here the halfwidth decreases with height. Table 3.1 shows the
variation of halfwidth expressed in terms of the broadening velocity in two

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

83

observed cases. The data point to the physical reason for the differences
between the two kinds of prominences. A change in So is to be interpreted as a
change in the activity (meaning now temperature and/or motion) of the
emitting gas. The data in Table 3.1 then show that the temperature and/or
motion is less in the upper part of surges than near their base. Since matter is
being shot up from below, the plasma will gradually relax as it rises and will
show less sign of activity the higher we observe it. On the other hand, loops
show decidedly more activity near their tops and are intimately connected
with the hot plasma above from which they derive their energy. We shall
return to the physics of these most interesting prominence types later.
TABLE 3.1

Variation of broadening velOcity, So' with height, h,
in different kinds of prominences

Loop

h(km)

17,000
21,000
25,000
29,000
34,000

38pOO

He I (So{km s-1»
38

40
50
50
53

40
52
48
56
43
48

Surge
h(km)
BOO)

12,500
17,000
21,500
25,000

57
61
59
49
47

90
81
69

We can now use the helium data to draw conclusions regarding the structure of
active prominences. The excitation of He II lines, like the He II, 4686 A line
(the n =4 --+ n =3 transition), the He II 1640 A line (3 --+ 2), and the resonance
line He II, 340 A (2 --+ 1), requires much higher temperatures of the order of 5 x
104 K than those pertaining to the hydrogen plasma, and we are hence faced
with a multicomponent thermal model for these prominences. It is uncertain

84

CHAPTER 3

how one should visualize this thermal fine structure. It may consist of rope-like
features in which the hydrogen emission comes from their cool cores and the He
II emission originates in the hotter outer layers of the ropes (compare Foukal,
1975). On the other hand there may be separate hot and cool ropes for the high
excitation lines and for the hydrogen-ionized metals-neutral helium lines. The
latter case was advocated for a loop prominence by Cheng (1980) who observed
lines from ions with temperatures ranging from 5 x IQ4 K to 3 x 1()6 K. The loops
observed in the relatively lower temperature lines (T < 106 K) like He II, Ne
VII, and Mg VII were systematically different from the picture of the loop as
observed in the higher temperature (T > 1()6 K) lines, like Si XII and Fe XV; see
Figure 3.1. We shall return to the question of the nature of the fine structure in
Chapter 5.
o

He 11304 A

o

Ne VII 465 A

o

Mg IX 368 A

A

o

Si XII 499 A

o

Fe XV 284 A

o

Fe XVI 335 A

Fig. 3.1. Loop prominence observed in lines emitted at different temperatures (after Cheng, 1980).

These very hot loops raise the question whether we really should consider
them prominences. They normally occur after flares, as the name post-flare
loop emphasizes, and are then an integral part of the flare phenomenon. However, they also exhibit many traits easily defining them as active prominences,
and we shall include them in our discussions.
Without going to the rather extreme temperature regime in some loops, we
can get important information from other active prominences. Vial et al. (1980)
studied an active region filament in the 0 VI, 1032 A line at about 3 x 105 K and

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

85

found a non-thermal velocity of about 30 km s-l, not unlike the ~ value from the
resonance lines of H I, Mg II, and Ca II at much lower temperature (Vial et al.,
1979). On the contrary, Kjeldseth-Moe et al. (1979) reported that the turbulence
in prominences observed by them increased with temperature from less than 10
km s-l to about 30 km s-l as they went from lines emitted at 1()4 K to lines at 3 x
lOSK.
Mein et al. (1989b) studied a surge in Ha and in the C IV, 1548 A line,
assumed temperatures of 2 x 1()4 and lOS K, respectively, and calculated the
microturbulence, ~, from Equation (2.50). Both lines gave high values of ~,
decreasing with time during the surge event from about 30 km s·l to 15 km s·l.
Even though uncertainties exist and details remain unsolved, a picture of
the thermal structure of both quiescent and active prominences has emerged,
where the fine structure of the prominence plasma is the dominant feature. This
fine structure, which, as we shall see later, is intimately linked with the magnetic field in the prominences, provides adjacent plasma elements of widely
different temperatures, thereby allowing emission from atoms and ions of
widely different degrees of ionization and excitation to be observed from the
same overall prominence body.
3.2. Spectroscopic Classifications
We have seen above that there are differences between the relative intensities
of spectral lines from a quiescent and from an active prominence. While all
prominences exhibit strong lines of H (especially early Balmer lines) and of
ionized Ca (in particular the K and H lines), the relative intensities of He I
and He II and metal lines vary strongly from one type of prominence to another.
This situation has been used to classify prominences, using the relative strength
of different spectral lines as criteria (see Figure 3.2). The quiet chromosphere
might provide the comparison basis, since line-intensity ratios are well known
there. The conditions in the chromosphere change with height, from the lower,
cooler part below about 1500 km, to the higher chromosphere and the lineintensity ratios change correspondingly (Thomas and Athay, 1961). For example, the low chromosphere is characterized by weak He I lines and strong metal
lines (from Ti II, Fe II, Ba II, etc.). Higher up, around 1500 km, the He I lines
attain the same strength as the metal lines, and in the high chromosphere, i.e.,
in spicules, the intensity of He I lines exceeds the metal-line intensity.
The first modern spectral classification is due to Waldmeier (1949, 1951,
1961) who used the b lines of Mg I (bl at 5184 A, b2 at 5172 A, and b4 at 5167 A)
and compared their intensities with that of b3 of Fe II at 5168 A (see Table 3.2).
Waldmeier found that while flares generally fall in classes IV and V (high excitation objects), prominences belong to class III, with a few in classes I and II.

86

CHAPTER 3

Feb. 19, 1962

Sept. 22, 1962

- HeII,4686

- FeII,4584
-TiII,4572
-TiII,4564
- BoII,4554
- Till,} 4 550
FeII,

- TiIT'}4534
Fen,

- TiII,4501

-TiII,4444

Fig. 3.2. Spectrum of a quiescent prominence (22 September 1962) and of an active prominence (19
February 1962) (courtesy HAO, Boulder, Colorado).

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

87

This classification distinguishes flares and prominences, but does not significantly subdivide prominences.
TABLE 3.2
Waldmeier's classification
Criteria
I

II
III
IV
V

I(b 3 )<I(b4 )
I(b 3 ) =I(b4)
I(b 4) <I(b 3 );S;I(b 2)
I(b 2 )<l(b3 );S;I(b 1 )
I(b 3 » I(b 1 )

Objects

Prominences

Flares

Another classification of solar atmospheric objects is due to Zirin and
Tandberg-Hanssen (1960) (see Table 3.3). This classification is based on the
multi-component model, i.e., on the idea that flares and active prominences
consist of different regions. Some regions are cool, less than 104 K, with strong H
and metal-line emission, while the He emission is weak. Other regions are
composed of a hotter plasma with temperatures in excess of 104 K, giving rise to
strong He and faint metal-line emission. A comparison between the classification lines gives a measure of which regions dominate a given object. As seen in
Table 3.3, such a comparison furnishes a convenient way of classifying different
aspects of solar atmospheric activity and of comparing it with the chromosphere (the quiet Sun). Flares and active prominences fall in the same class III,
characteristic of high chromosphere, indicating that, spectroscopically, active
prominences are more like flares than like quiescent prominences. The latter
fall in class II and may be considered similar to the middle chromosphere in
excitation. Extreme quiescent prominences may even border on class I.
The spectroscopic distinction between flares and active prominences, which
is brought out in Waldmeier's classification, is not present in the original
Zirin-Tandberg-Hanssen classification. To remedy this, the intensity of certain
metal lines that behave differently in flares and in active prominences was introduced as classification criteria (Tandberg-Hanssen, 1963). The distinction
can be made by using lines of Fe II, which are strong in flares, and comparing
them with lines of Ba II and especially Ti II, which characterize spectra of
prominences. Since the other classification lines in the easily observable bluegreen part of the spectrum lie around 4600 A, the 4584 A line of Fe II (multiplet
38) and the Ti II line at 4572 A (multiplet 82) were chosen as our flare/active
prominence discriminators. The ratio M = I (Fe II, 4584)!I(Ti II, 4572), where
I(Fe II, 4584) and I(Ti II, 4572) are the intensities of the two lines, respectively,

88

CHAPTER 3

will then classify an active limb event as a prominence when M < 1 and as a
flare when M ::? 1.
TABLE 3.3
Zirin-Tandberg-Hanssen classification
Class

Criteria

Objects

I(He I, 4026) « I(Sr II, 4078)
I(He I, 4713) « I(Ti II, 4572)
I(He II, 4686) « I(He I, 4713)

Low Chromosphere

II

I(He I, 4026) "" I(Sr II, 4078)
I(He I, 4713) "" I(Ti II, 4572)
I(He II, 4686) «I(He I, 4713)

Middle Chromosphere
(around h =1500 km)
Quiescent Prominences

III

I(He I, 4026) » I(Sr II, 4078)
I(He I, 4713) »I(Ti II, 4572)
I(He II, 4686) "" I(He I, 4713)

High Chromosphere
(spicules)
Active Prominences Flares

IV

Presence of
Fe X, 6374,
Fe XIV, 5303,
Ca XIII, 4086, or
Ca XV, 5694

I

Coronal Enhancements
Coronal Condensations

3.3. Density

If the temperature-at least in the cooler parts of quiescent prominences-is
known to within better than 50%, the density, on the other hand, is uncertain
by a factor of easily 5. The reason for this difference in our precise knowledge of
two of the main plasma parameters is to be found in the way we obtain the information from the prominence spectra. In the simplest case the halfwidth of a
spectral line can be interpreted directly in terms of a temperature. As we shall
now see, density does not follow as conveniently.
We owe to Hirayama (1972, 1979, 1985) some of the basic work on density
determinations in quiescent prominences. The best method seems to be via the
Stark effect of higher Balmer lines; see Section 2.1.4. Hirayama gives a mean
value for the electron density in five quiescent prominences of the hedge-row

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

89

type (Section 1.2) of 1011 cm-3, and ne = 3 x 1011 cm-3 for the average of two
curtain-like prominences. Higher values for ne have been reported using InglisTeller's formula (see Equation (2.56» (Ivanov-Kholodny, 1959). However, often
estimates based on this equation should be considered upper limits, since the
difficulty experienced in observing the higher Balmer lines may prevent detection of lines with principal quantum number n > 30. A careful analysis by
Jefferies and Orrall (1963) led to values for ne =5 x 1010 to 1011 cm-3, when they
resolved Balmer lines up to n =36.
One of the more powerful methods to determine number densities is by
studying line-intensity ratios that depend directly on the value of the density.
This method has been explored by Landman (1983a,b, 1984) in a series of papers
on quiescent prominences. He used observations of the resonance lines of Na I
and Sr II (later also Mg D, i.e~, the lines Na I, 5890 A and 5896 A (3 2p-3 25) and
Sr II, 4078 A and 4216 A (5 2p-5 25) (see Section 2.2.1.1 for the nomenclature),
and pointed out that for the atomic levels in question the dominant ionization
and recombination processes are photo-ionization and radiative recombination.
Under such conditions one has n(Na I, 3 2p) oc nen(Na 11). Landman assumed for
the prominence plasma that n(Na II»> n(Na I), so that n(Na II)=n(Natotal).
The resonance lines both of N a I and of Sr II are formed mainly by resonant scattering of disk radiation, and we write n(Sr II, 5 2p) oc n(Sr II, 5 25) =n(Sr II)
=n(Srtotal). Landman used such considerations to argue that by measuring the
ratio of the Na I to Sr II line intensities, one can determine ne, and he performed
the relevant statistical equilibrium calculations. His results give <ne> =1.6 x
1011 cm-3 as an average for a number of quiescent prominences.
With the advent of satellite observations, it became possible to also use
line ratios from spectral lines in the UV part of the spectrum, as well as to use
X-ray line ratios. However, one has to bear in mind that lines in the UV part of
the spectrum come mainly from the prominence-corona transition region, and
may not give relevant information as to the interior parts of quiescent prominences.
Wiik et al. (1993) used the three pressure-sensitive line ratios: (a) 1(0 IV,
1401)/1(0 IV, 1404), (b) [1(N V, 1238 + 1242) + I(C IV, 1548 + 1550»/I(N IV,
1486), and (c) 1(0 III, 1666)/I(Si IV, 1402). In ratio (a), the 0 IV lines belong to
the same multiplet, 2s22p 2p-2s2l 4 p (see Section 2.2.1.1), and the excitation
temperatures are practically the same. Furthermore, the excitation rates all
have the same temperature dependence (Dere et a1., 1982). In ratios (b) and (c),
the line ratios are made up of inter system and allowed lines. Such ratios
should be used for density diagnostics only when the density is high enough to
assure that the population of the upper metastable levels is in equilibrium
with the ground level population (Feldman et a1., 1977). The reliability of
using ratio (c) has been discussed by Doschek et al. (1978).
Wiik et al. (1993) found that all three ratios (a), (b), and (c) gave a mean
electron density ne "" 1011 cm-3 for a quiescent prominence observed with the High

90

CHAPTER 3

Resolution Telescope and Spectrograph (HRTS) (Bartoe and Brueckner, 1975).
This value is comparable to values derived by Poland and Tandberg-Hanssen
(1983) for a prominence using the Si IV, 1402/0 W, 1401 ratio (see Cheng et al.,
1982).

.
While ne = 1011 cm-3 may not seen unreasonable for the quiescent prominence
plasma, a number of authors, working with the Hanle effect (Section 2.2.1.2),
consider the value too high. Bommier et al. (1986) have analyzed the simul-

taneous measurements of the polarization parameters of the Na I, 03 line and
the H~ line. One gets four quantities, the degree of polarization and the direction for each line (Equations (2.68) and (2.69», and from these one can determine
the three coordinates of the field vector B, and the fourth can be the electron
density. This last determination is possible because for increasing densities, collisional depolarization becomes important, mainly due to electrons; the effect
of protons is smaller. This effect is important in hydrogen lines, but can be
ignored in studying the formation of the He I, 03 line. Consequently, it is the
simultaneous observations of the H~ and 0 3 lines that have permitted the
"Hanle determinations" of the electron density. The sensitivity of the H~
polarization to electron and proton collisions falls in the 109-1011 cm-3 range for
ne' For ne > 1011 cm-3 the depolarization is very efficient, and the degree of
polarization of H~ cannot be measured. A density ne > 1011 cm-3 is, therefore, not
compatible with the observed degree of polarization in H~. Sahal-Brechot
(1984) quotes ne = 7 x 109 cm-3 for quiescent prominences, Bommier et al. (1986)
give <ne> = 1.3 x 1010 cm-3 as the mean of 14 prominences observed at Pic-duMidi. Such low values are also indicated by recent non-LTE radiative Ha calculations which, when compared to observed intensities, are consistent with
electron densities in the range 1-5 x 10 10 cm-3 (Wiik et al., 1992); see Section
5.8.2.1. Furthermore, observations of a prominence in the millimeter wavelength range (Bastian et al., 1993) gave an electron density of 1010 cm-3. The best
we can say at the present time is that the cooler parts of quiescent prominences
probably have an electron density given by
(3.3)

If our knowledge of densities in quiescent prominences is somewhat uncertain, it is even more questionable for different kinds of active region prominences. Nevertheless, spectroscopic studies of surges and loops have been done,
and, in most cases, values of ne - 1011 to 1012 cm-3 are quoted. Again, like for quiescent objects, there is a tendency also for active prominences to be considered
less dense than they were quoted a decade or two ago. In a careful analysis of
the Ha emission from loops, as normally seen in absorption on the disk in tworibbon flares, Heinzel et al. (1992) give ne =5 x 1010_10 11 cm-3 (see also Svestka
et al., 1987). Hanaoka et al. (1986) deduced ne = 1011 cm-3 in a loop from the
electron scattering continuum; see Equations (2.28) and (2.34). Only when the

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

91

loops at times can be seen in emission on the disk would the density exceed 1012
cm-3 (Heinzel and Karlicky, 1987; Svestka et al., 1987).
3.4. Degree of Ionization
We shall see in Sections 3.5 and 3.6 that (1) all prominence plasmas are
pervaded by magnetic fields and (2) all prominence plasmas exhibit considerable motions. These important observations indicate that, depending on the degrees of ionization of the prominence plasma, strong electric currents may be at
work, and the material motions may be subject to stringent conditions as to
where they can flow. To understand the underlying physics we, therefore, need
to know to what extent we are dealing with a neutral gas or whether our
plasma is mainly made up of electron and protons, plus some other positive ions.
The radiation field that would ionize hydrogen in a prominence comes from
the chromosphere and the transition region between the chromosphere and the
corona. The penetration of this radiation into a prominence will depend on the
fine structure of the prominence (see Chapter 5) and on the optical thickness of
the Lyman continuous radiation. Landman (1983a,b, 1984) calculated the line
intensities to be expected under non-LTE conditions and derived a value of n (H
II) I n(H I) = 0.07 for the hydrogen ionization in quiescent prominences. Previous
values obtained by Hirayama (1979) are an order of magnitude greater, and
Kanno et al. (1981) estimated n(H II)ln(H I) :s; 1.9 for a prominence model,
building on information of the intensity of the Ly continuum and data from spectroheliograms of the Mg X, 625 A line.
With the higher temperature in active region prominences, as well as in
the outer parts of many quiescent prominences, conditions may be found where
n(H II) > n(H I). Heinzel et al. (1992) deduced n(H II) = 0.9 n(H I) for a loop
prominence. The best range of values for quiescent prominences is probably
0.05:S; n(H II)ln(H I) :s; 1.

(3.4)

In any case we can state that the conditions in prominences normally are such
that there always are sufficiently many electrons and ions present to make the
prominence behave like a plasma subjected to the influence of magnetic fields
on mass motions (see Equation (2.92».
3.5. Magnetic Field
In 1961 the first measurements of the longitudinal component of the magnetic
field in prominences were reported by Zirin and Severny (1961). Early measurements, all using the Zeeman effect (see Section 2.2.1.1), are summarized by

92

CHAPTER 3

Tandberg-Hanssen (1974) and Leroy (1979). By using more sophisticated instrumentation, all four Stokes parameters were subsequently measured and the
vector field determined. With the use of the Hanle effect (see Section 2.2.1.2),
a new technique was developed to measure the total (vector) field, and Leroy
(1985, 1989) has summarized the results.
For quiescent prominences the absolute field strength, I B I ,is generally
found in the range from a few Gauss to 10 Gauss, occasionally reaching 20 or 30
Gauss. Figure 3.3 shows examples of histograms giving the distribution of observed field strength (Tandberg-Hanssen, 1974; Leroy, 1989). A representative
value would be

(3.5)

IBI = 8 Gauss.

30

a
20
10

o

2

8

14

20

26

32

38

BII
30

r------------------------------.
b

20
10

o

32

38

Fig. 3.3. Histograms of magnetic field strengths measured in quiescent prominences. (a) Zeeman
effect measurements (Climax station of the HAO, Boulder, Colorado); (b) Hanle effect measurements (Observatoire du Pic-du-Midi; after Leroy, 1979).

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

93

While there is not unanimity, several authors find that the field strength
is slightly higher in the upper parts of a quiescent prominence. Furthermore,
there seems to be a weak correlation with the solar cycle: the prominence field
is somewhat stronger at solar maximum than at minimum, an effect first
pointed out by Harvey (1969).
Of the greatest importance for our understanding of prominence physics is
the question of how the magnetic field is oriented in and around the prominence
body. In the simplest case of a quiescent object, visualized as a plasma sheet
standing vertically in the solar corona above a magnetic neutral line, the magnetic field emanates in one polarity region, enters the prominence, exits from
the other side, and proceeds to an opposite polarity region. However, most indications are that the field vector B is not perpendicular to the prominence
sheet. Measurements show that the angle a between the vector B and the long
axis of the prominence is quite small (Tandberg-Hanssen and Anzer, 1970; Kim
et al., 1988). With Leroy et al. (1983, 1984), who studied the problem in considerable detail, we indicate that a has a strong maximum around 25°. Hence
we can give the representative value
(3.6)
As we shall see in Chapter 5 under prominence model, this small value of a
must be taken into consideration when we examine the role of the magnetic
field on prominence stability.
The structure of the magnetic field will determine its role in the support
and stability of prominence, and this will be discussed in Chapter 5. However,
from an observational point of view, we mention here that there are two main
field configurations found in and around quiescent prominences. The first, shown
schematically in Figure 3.4a, is the one associated with prominences where the
apparent polarity of the field across the prominence sheet is the same as the
polarity of the underlying photospheric field. We shall refer to these prominences as having a normal polarity field, N-type prominences. One of the early
investigations of such prominences is due to Kippenhahn and Schluter (1957),
and we often refer to these prominences as being of the Kippenhahn-Schliiter
type.
The other type of field configuration comes from the prominence model
initially proposed of Kuperus and Tandberg-Hanssen (1967) and developed by
Kuperus and Raadu (1974). This configuration is shown in Figure 3.4b and is
referred to as the inverse polarity field, I-type prominences, or the KuperusRaadu type. Leroy et al. (1984) have extensively studied both N- and I-type
prominences. As a rule I-type objects seem to be more numerous than N-types,
seem to be taUer, and to be typical of polar crown prominences. N-type prominences are lower than 30,000 km.

94

CHAPTER 3

a

+

+

Fig. 3.4. Two possible magnetic field topologies with embedded prominence. (a) Normal polarity; (b)
Inverse polarity.

While the magnetic field in quiescent prominences has been the subject of
numerous investigations and, therefore, is well known, less is known about the
field in active prominences. Nevertheless, good data have accumulated, and
we are starting to get a' fairly comprehensive view of the variety of magnetic
fields found in the different types of active prominences. As a matter of fact,
the first measurements made of magnetic fields in prominences (Zirin and
Severny, 1961) pertained to active objects, and Isohpa (1968) and Harvey (1969)
studied the fields in different types of active prominences.
One of the outstanding results from the major solar space research programs,
i.e., Skylab, Solar Maximum Mission, YOHKOH, is the realization that the
solar atmosphere is completely criss-crossed by electric currents that lead to
magnetic flux tubes of nearly all imaginable sizes, ranging from small loops in
and around plages to gigantic arches that span large distances between different active regions in which their feet are anchored. Some of these flux tubes
form the skeletons of the different kinds of prominences and constitute their
fine structure. Since the scale of this fine structure is much smaller than the
spatial resolution available in existing magnetographs, we always integrate
over many flux tubes when we measure the magnetic field in prominences. Depending on the filling factor, i.e., the ratio of the area of plasma-filled flux
tubes to the area of the inter-flux tube space observed in a given pixel, the
value of observed magnetic field is always a more or less good estimate of the
lower limit of the actual magnetic field in the flux tubes.
With these considerations in mind we find that observed fields in active
region filaments fall in the range 20 to 70 Gauss (Tandberg-Hanssen, 1974).
Since Doppler shifts have been seen in many cases, suggesting motions along the

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

95

filament, it seems likely that the magnetic field is nearly parallel to the long
axis of the prominence (Hirayama, 1985). The values quoted by TandbergHanssen are averages over many fine structures, and with higher spatial resolutions one would expect to see stronger fields from the flux tubes. Koutchmy and
Zirker (1990) found evidence of very strong fields in thin horizontal threads of
a low-lying active region filament observed above the limb with 0.6 arc sec
resolution. While the field between the threads was smaller than 150 Gauss,
the field in the threads showed values up to 1000 Gauss.
Surges show at times strong fields of the order of 150 Gauss, other times
hardly any field is measurable. More observations are required, but a fairly
obvious explanation is that the field is aligned along the surge (see, e.g., Roy,
1973a), and the measured field will depend on the angle between the surge axis
and the line of sight. Similar remarks apply to observations of the field in loop
prominences, which reveal field strengths generally between 20 and 100 Gauss.
3.6. Motions

We have referred to the motions of prominences in Chapter 1 and the nature
and degree of motions have been used in classification schemes. We have seen
that it is the speed of the prominence material that determines what we
describe as active or quiescent objects. In this section we shall investigate more
closely the nature of motions in different types of prominences.
It is useful to distinguish between the overall, gross motion of a prominence,
e.g., an oscillation or an eruption of the whole structure, or parts of it, and internal motions of the prominence plasma that leave the overall shape unaltered.
These internal motions may be different in different parts of the prominence
body.
3.6.1.

QUIESCENT PROMINENCES

Even though we normally think of quiescent prominences, filaments observed
away from active regions, as the least active type in the family of prominences,
they nevertheless show considerable motions, both internal and, during part of
their life cycle gross, overall motions.

3.6.1.1. Internal motions
We shall first discuss quiescent prominences well removed from any sunspot and
where there is no interaction between the ends of the prominence and active
regions. Under such quiet conditions it has been known for some time that on

96

CHAPTER 3

high-resolution Hu photographs one observes that the material is concentrated in thin ropes of diameter less than 300 km (Dunn, 1960), and from movies
one gets the impression that the material is slowly either streaming down or
rising up these more or less vertical ropes (e.g., Dunn, 1965; Engvold, 1976;
Anzer, 1979; Cui et al., 1985). Superimposed on this basically vertical velocity,
Vv, different knots or threads of the prominence plasma exhibit a random
motion, v. According to Pettit (1932), Newton (1934), and Engvold (1971) for
quiescent prominences, we have approximately:

v =5-10kms-1 .

(3.7)

During activations of quiescent prominences (see Section 3.6.2) velocities v =30
to 50 km s-l and even more can be observed (Larmore, 1953). However, the
thermodynamic velocity parameter (the line-broadening velocity ~) is always
much smaller, again of the order of 5 to 10 km s-l.
This, by now, nearly classical picture of the behavior of the prominence
plasma pertains fairly well to the most quiescent objects. However it is too simplified for most prominences. In particular, it does not take into account the
prominence feet, the connections from the prominence to the chromosphere and
photosphere. The complete function of these feet is obscure, but it is likely that
the prominence is anchored to the Sun by magnetic flux passing from the footpoints into and out of the prominence. When we observe these filaments on the
disk and record the Doppler shifts in their spectral lines, both blue and red
shifts are found (Mein, 1977; Kubota, 1981; Martres et al., 1981; Malherbe et al.,
1981; Kubota and Uesugi, 1986; Engvold and Keil, 1986; Simon et al., 1986b; You
and Engvold, 1989), indicating material motion in loop-like flux tubes, probably
connecting adjacent feet. According to Schmieder (1989, 1990) the blue shifts,
i.e., upward motions, dominate and are stronger for lines formed in hotter than
in cooler plasmas. Characteristic values for the upward motions are

=
=

Vy(Fe I) 0.3 km s-l
Vy(Hu) =0.5 km s-l
Vy(C IV) 5 km s-l.

(3.8)

Of great importance for the development of quiescent prominences are also
horizontal motions, VH' Such motions were recorded, e.g., by Mein et al. (1989a)
who recorded the velocity as Hu blue shift in a prominence above the limb and
found VH(Hu) =15 km s-l. Kulidzanishvili (1989) found Hu velocities around 10
km s-l, but with considerable spread, and stressed the importance of considering
a dynamic equilibrium for quiescent prominences; see also Kiryukhina (1990)
who found velocities 1-5 km s-l for quiescent filaments, and up to 60 km s-l for
active prominences. Schmieder (1989) has reviewed VH measurements pertain-

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

97

ing not only to Ha, but to lines of Mg IT (Vial et al., 1979) and "hotter"lines from
C IV and Si IV (Athay et al., 1985; Lites et al., 1976; Engvold et al., 1985).
The importance of these horizontal flows in and around filaments may also
be evidenced from analysis of center-to-limb observations of the mass motions,
and the magnetic field geometries may be deduced from the dynamic behavior
of the chromospheric plasma (Malherbe et al., 1983a) or of the transition
region plasma (Athay et al., 1985). We shall return to this use of flow information for model building in Chapter 5.
Summarizing observations of horizontal motions, we may give characteristic values
VH(Ha) =10 km s-l
(3.9)
VH<MgIT) =20kms-1
VH<C IV) =10 km s-l.
Velocities up to 30 km s-l were recorded by Engvold et al. (1978) in the outer
edges of a filament.
These motions, which are substantially greater than the more well-known
vertical motions, must have a profound impact on the formation and stability of
quiescent prominences. We shall return to this important question in the next
two chapters.
Another important streaming motion of the prominence plasma may be observed near sunspots. Initially, quiescent prominences form in pre-exising plage
areas (d' Azambuja and d' Azambuja, 1948) that may contain sunspots. As a result, we often observe a quiescent prominence, one end of which interacts with a
sunspot in such a way that material is streaming out of the prominence and
down into the spot. In this case the velocity, V, is much higher than in the case
of the slowly downward streaming motions of Equation (3.8). Due to the mass
losses involved, the quiescent prominences that interact with active regions
strongly call for a dynamic model.
In Section 3.6.2 we shall treat active region filaments that show streaming
motions along most of their length, not only out of the one end. They are not to be
considered quiescent prominences, but there may well be a generic relationship
between the two types, a question that will be discussed in Chapter 4.

3.6.1.2. Increased internal motions
At times quiescent prominences are subjected to external disturbances that result
in increased motions of the prominence plasma. The severity of the perturbations may range from a slight temporary "activation," manifested by enhanced
internal motions, to disturbances of such magnitude that the prominence is completely destroyed. The disturbing agents emanate for flares, growing sunspots,
or changes in the general magnetic field structure.

98

CHAPTER 3

These increased internal motions are characterized by a chaotic behavior of
knots and other fine-structure details of the different parts of the prominence
plasma and lead to velocities v = 30 to 50 km s-l. They are often accompanied by
a slow rising motion of the whole prominence (Bruzek, 1969). After the activation the prominence may return to its original quiescent state. However, in other
cases the increased internal motion is an indication that a major disturbance is
in progress. In other extreme cases the activation may impart very strong internal motions to the plasma, up to 300 km s-l, and then die down again
(Valnicek,1968). Such violent activities do not destroy the prominence, but significant changes occur. Of particular interest is the observation of a de-twisting
motion of the interwoven helices that make up the prominence.
Vrsnak (1984) observed a prominence that provides a typical case of destabilizing motions without leading to an eruption. He first observed matter flowing out of the prominence, and, as it rose, helical threads of material anchored
the prominence to the photospheric plasma. Similar destabilizing motions are
frequently observed on the disk (Malville and Schindler, 1981; Schmieder et
al., 1985b; Gaizauskas, 1985) and above the limb (Rompolt, 1975a; Athay et al.,
1987; Vrsnak et al., 1988); see also Vrsnak (1990a). We shall return to the importance of helical structure of prominences in Chapter 5.
In this connection we mention a dramatic motion reported by Liggett and
Zirin (1984). They observed material in quiescent prominences rising in a semicircular trajectory, turning over and descending in a continued rotating motion,
with velocities ranging from 15 to 75 km s-1, all within the prominence envelope. At times the whole prominence showed rotation. Even though the LiggettZirin observations may be the most dramatic reported, already Ohman (1968)
had drawn attention to rotational motions in quiescent prominences, and
Rompolt (1975a) has studied many cases of helical structures that imply rotation.
Observations of complex motions via Doppler shifts are of old lineage. In
Figure 3.5 we reproduce an old illustration summarizing velocity measurements
in a prominence observed above the southeast limb of the Sun on August 17, 1912,
at Meudon Observatory near Paris. The thick lines, leaning to the right,
correspond to the velocity of approach, i.e., negative Doppler shifts; the thin
lines, leaning slightly to the left, to positive Doppler shifts. The length of the
lines give the velocity in km s-l according to the scale on the left.
3.6.1.3. Oscillations

Sometimes quiescent prominences can be seen to move up and down, or sideways,
in a more or less oscillatory motion. When seen, e.g., on the disk, this leads to a
prominence spectrum whose lines are Doppler-shifted alternately to the red
and violet as the filament moves down and up in the corona. As a consequence

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

99

the image of the prominence is shifted alternately out of and into the passband
of narrowband Ha filters, and gives a ''wiriking'' impression. A winking filament
generally shows an initial downward motion followed by from one to four
damped oscillations. The phenomenon has been known since the spectrohelioscopic observations by Greaves, Newton, and Jackson (Dyson, 1930) and Newton
(1935), and the phenomenon has been studied by, e.g., Hyder (1966), Ramsey
and Smith (1966), Kleczek and Kuperus (1969), and Landman et al. (1977).
As we shall see later the winking is often caused by an explosive perturbation emanating from a flare, but in these cases the activation is a mild one,
never causing serious disruption of the prominence. An oscillatory behavior of
quiescent prominences, not related to flares, also has been recorded, and an
overview of the field has been given by Tsubaki (1988), who distinguishes between short-period and long-period oscillations.
The short-period oscillations are detected in their velocity signal, which
indicates a Doppler velocity, observed in the Ha, the Ca II K lines, or the He I,
10830 Aline, of the order of 1 km s-l and a period about
P(short) "" 3-15 min

(3.10)

(Vrsnak, 1984; Wiehr et al., 1984; Tsubaki and Takeuchi, 1986; Balthasar et
al., 1986; Tsubaki et al., 1987; Thompson and Schmieder, 1991; Yi et al., 1991; Yi
and Engvold, 1991). No oscillation has been detected in the line width or in the
line intensity, with the exception of a weak intensity modulation of the Ca II K
line in one prominence (Tsubaki et al., 1987).
It should also be noted that short-period oscillations are difficult to observe on the disk. Neither Engvold (1981) nor Malherbe et al. (1981, 1987) could
detect any clear evidence for such motions from observations of the Ca II
resonance or of the Ha and C IV resonance lines.
The long-period oscillations also are detected in their Doppler signals
which show velocity amplitudes of roughly 1-6 km s-l. The periods recorded by
Bashkirtsev et al. (1983), Bashkirtsev and Mashnich (1984), Wiehr et al.
(1984), and Balthasar et al. (1986) fall in the range
P(long) "" 50-80 min.

(3.11)

However, this distinction between short- and long-period oscillations is done
merely for observations reasons. From a physical point of view one might better
distinguish between eigenmode and forced oscillations (Vrsnak, 1993); see
Section 6.2.3.2.

100

CHAPTER 3

,

.. . . .",

"r._~ ~

i?tlh":

.-tp

ULt(UU

.1'-."'......-<1 .. If

'tdCllat:J d\u,",

Ao\,l

'glt "",
~

l,cot.lG(.:t."\I\.:i du.d.. L'Jl (J... r..""

'1iI·~~
. J._" -

I,,"\t: ..

_Cf.

k

'r"~'" ~.n.. « ~ ..

_ ~Jf \

..rtJjU

l:" •

~

.
/111 ,1 /

jl ... \1

~~

Ot

1

••

.•

~ ~ hr· f1
Fig. 3.5. Radial velocities in a prominence observed via the Doppler effect in the Ca K line, obtained
at the Observatoire de Paris-Meudon 17 August 1912. For details, see texl

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

101

3.6.1.4. Disparitions brusques
A gross velocity of particular interest results when a quiescent prominence
undergoes a sudden disappearance due to an ascending motion. To distinguish it
from other types of disappearances, we employ the often-used French term,
disparition brusque. The phenomenon was probably first observed by
Deslandres in 1889 using a spectroheliograph and studied extensively by
d' Azambuja and d' Azambuja (1948). The cause is generally not a flare-induced
activation (Raadu et aI., 1987), but the perturbation has a profound influence on
the stability of the filament. Prior to a disparition brusque Il:he prominence material exhibits increased random velocities, v ... 30 to 50 kIn s-1 (see Section
3.6.1.2), and then the whole prominence, or most of it, starts to ascent with increasing velocity Vv. This gross velocity attains values of several hundred km
s-l. The well-publicized observation of the nearly classical event of June 4,1946,
is still unsurpassed in beauty, and a sequence of photographs in Ha of this
prominence may still offer the best illustration of the disparition brusque
phenomenon (see Figure 3.6). As the disparition brusque phase commences the
quiescent prominence is ejected into the corona with steadily increasing velocity, resulting in a velocity curve (Valnicek, 1964) as exemplified by curve II in
Figure 3.7. The prominence material continues to be acted upon by a force, and
may reach velocities in excess of the escape velocity, Vesco The value of Vesc falls
from 618 kIn s-1 at the photosphere to about 400 kIn s-1 at a height of h =100,000
kIn in the corona, according to the expression
Vesc

=~2GMo ,
h+Ro

(3.12)

which is deduced from the vis viva (conservation of energy) equation, and
where Mo is the mass of the Sun, Ro it radius, and G the gravitational constant. Valnicek designates as group II those prominences that show velocity
curves like curve II in Figure 3.7, i.e., prominences that move out with increasing
gross velocity. Quiescent prominences during their disparition brusque phase
furnish the most important examples of this group.
In addition to the outward velocity of the erupting material, the disparition brusque phase is characterized also by a most interesting spiraling motion
and helical structure of the prominence material. Early observations are due,
e.g., to Severny and Khoklova (1953), Zirin (1968), Slonim (1969), and Anzer
and Tandberg-Hanssen (1970). Vrsnak et al. (1988) and Rompolt (1990) have
studied the phenomenon in greater detail; see also, e.g., RuSin (1989). As we
shall see in Chapter 6 the eruption seems to be due to the action of the
prominence-supporting magnetic field. The spiraling, therefore, reveals some of
the inherent nature of these prominences, being due to the interaction of the
rising magnetic field with the prominence plasma.

16!1~

U T

1620 U T

1742 uT

16~4

U T

2008 uT

1648 U T

Fig. 3.6. Disparition brusque of a quiescent prominence observed at the Oimax station of the HAO, Boulder, Colorado.

1604 UT

4 JUNE. 1946

......

10>

~

~

::r::

()

S

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

103

800~----------~----------------·

,
,,

I

,

,"

1/

600

I
I
I

I

I

,
,
-~ 400 ,
-..... ,,
,,
200 ,
,

: 1

M

o

I

I

I

I
I

I

I

O~------~------~------~~------~.
20
10
30
40
t (min)

o

Fig. 3.7. Height versus time plots for ten spray prominences, labeled 1 through 10 (after TandbergHanssen et ai., 1980). I and II are schematic Valnicekcurves for sprays and disparitions brusques,
respectively (Valnicek, 1%4).
1:
2:
3:
4:

5:

12 May 1969
18 February 1973
17 January 1974
5 July 1974
12 August 1972

6:
7:
8:
9:
10:

11 January 1973
11 January 1973
1 March 1969
30 April 1974
28 October 1972

104

CHAPTER 3

According to d' Azambuja and d' Azambuja (1948) nearly one-half of all lowlatitude filaments are seen to suffer a disparition brusque and disappear temporarily at least once. Consequently, the disparition brusque phase must be considered a "normal" experience for quiescent prominences; probably all go
through it. The disappearance lasts from a day up to a few weeks, after which
the prominence reforms in two-thirds of all cases. It is important to notice that
when the filament reforms, it appears in very nearly the same shape as before.
The rate of occurrence of disparitions brusques follows the solar cycle.
Vizoso and Ballester (1988) found that during maximum years there may be between 50 and 200 cases, versus 5-30 during years of minimum activity.
We shall return to the nature of these sudden disappearances in Chapter 6.
While the term disparition brusque was coined originally for the sudden
disappearance of quiescent prominences, it is now also used for active region
filaments; see Section 3.6.2.1. Furthermore, a filament (quiescent or active) may
abruptly disappear in Ha because of heating with no apparent dynamic process
involved. To distinguish this phenomenon from the classic disparition brusque
Mouradian et al. (1981, 1986) and Mouradian and Soru-Escaut (1989) call this
disappearance a thermal disparition brusque, DBt, while DBd is the dynamic
counterpart. It may then become visible, in situ, in spectral lines formed at
higher temperatures, e.g., from ions like C II, C III, 0 VI, Mg X (Malherbe et
al., 1983a). These thermal disparitions brusques might be flare related
(Malherbe and Forbes, 1986) or may somehow be related to neighboring hot
coronal arches (Schmahl et al., 1982; Mouradian et al., 1986). We also note that
a OBt and a DBd may take place simultaneously, and the combined effect of
these two processes has been studied by Fontenla and Poland (1989).

3.6.1.5. Sinking and shrinking filaments
The last types of activations that we shall discuss here lead, like the disparition brusque, to the disappearance of the whole prominence. But instead of ascending and disappearing, and then generally reforming, the sinking and
shrinking filaments are subjected to an activation that may look less spectacular, but that may have a more drastic outcome: these filaments disappear,
generally never to reform. In these cases we witness the whole prominence disintegrate by gradual dissolution, by falling into the Sun, or both. Again, the
gross velocity, VH, is considerable, easily attaining 100 km s-1 (see Kiepenheuer,
1953b).

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

105

3.6.2. ACTIVE PROMINENCES
All active prominences show considerable motion, and, as we have seen in
Section 1.3, most morphological classifications use these motions to define the
objects. The distinction between internal and gross motions is more blurred here
than for quiescent prominences and will not be very useful.

3.6.2.1. Active region filaments
The active region filaments (the sunspot filaments in Zirin's classification) are
characterized by motion along the filament. In high-resolution movies one can
see a continuous flow of material along the axis of most such prominences (S. F.
Smith, 1968). This flow is one of the strongest descriptive differences between
active and quiescent filaments. In the most quiet of quiescent prominence there
is hardly any mass motion observed along the long axis of the filament. As we
go toward more active conditions this situation changes. For example, if one end
of the quiescent filament is near an active region, we may observe significant
motion along part of the filament and out of one end into a sunspot or plage
(Martin, 1973; Schmieder et aL, 1985a). Finally, in active filaments the whole
prominence plasma is continuously flowing along the prominence axis, a situation that demands a completely dynamic model. More random and quite variable motions with velocities up to 30 km s·l have been analyzed by Nikiforova
(1990). This discussion points to the possibility that quiescent and active filaments somehow are related. We will return to this question in Chapter 4. Of
great importance also is the velocity pattern seen in the immediate surroundings of filaments in the so-called filament channels. We shall discuss other
characteristics of filament channels in Section 4.1; here we are interested in the
motions observed. As revealed, e.g., by Ha observations, the chromospheric
appearance is drastically different in the channel from its appearance away
from filaments, and in consort with this change goes the velocity pattern,
which reveals significantly reduced motions under, and immediately adjacent
to, the filaments; see Figure 3.8 for an example. The activation of an active
region filament, be it increased internal motion or a genuine disparition brusque,
normally precedes va flare, and hence furnishes a good flare precursor (Martin,
1980; de Jager and Svestka, 1985; Kaastra, 1985; Tandberg-Hanssen and Emslie,
1988). Particularly violent expulsions of active filaments may result in sprays;
Section 3.6.2.3.2.

106

CHAPTER 3

a

b

Fig. 3.8. Ha observations of a filament in its filament channel. (a) Une center data showing the
prominence and surrounding fibril structure; (b) Doppler observations in Ha ± 0.25 A showing the
velocity field; black regions correspond to redshifts, white to blueshifts (courtesy Observatoire de
Paris-Meudon and P. Mein).

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

107

3.6.2.2. Loop prominences
Some of the most beautiful examples of gracious motion of material in the
corona are provided by loop prominences and coronal rain, types A in the
Menzel-Evans classification. In these objects matter is observed to flow down
along curved trajectories into active region, seemingly after condensing out of
the corona, often at great heights.
These prominences are intimately connected with the flare phenomenon
and are, therefore, referred to as post-flare loops. Heinzel et al. (1992) suggest
that they be called cool flare loops to emphasize that they form part of the
flare. They occur after the disparition brusque of a filament (quiescent or
active), forming an arcade of loops connecting the two strands of emission
appearing as a two-ribbon flare. The systematic motion in these loops reveals
material streaming down the two legs of the loops from their apexes. The
streaming motion follows a single arc; one does not observe the spiraling motion
often seen in surges and in erupting quiescent prominences. During the next
several hours the loop system expands and reaches, typically, a height of
50,000 km. It is important to note that the individual loops do not grow or
expand much; rather, the system expands by generating higher and higher
loops, while the lower ones fade away. The resulting apparent velocity of expansion is quite small, ",5 km s·l. The real, downward flow motion in the two
legs of the loop is considerably faster. Free fall of 8-10 min would give V =Vg =
aot .. 130-160 km s·l, but much lower values have been observed (Heinzel et al.,
1992). Gu et al. (1990) showed that the density and the magnetic field in the
loops strongly influence the velocity of the downward flow.
Oscillations of loop-type prominences have been reported (Malville and
Schindler, 1981; Vrsnak, 1984) and may be associated with the flaring activity.
These post-flare loops should not be confused with the loop-like structures that
at times are seen during a flare display or other plage activity, when material
is ejected into radial flux tubes, giving surges, or into loop-shaped flux tubes
that bend over toward the surface. In this latter case the motion is along the
loop, moving up one leg and down the other (Loughhead and Bray, 1984;
Shilova and Starkova, 1987; Mogilevskij et al., 1988; Delone et al., 1989). The
velocity may reach 100 km s·l. In particularly large loop structures that reach
high into th~ corona, the ejected material may form so-called flaring arches
(Martin and Svestka,1988; Svestka et al., 1989; Fontenla et al., 1991).
In coronal rain, material is often seen to rain down into active regions from
coronal clouds suspended in the corona. The trajectories followed by the streaming material generally are strongly curved, apparently following the lines of
force of coronal magnetic fields. The velocities along these paths are comparable to the flow velocities observed in loop prominences, and it may be difficult to distinguish between the two types.

