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A Theoretical Investigation
Of Optical Emission in Solar
Flares

presented by

William P. Abbett

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A THEORETICAL INVESTIGATION OF OPTICAL EMISSION IN SOLAR
FLARES

By

William P. Abbett

A DISSERTATION
Submitted to
Michigan State University

in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1998

ABSTRACT

A THEORETICAL INVESTIGATION OF OPTICAL EMISSION IN SOLAR
FLARES

By

William P. Abbett

A dynamic theoretical model of a flare loop from its footpoints in the photosphere
to its apex in the corona is presented, and the effects of non-thermal heating of
the lower atmosphere by accelerated electrons and soft X-ray irradiation from the
flare heated transition region and corona are investigated. Important transitions of
hydrogen, helium, and singly ionized calcium and magnesium are treated in non-LTE.

Three main conclusions are drawn from the models. First, even the strongest of
impulsive events can be described as having two phases: a gentle phase characterized
by a state of near equilibrium, and an explosive phase characterized by large material
ﬂows, and strong hydrodynamic waves and shocks. During the gentle phase, one or
possibly two temperature “plateaus” form in the upper chromosphere. The line
emission generated in these regions produces profiles that are generally symmetric
and undistorted, in contrast to emission produced during the explosive phase, where
large velocity gradients that occur in the upper atmosphere produce line proﬁles
that are highly asymmetric and show large emission peaks and troughs. Second, a
signiﬁcant continuum (or “white light”) brightening results from increased hydrogen
recombination radiation in the upper chromosphere at the point where the accelerated
electrons deposit the bulk of their energy. Third, there exists a measurable time lag

between the brightening of the near wings of Ha and the brightening 0f the Paschen

continuum. This delay is controlled by the amount of time it takes for electron
densities in the upper chromosphere to become high enough, and the densities of
hydrogen atoms in high energy bound states to become low enough, to allow the

number of recombinations to dominate the number of photoionizations in the region.

To Andrew

iv

ACKNOWLEDGMENTS

First, I would like to thank my advisors, Bob Stein and Suzanne Hawley. Both
were always available to discuss issues, and were throughout the course of this work
encouraging and supportive. Special thanks to Suzanne and George Fisher, who are
allowing me to continue this exciting work with them out at UC Berkeley this fall.
Many thanks to Mats Carlsson and Viggo Hansteen for extending to me their ex-
pertise and hospitality during my two trips to Oslo Norway. Their help was crucial
in overcoming some of the computational issues involved in modeling the solar tran-
sition region. And of course, I am deeply appreciative to both Mats and Bob for
allowing me to use and modify their truly great radiation hydrodynamic computer
code, RADYN. I would also like to thank Dave Bercik. Dave has always been willing
at a moments notice to help me work through a difficult conceptual or computational
problem.

Most importantly, I would like to extend my deepest thanks and appreciation
to my family. Especially my wife Aimee, and my mother and father, Bill and Jackie
Abbett. Without their unquestioned support through some of the tough times, this

work would certainly not have been possible.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

1

2

INTRODUCTION

METHOD OF SOLUTION

2.1 The Conservation Equations .................
2.2 Complete Linearization ....................
2.3 The Adaptive Grid ......................
2.4 Non-Linear Advection ....................
2.5 The Transfer Equation ....................

THE INITIAL ATMOSPHERE

3.1 Atomic Data ..........................
3.2 The Pre—flare Model .....................
3.3 Accelerated Electrons .....................
3.4 Soft X-ray Irradiation .....................

FLARE ENERGETICS AND DYNAMICS

4.1 The Weak Flare ........................
4.2 The Strong Flare .......................

OPTICAL EMISSION

5.1 Continuum emission .....................
5.2 Line Emission .........................
5.3 Soft X-ray heating and the Gradual Phase .........

CONCLUSIONS

6.1 Summary ...........................
6.2 Limitations of the Model ...................
6.3 Future Work ..........................

vi

vii
viii
l

7
7
10
12
17
18

29
29
32
42
43

48
6O
67

76
76
94
106

118

120
121

LIST OF TABLES

Atomic Parameters for Hydrogen ([H/H]=12.0, A=1.01) ....... 29
Atomic Parameters for Helium ([He/H]=11.0, A=4.00) ........ 30
Atomic Parameters for Calcium ([Ca/H]=6.36, A=40.08) ....... 30
Atomic Parameters for Magnesium ([Mg/H]=7.58, A=24.12) ..... 31
b-b Transitions Calculated in Detail. .................. 32
b—f Transitions Calculated in Detail. .................. 33

vii

[\9

LIST OF FIGURES

An example of grid motion during the onset of a strong ﬂare ...... 16

The temperature stratiﬁcation of the PFI pre—ﬂare atmosphere. Shown

is the logarithm of the temperature (K) as a function of the logarithm

of the column mass (g cm ’2). The cross-hatches denote the preﬂare
position of the grid points ......................... 37