108

CHAPTER 3

3.6.2.3. Ejections

Some types of prominences owe their existence to the fact that at times material is ejected from certain active regions. When this material is seen projected
on the plane of the sky above the limb, we observe an ejected prominence. The
active region involved is nearly always flare-producing, and with increasing
importance of the triggering explosions we denote the ejections as "surges,"
"sprays," and "fast ejections" (Brozek, 1969). When the ejecta are observed
against the disk of the Sun, we may see them either in absorption or in emission, giving rise to dark or bright surges and sprays. The objects show the highest velocities observed among prominences.
3.6.2.3.1. Surges-In using the above nomenclature we have followed Giovanelli
and McCabe (1958) and Brozek (1969) who describe surges as straight or
slightly curved spikes which grow out rapidly of a small, round, luminous
mound associated with, or being, a flare (Gopasyuk and Ogir, 1963; Rust, 1968;
Westin, 1969). However at times surges occur far from spot groups (Kleczek et
al., 1971) and even at high latitudes (Okten and C;akmak, 1990), which poses a
problem when we look to strong magnetic fields to provide the acceleration
mechanism. Surges are shot out at velocities of 100 to 200 km s-l, typically last
10 to 20 min, and reach heights in the corona of 100,000 to 200,000 km. As the
material is ejected, it often performs a spiraling motion on its way out. After
the maximum height is reached, the material falls back into the chromosphere
along the trajectory of its ascent. This falling back often seems to trigger a new
surge (Zirin and Werner, 1967). Surges show a strong recurrence tendency, as
already observed by McMath and Pettit (1938) and Loden (1958); see also
Schmieder et al. (1983) and Kovacs and Dezso (1990). Recently the dynamic
nature of surges and their relation to active centers has been studied in detail
using, e.g., Ha, UV lines, and X-ray data, available from satellites; see, e.g.,
Schmieder et al. (1993).
The trajectory of the moving plasma and its collimation indicate that
surges are confined by more or less radial magnetic fields; see Section 3.5. For B
= 50 Gauss, the mass and ener~ contained in a large flare-associated surge is
1015_10 16 g (with ne = 101L101 cm-3) and 1030 erg, respectively (see also Roy,
1973b and Brozek, 1974). From a physical point of view, there may be similar
mechanisms at work in surges and in the flaring arches mentioned in Section
3.6.2.2.
From coordinated observations made in the Ha and C IV resonance lines,
Schmieder et al. (1984) derived the velocity field of ejected chromospheric
material along the magnetic field lines, and by also including X-ray data,
Schmieder et al. (1988) could study a surge in the temperature regime 1Q4 to 107
K. They found that the radiative energy released was lower than the mechanical energy (2.5 x 1()28 erg) and the potential energy (1028 erg) by at least two

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

109

orders of magnitude. Different parts of a surge may conceivably be ejected into
adjacent, but separate, flux tubes, and by combining Ha. and UV (C M data, one
finds that cold and hot material move upward following parallel lines of force,
independent of each other (Schmieder et al., 1983, 1990; Mein et al., 1989b). We
shall return to the implications of these velocity observations for surge models
in the next chapter.
3.6.2.3.2. Sprays-Sprays are violent, flare-associated ejections of plasma
which frequently are disrupted into bright clumps (Warwick, 1957; Bruzek,
1974; E.v.P. Smith, 1968; McCabe and Fisher, 1970; Tandberg-Hanssen et al.,
1980). Sprays reach high velocities (500-1200 km s-1) in a few minutes due to an
initial very high acceleration of a few km s-2. Valnicek (1964) called the
resulting height-vs-time, or velocity-vs-time, plot his type-I curve (see Figure
3.6)-quiescent prominences during their disparition brusque phase furnish the
type-II curve; Section 3.6.1.4.
When sprays originate on the visible disk one sees that the ejected material comes from a pre-existing active-region filament which undergoes increased absorption some tens of minutes prior to the abrupt chromospheric
brightening at the start of the flare (Tandberg-Hanssen et al., 1980). Most of
the spray material is confined within a steadily expanding, loop-shaped envelope with part of the material draining back down along one or both legs of the
loop; some material may escape the Sun as its velocity reaches and exceeds the
velocity of escape; Equation (3.12).
Some authors prefer to reserve the name spray for an ejection whose velocity exceeds Vesc (Smith and Smith, 1963). This may be an important characteristic from the point of view of the influence of the ejection on interplanetary
space. However, from the point of view of the physics of the prominence it is
probably not crucial whether or not the escape velocity is reached. We consider
as physically important the fact that the ejection is so violent that the material is no longer neatly contained in a surge-like configuration.
The reality of the observed clumpiness or fragmentation of the ejected
matter is not obvious. To a large extent, it may be due to the narrowness of the
instrumental passband used. With broadband observations at times one can see
massive prominences wh~re the spray appears fragmented as observed with
narrow passband filters (Ohman et al., 1967). Again we take the fragmentation
as an indication of the violence of the event, observable under certain types of
observations. Kurokawa et al. (1987) observed the expulsion of a helically
twisted filament and noted an untwisting motion as the filament quickly
ascended like a spray prominence.
The velocity of sprays and their height-vs-time, or velocity-vs-time curves
have been studied by Valnicek (1964), who defined his type-I curve as the
typical spray velocity curve. However, this seems to be an oversimplification.
Tandberg-Hanssen et al. (1980) studied 13 well-observed sprays and concluded

110

CHAPTER 3

that while the spray curves are different from the disparition brusque curve
(Valnicek's type-II curve), they cover a wide range, and the concept that the
motion of the ejecta may be represented by one particular curve is misleading;
see also Engvold (1980). Furthermore, the observed parts of the velocity curves
for many spray prominences are straight lines, indicating a fairly constant
velocity for the ejected material. However, if a spray prominence is seen to
stop, deceleration certainly ha!l set in, and Garczyt\ska (1991) observed a
number of sprays until they stopped and studied their maximum height as a
function of their velocity. Factors other than the initial velocity (e.g., magnetic field configuration) were found to be important in determining the maximum height. Garczynska found that most sprays reach heights of at least
250,000 km. The observed velocities ranged from about 200 km s·l to more than
1oookms·1.
Even though a substantial part of the height-vs-time space is more or less
filled with velocity curves for sprays (see Figure 3.5), it should be kept in mind
that even the lowest-lying curves (for the slowest-moving sprays) reveal conditions somewhat different from those of eruptive quiescent prominences, i.e.,
the "classical" disparitions brusques. The latter clearly indicates the increasingly important effects of accelerating forces during the observing period. This
is portrayed by the positive derivative, dh/ dt > 0, of the velocity curves for
disparitions brusques; for sprays, on the other hand, the acceleration phase is
very brief and occurs at lower heights. Hence, the acceleration phase would
seldom be seen in over-the-limb observations of sprays. However, it does not
necessarily follow that these differences between velocity curves for sprays and
disparitions brusques reveal fundamental differences in the basic physics
involved in the ejection mechanisms. The pre-existing quiescent prominence
(filament), giving rise to a disparition brusque, differs from the active region
prominence (filament) that gives rise to a spray, but this does not preclude a
basically similar physical mechanism operating in both cases.
3.6.2.3.3. Fast ejections-Bruzek (1969) has called attention to some fast
ejections which seem to form a class of ejecta distinguishable from sprays. These
very fast ejections consist of a compact portion of a flare which is ejected as a
whole, and is not fragmented. The velocities involved lie in the upper range of
the spray velocities, and these ejections will, therefore, influence conditions in
interplanetary space.
3.7. The Hvar Reference Model
During the deliberations by solar physicists at Colloquium No. 117 of the
International Astronomical Union, "Dynamics of Quiescent Prominences," at
Hvar, Yugoslavia in 1989, an attempt was made to create a table that would

PHYSICAL PARAMETERS OF THE PROMINENCE PLASMA

111

summarize our knowledge of the principal physical parameters in quiescent
prominences. Table 3.4 presents the essence of the Hvar Reference Atmosphere of
Quiescent Prominences (Engvold et al., 1990).
Values for many other parameters of importance in prominence physics
were also tabulated following discussions at the Hvar meeting and are published in the proceedings Oensen and Wiik, 1990).
TABLE 3.4
Typical observed values of physical parameters in solar prominences
Prominence
P-C Transition Region
Central Part

Te(K)

~t (km s-l)

ne (cm-3)
Pg (dyn cm-2)

n(H II)/n(H I)

B (Gauss)

V(kms-1)

8000-12,000
10-20
109.6
-0.02

4300-8500

3-8

1010-1011
0.1-1
0.2-0.9

Edges

lQ4-1Q6
30
3 x 101°-108
-0.2

4-20
±5

-10

CHAPTER 4

FORMATION OF PROMINENCES

The different processes that may lead to prominence formation may yield
stable or unstable objects, but we postpone the discussion of stability of prominences to Chapter 5. In their prominence classification Menzel and Evans (1953)
emphasized the important distinction between objects forming from above and
objects forming from below (see Section 1.3). We shall refer to the former case as
condensation, and, for the latter, we shall use the term injection.
A number of authors have advocated the view that some types of prominences form from material condensing out of the corona. The discussion of this
process goes back 40 years (Kiepenheuer, 1953a); Parker (1953) and Field (1965)
analyzed the different possible instability criteria for a theoretical interpretation of the formation as a condensation; see also Priest (1982), Hildner (1974),
and Heyvaerts (1974). The process has been applied to quiescent prominences,
as well as to active objects.
On the other hand, there obviously are prominences that originate from
below, active prominences being shot out of photospheric or chromospheric
layers. Foremost in this group are surges and sprays. Also spicules seem to have
a similar origin. No comprehensive theory exists to explain in detail the formation of these injections, but several interesting aspects of the processes that
might be involved have been discussed and can be applied also to quiescent
prominences (Schliiter, 1957b; Jensen, 1959; Gopasyuk, 1960; Warwick, 1962;
Jefferies and Orrall, 1965; Pikel'ner, 1971).
We may divide injection processes into surge-like and evaporation-like
models (Malherbe, 1989). An et al. (1986) treated material launched ballistically into an appropriate magnetic field configuration, and Poland and
Mariska (1986) showed that a sustained release of heat in a magnetic loop
structure would lead to evaporation from below; matter rising into the loop.
Several excellent reviews of different aspects of prominence formation
have been published in the last decade; see, e.g., Forbes (1986), Zirin (1988),
Malherbe (1989), Demoulin (1989, 1991, 1993), and Rompolt (1990).
Before we discuss different theoretical models of prominence formation, we
shall see what observations can tell us about conditions favorable for the
formation process.

114

CHAPTER 4

4.1. Filament Channels and Magnetic Arcades
In her comprehensive review of the observational evidence for the formation of
filaments, both quiescent and active, Martin (1990a) draws attention to the importance of the polarity inversion zone. From their study of photospheric magnetic fields Babcock and Babcock (1955) found that quiescent prominences occurred in the narrow lanes between extended areas of magnetic fields of opposite polarity. These lanes, where the observed longitudinal magnetic field
changes sign, constitute the polarity inversion zone. Also active region filaments are found between opposite polarity fields (Howard, 1959; Avignon et
al., 1964; Howard and Harvey, 1964; Martres et al., 1966). Since the magnetic
fields invariably exist before the prominences form, it seems that a necessary
condition for the formation of prominences of the filament type is the existence
of opposite polarity fields defining a polarity inversion zone (S. F. Smith, 1968;
Martin, 1973, 1990a,b). The converse is not true: such zones exist without prominences forming in them.
To better describe the observations in areas where filament-type prominences form, we need to include the concept of chromospheric fibrils (Figure 4.1).
These are one of the most conspicuous features seen in H<x in and around active
regions (Martres and Bruzek, 1977). The fibrils are long and thin; average
length 11,000 km, widths 725-2200 km, and lifetime 10-20 min. In the central
parts of active regions the fibrils are arranged in patterns connecting spots and
plages of opposite magnetic polarity. S. F. Smith (1968) observed that before a
filament would form in or near an active region between areas of opposite polarities, the fibrils would no longer connect these areas of opposite polarities, but
curve into the polarity inversion zone. In the middle of this zone the fibrils
would thereby be more or less aligned along the direction that would become
the long axis of the filament; see also Prata (1971), Foukal (1971), and Rompolt
and Bogdan (1986). This path of fibrils aligned along a polarity inversion zone
is commonly referred to as a filament channel and forms a subset of such zones
where prominences can form.
Figure 4.2 portrays prominence formations, both of a quiescent and of an
active region filament. Furthermore, we also witness the sudden disappearance
of the quiescent prominence (last frame of Figure 4.2), a subject to which we
shall return to in Chapter 6.
Seemingly intimately connected with filament channels are magnetic
arcades, consisting of closed field lines which span the channel high above it
and are anchored in the opposite polarity regions on either side of the channel.
If a prominence exists along and above the channel, the arcade is seen well
above both, and a dark coronal cavity may be observed between the prominence
and the overlying arcade (Waldmeier, 1970) (Figures 4.3 and 4.4). The arcade
can be seen not only in white light (von Kliiber, 1961; Leroy and Servajean, 1966;
Kawaguchi, 1967; Newkirk, 1971), but in X-rays (Vaiana et aI., 1973; McIntosh

FORMATION OF PROMINENCES

115

Fig. 4.1a. Chromospheric fibrils observed in Ha in and around active regions. Notice small active
region prominences (filaments) to the right of sunspots (courtesy Ottawa River Solar Observatory,
National Research Council of Canada and V. Gaizauskas).

et al., 1976; Serio et al., 1978; Davis and Krieger, 1982a) and in UV lines
(Schmahl et al., 1982). Magnetic arcades can be observed without an underlying
prominence (McIntosh et al., 1976); hence, also this concept-like the filament
channel-is not a sufficient condition for the formation of prominences, but it is
often overlooked that the cavity-arcade complex is crucial for an understanding
of prominence stability; see Section 6.5. Figure 4.5 shows different arcades
observed in X-rays.
The orientation of fibrils indicates the orientation of the magnetic field in
the chromosphere; whence the Hex pictures often is used as proxies for chro-

116

CHAPTER 4

Fig. 4.1b. Quiescent and active region prominence and surrounding fibril sbucture observed in Ha
(courtesy Ottawa River Solar Observatory, National Research Council of Canada and V. Gaizauskas).

FORMATION OF PROMINENCES

117

Fig. 4.1c. Same view of 4.1 b, but observed in deeper layers of the chromosphere, in the wing of the
Ha profile at Ha + 0.6 A.

Fig. 4.2. Formation and disappearance of quiescent and active region prominences; see text <courtesy Observatoire de Paris-Meudon and P. Mein).

1988, JUNE

~

:;:r::I

~

(")

::c

00

......
......

FORMATION OF PROMINENCES

119

Fig. 4.3. Negative print composed of two photographs of the solar corona observed at the 12
November 1%6 eclipse. The upper half was taken with a 10-sec exposure, the lower with a I-sec
exposure. Prominences, dark coronal cavities, multiple arches, and a well-developed helmet streamer
are seen (after Saito and Tandberg-Hanssen, 1973).

mospheric magnetograms (Foukal, 1971; S. F. Smith, 1971; Zirin, 1972) that are
difficult to obtain (Section 2.2.1). On the other hand, one finds that filaments
occur in magnetic configurations similar to those conducive to flares, i.e., where
the transverse magnetic field, as measured in the photosphere, is more or less
aligned with the filament channel (Moore et al., 1987; Venkatakrishnan et al.,
1989). This means that prominence formation is favored in field configurations
that have a maximum degree of magnetic shear, where the word shear does not
refer to motions, only to the configuration (Hagyard et al., 1984). Physically

120

CHAPTER 4

this implies that there is stored extra energy in the magnetic field, over and
above the energy of a potential field, whose lines of force would lie in planes
perpendicular to the filament channel. In flare theory, it is this extra energy
that provides the source for the flare display. It is not clear what the role of
this extra energy is in the theory of prominence formation.

Fig. 4.4. Sketch of the solar corona observed at the 12 November 1%6 eclipse. The twin arch system
and an overlying helmet streamer appear in the northwest quadrant. Quiescent prominences are
shown as dark filaments on the disk and under the arches (after Saito and Tandberg-Hanssen, 1973).

FORMATION OF PROMINENCES

121

Fig. 4.5. Arcades observed in X-rays 12 May 1992 with the Soft X-Ray Telescope on the YOHKOH
satellite <courtesy NAOJ, ISAS, and NASA).

4.2. Photospheric Motions and Filament Formation
Observations show that quiescent filaments form when the magnetic field
gradient across the filament channel decreases (Shelke and Pande, 1983;
Maksimov and Ermakova, 1986). The photospheric magnetic field is not strong
enough to withstand the influence of mass motions; i.e., we have a high-~
plasma, where

122

CHAPTER 4

p=

gas .pressure = 161rnkT
magnetic pressure
B2

(4.1)

Under such conditions, the energy of the plasma motions dominates the magnetic energy, and the quoted observations concerning the decrease in the field
gradients indicate that crucial motions take place prior to the filament formation. The motions may be diverging horizontal flows or a downward transport of
magnetic flux (Zwaan, 1978); either would accomplish the decrease in field
gradient. Since divergent motions seem to be rare, Martin (1990a) advocates the
existence of a cancellation of magnetic fields of opposite polarities to remove
the magnetic flux, a process observed by Martin et al. (1985) (see also Hermans
and Martin, 1986), and a process that may constitute another necessary condition for filament formation.
To accomplish the vertical, downward motion of flux through cancellation,
it is necessary to bring the opposite-polarity magnetic fields together in the
filament channel. This horizontal motion of converging magnetic fields has
been observed and reported by Martin et al. (1985) and Martin (1986), who consider it a necessary condition for prominence formation.
As Sections 4.1 and 4.2 show, considerable work has been done to observationally determine the conditions that would encourage the formations of
filament-type prominences-both quiescent and active. For active prominences
like surges and loops, somewhat less has been done in this regard, but then the
circumstances for their occurrence is probably more clear cut. Loops always follow the disparition brusque of a filament, exist as an arcade of loops spanning
the filament channel of the disappearing prominence, and have their feet in
opposite-polarity flare strands on either side of the channel. Surges are invariably shot out of active regions, probably into preexisting magnetic flux
tubes, which means out of areas with enhanced magnetic fields. However, more
observations are desirable as to the detailed structure of the magnetic fields
involved.
4.3. Condensations
Under quiet, unperturbed conditions, the solar corona is a stable plasma where
heating and cooling processes keep each other in check. However, situations can
arise when different kinds of instabilities develop and spoil the tranquil behavior of the plasma. The result may be a collapse of the atmosphere into condensations whose increased density and decreased temperature will constitute
the first signs of a forming prominence. The general theory for this process is
complicated as both gravity and magnetic forces, as well as thermal conduction
and radiation, all playa role in the development of the magnetohydrodynamic
(MHD) instabilities involved.

FORMATION OF PROMINENCES

123

4.3.1. CONDENSATION AS A THERMAL INSTABILITY
The fundamental MHO instability responsible for a plasma condensation is the
radiative thermal instability. If the thermal equilibrium of the plasma at
temperature Te and density ne (to be identified with the corona) is unstable,
formation of a condensation of lower temperature, Tp < To and higher density,
np > ne (to be identified with a prominence), becomes possible. This is a consequence of the shape of the cooling function (HUdner, 1974; Priest, 1982),
which has a maximum at temperatures around lOS K; Figure 4.6. As Malherbe
(1989) pointed out it is possible to show, from a discussion of the energy equation
(see Equation (2.95», with an appropriate heating term, that there should
exist two thermal equilibria, a hot equilibrium, to be identified with the
stable coronal plasma, and a cold equilibrium, corresponding to the prominence
regime. Field (1965), in a thorough analysis of this problem, points out that
Weyman (1960) seems to be the first to give the correct instability criterion for
the formation of a condensation, even though Parker (1953) did much of the
pioneering work. Heyvaerts (1974) worked out the mechanism in great detail
for a uniform plasma, and, later, Priest (1982) gave an up-to-date account. Field
analyzed the different possible instability criteria in an infinite uniform
medium in the absence of magnetic fields, ignoring thermal conduction and using
linearized equations. In such a medium the equation of continuity

ap +V.pv=O,

(4.2)

dv
p-+Vp=O
dt

(4.3)

at

and the equation of motion

are automatically satisfied. It is in the choice of the form of the energy equation

ae

-+V.ev=O,

at

(4.4)

where £ is the energy density of the plasma, that most treatments differ. Field
used a heat equation of the form
dp
pdp
- - r---p(r-1)(G-L)=O,
dt
pdt

(4.5)

where G and L are the energy gains and losses, respectively, per unit mass and
per second, exclusive of thermal conduction.

124

CHAPTER 4

-20

Do
0

0

•

-21

0

;;;-

Ie
0

Ien

It)

I

e
0

0>
~
CI>

-(!)

-22

(\I ..

Z

c:

:J
0
0

u
(!)

0

-23

...J

0

• Cox 8 Tucker (1969)
• Raju (1968)
o Pottasch (1965)
o Doherty Menzel (1965)
• Hirayama (1964)
- Present Fit

a

-24

_25~-L-L~~~_ _L-~~~_ _-L-L~LUll-~~-L~UW

3

4 5 6

7

LOG TEMPERATURE (K)
Fig. 4.6. Radiation losses as a function of temperature from tenuous solar plasma (Equation (4.18».
The parameters C2(T) and 0.(T) in Equation (4.18) that describe the solid line fit to the points are
given in the following table (after Hildner, 1971).
TOO
C2(T)
0.(T)
8xlcP<T

3xlcP<T<8xlcP
8xlo4<T<3xlcP
1.5 x 104 < T < 8 x 104
T < 1.5 x 104

5.5 X 10-17
3.9xl0~

-1
-2.5

4.7x 10-54

+1.8
+7.4

8.0x 10-22
1.2 X 10-30

o

FORMAnON OF PROMINENCES

125

The infinite plasma, e.g., the corona, may be perturbed (in density and
temperature), but in such a way that the pressure remains constant. If we
assume the validity of a perfect gas law, such an isobaric perturbation will
lead to instability if

or
[ 8(G-L] _&..[8(G-L)] > O.
Br p Tc
8p
T

(4.6)

As Field points out, this criterion is consistent with the equation of motion, and
condensations are governed by this isobaric criterion. A generalized criterion,
valid in the presence of thermal conduction was given by Hunter (1970).
The less restrictive isochoric perturbations lead to an instability criterion
(4.7)

and is the one given by Parker (1953). However, this criterion is not consistent
with Equation (4.3), since pressure variations due to the allowed temperature
variations will generate motions that can lead to non-constant density.
If entropy, s, instead of pressure, is kept constant during the perturbations, we
are faced with an isentropic perturbation, and the corresponding instability criterion is

or
[ 8(G-L)] +_I_PC[8(G-L)] >0.
Br
p
r- 1 Tc 8p T

(4.8)

The isentropic criterion applies to conditions governed by adiabatic motions,
i.e., to sound waves, and has been studied by Hunter (1966), while nonadiabatic motions are involved in the condensation mode governed by criterion
(4.6).
Most of the linear treatments of the condensation process have considered
only the energy equation (Equation (4.4» in one form or another. However, the

126

CHAPTER 4

heat equation appropriate to the condensation problem is nothing more than an
expression of the first law of thermodynamics
dU
dt

dQ
dt

pdp
p dt'

- = - + -2 -

(4.9)

where U is the internal energy and Q the energy fed into the plasma from external sources. With a perfect gas law, Equation (4.9) may be written
dp
pdp
dQ
--r---(r-l)-=O.
dt
pdt
dt

(4.10)

The rate at which energy is fed into the plasma, dQldt, is the net gain, i.e., the
difference between all gain terms, G, and all losses, L; compare Equation (4.5).
The fate of the condensation process depends then on the competition between the cooling and heating of the coronal plasma. If, e.g., the two competing
processes are radiative cooling and conductive heating, the condensation will
proceed if the time scales for the two processes satisfy the inequality
(4.11)

where trad and tcond are the time scales for radiative cooling and thermal conduction, respectively. However, the process will be applicable to prominence
formation only if the instability leads to a cooling down to the new equilibrium
on an acceptable time scale.
The radiative scale trad can be evaluated from linear theory and is given by
(see, e.g., Malherbe, 1989)
,..

Orad

p
--2--'
p Lrod

(4.12)

where the pressure, p, density, p, and radiative loss, L (the cooling function),
all are evaluated at coronal temperatures. Quasi-linear computations
(Kleczek, 1958; Uchida, 1963; Raju, 1968) indicate that the time necessary to
form prominences is about 10 days. To get more reasonable, i.e., shorter time
scales, it is necessary to inhibit the thermal conduction. This could be accomplished if a magnetic field, with the appropriate configuration, were present. Hildner (1974) considered this case and gave a non-linear treatment to
show that the time scale then would have a value of about 1 day.
In addition, Priest (1982) points out that even if the condensation still
seems to proceed too slowly, judged as an instability, i.e., trad is too long, the
formation may, in reality, go much faster. The reason has to do with the con-

FORMATION OF PROMINENCES

127

cept of non-equilibrium. If, as one approaches the situation where radiation
and conduction balance, the ratio radiative cooling/thermal conduction increases, a condition is met where thermal non-equilibrium sets in, rather than
instability. With no neighboring coronal equilibrium available for the plasma
it precipitously cools down to the thermally much lower, prominence equilibrium and at a rate much higher than the linear instability rate, 'trad (Equation
(4.12», would indicate.
Different authors have included different terms in G and in L, Grad =absorbed radiation energy, Gcomp =heating due to compression by external forces,
Gmech = dissipation of mechanical energy, Lrad =loss by emission of radiation,
and the effect of thermal conduction. Thermal conduction may lead to a loss or a
gain depending on the temperature gradient in the condensing pre-prominence.
For example, if compression temporarily has raised the temperature above the
ambient coronal temperature, conduction losses occur (Liist and Zirin, 1960; Olson
and Lykoudis, 1967). Later in the cooling process, the temperature gradient will
have changed sign, and conduction will try to destroy the condensation. It is
convenient to single out thermal conduction and write Equation (4.10) in the
form
dp
pdp
- - r---p(r-I)(G-L)-p(r-I)V -(KVT) =0,
dt
p dt

(4.13)

where K is the coefficient of thermal conductivity. We shall return to a discussion of the importance of heat flows below.
If the gain terms include external compression effects, and the losses are due
to radiation, i.e.,
G-L=Gcomp-Lrad'

(4.14)

we retrieve the form used by Kleczek (1957). Lust and Zirin used
G-L = Gcomp -Lrad - V -(KVT);

(4.15)

i.e., as stated above, they retained the conduction term V- (KVT) as a loss term,
since in their case the plasma became heated relative to the ambient corona
due to compression.
Kuperus and Tandberg-Hanssen (1967) included the terms
G-L= Gmech -Lrad

(4.16)

128

CHAPTER 4

and neglected the effect of thermal conduction because of the presence of magnetic fields. This form is also the one used in the more sophisticated treatments
due to Raju (1968), Nakagawa (1970), and Hildner (1971).
The exact form of the energy gain function, Gmech, is not obvious. If the
corona is heated mainly by the dissipation of wave energy from below, it is
likely that Gmech will depend linearly on the local density (Weyman, 1960;
Uchida, 1963), and we may write
G mech

=c.p,

(4.17)

where Cl is a constant to be specified at thermal equilibrium. Equation (4.17)
should hold also for heating due to corpuscular streams (Raju, 1966). The radiation losses, i.e., the divergence of the radiative flux F rod' Lrod =-V. F rod' are
due to emission in lines (bound-bound transitions) as well as in continua (freefree and free-bound transitions), i.e.,

The loss terms can adequately be approximated by a function of the form
(4.18)
where the temperature-dependent coefficients C2(T) and exponents «(1) can be
considered constants within certain temperature ranges; see, e.g., Cox and
Tucker (1969), Hildner (1974), and Figure 4.6. We notice that the loss function is
small for temperatures below a few thousand degrees, reaches a broad maximum between approximately 5 x lQ4 and 3 x lOS K, and then decreases again.
Extensive quasi-linear (Nakagawa, 1970) and non-linear treatments (Raju,
1966, 1968; Hildner, 1974) of the condensation process have been presented,
using the hydromagnetic equations for a perfect electric conductor. A macroscopic treatment is applicable since the crucial characteristic length in a
plasma pervaded by a magnetic field is the gyration radius. Even a weak field
of 10-3 Gauss gives gyration radii of less than 10 km in the corona.
4.3.2.

CONDENSATION OF PROMINENCES

As far as one can tell, magnetic fields seem to form an integral part of the
physics of any prominence, and in the previous section we mentioned that such
fields had been included in several of the treatments of the condensation process. The role of the magnetic field is at least twofold. As a condensation forms,
the density of the matter increases and one needs a force to counter balance the

129

FORMA nON OF PROMINENCES

increased gravitational pull. A more or less horizontal magnetic field will act
like a hammock in which the growing prominence can continue to form, and the
Lorentz force, jxB (see Equation (2.92», provides the counter balance to gravity.
The other main role of the magnetic field is to shield the forming prominence from the devastating heating from the surrounding corona. As the density
of the prominence increases, it will become more and more capable of radiating
away the excess heat that tries to destroy it, since the radiated energy is proportional to the square of the density. However, in the early stages the forming prominence is much more vulnerable. An appropriate configuration of the
magnetic field is then necessary to stop, or at least drastically reduce, the
otherwise overwhelming heat conduction that quickly would destroy any
embryonic condensation (Rosseland et al., 1958).
The flux of thermal energy is
F conti

= -KVT,

(4.19)

where K, the thermal conductivity, is a tensor. Compare the discussion of the
electrical conductivity (Equation (2.100» with a magnetic field, B, present.
The transfer of heat, i.e., the divergence of Fcond, can be split into a component
Vue(KuVuT) parallel to B, and another component V1. e(KJ. V loT) perpendicular
to B. For a fully ionized hydrogen plasma with conduction primarily done by
the electrons, Spitzer (1962) gives for Ku approximately
(4.20)
Perpendicular to the magnetic field the heat flow would be expected to be drastically reduced, and a properly oriented magnetic field could thereby save the
forming prominence from again heating up and be destroyed. The factor by
which the thermal conductivity K1. is reduced relative to Ku is given by ai/Vii,
where ai is the gyro frequency for the ions, i.e., protons, since these are more
efficient than electrons to transport the energy across the field lines, and vii is
the ion-ion collision frequency, in this case, for protons. For solar applications
(4.21)
and we find for the thermal conductivity KJ. (Spitzer, 1962) the expression
K1.= 10 -1S.7T-SI2(neT)2
ergcm -1 s-1 K- 1,
B

(4.22)

i.e., much smaller than Ku for all reasonable values of the parameters involved; see Braginski (1965) or Burgers (1969) for more general expressions for

130

CHAPTER 4

the transport coefficients. Note the interesting fact that KII depends on the
temperature to the +5/2 power, while Kl. shows a dependence on the -1/2
power.
Keeping in mind the concepts of condensations in a thermally unstable
plasma and the tensorial heat conduction in magnetic fields, we now investigate the possible configurations in the corona where a prominence could materialize.

4.3.2.1. Condensation in a current sheet
Historically, the first investigation of prominence condensation pertained to a
current sheet standing vertically in the solar corona. Kuperus and TandbergHanssen (1967) suggested that the thermal instability might take place in the
neutral sheet between two regions of oppositely directed magnetic fields, as one
would find above a filament channel. They further assumed that due to the
finite electrical resistivity of the plasma, filamentary structures would be
formed by the tearing-mode resistive plasma instability and these structures
would be thermally insulated from the hot surroundings by the magnetic field.
Smith and Priest (1977) investigated further the thermal behavior of a
current sheet and concluded that condensation is possible when the sheet elongates to a certain critical length Le. The value of Le depends on the strength of
the coronal magnetic field B and decreases as B increases. The reason for this is
that as B increases so will the pressure which again leads to increased radiative cooling.
The Kuperus-Tandberg-Hanssen model is crude, with the thermal instability inadequately treated, and later work has produced several improvements, including non-linear treatments. However, in its relative simplicity the
model incorporates the basic ingredients for the condensation process as applied
to quiescent prominences and relates them to well-observed parameters, and we
shall now briefly describe it.
Most quiescent prominences are associated with characteristic coronal
streamers that appear above them, and on either side of the filament one finds,
in the chromosphere, magnetic plages of opposite polarity. A model for the
history of the corresponding coronal field structure as used by Kuperus and
Tandberg-Hanssen is shown schematically in Figure 4.7. In the pre-active
phase of the active region Figure 4.7a shows a closed dipole-type configuration.
The corona overlying the center of activity is more intensely heated than the
quiescent corona (Kuperus, 1965); whence the gas pressure in region A becomes
higher than at the same level outside the active region. The active coronal
region may be heated until, at some height, B, matter is blown out, thus opening
the field lines (Figure 4.7b), the safety-valve mechanism (Parker, 1963a). In

FORMATION OF PROMINENCES

131

region B the field structure is determined by the outward flow, while region A
possesses a more or less potential field.
During the post-active phase (Figure 4.7c) the enhanced coronal expansion
ceases, and the field lines are no longer pushed sideways. The region near the
plane of symmetry has very weak fields; whence matter is compressed there to
maintain lateral equilibrium. The material near this current sheet has higher
density than the surrounding corona and loses more energy by radiation. The
magnetic field is radial and Gmech is assumed to be the same as in the surrounding corona. Hence, it is not possible to compensate for the enhanced radiation
loss, and the current sheet region becomes thermally unstable.

a

c

b

Fig. 4.7. A possible magnetic field configuration over an active region during (a) the early phase of
activity, (b) the main phase, and (c) post-active phase (after Kuperus and Tandberg-Hanssen, 1%7).

Kuperus and Tandberg-Hanssen assumed that the magnetic field configuration and the gas pressure in the corona near the neutral sheet do not change
during the condensation, and that thermal conduction is sufficiently inhibited
to be neglected. Equations (4.10) and (4.16) may then be written
cp

aT
iii
=Gmech -

Lrad ,

(4.23)

where cp = YR/(y -1) is the specific heat at constant pressure. For the radiation
losses they used Orrall and Zirker's (1961) expression, which gives too small a
value, Lrad = 6.3 X 1024 [p(z,t)]2 erg cm-3 s-l. The heat supply by dissipation of
mechanical energy just balances the radiative losses of the region far away
from the sheet, i.e., Gmech = 6.3 X 1024 [Pc (Z)]2 • The solution of Equation (4.23)
indicates that the temperature of the neutral sheet (the pre-prominence) drops

132

CHAPTER 4

by a factor of 10 and the density increases by the same factor, in about lOS s from
an initial coronal temperature of Tc =1()6K.
As the condensation process proceeds, the magnetic field in the current
sheet of widthlwill diffuse through the plasma on a time scale 'tdiff = .e2/'T1m'
where 'TIm is the magnetic diffusivity; see Equation (2.121). Unlessl is very
small this rate in solar cases is very slow. During this resistive diffusion, magnetic energy is ohmically converted to heat at the same slow rate. However,
the diffusion can drive plasma instabilities, and Furth et al. (1963) showed
that the resistive tearing-mode instability can convert magnetic energy into
heat and kinetic energy at a rate greatly surpassing 'tdiffabove.
It is this tearing-mode instability that Kuperus and Tandberg-Hanssen invoked in their model, thereby making it possible for the field lines to connect
across the current sheet. This would cause a filamentary fine structure in the
forming prominence, which is generally observed. Also, the lower-lying field
lines would become more or less horizontal and provide support for the denser
prominence material (see Figure 4.8).
The strong coupling of the system of equations that govern the complete condensation process made the general, nonlinear treatment mathematically
intractable. Hence, Kuperus and Tandberg-Hanssen omitted thermal conduction
and assumed a uniform magnetic field of unrealistically simple configuration.
Quasi-linear and non-linear numerical calculations have followed; not
necessarily all applied to current sheet geometries. In a one-dimensional treatment Raju (1966, 1968) considered a magnetic field parallel to the long axis of a
cylindrical configuration in the corona. He let the cylinder undergo a radial
compression and showed that instability will set in and lead to a condensation
for sufficiently weak initial fields. It is not obvious how a radial compression
can arise, and the difficulty associated with this initial compression, which,
in some form or another is also crucial to other treatments of the condensation
problem, is not solved. Raju showed that prominence conditions may be reached
after about 1()6 s, i.e., 10 to 12 days, which seems too long. Also, Raju could not
produce prominences with magnetic field strengths in excess of 0.1 Gauss, which
is too low. While his work constituted an important step forward, considering,
as it did, the effects of the intensification of the magnetic field when field
lines are swept into the condensation, the results indicate that condensation
must be along the field, since cross-field motions are too slow for any reasonable
field strength.
A two-dimensional treatment of the non-linear condensation process is due
to Hildner (1971, 1974). He considered an initial horizontal magnetic field, and
for computational reasons recast the energy equation in the form

iJe
a;+
V -S-(G-L)=O,

(4.24)

FORMATION OF PROMINENCES

133

v

~~~~~~~~~~-r~~~~~X

++ +

Photosphere

Fig. 4.8. Schematic representation of the magnetic field configuration during the formation of a
quiescent prominence in a current sheet (after Kuperus and Tandberg-Hanssen, 1967).

where the total energy density, e, is
P
1
B2
e =-- + - pv2 + - + pgz
r-l 2
Sir

134

CHAPTER 4

and the energy flow, S, the Poynting vector, is
S = E XB+(...1E....+.!.pv2)v.
r-1 2

In addition to Equation (4.24) Hildner needed the gas dynamical equations that
would govern the condensation process, i.e., the equation of continuity (Equation
(2.83'» or
(4.25)
the equation of motion (Equation (2.92», or
dv
1
p-+Vp-pg--jxB=O
dt
c

(4.26)

and appropriate forms of Maxwell's equations (Equations (2.107), (2.108),
(2.110», or
(4.27)

IdB
VxE+--=O
cOt

(4.28)
(4.29)

with the electric displacement D set equal to zero, and
1

E+-vxB=O;
c

(4.30)

see Equation (2.118').
He solved both one- and two-dimensional cases, and showed that an initial
density perturbation will grow to a condensation and will be supported against
gravity by the magnetic field. Instabilities in the computation prevented him
from reaching genuine prominence conditions, but the trend was clearly
established.
Kuperus and Tandberg-Hanssen (1967) and Raju (1968) assumed all variables inside the condensation to be spatially uniform (see also Uchida, 1963),
and this uniformity permitted the fluid motions to be neglected. Hildner in-

FORMATION OF PROMINENCES

135

eluded the fluid dynamics of the problem and showed that the response of the
fluid to a local density increase abruptly changes character when the initial
sound speed exceeds the Alfven speed. If the flow is small and perpendicular to
the magnetic field B, the system of equations is hyperbolic. If the flow is at an
arbitrary angle to B, the character of the governing equations depends strongly
on the value of this angle and has mixed features of both hyperbolic and elliptic equations. This leads to formidable mathematical problems that are not
solved in Hildner's work, but he pushed the non-linear treatment of the condensation process an important step forward.
Oran et al. (1982), in a one-dimensional analysis, showed that an initial
perturbation will develop on a linear time scale until the temperature has
fallen to about lOS K. At that stage the development is drastically accelerated,
and a dense, cool, T = 2 x 1()4 K, condensation forms; i.e., we have reached
prominence conditions. The prominence is surrounded by a prominence-corona
transition region; see Figure 4.9.
Let us return to the specific formation process in current sheets. The most
lasting contribution for quiescent prominences is due to Raadu and Kuperus
(1973), who built on the Kuperus-Tandberg-Hanssen model. Subsequently
Kuperus and Raadu (1974) considered the magnetic field configuration in more
detail and discussed support and stability conditions, and we shall return to
this work when we discuss different prominence models in Chapter 5. Here we
are mainly interested in the formation process per se, whether it leads to a
stable prominence or not. Raadu and Kuperus considered the effect of line tying
of the magnetic field, which is the stabilizing effect of anchoring the ends of
the (more or less vertical) field lines in the dense photosphere. Any perturbation of the current sheet should, therefore, vanish at its lower boundary. The
field lines will be held fixed in the photosphere, and magnetic stresses will
build up to resist the condensation process. However, higher up the effect of
line tying is small, the plasma in the current sheet will collapse in the thermal
instability, and Raadu and Kuperus concluded that the width of the condensation should increase with height above the photosphere. They refer to the
forming prominence as a wedge condensation, assuming that the wedge is sufficiently narrow for the dynamical time scale for horizontal adjustments to be
small. This means that the condensation, in their picture, proceeds through a
series of configurations with horizontal force balance, at least up to a certain
height.
Raadu and Kuperus commented that if the magnetic pressure in the cooleddown material increases, the condensation process will stop. However, in a
current sheet one can dissipate magnetic field or reconnect field lines across it,
thereby preventing the buildup of magnetic pressure. The annihilation of magnetic field leads to ohmic heating; Equation (2.106). Consequently, for material
to condense, both thermal and magnetic energy must be radiated away, and
Raadu and Kuperus estimated the corresponding time scale to be given by

CHAPTER 4

136

107r---,
----,
----T
----.-
---.-
---.-
--~

-

T

-

104~~____~1___~1__~1____~1_____~1___~1__~

10

20

30

40

50

60

70

x 103 = Position (km)
Fig. 4.9. Temperature profile showing the steep gradient in the prominence-corona transition region
(after Oran et al., 1982).

(4.31)

where Eth and EB are the thermal and magnetic energies, respectively, and LTad
refers to the radiative cooling; Equation (4.18). For LTad they defined the temperature dependence as q(t), simplified by using Cox and Tucker's (1969) value
for the cooling rate. Equation (4.31), which then becomes
(4.32)

FORMATION OF PROMINENCES

137

gives't =70 min for B =10 Gauss, T =1()6 K, and n =109 cm-3• It is interesting to
see that these crude estimates give a time scale that may not be incompatible
with prominence formation times.
Raadu and Kuperus pointed out that the set of equations that governs the
condensation process is similar to our set (Equations (4.24)-(4.30» but considered
their application only at the current sheet where v = B = 0 for reasons of symmetry. On the other hand, in a series of papers Forbes and Malherbe (1986,
1991), Malherbe (1987), Forbes et al. (1989), and Demoulin and Forbes (1992)
have solved the two-dimensional MHD equations in line-tied current sheets by
numerical methods and discussed their resistive-radiative nature; see also
Malherbe (1989) who points out that such a procedure is necessary to properly
account for the non-linear effects involved in the interaction of magnetic reconnection with radiation. Gravity is not included in these models; neither is
thermal conduction, which may not be a serious simplification with the correct
magnetic configuration. In the numerical computations the magnetic Reynolds
number (Equation (2.123» had values in the range 400-1000 and the plasma-~
(Equation (4.1» varied from 0.1 to 1.0, i.e., from conditions where the effect of
the magnetic field was dominant, to conditions where the dynamics had the
upper hand. The results indicated that for a low-~ plasma (~ < 0.2) condensations would arise with either a normal or an inverse magnetic field polarity;
see Figure 3.3. The prominence with inverse polarity, the Kuperus-Raadu
model, occurs above an X-type neutral line. It is unstable and breaks into
plasmoids, which, due to the reconnection flow, are ejected into the corona. The
normal polarity condensation, i.e., the Kippenhahn-Schliiter model, is stable
and forms at the top of the magnetic field with closed loop-like field lines that
form by reconnection below the X-type neutral line; see Figure 4.10.
This very interesting prominence-formation model, which we may call the
Forbes-Malherbe model, and to which Demoulin and Priest have contributed
significantly, addresses dynamically the interplay of forces in a radiative
magnetic reconnection process, but ignores gravity. Hence, the result that the
plasmoids will be ejected into the corona may be modified in a more realistic
regime, and, as a result, the prominence formed above the X-type neutral line
might be stabilized (Demoulin, 1989). This could then correspond to a RaaduKuperus (1973) formation process.
Even though the Forbes-Malherbe model first and foremost concerns the
formation of post-flare loop prominences, it also addresses the creation of lowlying, normal-polarity prominences of the Kippenhahn-Schliiter type. As the
magnetic reconnection takes place at the X-type neutral line, a supermagnetosonic (Section 2.2.2.4) jet forms, travels down the current sheet, and produces
a fast-mode shock when it encounters the tops of the closed loops that have
been formed by reconnection. The numerical calculations show that a cool condensation appears just below the shock which has bent down the loop tops to
form a hammock in which to receive the condensing prominence (Figure 4.10).

138

CHAPTER 4

I.

0.82
0.64
0.46
0.28
0.10
-0 .08
-0.26

-e. 44
-0.62

-e.se

Fast Mode Mach Nb

-e. 913

1

Fig. 4.10. Result of numerical calculations of radiative magnetic reconnection in a vertical current
sheet; ~ =0.1, R", =800. The figure shows field lines and, in gray scale, the fast-mode Mach number
at the end of the calculations; see text (after Forbes et al., 1989).

It should be noted that the more detailed theory for formation of post-flare
loop prominences goes back to Kopp and Pneuman (1976). We have seen in
Section 3.5.2.2 that the apparent upward motion of loops is caused by a
continual fading of old lower-lying loops, and the appearance of new loops
higher up. To account for this scenario, Carmichael (1964) and Sturrock (1966,
1968) had proposed that magnetic field lines from the photosphere extending
into the corona may continuously reconnect to form magnetic loops. The

FORMATION OF PROMINENCES

139

reconnection occurs at an X-type neutral line which slowly moves upward as the
region of the loops grows. Hirayama (1974) realized the important role of the
rising prominence material, and Kopp and Pneuman (1976) elaborated on the
overall mechanism, which is often refereed to as the Kopp and Pneuman model.
A hybrid formation process for prominences formed in a current sheet has
been proposed by Malherbe et al. (1983b). They adopt the view of condensation
of coronal material as described above, but add an interesting aspect of the
source material for the prominence. Building on observations of photospheric
motions below and around quiescent filaments (Malherbe et al., 1983c), they
propose that motions converging on the current sheet from both sides cause the
rising of magnetic field lines which, in tum, carries cool material up into the
forming prominence. We are, thereby, presented with a model in which the
prominence material may come both from above and from below. In Sections 4.4
we shall consider in detail the second possibility.

4.3.2.2. Condensations in loops and the effect of shear
Further development of the theory of formation of prominences via condensation has brought two new aspects into the discussion, viz. the existence of loops
and the often twisted and sheared magnetic field configurations. We have seen
in Chapter 3 that the plethora of electric currents flowing in-and below-the
solar atmospheric plasma is equivalent to having the corona criss-crossed by
magnetic loops and arcades. Since it is conceivable that prominences could form
in such structures, a number of authors have studied the development of the
thermal instability in flux tubes (Hood and Priest, 1979a; Priest and Smith,
1979; Chiuderi et al., 1981; Davis and Krieger, 1982b; Einaudi et al., 1984; An,
1984, 1985; She et al., 1986).
Second, in and near the footpoints of these loops at photospheric levels,
considerable plasma motions generally take place. Since the value of the
plasma-p at these levels is fairly high, the field lines in the loops may be
carried along, and one expects the magnetic field to be twisted and sheared. A
similar situation can also be found in the magnetic field of current sheets, even
though sheared fields were not considered in the early investigations of the
condensation process in such sheets. As it turns out, this shear may affect the
formation process of the prominences (as well as of flares), a situation first considered by Chiuderi and van Hoven (1979), while strong shear in a bipolar
region was studied by Aly and Amari (1988). Chiuderi and van Hoven ignored
thermal conductivity perpendicular to the magnetic field (K.d, as well as resistivity (1'\), but they could show the formation of typical "knife-blade" filaments. van Hoven and Mok (1984) included finite K.l which reduced the growth
rate of the condensation somewhat; see also Sparks and van Hoven (1985). The
effect of finite K.l and sheared fields was further studied by van Hoven et al.

140

CHAPTER 4

(1986) in an important investigation where they found new unstable condensation modes with growth rates fast enough to correspond to actual prominence
formation times.
A sheared magnetic field may be accommodated mathematically by an
equation of the form

as first used by Sparks and van Hoven (1985). Here I. s is the shear scale and i
and j are the unit vectors in the X- and Y-directions, respectively. To solve for
the dynamics of the thermal instability in this situation one needs further the
equation of continuity (4.25), the equation of motion (4.26) (van Hoven et al.
(1986) neglected gravity), an energy equation of the form (4.13), and the equation for the change in the magnetic field with no diffusion, i.e., Equation
(2.122). With 1.5 as the characteristic length scale, the hydromagnetic time
scale--determined by the Alfven speed-and to which all other time scales are
compared, becomes
(4.33)
van Hoven and his co-workers sought solutions to the linearized equations for
the perturbations of the form exp[mt + ikx). When they neglectedK.L, they could
show that solutions exist only if m < merit, determined by the magnetic field and
the loss term L =Lrad in Equation (4.13). However, by including a nonvanishing
K.L in the numerical calculations, with coronal values for temperature and
density and with ~ =10-2 and is =100 km, they could show the existence of new
modes with growth rates greater than merit.
From a physical point of view one needs to keep in mind that it is not the
inclusion of a finite K.L per se that helps the condensation process. Increased
conductivity will, on the contrary, slow the process as the temperature in a condensing blob increases (van Hoven and Mok, 1984; Sparks and van Hoven, 1985),
which is intuitively understandable. However, the inclusion of a finite K.L
permits the appearance of new condensation modes, and it is the existence of
these new modes with their faster growth rates that can speed up the prominence formation in this scenario.
To accomplish condensation in a sheared current sheet one needs to convert
the magnetic energy and then get rid of the created heat. The resistive tearing
instability and the thermal instability are normally involved for this phase of
the condensation. As we have seen these instabilities may be studied separately to investigate the effect of finite resistivity in the tearing mode and unstable radiation in the thermal mode. The question then arises whether the

FORMATION OF PROMINENCES

141

overall process proceeds fast enough to be of interest in the theory of prominence
formation. Steinolfson and van Hoven (1984) have suggested that the time
scale of the instability can be drastically reduced (by a factor of about 100) by
considering that the conversion of energy takes place in a tearing instability
modified by radiation, which they called the radiative instability. In their view
the tearing, i.e., the original Furth et al. (1963) dynamic tearing instability, is
altered by Joule heat transport and, in particular, by the unstable, optically
thin radiation; see also Steinolfson (1983). In other words they combined both
instabilities in the investigation and compared the modes of the co-existing
tearing and radiative instabilities. The results are that for solar atmospheric
conditions (n e = 109 cm-3, T =5 x 105 K, B = 20 Gauss, and is = 100 km) the
radiative instability growth rate, 'trad, is similar to the growth rate of the
thermal condensation mode (Chiuderi and van Hoven, 1979), and orders of
magnitude greater than the tearing-mode rate. Specifically, compared to the
hydromagnetic time scale 1:8 (Equation (4.33» we find
frad ==

10-4 f8'

(4.34)

Tachi et al. (1985) have generalized Steinolfson and van Hoven's (1984)
results by including the effects of compressibility and viscosity. They found
that compressibility affects the radiative mode, leading to a slight increase in
its growth rate, while viscosity modifies the tearing mode, leading to a decrease in its growth rate.

4.3.2.3. Whither condensation?
We have seen in sections 4.3.2.1 and 4.3.2.2 how numerous investigations have
explored different scenarios for the condensation of coronal material either in
current sheets or in loop structures. The basic physical processes involved are
fairly well understood, but the equations describing the formation process are
non-linear and strongly coupled. The mathematical difficulties encountered in
solving them are formidable, and often drastic simplifications are necessary. It
is in the choice of what simplifications one deems permissible that most investigations differ. Also, depending on what aspects of the condensation process
one finds most interesting, e.g., the conversion of magnetic energy, the role of
thermal conduction, the growth of the thermal instability, etc., one modifies
the set of equations accordingly. Finally, if one wants to incorporate what observations tell us about the final product (the prominence and its surrounding),
geometrical considerations come into play, viz. current sheets, loops, or arcade
structures. Often too little emphasis has been accorded these important restrictions that should serve as useful guidelines.

142

CHAPTER 4

On this background, we may say that even though the basic theory for the
formation of prominence material via condensation in current sheets or flux
tubes is fairly well understood, the application to actual prominences is still
problematic. In the case of loop prominences (post-flare loops), the condensation scenario as detailed, e.g., in the Forbes-Malherbe model fits well many
observed characteristics. However, when it comes to quiescent prominences,
detailed observations often contradict theoretical models. It is still an open
question whether these models can be modified sufficiently to accommodate the
increasingly detailed observational data now becoming available (see, e.g.,
Martin et al., 1992).
4.4. Injections
Under this heading we now discuss prominence formation in terms of mechanisms that provide material from below, i.e., matter of chromospheric and/or
photospheric origin somehow injected into the corona. This may be treated as a
macroscopic effect, carrying large blobs of plasma up from near photospheric
layers or consisting of a more gentle evaporation of matter, or it may be considered a basically microscopic effect, accelerating individual particles, e.g., in
electric fields in shock waves. Investigations along all these lines have been
published, and several mechanisms have been proposed as to how in detail the
material is raised to coronal height. It is customary to refer to these different
mechanisms by labels such as injection, evaporation, siphon, etc. In several of
the proposed models more than one of these mechanisms are at play, and there
is ambiguity as to under what heading a certain model should be treated. A
case in point is the formation model proposed by Poland and Mariska (1986), in
which the authors call it a siphon mechanism, but where the most efficient
part of the process takes place during an evaporation phase. Similar situations
occur in other models as well.
In addition, the process of condensation discussed in Section 4.3 occurs during
certain phases also in those models that account for the prominence material
from below. Therefore, strictly speaking, it is in these instances not possible to
separate condensation and injection as two distinct and self-sufficient mechanisms for prominence formation.
With these remarks in mind we now nevertheless proceed to treat proposed
models in a framework where we use the commonly applied terminology of
siphon, evaporation, etc., but we emphasize that the operationally important
distinction between models discussed in Section 4.3 and those discussed in
Section 4.4 is whether the prominence material originates from above, i.e., from
the hot corona, or from below, from the cool photosphere or chromosphere. In
the former case a relatively simple condensation process seems responsible; in

FORMATION OF PROMINENCES

143

the latter a mixture of several processes may be involved, often helped by condensation in certain phases.
4.4.1. SIPHON-TYPE INJECTIONS
In an attempt to account for the Evershed effect in sunspots, Meyer and Schmidt
(1968) studied the hydrostatic equilibrium conditions in a magnetic arch whose
feet in the photosphere were in regions of different gas pressures. This pressure
difference will drive a quasi-stationary flow along the flux tube. Consequently,
even though their work was not aimed at prominence formation, Meyer and
Schmidt's ideas are applicable to the injection category. They showed that the
flow velocity reaches the sound speed at the top of the arch, and this condition
determines the amount of mass that can be transported per second for a given
density. Furthermore, the authors claim that due to gravity the density
decreases vary nearly exponentially as a function of the maximum height of
the arch; whence, the total mass flow also decreases exponentially with the
maximum height of the arch. The mass transport from the photosphere,
therefore, is significant only for relatively low-lying flux tubes that may be
involved in active region filaments. Meyer and Schmidt found that the plasma
may reach supersonic velocities after it has passed the top of the arch and is on
its way down.
Pikel'ner (1971) was the first to study in more detail the steady state
siphon-flow model in which plasma, "evaporating" from the chromosphere, is
driven up in closed magnetic flux tubes and condenses near the top of the arcade.
The field configuration is depicted in Figure 4.11, and similar arches, placed
parallel to each other in the Y-direction would provide the three-dimensional
arrangement. Essential to Pikel'ner's theory is the pronounced bending down of
the top of the arches, whereby a "dip" is formed at A. Pikel'ner argues that the
heating of the gas at A is small. The main energy is transported to the coronal
regions by Alfven waves and slow waves (Pikel'ner and Livshitz, 1964), and
the magnetic field shields the gas at A from the heating from below. The
energy flux, F, that is put into the tube from chromospheric or photospheric
layers (at Band B'), is conducted along the tube toward A, but is gradually exhausted on its way due to radiative cooling, Lrtul. Because of this there will be a
temperature gradient along the tube, gas at A being at a significantly lower
temperature than at either B or B'. Hence, as in Meyer and Schmidt's (1968)
model, material will start to flow along the flux tube toward A as indicated by
the arrows in Figure 4.11. Pikel'ner argues that instead of supersonic flow, as
found by Meyer and Schmidt, his model probably calls for subsonic velocities,
since the scale height in the corona and the height of the arches are similar.
With the temperature decreasing upward along the tube, the thermal conductivity decreases and eventually the material, near A, becomes a dense, cool

144

CHAPTER 4

z

=:::> Moss Velocity

-

Field Line

Zm

I

I

I

I

~--~'--~/----------------------------~~~----~~X
Z

=0

(Photosphere)

Fig. 4.11. Schematic drawing of the magnetic field configuration and material motion in Pikel'ner's
(1971) model.

gas, which Pikel'ner identifies with part of a quiescent prominence. This condensation is not due to the thermal instability considered previously, but is the
result of the (macroscopically) decreased heating.
Pikel'ner assumes the following form for the shape of the tube from B (or
from B') to A:

where Zm is the maximum height of the arch and 2xo is the horizontal extent of
the arch. Flux constancy and cross-sectional area, A, of the tube are related by
AI Ao = BolB, where Bo is the magnetic field strength at the feet of the arch.
Rewriting, we find
A = AofB(Z),

where the function describing the change of the field with height is taken to be

Pikel'ner took Zm = lOS km, Xo = 7 x 10" km, and Wm = 10, and assumed steady
flow. The change of entropy, 5, along the flow is given by

FORMATION OF PROMINENCES

145

(4.35)

where ds = dQ/T. Since as/at = 0, Pikel'ner could write the energy equation
(Equation (4.5» in the form
ds
[C--+C
1 dn]
v dT
pTv-=pTv
(l-y)-dt
T dl
v
n dl '

(4.36)

where dl is a line element along the flux tube. In Equation (4.36) we have made
use of the expression s =So + Cv In (pp-Y/y - 1) for the entropy and assumed a
perfect gas law. Equations (4.35) and (4.36) combine to give the energy equation
dT
T dn
( KdT)
(l-y)-v-+L_.J-V.
- =0
Pcv v-+pc
v
ndl
rau
dl'
d1

(4.37)

since in the heat gain or loss per unit volume, p(dQ/dt), Pikel'ner included only
radiation losses, Lrad, and the effect of thermal conductivity, V.(KVT). Equation (4.37) was combined with the equation of mass conservation, nvA = const, or
nvW(z)

= Cl

(4.38)

and the equation of motion (Equation (4.26» in a steady state, or
dv 1 dp
dz
v-+--+g-=O.
dt p dl
dl

(4.39)

The system of Equations (4.37) to (4.39) was solved numerically with the
boundary conditions: at I =0, no =2 x 108 cm-3, To = 1.4 x 106 K and all energy
dissipated into the tube arises from thermal conduction; any mechanical wave
is assumed to have been absorbed lower down.
Values of Cl == 1014 to 1015 s-1 will give sufficient density increases at A to be
considered prominence conditions. With the corresponding flow, velocities
reach values of v == 10 km s-I, but decrease rapidly at z =Zc (see Figure 4.11) in
the cool gas.
The height, zc, where the condensing of the prominence takes place, depends strongly on the conditions at Band B', as well as on the energy flux. By
allowing the arch to start in areas with the appropriate value of the energy
flux F, Pikel'ner seems to be able to make the prominence formation take place
at reasonable heights. As indicated by the author, improved results may be
obtained with a non-stationary model allowing for some flux to be transported
by waves, i.e., by adding a gain term, Gmech, to Equation (4.37). However, other

146

CHAPTER 4

boundary conditions should be explored since To does not seem reasonable, and
improved expressions for Lrad should be used. Pikel'ner put LTad =n2f( T), where
f( is the function calculated by Pottasch (1965).
Several aspects of the siphon-type mechanism have been developed by
other authors. Soon after its publication Pikel'ner's model was further discussed by van Lyong (1974) and Sasorov (1975), and Engvold and Jensen (19m
evaluated in particular the choice of boundary conditions and presented additional solutions. Ribes and Unno (1980) described in detail the siphon mechanism under stationary conditions, while Uchida (1981) studied specifically
thermal effects.
A thorough mathematical treatment of siphon flows was given by Cargill
and Priest (1980, 1982) who considered the flow resulting from a difference in
pressure at the two footpoints of a hot coronal flux loop; see also Robb and Cally
(1992), who derived a steady subsonic flow under such conditions. Siphon flow
in smaller, low-lying cooler loops has been used to explain the observed flows in
and around sunspots-the Evershed effect-but might conceivably apply to surgetype prominences and perhaps spicules (Thomas, 1988; Thomas and Montesinos,
1990,1991, 1993; Montesinos and Thomas, 1989, 1993). See also McClymont and
Craig (1987), McClymont (1989), and Riiedi et al. (1992) who studied flows in
flux tubes connecting active regions of stronger and weaker magnetic fields.
Demoulin and Einaudi (1988) have touched on the central difficulty of the
siphon mechanism; namely, whether it is possible to affect a sufficient mass
flow up into coronal regions. They have shown numerically in a onedimensional case that during the evolution of a thermal instability at the top
of a loop it is possible to induce a mass flux from the photosphere to the corona.
However, as pointed out by Engvold and Jensen (1977), the boundary conditions
are crucial. Furthermore, non-linear treatments along the promising lines indicated by Demoulin and Einaudi are needed.
The injection mechanism may also be of a more ballistic nature (An et ai.,
1988a,b; Wu et ai., 1990), whereby matter is injected upward (by an unspecified
process) into a pre-existing magnetic arcade. By varying the injection velocity,
plasma density, and the strength of the magnetic field numerical calculations-with self-consistent, two-dimensional time-dependent MHD simulations (see Section 2.4)-shows that conditions can be found where material will
condense and accumulate in the magnetic field as this is bent down to provide a
dip for support of the forming prominence. Wu et al. (1990) showed in their
simulation that while the mass injection does supply most of the prominence
mass, it also triggers condensation of surrounding coronal mass to further supply
the prominence. The cited authors employed reasonable injection velocities of
10-20 km s-l, as found, e.g., in spicules, but the required strengths of the magnetic field may be somewhat restrictive. A weaker field favors the condensation in the dip, but a stronger field is necessary to support prominence mass den-

n

FORMATION OF PROMINENCES

147

sities. It is unclear whether these conflicting conditions leave enough latitude
for real cases.
In a comprehensive numerical two-and-one-half-dimensional MHO simulation Choe and Lee (1992) investigated the effect of shear motions of the footpoints of the arcade loops and found that the arcade expands with the result
that the plasma in the loop tops cools. Simultaneously dense material from the
lower parts of the loops is pulled up by the expanding field lines, radiative
cooling increases, thermal instability sets in (see Section 4.3.1), and condensation leads to prominence formation. In Choe and Lee's model the condensed
material grows vertically to form a sheet-like structure making dips in the
field lines and leading to the formation of a normal polarity-type prominence.
The mass is supplied partly by the corona via condensation, partly by the
chromosphere via siphon-type upflow, similarly as in the simulation by Wu et
al. (1990); see also Drake et al. (1993).
Another way to get material up into a pre-existing loop to form a prominence consists of more carefully evaporating chromospheric matter by heating,
letting it travel up the tube, and then condensing it as a prominence near the
loop top. This approach was studied by Poland and Mariska (1986), who solved
numerically, in a specified magnetic geometry, the time-dependent equations
for mass, momentum, and energy conservation, including conduction, radiation,
gravity, and heating. They found that it is quite difficult to form a sufficiently
dense prominence in the bent-down loop top without artificially injecting mass.
This can be accomplished by heating the chromospheric material in the loop
legs which then "evaporates" up to the loop top. Poland and Mariska showed
that a multistep process is necessary to form the prominence. First the heating
has to be reduced drastically in the loop, whereby a condensation starts to form.
Then a gradual energy deposition must follow, but only in the loop legs. One
way to accomplish this is to let the process take place in a twisted loop that
could provide both a gravitational well at the top for the condensation and
also affect the cutting off of the heating to the upper part of the loop. It is
during this second phase of the localized heating that material is siphoned
from the chromosphere. The calculations show that a condensation with a tenfold increase in density may form in about 5 hours during this latter phase, but
the first phase is much slower by a factor of about 4. Once the density of the
condensation has reached prominence values Poland and Mariska showed that
it is surprisingly stable. A large amount of heating would then be necessary to
overcome radiative cooling in the condensation.
Drake et al. (1993) used two-dimensional MHO simulations and considered
prominence formation at the apex of a coronal magnetic arcade following
siphon flow from the chromosphere. The condensing matter accumulates at the
apex where it bends down the field lines into a dip that supports the forming
prominence.

148

CHAPTER 4

As we shall see in Chapter 5, quiescent prominences are often modeled as
twisted, helical flux tubes into which prominence material might either condense or be injected, according to the processes discussed above. A third possibility would be that the flux tube ascends to prominence heights from the
chromosphere or below already filled with the cool prominence material in
place, trapped in the magnetic field. A promising scenario along these lines has
been advocated by Rust and Kumar (1994), and their model, which satisfies
many observational constraints (see Martin et al., 1992), deserves more scrutiny.
It may be further noted that rising helical flux tubes also play an important
role in Low's (1994) model of erupting prominences and coronal mass ejections;
see Section 6.5.2.
4.4.2. DIAMAGNETIC EFFECTS
Under certain conditions the solar atmospheric plasma may behave as a diamagnetic medium and this circumstance has been invoked to provide injection
mechanisms applicable to prominence formation. We shall here look at some
early work in this field, and then in Section 4.4.3 return to later applications.

4.4.2.1. The "melon-seed effect"
Severny and Khoklova (1953), Schluter (1957b), and Parker (1957) have suggested an interesting mechanism by which an aggregate of charged particles, a
plasma cloud, may be accelerated in a magnetic field. The plasma cloud behaves as a diamagnetic body, and it is the Maxwell tension of the magnetic
field that accelerates it, or stated differently, the cloud is subject to a Lorentz
force acting along the direction of divergence of the field. The mechanism is referred to as the melon-seed effect, since the plasma is being squeezed out between
the field lines like a melon seed between two fingers. It is made possible by the
high electrical conductivity of the plasma cloud. The force acting on the clouds
is given by V In B2, where B is the magnetic field in the absence of the unmagnetized clouds.
If effective, the mechanism may provide a means of explaining the surgelike ejections observed at times of flare. The outward acceleration is given by
dv/dt =-V[kT In B2 + 4>], where 4> is the gravitational potential.
4.4.2.2. Jensen's injection mechanism
Jensen (1959) has shown that when a plasma is not in thermal equilibrium, an
inhomogeneous magnetic field will cause the plasma to move in the field.

149

FORMATION OF PROMINENCES

Under certain conditions the plasma can be forced to move up into loop structures
in the field, thereby conceivably accounting for some loop prominences and coronal condensations. From the Vlasov equations, i.e., the collisionless Boltzmann equation (2.77), for either the ion or electron distribution functions f
(which relates to the microscopic velocities w; see Section 2.2.2.2)

df

df

df

wi (}B

df

WnW.l (}B

df

-+wn------g-+------=O
at
(}z 2B (}z (}wn
(}wn
2 B (}z (}w.l
'

where we have introduced

(}wr

at

=0

'

(}w.l

at

= wnw.l (}B

2B (}Z'

and (}wn

at

=_ wI

(}B _ g

2B (}z

,

Jensen derived a linearized equation of motion in vn, the macroscopic velocity
parallel to the magnetic field
(4.40)
where
00

00

00

v(r,t) = _1_ J J Jw/(r, w,t)dwxdwydwZ •
n(r,t)

W.l and Wn are the kinetic energies associated with motion perpendicular and
parallel to the magnetic field, respectively, and the field was assumed to have
the following components in cylindrical coordinates

The pressure Pn relates to the component of the random velocity un =wI! - VJI by
the assumption o(nmu 212) loz =0Pilloz. Equation (4.40) shows that if we have

150

CHAPTER 4

=2Wn.

thermal equilibrium (W.1
since there are two degrees of freedom in the
perpendicular direction, but only one in the parallel direction), the magnetic
field has no influence on the density distribution (van de Hulst, 1950). But if
there are deviations from equipartition, an inhomogeneous (CJBlCJz ~ 0) magnetic
field will change the density distribution of the plasma. This result was derived using the Vlasov equation, which means that the effect of collisions cannot be assessed. With CJBlCJz =-loBlozl, the hydrostatic equation becomes
(4.41)
Provided one can define a temperature Til that is independent of height, Equation (4.41) can be integrated to give the density distribution as a function of
height (4.41)
n(z) =

no exp{fZ [W.12WjIB
-2Wnl~l_ kTII
mg]dZ}

.

rT(.

o

Jensen assumed that the deviations from equipartition (W.1 ~ 2WII ) were caused
by a varying magnetic field, since only W.1, and not Wn, is altered by induction.
If aBfiJt> 0, W.1 is increased and W.1 -2WjI > 0, which means that the plasma is
diamagnetic and it will be pushed to regions where the magnetic field strength
has a minimum. (The condition oBlot < 0 leads to a paramagnetic plasma
which moves to places where the magnetic field is strongest; see Kiepenheuer
(1938).) To maintain a positive gradient of density against gravity, the ratio of
scale height, H =k T II I mg, to characteristic length, L, of the magnetic field
must satisfy the condition
(4.42)

HIL> 2WII /(W.L -2WII ).

One can find the time scale for the variation in the magnetic field necessary to
maintain deviations from thermal equilibrium large enough to be of interest by
writing for the variation of the ion component of W.1 -2WII with time (see
Schliiter, 1957a):
a( Wi,.L - 2 Wi,lI)

iJt

=_ Wi,.L - 2 Wi,lI + Wi,.1
'ri

B

aB

iJt'

where 'tj is the mean time between collisions for ions. Any deviation from an
isotropic distribution of velocities will be smoothed out in time intervals of the

FORMATION OF PROMINENCES

151

order of

'tj. We require that d(Wj,.L -2Wj,IIYdt = 0 to maintain the difference
Wj,.L -2Wj ,lI' Then defining the time scale for the variation of the magnetic
field t B I(lIB)(dBldt)r l , we find

t B -_

Wj,.L
Wj,.L -2Wj ,1I

'rj.

(4.43)

For the electrons we may assume thermal equilibrium, We,.L = 2 We,ll , since the
relaxation time for the electrons is much shorter than for the ions. This gives,
with Equations (4.42) and (4.43), the condition for the density gradient
(4.44)
For example, for a scale height 10 times greater than the characteristic length,
we find from Equation (4.42) that W.L > 2·2 WII , and if 'tj is of the order of 10 to
100 s, Equation (4.44) requires tB to be less than a few minutes.
Consequently, dense regions can be formed as a result of this diamagnetic
effect if the magnetic field undergoes changes on a time scale of the order of a
few minutes. Such rapid changes may not be unreasonable in certain active
regions.
4.4.3. SURGES, SPICULES, AND FIBRILS
The formation of surges as well as spicules and fibrils deserves special mention,
even though the general injection mechanisms discussed in the previous sections
may largely be applicable to them also.
4.4.3.1. Surges

Surge prominences are invariably found in active regions, and their formation is
closely linked to the extra forces created there. These may be of a hydrodynamic or magnetohydrodynamic nature, and theories for surge formation consequently fall into two classes. In addition, the cool, chromospheric surge material may be accelerated into magnetic field configurations that are either open
to the corona or closed, i.e., consist of loop-like flux tubes. In the former case the
surge material will rise in the magnetic field, reach a certain height, and fall
back to the chromosphere. The energy flux is not expected to build up to a high
enough value to heat the plasma to temperatures sufficient for significant Xray emission (Rust et al., 1977; Schmieder et al., 1988).

152

CHAPTER 4

On the other hand, if the surge material is trapped in loop-like flux tubes,
sufficient heating may occur before the material either falls back or is transported over the loop top and down the other leg. Under such conditions UV and
X-ray emission followed by Ha. emission may accompany the surge phenomenon, and we observe a flare-related surge (Schmieder et aI., 1993).
4.4.3.1.1. Pressure-pulse acceleration-Steinolfson et al. (1979) suggested that
surges are created by a sudden pressure increase (an explosion) in the chromosphere whereby material is injected into flux tubes; see also Schmahl (1981),
Shibata et al. (1982), and Schmieder et al. (1983). For one-dimensional flow
along a magnetic field, the MHO equations reduce to

p( dv + v dv) = _ iJp _ P a(/J
at

dz

az

ik '

where z is directed along the surge flux tube, and <I» is the gravitational potential. Before the explosion pressure is Po and the two terms on the right-hand
side of the equation cancel when there is no flow. If Pl is the pressure due to the
sudden increase at t = 0, then the initial flow acceleration is

Schmahl (1981) compared the parameters in this equation with observations of
surges observed in the EUV and concluded that flow velocities of the order of
100 km s-l could be reached in less than 30 s, compatible with surge observations. In the Steinolfson et al. (1979) model it is the pressure-gradient force that
plays the fundamental role in the formation of a surge, and the authors showed
that the pressure pulse created must be thermal to keep the density constant.
Schmieder et al. (1994) studied the nature of the driver gas in the pressuredriven models and developed constraints that can be used to decide whether or
not X-ray emission should accompany the surge phenomenon.
It is not obvious what triggers the explosions in this pressure-pulse mechanism for surges, but the magnetic field in the active region involved may be
responsible. In such regions one observes rapid changes in magnetic field configurations; e.g., emerging flux, and a sudden temperature increase might follow
due to reconnection. Magnetic fields may play an even more crucial role as we
shall see in the next section.

FORMATION OF PROMINENCES

153

4.4.3.1.2. Magnetic force acceleration-The idea that material can be injected
into the corona via the melon-seed effect (Section 4.4.2.1) or jensen's injection
mechanism (Section 4.4.2.2) is based on analyses of diamagnetic effects in the
solar plasma. For the mechanism to work the formation of a closed, magnetized
body, called a plasmoid, is required. When such a body is located in a nonuniform magnetic field, it will tend to move in the direction of the field
gradient toward regions of weaker field. After the early investigations in the
1950's Roy (1973b) adopted the proposed scenario and compared predictions
with observations of surges.
Pneuman (1983) and Cargill and Pneuman (1984) considered in more detail
the physics involved in diamagnetic propulsion. The Lorentz force acting on the
plasmoid can be found by integrating the magnetic stresses around the surface 5
of the body, i.e.,

J

1 [(BwedS)Bw--BwdS],
1 2
F=4n2

s

where Bext is the external field. The plasmoid is assumed to be magnetically
isolated; i.e., Bext is everywhere tangential to the plasmoid, whence-by
Gauss' theorem-we have

where V is the volume. Since it is only the radial component of F that acts to
move the plasmoid we find
F = -v

~(B;xt
Jr,
dr 8n-

where r is the unit vector in the radial, r, direction. Cargill and Pneuman considered also the possibility that a pressure gradient might be present and act
together with the magnetic force, in which case one has

Bixt)A
8n-

F =-V -d ( Pxt+-- r.
dr

e

Furthermore, if the plasmoid maintains pressure balance with the exterior,
i.e., if

154

CHAPTER 4

and if we write for the mass of the plasmoid M
following expression for the force

= Vnint m p'

we obtain the

If Pext = Bint = 0, the expression reduces to the case studied by Schluter
(1957b) and Parker (1957); see Section 4.4.2.1. Cargill and Pneuman studied both
gas-pressure dominated and magnetic-pressure dominated cases and found, for
the latter, accelerations giving speeds of 200-400 km s-l when T = loS K.
In a comprehensive analysis of ejection mechanisms applicable to acceleration of prominence material Raadu et al. (1987) concluded that "the most
promising model explains the motion as the consequence of magnetic stresses
acting on an isolated magnetized plasmoid in a diverging flux tube."
4.4.3.2. Spicules and fibrils

While surges are manifestations of solar activity and are not found outside the
activity zones on either side of the equator, spicules and fibrils form an integral
part of the quiet Sun's chromosphere and exist at all latitudes. A discussion of
their physics belongs in a treatise on the chromosphere, yet they may also be
considered "mini-surges" and treated as prominences. We shall not go into great
details, but advocate the view that mechanisms that can explain formation of
surges should be studied in conjunction with spicules and fibrils as well.
We discussed observational aspects of fibrils in Section 4.1 and indicated
their importance in the early phase of filament formation. Many of the physical aspects of spicules that we are now going to explore may also apply to the
formation of fibrils.
4.4.3.2.1. The characteristics of spiculeS-Roughly speaking, spicules are
cylindrical or cone-shaped objects with a diameter of about 1000 km reaching
from the low chromosphere 6000 to 10,000 km into the corona. Their lifetime is
about 5tol0 min and the whole spicule shows an upward motion of 20to30kms- 1.
After their ascent many spicules diffuse and fade out, while others are seen to
descend, seemingly following in reverse the path of ascent, e.g., Beckers (1968).
Recent high-spatial observations of spicules on the disk (historically referred

FORMATION OF PROMINENCES

155

to as dark mottles) confirm previous results; see, e.g., Tsiropoula et al. (1993)
who used the cloud model (Section 5.8.2.2) to derive the physical parameters.
Not only the diameter and lifetime but also the birth rate of spicules are
similar to those of the photospheric granulation. At anyone time there may be
about lOS spicules present, or a few times this value.
Studies of emission-line profiles from spicule spectra indicate that the
electron temperature is of the order of 1.5 x 1()4 K and the electron density about
1 to 2 X 1011 cm-3, much cooler and denser than the interspicular region, as found
for surges and other prominences. The values quoted are averages pertaining to
heights around 4000 to 6000 km, but line-width measurements indicate that ne
and Te vary both across a spicule and with height. The metal-line emission
comes from the cooler regions, He lines from the hotter parts of the spicule
plasma.
Spectroheliograms of the solar disk taken in strong lines show a bright pattern, called the chromospheric network. This network coincides with the boundaries of supergranulation cells (Simon and Leighton, 1964), and the boundaries
lie in regions of enhanced magnetic field in the photosphere. Although the
photospheric field is generally weak in other parts of the quiet Sun (a few
Gauss), it exceeds 30 Gauss along boundaries of supergranulation cells. Since gas
motion in the cells is (horizontally) from the center toward the boundaries, it is
believed that the enhanced magnetic fields are caused by the supergranulation
(Parker, 1963b).
It is along these boundaries that we find spicules. Therefore, it looks as if
the formation of spicules is intimately connected with enhanced magnetic
fields in the photosphere, and that, in turn, the magnetic fields owe their enhancement to the supergranulation. If this picture is correct, the spicules have
at their disposal at least part of the vast energy reservoir of the subphotospheric convection. In this way the spicule phenomenon is much more systematically tied in with the structure of the quiet Sun than all the prominences
hitherto considered.
At times, however, very large spicules are produced that hardly qualify as
a common characteristic of the quiet Sun. These structures are referred to as
macrospicules and were first described from Ha observations by Waldmeier
(1955). However, it was the availability of EUV data that spurred new interest in the phenomenon (Bohlin et al., 1975; Withbroe et al., 1976), while
laBonte (1979) summarized the results from Ha observations. High-resolution
UV observations (Dere et al., 1989) later have confirmed many of the earlier
conclusions regarding the characteristics of these interesting structures.
The length of macrospicules may reach values of 18,000-20,000 km; in extreme cases, twice those values (Bohlin et al., 1975). Their width is around 7000
km with a large spread, and their lifetime varies from several minutes (like
normal spicules) to 20 or even 40 min. The radial velocity of macrospicules
ranges from values found in normal spicules, i.e., 10-20 km s-1 to 50 km s-1 or

156

CHAPTER 4

more. We notice from the above discussion that in extreme cases, macrospicules
are little more than small surges.
4.4.3.2.2. Origin of spicules-From the observations cited above we may picture
the spicules as those regions of the chromosphere through which the strongest
magnetic fields are channeled. The field lines run more or less along the axis of
the spicules and prevent them from dispersing quickly into the coronal environment. Equality of the kinetic energy density of the spicular plasma, nkT,
and the magnetic energy density requires, for Te =1.5 x 1()4 K and ne =2 x 1011
cm-3, a magnetic induction of only B == 5 Gauss.
Osterbrock (1961) was the first to consider spicules the manifestation of
hydromagnetic waves carrying energy up into the corona from subphotospheric
layers. In this picture the spicules are the chromosphere, constituting the
transition region between the photosphere and corona and providing the
coupling between the two. Osterbrock discussed in some detail the enhanced
wave generation possible in plage regions, and the idea was further developed
by Parker (1964) and Kuperus (1965). Basically, it rests on the fact that the
mechanical energy flux, Fmech, increases when the vertical magnetic field increases.
In Parker's model magnetoacoustic waves develop into shocks as they
travel up into the atmosphere, and the shocks are identified with the spicules;
see also Uchida (1961). However, no detailed calculations explain how this
shock-to-spicule mechanism takes place. Parker's work has been developed
further by Wentzel and Solinger (1967). They started out with an initial nonthermal photospheric motion of 1.5 km s-1 and studied the development of the
shock. With a magnetic field of 25 Gauss they showed that the resulting gas
behind the shock would be heated to between 6000 K and 31,000 K, depending on
The density for
whether the gas was isothermal (y = 1) or adiabatic (y =
these two extreme cases came out to be 2 x 1010 cm-3 and 4 x 1010 cm-3, respectively.
Of early investigators we also mention Kuperus and Athay (1967) who used
the dissipation of the mechanical flux, Fmech, to heat the chromosphere. The
corresponding term in the energy equation, Equation (4.13), Gmech =V -Fmech, is
balanced by radiation and conduction losses, Lrad and Lcond (= V- [KVT». Since
they include Lcond, instead of Equation (4.23), Kuperus and Athay used, in a
steady state,
iJr
(4.23')
cpa;=
Gmech -400 - V . (KVT) =o.

t

i).

In the expression for Gmech (Equation (4.17», they took C} =K} vl,(M), where K1
is a constant, Vs the speed of sound, and ,(M) a slowly varying function of the

FORMAnON OF PROMINENCES

157

Mach number M (Kuperus, 1965). The radiation losses they approximated by
Lrad =p2 f(T), and used for fiT) Athay's (1966b) values.
It turns out that Equation (4.23') leads to unacceptably large conduction
losses in a shallow layer at the top of the chromosphere. Kuperus and Athay
remedied this by including energy losses due to acceleration of matter. With
this kinetic energy loss, Equation (4.23') takes the form
Gmech -4ad -Llcin - V.(KVT)=O,
and Kuperus and Athay argued that the shallow layer in the chromosphere
which becomes unstable is forced to a dynamic behavior. They identify this
dynamic configuration, in which matter is accelerated, with spicule formation.
It is interesting that this approach, which differs radically from the shock
wave picture discussed above, and which considers the spicules as jets caused by
an instability of the Rayleigh-Taylor type in the chromosphere, gives reasonable predictions for the spicule velocities.
The kinetic energy losses correspond to the kinetic energy flux in spicules,
Llcin = V. Fsp, where

Taking for the fractional area of the Sun covered by spicules, A =0.1, and assuming a density corresponding to n =2 x 1011 and a flux Fsp =3 x lOS erg cm-2 s-l,
we find with Kuperus and Athay a spicule velocity of vsp= 20 km s-l.
Acceleration due to magnetic forces was also invoked earlier for spicules, as
it was for surges (e.g., Uchida, 1969; Pickel'ner, 1969, 1971); see discussion in
Section 4.4.3.1.2.
An interesting time-dependent model has been developed by Suematsu et
al. (1982), incorporating the important observation that bright points are normally seen at the root of spicules, and benefiting from more sophisticated numerical computation. The authors assume that the acceleration of the spicule
material occurs as a result of the sudden pressure increase created by the bright
point. The pressure gradient will generate a shock wave that strengthens as it
passes up through the chromosphere. Eventually the shock (a slow-mode
shock) collides with the chromosphere-corona transition region, which
Suematsu et al. consider as a contact discontinuity. The shock then moves this
interface upward, and the matter which follows behind is identified as a
spicule.
While Suematsu et al. (1982) treated a hydrodynamic situation, Hollweg
et al. (1982) performed a magnetohydrodynamic simulation. They studied the
non-linear propagation of Alfven waves in open, vertical magnetic flux tubes
and found that the waves steepen into shocks (in this case fast shocks) in the

158

CHAPTER 4

chromosphere. Hollweg et al. used this circumstance to heat the corona, but the
shocks will also lift up the chromosphere-corona transition region to drive upward flows which the authors identify with spicules.
4.4.4. PARTICLE ACCELERATION
As mentioned in the introduction to Section 4.4 an injection may, at times, be a
microscopic process, in which individual particles are accelerated into the
corona to form the prominence matter. Probably the only prominence-formation
model using such a microscopic process is due to Jefferies and Orrall (1965),
whose idea is that the mass of loop prominences is fed into the magnetic loop
system in the form of energetic protons at the feet of the loops. Before we look a
little closer at this model, we shall first consider the problem of particle acceleration in the solar atmosphere.

4.4.4.1. Some acceleration mechanisms
The ability of the Sun to eject energetic particles was conjectured already 50
years ago from solar cosmic ray events (Lange and Forbush, 1942). Acceleration
of particles to cosmic ray energies (say, >107 eV) may be confined to flare
regions, but less energetic particles could come from other, less violent active
regions, and be involved in the development of certain active prominences.
Warwick (1962) drew attention to the "particle aspect" of solar flares, and some
of his analysis may be applicable in prominence considerations. The question is
whether the high-energy particles have mean free paths of sufficient length to
permit their transport. Following Warwick, we may state that the source of
the particles can be no farther from the region of deposition than the distance
in which protons of 3 x lOS eV would be stopped by ionization processes. The
relation between the vertical range, h, of a 3 x lOS eV proton and its velocity, v,
in a neutral H gas is (see Mott and Massey, 1949):
dh=

(4.45)

where X(H) is the ionization energy of Hand n the number density given by n =
no e-hlH, where H is the scale height of photospheric gases; that is, =110 km,
and no = 1016 cm-3• To integrate Equation (4.45) from the initial velocity Vo = 0.8
c to zero at height hI, we assume that above hI there is a distance less than H
where the atmosphere has the same density as at hi. For h > hI, Warwick
assumed a negligible density. Integration of (4.45) then gives

FORMATION OF PROMINENCES

159

or hI ... 480 km, which is the greatest depth from which protons of energy 3 x 108
eV can emerge from the photosphere.
Warwick's picture offers interesting suggestions concerning certain lowlying flares, but for most prominences we would need an acceleration mechanism
in the chromosphere or even in the corona.
Considerable work has been done in order to answer the question of how the
particles are accelerated in the solar atmosphere. An electric field is necessary
to accelerate charged particles, and one way to create an electric field is by
changing a magnetic field according to Faraday's law (Equation (2.108» V x E =
-1/ ce(aB/ilt). In the early 1930's Swann (1933) considered the effects of the
changing magnetic field in sunspots and showed that particle acceleration
might occur. The mechanism is similar to that applied in betatrons, and it is referred to both as the Swann mechanism and as betatron acceleration The force
experienced by particles of charge Ze is then given by ZeE.
About 40 years ago Fermi (1949, 1954) published a far-reaching theory for
particle acceleration which bears his name. Basically the mechanism involves
the relative motion of a magnetic region and the particle in question. Of special
interest is the case where acceleration may occur when the charged particle is
reflected repeatedly between two regions with strong magnetic fields (magnetic
mirrors) moving toward each other. For such a periodic motion the integral of
the particle momentum along a magnetic line of force through an entire period
between the mirrors
J II = _I_I PII ds
2rrt

(4.46)

is constant, so long as the field does not change appreciably through one period,
i.e., JII is an adiabatic invariant, called the longitudinal invariant. Under these
conditions the energy gain of the particle of mass m, between two mirrors whose
distance originally was L, is

LiW=-l-[Pl
2m + (_L_)2
L-LiL PIT] ,
when the distance has decreased by AL. The Fermi mechanism acts as the vehicle whereby part of the energy of a large number of particles-responsible for
the motion of the magnetic fields-is transferred to a single particle.

160

CHAPTER 4

Hayakawa et al. (1964) discussed the Swann and Fermi mechanisms and
brought out the similarities between them. We shall briefly look at this interesting coupling of the two mechanisms.
Consider the motion of charged particles in a magnetic field in the guiding
center approximation. The motion can be divided into three components, the
gyration around the magnetic lines of force, characterized by the frequency co.1,
the longitudinal motion along the lines, characterized by the frequency COli' and
the drift motion over the surface of the magnetic flux tube, given by the frequency COD'
For periodic motions we introduce the angle variable ., and the adiabatic
invariants, or the action variables J. For the gyration

cpr

W

J.1 =--=J,l2ZeB
Zec'
where P.1 =P sin (X, (X is the pitch angle, ~ the gyroma~etic moment, Ze the
charge of the particle, and the total energy is W =c"m~z + pZ. We shall denote by VB the velocity of the magnetic mirrors, and by v the velocity of the particle, relative to the observer. The angle variable, • .1, is the phase of the
gyration. The second component, the longitudinal motion, is also often periodic,
as, for example, when the particle is reflected repeatedly between two magnetic mirrors. Then JII =1/21tt PII ds, as given in Equation (4.46). Finally, for the
drift motion component, JD is the flux invariant, which reduces to the total angular momentum in an axially symmetric case. The frequencies of these three
components are
gyro frequency

ZecB

co.1=-W

transit frequency co 11= 2mfvij"lds
and
drift frequency

coD'

and they generally satisfy the inequalities CO.1> COil> COD'
The energy of a particle is constant in a static magnetic field; hence, time
variations of the field B(s,t) are essential for acceleration. We have

FORMATION OF PROMINENCES

161

where VB is the velocity of the magnetic region, and the energy of the particle
in the observer's frame of reference is

dW =(dW) +(dW) +(dW) +(dW) ,
~

~

s

~ F

~ I

~ T

where the terms are,

.
(dW)
the Swann mecharusm:
~

ZecJl.
dB =---c2Pl dB
=
s
W ~ 2WB ~'

.
(dW)
Zech
dB c2Pl
dB
the Fernu. mecharusm:
=-VB - = --VB-'
dt F
W
dS 2WB
~

the induction effect:

(1

. .
c2Pu ds
d Bath
dB)~ (h +JD ) cos'l.'
the tranSlt-tlme
effect: (dW)
dt T =--W

and where we have used the relation 2ZechB(s,t)=c2pl. We note that the
Swann term comes from CJBfiJt, the partial time derivative of 8, while the
Fermi term is due to the space derivation vB(CJBlCJs), and they add to give

2
( dW) +(dW) = c pl.!. dB.
dt S
~ F
2W B dt
In other words, these two effects combined give the energy change of the particle due to the total time variation of the field.
To lowest order, when gyration is very fast, we may take the average over
ch (indicated by ( )), and the terms corresponding to the induction effect and
the transit-time effect vanish, leaving

162

CHAPTER 4

(4.47)

Similarly, if the longitudinal motion of the particle is sufficiently fast, we
may further take the average over pitch angle variations along the line of
force, weighted proportionally to the time At a particle of given momentum
spends on a line element ds. If we indicate this average by ( }u' we find for the
Swann term

I(aw) ) =c (Sin a aB) ,

\

at

S II

2 p2

2

2W

2B

which shows that for this mechanism fl.p/fl.t

oc

at

(4.48)

II

p. For the Fermi term, we obtain

(4.49)

We are interested in the case where the particle is reflected by a magnetic
mirror, changing the pitch angle from an initial value (Xl to, say, (X2' It can then
be shown that to first order

aw)) =--2-cos
W 2VBV
(( --;a,
ut
Lit c

(4.50)

FU

which shows that for this so-called Fermi II acceleration (oW/ot) W. Finally,
Equation (4.50) can be averaged with respect to the initial pitch angle (Xl,
resulting in
DC

)
(( aw)
at

F lI,a!

2W(VB)2

= Lit 7" .

(4.51)

FORMATION OF PROMINENCES

163

Equation (4.51) gives the well-known result that the energy gain is proportional to energy, or Woc
Even though the Swann and Fermi mechanisms are physically well understood as the preceding discussion indicates, it is not so easy to apply them to the
solar plasma (Dungey, 1958; Parker, 1958); the difficulty being the very high
conductivity reigning there. Parker tried to mitigate this criticism by proposing
that the particles are generated between shocks which cross each other, and
Wentzel (1963) elaborated this idea and showed that the shocks need not be
strong to accomplish significant acceleration. Most work on solar particle acceleration has since then involved different kinds of shock interactions. One of
the earliest and most important studies is due to Schatzman (1963) who considered the case of perpendicular shocks in which the magnetic field is perpendicular to the shock normal. He then extended the time of interaction of a
particle with the shock by invoking a stochastic scattering process, and as a
result of repeated shock crossings the particle could gain energy. Shabanskii
(1962), Wentzel (1964), and Hudson (1965) applied the Fermi mechanism