The temperature stratiﬁcation as a function of log column mass (g
cm '2) for the three model atmospheres PFI, VAL3C, and MFIME. . 38

The electron density stratiﬁcation (cm ‘3) as a. function of log column
mass (g cm '2) for the three model atmospheres PFI, VAL3C, and
MFIME ................................... 39

The Ca 11 K resonance line proﬁles for the three model atmospheres
PFl, VAL3C, and MFIME. Intensity is relative to the continuum,
and the wavelength from line center is expressed in Angstroms. . . . . 40

The Ha line core for the three model atmospheres PF 1, VAL3C, and
MFIME. Intensity is relative to the continuum, and the wavelength
from line center is expressed in Angstroms ................ 41

Loop geometry used in the calculation of soft X-ray heating. Symbols
are deﬁned in the text. .......................... 46

The temperature structure of the F10 chromosphere during the ﬁrst
two seconds of ﬂare heating. Shown is the log of the temperature (K)
as a function of height (Mm) and time (s). A temperature plateau at
T m 104 K between z 1.0 and 1.5 Mm is well developed after z 0.23.
Two plateaus (one at T w 104 K and one at T m 105 K are well
developed after 2.08. The motion of the grid is also apparent ...... 49

The density structure of the F10 chromosphere during the ﬁrst two
seconds of ﬂare heating. Shown is the log of the gas density (g cm ‘3)
as a function of height (Mm, 1 Mm = 106 meters) and time (s).
The lack of large peaks near the point of maximum ﬂare heating is
indicative of the gentle phase. ...................... 50

viii

10

11

12

13

14

15

16

17

The proton density stratiﬁcation of the F10 chromosphere during the
ﬁrst 1.2 seconds of ﬂare heating. Shown is the log of the proton

density (cm '3) as a function of height (Mm) and time (s). ......

The velocity structure of the F10 chromosphere during the ﬁrst 1.2
seconds of ﬂare heating. Shown is the gas velocity (km s ’1) as a func-
tion of height (Mm) and time (s). The modest ﬂows are characteristic

of the gentle phase. ............................

The non-thermal heating proﬁle during the ﬁrst 1.2 seconds of the
F10 run. Shown is the log of the heating, Qe— (ergs cm ’3 s") as
a function of height (Mm) and time (3). Note the slow progression
of the peak during the ﬁrst 023 of ﬂare heating, and the subsequent

broadening of the proﬁle ..........................

The F10 atmosphere during the time of explosive evaporation. Shown
is the log of the temperature (K) as a function of height (Mm) and
time (8). Note how the grid moves in response to the temperature
and density structure. The narrow local temperature minimum marks
the condensation, and is located at z z 1.3 Mm after 50 seconds. A
chromospheric temperature plateau located between % 0.7 and 1.1

Mm persists through the explosive phase .................

The F10 atmosphere during the time of explosive evaporation. Shown
is the log of the gas density (g cm ’3) as a function of height (Mm)
and time (s). The upward moving density peak is the evaporated
material, and the downward moving peak marks the chromospheric

condensation. ...............................

The velocity structure of the F10 atmosphere during the time of ex—
plosive evaporation. Shown is the gas velocity (km s ‘1) as a function
of height (Mm) and time (5). Negative velocities denote downward

moving plasma (ie. toward the photosphere) ...............

The non-thermal heating proﬁle of the F10 run. Shown is the log
of the heating, Qe— (ergs cm '3 s") as a function of height (Mm)
and time (s). The “split” in the heating peak denotes the onset of
the explosive phase. The local maxima in the heating proﬁle reﬂect
the relatively high densities of material within the upward moving

evaporation and downward moving condensation. ...........

The F9 atmosphere. The solid line represents the log of the electron
density as a function of height, 2 (Mm), at timet (log 72., , [ne]=cm '3 ).
The dotted line denotes the pre—ﬂare electron density, and the dashed
line is the log of the non-thermal electron heating, scaled so that it

will ﬁt on the plot (log Qe-+I2.5 , [Qe—]=ergs cm ‘3 s '1) ........

51

52

53

56

58

61

18

19

20

21

22

23

The F9 atmosphere. The solid line represents the log of the temper-
ature, T (K), as a function of height, 2 (Mm), at time t. The dotted

line denotes the pre—ﬂare temperature structure. ............

The F9 atmosphere at 23 (top panel) and 608 (bottom panel). The

temperature (solid line), total hydrogen number density (cross-hatches),

and proton density (dashed line) are shown in relation to the position
of the normalized heating proﬁle of the non-thermal electrons (dotted

line). The densities are expressed per cubic centimeter. .......

Energy balance for the F9 run after 60 seconds of ﬂare heating. Shown
in the bottom panel is the non-thermal electron heating (solid line),
the radiative heating due to optically thick transitions calculated in
detail (dashed line), and radiative heating due to optically thin metal
cooling (dotted-dashed line). In both panels, the (cross—hatches)
indicate the total gain in material energy. The top panel shows
the amount of non-thermal energy that is not radiated away (solid
line), the work done by pressure (dashed line), the work done by
gravity (dotted line), and the conductive heating (thrice dotted, once
dashed line). The conductive heating is non-negligible only in the
coronal transition region, and to a much lesser extent at 1.7Mm. The
viscous dissipation is everywhere negligible. Note the difference in

scale between the top and bottom panels .................

The FII atmosphere. The solid line represents the log of the electron
density as a function of height, z (Mm), at timet (log n6 , [ne]=cm "'3 ).
The dotted line denotes the pre—flare electron density, and the dashed
line is the log of the non-thermal electron heating, scaled so that it

will ﬁt on the plot (log Qe-+10.5, [Qe—]=ergs cm ‘3 s ’1) ........

The F11 atmosphere. The solid line represents the log of the temper-
ature, T (K), as a function of height, 2 (Mm), at time t. The dotted

line denotes the pre-ﬂare temperature structure .............

Energy balance for the F11 run after 0.5 seconds of ﬂare heating.
Shown in the bottom panel is the non-thermal electron heating
(solid line), the radiative heating due to optically thick transitions
calculated in detail (dashed line), and radiative heating due to opti—
cally thin metal cooling (dotted—dashed line). In both panels, the
(cross-hatches) indicate the total gain in material energy. The top
panel shows the amount of non-thermal energy that is not radiated
away (solid line), the work done by pressure (dashed line), and the
viscous work (dotted line). The conductive heating, buoyancy work,
and soft X—ray heating are negligible by comparison. Note the differ-

ence in scale between the top and bottom panels. ...........

62

66

69

70

24

25

26

27

29

30

31

32

33

The temperature structure (solid line) and the electron density strat-
iﬁcation (dashed line) after 4.08 of non-thermal heating in run F11. . 74

Normalized Balmer and Paschen continua for the F11 run. The dotted
line is the continuum emission of the pre—ﬂare atmosphere. The early
impulsive phase is dominated by hydrogen photoionization, and the
later impulsive phase by hydrogen recombination radiation ....... 77

Normalized Balmer and Paschen continua for the F10 run. The dotted
line is the continuum emission of the pre—ﬂare atmosphere. ..... 78

Normalized Balmer and Paschen continua for the F9 run. The dot-
ted line is the continuum emission of the pre-flare atmosphere. The
changes in continuum ﬂux in this case are barely noticeable. ..... 79

Continuum intensity (normalized between the preﬂare value and the
maximum value at 2.08) at 5000 A (solid line) for the F11 run. The
normalized Ha line wing intensities are shown at +2.5 A (dashed line)
and -2.5 A (dotted-dashed line). For this case, the lag between the
5000 A continuum brightening and the Ho wing brightening is z 0.1
seconds. .................................. 80

Continuum intensity (normalized between the preﬂare value and the
maximum value at 25.08) at 5000 A (solid line) for the F10 run. The
normalized Ha line wing intensities are shown at +2.5 A (dashed line)
and -2.5 A (dotted-dashed line). For this case, the lag between the
5000 A continuum brightening and the Ha wing brightening is z 1.08. 81

Continuum intensity (normalized between the preﬂare value and the
maximum value at 60.08) at 5000 A (solid line) for the F9 run. The
normalized Ha line wing intensities are shown at +2.5 A (dashed line)
and -2.5 A (dotted-dashed line). For this case, the lag between the
5000 A continuum brightening and the Ha wing brightening is Rs 60.08. 82

The F10 atmosphere. The solid line represents the log of the electron
density as a function of height, 2 (Mm), at timet (log n8 , [ne]=cm '3 ).
The dotted line denotes the pre—ﬂare electron density, and the dashed
line is the log of the non-thermal electron heating, scaled so that it
will ﬁt on the plot (log Qc—+11.5 , [Qe—]=erg8 cm ‘3 S ‘1) ........ 83

The F10 atmosphere. The solid line represents the log of the temper-
ature, T (K), as a function of height, 2 (Mm), at time t. The dotted
line denotes the pre—ﬂare temperature structure. ............ 84

The fraction of the total number of hydrogen atoms undergoing pho-
toionizations after 0.2 seconds of ﬂare heating in run F10. ...... 85

xi

34

35

36

37

38

39

40

41

42

Number Densities per unit volume of the hydrogen ground state
(cross-hatches), n22 energy level (solid line), n=3 energy level (dotted
line), n=4 energy level (dashed line), n=5 energy level (dotted-dashed
line), and the proton density (thrice dotted, once dashed line). The
left panel shows the pre-ﬂare density stratiﬁcation, and right panel

shows the density proﬁles after 0.28 of run F10. ............