(reflection at a moving mirror) by considering reflection and transmission at the
magnetic discontinuity.
Of considerable interest is the so-called shock drift acceleration theory
(Chen and Armstrong, 1973), where the particles drift in the direction of the
induced electric field. The equation governing the energy gain of a particle
with charge Ze and velocity v expresses the Lorentz force (Equation (2.72» and
can be written

Woet.

F=Ze('!"VXB+E)= dp .
c
dt

(4.52)

If the fields Band E = -11 c • (v x B) are homogeneous for several gyro radii
on either side of the shock, the interaction is said to be "scatter-free."
Armstrong et a!. (1985) have analyzed this process and find that at perpendicular or quasi-perpendicular shocks particles tend to drift parallel to E during
their interaction with the shock. The particle energy, thereby, increases an
amount Ze .Ar, where Ar is the distance the particle drifts; see also Forman and
Webb (1985).

4.4.4.2. Jefferies and Orrall's loop model
The basic idea in this work Qefferies and Orrall, 1965) is that the mass of loop
prominences (post-flare loops) is fed into the system in the form of energetic
protons at the feet of the loops. The authors did not go into details of how they
envisioned the particles to be accelerated and travel up into the loops. It is con-

164

CHAPTER 4

ceivable that some of the particle-acceleration mechanisms discussed in the
previous section could be applicable.
Once the particles are fed into the loops they follow the magnetic lines of
force until they give up their ordered motion by Coulomb collisions with the
ambient gas. This thermalization process is supposed to take place mainly near
the top of the loops. The energy thereby released creates a dense hot region,
and as the density increases, more and more energetic particles can be trapped.
Ultimately, this hot region becomes so dense that it explodes, and matter will
stream out along the magnetic lines of force and flow down the two legs of the
loops. The expanding plasma will cool enough to be visible in Ha and other
optical radiations.
Jefferies and Orrall's model has a number of attractive features and we
shall briefly discuss its main points. As the particles become thermalized near
the top of the loop, they produce heat, much of which will be lost by radiation.
The authors made an estimate of the net radiant energy loss by using an expression for an optically thin solar plasma derived by Orrall and Zirker (1961),
(4.53)
The total energy loss from the whole loop system due to radiation is then
4ad M
Lrad,tot ... ----tloop '
ne mH

(4.54)

where tloop is the lifetime of an individual loop in the system. Also, tloop ... 103 s
and M/mH =6 x 1039 cm-3 for a total mass M =1016 g. Equations (4.53) and (4.54)
give
Lrad,tot ... 2 x 10

20 2

ne.

The electron density is difficult to estimate. It is probably greater than 1011
cm-3, which means that the total radiative loss is several times 1031 erg. The
model implies that there is energy balance between the radiative loss and the
kinetic energy, Gkin, delivered to the loop by the fast particles. If all the particles have the same initial velocity, v, the total kinetic energy is Gkin =
1/2. (Mv2) and v2 = 20 ne tloop. Jefferies and Orrall assumed ne = 5 x 1011 em-3 and
concluded that the energy balance of this type of active object can be maintained by protons with velocities lOS em s-l, or energy about 1()4 eV.
To get these particles up into the corona to form loops they can hardly be
accelerated at photospheric levels. In their treatment of the injection mechanism Jefferies and OrraH relied on the concept of particle storage in coronal

FORMATION OF PROMINENCES

165

magnetic fields. The particles are assumed to be generated during those flares
that always precede loops.
As the cooled prominence material flows down into the photosphere, it
emits a spectrum, the lines of which should reveal characteristic profiles. For
instance, the wings of Ha should be quite pronounced and be different from
Stark-broadened lines, and such profiles are actually observed in loops.
Jefferies and Orrall drew attention to the possibility that other active
prominences, and even quiescent objects, may be produced by the injection mechanism. If the thermalization of the injected particles takes place near the top
of the loop of the magnetic field, a loop prominence should be formed as
described. But if it occurs near the bottom of one side of a loop-formed flux tube,
the result might be a surge-like object or a loop in which matter is seen to
stream up one of the legs and down the other.

CHAPTERS

PROMINENCE MODELS

Regardless of how a prominence is formed, once it exists, we are faced with the
problem of devising a model that can describe its appearance and behavior.
Closely linked to this problem is the question of the stability of the prominence
and of how the prominence is maintained or developed.
The distinction between formation, discussed in Chapter 4, and models, discussed in this chapter, is often blurred. When we consider certain formation scenarios, we often imply the resulting model. Similarly, as we approach the
problem of stability of certain prominence models in this chapter, we shall see
that the discussion of the formation process has already anticipated the answer. Nevertheless, we shall find it useful in bringing out the basic physics involved to have subdivided our overall discussion into formation and models.
Before we present different prominence models, we shall first look at some
global aspects of prominence location, and then discuss a fundamental characteristic of all prominence plasmas, viz. their fine structure. In discussing the fine
structure we are naturally led to consider the prominence-corona transition region (PCTR), which prOvides the link between the cool prominence plasma and
the hot corona and which is a highly heterogeneous region with its own fine
structure.
5.1. The Location of Prominences
We may be able to gain considerable insight into the nature of prominences from
a study of their location on the solar surface in relation to active regions. Loops
and surges generally occur in or close to active regions with developing sunspots,
accompanied by flare activity, locations that strongly suggest the important
role played by magnetic fields in these types of prominences.
Also active region filaments and quiescent filaments (quiescent prominences) owe their existence to the effects of a magnetic field. Martin (1973) and
Tandberg-Hanssen (1974) independently considered filaments to form either in
an active region or between two active regions. We shall define two types, A

168

CHAPTER 5

and B, of low-latitude filaments, according to their positions in the magnetic
structure and with respect to active regions (Tandberg-Hanssen, 1974). Type A
filaments separate areas of opposite magnetic polarity belonging to one and the
same bipolar region; i.e., the filament is found between the preceding and following part of the active region; see Figure 5.1. The filament occurs over an Atype neutral line. Type B filaments are situated between two active regions,
i.e., between the following part of one active region and the preceding part of
the other region, in other words, over a B-type neutral line.
In the figure we have sketched the prominences as N-type prominences,
i.e., having a normal polarity field (see Section 3.5 and Figure 3.3), but I-type
prominences would be equally appropriate for illustration. Leroy (1989) added
a type C: filaments that are found over the C-type neutral line which is associated with the polar crown. This is an important addition since the evolution of
the type C neutral line is associated with aspects of the global solar magnetic
field (Hansen and Hansen, 1975; McIntosh, 1980). In addition Leroy noted that
polar crown filaments have a unique location, viz. on the polar side of expanding active regions (Bumba and Howard, 1965), and they drift toward the poles
(Waldmeier, 1957; Hyder, 1965; Topka et al., 1982).
Mouradian and Soru-Escaut (1994) have used the three-type classification
in a study of the evolution and motion of filaments, constructing butterfly diagrams, as for sunspots, and have shown that the time evolutions of the filaments consist of a 22-year cycle. This cycle is made up of two II-year cycles in
which type B filaments of the first II-year cycle become type A of the following II-year cycle. This study points again to the importance of developing a
global picture of the Sun's magnetic activity, and will be addressed further in
the next chapter.
Tang (1987) has studied whether quiescent filaments belong mainly to one
or the other of the types A and B. She used data from 1973 and 1979 and found
substantially more B-type than A-type filaments in her sample.
Quiescent prominences may be formed anywhere on the solar surface, while
active prominences are confined to the activity zones on either side of the equator. Since the solar surface exhibits a differential rotation, and since quiescent
prominences have their feet anchored in the Sun, one expects the prominences
also to partake in the general differential rotation. d' Azambuja and
d' Azambuja (1948) and Bruzek (1961) were the first to measure rotation rates
from latitude cjI ... 40° to close to the pole using quiescent prominences as tracers.
Brajsa et al. (1991) performed new measurements, avoiding limitations of previouswork.
We can approximately represent the Sun's differential rotation by an
expression of the form
.cl(tf»=A+Bsin2 tf>+Csin 4 tf>,

PROMINENCE MODELS

169

where O(eII) is the sidereal angular rotation rate in degrees per day. Spectroscopic
measurements reveal for the solar photosphere a differential rotation given by
(Howard and Harvey, 1970)
Dph(l/» = 13.76°-1.74sin2 I/> -2.19sin 4 1/>,

which corresponds to a rotation period for a point on the equator (ell =0) of about
26 days. On the other hand, for quiescent prominences their differential rotation is better represented by (Brajsa et al., 1991)
Dprom(l/»

=14.45°-0.l1sin2 I/> - 3. 69 sin4 1/>.

Some prominences deviate in their rotation rate from the expression for
Oprom(eII> given here, and instead tum slowly around so-called pivot points (see
Section 6.2.1), which themselves rotate rigidly, i.e., rotate on the Sun with the
so-called Carrington rotation velocity which is independent of latitude and has
a sidereal rotation rate of 14.3' per day, or a period of rotation of about 25 days.

Fig. 5.1. Schematic representation of two possible field configurations, type A and type B, in bipolar
magnetic regions.

170

CHAPTER 5

5.2. Radio Waves and Prominences

Radio waves furnish a unique way of investigating the outer parts of prominences as well as their immediate surroundings. Direct observations of the
prominence plasma is possible only in the millimeter-wavelength range. Longer
waves, in the dm and meter range, are not able to escape from the chromosphere
and low corona, and they give information from the corona. This behavior of
radio waves is due to the fact that the refractive index n of the solar
atmosphere is given by

n=[-tJ ,

(5.1)

where v is the frequency of the radio wave, and the critical frequency ve, the
plasma frequency (Equation (2.140», is

(5.2)
Since the refractive index is less than unity, we notice the radio waves are refracted in the opposite sense to light waves when they pass from a tenuous to a
dense medium. When 1 - (vel v)2 becomes negative, propagation is no longer possible. This cut-off occurs when the density of the plasma increases so that Vc =
v. Using a model for the solar atmosphere, we may construct ray paths for radio
waves of different frequencies and explore the region of the atmosphere from
where the rays can escape; see, for instance, Pawsey and Bracewell (1955) and
Kundu (1965).
The observed temperature at a given frequency and along a ray path is related to the radiation temperature TR at the optical depth 'tv by the equation
Tb

=

r

'f

o

TR e-Tvdr v .

(5.3)

The observed temperature is referred to as the brightness temperature, and for an
isothermal atmosphere it is simply
(5.4)

We see that TR is an upper limit for the brightness temperature, a limit that is
reached for very large optical depth, i.e.,

PROMINENCE MODELS

171

(5.5)
For small optical depths, on the other hand, we have

(5.6)
The optical depth, d't y = -kv ds, where ds is a ray path element and ky the
absorption coefficient, is determined mainly by free-free transitions. Generally,
the absorption or emission, due to the acceleration, or deceleration, of electrons
in the Coulomb field of a charge Ze, is given by

(5.7)
where v is the velocity of the free electrons and gff is the Gaunt factor; see
Section 2.1.4.
By choosing an appropriate wavelength, which for reasonable models of
the solar atmosphere is A. < 1 cm, one can explore quiescent prominences and
their immediate surroundings. Longer radio waves, in the dm and m range, are
often associated with active, eruptive prominences. During coronal mass ejections (CMEs) (see Chapter 6), meter wave type II and type IV radio bursts are
observed. However, these waves are generated in the coronal plasma where
either particles from a flare or an eruptive prominence have triggered longitudinal plasma oscillations at the local plasma frequency (Equation (5.2». These
oscillations are subsequently converted into the observed electromagnetic
waves, which consequently can give information only about the corona.
The first observation of a bona fide "radio prominence" at millimeter wavelength was made by Khangil'din (1964), who recorded radio depressions above
H<x filaments. The observation can be explained if the prominence is optically
thick with a temperature T pr and if the prominence then absorbs the radiation
coming from the underlying chromosphere whose temperature Tch> Tpr.
Apushkinskij and Topchilo (1976) constructed a prominence model from observations in the 4 to 8.6 mm range and deduced temperatures that increase with
wavelength from Tb = 6300 K at A. = 4 mm to Tb = 8300 K for A. = 8.6 mm. Brajsa
(1993) studied the absorption by prominences for radio waves with wavelengths 8 mm and 14 mm, and with free-free absorption and assuming T =6400 K,
ne =5 x 1010 cm-3, he found that optical depth 't = 1 is reached at a length scale
of about 10 km. These values of the radio brightness temperature are to be compared with the similar values deduced from data obtained in the visible part
of the spectrum; see Section 3.1. Subsequently many radio observations in the
millimeter range have been made of quiescent prominences, and an excellent review of salient factors is given by Chiuderi Drago (1990).

172

CHAPTER S

When we go to slightly longer wavelengths, to the centimeter range, sufficient angular resolution is more difficult to attain. The first solar radio maps
showing radio wave depressions due to filaments were obtained by Chiuderi
Drago and Felli (1970) at A. =1.95 em. The reduction in observed radio emission
over prominences at this wavelength is not due to absorption in the prominence
(as in the case for A. < 10 mm), since optical depth t > 1 is reached above the
prominence. The depression may be due to decreased emission from the corona
above the prominences, the coronal cavity (Chiuderi Drago and Felli, 1970;
Straka et al., 1975), but observations regarding the size of radio versus optical
filaments contradict this interpretation (Raoult et al., 1979; also Schmahl et
al., 1981). Lantos and Raoult (1980) discussed observations of prominences
recorded in the 3.5 mm to 6 cm region of the radio spectrum. They argued that
the radiation reveals the conditions in the outer layers of the prominences adjacent to the corona. For further discussion of this problem, see, e.g., Chiuderi
Drago (1990). The centimeter observations provide a powerful means of investigating the coronal environment of a prominence, while by going to shorter and
shorter wavelengths we penetrate the prominence plasma through the prominence corona transiti.on region (see Section 5.3.1). Lang (1990) observed quiescent
filaments in emission on the disk at a wavelength of A. =91.6 cm, with a brightness temperature (Equation (3.5» of 3 x lOS K. The emission may be attributed to
thermal Bremsstrahlung from the hot prominence-corona transition region. This
region was observed in emission because of the lower optical depth of the intervening coronal plasma at A. =91.6 em.
It is also possible that the millimeter observations can tell us something
about the formation process of filaments. Buhl and TIamicha (1970) found that
in several regions with depressed radio signals but with no Hex filament, an Hex
filament would form within a few solar rocations. Similarly, Hiei et al. (1986)
observed a radio depression which increased in depth, i.e., Tb decreased, the
following day when an Hex filament appeared in the same position. Cool
material is apparently present and can absorb observable amount of microwave
radiation before it lets its existence be known by Hex observation.
In a statistical study Schmahl et al. (1981) found that about two-thirds of
the radio depressions were associated with observable Hex filaments. Also,
nearly all of the depressions lie above filament channels (Section 4.1), and the
authors claim that ''microwave observations can, therefore, supplement optical
observations in identifying neutral lines."

5.3. Fine Structure of the Prominence Plasma
As we discuss the different models that have been proposed for prominences, we
shall see that many of them do not explicitly treat the fine-structure elements
(FSEs), that have been labeled threads, filaments, blobs, etc., by different au-

PROMINENCE MODELS

173

thors, and that make up the prominence body. This may not necessarily be a fatal flaw if the model does not preclude FSEs, but barely ignores them to give a
sort of average model, that still could allow a fine structure in a higher approximation. These models may, e.g., provide insight into the role of magnetic
fields regarding prominence support. However, they are ill-equipped to answer
questions related to the radiation from the prominence plasma.
5.3.1.

QUIESCENT PROMINENCES

As far as one can tell prominences never exist as amorphous blobs; they all exhibit a more or less well-defined fine structure. This applies to quiescent as
well as active objects, and in all probability the fine structure is determined by
the magnetic fields that invariably form the skeleton of a prominence. For a
precise overview, see Schmieder (1992).
It has been known for a long time that quiescent prominences possess a fine
structure, but its nature still poses one of the most challenging questions in contemporary prominence research. The problem may simply be stated as follows:
Why do we observe an essentially vertical fine structure in basically horizontal magnetic fields?
Limb observations leave little doubt that the prominences consist of
threads and knots; typical dimensions of the former are 5000 km in length and
300 to 1000 km in width. However, the threads may be thinner, since the
smallest observed widths are of the same size as the instrumental resolution
limit. The threads are predominantly vertical, but may exist in any orientation, including horizontal.
A question that quickly poses itself regards how densely the threads fill
the overall prominence body. This question was mentioned in Section 2.1.3 in
terms of the porosity of the plasma which was characterized by the filamentary degree y. A more common practice in solar physics research is to analyze
the porosity in terms of the fraction of the overall volume filled by fine structure, referred to as the filling factor f. Values of f between 0.01 and 0.1 are commonly quoted (Engvold, 1976; Simon et aI., 1986b; Engvold et al., 1990).
Many attempts have been made to determine the number of threads along
the line-of-sight in a prominence. Basically one of two methods is used, viz.:
a. Analyses of emission line profiles, or
b. Non-LTE modeling of hydrogen lines.
When the first method is used one finds that there is a fairly small number
of threads, say 5 to 15, in the line-of-sight. To derive such numbers Engvold et
al. (1989) studied the distribution of Ca II, K line intensities, and line widths

174

CHAPTER 5

versus line shifts; Zirker and Koutchmy (1990, 1991) analyzed the film contrast
in terms of a random clustering of threads; while Mein et al. (1989a) compared
Hex profiles with stochastic distributions of velocity threads.
The situation is different when the second method is used. When one explores non-LTE modeling, 50 to several hundred threads are needed to get theoreticalline profiles (Fontenla and Rovira, 1985) or line intensity ratios (Vial et
al., 1989) that are comparable to observed hydrogen lines and their ratios. The
resolution of this seeming discrepancy is probably simple enough, even though
it requires a certain model for the arrangement of the fine structures. When we
use the non-LTE modeling approach, we explore a characteristic length, LT, related to the temperature (and density) fine structure, and we see all the fine
threads in the prominence plasma of which there may be hundreds in the lineof-sight. However, they are clumped together in such a way that many of them
share the same dynamical nature; i.e., they have the same velocity relative to
an observer, and when a stochastic distribution of velocity elements is studied,
we find a fairly small number pertaining to a characteristic length, Lv, for velocities. Consequently, each of these velocity elements is, therefore, made up of
a large number of smaller threads. The final conclusion is that quiescent prominences show a fundamentally important fine structure, and that the radiation
from the prominences reflects the nature of this fine structure. All these finestructure elements are found in the ubiquitous magnetic field, which itself has a
fine structure, characterized by a length, LB. In a lucid review of prominence
fine structures, Mein (1994) has summarized the above discussion by stating
that observations seem to indicate that LT < Lv < LB.
Since it is generally believed that magnetic fields play the dominant role
in the stability of prominences, most attempts to model them invoke the effects
of such fields. This applies also to the details concerning the fine structure, and
the FSEs, the threads, may be considered as flux tubes. However, as we have
implied above this view is not without difficulties. Since the magnetic field in
prominences is more or less horizontal (e.g., Leroy, 1989) while the threads may
take on any orientation including vertical, which is even the most prominent.
We shall consider attempts to solve this discrepancy later in this chapter.
Basically two different models have been envisioned to account for the observed radiation from the fine-structure elements. Poland and TandbergHanssen (1983) advocated the view that there are FSEs with different temperatures, some hot, some cold, and that the different emissions, e.g., lines from H
I, He I, He II, C IV, 0 V, etc., come from flux tubes with plasmas at the appropriate temperature for the ion in question. The other model, discussed by
Engvold (1989), pictures all flux tubes as having cool cores with a transition
shell around the core in which the temperature increases from the 1Q4 K regime
of the core to the million degree coronal plasma; see also Pojoga (1994).
The shell or layer where we find the abrupt temperature change is called
the prominence-corona transition region (PCTR) and may be considered a thin

PROMINENCE MODELS

175

skin on the fine-structure elements, characterized by a strong temperature gradient. The conditions are similar to those found in the more familiar
chromosphere-corona transition region (CCTR).
A closer study of the transition regions, both the one between the chromosphere and the corona as well as the one between prominences and the
corona, becomes possible with the availability of data on spectral lines formed
at temperatures up to 1()6 K and situated in the ultraviolet and extreme ultraviolet parts of the spectrum. Analyses of such lines from quiescent prominences
show that the PCTR is very thin and similar to the CCTR (Yang et al., 1975;
Orrall and Schmahl, 1976, 1980; Schmahl and Orrall, 1986; Rabin, 1986).
Smartt and Zhang (1984) observed a ~uiescent prominence in Ha as well as in
the forbidden coronal lines Fe X, 6374 A and Fe XIV, 5303 A, and found that the
coronal emission came from locations corresponding to the outer regions of the
prominence, i.e., from the PCTR. The physics of the transition regions has been
discussed recently in two excellent reviews by Engvold (1989) and Vial (1990),
while Chiuderi Drago et al. (1975) and Chiuderi Drago (1990) have shown the
importance of considering the radio wave emission from the PCTR plasma.
It is customary to discuss both the CCTR and the PCTR in terms of the differential emission measure, Q(T), and a number of investigators have elaborated
this method (Athay, 1966a; Jordan and Wilson, 1971; Dupree, 1972; Withbroe,
1975; Raymond and Doyle, 1981; Dere and Mason, 1981; Nicolas et al., 1982;
Schmahl and Orrall, 1986; Kjeldseth-Moe et al., 1984). We may arrive at this
useful concept in the following way. First recall that the line emissivity, i.e.,
the power per unit volume, is given by (see Equation (2.7»
(5.8)
The lines one observes and uses in the differential-emission-measure
method are optically thin and allowed (electric dipole transitions obeying the
rules 2.57); whence the excited levels are populated by electron collisions and
depopulated by radiative decay. Furthermore, the populations of the excited
levels are negligible compared to that of the ground level, and the statistical
equilibrium equations (Section 2.1.2) can be solved as a two-level system for
each transition
(5.9)
where Cij is the collisional excitation rate (Equation (2.21», and the emissivity
can be written as

176

CHAPTER 5

(5.10)
This expression is often rewritten in the form

e(A) ={3G(T)ne 2 ,

(5.11)

where the density and temperature dependencies are shown explicitly, and
where ~ contains atomic parameters and abundances. The function G(T) is called
the contribution function and is strongly peaked in temperature (Pottasch,
1964). We may now write for the power in a line radiated from a volume V

J

P={3 G(T)n/dV.

(5.12)

If the plasma in volume V is isothermal at temperature T, the power will be
given by

(5.13)
where the integral of ne 2dV is known as the emission measure. This quantity
can be determined from the observed power, provided the temperature is
known. Since, in general, the function G(T) is sharply peaked, we may assume
that the temperature is the one that maximizes G(T), and we can then derive
"isothermal" emission measures as a function of temperature from a set of observed line intensities.
In the more general case we write, instead of Equation (5.13), the expression
for the power in a spectral line

J

P ={3 G(T)Q(T)dT,

(5.14)

where we have defined the differential emission measure Q(T). Different authors have used somewhat different expressions for the differential emission
measure (OEM). With a sharply peaked function G(T) at T =T maxt Equations
(5.13) and (5.14) will define Q(T) by the equation

J

J

Q(T)dT = ne 2dV,

(5.15)

PROMINENCE MODELS

177

which gives the emitting power at temperatures between T and T + dT contained in the volume V. For the OEM we may use the expression
(5.16)
where dV(T) is the volume of the radiating plasma in the logarithmic temperature interval dT.
Craig and Brown (1976) changed variables and defined Q(n by the expression
(5.17)

where the integration is performed over all surfaces S at temperature T in the
volume and dTldh is the gradient along the line-of-sight. This expression is
valid so long as the contribution function G(n is either independent of density
or, if dependent, provided the density is a function only of temperature. Nicolas
et al. (1982) expressed the OEM by

(dT)-1 '

Q(T) =AeffPe 2 dh

(5.18)

where Aeff is a filling factor, viz. the effective radiating area and Pe the electron pressure.
Formally, we can combine Equations (5.12) and (5.15) and relate the intensity of a spectral line to the differential emission measure

J

I(,t) = B(,t,T)Q(T)d(T),

(5.19)

where the function B(A,n contains all the other parameters. Equation (5.19) is
a Fredholm equation of the first kind and can be solved by quadratures.
However, the equation is known to be unstable, and unphysical solutions usually are found (Craig and Brown, 1976). With the determination of Q(n neither well posed nor unique-only the integral over temperature is well defined
observationally-many different Q(n can have the same JQ(T)dT. In practice
one makes certain assumptions regarding smoothness, monotonicity, etc. First,
one measures a number, N, of line intensities Ij(A) (i, = 1,2 ... N) and derives a
set of M (M S N) parameters to define Q(n. As discussed by Dere and Mason
(1981) a popular choice is

178

CHAPTER S

(5.20)
and the solution is sought as a linear least-squares problem.
In the case of both the CerR and the PCI'R we need data pertaining to spectrallines that cover temperatures from lQ4 K to roughly 1()6 K, i.e., observations
of lines mainly in the UV and EUV regions of the spectrum.
However, we note that also emission in other parts of the spectrum canand have-been used. X-ray data (e.g., Ca XIX, 3.2 A) have been extensively
studied by Antonucci et al. (1982), while, in the other extreme part of the spectrum, radio waves have proven very useful. Chiuderi Drago et al. (1975)
showed that radio observations demanded the existence of a transition region,
and its physical parameters were deduced by, e.g., Butz et al. (1975) who
showed that the PCTR as observed at millimeter wavelengths is quite thin, of
the order of a few hundred km; see also Kundu et al. (1978). However, discrepancies remained between PCTR observations in EUV and at radio wavelengths. These were reconciled by Chiuderi and Chiuderi Drago (1991); see
below.
The many investigations pertaining to the OEM, both for the CCTR and the
PerR, agree on the basic shape of curve for OEM as a function of temperature,
viz. a broad minimum in the range lOS K < T < 5 x lOS K and a somewhat steeper
rise to lower temperatures than to higher temperatures; see Figure 5.2 for an
example (Engvold, 1988). Since one generally considers the transition region as
a thermal connection between the corona and the chromosphere (CCTR) or between the corona and a prominence (PCTR), the standard picture accounts for
the thermal structure of these regions in terms of thermal conduction from the
corona along the magnetic field, balanced by radiative losses (Giovanelli, 1949;
Athay, 1966a). The energy equation, Equation (4.10), may be written for the
balance in the PCTR
(5.21)
where the gain terms include mechanical and ohmic heating and the losses are
due to radiation, and Fcond and Fvare, respectively, the conductive energy flux
and the flux associated with mass flows. In the standard picture we retain only
thermal conduction as the dominant heating term and if we assume a constant
flux (Engvold, 1988),
F

cond

=1.1 x 10-6 T SI2 dT
dh'

we may combine Equations (5.18) and (5.22) to find

(5.22)

179

PROMINENCE MODELS

5
2

log Q(T) = const + -log T.

(5.23)

This expression is the straight line pictured in Figure 5.2 and shows that for T
~ 5 x lOS K, the standard picture seems an adequate approximation. However,
it fails miserably for lower temperatures, which means that the cooler part of
the transition region is so thin that it cannot radiate enough to account for the
rise in OEM for T < lOS K. At these lower temperatures, deposition of energy is
necessary to account for the empirically-derived temperature gradient. These
remarks are valid for the CCTR as well as for the PCTR.

1.5
1.0

a

OJ
0

-l

0.5

t

0.0
-0.5

t

-1.0
-1.5
4.2

4.4

4.6

4.8

5.0

5.2

5.4

5.6

LogT
Fig. 5.2. Differential emission measure of prominences as a function of temperature. The straight line
shows the relation predicted by thermal conduction above (Equation (5.23» (after Engvold, 1989).

Many attempts have been made to model the low temperature part of the
OEM curve by invoking different heating mechanisms, but none has proven completely satisfactory. One can always postulate a heating term in Equation
(5.21) that will produce a rise in the OEM curve for T < lOS K, but closer scrutiny
often reveals flaws in the overall physical picture of the PCTR, or sufficient insight cannot be gained. Engvold (1988) considered dynamic effects and commented that if the enthalpy flux is comparable to the conductive flux, we
should expect that flows will dominate the plasma. He estimated this regime
at a temperature T = lOS K by setting

CHAPTER 5

180

(5.24)

where v is the flow velocity and Ether the thermal energy density of the
plasma. We can use Equation (5.18) and replace the temperature gradient in
Equation (5.22) to obtain
Fcond

=3.48x 106 Ae!!

2

( Pe 5 ).
Q T=lO K

(5.25)

Engvold took the value for Q(T = lOS K) from the empirical curve in Figure 5.2,
put Ether = 3 Pe, used Equations (5.24) and (5.25), and found a
(5.26)

The effective radiating area Aet! falls in the range 0.05 S; Aet! < 1 and Pe == 0.01 to
0.1 dyn cm-3. From these data Engvold found that the critical flow speed in the

PCTR would be v S; 5 km s-l. Such velocities are often seen in quiescent prominences, and if they can be expected to occur in the transition region, one is led to
conclude that plasma flows should be included in modeling of the PCTR.
Whether the effects can be important enough to provide the necessary heating
of the transition region for T < lOS K is still an open question, but we are again
reminded of the basically dynamic nature of even the more quiescent prominences and their immediate surroundings.
In addition to realizing the dynamic nature of prominences and their transition region, we judge from their fine structure that the plasma involved is in a
very fragmented state. Under such conditions filamentary electric currents
might be expected, and extra energy would be produced by Joule heating. This
interesting scenario was proposed by Rabin and Moore (1984) to successfully explain the heating of the lower, i.e., cooler chromosphere corona transition region. The hotter, outer parts of the CCTR are still, in this scenario, heated in
the standard way by parallel thermal conduction.
The likelihood of copious electric currents also flowing in the PCTR
prompted Rabin (1986) to apply the Rabin-Moore heating mechanism also
there. However, he abandoned the idea since such a theory could not explain
how the necessary currents are produced. As he points out the physical reason
that the cooler part of the OEM curve can be accounted for in the Rabin-Moore
(1984) model is the temperature dependence of the thermal conductivity. For
constant pressure Equations (4.20) and (4.22) give KII oc T5/2 and K1. oc T-5/2,
which means that when cross-field conduction K1. is important, it will lead to
a negative slope in the curve for OEM vs. temperature. Rabin (1986) then postu-

PROMINENCE MODELS

181

lates that the extra heating also for the cooler transition region plasma comes
from thermal conduction, this time across the field lines. For this scenario to
work, the plasma must be highly fragmented, such that the area exposed to the
hot corona is orders of magnitude greater than the projected surface area.
In a thorough analysis of the energy balance in the PCTR Chiuderi and
Chiuderi Drago (1991) and Chiuderi Drago et a!. (1992) focused the attention on
the effect the angle 9 between the magnetic field B and the temperature gradient VT will have on the heat conduction. If the fine-structure elements are more
or less cylindrical magnetic flux tubes, each of these threads will be wrapped in
a transition region, and the angle 9 would be expected to be large--approaching
90 Chiuderi Drago et al. make the point that for the outer parts of the PCTR
where T > lOS K the thermal conduction may be treated as a modified longitudinal case, and the derived differential emission measure correctly reproduces the observed OEM. However, in the inner parts of the PCTR, corresponding to the lower CCTR, where T < lOS K, VT 1.B, and we are faced with transverse thermal conduction. This conduction is so reduced (Equation (4.22» that
the resulting OEM falls orders of magnitude below the observed value.
Chiuderi Drago et a!. suggest that the extra needed heating of these layers
may come in the form of dissipated Alfven wave flux, since Yi and Engvold
(1991) have found evidence for Alfven waves in their observations of oscillations of quiescent prominences. We shall now look briefly at their arguments.
The Alfven waves would run parallel to the magnetic field and be damped exponentially, and the calculated damping length A.A in the PCTR is smaller than
the length,i, of the threads, indicating that all the energy carried by the
waves can be dissipated in the thread and heat them.
If the Alfven wave has an amplitude ~B, the energy flux of the wave will
be
D.

(5.27)
where VA is the Alfven velocity. If the waves are linear, i.e., ~B « B, the
damping length may be calculated from the theory of Califano et a!. (1990),
provided the dissipation occurs due to ohmic heating, and provided the planar
geometry used by Califano et al. can be applied to the flux tube considered by
Chiuderi Drago et at The Alfven velocity will then satisfy
(5.28)
where r is the radius of the flux tube, a the length scale of the variations of the
Alfven velocity, and ~ =V~ (0) I V~ (00) -1. If the pressure is constant one may
write Equation (5.28) in the form

182

CHAPTER 5

(5.29)
where we have put s =(T - TO )la, and where Bo is the magnetic field at the border of the cool flux tube cores defined by r =roo The temperature at this border is
To =1Q4 K. Under such assumptions the minimum damping length is given by
2
~1+L1
A.A =--rVA(O)
-,
1C
L1

(5.30)

where 't is the period of the wave. Chiuderi Drago et al. used the Yi et al.
(1991) value for 't =5-15 min (see Equation (3.8» and found that for AA < £, 85%
of the total amount of energy carried by the Alfven waves is dissipated along
the threads and may furnish the required extra heating of the inner, lower,
transition region. This use of Alfven-wave energy has also been used by Jensen
(1990) to support prominences against gravity. We return to this model in
Section 5.6.
5.3.2. ACl1VE PROMINENCES

While we understand that the fine structure of quiescent prominences is intimately linked with the structure of magnetic fields that threads these objects,
serious questions remain unsolved regarding the detailed orientation of the
threads and the field. The situation seems less complicated in active prominences like surges and loops, even though in other types, like coronal clouds,
little is still known.
In the case of loop prominences, i.e., post-flare loops and arcades, the fine
structure is itself loop-shaped and is made up of the magnetic flux tubes in
which prominence material rains down the two legs from the loop apexes.
Loops with cool (Ha) and hot (C IV, 0 VI, Fe X, etc.) material may be found
juxtaposed, and a simple model calls for more or less parallel magnetic flux
tubes to control the phenomenon (see, e.g., Cheng, 1980; Hanaoka et al., 1986).
In surges we can also account for the fine structure in terms of magnetic flux
tubes into which material is injected from below, and again hot and cool material may be found in neighboring, but separate, nearly parallel flux tubes.

PROMINENCE MODELS

183

5.4. Early Models, Historical Notes
Before the crucial role played by magnetic fields was fully realized, prominences were regarded as cool objects in hydrostatic equilibrium with the hot
corona. Pressure equilibrium must then reign at any height, i.e.,
(5.31)
where the subscript pr means prominence and cor means corona.
The prominence material was supposed to be supported against gravity by
the hydrostatic pressure:
dp
-=-pg.

(5.32)

dz

For an ideal gas p =mp/kT and Equation (5.32) integrates to
p(z) =Po

exp[-Jo~l'
Z

H(z)

(5.33)

where

kT

H(z)=-,
mg

(5.34)

is the pressure scale height and gives the vertical distance z over which the
pressure falls by a factor e. In prominences Hpr"" 300 km, while the hot, surrounding corona has a scale height Hcor "" 50,000 km, comparable to the height,
hpr, of many quiescent prominences. We shall shortly return to the importance of
the inequality:
(5.35)
Early models based on the assumption (Equation (5.31)) were presented to
give values of the important prominence parameters, e.g., temperature and
density. An excellent review can be found in the comprehensive article by de
Jager (1959).
However, by that time Menzel had argued that coronal magnetic fields
could support quiescent prominences in a static equilibrium (Bhatnagar et al.,
1951), and the importance of magnetohydrostatics was realized.

184

CHAPTER S

5.4.1. SOME GENERAL COMMENTS ON MAGNETOHYDROSfATIC MODELS
When a magnetic field is present in the prominence plasma, Equation (5.31) is
no longer valid. However, it cannot simply be modified by adding magnetic
pressure terms, V(B2/81t), to gas pressure gradients, Vp. This would assume that
the fields had no twist, and in actual cases the tension, (B pr • V)B pr for instance, resulting from a twist, may cancel part of the V(B;r/81t) term.
The structure of magnetic fields in prominences raises several interesting
problems. In a steady state the equation of motion (Equation (2.92» reduces to
(5.36)
and shows that when pressure equilibrium reigns, the gravitational pull can be
balanced by a Lorentz force. In other words, for certain magnetic field configurations one might expect the prominence material to be supported against gravity
by the action of the magnetic field.
Magnetohydrostatics, as described by Equation (5.36) means that the flow
speed v of the plasma is much smaller than the sound speed V 5 and the Alfven
speed VA. From Equation (2.121) we find for the sound speed in prominences Vs 10 km s-l and Equation (2.117) gives for Alfven waves a velocity VA - 100 km s·l.
On the other hand we have seen in Section 3.5.1 that the prominence plasma at
times exhibit velocities that approach, and even exceed, the quoted sound
speed. Consequently, in these cases, one should consider magnetohydrodynamic
models.
An order of magnitude analysis of the terms in Equation (5.36) shows that
for reasonable values of the parameters, the (l/c)j x B term will dominate.
Consequently, we conclude that in quiescent prominences a large part of the
current j is parallel to the field B, or, stated differently, a large part of the
prominence field must be force free.
When the electrical conductivity is sufficiently high, the condition that a
field is force free can be written (see Equation (2.126»:
VxB=a(r,t)B,

(5.37)

where we have indicated that in the general case the quantity (X may be a function of both space and time. In a steady state this scalar function of space is
called the reciprocal pitch of the field. Equation (5.37), and the condition
V.B=O

(5.38)

defines the force-free field. When (X = constant, general solutions of Equations
(5.37) and (5.38) are possible (Chandrasekhar and Kendall, 1957), and for sim-

PROMINENCE MODELS

185

pIe cylindrical symmetry (Lust and Schluter, 1954; Schluter, 1957a), the solution is relatively straightforward (see also Schatzman (1961». However, there
is no reason to assume that (X should be constant for magnetic fields in and
around prominences. For the case of non-constant (X, no general method for solving the equations is known, even though solutions have been found for some
simple geometrical situations (Ferraro and Plumpton, 1966). Grad and Rubin
(1958), Gold (1964), Molodensky (1966), Schmidt (1966, 1968), Sturrock and
Woodbury (1967), Jette and Sreenivasan (1969), Nakagawa et al. (1971), and
Raadu and Nakagawa (1971) contributed to the early development of this
field.
5.4.2. FIELD CONFIGURATIONS CAPABLE OF SUPPORTING PROMINENCES
The basic idea in these models is that the prominence material is supported
against gravity by the Lorentz force, j x B, where the field lines lie in planes
perpendicular to the prominence sheet, and the current flows along the prominence (the Y-direction). The models describe the prominence material in
mechanical equilibrium under the combined actions of gas pressure, gravity, and
the Lorentz force, with other forces being neglected. Equation (5.36) with j =
(1/41t)VxB then reads (see Equation (2.124»:
VP - pg - -

1

4n

(V x B) x B = o.

(5.39)

Equations (5.38) and (5.39) and an equation describing the atmospheric model
will give us five equations for the five unknowns, B, p, and p. Menzel was the
first to try to show that coronal magnetic fields can support quiescent prominences in a static equilibrium model (Bhatnagar et al., 1951). The magnetic
field lines were supposed to be in planes parallel to the long axis of the prominence. We shall apply Menzel's analysis to a field perpendicular to the long
axis. Somewhat different formulations of the problem are due to Dungey (1953)
and Kippenhahn and Schluter (1957). A particularly lucid treatment by Brown
(1958), which shows the interrelation between these early models, will be
partly followed here. Even though the details of the models are largely obsolete, a discussion of the basic ideas has more than historical interest, as it
brings out some general principles of lasting importance.
In our coordinate system all these models assume that By =0 and that p, p,
and B are independent of y. Furthermore, the atmosphere is isothermal so that
the last equation we need reads

p = nkT = pgHo,

(5.40)

186

CHAPTER 5

where Ho= kT /mg is the scale height in the absence of a magnetic field. Let F
be a scalar function, F =F(x,z); then we may state
(5.41)
This means that Equation (5.39) may be written
1
2
Vp-pg=--(V F)VF.

4n

(5.42)

If we combine Equations (5.40) and (5.42) we can write

This shows that pezlHo is a function of F, p(F) say, and that

~[p(F)] =__
1 V2 Fezl Ho.
dF
4n
Then V2 Fe zl Ho must be a function of F and the scalar function satisfies the equation
(5.43)
where «\>(F) is any arbitrary function of F.
Brown (1958) considered Equation (5.43) the basic equation for static equilibrium in this two-dimensional case. Any function F that is a solution of
Equation (5.43) will give a possible model for the magnetic field, Equation
(5.41). The ensuing pressure distribution is then given by
p

=__l_e-zIHojtP(F)dF.
4n

The solutions due to Menzel, Dungey, and Kippenhahn and Schluter are shown
in Table 5.1 in terms of the choice of function «\>(F) and the form of the scalar
functionF.

187

PROMINENCE MODELS

TABLE 5.1
Models of magnetic field in quiescent prominences, By = 0
c!>(F)
F
Author
Menzel (modified)

= AF(1-2HI Ho)

c!>(F)

F = FI(x)e-zj2R

A=const
Dungey

c!>(F) = D = const

F= DH6e-a1Ho +F2
F2

Kippenhahn and
Schluter

=l_e- zIHo _2e-z12Ho cos~
2Ho

cp(F) = CeFIGHo

C=const<O
G=const

In the framework of Equation (5.43) Menzel's model corresponds to the
choice
rp(F)

= AF(1-2HIHo).

(5.44)

where A is a constant. In the original paper, Menzel remarked that Equation
(5.42) is separable if the function p (F) has the form PI (x )e-z/H and if F can be
written as F = Fl (x)e- z/2H, which is equivalent to the choice (5.44) and leads to
the following differential equation for FI,
2

d Fl
dx2

+~=AF(1-2HIHo)
4H2

1

.

This equation integrates to

where c is a constant, and where H/Ho = q + 1 > 1. If we assume that the lines of
force are horizontal as they traverse the prominence sheet, i.e., dFt/ dx I x = 0 = 0,
we find

188

CHAPTER 5

( -dF} )

2

dx

2

2

F}
F} (0) A [F (0)-2q - I '~2a]
+-;;';2=-:;;r+-}
}
4H
4H
q
,

and the corresponding pressure is
p =A- F-2 q e-1.IH .
81rq

In terms of the ratio

P between

gas and magnetic pressure,

(qFl(O)2q + 2), the equation for Fl takes the form

p =4H2A/

and the pressure can be written as
(5.46)
Equation (5.45) must be solved numerically, but some general physical properties may be inferred without solving it. The lines of force, along which F =
const, form a set of curves that may be described by the equation
F } -- e(z-1{)I2H ,

where Zo is a constant that varies from line to line. For pq > I, Fl has a minimum
at x = 0, which means that the lines of force are bowed to a minimum height in
the prominence sheet. Furthermore, the pressure has a maximum at x = 0, and
the model consequently gives a rough picture of how prominence material is
denser than the surroundings and causes a dip of the lines of force in the prominence (see Figure 5.3).
An obviously simple choice for the function cjl(F) is a constant, cjl(F) = D.
Then

is a solution for F. Dungey gave his solution for

PROMINENCE MODELS

F2 = 1+ e-zl Ho _ 2e - z/2Ho cos _X_.

2HO

189

(5.47)

In this case the lines of force are closed loops for F < 1 and infinite wavy lines
for F > 1. The magnetic field is assumed to be zero except between two closed
loops, F = a and F = b, so that the pressure outside the outer loop F = a is not
affected by the field. Inside the loop F = b, the pressure is greater than outside
F = a at the same height. By redistributing the matter inside F = a (the prominence), the magnetic field may ensure that a narrow horizontal filament floats
in equilibrium, surrounded by material of considerably smaller density.
Cowling (1957) criticized this model and drew attention to the fact that
Dungey's solution requires currents running in opposite directions near F =a and
F = b, which hardly seems a likely situation. However, by letting the prominence exist inside closed loops of magnetic field, Dungey introduced a field configuration that is of importance in the study of prominence stability, and may
have a bearing on the question of twisted magnetic fields.
A third model, due to Kippenhahn and Schluter, corresponds to the choice
(see Table 5.1)
4J(F) =CeFIGHo

(5.48)

in Equation (5.43), where C and G are constants and the trial solution has the
form

We obtain an equation for F3(X) that can be integrated directly.
This choice leads to the celebrated Kippenhahn-Schluter model. In their
original paper, Kippenhahn and Schluter (1957) combined Equations (5.39),
(5.40), and (5.41) to derive the following equation for the field
(5.49)

The density distribution p(x,z) follows from the x-component of Equation (5.39):

ap = _ mBz [aBz _ aBx ].
ax 4nkT ax az
Bx

(5.50)

The authors argued that
will vary slowly with x in a thin prominence, and
took Bx independent of x. They further assumed Bx to be independent also of

190

CHAPTER 5

height; in other words, Bx = const. By Equation (5.38), Bz will not depend on z.
These conditions simplify Equation (5.49) to a differential equation for Bz
iPBz ~aBz_O
2 +
- .
ax
HoBx ax

The boundary conditions Bz(x = O}
following solution of Equation (5.51),

= 0 and Bz(x-+ oo} = Bz(oo}

(5.51)
lead to the

(5.52)
where Bz(oo)/Bx > O. The corresponding density distribution (Equation (5.50»,
becomes
ap __ ~aBi
ax - 8nkT ax

and yields the solution
(5.53)
for the boundary condition p(x-+oo) =O. Equation (5.52) indicates that the field
lines bend down as they traverse the filament, in a manner not unlike the
central part of the lines in Menzel's model. The density distribution (Equation
(5.53» shows a fairly sharp maximum in the prominence sheet, (x = 0), and
falls to half its maximum value at x =±1. 8Ho[ BxlBz (00)]. The maximum value is
Pc =mBz (00)2 187tkT, which, for a 5000 K prominence whose supporting field
lines correspond to Bz(oo) equal to 2 or 3 Gauss, amounts to about 5 x 10-13 g cm-3,
or a hydrogen density of n(Ff) "" 3 x 1011 cm-3.
For many years the Kippenhahn-Schliiter model provided a valuable
frame of reference in which to discuss prominence support and stability, and it
is basically one of the two models referred to in discussions of topologies for
supporting prominence fields (see Section 3.5).
Brown's (1958) analysis of the prominence models from the 1950's shows
that these models can all be discussed in the framework of Equation (5.43). Now
let us recall that two-dimensional equilibria, in general, are modeled in terms
of a flux function F(x,z) that satisfies a Grad-Shafranov equation (see, e.g.,
Dungey, 1953; Brown, 1958),

PROMINENCE MODELS

V2 F+ ~
: [p(F)exp(-i H(~z)
dz )+.!B;(F)]=O.
2

191

(5.54)

If By(F) = 0 and the atmosphere is isothermal so that the scale height
H(F,z) = const, Equation (5.54) reduces to the formulation given by Brown. The

models of Menzel, Dungey, and Kippenhahn and Schluter portrayed in Table
5.1 are all special solutions of the Grad-Shafranov equation.
We shall now look at more contemporary magnetohydrostatic models that
have been proposed in recent years.
5.5. Global Magnetohydrostatic Equilibrium
During the last 20 years a large number of investigations of prominence support
and structure has been published, and the stability of the KippenhahnSchluter model as well as many later models has been studied. The problem of
prominence equilibrium falls really into two parts, the external equilibrium,
i.e., the global, overall magnetic field configuration that supports the prominence material in these models, and the internal equilibrium, including the
thermal structure of the prominence plasma. These two parts can be treated
separately, and the reason for this decoupling of the external and the internal
equilibria is due to the inequality (Equation (5.35»: the density scale height of
the prominence plasma is orders of magnitude smaller than the overall dimension, the height, of the prominence itself (-300 km versus -50,000 km). Most investigations have been concerned with two-dimensional models, since the
prominences have been verified as being very long and straight and fairly uniform along their axis, which we also in the following shall take as the Y-axis.
These models, therefore, ignore the presence of the prominence feet, a phenomenon we shall return to later. Under these conditions the prominences may
be represented by current sheets or line currents.
Some of the models address both parts of the problem, both the external
supporting magnetic field and the interior conditions of the prominence plasma.
Other investigations have been mainly concerned with the stability of the configurations. In this section we shall look at some of the many external, global
equilibrium models proposed and refer to Section 5.8 for internal equilibrium
discussions.
In current sheet models, like in the original Kippenhahn-Schluter prominence, the idea is that the prominence thickness is so small, in comparison with
coronal dimensions, that it may be represented by a current sheet. In this sheet
one may have a jump in the vertical component of the magnetic field, and with
a potential external coronal field complex variable theory allows us both to
describe the field and to treat the current sheet as a cut in the complex plane. In

192

CHAPTER 5

Section 5.5.1 below, we shall see how this method has been developed and applied to prominence research during the last 20 years.
Instead of using a potential magnetic field in the corona, prominence models
can also be studied when the coronal field is force-free. One may then easily incorporate the effects of magnetic shear, a condition that seems to play an important role in the stability of prominences. Shears and twisting motions of flux
tubes have also been studied as possible ingredients in prominence models, and
we shall return to these questions in Sections 5.5.2 and 5.5.3 below.
5.5.1. SUPPORT IN NORMAL POLARITY FIELDS
The early models by Menzel and by Kippenhahn and Schliiter discussed in the
previous section belong to this category, but they were conceived long before the
distinction between normal and inverse polarity fields (Section 3.5) was made.
The configuration of the supporting magnetic field in Menzel's model is shown
in Figure 5.3. Other possible configurations have been proposed and their stability has been explored.
In particular, Kippenhahn and Schliiter constructed the current sheet by a
mirror-imaging technique of a potential field at the plane of the prominence, x
= 0; see Figure 5.4. This field may be thought of as produced by lines of dipoles
in the XY plane; for x < 0 the line of dipoles is at x = -1 with a line of fictitious
dipoles at x = cx; for x > 0 a line of dipoles at x = 1 and at x = -cx. The resulting
magnetic field may be written
(5.55a)

B

z

=±(

1 _ a2 )
~ a 2 +z 2 •

(5.55b)

Kippenhahn and Schliiter's method leads to an infinite current sheet in the x =
oplane. Any kind of a two-dimensional symmetric potential field can be used in

this method and only afterward can one check to see whether the resulting
field geometry is reasonable.
A method to construct a prominence model as a current sheet of finite vertical extension was accomplished by Malherbe and Priest (1983). Their twodimensional configuration was obtained with complex-variable functions
which allow a discontinuity of the vertical field component at the location of
the prominence. In other words, the current sheet is described by these complex
functions which have a cut along the Z-axis, say from a point P to a point Q
(Figure 5.5), so that the prominence exists at heights hpr such that

193

PROMINENCE MODELS

z

J

x=O (prominence sheet)

--------------------~~--~------------------~x

z =0 (photosphere)
Fig. 5.3. Supporting magnetic field in Menzel's model (after Bhatnagar et aI., 1951).

P<hpr <Q.

(5.56)

The magnetic field is expressed by a complex function
(5.57)

of the complex variable
potential field when

~

=x + i z, and any harmonic function CI>(~) generates a
Bx = Im(4J), Bz = Re(4J).

(5.58)

Figure 5.5 shows two different field configurations with current sheets defined
between P and Q obtained with Malherbe and Priest's method.

CHAPTER 5

194

z

......... ...

.... ....

....

,,

\

\

\

~---_\

....

\

\
',\

II

,III,,"

I

\1 II,,"

-1

+1

x

Fig. 5.4. Supporting magnetic field constructed by the Kippenhahn and Schluter (1957) method; see
text.

(a)

(b)

z

+

x

z

+

x

Fig. 5.5. Supporting magnetic fields obtained by Malherbe and Priest (1983) using complex functions.

PROMINENCE MODELS

195

As emphasized by Anzer (1989) in his review article, the two methods just
described to construct two-dimensional field configurations capable of supporting prominences cannot start from observed field distributions. However,
several years earlier in an important paper, Anzer (1972) had inaugurated the
next phase of prominence modeling by using observed magnetic fields in developing a method to calculate the electric currents in quiescent prominences.
He assumed the coronal field to be current free and allowed currents to flow in
the photosphere and inside the prominence. As in most of the models the
prominence is infinitely thin, the YZ plane. Anzer then used the observed
normal field components both in the photosphere (vertical component from disk
observations) and in the prominence (horizontal component from limb observations) and solved this mixed boundary value problem to calculate the sheet currents inside the prominence and the resulting Lorentz force. A numerical example showed that a positive, upward Lorentz force could provide support for
material above a certain height z =Hmin "" 17,000 km. Below this height the
curvature of the field is downward, i.e., the field lines are stretched upward,
the Lorentz force becomes negative and no support exists.
Anzer used a flux function F =F (x,z) to describe the field, Bx =iJFlik, Bz =
-iJFliJx, where the flux function is completely defined by assuming F(x,O) -+
for x -+ + 00. He then obtained the solution via complex-variable functions; see
Figure 5.6.

°

Fig. 5.6. Supporting magnetic field obtained by Anzer (1972).

CHAPTER 5

196

Anzer realized that to avoid the region of negative Lorentz force, i.e., to assume that the prominence exists only for Z > Hmin, one needs a more complicated
flux function to also describe the field that passes underneath the prominence.
This generalization of Anzer's model was accomplished in a thorough analysis
of the problem by Demoulin et al. (1989). Since they worked with a finite current sheet at a fixed height they found it easier to use the complex magnetic
field (Equation (5.57» rather than the potential used by Anzer that defines the
field via the flux functions. Demoulin et al. (1989) were able to produce supporting fields both with normal and with inverse polarity.
Hood and Anzer (1990) generalized Menzel's model by including also the internal structure of the prominence abandoning the strict isothermal case, and
allowing for a longitudinal component, By, of the magnetic field. In our simple
geometry By =Bx cotg a, where, according to Equation (3.6), a'" 25". In an actual
case worked out by Hood and Anzer a =22.6". For the pressure and magnetic
field they assumed the forms (compare Equation (5.44»
p

=Pi (x)e-2kz ,

0
0
0 ] -kz ,
B= [ Bx(x),By(x),Bz(x)e

(5.59)

(5.60)

and let the temperature be a function of the horizontal distance, x, from the
prominence only; i.e., T = T(x). The pressure scale height becomes H =kTlmg =
H(x),and Hood and Anzer could not easily use the Grad-5hafranov equation
(5.54) in which the flux function F(x,z) should be known in order to express the
scale height as H(F,z). Instead they developed their equations directly from
the magnetohydrostatic equations
1
-(VxB)xB- Vp+pg= 0
Jl

(5.61)

V.B=O.

(5.62)

and

Equation (5.62) gives the relationship

B

z

=.! dBx

k dx '

(5.63)

PROMINENCE MODELS

197

which, with the x-component of Equation (5.61), leads to
(5.64)

indicating at equilibrium a constant total pressure in the horizontal direction.
The vertical component of Equation (5.61) gives

(5.65)
Hood and Anzer solved Equations (5.63) and (5.65) numerically for what
they considered a typical prominence: rpr = 6 x 103 K, mp = 2 x 1011 cm-3, Bx (z =
0) = 5 Gauss, By (z = 0) = 12 Gauss, giving a plasma beta P= 0.025. In an extension
of the basic model they put a lower boundary on the prominence, assuming
coronal plasma below it pervaded by a potential magnetic field.
The overall shape of the resulting magnetic arcade that supports the
prominence material in the dip formed by the field lines is shown in Figure 5.7.
By including a B component of the magnetic field along the axis of the promiAnzer could show that as this component is increased the innence, Hood
ternal structure of the prominence would be affected. In particular, the magnetic
pressure would increase and the dip in the field lines would decrease and even
disappear for small values of a. We shall return to the importance of this dip
in Section 5.5.3.
Fiedler and Hood (1992) have presented a model that may be considered an
extension to the Hood-Anzer model. Their two-dimensional model considers an
isothermal slab of finite width and height in a hot isothermal coronal arcade.
They considered the magnetohydrostatic equilibria as given by solution to the
generalized Grad-Shafranov equation. The external magnetic field is matched
smoothly to the internal prominence field, thereby realizing the condition of no
extra current sheets at the prominence sides.
Fiedler and Hood found a range of values for the coronal plasma-p which
can support normal polarity prominences. However, if the value of Pis greater
than a certain critical value, prominence plasma will cause the field to sag.
Therefore, if there would exist regions along the prominence where the field
strength were less than the critical value, the field would bend down toward
the photosphere and plasma would appear like column or feet below the main