The F10 atmosphere after 0.4 seconds of ﬂare heating near the time
of maximum Paschen dimming. The top left panel shows the ratio
of the proton density 72C to the number density of the n=3 energy
state of hydrogen (log nC/ng). The top right panel shows the ratio
of the recombination rate to the photoionization rate from the n=3
excited state (log 1203/ R36) The bottom left panel shows the ratio
of the number of recombinations to the number of photoionizations
(log ncch/n3R36), and the bottom right panel shows the ion density
(log np, solid line) and the number density of hydrogen in the n=3

energy state (10g 713, dashed line) per unit volume. ..........

The same as Figure (35), but after one second of ﬂare heating in run
F10. This corresponds to the onset of continuum brightening.

The grey scale background contour denotes the continuum emer-
gent intensity contribution function plotted against height (Mm) and
wavelength (nm) for the F10 run 508 after the onset of non-thermal
heating. Also shown is the temperature stratiﬁcation of the atmo-
sphere (log T (K)). The chromospheric plateau is the region between
z 0.7 Mm and 1.15 Mm, and the condensation is located between z

1.3 and 1.4 Mm ...............................

The time evolution of the Ca II K line proﬁle for run F10. The vertical
axis is normalized intensity with respect to the continuum, and the
horizontal axis denotes the number of Angstroms from line center.

The time evolution of the Ca II K line proﬁle for run F11. The vertical
axis is normalized intensity with respect to the continuum, and the
horizontal axis denotes the number of Angstroms from line center.

The components of the intensity contribution function for Ca II K
after 50 seconds of ﬂare heating in run F10. See text for details.

The components of the intensity contribution function for Ca II K
after 4 seconds of ﬂare heating in run F11. See text for details.

The time evolution of the Ha line proﬁle for run F10. The vertical
axis is normalized intensity with respect to the continuum, and the
horizontal axis denotes the number of Angstroms from line center.

xii

90

93

95

96

98

100

102

43

44

45

46

47

48

49

50

The time evolution of the Ha line proﬁle for run F11. The vertical
axis is normalized intensity with respect to the continuum, and the
horizontal axis denotes the number of Angstroms from line center.

The components of the intensity contribution function for Ha after

50 seconds of ﬂare heating in run F10. See text for details. .....

The F10 atmosphere during the gradual phase, when the non-thermal
heating has come to an end. The solid line represents the temperature
T (K) as a function of height, 2 (Mm), at time t. The dotted line

denotes the pre-ﬂare temperature structure. ..............

The F 10 atmosphere during the gradual phase, when the non-thermal
heating has come to an end. The solid line represents the electron
density as a function of height, 2 (Mm), at timet (log n.3 , [ne]=cm ‘3 ).
The dotted line denotes the pre—ﬂare electron density, and the dashed

line is the soft X-ray heating (log Q$+14.5 , [Q$]=ergs cm ‘3 8 "1). . .

Chromospheric energy balance for the F10 run during the gradual
phase. Shown in the bottom panel is the soft X—ray heating (solid
line), the radiative heating due to optically thick transitions calcu—
lated in detail (dashed line), and radiative heating due to optically
thin metal cooling (dotted-dashed line). In both panels, the (cross-
hatches) indicate the total gain in material energy. The top panel
shows the net radiative energy gain (solid line), the work done by
pressure (dashed line), and the viscous work (dotted line). The con—
ductive heating and buoyancy work are shown, but are negligible by

comparison. ................................

The soft X-ray heating proﬁle after 90.0 seconds of run F10 (during
the gradual phase). Shown, are the total X-ray heating (solid line),
the contribution to the X—ray heating due to hydrogen (dashed line),
the contribution due to helium (dotted—dashed line), and the contri-

bution of other metals (dotted line). All quantities are expressed as
1 —1

log Q3, where [Qx]=ergs g‘ s . ....................

The time evolution of the Ha line proﬁle for run F10 during the
gradual phase. The vertical axis is normalized intensity with respect
to the continuum, and the horizontal axis denotes the number of

Angstroms from line center. .......................

The time evolution of the Ca 11 K line core for run F10 during the
gradual phase. The vertical axis is normalized intensity with respect
to the continuum, and the horizontal axis denotes the number of

Angstroms from line center. .......................

xiii

108

110

114

115

51

52

The components of the intensity contribution function for Ha during
the gradual phase after 90 seconds in run F10. See text for details. . 116

The components of the intensity contribution function for Ca II K
during the gradual phase after 90 seconds in run F10. See text for
details. .................................. 117

xiv

Chapter 1

INTRODUCTION

A solar ﬂare is believed to be the result of a sudden release of free magnetic energy
at a localized point in the solar corona. Exactly how this energy is liberated into the
solar atmosphere is not well understood, but its effects include the rapid acceleration
of charged particles along lines of magnetic ﬁeld. As these accelerated particles
propagate toward the photosphere, they interact with the surrounding atmosphere.
In the standard model, the accelerated particles are electrons which emit hard X-
ray bremsstrahlung and gyrosynchrotron microwave radiation as they lose energy via
Coulomb collisions.

The peak energy deposition of a strong ﬂux of non-thermal electrons will occur
in the upper chromosphere, triggering downward moving condensations along with
explosive evaporation of chromospheric material to coronal temperatures (Fisher et
al. (1985a), Zarro et al. (1988), Fisher (1987)). White light continuum emission is
observed to emanate from localized areas around the footpoints of ﬂare loops, and
this emission tends to be both spatially and temporally correlated with the hard
X-ray radiation (Hudson et al. (1992)). This suggests that the emission occurs
at the point where the accelerated electrons impact the pre—ﬂare chromosphere. A

signiﬁcant number of protons may also be accelerated to high energies, as evidenced

by gamma ray emission in strong ﬂares (Chupp (1984), Ramaty et al. (1995), Share
& Murphy (1995)), but the protons tend not to be temporally correlated with the
hard X-ray bursts (Chupp (1984), Gardner et al. (1981)) and do not account for
the observed atmospheric response (Emslie et al. (1998)). The model presented here
investigates only the effects of non-thermal accelerated electrons.

As the electrons deposit energy in the solar atmosphere, they give rise to a variety
of thermal effects, including enhanced soft X-ray, EUV, and hydrogen continuum
radiation. Observed enhancements in the Balmer and Paschen continua originating
from the upper chromosphere can provide an additional source of heating below the
temperature minimum region (Metcalf et al. (1990)), where only a few of the highest
energy non-thermal electrons are able to penetrate. Similarly, soft X-ray irradiation
emanating from the electron-heated upper chromosphere can reach into the lower
atmosphere, likely providing an additional source of heating to the region (Hawley
& Fisher (1994)). These effects may be responsible for the persistence of optical
emission long after the the primary source of impulsive heating (the non-thermal
electrons) has died away.

It is the response of the chromosphere and transition region to ﬂare energy de-
position that ultimately determines the structure of the lower atmosphere and the
characteristics of the optical line and continuum radiation observed during the course
of a ﬂare. Much of the observable line radiation originating in the chromosphere is
signiﬁcantly decoupled from the local temperature of the medium at the point of its
formation, thus it is necessary to use the formalism of non-LTE (non—local thermo-

dynamic equilibrium) radiation hydrodynamics in order to properly understand the

structure, energetics, and optical emission of the atmosphere.

Previous static, semiempirical models of Mauas et al. (1990), Avrett et al.
(1986), and Machado et al. (1980) were generated by guessing an atmospheric tem-
perature and density stratiﬁcation, solving the equations of statistical equilibrium
and radiative transport, and comparing the results to observations. The initial guess
was then reﬁned, and the process repeated until a solution was obtained that matches
a given observation. The difficulty with this approach is that the end result is hard to
interpret, since the method of solution ignores the physical processes that led to the
ﬁnal atmospheric state. The synthetic models of Canﬁeld, McClymont, & Puetter
(1984), Canfield & Ricchiazzi (1980), and Hawley & Fisher (1994) do not suffer from
this limitation, as they specify the form of heating associated with a ﬂare event, and
solve simultaneously the equations of energy conservation, statistical equilibrium and
radiative transport at a given time during a ﬂare. With this approach, the effects of
different heating mechanisms on the atmosphere can be interpreted physically, and
the results can then be compared with observations. We follow the synthetic approach
here, but with a much more rigorous treatment than was previously possible.

There have been a number of theoretical investigations of the dynamic response
of the lower atmosphere to ﬂare energy input. Notable examples of this type of
analysis include the work of Fisher et al. (1985a), Fisher et al. (1985b), Ricchiazzi &
Canﬁeld (1983), Canﬁeld, Gunkler, & Ricchiazzi (1984), Jakimiec et al. (1992), and
Emslie & Nagai (1985). Each modeled the atmospheric response to rapid inﬂuxes of
ﬂare heating, all but the latter taking into account the radiative transport properties

of the atmosphere using an escape probability formalism.

3

This escape probability treatment of the radiation is a shortcoming in all but
the static models of Hawley & Fisher (1994), and was used in the past because of
limitations in the speed and memory of computers available at the time. With the
current generation of multi—processor, massively parallel supercomputers it is possible
to calculate the radiation ﬁeld in detail. The static models of Hawley & Fisher (1994)
treat the radiation in detail, and consider the effects of non-thermal electron ﬂux and
thermal soft X—ray irradiation of the lower atmosphere. However, they were only able
to consider the stages during the evolution of a ﬂare where a static approximation is
adequate to describe the energetics and emergent radiation. Further, static models
are unable to account for departures of populations from equilibrium due to the
relatively slow transition rates found in the chromosphere.