body of the prominence. We shall return to the phenomenon of prominence feet
in Section 5.7.

and

198

CHAPTER 5

z

~~----~--~------~------~--~----~~ x

-a
Fig. 5.7. Prominence in supporting arcade field with dip (after Hood and Anzer, 1990).
5.5.2 . SUPPORT IN INVERSE POLARITY FIELDS

In discussing how magnetic fields could best protect the cool prominence plasma
from being heated by the hot corona Kuperus and Tandberg-Hanssen (1967) and
Kuperus and Raadu (1974) concluded that neutral magnetic sheets in which a
vertical magnetic field reverses direction would be good candidates. They then
assumed that reconnection of the field would occur leading to a prominencesupporting field configuration of the form sketched in Figure 4.8. By tearingmode instability "islands" of magnetic field form, coalesce, and form a line
current (see Figure 5.8), which is stabilized against gravity by line tying of the
field lines which are anchored in the dense photosphere (Raadu and Kuperus,
1973).

To produce a vertical, upward Lorentz force capable of supporting the
prominence plasma, Kuperus and Raadu proposed that the formation of the line
current, I, would induce surface currents in the photosphere, and these currents
would prevent the field from penetrating into the Sun. This induction process
may be modeled by an image line current, -I, below the photosphere. This virtual current will produce an upward Lorentz force

199

PROMINENCE MODELS

(5.66)
where h is the height of the line current above the photosphere. However, the
prominence support against gravity is not simply given by the force Fl in
Equation (5.66). The configuration in Figure 5.8 is assumed to have a horizontal
magnetic field Bx at the position of the line current giving rise to a downward
Lorentz force
(5.67)
Only if F1 + F2 > 0 will a resulting, supportive Lorentz force be available in this
model.

Fig. S.S. Formation of a line current, representing a prominence, in an inverse polarity field (after
Kuperus and Raadu, 1974).

The supporting magnetic field has an inverse direction at the location of
the prominence compared to the normal orientation of the KippenhahnSchluter type models, and the Kuperus-Raadu model is the original representative of inverse polarity models; see Section 3.5. These models have been extended by van Tend and Kuperus (1978), Anzer (1984), and Anzer and Priest
(1985). The existence of a potential field in the corona was included, a field
that exists before the prominence forms and provides a logical sequence of
events for the establishment of the model. However, difficulties remain, chief
among them the question regarding the force that drives the line current I of
Equation (5.66). Following Anzer's (1989) analysis of the problem we note that
the current should flow in a direction opposite to the one found in normal polar-

200

CHAPTER 5

ity models, where the current is driven by the downward gravitational force.
The upward force necessary for the inverse model could come from the outwardflowing solar wind (Anzer, 1984), and reconnection could lead to the desired current. Even though this somewhat artificial scenario might work, it shows that
the line current initially would be pulled down until the repulsive force of the
mirror current, -I, becomes strong enough to overcome gravity. Consequently, the
resulting prominence model calls for low lying objects, while observations show
them to exhibit considerable vertical extension. Also, line currents do not represent prominence sheets very well, and Anzer (1984) extended the models to vertical current sheets and calculated the magnetic field produced by such a current
distribution. However, he found that for some parts of the prominence the
Lorentz force was directed downward and equilibrium did not become possible.
In a re-examination of inverse polarity models Anzer (1993) concluded that "at
present no variable models for prominences with I polarity exist." Since observations show that such prominences exist abundantly, one of the pressing problems in contemporary prominence research is the establishment of a comprehensive model for the supporting magnetic field.
5.5.3. THE IMPORTANCE OF DIPS IN THE FIELD LINES
In an arcade of magnetic field lines the prominence material is supported
against gravity in dips in the field lines. While some formation models of the
injection type (Section 4.4) let the prominence material itself create the dips as
this heavy material is deposited on the field lines, it is believed that the preexistence of dips is necessary for the material to accumulate and form the
prominence. This was first pointed out by Kippenhahn and Schluter (1957) and
emphasized by Priest et al. (1989). Different field configurations have been
considered as possible candidates for arcades with dip-forming field lines.
Shearing motions are often invoked to produce fields with certain characteristics, but Amari et al. (1991) proved that in a bipolar region a two-dimensional
force-free arcade field cannot have a dip. However, prominences above a Btype neutral line (Section 5.1} occur in quadrupolar regions and Demoulin and
Priest (1993) constructed a model for an inverse polarity prominence supported
in a dip of such a region.
If the magnetic field has a helical structure, prominence material might
conceivably collect in the lower parts, the dip as it were, of the basically horizontal helix. Models by Ioshpa (1968), Nakagawa and Malville (1969), Anzer
and Tandberg-Hanssen (1970), Pneuman (1983), and van Ballegooijen and
Martens (1989) have all treated different aspects of this scenario.
Dips capable of supporting prominence material may also be created in flux
tubes by twisting the legs of the tubes near the photosphere, whereby a dip

PROMINENCE MODELS

201

may form near the summit of curved flux tubes (Priest et al., 1989). We shall
now take a closer look at some of these models.

5.5.3.1. Helical field configurations
Under good seeing conditions Ha pictures of prominences often reveal a helical
structure which becomes easier to observe during the disparition brusque phase.
Extensive studies of quiescent prominences were made at HAO's Climax station
in the 1960's in search of helical structure, and the conclusion was that this is
not an uncommon characteristic. Since the electrical conductivity of the prominence plasma is very high, it seems natural to assume that the fine structure of
material as outlined in Ha pictures is the same as the fine structure of the magnetic field. This then led to the study of helical magnetic fields in prominences
(Rompolt, 1971; 6hman, 1972).
On the other hand, quiescent prominences owe their support to a field that
is anchored in the photosphere on both sides of the filament. Consequently, two
magnetic field components seemed involved: one, the previously discussed supporting field, which we designate Bo, and another, which we shall call the internal field and designate Bl' The field Bo is produced by photospheric or subphotospheric currents as assumed previously. For the latter field two cases
have been considered. Ioshpa (1968) studied a Bl field directed along the
prominence (i.e., along the Y-axis) and bent down into the photosphere at the
ends of the filament. Anzer and Tandberg-Hanssen (1970) considered the internal field due to currents flowing along the prominence. As we shall see below,
this may result in helical field configurations. Also Nakagawa and Malville
(1969) have considered a Bl field along the prominence and used it to study the
stability of prominences.
Ioshpa argued that if the internal field is attached to the photosphere at
the ends of the prominence, this will lead to great stability of the filament. He
estimated the strength of the field necessary to produce stability by comparing
the tension of the magnetic lines of force arising from the bending of the filament, B[141tRc, with the gravitational force responsible for the bending (pg). Rc
is the radius of curvature of the lines of force, and Rc'" ').J4 for small perturbations, where)' measures the scale of the perturbation [sin (2ny/).)]. Ioshpa
took), equal to the thickness X of the prominence =109 cm, and n = 3 x 1011 cm-3,
to arrive at Bl = 10 Gauss. This value seems reasonable, but it is not obvious
where the currents, responsible for the Bl field, flow in Ioshpa's model, nor
what the effect will be of the actual superposition of the two fields in the
prominence.
If the current flows along the filament, assumed to be infinitely long, it = (0,
iy, 0), the internal field will be given by (Anzer and Tandberg-Hanssen, 1970)

202

CHAPTER S

(z,O,-x) for rSR,
1l.
'1

1.

=- Jo
2
R2

~

"'Tz,O'-"'Tx for r> R,
r
r

iO

where i y = { 0

for rS R
for r> R

=

and io const. R is the radius of the circularly cylindrical model prominence and
r is the distance from its axis. The supporting field is assumed to be initially
horizontal, Bo (Bo,x, Bo,y, 0). Anzer and Tandberg-Hanssen measured Bl in
units of Bo,%, and defined the dimensionless parameter C, giving the ratio of the
magnetic field produced at r =R by the currentiy to the field component Bo,x existing in the absence of a prominence,

=

C5

ioR =

280,%

J

2lrR8o,x

,

(5.68)

where J =1tR2io is the total current in the prominence.
The total field is given by the superposition of the fields; i.e., B =Bo + Bl.
Let us study the shape of the field lines projected on planes y = const. The field
lines are calculated from the flux function of the field, F(x,z). We have

J

F(x. z) = Bo • ndl,

(5.69)

L

where the integration is along a path L and where n is the normal to the line
element of L. The curves F = const then represent the field lines, and the components of the field vector are given by Bx =of/i}z, Br. =-oF/ox.
When the supporting field dominates the internal field; i.e., when C is
small, the field lines are open (see Figure 5.9 where the density of the lines
indicates the strength of the field). As the relative importance of the internal
field increases (C increases beyond unity), a larger and larger part of the
prominence is filled with closed field lines. Since Bo has a Y-component, these
closed field lines represent in reality helices along the long axis of the prominence.

203

PROMINENCE MODELS

~__z+--_- C=O.6

z

C=1.2

Fig. 5.9. Helical prominence in its supporting magnetic field (after Anzer and Tandberg-Hanssen,

1970).

Anzer and Tandberg-Hanssen made some numerical estimates for a quiescent
prominence with Bo,x = 5 to 10 Gauss, ne = 101°-5 x 1010 cm-3, and R = 30,000 to
60,000 km, considered to be in equilibrium such that the Lorentz force per unit
len¥th of filament, (V41t)JBo,x =V2CRB6,x, balances the gravitational force,
1tR pg. The equilibrium condition gives

204

CHAPTER 5

c-

tRpg

- (lI8)1rBc!,x '

(5.68')

which states that the parameter C is also the ratio between the potential
energy of prominence material lifted to a height of 1I4R and the energy associated with the magnetic field 80,x in the prominence. For the values of the
physical parameters listed above, C lies in the range 0.1 < C < 15. Such values
of C, and the high electrical conductivity of plasma, indicate that in prominences the material distribution and the magnetic field are strongly coupled.
A value of C = 3, which leads to a helical magnetic field in most of the
prominence, can be realized with the following values of the parameters:

Bo,x =8 Gauss, n =1.4 x 1011 cm-3 , R =50,000 kIn.
It has become increasingly clear in the last several years that shear (antiparallel) motions at photospheric level and their interaction with magnetic
field lines play an important role in solar flares and probably in many
prominences. In his important paper on prominence formation Pneuman (1983)
studied the formation of a helical field in a rising, sheared bipolar field invoking reconnection of the field lines. The prominence material in this model is
supported in the bottom central parts of the helix.
Helical field configurations were also studied by van Ballegooijen and
Martens (1989), whose model is similar to Pneuman's in that helical fields are
produced by reconnection below the prominence. But van Ballegooijen and
Martens differ in that they invoke flux cancellation in a sheared field leading
to a reconnection process which decouples transverse and longitudinal field
components. The lower loops can then submerge below the photosphere, and a
helical field is created above the neutral line where the prominence material
is supported; see Figure 5.10.
From a discussion of helical magnetic fields it is but a short step to consider
sheared or twisted flux tubes.

5.5.3.2. Twisted flUX tubes
The early models of prominence magnetic fields were basically twodimensional and considered the shape of the supporting field in the X-Z plane,
perpendicular to the long axis, Y, of the prominence. Eventually, a Y-component
of the field was introduced, and an advantage of the use of the helical field
lines is that we are gradually considering three-dimensional models.

PROMINENCE MODELS

(a)

(d)

(b)

(e)

205

(e)

(f)

Fig. 5.10. Development of sheared prominence-supporting field due to flux cancellation leading to
reconnection and decoupling of transverse and longitudinal field components (after van Ballegooijen
and Martens, 1989).

To accomplish the formation of helical field lines, one may consider
sheared motions, i.e., closely spaced, oppositely directed, mass motions in
planes at different angles to the original field lines, to facilitate reconnection
processes. Finally, twisting motions of field lines, e.g., rotational motions of
footpoints of field lines, are being brought into the picture, and fully threedimensional models are conceptually, albeit not always mathematically,
achieved.
Evidence for shear and twist abounds (e.g., Schmieder et al., 1985a; Mein
and Schmieder, 1988; Rompolt and Bogdan, 1986; Rompolt, 1990; Vrsnak et al.,
1991), and may be created in several ways, viz. by evolutionary motions of the
field line footpoints in the photosphere or by cancellation of magnetic flux
(e.g., van Ballegooijen and Martens, 1989; Inhester et al., 1992). The Coriolis
force would twist a flux tube and produce one complete twist in about 35 days.
The differential rotation of the Sun's surface features would also lead to twisting, but on a time scale that seems too long to be of interest here.
Priest et al. (1989), Demoulin and Priest (1989), Amari et al. (1991), and
Ridgway et al. (1991) have studied the formation of prominence-supporting
dips in twisted or sheared flux-tube models. We shall briefly consider the
twisted flux-tube model by Priest et al. The basic geometry is a curved flux tube;
see Figure 5.11. Starting from no twist, we subject the footpoints of the flux tube
to twisting motions as indicated by the arrows, Figure 5.11a. As the twist, +, is
increased a dip in the field lines occurs and the idea is that prominence mate-

206

CHAPTER 5

rial now can start to condense into the dip, Figure 5.11b. There is a critical value
of the twist, C\lcrit, for the dip to occur, given by
(5.70)
where R and a are major and minor radii of curvature of the flux tube and C\lo is
the angle subtended by the tube as seen from the center of curvature. When the
twist increases beyond C\lcrit, the length of the dip increases (Figure 5.11c) and
the prominence can continue to form along the dip. Finally, when the twist becomes too large, the configuration becomes unstable, and the prominence erupts.
We shall return to this scenario in Chapter 6. A large number of authors have
studied the general behavior of twisted flux tubes, and different analyses give
different values for C\lcrit in Equation (5.70), varying from 2.5 n to 20 n; see, e.g.,
Hood and Priest (1980), Einaudi and van Hoven (1981), Birn and Schindler
(1981), and Hood (1983, 1984).
While Priest et al. (1989) did not include shear in their twisted flux model,
shear alone and shear with twist have been considered by Amari et al. (1991).
They showed that shear alone cannot create dips in a two-dimensional forcefree magnetic arcade of a bipolar field, even though the shearing produces very
flat field lines. When both twist and shear are present, Amari et al. (1991)
showed that the value of C\lcrit decreases as the shear increases. Photospheric
shear would then facilitate prominence formation.
A solution in terms of a twisted flux-tube model was also realized by
Ridgway et al. (1991), who found support for a current sheet prominence using a
constant axial current density.
Little attention has been paid to the handedness of the twisted magnetic
flux ropes used in flux-tube models. However, recent work by Rust (1994) indicates that the handedness is directly related to, e.g., the dextral and sinistral
characteristics of prominences as described by Martin et al. (1994); see Section
5.7.1. This means, from a basic physics point of view, that the direction of the
responsible currents needs to be taken into account.
More work is needed in this area and may lead to a better understanding of
the role this magnetic helicity (defined by Woltjer (1958) as JA. V x AIlV, where
A is the magnetic vector potential) plays during the development of prominences; see also Section 6.5.2.
In most current sheet models the internal structure is not considered, and the
plasma parameters, like density, are determined from the horizontal and vertical force balance. Conversely, models of the internal structure often consider
only the local behavior without treating the matching onto the external magnetic field. Cartledge and Hood (1993) examined a combination of these two
approaches and showed how their internal solution for the magnetic field can

PROMINENCE MODELS

(a)

(c)

207

(b)

(d)

Fig. 5.11. Evolution of twisted flux-tube model (after Priest et aI., 1989).

match smoothly onto an exterior force-free equilibrium solution, using the
twisted flux-tube model proposed by Ridgway et al. (1991).
Due to the mathematical complexity of treating three-dimensional magnetic field structures, most investigations are restricted to two-dimensional
cases. However, recently significant progress in our understanding of threedimensional magnetohydrostatic equilibria has been made by Low (1991), and
his solutions may lead to modeling of more realistic magnetic field configuration around prominences. We shall return to Low's important investigations in
Chapter 6, when dealing with the onset of instability leading to the eruption
of prominences.
Another interesting investigation of three-dimensional fields has been presented by Finn et al. (1994), who considered force-free MHO equilibria in loop
structure that may be applicable to prominences. The authors showed that as

208

CHAPTER 5

the parameter a in the force-free equation V x B =aB (see Equation (2.126» increases, the flux loops first become kinked, and-for sufficiently large values of
a-they develop magnetic knots. Further exploration of the magnetic reconnection that must be present for an unknotted equilibrium to become knotted may
lead to interesting applications to prominence physics.
Another important contribution to the study of the role of threedimensional magnetic fields in prominence support comes from Antiochos et al.
(1994). They have presented a model where the seemingly all-important dip is
created in a truly three-dimensional field which is strongly sheared in the
photosphere. For the source of the field they use a point dipole located below
the photosphere. The initially force-free prominence flux tube lies in a bipolar
field and has footpoints in areas of strong shear. This shear causes the footpoints to move parallel to the neutral line to regions of weaker fields far away
from the dipole source. At the same time the shearing motion will cause the
flux tube to expand upward; see, e.g., Yang et al. (1986), Klimchuk (1990),
Dahlburg et al. (1991), and Wu et al. (1991). Antiochos et al. (1994) argue that
since the magnetic field varies along the neutral line, the degree of upward expansion will vary along the flux tube. Since the shear has moved the footpoints
to areas of weaker fields, while the midpoint of the flux tube has remained in a
region of strong field, the expansion in this low-~ plasma will be less near the
midpoint and a dip is formed. These ideas were verified by numerical simulations using time-dependent and three-dimensional codes.

5.5.3.3. Dips in quadrupolar regions
While some prominences occur in bipolar regions above an A-type neutral line,
others could conceivably be found between two close active regions, i.e., above a
type B neutral line (see Section 5.1). Anzer (1990) investigated the magnetic
support of prominences in such quadrupolar configurations where dips naturally
form in the field lines; see also Demoulin and Priest (1990). The idea of using a
quadrupolar configuration for inverse polarity prominences was further developed by Demoulin et al. (1992) and Demoulin and Priest (1993). Figure 5.12a
shows the magnetic field configuration of two close bipolar regions separated
by a current sheet at X =o. As the two regions move toward each other, the current density increases and finally reconnection sets in and the two regions become topologically connected (Anzer, 1990); see Figure 5.12b. Dips form in the
arcade lines, and prominence material may be supported between two heights,
P and Q say. Below the prominence we have an X-type neutral line, and the
prominence is of the inverse polarity type.
Mathematically the treatment has been in terms of force-free coronal fields
(Equation (2.126» or, in terms of the vector potential A:

209

PROMINENCE MODELS

z

+

(a)

x

o +

z

+

(b)

x

o +

Fig. 5.12. Evolution of magnetic field configuration of two close bipolar regions (after Anzer, 1990).

[ aA

B= - ik ,F(A),

aA]
ax '

(5.71)

210

CHAPTER 5

where the flux function F(A) can be determined by the shear in the photosphere
(Anzer, 1993), and which leads to the equation for A:
(5.72)
It was, for instance, by studying the numerical solutions of Equation (5.72)
that Inhester et al. (1992) constructed two-dimensional magnetic arcades for
prominence support after having subjected the footpoints of the field lines to
shear motions; see Section 5.5.3.2.
For a force-free field (Equation (2.126» with constant ex, Anzer pointed out
that Equation (5.72) reduces to
(5.73)

which is the equation that Demoulin et al. (1992) solved numerically to study
the quadrupolar configuration that gives rise to the prominence supporting dips
in the field lines.
5.6. Dynamic Support

As outlined in Section 5.5 many papers and an enormous amount of research
have been devoted to understanding static support of quiescent prominences. We
have learned a lot from these investigations, but problems remain. These are,
on one hand, due to the complexity of the non-linear systems under study, where
simplifying assumptions (two-dimensional geometry, constant-ex force-free
fields, etc.) often make one feel uncomfortable with the validity of the obtained results. On the other hand, we have stressed in, e.g., Chapter 3 the dynamic nature of even the most quiescent prominence, and it seems logical that
truly dynamic conditions should be included in any complete model. However,
from a mathematical point of view we are not ready for all the intricacies of a
three-dimensional magnetohydrodynamic model describing the stable nature of
prominences.
Furthermore, one should keep in mind that to a first-and probably quite
good approximation-a magnetohydrostatic model can describe the nature of a
quiescent prominence in its long lasting, stable configuration. On the other
hand, the formation process, when chromospheric or coronal plasma is brought
into appropriate magnetic fields, is by definition a dynamic process, and dynamic effects are naturally studied. Such dynamic models were discussed in
Chapter 4.

PROMINENCE MODELS

211

In an attempt to develop a dynamic model for the stable phase of a quiescent prominence, Jensen (1983, 1986) argued that the main characteristics of the
prominence plasma is the presence of MHO turbulence, which he considers to be
driven by a flux of Alfven waves from below. The value of prominence magnetic
fields and the densities observed indicate that the Alfven waves would
undergo mode conversion and would dissipate, leading to the turbulent state of
the plasma. Jensen argued that this would account for the rather chaotic velocity fields observed even in quiescent prominences. As the waves dissipate
the prominence plasma acquires an upward momentum that balances the force
of gravity and thus supports the prominence. The dissipation of the Alfven
waves will lead to heating of the plasma, and Chiuderi Drago et al. (1992)
used, as we have seen in Section 5.3.1, this mechanism to explain the extra
heating needed in the prominence-corona transition region.
In jensen's model the supporting force is created by the waves. This force,
K,A, per unit volume can be written
(5.74)
where F A is the flux of Alfven waves and L is the dissipation length for the
waves in the prominence plasma. FA is given by Equation (5.27). Jensen used the
value FA =3 x 105 erg cm-2 s-l, and if we assume L = 107 cm, we find from
Equation (5.74) that KA =6 x 10-9 drn cm-3. The opposing force of gravity per
unit volume is pg =3 x 10-9 dyn cm- , indicating that an Alfven wave flux of 3 x
lOS erg cm-2 s-l can balance the force of gravity if the waves can be dissipated
over a length of L =100 kIn.
jensen's work is interesting and introduces another aspect than magnetic
field support in the theory of prominence modeling. A possible difficulty may
be the effect of the heating that would result from the diSSipation of the
waves. This could indicate that the prominence plasma would be hot, while we
really are seeking a cool plasma in the hotter environment. However, Jensen
(1990) showed that while the heating by wave dissipation is important for the
energy balance, it will not dominate the energy budget. Another, maybe more
serious, difficulty is that the momentum transport must be along the magnetic
field lines, while observations indicate that the field is basically horizontal
in the prominence. It is not clear how to resolve this problem.
A different approach to the problem of describing the stable quiescent
prominence plasma in a dynamic model was adopted by Sakai and Washimi
(1984) and Bakhareva et al. (1992). Both investigations considered a
Kippenhahn-Schliiter type current sheet and Bakhareva et al. also included
the effects of partial ionization and non-steady state plasma motions on the
magnetic diffusion equation (Equation (2.119» by using a generalized Ohm's
law (Equation (2.98». Their diffusion equation takes then the form

212

CHAPTER 5

as
=VX(VXB)+L.V2B+f2 vx[(L)dV XB]
at
41r0'
nVja dt
mj

where 1= mana/ (mana + mini) is the relative density of neutral particles, Via is
the collision frequency, and subscripts a and i refer to neutrals and ions, respectively.
We see in this equation-in addition to the two first familiar terms,
Equation (2.120) and Equation (2.122)-the influence of the partially ionized
gas and the time-dependent flow. The last term is the Lorentz force affecting
the electric current via the generalized Ohm's law.
Bakhareva et al. solved the MHD equations, i.e., Equations (2.83'), (2.92),
and (2.93) for an adiabatic case, together with their diffusion equation. They
showed that the partial ionization causes instability of a prominence considered in the Kippenhahn-Schliiter magnetohydrostatic model, in that an
oscillatory regime may set in, endangering the stability of the model. The
buildup time, t, for oscillations, as well as the period, P, depend on the strength
of the transverse magnetic field component Bx, the ratio I, temperature T, and
the equilibrium density. As examples, the authors give buildup times ranging
from t = 1.6 hr when I = 0.9, Bx = 0.1 Gauss, to t = 7600 hr when I = 0.1, Bx = 1
Gauss, all for a 1()4 K plasma of density 1012 cm-3. The period of oscillation was
found to range from P = 1.1 min when Bx = 0.1 Gauss, to P = 17 hr when Bx = 100
Gauss, both for I = 0.9, t = 1()4 K, and ntot = 1012 cm-3.
The physics behind this dynamic behavior of the prominence plasma is
dictated by its neutral component which cannot be supported by the magnetic
field, and begins to flow down. Collisions between neutral particles and ions
will then lead to motion of the entire plasma. This motion in the magnetic field
generates currents along the prominence and the current in the current sheet increases. This stronger current, in turn, leads to an increased Lorentz force, which
provides added support for the plasma and moves it up. However, when that
happens, the current will decrease as will the Lorentz force with the result
that the prominence plasma again will start to flow down, and the whole process may repeat itself. We notice that in this dynamic model, the energy for the
oscillations come from the potential energy of the prominence plasma in the
gravitational field.

PROMINENCE MODELS

213

The Bakhareva et al. model is one-dimensional, adiabatic, and greatly
simplified. However, it is an important step in the direction of providing dynamic models for even the most quiescent examples of prominences.
5.7. Prominence Feet
When a quiescent prominence is seen above the solar limb, a series of fairly
regularly spaced arches and columns-often called feet-is generally seen
(Figure 1.2). Observed on the disk, the quiescent prominence displays its feet as
appendages stretching out from the main prominence body; see Figure 5.13. The
physical nature of these feet, their role in the formation and development of
the prominence, and their connection with the prominence fine structure have
not been well understood, but it is recognized that they play an important role
in the stability of prominences (Bhatnagar et al., 1992). We shall in this
section discuss these problems in some detail.

Fig. 5.13. Prominences with their feet, observed in Hu on the disk, and surrounding fibril structures
(courtesy Big Bear Solar Observatory, California and S. F. Martin).

214

CHAPTER 5

5.7.1. OBSERVATIONS, EMPIRICAL MODELS
Time-lapse photography shows that mass is flowing down through the legs
from the top parts of the prominence to the chromosphere, and maybe photosphere, below. Between the feet, large arches form the prominence and the
direction of flow is horizontal and downward along these arches.
We shall now combine this information with our knowledge of the structure
and velocities in the chromospheric fibrils surrounding the prominence. Foukal
(1971) interpreted the fibrils as representing the direction of the magnetic field
in the chromosphere, and noticed an asymmetry in their pattern around the
prominence. The fibrils are more or less parallel to the long axis of the prominence as long as one observes them dose to the filament channel in which the
prominence is formed. Many of the fibrils come from small bright plages that
are called plagettes, and that correspond to patches of the magnetic field network observed in magnetograms. Let us consider, e.g., a filament oriented eastwest on the Sun. The asymmetry arises in that on one side (e.g., south) of the
filament the fibrils point in one direction, e.g., to the right (west) from the
plagettes, while the fibrils on the opposite side (north) of the filament point in
the opposite direction, e.g., to the left (east) from their plagettes. Foukal referred to this pattern as "anti-parallel." This orientation indicates that the
magnetic field near the prominence and at chromospheric levels is more or less
parallel to the long axis of the filament. Observations of magnetic fields in
prominences show, however, that higher up the magnetic field traverses the
prominence, albeit often at a small angle with its long axis. Martin et al. (1994)
combined the above information and deduced a rotational magnetic field configuration in the corona around the middle part of a long filament located in
the plane X =0; see Figure 5.14. At a certain distance (+XI) from the prominence
the magnetic field comes vertically, +Z, up from the chromosphere, crosses
above the prominence sheet, and goes down vertically into the chromosphere
(at -Xl) on the other side of the prominence. However, inside the prominence at
X = 0 the field is horizontal and more or less aligned with the long axis, Y, of
the prominence (Leroy, 1988) and is anchored "in front of" the prominence, e.g.,
at +Y and at -Y, "behind" the prominence. Consequently, for intermediate distances, i.e., for -Xl < X < Xl, Martin et al. (1994) envision a rotational configuration for the coronal field, where the lines are rotated more and more from the
vertical as one approaches the filament from X = ±Xl' If we assume that fibrils
delineate magnetic field structures, the fibrils near the filament channel will
indicate the direction of the field lines near the prominence. Imagine that we
look at the prominence from the point Xl on the X-axis, in the positive plage
area where the field lines come up from the chromosphere. As we consider the
field vectors closer and closer to the prominence, they will seem to rotate,
either clockwise or counter-clockwise. Close to the prominence the fibrils---outlining the field vectors-will then either seem to point slightly upward and to

215

PROMINENCE MODELS

the left of the plagettes from which they originate, or they will point upward
and to the right of their plagettes. Martin et al. refer to these configurations as
sinistral or dextral, respectively. They further deduced, from studying the
polarity of the areas where the ends of prominences lie, that the direction of
the magnetic field component along the long axis of prominences is the same as
the direction of the horizontal component of the field on both sides of the
prominence, the latter being outlined by the fibril structure. Martin et al.,
therefore, also divides filaments into sinistral and dextral prominences.

z
,,

,
,,

,,

,,

,
I
I

I

-

-

I

I

+

+
x

Fig. 5.14. Carton indicating possible magnetic field configuration in the chromosphere and corona
around a quiescent prominence, and the relation to plagettes (see Martin et a!., 1994).

The above mentioned findings have consequences also for our study of
prominence feet. Observations show that all feet observed on one side of filaments depart from the main (Y) axis of the filaments and extend to the chromosphere at a certain angle. In some cases the feet curve away from the axis to the
right-in other cases to the left. This "right bearing" or "left bearing" characteristic is found independently of whether we observe the prominence from
above or from the positive or negative polarity side of the filament channel. It
is interesting and of considerable importance for our view of prominence feet

216

CHAPTER S

that filaments with left-bearing structure are sinistral, while dextral prominences are right-bearing (Martin et a1., 1992, 1994). These authors also looked
at the location of the feet in the chromosphere, a difficult observation that
both Plocieniak and Rompolt (1973) and Martin (1986) have addressed before.
The result seemed to be that the feet are rooted at the intersection of several
supergranule cells, close to the neutral line. Martin et al. found that they are
probably rooted in weak magnetic field patches whose polarity is opposite to
the dominating polarity of the field on the same side of the prominence; see
also Martin and Echols (1994). Furthermore, they discovered a global pattern in
the magnetic field and structural pattern of quiescent prominences; during the
time period that they made their observations dextral prominences dominated
the northern solar hemisphere and sinistral prominences the southern. A similar pattern did not apply to active region filaments.
These empirical models of prominences and their feet open new and exciting
ways to consider the formation and structure of quiescent as well as active filaments and put constraints on theoretical models. In particular, revisions may be
required on some models using magnetic arcades in which to form and support
prominences.
5.7.2. PHYSICAL MODELS
The empirical model of prominence feet discussed in Section 5.7.1 helps us
visualize their relationship to the magnetic structure of the prominence and of
the surrounding atmosphere. However, it does not deal with the physical processes involved in the formation of the feet.
Nakagawa and Malville (1969) suggested that the periodic structure of
prominence feet could be identified with the Rayleigh-Taylor plasma instability
of the interface between the cool prominence plasma of density PPT supported
against gravity by the magnetic field. With the MHO equations they studied
the interface allowing small perturbations with a space-time dependence of
the form exp[i(k. r - cot)]. The basic idea of their investigation can be expressed
in terms of the growth rate of the perturbation (see also Priest, 1982) which is
given by:
(5.75)
where PeoT is the density of the coronal plasma below the prominence and 80 is
the strength of the supporting magnetic field. For the fastest growing mode the
wave number is

PROMINENCE MODELS

k- Ppr -Peor
-g

2B6 '

217

(5.76)

which means that for a prominence plasma with ne =2 x 1011 cm-3 and a field of
Bo = 10 Gauss, the wavelength is 30,000 km, not unlike the observed spacing of
prominence feet. Nakagawa and Malville (1969) further showed that the
wavelength of the instability depends on the direction of the magnetic field
with respect to the long axis of the prominence. Thus the spacing of the feet can
range, in their example, from 17,000 km to 90,000 km, the longer distances corresponding to fields that make greater angles with the long axis of the prominence.
Milne et al. (1979) computed a one-dimensional model for a prominence in
both magnetohydrostatic (Equation (5.36» and thermal equilibrium; the latter
describing the balance between thermal conduction, radiation, and wave
heating; compare Equation (4.15). They studied the effects of changing the
pressure, p, the magnetic field, Bx, and the magnetic shear, ell, expressed as the
angle between the horizontal magnetic field and the normal to the prominence
sheet. The results indicate that there are upper limits to both the plasma-~
and the magnetic shear, above which no equilibrium state exists. The limit on ~
was interpreted by Milne et al. as being due to the magnetic field being too
weak to support the plasma, which then drags the field down with it and
forms prominence feet.
The two models just presented introduce basic physical processes to explain
the existence of prominence feet, and discussions of this kind are needed in order
to arrive at an understanding of the phenomenon. However, in their simple
forms the models cannot explain all the intricacies of the empirical picture we
have of the prominences and their relationship to the corona as well as to the
underlying filament channel (Martin et al., 1994).
In a thorough analysis Demoulin et al. (1989) constructed a threedimensional model to account for prominence feet by superposition of two magnetic fields. Prior to the prominence formation the coronal field is threedimensional and force free. The prominence is then represented by a line current
which interacts with the coronal field, but the linear force-free field of the
line current was neglected by Demoulin et al. By taking the curl of the linear
force-free field the authors studied the equations
(5.77)
and considered periodic variations along the prominence axis (the Y direction).
They used a Fourier series with components of the form Bo exp[i(kxX + kyy + kzz»),
where, from Equation (5.77), ki + kl + kz2 =(X2. To avoid spatial oscillations
the vertical dependence was assumed to be of the form exp( -lz). The horizon-

218

CHAPTER 5

tal field components can then be expressed as functions of the vertical component, i.e.,
(S.78a)
and
(S.78b)
and the z-component will satisfy the equation
(5.79)
With the solution of Equation (5.79), e.g., by using a Fourier series,
Equations (5.78) will give the horizontal co~ponents. The periodic variation
along the prominence was studied by choosing'
Bz = Bo cos(kyy)sin(kxx)exp(-iz)

(5.80)

By inserting Bz from Equation (5.80) into Equation (5.78) one finds expressions
for Bx and By. Demoulin et al. added a small number of harmonics to describe
the field around a prominence and obtain a reasonable flux concentration in the
photosphere, and then the line current representing the prominence was added.
The prominence is located between areas of positive and negative photospheric
flux concentrated in the supergranulation pattern and the current representing
the prominence may have one of two directions. With a vertical magnetic field
positive for X > 0 a prominence current in the positive Y-direction would give an
upward, supporting Lorentz force. This corresponds to a normal polarity prominence since the current creates a magnetic field in the same direction as the
photospheric pattern. Similarly, in this model a current in the negative Ydirection could support an inverse polarity prominence.
The final field configuration in the Z = 0 plane is sketched in Figure 5.15
where arrows indicate the direction of the horizontal field (Bx, By) and where
contours give positive (dashed) and negative (solid) Bz values. From such plots
Demoulin et al. showed that the feet of a normal polarity prominence are found
near the centers of supergranulation cells, while an inverse polarity prominence
has its feet at the boundaries between such cells. There is still observational
uncertainties regarding the location of prominence feet in the chromosphere, but
we have seen in Section 5.7.1 that the work of Martin et al. (1994) indicates
that the feet are rooted in weak magnetic fields of polarity opposite to the

219

PROMINENCE MODELS

dominant supergranulation field. It is not yet clear how theory and observations can best be combined in these cases.

-30

-20

-10

o

10

20

X
Fig. 5.15. Magnetic field configuration in the Z = 0 plane in relation to prominence feet; coordinates
in 103 Ian; see text (after Demoulin et al., 1989).

S.S. Internal Equilibrium
Of the two parts that constitute the realm of prominence models, i.e., the
global magnetohydrostatic equilibrium describing the overall support, and the
equilibrium of the prominence plasma, that we shall refer to as the internal
equilibrium, we have treated the former in Section 5.4. When we now treat the
second part, we notice that two aspects of the equilibrium are involved and
they are closely linked. One is the question of how the internal magnetic field
configuration supports the plasma and how it matches on to the external, global
field that provides this local support. The other aspect concerns the energy
budget that leads to a stable thermal equilibrium for the prominence plasma.
In trying to build these internal equilibrium models, difficulties arise because the forces that affect the distribution of temperature and density in a
prominence are coupled in such a complicated way that significant simplification of the problem is, in general, necessary to solve the equations involved. For
example, heating and cooling of the prominence plasma depends on density,
while the local temperature affects the pressure equilibrium, which, in turn,
determines the density and its variation along the magnetic field lines that
support the prominence in the first place. Consequently, the goal is a prominence model where the internal structure, as given in magnetohydrostatic equi-

220

CHAPTER 5

librium, and the energetics, as expressed in thermal equilibrium, are consistently combined. The first steps toward this goal have been taken, but most of
the existing models describe either the magnetohydrostatic equilibrium or the
thermal balance. Even though we shall divide this section into models for
magnetohydrostatic equilibrium and models for energy balance or thermal equilibrium, some investigations bridge the gap and contribute to both aspects of the
problem. In his review article Anzer (1989) considers many of these models, and
we shall partly follow Anzer's excellent expos~.
5.8.1. MAGNETOHYDROSTATIC EQUILIBRIUM
The equilibrium structure inside prominences was first seriously discussed by
Kippenhahn and Schliiter (1957). We have in Section 5.4.2 considered their
model, mainly from the point of view of how its magnetic field provides the
support (Equation (5.52)), but the model (which is isothermal) also provides an
expression for the density distribution of the prominence material under these
simplified conditions (Equation (5.53». This early model, therefore, is one of
those that bridges the gap between the two aspects of internal equilibrium.
Figure 5.16 shows the magnetic field structure inside the prominence in the
Kippenhahn-Schliiter model, according to Equation (5.52). The coronal part of
the supporting field is sketched in Figure 5.4. The field lines are bent down
symmetrically about the current sheet (at X =0), providing a "hammock" for
the prominence plasma, whose density distribution is shown in Figure 5.17,
according to Equation (5.53).

1.0

...-----.--.---m--.,---,----,

P o.sl----jf-----!I--+--\l---+--I

x

Fig. 5.16. Structure of magnetic field inside
prominence in the Kippenhahn-5chliiter
(1957) model.

xFig. 5.17. Density distribution in a Kippenhahn
and Schliiter (1957) prominence model.

PROMINENCE MODELS

221

Over the years many models have been suggested to generalize the work of
Kippenhahn and Schluter. Poland and Anzer (1971) considered the temperature
as a function of the distance from the prominence sheet, i.e., T = T(x), which
changes Equation (5.48) to an equation of the form
(5.48')
Low (1975, 1984) let the temperature and the magnetic field be coupled
through the flux function F, and the field is given by
(5.81)
He assumed a functional form
F(x,z) = f/I(x)+z,

(5.82)

and considered the temperature as a function of 'If, i.e.,
(5.83)
The three equations (5.81), (5.82), and (5.83) can be combined to give an equation
for 'If that must be solved numerically and will give an expression for the supporting magnetic field in this model.
We have seen in Section 5.7.2 that Milne et al. (1979) coupled magnetohydrostatics and energy balance to develop a model that also implied the existence of prominence feet. Of later models we mention Osherovich's (1985) twodimensional field of the form
oF
OF]
B = [ a;,G(x,z),- ox '

(5.84)

where he chose F(x,z)=Fo~2exp(-1(x2+z2»+Hoz, and G(x,z)=GoF(x,z),
and where Fo, 1(, Ho, and Go are constants.
These somewhat arbitrary assumptions lead to a completely determined
supporting field B(x,z), and density and pressure can be deduced from the magnetohydrostatic equilibrium along and perpendicular to the field lines, both as
function of F(x,z) and G(x,z). The equation p = kTlm then gives the temperature
distribution T(x,z).

CHAPTER S

222

Ballester and Priest (1987) modified the Kippenhahn-Schliiter model by
allowing a slow variation with height for the field, i.e.,
B =Bo(x)+ ED} (x,z),

(5.85)

where Bo(x) is the Kippenhahn-Schliiter solution (Equation (5.48». This modification leads to the result that the width of the prominence sheet decreases
slightly with height, while the field lines become less curved and the field
strength increases (Figure 5.18) in agreement with observations; see Equation
(3.5) and the subsequent discussion.
Hood and Anzer's (1990) model has been treated in Section 5.5.1; it combines
both the external and internal configurations. This remark also applies to the
interesting two-dimensional model by Fiedler and Hood (1992), who wanted to
obtain more realistic models that are not infinite in extent and that can be constructed without current sheets.
The most thorough analytic treatment of prominence-supporting magnetic
fields is due to Low (1982a, 1984, 1993a). In the earlier works he prescribes the
magnetic field topology and deduces the gas pressure from the horizontal force
balance. The vertical balance gives the density and the equation p = nkT provides the temperature. Again we have a model that bridges the gap between
magnetohydrostatics and energy balance. The later work furnishes new insight
into the configuration of coronal fields; see Chapter 6.

QS

........

-

-

~
">

~

~

Ii

~

~

~

~

a)

(5.

~
b)

""

Fig. 5.18. Modification of the I<ippenhahn-Schliiter model due to Ballester and Priest (1987).

PROMINENCE MODELS

223

5.8.2. THERMAL EQUIUBRIUM

A complete model for a quiescent prominence in equilibrium should solve not
only the equations for magnetohydrostatic equilibrium, i.e., Equation (5.39)
1
Vp-pg=-(VxB)xB,
41r

and

p =nkT,

(5.39)

but combine them with the energy equation (Equation (4.13», which we write in
the form
Q= V • (KVT) = L-G,

or

(B.V{:~ B.VT)=L-G,

(5.86)

where now the left-hand side represents thermal conduction parallel to the
magnetic field, with KII being the coefficient of conductivity (Equation (4.20».
The term L =L,adgives the loss due to radiation, and G is the gain due to
heating.
Different gain functions have been discussed in Section 4.3.1. The exact form
of a heating function is not known; often it is taken to be proportional to the
density, i.e., G = hp. If the heating is purely mechanical, the adjustable
parameter h may be considered a constant; Equation (4.17). If the heating depends on electric current or magnetic fields, the parameter h should depend on
these quantities, e.g., h =h(B), but no detailed model exist for these heating
processes. If the heating is due to dissipation of Alfven waves, the work of
Jensen (1983) and Chiuderi Drago et al. (1992) applies and is discussed in
Section 5.6.

5.8.2.1. Thermal equilibrium models
One of the first successful attempts to model the energy balance in a prominence
in a reasonably realistic way using radiative transfer theory was made by
Poland and Anzer (1971); see also Poland et al. (1971). Since their investigation
illustrates well the methods that can be applied to studies of the energy budget
in prominences, we shall discuss their model in some detail. They considered a
prominence consisting of an interior isothermal plasma slab surrounded by a
prominence-corona transition region, and neglected mechanical heating in both

224

CHAPTER 5

regions, i.e., Gmech =O. The density in the slab decreases exponentially outward,
and the slab is irradiated on both sides by photospheric, chromospheric, and
coronal radiation fields. Poland and Anzer solved the radiative transfer problem for the Lyman continuum (Lye) and the Ha line, assuming Lya to be saturated. They determined the net radiative energy loss for hydrogen, considered
to be the dominant part of L =Lrad. This loss was then balanced by the conductive energy gain, G =Gcond, from the corona. The external radiation field (see
Equation (2.13» was treated explicitly by separating it into two parts:
Jv(tv)=J/nt(tv)+J/xt(tv). The external field J/xt(tv) in the prominence
plasma, due to radiation from outside, is reduced by a factor exp(-t v) as it penetrates into the prominence at optical depth tv- Therefore, if Jvimp(tv) denotes
the radiation field impressed on the prominence slab from outside, we may
write
J vext ('rv) =exp( -'rv)J /"'P( 'rv)'

(5.87)

Poland and Anzer modeled the hydrogen atom as a two bound levels plus
continuum system and solved simultaneously the radiative transfer equation
(Equation (2.3» and the statistical equilibrium equation (Equation (2.15» to
find the energy flow in the Lye, the degree of hydrogen ionization (see Section
3.4) and the population of the n = 2 level in the hydrogen model atom. The
authors assumed for the hydrogen Ha line
.

1

Jv'tnp =2'Bv (TR )exp(-'r v )

(5.88)

and the integration over frequency was accomplished for a Doppler profile (see
Equation (2.48') for Y =Z « X) so that, from Equation (5.87) the mean intensity
becomes

J
00

Jext

=~B(n =3 -+ n =2)(TR )- C

exp(-v 2 -exp[-'ro exp(-v 2 )])dv,

o

where C is a normalization factor.
The Balmer continuum is optically thin (to « 1) and the Lya line is so
thick that it can be explained by pure scattering. Consequently, Poland and
Anzer considered the transport of radiation for hydrogen to be determined to a
very good approximation by the Lyc and the Ha line.
With the solution for transfer problem for the Lye and Ha in hand, the
authors then calculated the energy lost or gained, AE, in these transitions. For
the Lye they evaluated the expression (Thomas and Athay, 1961)

22S

PROMINENCE MODELS

J J
-

Tv(max}

L1E(Lyc)=21r dv
~

pSVd'rV =

2~Te

0

where p is the net radiative bracket

J

TO (max)

pSOd'rO'

(5.89)

0

J

p=nj Aij(1-Sijl r/JvJvdv),

(5.90)

and 5 is the source function (see Equation (2.14» and a suffix zero indicates the
value at the head of the continuum, while 'tv(max) is the maximum optical
depth reached in the prominences at frequency v. For Ha, which they assumed
effectively optically thin, all created photons in the line will escape, whence
the loss in the line is given by
(5.91)
where n2 is the population of level n =2, C23 is the collisional excitation rate
(Equation (2.21)), hv the energy of a photon, and X the prominence thickness.
Poland and Anzer found for reasonable values of temperature (6000 K) and
density (nH =1011 cm-3) in the prominence a net gain in the Lyc; i.e., from
Equation (5.89) AE(Lyc) = 9 x 104 erg cm-2 s-l. The Ha line provides a loss
(Equation (5.91)) AE(Ha) =2.5 x lOS erg cm-2 s-l. Consequently, for the radiation
loss term in Equation (5.86) we have L =L,"d"" lOS erg cm-2 s-l.
The thermal conductive flux to the prominence from the hot corona must
take place along the mafnetic field lines. Poland and Anzer showed that a
heat flow of ",,1()5 erg cm- s-l may be accomplished to balance Equation (5.86),
which they wrote in the form
(5.92)
and where a is the angle between the prominence and the magnetic field vector
and D is the distance over which the temperature drops from the milliondegree corona to the assumed 6000 K prominence plasma. In particular, they
quoted as a good approximation for the transition region a model with T =0.5 x
1()6 K and D =500 km, giving Q =1.1 x lOS erg cm-2 s-l.
The radiative transfer problem was studied in much greater detail by
Heasley and Mihalas (1976), who added 10% helium to the prominence plasma
and considered a hydrogen atom with five bound levels plus continuum, while
the He I and He IT systems had two levels plus continuum. The authors showed
that reasonable agreement with observations of the relative intensities of the
three first Balmer lines-which for the first time could be predicted due to the
five bound level model atom-was pOSSible if one postulates a highly fila-

226

CHAPTER 5

mentary prominence plasma. In such a prominence the Lyc radiation can
penetrate deep into the prominence interior and establish a temperature
structure that gives acceptable values for the relative intensities of the Balmer
lines, i.e., Balmer decrement. Heasley and Mihalas' (1976) work shows the importance of the fine structure of prominences; they operated with filling factors
of the order of 0.1-0.2; see Section 5.2.1.
Figure 5.19a shows the temperature drop from the surface of the prominence
toward its center for one of the models computed by Heasley and Mihalas. The
abscissa is the column mass density m given by
dl2

m=

Jpdx,

(5.93)

x

where p is the mean density of the filamentary prominence plasma. Since the
model is isobaric, the temperature drop from surface to center implies that the
density increases toward the center. Figure 5.19b portrays the variation of hydrogen level population nj (i = 1,2...5) with depth in the prominence, while
Figure 5.19c gives the corresponding variation for electron density ne and total
density ntot. For the Balmer decrement this model gives [(Hex): (HP): [(Hy)
.. 3.8:1:0.3 which should be compared to observed values (Tandberg-Hanssen,
1967)
[(Hex): (HP): [(Hr) =4.5:1:0.45

(5.94)

Heasley and Mihalas recognized that instead of increasing the exciting
radiation field inside the prominence by assuming a filamentary structure, an
appropriate temperature structure can also be obtained if there is a source of
non-radiative energy input. The internal heating may be due to a heating function of the form G = hp. The authors tried h = 1.5 x 105 W kg-I, and found a
prominence with temperature dropping from 8300 K at the surface to 7000 K in
the center and giving a Balmer decrement of [(Hex): (HP): [(Hy) .. 4.3:1:0.3. The
heating may be due to conduction, waves, or maybe a combination, and may produce acceptable models.
Lerche and Low (1977) constructed an infinite sheet prominence by solving
the equations of magnetohydrostatic and thermal equilibrium, where the
thermal flux is everywhere balanced by a hypothetical heat sink (Low, 1975).
This heat sink is proportional to the local density, so that the right-hand side
of Equation (5.86) becomes
L - G =Cp,

PROMINENCE MODELS

227

where C is a positive constant. With this simplifying condition, and assuming
a thermal conductivity that varies linearly with temperature, i.e., Ku =KoT,
the authors derived models for the distribution of temperature and density
through the prominence slab. Their work is of particular interest as it shows
the effects on the modeled prominence structure by varying different physical
parameters, e.g., the plasma-~ or the parameter a. in the cooling function Lrad'

7500

1="------------...

7000

T
6500
6000
5500
5000

10- 14

Fig. 5.19a. Prominence model due to Heasley and Mihalas (1976). Variation of temperature with
depth, i.e., from surface toward prominence center; m is the column mass density, given by Equation
(5.93), in units of g cm-2•

We mentioned in Section 5.7, while discussing prominence feet, that Milne
et al. (1979) also solved the full energy equation, i.e., Equation (5.86). They developed interesting models, albeit for fairly simple heating and cooling functions, and presented temperature profiles for the prominence plasma.

228

CHAPTER 5

--

n1

n2
/

n3

n4

nS
10 10 L...-.l...-.l..-...l...-....L-....1....---L--L--'----'---'
10-14 10-12 10-10 10-8 10-6 10-4

m

m

Fig. 5.19b. Variation of hydrogen level populations, n.{i = 1,2...5), with depth, mj see caption
Fig.5.19a.

Fig. S.l9c. Variation of electron density, net and
total density,
with depth, mj see caption
Fig.5.19a.

n,o"

Wiik et al. (1992) considered a five bound level plus continuum hydrogen
atom-as had Heasley and Mihalas (1976)-and performed the non-LTE calculations on the statistical equilibrium equations for a one-dimensional, slabprominence model with isobaric and isothermal structure. The vertical slab of
thickness D is irradiated with photospheric and chromospheric radiation pertinent to all line and continuum transitions in the model atom. The thickness D
is the effective thickness along the line of sight and depends on the filling
factor in the filamentary prominence plasma. Values of D between 750 and 3000
km were used as input parameters. The authors assumed a constant Ha source
function; whence the intensity of the normally emerging Ha line is
Iv

=S[l-exp(-'ro(v))],

where 'to(v) is the optical depth of the slab. The numerical code used
(NovockY, 1989) permits the determination of the maximum value 10 of Iv (at
disk center) as a function of election density ne; see Figure 5.20a. The three
curves correspond to three values of the assumed temperature. Figure 5.20b
shows the 10 = fine) dependence with the thickness D as a parameter. Wiik et
al. observed Ha profiles in a quiescent prominence and compared the observed
absolute intensities with the computed values of Figure 5.20 to derive the
electron densities in different parts of the prominence. They constructed two-

PROMINENCE MODELS

229

dimensional electron density maps (see Figure 5.21) for two assumed values of
the thickness D, i.e., 750 and 3000 km. For the outer parts of the prominence the
authors found electron densities in the range 1010 cm-l to 5 x 1010 cm-3•
20
_
15

T= 6000K
T = 7500 K
T=9000K
0= 1500 km

(a)
15

0=
0=
0=
0=
T=

750 km
1500 km
2250 km
3000 km
7500 K

(b)

Imax

Imax
10

10

5

5

0

4
2
3
ne in units of 1OlD cm-3

5

ne in units of 10 10 cm- 3

Fig. 5.20. Prominence models due to Wiik et al. (1992). (a) Intensity vs. electron density for three
temperatures; (b) Intensity vs. electron density for various slab thicknesses D.

From a detailed study of the observed profiles Wiik et a1. detected both
red and blue shifted regions, indicating sheared velocity fields or twisting motions in the prominence fine structure. It is interesting that regions with higher
velocities correspond to lower intensities, which would mean that these highvelocity, fine-structure elements cannot contain many threads along the line of
sight; see also discussion of fine structure in Section 5.3.1.
Recently highly sophisticated extensive numerical modeling of non-LTE
conditions in prominences has been accomplished by Gouttebroze et a1. (1993)
and Heinzel et a1. (1994). The 140 models calculated are represented by slabs
with uniform gas pressures, in the range 0.01 to 1.0 dyn cm-2, and temperatures in
the range 4300 to 15,000 K. The slab thickness, D, ranges from 200-10,000 km,
while the microturbulent velocity (Equation (3.1» is 5 km s-l. The authors used
a twenty bound level plus continuum model for the hydrogen atom and considered partial frequency redistribution; see Section 2.1.3. Several important
correlations between the plasma parameters and between plasma and radiation
properties were discovered. Figure 5.22 portrays one of these correlations, viz.
the relationship between the integrated Ha intensity, I(Ha) for radiation observed normal to the slab, and the emission measure,EM=iie2 D,compare
Equation (5.6), where iie is the mean electron density along the line of sight.

230

CHAPTER 5

This almost unique correlation between I(Ha) and EM provides an important
diagnostic tool for electron density determinations.

Fig. 5.21. Electron density determination in a prominence; see text (after Wiik et aI., 1992). Left:
Intensity map; Upper Right: Electron density contour map for thickness D =750 km; Lower Right:
Electron density contour map for thickness D = 3000 km.

In Figure 5.23 we show another important correlation from the work of Heinzel
et al. (1994), viz. the variation of the electron density versus total hydrogen
density nH in the center of the prominence slab. The degree of ionization (see
also Section 3.4) as given by the ratio ne/nH is almost model independent for
low-density models. For a given temperature, i.e., same symbol in Figure 5.23,
the scatter is relatively small and due to the variation in the thickness D.
Heinzel et al. could also study the variation with optical depth t of the source
functions, Sit); see, e.g., Equation (2.14). In the case of the Ha line the source
function is nearly model independent and constant through the slab, i.e.,
S(-r< 1) "" O.06ldisk '

231

PROMINENCE MODELS

where Idisk is the intensity of the solar continuum at disk center. In these outer
layers of the prominence the quoted value of the source function is determined
by the incident solar radiation. As we go further into the prominence, i.e., for 't
> I, the source function becomes model dependent and increases with 'to This behavior is reflected in the cloud-model analysis; see Section 5.8.2.2 below.
Important relationships like the ones just quoted can be of great diagnostic
value in studying prominences, and they can probably be found only with the
help of extensive modeling of the type described.

6.5

~ ~

6.0

¥

5.5

-

4.5

I

4.0

~

JO

3.5
3.0
2.5

0

,i'

5.0

'3'

~~~

~O.ti

,I

I

2.0
26.0

l::,.

DO~k>

0
27.0

I

-j

I

I

I

28.0

29.0

30.0

I

31.0

32.0

33.0

log 10 (EM)
Fig. 5.22. Integrated Ha intensity vs. emission measure for different temperatures of the prominence
plasma (after Heinzel et aI., 1994).

232

CHAPTER S

12.0

,-----r-----r-----r---.,.----...,---.....----,

11.5

11.0

--

Q,

~

c

tl

~ 10.5
Ol

o

§
0

0

o

"* o

10.0

t~

0

o
9.5

9.0 '---_ _...I.-_ _--l..._ _----'-_ _ I
9.0
9.5
10.0
10.5
11.0

~'---

_ _- ' - -_ _- ' -_ _---'

11.5

12.0

12.5

Fig. 5.23. Correlation between total hydrogen density and electron density for different temperatures
of the prominence plasma (after Heinzel et aI., 1994).

5.8.2.2. Cloud models
When a prominence is observed on the disk, the profile of a strong line like,
e.g., Ha will depend on the optical depth in the line. For an optically thick
filament the information in the line comes from the prominence plasma.
However, if the optical thickness is relatively small, the observed profile will
have in it contributions from the underlying chromosphere in addition to the information from the prominence itself. Such profiles may be interpreted using
the so-called cloud model (Beckers, 1964), which has been used extensively to
analyze the Ha line. The model assumes that the source function S(AA.) is constant along the line of sight, a condition that also must hold for the temperature, microturbulent velocity, and radial velocity of the plasma. These quanti-

PROMINENCE MODELS

233

ties are then derived by comparing the line profile from the filament, I(aA),
with the profile of the line emitted by the chromospheric background, Io(aA),
supposed to be the same over the field of view.
The comparison is done through the contrast profile C(aA) which Beckers
proposed in the form
(5.95)
and which can be written as
c(.1.t)=[ S(.1.t) -l][l-exp(-'t'(.1.t»)],
10 (.1.t)

with
.11) = 1'0 exp[ 't'( Lll\.

(.1.t-~.t(Ha)/c)2l.
.1.tD

(5.96)

is the velocity in the filament; aAD the Doppler width of the Hex line.
Great caution should be exercised in using this method because of the many
simplifying assumptions (e.g., Mein et a1., 1985; Schmieder, 1989). In particular, one assumes the background profile to be constant over the field of view.
However, it is immediately obvious from observations that the chromospheriC background is not uniform, and the standard cloud model has been improved to account for this fact. Mein and Mein (1989) and Mein and Schmieder
(1988) developed differential cloud models. The differential cloud model of first
order allows the chromospheric background Io(aA) to vary over the field of
view, while the differential cloud model of second order can be used to study
variation along the line of sight of the source function, radial velocity, and
microturbulence associated with a filament. Cloud models also have been
successfully used to study the dynamic nature of prominence fine structure at the
onset of a disparition brusque (Schmieder et al., 1991b).
~

5.9. Fine Structure Revisited

The physical characteristics of developed fine structures have been discussed in
Section 5.3, and we have seen throughout this chapter how the fine structures
enter nearly all aspects of prominence modeling. Nevertheless, the strongly
dynamic behavior of the fine structure elements, e.g., in quiescent prominences,
is not well understood.

234

CHAPTER S

Magnetic fields playa dominant role in all types of prominences, active as
well as quiescent. It is, therefore, assumed that the fine-structure threads outline magnetic flux tubes, and Wiehr and Stellmacher (1991) have measured
magnetic field strengths of the order of 10 Gauss in structures as thin as 1 arc sec
in diameter that may reflect the field in fine-structure threads. By properly
orienting the flux tubes one may be able to explain the appearance of fine structures in a prominence, but a convincing theory for their formation is not readily
available. Nevertheless, considerable work has been done in this regard, and
the way has been pointed to possible solutions, at least for some types of fine
structures. One should keep in mind that different processes may be at work to
form threads in, e.g., a post-flare loop, from those leading to threads in an
active region filament or in a quiescent prominence. We shall now discuss the
basic ideas behind some of the proposed formation mechanisms. This is a case
where the distinction between formation and model largely vanishes.
Schmieder et al. (1991a) proposed a dynamic interpretation for the threads
in a rising active region filament. In their "leaky-bucket" model nearly horizontal flux tubes rise slowly carrying material inside with them and experiencing
only slow loss rate along the tubes; see also Georgakilas et al. (1990) and
Schmieder et aI. (1991b). In this type of model the fine structure is linked to the
appearance of magnetic flux tubes in the chromosphere, and the actual formation process is given by the buoyancy of the magnetic field from below the
photosphere. Compare Rust and Kumar's (1994) rising flux tubes; Section 4.4.1.
Kuperus and Tandberg-Hanssen (1967) suggested that the fine structure of
quiescent prominences, modeled as a vertical current sheet (Section 4.3.2.1), is
formed by the interaction of the tearing-mode instability with the thermal instability that leads to the condensation of the prominence plasma. Further
developments of this basic idea are due to van Hoven and Mok (1984) who
presented model calculations of the structure and growth rate of a condensing
prominence, and showed the importance of having a sheared magnetic field in
which the thermal conductivity perpendicular to this field can act. The numerical calculations of Forbes et al. (1989) (see Figure 4.10) also clearly demonstrate the formation of elongated "islands," that may constitute fine-structure
elements.
The influence of perpendicular thermal conduction on the thermal instability was studied by van der Linden and Goossens (1991). Even though this influence was found to be negligible as far as the equilibrium is concerned, a rapid
spatial oscillation may result on a scale that is a function of the coefficient of
perpendicular thermal conduction. The rapid oscillation is confined to the most
unstable part of the equilibrium, and the authors suggest that it may be
relevant to the formation of the fine-structure thread. Further work along the
lines of this interesting paper may prove fruitful.
We have seen in Section 5.7 that Nakagawa and Malville (1969) invoked
the Rayleigh Taylor instability to account for prominence feet. It would be tempt-

PROMINENCE MODELS

23S

ing to assume that also fine-structure thread, in general, could be due to this
instability. However, in a review of the possible roles of different instabilities
in the formation process of such threads, Priest et al. (1991) concluded that
either the growth times or the resulting dimensions preclude both Rayleigh
Taylor and tearing-mode-as well as ballooning instabilities-to be seriously
considered. The authors suggest that the threads are most likely formed during
the condensation process by a thermal instability where anomalous perpendicular viscosity sets the scale.
To account for the strictly vertical fine-structure thread in quiescent prominences it is logical to think of gravity as a major factor. Poland and Mariska
(1988) followed this idea and suggested that the threads would form when the
prominence plasma locally becomes so dense that gravity causes the supporting
field lines to sag. This will cause a compression of the magnetic field below;
thermal instability will result and threads will form. Low (1982b) presented a
magnetostatic model of a thin, vertical prominence sheet and found equilibrium
solutions where the sheet is broken up in long vertical structures whose weight
is supported in sagging magnetic field lines. Because a free function is available, Low could produce vertical fine structures of different sizes and spacings.
As can be seen from the above discussions there is no lack of ideas, and many
good ones, of how filamentation of the prominence plasma may be achieved.
The difficulty is to show that the time scale for the process fits prominence
formation time scales, that the spatial scales and the orientation of the resulting threads are in accordance with high resolution observations, and, above all,
that one can account for the dynamic nature of the fine structure. To date these
aspects of the problem have not been treated adequately.

CHAPTER 6

THE DEATH OF PROMINENCES

Whether prominences, like surges, exist for only hours, or, like quiescent filaments, live for many months, sooner or later they disappear, and we are faced
with the task of explaining the physical processes that bring about their
demise. The disappearing act may take any of several forms, ranging from spectacular displays of violent disruptive motions to slow dimming and disappearance in situ. In some cases the disappearance coincides with changes of the
above lying corona; in other instances hardly any effect is noticed in the prominence's environment.
In discussing the prominence models in the previous chapter we have repeatedly alluded to the stability of these models, and some of the investigators have developed criteria to determine when an instability might set in and
lead to the disappearance of the prominence. In this chapter we shall discuss
further such instabilities and explore the different ways prominences die.
6.1. Comments on Active Prominences
By definition active prominences are dynamic in nature, meaning here that
material is constantly moving in or through them. In the case of surges, which
are being injected from below into magnetic flux tubes, the material ascends
until gravity stops it, and it flows back toward the chromosphere. There is no
onset of an instability involved, and the end of the prominence is the natural
outcome of material falling back into the Sun.
Similar remarks also apply to loop prominences that form after the disappearance of a filament and the accompanying flare display, as post-flare
loops. The physics involved is probably somewhat more complicated than in
the case of surges, but the material also here is constantly flowing from where
it condenses at the top of loop-shaped flux tubes, along and down the legs of the
flux loops toward the chromosphere. In this case onset of instabilities is involved, but in creating the prominence. The demise of the prominence is not
unlike the demise of a surge; its very nature is a display of down flowing
material, and when that display is over, that is the end of the loops.

238

CHAPTER 6

This does not mean that instabilities cannot occur during a surge or loop display, e.g., in the up- or downflowing plasma, but the instability is not the cause
of the prominence demise. As an example we mention a surge model proposed by
Carbone et a!. (1987) in which the surge plasma becomes unstable and develops
a turbulent energy cascade. They further showed that this may produce electron
acceleration. With the accelerated particles, one may expect, e.g., type III
radio bursts, and Chiuderi Drago et a!. (1986) found a correlation of such bursts
with Ha structures that probably were surges. The nature of the surge plasma
has been studied further by Mein and Mein (1989) who analyzed the data using
the differential cloud model (Section 5.8.2.2) and derived values for the
velocity and microturbulence. Situations like this with turbulence of surge
plasmas can probably be found in other types of active prominences; however, in
this chapter, we are mainly exploring instabilities that cause, more or less
directly, the end of the prominence.
There are other, intermediate, types of prominences, like coronal rain and
clouds, to which the previous comments apply only approximately. The down
flowing material from such prominences behaves like the material in postflare loops, but no detailed model of the formation of the clouds exists, even
though a condensation process is probably involved.
Since the disappearance of the above mentioned active prominences is the
consequence of their. inherent dynamic nature, we shall not treat this aspect
further, but turn to the disappearance of quiescent and active filaments, i.e., to
prominences that during part of their life exhibit equilibrium conditions of
their plasmas. However, loop structures form parts of quiescent prominences,
and the stability of such loops must be considered in this context.
6.2. Thermal Equilibrium Breakdown

We have seen in Chapter 5 how two classes of equilibria are involved in prominence stability, viz. thermal equilibrium and magnetohydrostatic equilibrium.
If the former breaks down, the plasma can heat up; if the latter fails, the support is drastically changed.
The non-linear equations governing both the thermal and the magnetohydrostatic equilibrium are strongly coupled, and it is, therefore, an oversimplification to treat the two equilibria separately. Nevertheless, in so
doing, we shall still gain insight into the condition we find in prominences that
are on the verge of instability. The complete, coupled, non-linear treatment is
presently beyond reach, but that fact should warn us that the results at which
we arrive are not the final answer. Since quiescent prominences remain nearly
stationary for long periods of time, we conclude that all the energy flowing into
the prominence must be transported out of it at the same rate during the equilibrium phase of their life.

THE DEATH OF PROMINENCES

239

Prominences get rid of energy mainly by radiation, and since radiative
losses, Lrad, are proportional to the density squared (Equation (4.18», the dense
prominence plasma can effectively loose energy. However, if the radiation
losses were too small, the prominence would quickly heat up. Thermal conduction would be the main culprit, and we can find a lower limit to the time it
would take to heat the plasma from an initial temperature, T pr' to coronal
values, TeoT, due to thermal conduction alone, by neglecting radiative losses,
i.e., by writing Equation (4.13) in the form (Rosseland et al., 1958; Ioshpa, 1965)
cV

err

at =V(KVT),

(6.1)

where Cv is the specific heat per unit volume. Following Ioshpa, we consider
isobaric heating, put K = Ay512, and obtain in the one-dimensional case the
equation
(6.2)
with the boundary conditions T(O,t) = Teor' T(x,O) =0.
The approximate solution found by Ioshpa represents a thermal wave with
a steep front. At any time t, the width of the zone in which the temperature
drops from 0.4 T to 0 is approximately 1/10 the thickness of the already heated
region. If, for example, the prominence has a thickness of 109 cm, it will be
heated in t =2.5 x 1()3 s, and if the prominence consists of fine structure threads
of thickness 108 em, the time to heat it to coronal temperatures is 25 s. Similar
results were already published by Severny and Khoklova (1953) and
Shklovsky (1965) (see also Zel'dovich and Raizer, 1967).
Since quiescent prominences reveal nearly static conditions, these estimates
indicate the efficiency with which radiation losses strike a balance. In addition, we have seen how the existence of a magnetic field in the prominence with
a significant component transverse to the heat flow will drastically alter these
estimates. While heat conduction along the field is the same as in the absence
of the field, the conduction perpendicular to the field is reduced by the factor 1
+ (COBt)2, where COB = eBjmc is the cyclotron frequency. Therefore, to increase
the time scale above by a factor of 104, for example, it is sufficient to have a
transverse magnetic field of less than 1 Gauss. However, even though the
transverse conductivity is greatly reduced, there will be an additional heat
flow in a direction perpendicular to both the magnetic field and the
temperature gradient, i.e., in the direction B x VT (the Righi-Leduc effect). The
corresponding conductivity is 1/ coBt times the direct conductivity and may be of
importance for some geometrical configurations in prominences (TandbergHanssen, 1960; Orrall and Zirker, 1961).

240

CHAPTER 6

We also have to take into consideration the fact that the prominence is
bathed in a UV and X-ray radiation field from the chromosphere and the
corona (see Equation (5.87» that will tend to heat up the prominence plasma.
The possible results of these different heating effects will be discussed in the
next section.
6.2.1. THERMAL DISPARITIONS BRUSQUFS

The concept of a thermal disparition brusque (DBt) was first introduced by
Mouradian et al. (1981); see Section 3.6.1.4. A filament that undergoes this precess starts to fade in Ha and then becomes visible successively in spectral lines
formed at increasingly higher temperatures, e.g., C III, 0 VI, Mg X and in Xrays (Mouradian et al., 1986; McAllister, et al., 1992; Watanabe et al., 1992).
After a certain time the prominence plasma cools again, and the filament reappears in Ha, referred to as an "Apparition Brusque" (Malherbe, 1989). These
disappearances are, therefore, temporary disparitions brusques (Mouradian
and Soru-Escaut, 1989; Soru-Escaut and Mouradian, 1990) and do not lead to the
final demise of the prominence.
The concept of a thermal type of disparition brusque raises some interesting
questions concerning heating and cooling processes in the solar corona, and we
shall shortly discuss this aspect. However, we should bear in mind that the
filaments in question seem to disappear only because we normally observe them
in "cool" lines like Ha. If our instruments had been very broadband, we would
have seen them still in their hot phase, in "hot" lines in the UV or X-ray demain.
The reason for the heating is not clear. Schmahl et al. (1982) pointed out
that hot coronal arches generally overlie filaments that undergo a DBt, and it
has been suggested (Mouradian et al., 1986) that the heating is due to irradiation of the filament plasma from these hot arches.
Malherbe and Forbes (1986) tried to answer the question of how to evaporate a prominence, and we shall look briefly at their analysis, which describes
the thermal equilibrium of the prominence by the equation
0mech + 0 louie + 0visc

+ 0cond - 4wI = 0,

(6.3)

where
Omech = qp is heating due to, e.g., wave dissipation or magnetic heating;
see Equation (4.17)
Oloule

= j21a is the ohmic heating; see Equation (2.106),

241

THE DEATH OF PROMINENCES

Gvisc
Geond

=llV2/1.2

is viscous heating,

=KoT 7/21 L2 is heating by thermal conduction,

4ad =p2 C2 (T)T a (T)

is the radiative loss, Equation (4.18),

which the authors wrote in the form Lrod =p2Q(T). Malherbe and Forbes
assumed that the electrical conductivity has the form (J =(Jo"(3/2 and the coefficient for viscosity the form 1'\ =1'\0 T5/2. (See the text leading to Equation
(2.123) for a discussion of viscosity and magnetic diffusivity.) For L",d they used
the form discussed in Section 4.3.1. The authors then considered separately the
balance between radiation losses and the four heating mechanisms and concluded that the mechanism most likely to destroy the cool prominence equilibrium is thermal conduction, even though enhancement of the largely unknown
ambient coronal heating, that we have called Gmeeh, cannot be ruled out.
With conduction balancing radiation losses Malherbe and Forbes write
P2Q(T)

2
=K oT 712
eor /L '

(6.4)

where L is a thermal length scale along magnetic field lines from the photosphere to the prominence. If we take L =3 x 104 km, p =10-12 g cm-3 , and Teor =
106 K, Equation (6.4) gives Tpr =4800 K. The authors then perturbed this equilibrium keeping the gas pressure constant, obtained
Q(T)/T 2

=const T7/2
eor'

(6.5)

and made the point that since no cold solution exists for T> T m =1.5 x 1()4 K, the
prominence will disappear when
Teor > Teor,m

=

217(T )417

=Teor [gg;~ ]

:~

()2(a-2)17

=Teor ~;

=

•

(6.6)

With the value L 3 x 102 km, we get Teor,m 1 Teor 9.21, indicating that a hot
region in the neighborhood of a prominence could heat and evaporate the
prominence. Malherbe and Forbes (1986) suggest that such a hot region could be
created by a flare. It might also be the hot coronal arch seen above prominences
that undergoes DBt's (Schmahl et a!., 1982). However, they did not rule out
wave heating, but little is known about the form of Gmeeh, so estimates become
little more than guesses.
When we discuss dynamic disparitions brusques in Section 6.3.3, we shall
explore further some differences between thermal and dynamiC DBs, as well as

242

CHAPTER 6

their common features. Here we shall now present a typical OBt feature not
found in OBd events, namely the concept of a pivot point.
If one follows the shape and orientation of a long-lived quiescent prominence for several months and records its locations one solar rotatio~ period
apart, one will notice that an originally north-south oriented filament will,
after several rotations, be stretched out in a more or less east-west direction; see
Figure 6.1a. This is due to the differential rotation of the Sun's atmospheric
layers, and was discussed already by d' Azambuja and d' Azambuja (1948); see
Section 5.1. However, Mouradian et al. (1987) discovered that there exists
filaments that do not respond to the differential rotation, but seem anchored in
so-called pivot points which show rigid rotation. Figure 6.1b shows the successive orientations of another long-lasting filament, and one observes that the
filament traces made one solar rotation period apart this time seem to intersect
in one area, the pivot point.
These pivot points are located in or near small active centers, and the
prominence in question either has a foot or one end anchored in the point.
Mouradian and Soru-Escaut (1989) found that filaments that undergo a thermal
disparition brusque have a pivot point, while dynamic disparitions brusques
occur for prominences without pivot point, and which, therefore, exhibit
general differential rotation. Consequently, if a filament with a pivot point
undergoes a OBt, it is a temporary disappearance in "cool" lines like Hex and it
reappears as it cools; no new condensation of coronal matter may be involved.
Conversely, when a filament without a pivot point undergoes a OBd, material
and field are expelled, and if the prominence reappears, it reforms by, e.g.,
condensation of coronal matter in a magnetic field that has readjusted to again
support the prominence.
The pivot points may furnish another answer to what is the reason for the
heating that leads to a OBt. Mouradian and Soru-Escaut (1989) found that
pivot points occur in or near active regions, meaning areas with enhanced magnetic fields. They suggest that the energy is transported into the prominence
from the pivot point along flux tubes in the prominence feet, in form of waves;
see Equation (6.4) and the analysis of Malherbe and Forbes (1986).
6.3. Magnetohydrostatic Equilibrium Breakdown
The loss of magnetohydrostatic equilibrium can have a far more devastating
effect on a prominence than the temporary breakdown of thermal equilibrium.
This loss can lead to the permanent destruction of the prominence, even though
in many cases the supporting magnetic field may still be available for a new
prominence to form in it
The permanent disappearance of prominences may-from an observational
point of view-occur in one of two ways. On the disk we may witness a gradual

243

THE DEATH OF PROMINENCES

+60 0

q>

fl-

(a)
____ ...

f-

~__

5

.. -......

4

+30 0 r

-

• ••• 3
,,',

~_._.

'---

.... ......2,,~
.......,. .... "'''.

'.,....

'0

o

J•• (1693)

\

L
I

I

(b)

"
1601,
1602

-

"\

,... ~
.......::u ..~

.......

1605

~

!i
:'1

...

... ,
.;- I

1607 --

\.

~

\

\

,

\

.,

-

11603

I

I

Fig. 6.1. Change of filament orientation with time. Five rotations are superimposed using the
Carrington rotation velocity to relate them. (a) Example of quiescent prominence whose change in
orientation and location agrees with the general differential rotation; (b) Example of a prominence
whose successive orientations reveal a region, called a pivot point, marked by a circle, with rigid
rotation, i.e., rotation independent of latitude (after Mouradian et aI., 1987).

decline in the size of the filament as well as in its visibility. It looks like the
evanescence is due to a gradual submergence of the magnetic support (Mouradian
and Soru-Escaut, 1989). We have discussed this phase in Section 3.6.1.5 where

244

CHAPTER 6

it was referred to as the sinking and shrinking filament, and in these cases the
magnetic field is no longer available for another prominence formation.
The other way for a prominence to react to a breakdown of its magnetohydrostatic equilibrium is to erupt up into the corona in the type of a sudden
disappearance that we call a dynamic disparition brusque (DBd), or just DB. A
subset of these prominences does reform, meaning that the supporting field is
still available for prominence formation.
Before we discuss further this most interesting form for prominence activity,
we shall first look at the inherent stability of prominences and then consider
situations that may bring about the loss of the static equilibrium.
6.3.1. PROMINENCE STABILITY

The development of certain fields of solar physics has often gone hand-in-hand
with investigations in plasma physics, and maybe nowhere is this statement
more applicable than when it comes to understanding prominence stability.
Bernstein et al. (1958) derived a now famous energy principle that bears their
name (see also Lundquist, 1951), and which addresses the problem of a plasma
surrounded by a vacuum magnetic field, that has no component perpendicular to
the plasma surface. Anzer (1969) modified the Bernstein et al. stability criterion to fit the situation of a prominence of the Kippenhahn-Schliiter (slab)
type in an appropriate magnetic field, i.e., no field component along the long,
Y-axis of the prominence, while the current in the slab, of thickness d, only has
a V-component. We shall briefly follow Anzer's analysis.
The set of equations that governs the physics involved is given by the
moments of the Boltzmann equation, or Equations (2.83'), (2.92), (2.93);
Maxwell's equations, i.e., Equations (2.107)-(2.110) with D = 0; and Equation
(2.118'), assuming infinite electrical conductivity. Anzer studied the stability
of the equilibrium conditions of this system against small perturbations; of the
prominence plasma. The set of equations gives for; in first order
(6.7)

where

is the linearized force per unit volume acting on the plasma. A subscript zero
indicates equilibrium value and

THE DEATH OF PROMINENCES

245

The perturbation ~ will lead to a change in potential energy which is minus
the work done by the force during the displacement, integrated over the prominence plasma, i.e.,
(6.9)

t

The factor arises since the mean force during the displacement from 0 to ~ is
tF(~). The plasma is unstable if and only if there exists a displacement ~
which leads to a negative value of ~W. From an analysis of Equation (6.9)
Anzer showed that there exists two necessary and sufficient stability criteria
that may be written in terms of the jump of Bz over the prominence slab. He
defined this jump by
(6.10)
and derived the stability criteria
[Bz(z)] dBx ;;:: 0
dz

(6.11)

and
(6.12)

When the electric currents are small, condition (6.11) was already found by
Kippenhahn and Schluter (1957). As mentioned in Section 3.5, observations
indicate that this condition is fulfilled in stable filaments. Condition (6.12),
which states that the Bz component at the border of the filament should
decrease with height, has not been confirmed observationally.
Anzer's original work pertains to an infinitely thin slab-a sheet, d--+O,
and small perturbations. A more rigorous investigation by Galindo-Trejo and
Schindler (1984) established the stability of the Kippenhahn-Schluter model
for arbitrary perturbations. Galindo-Trejo (1987) also analyzed the stability of
Menzel's model (Bhatnagar et al., 1951), Dungey's (1953), and Lerche and Low's
(1980) models. Note that if the displacement is of the special form ~(ro,t) =
~(ro)·exp(imt), the linearized equation of motion, Equation (6.7), takes the
form
(6.7')

246

CHAPTER 6

where the force as before is given by Equation (6.8).
In this case Equation (6.7') reduces to an eigenvalue problem, and the solution is referred to as the normal mode method. Equation (6.7') is, in general, a
complicated set of equations, and analytic solutions are known only for simple
equilibria.
If we take the scalar product of Equation (6.7') with ~ and integrate over
volume, we find
(6.9')
For a given displacement we can use Equation (6.9') to determine either the
frequency of the oscillation resulting from the perturbation, i.e., the case where
002 > 0, or the growth rate of the instability resulting if 002 < O.
Besides slabs or sheets, loops constitute the other major class of prominence
models that has been explored in terms of their stability. Again the energy
principle of Bernstein et al. (1958) or the normal mode method has often been
used. At other times, general discussions of plasma instabilities in the framework of plasma physics have implied the probable or possible presence of one
or more of the instabilities in certain prominences. For excellent discussions and
reviews of prominence stability and possible instabilities, see e.g., Priest (1982)
and Hood (1989).
We shall mention a few of the instabilities that may affect prominence
structures like loops. In laboratory studies a cylindrical plasma column forms a
linear discharge confined (pinched) by an azimuthal magnetic field BIP due to a
current jz along the column. This pinched discharge is known to be subjected to
kink and sausage instabilities, and since the discharge is similar to magnetic flux
tubes, assumed to make up the prominence fine structure, it is also assumed that
kink and sausage instabilities may play a role in prominence equilibrium.
Other possible instabilities of interest in prominence physics include interchange instabilities, where two neighboring bundles of magnetic field lines in a
certain volume are interchanged in such a way that the magnetic energy
decreases, thereby leading to instability.
Specifically, Rayleigh-Taylor and Kelvin-Helmholtz instabilities have been
invoked. The former occurs when a denser plasma rests on a less dense plasma;
see Equation (5.75). The latter occurs when a uniform, inviscid fluid rests on
another fluid and is in horizontal motion relative to this other fluid. Resistive
instabilities will occur due to the finite electrical conductivity of the plasma,
e.g., in form of a tearing-mode instability; see Section 4.3.2.1.
Hinata (1987) considered the possible existence of large-scale electric
fields parallel to the magnetic field in post-flare loops. He showed that the
finite electrical conductivity in non-ideal MHO approximation allows strong

THE DEATH OF PROMINENCES

247

enough electric fields to drive runaway electrons, which, in turn, may induce
plasma instability. That strong macroscopic electric fields may exist in postflare loops was advocated by Foukal et al. (1983); see Section 2.1.5.
Redcoborody (1990) studied the instability of internal gravity waves (see
Section 2.3.4) and its possible effect on prominence destabilization.
The application of some of these plasma instabilities to the study of
prominence disappearance is often little more than suggestions that one of the
instabilities may be involved. Nevertheless, these suggestions can be very
valuable, as they may reveal physical insight and lead to further quantitative
analysis. On the other hand, there also have appeared in the literature investigations that treat the application to a prominence plasma at considerable
depth.
We may model a loop-like prominence feature by a cylindrical magnetic
flux tube, where the field has the components B[O,B(e),B(z)] in terms of cylindrical polar coordinates (r,e,z). If the flux tube is twisted; i.e., the field lines
twist about the tube axis in going from one end along the tube of length 2L to the
other end, we can measure the twist <I> given by

lP = 2LB(O) .
rB(z)

(6.13)

Kruskal (1954) and Shafranov (1957) demonstrated that an infinitely long flux
tube is unstable to a helical perturbation of the form
~ = ~(r)exp[i(O+kz)]

(6.14)

when
B( 0)/ r

+ kB(z) ~ o.