Intense non-thermal heating in the ﬂaring atmosphere drives signiﬁcant quanti-
ties of mass through the chromosphere and corona. To properly describe the ener—
getics and emission characteristics of the chromosphere during each stage of a ﬂare
requires the formalism of non-LTE radiation hydrodynamics. This is the major im-
provement of the new models reported here. A modiﬁed version of the non-LTE
radiative hydrodynamic code of Carlsson & Stein (1997) (hereafter referred to as
RADYN) is used to model a magnetically conﬁned ﬂare loop from the coronal apex
to its footpoints in the photosphere. A ﬂux of energetic, non-thermal electrons is
introduced at the top of the loop, and the dynamic, energetic, and radiative response
of the atmosphere is computed as the ﬂare evolves.

The goal of this research is to produce models that provide detailed atmospheric

diagnostics for comparison with the latest high resolution optical observations (eg.

Johns-Krull et al. (1997), and Neidig et al. (1993)), and high resolution X-ray
spectra obtained by space based platforms such as Yokhoh. Three major conclusions
are drawn from the simulations.

First, each strong impulsive event can be broken down into two distinct phases,
each with deﬁnite observational signatures. The ﬁrst state is one of gentle evapora-
tion, and is characterized by the formation of one (or two) temperature “plateaus” in
the upper chromosphere. This gentle phase is further characterized by a state of near
energetic and radiative equilibrium. Velocity gradients in the atmosphere are small,
and line emission that originates in the chromospheric plateau is not signiﬁcantly
Doppler shifted, and the proﬁles remain essentially symmetric about their nominal
line—center frequencies. The second state is one of explosive evaporation. This phase
is characterized by large mass motions and steep atmospheric velocity gradients at
the boundaries of rapidly moving hydrodynamic waves and shocks. Line emission
during this phase can be highly Doppler shifted and distorted, reﬂecting the velocity
ﬁeld of the atmosphere at the depth of their formation.

Second, the white light continuum emission originates from elevated levels of hy-
drogen recombination radiation within the broad chromospheric plateau. The chro—
mospheric condensation also contributes strongly to this emission once it has fully
developed in the latter part of the impulsive phase. Recombination radiation from
the plateau backwarms the temperature minimum region as the condensation pushes
downward into the chromosphere.

Third, there is an initial reduction in continuum intensity immediately after ﬂare

onset which results in a noticeable time delay between the brightening of the Ha line

wings and the Paschen continuum. The non—thermal electrons deposit energy into
the upper chromosphere, and there is a corresponding increase in the populations of
the high energy bound states of neutral hydrogen. The number of photoionizations
throughout the region increases, thus reducing the number of photons from the pho-
tosphere that are able to escape the solar atmosphere. This situation persists until
the photoionization sufﬁciently elevates the electron density throughout the plateau
allowing recombination to become the dominant process. This effect is consistent
with the recent observations of Neidig et al. (1993) who observed a similar time lag
in the near wings of Ho and the 5000 A continuum in the white light ﬂare of March
7, 1989.

The remainder of this thesis is organized as follows: In Chapter (2) the theory
of non-LTE radiative transport in a dynamic atmosphere is summarized, along with
the numerical techniques involved in obtaining a solution. Chapter (3) describes the
atomic data and atmospheric model, and gives details of the assumed mechanisms
of ﬂare energy deposition considered here. Chapter (4) discusses the results of the
simulations as they pertain to the energetic and dynamic state of the ﬂare atmo-
sphere, and Chapter (5) describes the formation of the emergent line and continuum

radiation. Chapter (6) summarizes the results, and provides a sketch of future work.

Chapter 2

METHOD OF SOLUTION

A quantity of material is deﬁned to be in LTE if its atomic and molecular level
populations are given by Saha—Boltzmann statistics; that is, if they are coupled to
local conditions within the material. In the solar chromosphere this assumption
fails, as the densities of many atomic species are largely determined by the non-local
radiation ﬁeld which may, in general, be entirely unrelated to the local temperature
of the atmosphere. Further, the relatively low density and weak radiation ﬁeld in the
chromosphere results in slow transition rates. To properly understand the emission
and energetics of a ﬂare heated solar chromosphere, a modiﬁed version of the code of
Carlsson & Stein (1997) (RADYN) is used to obtain the simultaneous solution of the

highly non-local, non-linear, coupled equations of non-LTE radiation hydrodynamics.

2.1 The Conservation Equations

If a material quantity is conserved, its time rate of change in a volume element must
be equal to the sources or sinks acting on it within that volume, less its ﬂux out the
boundaries. Each of the hydrodynamic equations can be written in this manner. For

example, the statement of the conservation of mass (where there are no sources or

K]

sinks) can be expressed as

3r)

-8—t+V-pv( v:) 0. (1)

The plane-parallel approximation is assumed here, thus material properties depend

only on depth, and the continuity equation reduces to

 

 

 

0p 8,02)
— = 0 . 2
(9t + 82 ( )
The equation of momentum conservation
apv 8pv2 a
—— —_ v = 0 , 3
at 8: +az(p+q)+pg ()
the equation of internal energy conservation
ape pre (91) 8
—- v FC F, = , 4
a,+,g+(p+q),— +5; + )— Q o ()

and the level population equations

 

8m+ an 2) N
8t+ 37 (Zan ii MEAL) =0 (5)

” J75 196i

are similarly expressed. In these equations 2, t, p, v, e and p have the usual deﬁni-
tions of height, time, density, velocity, internal energy per unit mass, and pressure
respectively. The qv that appears in equations (3) and (4) is the viscous stress, and g

is the acceleration due to gravity. Q represents the additional non—radiative heating

(including that due to the ﬂux of non~thermal electrons), and FC and F, refer to
the conductive and radiative ﬂuxes. Within the level population equations are the
number densities in a given state, 72,-, along with the transition rate per atom, PU,
from state i to state j. N is the total number of atomic states calculated in detail.

Although it is desirable to avoid excessive amounts of numerical diffusion, a
certain amount is necessary so that the solution remains numerically stable over
successive iterations. To achieve this stability, the viscous stress, qv of equation (3),
includes two terms: a term that is proportional to the velocity divergence to stabilize
shocks, and a term that is proportional to the sound speed to stabilize low amplitude
waves. The conductive heating found in the equation of internal energy conservation
( equation (4)) can be taken as the divergence of the classical Spitzer (1962) value of

the conductive ﬂux,

Fe, = wit/23; . (6)

Here, T refers to the gas temperature, and KITS/2 is the conductivity which in gen-
eral, can be a function of electron density as well as temperature. However, this
assumption allows for unphysically large values of conductive ﬂux if the gradient of
the temperature is locally steep. This situation can occur in the solar transition re-
gion, where temperatures can increase two orders of magnitude over a narrow span of
only a few hundred kilometers. To alleviate the possible overestimation of conductive

heating in this region, the saturation ﬂux of Smith & Auer (1980) is used to place a

physical upper limit on Fe:

 

F58, = sgn(Fc1) (kT)3/2 . (7)

ne
4,/m,3
The coefﬁcients n6, me, and I: refer to the electron number density, the mass of the

electron, and Boltzmann’s constant respectively. Following Fisher et al. (1985a) the

conductive ﬂux is expressed as

Fcl
Fe: 9
(1+Fcl/Fsat)

 

(8)

which tends to the saturation limit when the classical ﬂux approaches or exceeds this
limit, and tends to the classical value when FC << Fsat . The subtleties involved in
the determination of the radiative ﬂux and the calculation of the level populations

and transfer probabilities of equation (5) are discussed in section (2.5).

2.2 Complete Linearization

The system of equations (2) - (5) is solved implicitly via multi-dimensional Newton—

Raphson iteration, a technique based upon the Taylor expansion:

6 z
f()=x+6x x)+:a fie-6 ,(2.+06x) (9)
j=la
The functions f,- of the variables x=(p, v, 6, n1, n2, nN) are the N’=N+3 con-

servation equations to be solved. For an x that is not the solution vector, setting

f,(x+ 6x) = 0 implies that 6x is the correction vector which, when added to x, gives

10

the zero of each f,. In practice, an initial guess of x is made, terms of order 6x2 and
higher are neglected (thus this method is often referred to as “complete lineariza—
tion”), and an approximate correction vector 6x is determined by the solution to the

matrix equation:

ft(XI :- — Z 8—,, x.» (10)

This correction vector is then added to the initial guess, and the process is iterated
until the error is suitably small. Unfortunately, if the initial guess is far from the
true solution, higher order terms in the Taylor expansion cannot be neglected, and
this approach will diverge. On the other hand, if the initial guess is a good one, then
the method is very efﬁcient, and converges rapidly (quadratically, under the best of
circumstances) to the solution vector.

The correction vector 6x is expressed in terms of the conservation equations and
their ﬁrst derivatives. In order for this procedure to be implemented on a computer,
each derivative within f, is expressed as a ﬁnite difference centered in a computational

zone. For example, a spatial derivative of the velocity is cast in the form

0'0 n vii—1"”?
,_ = 11
(02). 27-1 — z? I ( )

where 2;-1 refers to the value of the height variable at grid point j and timestep n.
Since 2: is taken to increase along a descending grid, the denominator in equation

(11) is positive. Each timestep takes as the initial guess the solution of the previous

timestep. In general, this provides a guess that is sufﬁciently close to the solution

11

of the matrix equation, and the iterative process quickly converges to a solution. If,
however, the iteration diverges, as may happen if the guess is too close to a local
extremum in one of the ﬁrst derivatives of equation (10), the timestep is reduced,
and the initial guess is then reﬁned. Thus, the size of the timesteps is controlled by
the rate of change in the variables, not by the Courant condition (At = Az/s, where

s is the sound speed in the medium) as is necessary in explicit differencing schemes.