(6.14')

Applying this instability criterion (6.14') to our flux loop, we find that it is
kink unstable for wavenumbers greater than a critical wavenumber kcrit given
by kcrit = -f<l>1 L, or, stated differently, instability occurs for a given wavenumber k when the twist exceeds a critical twist

lPcrit = -2kL.

(6.15)

This value of the twist should not be confused with the other critical value of
the twist, necessary to from a dip in a flux loop as discussed in Section 5.5.3.2,
and given by Equation (5.70). The latter critical twist concerns the formation of
a prominence, while the critical twist in Equation (6.15) concerns the disappearance of a prominence.

248

CHAPTER 6

Of other early investigations we mention the analysis of Newcomb (1960)
who studied the flux tube behavior in terms of a linear pinch using the energy
method of Bernstein et al. (1958) and developing his stability criterion, and
Callebaut and Voslamber (1962) who considered a force-free flux tube; see also
Anzer (1968) and Sakurai (1976) where the kink instability is considered as the
cause of filament eruptions.
The instability criterion (Equation (6.15», given by the KruskalShafranov et al. analyses, is too restrictive and leads to the conclusion that
nearly all flux loops are unstable to kink perturbation (Anzer, 1968). The reason
for this is that the all-important effect of line tying has been neglected. When
the feet of the loop are anchored in the photosphere, the flux tube experiences
a strong stabilizing effect. This comes about because the perturbations are
restricted by the rather rigid, high-inertia, boundary conditions in the
photosphere where the feet of the flux tube reside. This phenomenon is referred
to as line tying in laboratory situations (Krall and Trivelpiece, 1973) where it
describes the inability of the field lines to move through conducting end plates.
The name line tying has been fairly generally adopted also in solar physicS.
With line tying it is assumed that all plasma perturbations disappear in the
photosphere. However, photospheric mass motions will still be able to twist
the field lines in the flux tube, since we are in a high-~ plasma regime, and the
twist, may be high enough to lead to instability, but its value is greater than
that given by Equation (6.15).
Raadu (1972) included line tying by modifying the form of the perturbation
(Equation (6.14» to read

i.e., modifying the previous form by multiplication with a function of z that
vanishes at the feet of the loop in the photosphere. Raadu then used the
Bernstein et al. energy method to find the minimizing function f(z), and showed
that the flux-tube loop is stable for twists less than a critical value, 'eTit ... (2 to
4) 1t, i.e., not precisely determined, but, < 21t should mean stability.
Further developments and refinements of the theory for flux loop stability
are due to Hood and Priest (1979b, 1980, 1981) who have conducted full stability
analyses of line-tied perturbations, allowing pressure gradients in the tubes,
and not restricting the field to be force-free. However, in the force-free case
they find for the critical twist
fPerit = 2. 51t',

(6.16)

in general agreement with Raadu (1972). Regardless of the details it is clear
that line-tied flux tubes may increase the possibility of stable support for

THE DEATH OF PROMINENCES

249

prominence structures. Nevertheless, it is worth emphasizing that line tying
cannot save a flux tube if the twist becomes too large; in other words, there is
also here for a· given twist a critical length of the loop that may not be exceeded. Several investigations have been concerned with the limitations of the
line-tying effect. As an example An (1982) studied the two effects of line tying
on the MHO stability of a cylindrical loop, viz. a line-bending effect which is
always stabilizing but also a shear effect. The latter is stabilizing or destabilizing depending on the sign of the gradient of the magnetic field. We also mention an investigation by Longcope and Strauss (1994) who considered an array of
closely packed parallel line-tied flux tubes with alternating senses of twist.
They showed that the system is unstable as it develops current sheets between
the flux tube tops and spontaneous reconnection sets in. See also the analysis of
the stability of line-tied coronal loops by Einaudi and van Hoven (1983) and de
Bruyne and Hood (1992). Finally, the important effect of line tying ill currentsheet models has been demonstrated by Longbottom et al. (1994). We have
reached a degree of sophistication in describing the physics of evolving magnetic flux tubes where hitherto neglected MHO and plasma physical effects
need to be included in the discussion of flux tubes in the solar atmosphere.
6.3.2.

OESTABILIZAnON OF PROMINENCES

The destabilization of the supporting magnetic structure in a prominence may be
due to either an exterior agent or be the result of a perturbation excited by
interior conditions. For example, some of the oscillations discussed in Section
3.6.1.3 may be caused by MHO waves emitted by a distant flare, hitting the
prominence sheet, and setting it in oscillatory motions. In other prominence
plasmas, different types of waves may be generated locally and cause the
plasma to vibrate.
The destabilizations that manifest themselves as oscillations are, in
general, not fatal to the prominence involved; it regains stability as the oscillations are damped and die away. However, other exterior agents may interact
with the supporting magnetic field in such a way that the destabilizing effect
grows and severely alters the supporting field. In the framework of stability
criteria one would expect, e.g., conditions (6.11) and (6.12) to be violated, or, in
the framework of prominence flux tubes, we would expect, e.g., the twist to
exceed the critical value in (6.16). Finally, and maybe most importantly, the
supporting field, as part of the overall, global magnetic structure in the corona,
may become unstable and cause the whole prominence, whether slab- or loopshaped, to erupt in a disparition brusque.

250

CHAPTER 6

6.3.2.1. Early investigations-historical notes
Bruzek (1952) considered the possibility that the destabilizing disturbance
may come from developing sunspots. He found that there exists a correlation
between the time, at, from the first appearance of a sunspot to the activation of
a nearby filament, and the distance, d, of the filament from the spot. From the
at, d correlation, Bruzek concluded that there is a perturbation propagating
from sunspots with a velocity of about 1 km s-l. Using observations of photospheric magnetic fields, we can link this disturbance to the development and
propagation of the magnetic field around forming sunspots. The observations
show that there is an expanding elliptic area around the developing sunspot
group in which the chromospheric fibrils become aligned by the magnetic field.
The growth of the outer boundary of this area proceeds at about 0.2 km s-l
(Bumba and Howard, 1965), and when the disturbance reaches a filament, the
latter breaks up and disappears. This sunspot-induced disappearance should be
clearly distinguished from flare-induced phenomena, which are triggered by a
much faster disturbance; see below. Bumba and Howard's observations show
that the disturbance is to be linked with the electromagnetic field propagating
away from developing sunspots. Giovanelli (1947) compared the growing
magnetic field of a sunspot with the field resulting from a continuously oscillating magnetic dipole. The electromagnetic field set up around the dipole is given
by (Stratton, 1941)
(6.17)

(6.18)
where co is the frequency and A is the amplitude of the magnetic moment of the
dipole, ).1 is the permeability of the plasma, £' = £ - i(41ta / co) is the complex
dielectric constant (Equation (2.112», k1 = (£).1C02 - i41tcoa)1/2 and r is the distance from the dipole. The frequency co is very small, that is, 10-6 to 10-5 s-l for
typical sunspot growth times. Giovanelli showed that the magnetic field
would grow as if propagated with a velocity
2

Vdist "" 3( 2 roc /a)

112

.

(6.19)

THE DEATH OF PROMINENCES

251

To make a rough estimate of this velocity, we take ro =5x 1()-6 s-l and a =lOS s-l.
This gives Vdist" 0.3 km s-l, not unlike the velocities inferred by Bruzek (1952)
and by Bumba and Howard (1965).
If the propagating magnetic field encounters a filament, the interaction between the filament magnetic field and the disturbance may possibly be quite
different from the interaction between a filament and a flare-induced disturbance. Bumba and Howard's observation showed that the filament that was
hit by the slow disturbance broke up over a period of 2 days, indicating that the
two magnetic fields involved interacted in such a way that the filament field
lost its supporting capability.
It became clear many years ago that disturbing agents are also generated in
flares, and we have, in Section 3.6.1.3, seen how oscillatory motions of quiescent
prominences result from the destabilizing effects of such agents. The work on
these activations was continued by Dodson-Prince (1949), Bruzek (1951, 1958),
and Becker (1958) and especially by Moreton (1960, 1965), whose refined photographic technique permits the actual observation of the propagating disturbance as it travels from the flare to the prominence with a velocity in the range
500 to 1500 km s-l. For other significant contributions to the observations of
these activations, see papers by Moreton and Ramsey (1960), Dodson-Prince and
Hedeman (1964), and Smith and Angle (1969). Theoretical investigations were
made by Athay and Moreton (1961), Anderson (1966), Hyder (1966), Meyer
(1968), Uchida (1968, 1970), and Kleczek and Kuperus (1969).
The prominence oscillations were analyzed in two ways, as vertical or as
horizontal oscillations. In both cases the motions may be treated with the
equation for a damped harmonic oscillator. The observations of Ramsey and
Smith (1966) were used to infer the frequency of the oscillation, vose (or the
period P) and the damping time 't (or the decay constant y). Ramsey and Smith
used narrow-band observations of the filament in the center of the Ha line, and
at +0.5 A and -0.5 A from the line center, and deduced the above-mentioned
parameters from the relative visibility of the filament at these three wavelengths.
For vertical oscillations Malville (1961) suggested that the magnetic field
in the filament must be crucial, and Hyder (1966) developed a model where an
electromagnetic wave disturbance causes the filament to oscillate under the influence of its own magnetic field (see also Anderson, 1966). Hyder considered
the frequencyofoscillation tobevose =1O-3 s-1 and the decay constantyz = 10-3 s-l.
In the model the mass of the filament is supposed to be suspended in the
magnetic field supporting the quiescent prominence, in a manner similar to the
model of Kippenhahn and Schluter. The vertical component of the field is Bz,
the scale height of the filament is H, and a small downward displacement Az
of the filament is supposed to lead to a linear increase in Bz,
(6.20)

252

CHAPTER 6

Under these conditions, one can analyze the filament oscillations in terms of a
damped harmonic oscillator, and the frequency is determined from the equation
of motion
d 2z
dt 2

J.l dz
Mdt

K
M

-+--+~z=O

(6.21)

and from
(6.22)
where Vo is the frequency of an undamped harmonic oscillator, Vo = (1/21t)
~ K z 1M, M is the mass of the filament, K z the restoring force, and J.l the coefficient of friction in the corona in which the filament moves. The ratio KzI M
can be expressed in terms of the magnetic field in the prominence (through the
magnetic tension, B2 141tH) and the physical parameters describing it (density
p, volume V, scale height H):

& =2!!...(~)~_1_.
dz

M

4

H pV

(6.23)

Combining Equations (6.20) and (6.23), we find

& = (!!z..)2 _1_.
M

H

(6.24)

trp

This equation may be combined with the following expression for KzI M in terms
of the frequency of oscillation and the decay constant, that is,

rz

2 2
2
K z I M = 4 tr Vose +

to obtain an expression for the vertical field

The frequency vase as given by Equation (6.22) depends on the friction experienced by the oscillating filament. The coefficient of friction may be defined
in terms of the coefficient of viscosity in the corona 11 =2M dyl Ac, where A is
the area of the vertical surface of the filament and d the effective distance,
perpendicular to A, over which shears exist in the coronal plasma as a result of

THE DEATH OF PROMINENCES

253

8

the vertical oscillations. Hyder inserted the followin values for the parameters: M = 1015 Gauss, H = 3 X 109 em, d = 109 em, A = 1()2 cm2, which leads to
Br. ... 10 Gauss, 71'" 10-9 poise.

Linhart (1960) has given a relation between the strength of a coronal magnetic
field, Bear, and the coefficient of viscosity in the corona
1l2
71 -- 1. 6 x 10-26 ne21B2cor rcor'

Using the value for 11 determined by Hyder and inserting for ne =109 cm-3 and for
r cor =1()6 K, we find Boor'" 0.13 Gauss.
Hyder implied that the disparition brusque and the oscillating filament
are similar phenomena. The property that distinguishes between them may be
the direction of the initial displacement AZ of the filament. If the displacement takes place along the positive Z-axis, the filament may completely
disappear, but if the displacement is downward (along the negative Z-axis),
the disturbed filament may undergo oscillations. As the filament is pushed
down initially, the density increases rapidly and the motion becomes highly
damped.
Observations indicate that many prominence oscillations are horizontal,
and Kleczek and Kuperus (1969) developed a model where the sheet-like
prominence is hit broadside by the flare disturbance and displaced horizontally from its equilibrium position. Kleczek and Kuperus approximated the
supporting magnetic field by an effective field along the long axis of the prominence (the Y-axis) and anchored in fixed positions. The horizontal restoring
force, K x, then is given by the magnetic tension-(l /41t)(B • V)B. If 2L denotes
the length of the prominence and x the displacement at the middle of it, the
restoring force is to first order proportional to x and given by (B;/41tL2 )x. The
effect of this force is to induce damped oscillations of the prominence sheet
according to the equation
(6.25)
where R is the damping constant. Equation (6.21) is mathematically the same
equation that Hyder used (Equation (6.21», but in Hyder's case of vertical
oscillations the damping is due to magnetic viscosity. For a sheet-like prominence the effect of viscosity is negligible under horizontal oscillations.
However, as pointed out by Kleczek and Kuperus, the coronal plasma will
alternatively by compressed and rarefied during the horizontal oscillations,
and compression waves will be generated. The authors analyzed this situation

254

CHAPTER 6

in terms of the theory for radiation of acoustic waves from a circular piston
(Lindsay, 1960). The emission of waves of the type (l/r)ei(mt-kr) from a circular
piston of area 1ta 2 will affect the motion of the piston, and the effect can be
described as a radiation reaction force R. If we neglect the acoustic reactance
part of this force-which basically results in an increase in the effective mass
of the prominence, and therefore in a relatively small increase in the oscillation period-R may be written
R -- n:a 2Pcorc dx
dt

[1- I}(21ea)]
,
lea

(6.26)

where Pcor is the coronal density and II (x) the Bessel function of first order.
Equations (6.25) and (6.26) combine to give the following equation of motion for
the damped oscillating prominence
(6.27)
where 1ta2 =2ZL is the surface area of the prominence sheet of height Z and
thickness X. Equation (6.27) gives the oscillation period (P = 21t I CI) =2~MIKx)
(6.28)
where Ppr is the density of the prominence plasma (= M/2LZX). Kleczek and
Kuperus took 2L =1010 km, Ppr =10-14 g cm-3 and B = 9 Gauss to find a period of
about 20 min. The decay constant for the oscillations, Yx =R/ M, is given by
(6.29)
For two cases studied by Kleczek and Kuperus, with P =20 min, Equation (6.29)
indicates a decay constant of about 4 x 10-4 s-l.

6.3.2.2. From oscillations to loss of equilibrium
The nature of the perturbations that emanate from flares and cause the prominence disturbances has been the subject of numerous investigations. The perturbations propagate with velocities of the order of 1000 km s·l; however, the
spread is large, between less than 400 km s·l to more than 1800 km s·l. The
propagation takes place in the chromosphere or the corona at speeds much

THE DEATH OF PROMINENCES

255

greater than the velocity of sound, Vs' For comparison, Vs =23 km s-l for a temperature of T =2 X 104 K and in the corona Vs 200 km s-l, while the Alfven
velocity V A ranges from 1000 to 3000 km s-l. The perturbations are probably
hydromagnetic shocks generated during the impulsive phase of solar flares.
Such external disturbances like MHO blast waves cause the prominence to oscillate with eigenmode frequency due to the restoring forces; see, e.g., Kuperus and
Raadu (1974), van Tend and Kuperus (1978), and Vrsnak (1984).
We have seen in Chapter 5 that during the last several years increasingly
more sophisticated models of prominence magnetic fields have been analyzed.
Theoretical work on prominence oscillations have benefited from this development. Furthermore, Vrsnak (1993) has classified prominence oscillations
into membrane-type (e.g., horizontal oscillations of sheet-type prominences),
string-type (oscillations of the prominence axis in cylindrical, loop prominences), and sausage-type (oscillations of the magnetic flux tube supposed to
constitute the prominence).
From developments like these different oscillation modes triggered by
internal MHO instabilities have been studied in great detail (Joardar and
Roberts, 1992a,b; Oliver et al., 1992, 1993; see also Sakai and Nishikawa,
1983), and in some cases the observed oscillations may be accounted for in terms
of internal magnetoacoustic gravity waves (see Section 2.3). Periodic motions in
the photosphere under, or adjacent to, the prominence may also shake the
prominence (probably mainly the feet), and cause forced oscillations of the
prominence (Bashkirtsev and Mashnich, 1984; Balthasar et al., 1988). In these
cases the frequency is determined by the external source, e.g., the 5-min oscillation of the photosphere. Photospheric motions may in other cases lead to twisting of the flux tubes that make up a prominence structure and cause instability;
see Section 6.3.1.
There is little doubt that some of the many possible modes of oscillation of
a prominence at times can lead to serious destabilizations. However, by and
large, the disturbing agents we have looked at above can only accomplish a
fairly benign, temporary departure from equilibrium.
Other forces than the ones just considered may also cause prominences to rise
in corona and possible erupt. Yeh (1985, 1989) considered the effect of hydromagnetic buoyancy on a prominence, modeled as a magnetic flux tube in the
coronal field, and the model has been further developed by Yeh and Wu (1991),
who considered the rising motion of a prominence loop. Central to the treatment
is the assumption that the time scale for interaction between the two equivalent current systems is so long that the systems are not magnetically connected.
This magnetic separation is maintained by currents induced at the interface between the flux tube and the coronal field. There is a stress on the flux tube
caused by changes in the stress on the interface, and spatial gradients of this
stress create a force density that manifests itself as the hydromagnetic buoyancy force. This force is made up of three parts, viz. hydrostatic, hydrolOS

256

CHAPTER 6

dynamic, and magnetic pressures (without any shear stress). The three pressures then give rise to the hydrostatic buoyancy, the hydrodynamic buoyancy,
and the magnetostatic diamagnetic force, respectively. We have in Section
4.4.2 seen how the third part, the diamagnetic force, has been invoked in the
formation of prominences and spicules. Yeh (1989) and Yeh and Wu (1991) considered a helical flux loop in a bipolar coronal field. They showed that the
hydromagnetic buoyancy force, given by the surface integral of the ambient
hydromagnetic pressure over the interface, viz.
(6.30)
will overcome gravity when one strengthens the magnetic monopoles responsible for the source currents for the bipolar field. If there were no ambient
(coronal) field, the force would reduce to Fl = 1- psdS. With the magnetic part
present, large changes in the source currents will force the prominence flux tube
out into the corona in an eruption-like manner.
Of a particularly sinister nature for the stability of prominences are the
twisting or torsional oscillations observed in activated or erupting prominences
(Schmieder et al., 1985a; Vrsnak et al., 1989; Vrsnak, 1990a,b). These oscillations probably take place in the magnetic flux tubes that form the fine structure,
i.e., the skeleton, of the prominence. The oscillations observed in loops (Section
3.6.2.2) may be similar in nature. There seems, furthermore, to be a connection
with the helical fine structure often seen in quiescent prominences during their
disparition brusque phase (Section 3.6.1.4). This fine structure may exist-often
unnoticed-before the DB, and may be the location of various modes of oscillation, e.g., string-like oscillations of individual field lines, or sausage-like
oscillations of the whole flux tube. From such more violent oscillations there is
but a step to situations where plasma instabilities take over and dictate the
outcome.
Vrsnak (1990a,b) and Vrsnak et al. (1991) have studied in detail the
destabilization of cylindrical prominences, i.e., magnetic flux tubes, with helical fine structure, both from an observational and theoretical point of view.
Destabilization, leading to eruption, sets in when the twist in the fine structure
or the height of the helix exceeds critical values; see also Section 6.3.1. From
winking filaments to erupting helical structures we have progressed from mild
disturbances to bona fide destabilizations and loss of equilibrium.
Before we leave this discussion of destabilizing agents, we shall highlight
a few of them that often seem to play the dominant role, chief among them the
change in magnetic flux through the photosphere close to the prominence. In
most cases the change is observed as the appearance of new flux, but the disappearance (canceling) of flux and the motion of flux tubes-in the form of mag-

THE DEATH OF PROMINENCES

257

netic pores ("mini-sunspots")-also are involved (Martres and Soru-Escaut,
1977; Martres et al., 1982; Gesztelyi and Kondas, 1983; Simon et al., 1984, 1986a;
Schmieder et al., 1985b; Hermans and Martin, 1986; Apushkinskij, 1988; Raadu
et al., 1988; Uralov, 1989; Martin and Livi, 1992). Closely linked with the
change in flux are photospheric mass motions, e.g., shearing motions around the
flux tubes; see, e.g., Martin et al. (1983) and Wu et al. (1991). These motions may
pOSSibly lead to shears or twists in the adjacent fine-structure elements of a
prominence and result in the observed helical structure which, as we have seen,
becomes kink unstable if twisted too much.
Of particular interest in connection with erupting motions of prominences is
the case where the flux tube responds to shear motions by moving out into the
corona (see Section 5.5.3.2).
Wu et al. (1991) have considered such a case where a bipolar magnetic
field, simulating an arch filament, undergoes shear motion at the footpoints.
They found that the vertical plasma flow velocities grow exponentially leading to a global MHO instability (which they call dynamic shearing instability)
with growth rate about 3 aV A, where 1/a is a characteristic length scale and VA
is the average Alfven speed. Nonlinearity sets in, which would mean that the
arch filament would erupt.
A similar case was investigated by Roumeliotis et al. (1994), but their initial unsheared field was due to a dipole in the center of the Sun, and the fields
they studied are large coronal loops. Initially this field will expand steadily
as the footpoint displacements increase. Then, when the displacement exceeds
a critical value, the behavior of the evolving field suddenly changes to a new
regime, where the outward expansion of the loops becomes a much more rapid
increasing function of the footpoint displacement. Roumeliotis et al. identify
the transition to this new regime with the onset of eruptive solar phenomena.
See also Sakai and Koide (1992) who considered the destabilizing effect of
shear motions on a prominence modeled as a current sheet.
The details are not clear of how the emerging, or subsiding or moving, flux
interacts with the prominence supporting field and triggers the destabilization, but in the case of emerging flux a scenario has been involved for flare
triggering. Heyvaerts et al. (1977) developed a very interesting emerging flux
model in which the instability is started by the onset of turbulence as the
current sheet, which is formed between the new and old flux tubes, rises out of
the photosphere. Tur and Priest (1976) had already shown that such a current
sheet will form when a small dipolar flux system emerges and appears against
a larger dipolar field. It is also possible that certain prominence eruptions may
be attributed to this Heyvaerts et al. flare triggering mechanism, where parts
of the prominence-supporting magnetic field play the role of the larger dipolar
field in the flare scenario.
In a series of papers Moore and LaBonte (1980), Hagyard et al. (1984),
Moore and Rabin (1985), and Moore et al. (1984) developed an interesting model

258

CHAPTER 6

for the role of magnetic shear in prominence eruption and flare triggering; a
model that has several features in common with Heyvaerts et al. (1977) emerging flux mechanism. We shall briefly review this Moore et al. model to the
extent it pertains to prominence eruptions; in this case the eruption of active
region filaments.
Figure 6.2 shows the development of an erupting filament and the onset of a
two-ribbon flare, both caused by the reaction of the magnetic field to the destabilizing action of the shear. In the model the magnetic shear is concentrated
in the filament channel where the active region filament resides (Figure 6.2a).
Further away from the prominence the sheared field merges into a potential
coronal field rooted in the photosphere at some distance on either side of the
channel. Hagyard et al. (1984) enumerate several processes that might have
produced the magnetic shear, viz. shear flow in the photosphere in the filament channel, emergence of magnetic field (compare Heyvaerts et aI., 1977),
reconnection of field lines across the filament channel, and submergence of magnetic flux (compare van Ballegooijen and Martens (1989) and Rabin et al. (1984».
If the shear reaches an-as yet unspecified-critical limit, the field will
start to reconnect. This will first occur in the low-lying, strongly sheared, parts
of the field below the filament. When reconnection starts the overall field configuration becomes unstable as the field in and around the filament looses its
connection to the photosphere (Figure 6.2b). The result is further reconnection
below the filament, and reconnected field lines are expelled upward with the
prominence. If the eruption of the sheared field is violent enough to open the
overall closed coronal field above, the filament will be ejected in a disparition
brusque event. Subsequent reconnection of the opened coronal field will result in
the production of the two-ribbon flare normally observed after such ejections
(Figures 6.2c and 6.2d). The authors make the point that if the erupting
sheared field is not expelled very far into the outer corona, neither will the
filament be expelled; the overall field configuration then remains closed, and
there is, in this case, no final reconnection to provide further energy for an
extended, late flare phase.
It is becoming increasingly clear that shear motions, which-since we are
dealing with a high-~ plasma-imply sheared magnetic fields, play a
decisive role in the destabilizing of prominences. Athay (1990) showed that
active region filaments are found above long-lived shear lines, and Schmieder
et al. (1990) found significant shear (from Hex and C IV spectra) in a filament
prior to its eruption.
6.3.3. DYNAMIC DISPARITIONS BRUSQUES
The main features of a disparition brusque event are summarized in Section
3.6.1.4, and this often spectacular activity display has been the subject of many

THE DEATH OF PROMINENCES

a

b

c

d

259

Fig. 6.2. Sketch of the magnetic field configuration during a prominence eruption and the
accompanying two-ribbon flare; see text (after Hagyard et al., 1984).

reviews, see, e.g., Malherbe et al. (1983a), Soru-Escaut et al. (1985), Tang
(1986), Mouradian and Soru-Escaut (1989), Forbes (1990), Rompolt (1990), and
Demoulin and Vial (1992).

6.3.3.1. General characteristics
In his thorough review of DBs, Rompolt (1990) distinguishes two types of eruptions, viz. (when observed above the limb):
(i) prominences in the shape of a large arch with both ends remained
anchored in the chromosphere, and
(ii) prominences with only one end attached to the chromosphere while the
other end ascends and the prominence becomes nearly vertical.

260

CHAPTER 6

An example of the first type is furnished by the now classical case shown in
Figure 3.6, while an example of the second type is provided in Figure 6.3 (first
published by Kleczek and Hansen, 1962).
The significance of the distinction between the two types is not obvious, but
the different behavior may be due to the way the prominence destabilization
occurs. In the first type, that we may label a symmetrical DB, the interaction of
the supporting magnetic field with the global coronal field is somewhat different from the reaction of the prominence field to disturbances affecting
mainly one end of the structure, possibly via emerging flux, twisting photospheric motions, etc.
Tang (1986) has studied the eruption of filaments and also finds two types,
albeit with characteristics quite different from Rompolt's types. Tang distinguishes between: (i) prominences whose lower part remains intact above the
magnetic polarity inversion line, but whose upper layers separate and are
peeled off and then ascend into the corona, and (ii) the "classical" type of eruption where the whole promineIl;ce undergoes a disparition brusque. The first
type, whose lower part Tang refers to as the inversion-line filament, should be
studied further as it may reveal important information on the way the magnetic field can readjust during prominence activations.
It should be kept in mind that both quiescent prominences as well as active
region filaments can be destabilized and erupt: quiescent prominences in a
classical disparition brusque, active filaments in an eruption that is
disparition-brusque like, but that can be violent and lead to a spray; see Section
3.6.2.3.2.

6.3.3.2. Helical structure
Eruptive prominences frequently exhibit helical fine structure that has been
studied extensively; see, e.g., Rompolt (1975a,b, 1990), House and Berger (1987),
Moore (1988), Vrsnak et a!. (1988, 1991), Rusin (1989). The helical pattern
may be noticeable in the pre-eruption phase, although it is more discernible
after the eruption has been initiated. Such patterns can show up in a variety of
magnetic field configuration, as has been discussed, e.g., by Hirayama (1985)
and van Ballegooijen and Martens (1990). All these configurations can be
thought of as arising from an electric current along the prominence, which then
is represented by a twisted, helical, magnetic flux tube. Since matter should
follow the field lines if and when it moves, one should occasionally observe
rotational motions in such prominences, and such motions have been reported
(Liggett and Zirin, 1984; Schmieder et at., 1985a).
As the helical flux tube erupts its magnetic energy should decrease. Moore
(1988) considered such a twisted flux tube to find the dependence of its energy on
the field expansion and untwisting, and then evaluated the decrease in energy.

Fig. 6.3. Adisparition brusque event where one end of the prominence remains attached to the chromosphere; see text (courtesy HAO, Boulder,
Colorado).

-I

N
0\

-

en

n
tTl

Z

Z
tTl

E:

o

:=:l

"1:1

o"!1

~

otTl

tTl

:I:

262

CHAPTER 6

While the model is simple, it brings out the basic physics involved, and we
shall briefly discuss Moore's work.
The twisted flux tube has a length L, a circular cross section of radius r, and
magnetic field strength B. The energy of the magnetic field is, roughly, the
product of the magnetic energy density and the volume of the tube, i.e.,
(6.31)
If we consider the two orthogonal components of the magnetic field, one parallel, ~,' and the other perpendicular, Bol' to the axis of the tube, we can split the
magnetic energy into the two corresponding parts

(2 2) 2

1
E=EII+E.L""S1r
~I +B.L 1rr L.

(6.32)

The magnetic flux through the tube is
(6.33)
Equations (6.32) and (6.33) then give
(6.34)
The ideal MHO frozen-in flux condition (Equation (2.122» should hold, and the
flux F should remain constant as the tube expands and untwists. Consequently
the energy Eil must evolve as the ratio of the length to the cross sectional area
of the tube; i.e.,
(6.35)
The pitch of the twisted field lines is given by the ratio B.L I ~I' whence
Equations (6.32) and (6.35) give
(6.36)
Moore used this model in conjunction with observations of erupting prominences to estimate the decrease in energy in the untwisting erupting tube. First,

THE DEATH OF PROMINENCES

263

observations must yield an initial aspect ratio(rlL)o and the initial pitch
(B1J~I)O of the field in the tube. To do this Moore assumed that the erupting
field is traced by the observed filament material. Then, by following the eruptions, the quantities Bl./~l,rlro,LILo, etc. can be estimated and the change in
energy flE =Eo - E evaluated. As an example, for the eruption of a quiescent
filament in a disparition brusque event, viewed as an erupting helix, Moore
estimated a decrease in energy flE == 1()30 erg.
The twisted flux-tube model has been used in a number of theoretical
studies of erupting prominences. In an interesting analysis Priest and Forbes
(1990) considered a horizontal flux tube being expelled by magnetic pressure
forces. They approximated the tube either by a line current subjected to the
repulsive force due to an image line current below the photosphere, or by a
vertical current sheet and its image current sheet. The height and the velocity
of eruption could then be determined by the equation of motion for the flux tube;
see also the analysis of Srivastava et al. (1991), who considered the evolution
of a twisted force-free magnetic field in a model of prominence eruption.
A comprehensive and interesting model for prominence eruption and the
associated two-ribbon flare has been developed by Martens and Kuin (1989),
who built on previous work by Syrovatskii (1971), van Tend and Kuperus (1978),
and Kaastra (1985). Martens and Kuin consider the filament as a line current
and derive a circuit model for its eruption. The free magnetic energy in the system is concentrated in the prominence current, a current through an underlying
current sheet and in the appropriate return currents. Two circuits are involved,
one that of the filament and its return current, and the other the circuit of the
current sheet and its return current. Since these two circuits are coupled inductively, free energy originally stored in the filament is transferred to the
current sheet during the impulsive phase and rapidly dissipated. Of interest
for the prominence eruption is that a comparable amount of magnetic energy is
converted to kinetic energy of the ejected filament.
In Section 5.5.3.2 we have seen how twisted flux tubes have been invoked in
the models of prominences to form a dip for support of the material. If the twist
becomes too large, instability sets in and the flux tube may be ejected; see, e.g.,
Vrsnak (1990a,b); Hood (1991). The models are not detailed enough to predict
well the critical twist for expulsion of the flux tube to set in, but values of 2n-4n
are quoted. Hirayama (1985) notes that few examples are known from observations with twists greater than 2n. More work where theory is combined with
high spatial resolution observations is needed to firmly understand the physics
of twisted flux tubes and their applications to prominence eruption (e.g.,
Vrsnak,1990a,b; Vrsnak et al., 1991, 1993).

264

CHAPTER 6

6.4. Coronal Mass Ejections
As the mass of an eruptive prominence travels out through coronal material, one
would expect to see changes in the coronal structure, and a causal
"prominence~coronal change" scenario has been advocated. Great changes are
certainly observed in the corona at times when a spray or a disparition brusque
event takes place, but the connection between the two aspects of the activity is,
in general, not the simple causal one just mentioned. Before we consider the
relationship of eruptive prominences to changes in the corona, we shall first
discuss the major type of observed coronal change, the well-studied coronal
mass ejections (CMEs). Then, by exploring their relationship to erupting prominences, we may explain the spectacular way in which these prominences meet
their demise and be able to understand the physics behind the activity.
Short-lived changes in the brightness or structure of the corona have been
studied for many years and are often lumped together under the heading coronal
transients. The observations are made with a coronagraph recording the
electron-scattered light, the K-corona, and, therefore, give information on the
electron density; see Section 2.1.4. The transients may consist of fast changes in
the apparent filamentary fine structure of the coronal plasma, as, for instance,
in a coronal whip when one end of a loop structure suddenly undergoes a violent
release and whips outward. The velocity of the disturbance indicates the
action of a wave front rather than material ejection. Other types of coronal
transients entail structural changes or brightness increases in part of otherwise
long lasting coronal streamers that constitute some of the building blocks of the
corona. These events form a subset of coronal transients called coronal mass ejections. From an observational point of view a coronal mass ejection may be
defined as a change in the coronal structure that occurs on time scales up to
several hours and involves the appearance of new bright features seen in the Kcorona (Munro et al., 1979; Hundhausen et al., 1984; Howard et al., 1985). To the
extend that part of the ''new bright features" may be due to erupting prominence
material, a study of coronal mass ejections forms part of prominence research,
particularly as this applies to the disparition brusque phenomenon.
Coronal mass ejections have been observed from the ground for many years
and later, since the Skylab mission 1973, from space, and excellent reviews are
available (e.g., MacQueen, 1980; Dryer, 1982, 1994; Fisher, 1984; Hundhausen et
a1., 1984; Wagner, 1984; Howard et a1., 1985; Low, 1986; Hildner, 1986;
Hundhausen, 1988; Kahler, 1987; Harrison, 1991a,b). Figure 6.4 shows an example of a well-observed CME and the fate of the erupting prominence associated
with it. The prominence was situated under a big helmet streamer in the corona
(composite picture at 2247 UT). As the CME progressed, the streamer broadened
and expanded (1154 and 1215 UT), while part of the CME, in the form of a giant
arch-or bubble-quickly moved out (1215 UT) and disappeared from the field
of view. The prominence followed behind, revealing a typical disparition

THE DEATH OF PROMINENCES

265

Fig. 6.4. Development of coronal mass ejection and associated eruptive prominence (courtesy HAD,
Boulder, Colorado and A. J. Hundhausen).

brusque appearance, and broke up displaying its thread-like fine structure as it
moved out through the field of view. Following some CMEs one may see that
the coronal streamer involved is completely blown away; in other cases it is
merely pushed aside or only part of it has disappeared. A crucial pair of observations is that (a) the initiation of the CME takes place before the erupting
prominence takes off, and (b) the outward speed of the CME is often greater
than the speed of the ascending prominence material.

266

CHAPTER 6

A large number of investigations has been concerned with the question of
which solar activity manifestations most frequently are associated with the
occurrence of coronal mass ejections. It is illuminating to look at the appropriate
statistics to answer the question. According to St. Cyr and Webb (1991), who
looked at 73 CMEs in the time interval of 1984 to 1986, erupting prominences
were involved in 67% of the cases. Other forms of activity, e.g., flares, type II
and IV radio bursts, were less often observed. One may, therefore, conclude that
in the case of roughly two-thirds of observed coronal mass ejections the cause of
the ejection is to be traced to a physical process that is also responsible for the
prominence eruption.
The magnetic field in and around prominences in the corona exerts the dominating force that determines the dynamics of the plasmas involved. Therefore,
when a prominence is destabilized and erupts and, at about the same time, a
coronal mass ejection propagates outward in the corona, one is led to seek the
explanation in a common cause, viz. the reorganization (including reconnection)
of the global magnetic structure, one part of which has been the supporting
prominence field. We shall return to this global instability concept in the next
section.
The statistics quoted above (St. Cyr and Webb, 1991) tried to answer the
question concerning how often a CME is associated with an erupting prominence.
We can tum the question around and ask how often an erupting prominence is
associated with a CME. However, because disparition brusque events are often
observed without accompanying data on the state of the corona, few studies
have been done on this latter question. For one sample, Munro et a1. (1979) found
that the correlation with a CME was higher the greater the height to which
an erupting prominence could be traced. Of the 68 eruptive prominences studied
by Munro et a1. from the 1973-1974 Skylab period, those that reached altitudes
of at least 200,000 km were all associated with a coronal mass ejection; see
Figure 6.5. The percentage association fell to about 60 when considering all
prominences observed above 70,000 km. Finally, for small eruptive prominences
that could not be traced beyond 70,000 km, the association was less than 10%.
Using the information from Munro et al. and drawing on our previous discussion of prominence destabilization, we seem to be faced with two different
scenarios as to how quiescent and active region filaments disappear. Scenario
#1 would concern prominences that are locally destabilized and undergo minor
eruptions, and in Section 6.3 a number of ways to destabilize a prominence have
been considered. No associated CME would then be expected. Scenario #2 concerns prominences that are subjected to the effects of a global restructuring of the
coronal field, leading to prominence destabilization, a major eruption, and
instigation of a CME. We shall in the next section discuss the physical conditions that may be responsible for the postulated changes taking place in the
global magnetic field configuration.

267

THE DEATH OF PROMINENCES

60

N

40

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Height Above Limb (R®)
Fig. 6.5. Distribution of eruptive prominences rising above a given height and their association with
coronal transients (after Munro et al., 1979).

6.5. Global Magnetic Field Restructuring
We have seen how observational data on eruptive prominences and their
associated coronal mass ejections favor an explanation in terms of a common
cause. Nevertheless, since an ascending prominence or an impulsive flare
triggered locally would impart momentum and energy to the coronal plasma
above, we cannot rule out the possibility that events occur where a causal
relationship exists. Dryer (1994) advocates the view that there is a "spectrum"
of events from cases where the energy in a flare causes the CME to cases where
the energy is provided by the destabilized magnetic field.
6.5.1.

FROM LOCAL TO GLOBAL DESTABILIZAnON

Several models have been proposed to account for coronal mass ejections under
the assumption that the material is accelerated by a compression wave whose
energy derives from the explosive energy release in a flare or erupting prominence. In these compression-wave models both gas dynamic shocks (e.g., Parker,
1963c; Dryer and Jones, 1968; Burlaga, 1971; Hundhausen, 1972; Dryer, 1974) and
MHD shocks (Nakagawa et aI., 1978b; Nakagawa and Steinolfson, 1976;
Steinolfson and Nakagawa, 1976; Steinolfson et aI., 1978, 1979; Wu et al., 1978,

268

CHAPTER 6

1982; Dryer and Maxwell, 1979; Dryer et al., 1979) were considered. As a result
of these studies we have a good idea of how shock waves will interact with the
ambient coronal plasma and the solar wind (D'Uston et a1., 1981). However,
when one compares the details of the observed shape of CMEs with the shapes
predicted by the MHD or gas dynamic shock models, one often finds discrepancies, and those CMEs whose detailed shapes are well explained by the shock
wave models may be fairly rare (Sime and Hundhausen, 1987). Nevertheless,
our understanding of the response of the solar coronal plasma to explosive injection of energy is, to a large extent, due to the theoretical studies cited above.
From the standpoint of prominence physics the relative start times of a
prominence eruption and the associated CME pose a particular problem for the
erupting prominence-+CME scenario. In several well-observed cases the CME
seems to be already underway when the eruption (or flare) is triggered (e.g.,
Harrison, 1991a,b). Consequently, while there may be no theoretical objection
to an erupting prominence causing a coronal mass ejection, e.g., via the
compression-wave model, observations seem to require that we look for other
explanations in many cases.
In moving away from the compression-wave models for the cases where
erupting prominences are involved, several investigators have considered the
effect of a slow evolution of large coronal structures through metastable states
until destabilization or equilibrium breakdown sets in, leading to a CME (see
e.g., Low, 1990, 1993a; Steinolfson, 1991). The reason for this destructive evolution of the coronal structures, especially coronal streamers, may be motion of
loop footpoints (Low et al., 1982), emerging magnetic flux (Steinolfson, 1992;
Guo et al., 1992), changes in magnetic arcades (Inhester et a1., 1992), or shear of
field lines across neutral lines (Steinolfson, 1991; Wolfson and Low, 1992). We
recognize many of these destabilizing agents from our study of the breakdown of
prominence stability (Section 6.3.2), and we are again pointed in the direction
of seeking a common cause for erupting prominences and their associated CMEs.
The idea that a prominence eruption, like a disparition brusque event, or a
flare and the associated coronal transients of the CME variety are both due to a
common cause, viz. large-scale magnetic field rearrangements, has been advocated or alluded to by several authors (see, e.g., Tandberg-Hanssen et a1., 1980;
Fisher et al., 1981; Low et al., 1982; Pneuman, 1984; Rompolt, 1984; Illing and
Hundhausen, 1986; Hundhausen, 1988; Kahler et al., 1988; St. Cyr and Webb,
1991; Demoulin and Vial, 1992; Smith et a1., 1992; Hiei et a1., 1993;
Hundhausen,1993).
Few detailed analyses are available of how this global rearrangement
may take place so that both a prominence filament will be ejected and, more or
less simultaneously, a CME can be expelled. An interesting approach to this
problem has been presented by Low (1994), and even though observational confirmation of some of the consequences of his model is lacking, we shall briefly
discuss the physics involved.

THE DEATH OF PROMINENCES

269

6.5.2. Low's MODEL
Low (1993b, 1994) considered an inverse polarity model (Section 3.5) for a quiescent prominence sheet in its usual location under a helmet streamer with its
cavity; see Figure 6.6. Central to Low's model is that the cavity contains, or consists of, a twisted magnetic flux rope oriented in the azimuthal direction around
the axis of the streamer. In Figure 6.6 the field lines of this flux rope would be
perpendicular to the plane of the figure and project onto the dosed lines indicated in the cavity. Low suggests that the twisted flux rope had its origin
below the photosphere and survived the transport through the turbulent high~ plasma in the form of some net magnetic helicity (Section 5.5.3.2). In this
scenario the filament channel can be taken as the manisfestation in the chromosphere of the flux rope. To affect the transport upward of the flux rope, Low
applied the theory of Taylor relaxation for laboratory pitch devices (Taylor,
1974, 1986). This relaxation theorem states that for a perfect electric conductor
the magnetic helicity, I A . BdV, is conserved in any volume V bounded by a magnetic flux surfaces. A is the magnetic vector potential and B the magnetic field
strength. We notice that Norman and Heyvaerts (1983) and Heyvaerts and
Priest (1984) had previously suggested the application of the conservation of
helicity to solar plasmas, and Rust (1994) has studied the removal of helicity
associated with mass ejections related to flux tube eruptions. Low now assumes
that the magnetic helicity is sufficiently conserved to accomplish the transport
of the flux into the cavity above the prominence. The resulting flux rope is
force-free and exerts no force, except at the prominence sheet. It is the attraction between the electric current flowing in the prominence and the current of
the force-free flux rope above it that provides the support for the prominence
material against gravity in Low's model. Conversely, the prominence may be
considered the anchor for the cavity flux rope, which then provides the equilibrium conditions for the whole helmet streamer. Over time helicity will be
accumulating in the cavity as more and more magnetic flux is ejected from
below, and Low (1994) suggests that the logical way for the corona to get rid of
the helicity and the trapped energy is by ejecting the helmet streamer as well
as the prominence as an ideal MHO process. The mass ejection opens the coronal
magnetic field where little or no helicity remains. Low's model is interesting in
that it provides a common process for the erupting prominence and the CME,
and it ties the process to a global change in the coronal magnetic field. We also
notice the important role played by the coronal cavity and arcades.
6.5.3. THE DISPARITION BRUSQUE REVISITED
Our understanding of the underlying cause of a dynamic disparition brusque
event, the most spectacular of prominence activities, is still wanting, but new

270

CHAPTER 6

Prominence
Sheet

<

SOLAR WIND

~

Fig. 6.6. Sketch of helmet streamer with prominence, coronal cavity, and magnetic field lines (after
Low, 1994).

THE DEATH OF PROMINENCES

271

insight, both from observation and theory, is gradually filling in the gaps. If
the prominence is not pushing the magnetic field lines out as it erupts, the
opening of the field that takes place as the accompanying coronal transient is
ejected may be due to the instability in the field itself. However, Aly (1991)
pointed out a basic problem in this context; viz. the energy in an open field
configuration exceeds the energy of any closed configuration, and this situation
holds for potential as well as for force-free fields. A way around this difficulty
was pointed out by Low and Smith (1993) who found that a helmet streamer
may have enough free energy to open the field if cross-field currents flow in a
magnetic bubble within the core of the streamer. It is interesting that such an
arrangement may be accommodated in Low's model, presented in Section 6.5.2,
where the bubble could reside in the coronal cavity above the prominence. More
work needs to be done on this aspect of the prominence eruption and its association with a CME, but we may be closer to a correct interpretation when we say
that as the magnetic field becomes unstable it opens and lifts the prominence up
and out in a dynamic disparition brusque event.
We are again reminded of the seemingly close connection that must exist
between the prominence and the accompanying coronal cavity and arcades.
Only recently has it become more easy to observe the arcades in X-rays, and a
beautiful example of a disparition brusque event and hot coronal structures is
shown in Figure 6.7. The prominence is seen in Figure 6.7a on the east limb; the
coronal structures are seen in Figure 6.7b, which also reveals arcades in other
locations on the Sun at that time.
With the opening of the magnetic field we encounter a configuration with
field lines running in opposite direction in the corona above the location of the
erupted prominence, and a current sheet forms there. One would then expect
reconnection to take place /across the sheet (compare, e.g., the situation
depicted in Figures 4.8 and 4.10). If this is the case, two interesting consequences
may follow, viz. particle acceleration and reformation of the prominencehelmet streamer configuration.
Evidence of electron acceleration (see Section 4.4.3.1) comes from observation of radio bursts associated with erupting prominences. Gopalswamyand
Kundu (1989) observed a moving type IV burst associated with a filament
eruption. Type IV bursts are interpreted as gyro synchrotron emission from a
plasmoid containing a magnetic field and non-thermal (i.e., accelerated) electrons. Chiuderi Drago et al. (1986) have studied the association of type III
radio bursts in the corona and ejection of chromospheric material. The electrons
in the corona responsible for the radio emission may be accelerated in several
ways of which a catastrophic field reconnection is one.
Zodi et al. (1988) observed millimeter and centimeter radio emission
following a disparition brusque event seen on the limb, and they interpreted the
emission as gyrosynchrotron radiation from non-thermal electrons trapped in
closed magnetic loops formed after the prominence ejection. Also particleS' nf

272

CHAPTER 6

Fig. 6.7a. Eruptive prominence observed in Ha at 0128 tIT (courtesy Norikura Solar Observatory,
Japan).

lower energy (35-1600 keY) seem to originate in the disparition brusque phase,
after which they may be detected by satellite-borne instrumentation (Sanahuja
et al., 1991).
We seem to be drawn to the conclusion that reconnection of magnetic
fields plays the crucial role in the development of prominences. This physical
process, whose details may still be obscure, enters prominently in the theory of
loop prominence formation as advocated, e.g., in the Forbes and Malherbe
model (Section 4.3.2.1), it may be at the root of surge prominence ejections, and
we find it in the course of the development of quiescent filaments. In this latter
case, if and when reconnection of the magnetic field takes place, conditions may
then again be ripe for the formation of a new streamer and a new prominence, a
scenario advocated by Hiei et al. (1993). We have then come full circle from

THE DEATH OF PROMINENCES

273

N

I

JuI 30-31, 1992

Soft X-ray (YOHKOH SXT )

Fig. 6.7b. Soft X-ray images observed by the YOHKOH satellite at the time of the disparition brusque
event of Figure 6.7a (courtesy NAO], ISAS, and NASA).

the formation of a quiescent prominence in a filament channel under a coronal
cavity in a helmet streamer, through its mature stable phase, to its destruction
as a disparition brusque, a process that nevertheless harbors the possibility for
the birth of a new generation quiescent prominence.

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296

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AUTHOR INDEX

Acton, L. W. 178,240,275,286
Akhiezer, A. Z. 77, 275
Akhieze~I.)\. 77,275
)\lissandrakis, C. E. 109, ISS, 234, 258,
280, 291, 293
)\Iy, J. J. 139, 271, 275
)\mari, T. 139, 200, 205, 206, 207, 275, 290
)\mbastha, A 213, 263, 276, 292
)\n, C.-H. 113, 139, 146, 147,249,275,295
J\nderson, C. F. 251, 275
)\ndreassen, 0.175,283
J\ngle, K. L. 251, 292
)\ntalova, A 257, 286
)\ntiochos, S. K. 152, 208, 257, 275, 278,
290, 291, 295
)\ntonucci, E. 178,275
)\nzer, U. 93, 96, 101, 195, 196, 198, 199,
200, 201, 203, 205, 206, 207, 208, 209,
210, 220, 221, 222, 223, 235, 244, 248,
275, 281, 289, 293
)\pushkinskij, C. P. 171,257,275
Mmstrong, T. P. 163,275,277
Mtzner, C. 84, 85, 97, 294
)\thay, R. C. 27, 85, 97, 98, 156, 157, 175,
178, 223, 224, 251, 258, 275, 276, 284,
285, 289, 293
)\uer, L. H. 27, 288
)\vignon, Y. 114,276
)\vrett, E. H. 29, 279
Babcock,H. [). 13, 114,276
Babcock,H. W. 13,114,276
Bakhareva, N. M. 211, 276
Ballester, J. L. 104, 222, 255, 276, 288, 294
Balthasar, H. 99, 255, 276, 295
Bao, J. J. 113, 146, 147, 275, 295
Bartoe, J.-[).F. 89, 90, ISS, 175, 177, 276,
278, 283, 288
Bashkirtsev, V. S. 14,99,255,276
Bastian, T. S. 90, 276
Baur, T. C. 14, 48, 276
Becker, U. 251, 276
Beckers, J. M. 48, 154, 232, 276
Berge~M.)\.260,282

Bernstein, I. B. 244, 246, 248, 276
Bhatia, A K. 89, 278
Bhatnagar, A 213, 263, 276, 292
Bhatnagar, P. L. 183, 185, 193,245,276
Bilimoria, R. 142, 148, 206, 214, 215, 216,
217, 218, 286
Billings, [). E. 31, 34, 185, 276, 288
Birn, J. 205, 206, 210, 268, 276, 282
Bobrowsky, M. 172,291

Bogdan, T. 114,205,290
Bohlin, J. [). ISS, 276
Bommier, V. 14, 54, 90, 93, 276, 284, 285,
290
Boris, J. P. 135, 136, 288
Bracewell, R. N. 170,289
Braginski, S. I. 129, 276
Brajsa, R. 98, 101, 168, 169, 171, 256, 260,
276,294
Bray, R. J. 107, 285
Breit, C. 51, 276
Brown, A 185, 186, 190, 276
Brown, J. C. 177, 277
Browning, P. 200, 205, 206, 275
Brueckner, C. E. 89,90, 155, 175, 177, 276,
278, 283, 288
Bruner, E. C. 90, 97, 277, 285
Bruner,M. E.240,286
Bruzek, A 98, 108, 109, 114, 168, 250, 251,
276, 277, 286
Brynildsen, N. 81, 173, 279
Buhl, [). 172,277
Bumba, V. 168,250,251,277
Burgers,J. M. 129,277
Burlaga, L. F. 267, 277
Butz, M. 178,277,284
~akmak,H. 108,288
Califano, F. 181,277
Callebaut, [). K. 248, 277
Cally, P. S. 146,290
Canfield, R. C. 41, 277
Carbone, V. 238, 277
Cargill, P. J. 146,153,277
Carmichael, H. 138, 277
Cartledge, N. 206,277
Celsius, A 11,277
Chandrasekhar, S. 22, 46, 184, 277
Chapman,S. 57,277
Chen, C. 163, 277
Cheng, c.-c. 84, 90, 182,277
Chipman, E. C. 97, 285
Chiuderi [)rago, F. 14, 171, 172, 175, 178,
181, 211, 223, 238, 271, 277
Chiuderi, C. 139, 141, 178, 181, 277, 279
Choe, C. S. 147,277
Cook,J. W.85, 155,278,283
Condon,E. V.43, 277
Correia, E. 271, 296
Costa, J.E.R. 271, 296
Cowling, T. C. 57, 189, 277
Cox, [). P. 128, 136, 277
Craig, I.-J.[). 146, 177,277,286

298

THE NATURE OF SOLAR PROMINENCES

Cui, L.-S. 96, 277
Culhane, J. L. 178,275
d' Azambuja, L. 97, 101, 104, 168, 242, 278
d'Azambuja,M.97, 101, 104, 168,242,278
D'Uston, C. 268, 278
Dahlburg, R. B. 208, 275, 278
Davis. J. M. 115, 119, 139, 278, 287
de Bruyne, P. 249, 278
de Jager, c. 16, 105, 183, 278
Decker, R. B. 163,275
Delone, A. 107, 278
Demoulin, P. 96, 109, 113, 137, 146, 173,
196, 200, 205, 206, 208, 210, 217, 219,
258, 259, 268, 275, 278, 291, 292
Dere, K. P. 89, 109, 155, 175, 177, 258, 278,
291,295
Deslandres, H. 12, 15, 278
DezsO, L. 108,257,283,286
Dodson-Prince,H. VV.251,278
Domingo, V. 272, 290
Doschek, G. A. 89, 278, 279
Doyle, J. G. 175, 178,275,289
Drake,J. F. 147,278
Dryer, M. 78, 208, 257, 264, 267, 268, 278,
292,295
Dungey,J. VV. 163,185,190,245,278
Dunn, R. 96, 278
Dupree, A. K. 175,278
Dyson, F. 99, 279
Dzubur, A. 98, 101, 260, 294
Echols, C. R. 216, 286
Edberg, S. J. 99, 284
Einaudi, G. 98, 101, 260, 277, 278, 279
Ellerman, F. 12, 15, 280
Elmore, D. E. 14,276
Emslie, G. A. 105,293
Engvold, O. 81, 96, 97, 99, 110, 111, 146,
173, 174, 175, 178, 179, 181, 182, 211,
223,277,279,295
Ermakova, L. V. 121,285
Evans,J. VV. 16, 113,287
ElVell,M. VV.90,276
ElVing, J. VV. 155,278
Fang, C. 26, 81,96,277,279,295
FArnik,F.I07,279,292
Feldman, U. 89, 278, 279
Felli,M.172,277
Fermi, E. 159, 279
Ferraro, V.C.A. 185, 279
Fiedler, R.A.S. 197,222,279
Field, G. B. 113, 123, 279
FiIipolVski, S. 27, 29, 279

Finn,J.M.207,279
Fisher, R. R. 109, 264, 268, 279, 285, 286
Fontenla, J. M. 27, 28, 29, 30, 31, 90, 91,
104, 107, 174, 279, 292, 294
Forbes, T. G. 104, 113, 137, 138, 139, 234,
240, 241, 242, 259, 263, 278, 279, 280,
285,289
Forbush,S.E.158,284
Forman, M.A. 163,280
Foukal, P. 41, 84, 114, 119, 155, 214, 247,
280, 295
Frieman, E. A. 244, 246, 248, 276
Furst, E. 172, 175, 178, 277, 284, 289
Furtb,H.P. 132, 141,280
Gabriel, A. H. 178, 275
Gaizauskas, V. 98, 280
Galindo-Trejo, J. 245, 280
GallolVay, D. 157, 281
Garcia, C. J. 268, 279
Garczylu;ka, I. N. 110,280
Gary, D. E. 14, 280
Georgakilas, A. A. 234, 280
Gerlei, O. 108, 151, 291
Gesztelyi, L. 257, 280, 286, 289, 292
Gilliam, L. 41, 247, 280
Giovanelli, R. G. 108, 178,250,280
Godoli, G. 115, 291
Gold, T. 185,280
Golub, L. 152, 291
Goossens, M. 234, 294
GopalslVamy, N. 271, 280
Gopasyuk, S. I. 108, 113, 280
Gosling, J. T. 264, 266, 267, 287
Gouttebroze, P. 26, 27, 28, 29, 30, 31, 84, 85,
97,174,229,230,231,232,280,281,294
Grad, H. 185, 280
Grant, R. 11, 280
Gu, X.-m. 107,280
GunkIer, T. A. 41, 277
Guo, VV. P. 268, 280
Guzdar, P. N. 207, 279
Hagyard, M. J. 78, 119, 257, 258, 259, 280,
287,289,294,295
Hale, G. E. II, 15, 280
Han, S. M. 267, 268, 278, 288, 295
Hanaoka, Y.9O, 109, 182,280,284
Hanle, VV. 50, 51,280, 281
Hansen, R. T. 103, 109, 168, 260, 268, 281,
283,293
Hansen,S.F.168,281
Harris, D. L. 38, 281
Harrison, R. A. 264, 268, 281

AUTHOR INDEX

Harvey, J. 13, 15, 45, 48, 93, 94, 114, 169,
281,282
Harvey, K. L. 240, 257, 286, 294
Hathaway, D. H. 119,294
Hayakawa, S. J. 160, 281
Heasley, J. N. 225, 226, 227, 228, 281
Hedeman, E. R. 251,278
Heinzel, P. 26, 27, 28, 30, 90, 91, 107, 228,
229, 230, 231, 232, 280, 281, 295
He~enberg, VV.52,281
Henze, VV. 90, 277
Heras, A. M. 272, 290
Hermans, L. M. 122,257,281
Hesse, M. 205, 210, 268, 282
Heyvaerts, J. 113, 123, 139, 257, 258, 269,
281, 285, 288
Hiei, E. 14, 172, 240, 268, 272, 281, 286
Hildner, E. 113, 124, 125, 126, 128, 132,
264, 266, 267, 268, 281, 287, 292
Hinata, S. 246, 281
Hirayama, T. 81, 88, 91, 95, 111, 139, 173,
260,263,279,281
Hirth, VV. 175, 178, 277, 284
Hollweg, J. V. 157,281
Holtsmark, J. 40, 281
Hood, A. VV. 139, 196, 197, 198, 200, 201,
205, 206, 207, 222, 235, 246, 248, 249,
255, 263, 275, 277, 278, 279, 281, 282,
285, 288, 289
House,L. L. 14,50,53,260,264,276,282
Howard, R. A. 264, 282
Howard, R. F. 114, 168, 169, 250, 251, 277,
282,293
Hu,J. 81, 96,277,295
Hu, Y. Q. 78, 79, 282, 288, 295
Huang, Y.-R. 96, 277
Huber, M.C.E. 155, 295
Hudson, P. D. 163,282
Huggins, VV. 281,282
Hull, H. K. 14, 276
Hundhausen, A. J. 264, 267, 268, 272, 281,
282,291
Hunter, J. H. 125,282
Hurford, G. J. 257,287
Hyder, C. L. 13, 50, 53, 99, 168, 251, 282,
294
Illing, R.M.E. 264, 268, 282
Inglis, D. R. 41, 282
Inhester, B. 205, 210, 268, 282
Ioshpa, B. A. 13, 94, 200, 201, 239, 282
Ishiguro, M. 14, 172, 281
Ivanov-Kholodny, G. S. 88, 282

299

Jackson, S. 157, 281
Jaffee, D. T. 155, 295
Janssen, P. J. 12,282
Jefferies, J. T. 25, 39, 89, 113, 158, 163, 282
Jensen, E. 111, 113, 129, 146, 148, 173, 181,
182, 211, 223, 239, 277, 279, 282, 290
Jette, A. D. 185, 283
Ji, G.-P. 96, 277
Joardar, P. S. 255, 283
Jones, D. L. 267, 278
Jones, H. P. 97, 257, 276, 287
Jordan, c. 175,283
Joselyn, J. A. 240, 272, 290, 294
Kaastra, J. S. 105,263,283
Kahler, S. VV. 264, 268, 283
Kalikhman, L. E. 78, 283
Kan~S.R.257,268,283,287

Kanno, M. 91, 283
Karlicky, M. 91, 281
Kaufmann, P. 271, 296
Kawaguchi, I. 114,283
Keil, S. L. 96, 99,182,279,295
Kendall, P. C. 184,277
Kenny, P. J. 90, 277
Khangil'din, U. V. 171,283
Khodachenko,M. L.211,276
Khoklova, V. L. 16, 101, 148,239,291
Kiepenheuer, K. O. 104, 113, 150,283
Killeen, J. 132, 141, 280
Kim,I.S.14,93,280,283,288
Kippenhahn, R. 93, 185, 189, 194, 200, 220,
221, 245, 283
Kiryukhina, A. I. 96, 283
Kitai, R. 152, 157, 291, 292
Kjeldseth-Moe, 0.85,175, 177, 283, 288
Kleczek, J. 99, 108, 126, 127, 251, 253, 260,
283
Kleczekova, H. 108, 283
Klimchuk, J. A. 208, 240, 275, 283, 294
Koide, S. 257, 290
Knolker, M. 99, 276
Kobanov, N. I. 99, 276
Kogut, J. A. 172,292
Kojima, M. 240, 294
Kondas, L. 257, 280
Koomen, M. J. 264, 282
Kopp, R. A. 138, 139,283
Kosugi, T. 14, 172,281
Kot, C. H. 78, 291
Koutchmy, S. 14, 30, 93, 95, 174, 283, 288,
296
Kovacs, A. 108, 283

300

THE NATURE OF SOLAR PROMINENCES

Kozuka, Y. 240, 294
Krall, K. R. 78, 295
Krall, N. A. 248, 283
Kramers, H. A.33, 283
Krieger, A. S. 114, 115, 139, 278, 286, 293
Krook,~. 183, 185, 193,245,276
Kruskal, ~. D. 244, 246, 247, 248, 276, 284
Kubota, J. 96, 284
Kucera, A. 257, 286
Kuin, N.P.~. 26, 263,284,286
Kulidzanishvili, V. I. 96, 284
Kulsrud,R.~.244,246,248,276

Kumar, A. 148,290
Kundu, ~. R. 14, 170, 172, 178, 271, 277,
280, 284, 291, 296
Kuperus, ~. 93, 99, 127, 130, 131, 133, 134,
137, 156, 198, 199, 200, 234, 251, 253,
255,263,283,284,289,294
Kurochka, L. N. 41, 284
Kurokawa, H. 90, 109, 182,280,284
Kusoffsky, U. 109, 288
Kvicala, J. 108, 283
LaBonte, B. J. 155, 168, 257, 284, 287, 293
Landi Degl'Innocenti, E. 14,54,284
Landman,D.A.89,91,99,284
Laney, C. D. 99, 284
Lang, K. R. 172, 284
Lange, I. 158, 284
Lantos, P. 172, 175, 178, 277,284,289
Larmore, L. 96, 284
Lee, L. C. 147,277
Lee,R.H.14,276,284
Leibacher, J. W. 178, 275
Leighton, R. B. ISS, 292
Lemaire, P. 84, 85, 97, 294
Lerche, I. 226,245,284
Leroy, J.-L. 14,48,50,54,90,92,93, 111,
114, 168, 173, 174, 214, 276, 279, 284,
290
Li, Q.-s. 107,280
Liang, B. X. 268, 280
LiggeH,~.98,26O,285

Lin, J. 107, 280
Linhart, J. G. 253, 285
Lindsay, R. B. 254, 285
Lites, B. W. 97, 285
Livi, S.H.B. 122, 257, 286
Livingston, W. C. 26, 81, 97,279, 295
Livshitz, ~. A. 143, 289
Lockyer, J. N. 12, 285
Loden, K. 108, 285
Loeser, R. 29, 279

Longbottom. A. W. 249, 285
Longcope, D. W. 248, 285
Loughhead,R.E.I07,285
Low, B. C. 98, 148, 207, 221, 222, 226, 235,
245, 264, 268, 269, 270, 271, 276, 284,
285, 295
Lundquist, S. 244, 285
Liist, R. 127, 185, 285
Lykoud~,P.S.12~288

Lyot, B. 13, 285
~cCombie, W. 151, 290
~achado,~.E.90,91, 178,275,292
~acQueen,R.~.264,266,267,285,287
~akarova, E. 107,278

257, 286
V. P. 121, 285
J. ~. 96, 97, 98, 99, 101,
105, 113, 123, 126, 137, 138, 139,
192, 194, 196, 205, 208, 210, 217,
233, 234, 240, 241, 242, 256, 257,
260, 278, 280, 285, 287, 289, 291
~altby, P. 175, 283
~alville, J. ~. 13, 97, 98, 107, 200,
216, 217, 234, 251, 279, 285, 287
~akita, ~.
~ksimov,
~alherbe,

~ango,S.A.85,283
~ariska,
T. 113, 135,

J.

288, 289

~arquette, W. H.
~artens, P.C.H.

104,
154,
219,
259,
201,

136, 142, 147, 235,

142, 148,216,286
200, 204, 205, 208, 257,
258,260,263,285,294,295
Martin, S. F. 90, 91, 105, 107, 109, 114, 122,
142, 148, 167, 206, 214, 215, 216, 217,
218, 257, 268, 271, 281, 286, 292, 293,
296
~artres, ~.-J. 96, 104, 108, 114, 115, 240,
241, 243, 257, 259, 276, 286, 287, 291,
292
~shnich, G. P. 14,99, 255, 276
~slennikova, L. B. 41, 284
~ason, H. E. 89, 175, 177, 278
~ssey, H.5.W. 42, 158, 286, 287
~axwell, A. 267, 278
~cAllister, A. 240, 286
~cCabe,~.K. 108, 109,280,286
~cClymont, A. N. 146,286
~cIntosh, P. S. 114, 115, 168,286
Mc~th, R. R. 108, 286
~cNamara, E. 185, 288
~ein, N. 30, 81, 96, 97, 104, 174, 233, 238,
257, 259, 271, 277, 285, 286, 287, 289,
292

AUTHOR INDEX

Mein, P. 30, 81, 85, 90, 91, 96, 97, 99, WI,
104, 107, 108, 109, 139, lSI, 152, 153,
154, 174, 205, 233, 234, 238, 257, 259,
281, 285, 286, 287, 289, 291, 292
Melville, J. P. 249, 285
Menzel, D. H. 16, 113, 183, 185, 193, 245,
276,287
Meyer, F. 143, 251, 287
Michard, R. 47, 114, 286, 287
Michels, D. J. 264, 281
Mihalas, D. 27, 225, 226, 227, 228, 281, 287
Miller, P. 41, 247, 280
Milne, A. M. 217, 221, 227, 287
Mogilevskij, E. J. 107,287
Moiseiwitsch, B. L. 42, 286
Mok, Y. 139, 147, 234, 278, 294
Molodensky,M. M. 185,287
Montesinos, B. 146, 287, 293
Moore, R. L. 119, 168, 180, 257, 260, 268,
283, 287, 289, 293
Moreton, G. E. 251, 275, 287
Morgan, F. J. 175, 295
Moriyama, F. 257, 286
Morozhenko,N.N.27,28,287
Mott, N. F. 158,287
Motta, S. 115, 291
Mouradian, Z. 104, 115, 168, 240, 241, 243,
259, 287, 291, 292
Munro, R. H. 264,266,267,268,285,287
Nakagawa, Y. 78, 79, 128, 185, 200, 201,
216, 217, 234, 267, 287, 288, 289, 292,
295
Nakagomi, Y. 81, 281
Neidig, D. F. 90, 91, 292
Newcomb, VV.A.248,288
Newkirk, G., Jr. 114, 288
Newton, H. VV. 16,96,98,288
Ni, X.-B. 96, 277
Nicholls, R. VV. 175,295
Nicolas, K. R. 175, 177, 283, 288
Nikiforova, T. P. lOS, 288
Nikolsky, G. M. 14,288
Nishikawa, K.-1. 255, 290
Nishikawa, T. 152, 157, 291, 292
Nishimura, H. 160,281
Noens, J.-c. 30, 96, 174,287
Nolte, J. T. 114, lIS, 286
Norman, C. A. 269, 288
Novocky, D. 228, 288
Noyes,R. VV.91, 155,283,295
Obayashi, H. 160,281
Obridko, V. N. 47, 53, 288

301

9gir, M. B. 108, 280
Ohman, Y. 98, 109,201,288
Ohnishi, Y. 99, 293
Ohyama, M. 240, 294
Okamoto, T. 81, 281
Okten, A. 108,288
Oliver, R. 255, 288
Olson, C. A. 127, 288
Oran, E. S. 135, 136,288
Orrall, F. Q. 27, 89, 113, 131, 158, 163, 164,
175, 239, 282, 288, 291
Orwig, L. E. 178, 275
Osherovich, V. 221, 288
Oster, L. 33, 34, 288
Osterbrock, D. E. 156, 288
Paletou, F. 27, 288
Pande, M. C. 121, 291
Papagiannis, M. D. 172,292
Parker, E. N. 113, 123, 125, 130, 148, 154,
155, 156, 163, 267, 288
Pawsey, J. L. 170, 289
Pesses, M. E. 163,275
Pettit, E. 15, 96, 108, 286, 289
Pick, M. 114, 238, 271, 276, 277
Pikel'ner, S. B. 113, 143, 144, 157, 289
Pirronello, V. lIS, 291
Plocieniak, S. 216, 289
Plumpton, C. 185, 279
Pneuman, G. VV. 138, 139, 153, 200, 204,
268, 277, 283, 289
Pohjolainen, S. 168, 169, 276
Pojoga, S. 174,289
Poland, A. I. 26, 82, 90, 96, 104, lOS, 113,
142, 147, 173, 174, 205, 207, 221, 223,
235, 256, 260, 264, 266, 267, 279, 284,
287, 289, 291, 292
Poletto, G. 90, 91, 277, 292
Polovin, R. V. 77,275
Porfir'eva, G. 107,278
Pottasch, S. R. 146, 176, 289
Prata, S. VV. 114, 289
Priest, E. R. 71, 111, 113, 123, 126, 130, 137,
138, 139, 146, 173, 192, 194, 196, 199,
200, 201, 205, 206, 207, 208, 216, 217,
219, 221, 222, 227, 234, 235, 246, 248,
255, 257, 258, 263, 269, 275, 276, 277,
278, 279, 280, 281, 282, 285, 287, 288,
289, 290, 292, 293
Purcell, J. D. 155,276
Querfeld, C. VV. 14,276
Raadu, M. A. 93, 98, 101, 109, 135, 137,
154, 185, 198, 200, 208, 210, 233, 234,

302

THE NATURE OF SOLAR PROMINENCES

248, 254, 257, 258, 278, 284, 288, 289,
291
Rabin, D. 146, 175, 180, 258, 287, 289, 290
Rachovsky, D. N. 47, 289
Raizer, Yu.P. 239, 295
Raju, P. K. 126, 128, 132, 134, 289
Rarnsey,H. E. 99,251, 287, 289
Raoult, A. 172,284,289
Rapley, C. G. 178, 275
Ratier, G. 14, 54, 285, 289
Raymond, J. C. 175,289
Rayrole, J. 257, 286
Redcoborody, Y. N. 247, 290
Reeves,E. M. 155,295
Rei~,E.J.27,29,97,279

Flibes,E. 139, 146,285,290
Flicchiazzi, P. J. 41, 277

Flid~ay,C.205,206,207,290

Robb, T. D. 146, 290
Roberts, B. 217, 221, 227, 255, 283, 287
Rogers, S. R. 14, 276
Rompolt, B. 27, 98, 101, 113, 114, 201, 205,
216, 256, 259, 260, 263, 268, 276, 281,
289,290,294
Rosa, D. 263, 294
Roschina, E. 107,278
Rosenberg, F. D. 89, 279
Rosenbluth,M.N. 132, 141,280
Ross, C. L. 264, 266, 267, 287
Rosseland, S. 22, 24, 33,129,239,290
Rottman, G. J. 97, 285
Roumeliotis, G. 257, 290
Rovira, M. 27, 28, 29, 30, 31, 174, 279, 294
Roy, J.-R. 95, 108, 153,290
Rubin, H. 185, 280
Ruedi, I. 146, 290
RU§in, V. 101, 260, 290
Rust, D. M. 13, 108, 148, 151, 206, 257, 258,
269,281,284,290
RuMjak,V. 98, 101, 168, 169, 205, 256,
260, 263, 276, 294
Saha, M.N. 21,290
Sahal-Brechot, S. 14, 50, 54, 90, 93, 276,
284,290
Saito, K. 119, 290
Saito, S. 90, 182, 280
Sakai, J. I. 211, 255, 257, 290
Sakurai, T. 248, 290
Sanahuja, B. 272, 290
Sasorov, P. V. 146,290
Sato, H. 160, 281
Sauerwein, H. 77, 290

Sawyer, C. R. 264, 282
Schatzman, E. 163, 185,290
Schindler, K. 206,245,280
Schindle~M.98, 107,276,285
Schluter, A. 93, 113, 148, 150, 154, 185,
189, 194, 200, 220, 221, 245, 283, 285,
290
Schmahl, E. J. 27, 104, 115, 152, 155, 172,
175, 240, 241, 267, 288, 290, 291, 292,
295
SchmidtH. 1I. 143, 185,287,291
Schmieder, B. 30, 85, 89, 90, 91, 96, 97, 98,
99, 101, 104, 105, 107, 108, 109, 139, 151,
152, 155, 173, 174, 205, 228, 229, 230,
233, 234, 256, 257, 258, 259, 260, 281,
285, 286, 287, 289, 291, 292, 293, 295
Schroll, A. 168, 169, 276
Schwarzschild, K. 12, 291
Seagrave~P.268,279

Secchi, A. 3, 11, 12, 14,291
Semel, M. 257, 286
Serio, S. 115, 291
Servajean, R. 114,284
Sevemy, A. B. 13, 16, 91, 94, 101, 148, 239,
291,295
Shabanskii, V. P. 163, 291
Shafranov, V. D. 247, 291
She, Z. S. 139,291
Sheeley, N. R. 155, 264, 276, 282
Shelke, R. N. 121, 291
Shibasaki, K. 14, 172, 281
Shibata, K. 109, 152, 157, 240, 284, 286,
291,292
Shih, Y. W. 78, 291
Shilova, N. S. 107,287,291
Shine, R. A. 90, 97, 277, 285
Shklovsky, I. S. 239,291
Shortley, G. H. 43, 277
Shurcliff, W. A. 45, 291
Sime, D. G. 268, 272, 281, 291
Simnett, G. M. 108, 151, 152, 234, 291
Simon, G. 96, 97, 104, 105, 173, 205, 256,
257, 259, 260, 285, 291, 292
Simon, G. W. 155, 292
Sitenko, A. G. 77, 275
Skumanich, A. 223, 289
Slonim, Yu.M. 101,292
Smartt, R. N. 175, 292
Smith,D.F.268,271,285,292
Smith, E. A. 130, 139, 289, 292
Smith, E.v.P. 109, 292
Smith, H. J. 109,292

AUTHOR INDEX

Snrlth,J. B.78, 119,257,258,259,280,295
Snrlth, S. F. 99, lOS, 114, 119, 251, 257, 289,
292
Sacker, D. G. ISS, 278
Solanki, S. K. 146,290
Solinger, A. B. 156, 295
Song, M. T. 208, 257, 295
Soru-Escaut, I. 96, 104, 115, 168, 240, 241,
243, 257, 259, 286, 287, 291, 292
Soru-Iscovici, I. 114, 286
Sparks, L. 139,292,294
Spitzer, L. 66, 129, 292
Sreenivasan, S. R. ISS, 283
Srivastava, N. 213, 263, 276, 292
St. Cyr, O. C. 266, 268, 292
Stark, J. 40, 292
Starkova, L. J. 107, 287, 291
Steinolfson, R. S. 141, lSI, 152, 267, 268,
278, 288, 292, 293
Stellmacher, G. 93, 99, 234, 255, 276, 283,
295
Stepanov, A. I. 93,283
Stepanov, K. N. 77,275
Stepanov, V. E. 47, 292
Stiber, G. 109, 288
Straka, R. M. 172,292
Stratton, J. A. 64,250,292
Strauss, H. R. 249, 285
Strong, K. T. 240, 286
Sturrock, P. A. 138, 185, 208, 257, 290, 292,
295
Suematsu, Y. 99, 152, 157,291,292,293
Suess, S. T. 146,275
~vestka, Z. 41, 90, 91, lOS, 107, 278, 279,
286,292
Swann, W.F.G. 159, 293
Syrovatskii, S. I. 263, 293
Tachi, T. 139, 141, 293, 294
Takeuctrl,A. 99, 293
Tanaka, K. 257, 286
Tandberg-Hanssen, E. 13, 27, 29, 78, 82, SS,
87, 90, 92, 93, 94, 97, 99, 101, 103, 105,
108, 109, 111, 119, 127, 129, 130, 131,
133, 134, 146, 147, 151, 152, 167, 173,
174, 198, 200, 201, 203, 223, 226, 234,
239, 267, 268, 275, 277, 279, 281, 284,
2SS, 287, 289, 290, 291, 292, 293, 295
Tang, F. Y. 107, 168,259,260,279,293
Taylor, J. B. 269,293
Teller, E. 41, 282
Terasranta, H. 168, 169, 276
Teube~D. 119,258,259,280

303

Thomas,J.H.SS,146,287,293
Thomas, R. N. 22,25,224,282,293
Thompson, W. T. 99, 293
Timothy, A. F. 114,293
Timothy, J. G. 155,295
Tlanrlcha, A. 172, 277
Topctrllo, N. A. 171,275
Topka, K. 168, 293
Torricelli-Ciamponi, G. 139,277,279
Tousey,R.155,276
Tracadas, P. W. 206, 214, 215, 216, 217,
218,286
Trivelpiece, A. W. 248, 283
Tsiropoula, G. 155,293
Tsubaki, T. 99, 293
Tsuneta, S. 240, 286
Tucke~ W.H. 128, 136,277
Tur, T. J. 257, 293
Tuseta, S. 240, 294
Uchida, Y. 109, 126, 128, 134, 146, 156,
157, 240, 251, 284, 286, 293
Uesugi, A. 96, 284
Ulloa, A. 11,293
Unno, W.45,47, 146,257,286,290,293
Uralov, A. M. 257, 293
Urpo,S. 168,169,276
Usikov, D. 207, 279
Vaiana, G. S. 114, 115,286,291,293
Valnicek, B. 98, 101, 103, 109, 110, 293,
294
van Ballegooijen, A. A. 200, 204, 205, 258,
260,294
van de Hulst, H. C. 33, ISO, 294
van der Linden, R.A.M. 234, 294
van Driel-Gesztelyi, L. 108, 151, 291
van Hoosier, M. E. ISS, 276
van Hoven, G. 139, 140, 141, 147, 206, 234,
249, 277, 278, 279, 292, 293, 294
van Lyong, L. 146,294
van Tend, W. 199,255,263,294
Vassenius, B. II, 294
Veltri, P. 238, 277
Venkatakrishnan, P. 119, 294
Vernazza, J. E. 155, 295
Vial, J.-C. 26, 27, 28, 29, 30, 31, 84, SS, 97,
104, 108, 109, 152, 174, 175, 229, 230,
231, 232, 259, 268, 278, 280, 281, 285,
288, 291, 294
Vizoso, G. 104,294
Vogel, S. N. 155,276
von Kliiber, H. 114,294
Voslamber, D. 248, 277

304

THE NATURE OF SOLAR PROMINENCES

VrSnak, B. 98, 99, 101, 107, 168, 169, 205,
255, 256, 260, 263, 276, 294
Vyssotsky, A. N. 11,294
VVagne~ VV.J.264,282,294
VValdmeier, M. 85, 114, 155, 168, 294
VVang, A. H. 78, 295
VVang,J.F.77,122,268,28O,286,295
VVang, S. 78, 295
VVarwick,J. VV.53, 109, 113, 158,294
VVashimi, H. 211, 290
VVatanabe,T.24O,286,294
VVatari, S. I. 240, 294
VVebb, D. F. 151,266, 268, 290, 292
VVebb, G. M. 163, 280
VVeisskopf, V.F.I. 35, 294
VVentzel, D. G. 156, 163, 294, 295
VVerne~S. 108,295
VVest, E. A. 119, 258, 259, 280
VVestin, H. 108,295
VVeyman,R. 123, 128,295
VVhite, O. R. 24, 97, 285, 295
VViehr,E.99,234,255,276,295
VVigne~E.35,294

VViik, J. E. 89, 90, 111, 228, 229, 230, 233,
234, 282, 291, 295
VVilson, R. 175,283
VVilson, R. M. 268, 278
VVithbroe, G. L. 91, 155, 175, 283, 295
VVolfson, R. 268, 295
VVoltjer, L. 206, 295
VVoodbury,E.T.185,292
VVoodgate, B. E. 90, 277
VVu, S. T. 77, 78, 79, 113, 146, 147, 152,208,
255, 256, 257, 267, 268, 275, 278, 280,
282, 288, 292, 295
Yakunina, G. 107, 278
Yamagnchi, K. 240, 294
Yang, C. Y. 175, 295
Yang, VV.-H. 208, 295
Yeh, T.255,256,295
Yi, Z. 99, 173, 181, 182, 279, 295
Yin, S. 26, 279
You, J.-q. 96, 295
Young, C. A. 14, 295
Zachariadis, Th.G. 234, 280
Zaitsev, V. V. 211, 276
Zang, T. A. 208, 278
Zappala, R. A. 115, 291
Zel'dovich, Ya.B. 239, 295
Zhang, Q. Z. 26, 81, 175, 279, 295
Zhang, Z. 175,292
Zharkova, V. V. 27, 295

Zirin, H. 13, 17, 87, 90, 91, 94, 97, 98, 101,
108, 113, 119, 127, 131, 174, 260, 268,
276,283,284,285,295
Zirker, J. B. 30, 90, 164, 239, 283, 288, 296
Zlobec,P.263,294
Zloch, F. 256, 294
Zodi, A. M. 271, 296
Zwaan, C. 122,296

SUBJECT INDEX
Abel's integral equation 33
absorption coefficient 20,37,47,171
acceleration of particles 158, 163,238,271
active region filaments 4, II, 84, 95, 104,
105,234
active regions 7
adiabatic invariant 159
Alfven waves 66, 71, 181
allowed lines 42, 89, 175
ambipolar diffusion 29,31,75
Ampere's law 64,68
Balmer decrement 226
Bessel function 254
Boltzmann equation 57, 149
Boltzmann's law 21, 24
boundary conditions 76
Bremsstrahlung 31,172
bright pOints 157
Brunt-Viiisalii frequency 73
butterfly diagram 168
caps 11
Carrington rotation velocity 169
chromosphere-corona transition
region, CCI'R 157, 175
chromospheric faculae 5
chromospheric fibrils 114, 154, 214, 250
chromospheric network 155
classifica tions
de]ager16
Newton 16
Menzel and Evan 16
Pettit 15
Sevemy16
Waldmeier 85
Zirin 17
Zirin-Tandberg-Hanssen 87
cloud model 232
cloud-model analysis 231, 238
column mass density 226
compatibility equation 77
complete (frequency) redistribution, CRD
27
condensation 113, 122, 128, 141
condensation in a current sheet 130, 140
condensation in loops 139
contribution function 176
cool flare loops 107
coronal cavity 114, 115, 269, 271, 273
coronal clouds 8
coronal mass ejections 171, 264
coronal rain 9, 107
coronal streamers 264, 271

coronal transients 264
coronal whip 264
cosmic rays 158
damping constant 36
damping profile 36
dark mottles 155
Debye length 55, 75
degenerate levels 43, 53
dextral filaments 206, 215
diamagnetic effects 148
dielectric constant 64, 250
dielectric pennittivity 64
differential cloud model 233
differential emission measure 175, 176
differential rotation 168, 242
disparitions brusques
dynamic 11, 101,206,244,253,258,269
thermal 104, 240
Doppler brightening 26
Doppler dimming 26
Doppler profile 36
Doppler width 38, 82, 233
dynamic support 210
Eddington approximation 25
eigenmode oscillations 99
Einstein coefficient 21
ejections 108
electrical conductivity 62, 201, 247
electric displacement 64, 134
electric current 58, 61, 139, 180, 201, 208,
217, 260, 263, 271
electric fields
macroscopic 41,246,248
microscopic 40
emission coefficient 20, 47
emission measure 176, 229
energy density 123, 133
energy level diagram 42
energy principle 244
enthalpy 179
entropy 60, 144
eruptive prominences 11
escape velocity 101
Euler's constant 34
evaporation 113, 143, 147, 240
Evershed effect 146
Faraday's law 159
Fermi mechanism 159, 161
fibrils 9, 154, 214
filament channel lOS, 114,215,269, 273
filamentary degree 28
filling factor 28, 94, 173, 226

306

THE NATURE OF SOLAR PROMINENCES

fine structure 167,201, 233
fine-structure elements 27
first law of thermodynamics 126
flaring arches 107
flux function 190, 195, 196,202,210,221
forbidden lines 42, 175
force-free field 184, 207, 217, 269
forced oscillations 99
Fraunhofer lines 45
Fredholm equation 177
friction 252
Gaunt factor 34, 171
Gauss' theorem 153
global instability concept 266
Grad-Shafranov equation 190, 196
guiding center approximation 160
gyro frequency 56,160,239
Hall current 63
handedness 206
Hanle effect 50
harmonic oscillator 252
hedgerow, tree, and tree-trunk 11
Heisenberg's principle of spectroscopic
stability 52
Heisenberg's uncertainty principle 35
helical configurations 109, 200, 201,247,
256, 260, 262
helicity 206, 270
High Resolution Telescope and
Spectrograph 89
Hvar reference model 110
hydromagnetic buoyancy 255
induction effect 161
Inglis-Teller formula 41, 89
injection 113, 142
instabilities
ballooning 235
convective 73
dynamiC shearing 257
Kelvin-Helmholtz 246
kink 246, 257
radiative 141
Rayleigh-Taylor 157, 216, 234, 246
sausage 246
tearing mode 130, 140, 234, 246
thermal 123, 140, 147, 240
inter-system line 42, 89
internal motions 95
inverse polarity field 93, 196, 198, 199,
208,269
ionization
degree of 91, 230

equilibrium 21
isentropic perturbation 125
isobaric criterion 125
isochoric perturbations 125
Jensen's injection mechanism 148
Joule diffusion 67
Joule (ohmic) heating 61, 64, 141, 178, 180,
240
K-corona, 30, 264
Kirchhoff's law 20
Kruskal-Shafranov's instability criterion
247
Lande g-factor 43, 51
leaky-bucket model 234
line profiles 35
line tying 135, 198,248
local thermodynamic equilibrium, LTE 20
loop prominences 107, 122, 137, 139, 142,
163, 237, 272
Lorentz force 65, 153, 184, 185, 195, 198,
203, 212, 218
Mach number 157
macrospicules 155
magnetic arcades 114, 147,210
magnetic buoyancy 255
magnetic diffusion equation 67, 212
magnetic diffusivity 67, 132
magnetic dip 143, 188,200,205,208
magnetic helicity 206, 269
magnetic knots 207
magnetic mirrors 159
magnetic shear 119, 258
magnetohydrodynamic models 184, 220
magnetohydrostatic equilibrium 207, 210,
221, 238, 242
magnetohydrostatics 183
Maxwell's equations 64, 134
melon-seed effect 148
metastable levels 42, 89
method of characteristics 77, 78
microturbulence 39, 81,238
models
Dungey 187, 246
Forbes-Malherbe 137, 142,272
Heyvaerts 257
Jefferies-Orrall 163
Menzel 187, 193, 246
Kippenhahn-Schliiter 137, 187, 192, 194,
220,244,251
Kopp-Pneuman 139
Kuperus-Raadu 135, 199,200
Low 269

SUBJECT INDEX

Pikel 'ner 143
multiplet 42, 89
multiplicity 42
net radiative bracket 225
normal mode method 246
normal polarity field 93,192,196
Ohm's law 61, 66, 212
oscillator strength 37
partial (frequency) redistribution, PRO 27,
279
partition function 21
Pauli's exclusion principle 41
permeability 69, 250
permittivity 69
photospheric faculae 5
pivot point 169,242
plage faculaire 5,156
plagettes 214
Planck function 20
plasma 55
plasma-p 122, 137, 188, 197,258
plasma frequency 74,170
plasma oscillations 74
plasmoid 137, 153, 272
point dipole 207
polarity inversion zone 114
Poynting vector 134
pressure scale height 183, 185, 191, 196,
252
prominence feet 96, 197,215,219
prominence location 167
prominence-corona transition
region, PCfR 82, 135, 167, 174
prominence oscillations 98, 252
quadrupolar region 200
radiative cooling 126, 136
radio bursts 171, 238, 266, 271
Rayleigh-Jeans law 20
reconnection (magnetic) 135, 138, 152, 198,
204,208,249,258,271
refractive index 170
resistivity 63
Reynold's number 68,137
Righi-Leduc effect 239
runaway electrons 247
Russell-Saunders coupling 42
Saha's equation 21
Schwarzschild's criterion 73
shock drift acceleration 163
shock wave 156, 162, 267
sidereal angular rotation rate 169
similitude principle 78

307

sinistral filaments 215, 206
sinking and shrinking filaments 104,244
siphon-type injections 143
Skylab 94
Solar Maximum Mission 94
source function 20, 225, 228, 230, 232
specific heat 70, 239
spicules 154
sprays 8, 105, 260
stability
convective 73
magnetohydrostatic 167, 244
thermal 123
Stark effect 40
statistical equilibrium 22, 175
statistical weight 43
Stokes parameters 48
stress tensor 59
surges 7, 108, 122, 148, 151, 237, 272
Swann's mechanism (betatron acceleration)
159, 161
Taylor relaxation 269
temperature
brightness 170
electron 23, 75, 81
excitation 21, 89
ionization 21, 75
kinetic 20, 81
radiation 170
term (spectroscopic) 42
thermal conduction 126, 178, 225, 239, 241
thermal conductivity 61, 127, 129, 139, 145,
180, 223, 227, 239
thermal equilibrium 126, 150, 223, 226, 238,
240
thermal non-equilibrium 127
Thomson cross section 32
transit-time effect 161
turbulence 211, 257
twisted flux tubes 148, 204, 247, 257, 260,
262, 264, 269
two-ribbon flare 107, 263
vector potential 208
velocity
Alfven 66, 135, 140, 181, 184, 211, 255,
257
Doppler 37

escape 101, 109
group 70, 74
line oroadening 82

macroscopic (flow) 58, 149, 152
microscopic 58,149

308

THE NATURE OF SOLAR PROMINENCES

phase 70, 72
sound 70, 143, 156, 184, 255
viscosity 68, 241, 253
viscous heating 241
Vlasov equation 57,149
Voight profile 38
wave euation 65
waves 70, 156, 239, 254, 267
winking filament 99
YOHKOH 14, 94, 121, 273

Zeeman effect 43, 45