2.3 The Adaptive Grid

Another equation must be added to the system in order to allow the grid itself to
respond to rapid changes in the variables or their derivatives. This is crucial in
the effort to model ﬂares, as features propagate through the atmosphere that require
dense spatial distributions of grid points in order to be properly modeled. For example
during a strong ﬂare, very narrow (100—200 kilometer) regions of elevated density
form in the upper chromosphere. If these regions are not sufﬁciently resolved, their
contributions to the emergent radiation ﬁeld are lost. Additionally, the transition
region can move thousands of kilometers from its original position in only a few
seconds of solar time. The grid must be able to resolve this and other regions of rapid
change, otherwise the system of equations will not converge to a solution. With an
adaptive grid, it is possible to limit the total number of spatial points needed in a
computation, since the grid points will collapse in regions where high resolution is
required, and spread out in areas where such resolution is not crucial.

Dorﬁ & Druri (1987) deﬁne the point concentration n,- to be the number of grid

12

points per length scale

71., = Z/(Z,_1— 2,) . (12)

Z is a natural length scale that in general, can be a function of height 2, and time,
t (in the ﬂare simulations, Z is taken to be the size of each zone if all available grid
points are equidistant). They further deﬁne a quantity R as the desired resolution,
then develop an equation that rearranges the grid such that n o< R. However, if
n changes too rapidly in space or time, numerical instabilities will result. Thus in
practice, it is rearranged such that it is proportional to the result of applying spatial

and temporal smoothing operators to R.

 

The choice of R = ds/dz : \/l + (df/dz)2 is motivated by the desire to dis—
tribute grid points uniformly in path length along a given function f. Dorﬁ & Druri
(I987) extend this idea to a set of functions (taken to be the functions x deﬁned in

Section (2.2)) and express the desired resolution as

, 1/2
N Z Cll‘j 2
“—(IJIgIXjEE) . (13)

Here, X, is a scale factor associated with each function 13,, and is taken to be the

 

value of the function at, at the height at which R is being evaluated. This particular
choice of R resolves regions where there are steep gradients in the components of
x. Carlsson & Stein (1997) multiply Z by a weighting factor, 10,-. This factor is
a constant parameter assigned to each function so that preference can be given to

those that require a high level of grid sensitivity. A8 a practical matter, the ﬂare

13

simulations required large weights on temperature, velocity, and pressure in order
to properly resolve the rapidly changing position of the transition region and the
development of shocks. Moderately strong weights were also assigned to each of the
ground state population densities, as this greatly enhanced the convergence properties
in some of the more energetic ﬂare simulations. The remaining variables were not
critical to either the resolution or convergence properties of the atmosphere, thus
their associated grid weights were set to zero, and their changes had no effect on grid
motion.

In order to resolve regions where there are large second derivatives (that is, where
there are sharp corners in .r,), the desired resolution can be modiﬁed to include an

additional term,

I I 1/2
N w-Zdr- 2 N w’Z2 dzx' 2
R:(1+-1()Ij 42]) +21( 3‘3 422]) I (14)
J: .7:

However, in all but the weakest ﬂare simulations, small non-zero second derivative

 

 

weights (103) on temperature caused the grid to be oversensitive to peaks near the
points of ﬂare energy deposition. In general, numerical disadvantages outweighed the
advantages of non-zero second derivative weights, thus to;- was set to zero for all j.
Numerical stability requires that the grid separation should not vary by more
than thirty percent from point to point. Dorﬁ & Druri (1987) achieve this by intro-

ducing a rigidity constant 0 (taken as 2 in the ﬂare simulations) and rewriting the

14

proportionality n,- o< R,- as

ii, E n,- — a(a +l)(n,-+1— 2n,- + n,_1) o< R,- . (15)

This formulation results from the choice of a spatial smoothing operator acting on R,-
that is the Green’s function corresponding to the difference operator 1 — a(a + 1)A2.
Similarly, a temporal smoothing operator acting on R,- is chosen such that the left

hand side of equation (15) can be expressed as

a,- E m + —(a’.‘ — arr-1) oc 12,-. (16)

Here, the superscript n refers to the discrete timestep, and At to the time interval.
Details on the derivations of these expressions can be found in Dorﬁ & Druri (1987).
In practice, the smoothing parameter 7‘ had to be quite small (x 10‘5 8), since
explosive events during the ﬂare simulations required the grid to respond very quickly.

Elimination of the proportionality constant gives the ﬁnal form of the grid equa-

tion,
72,-1 ii,-
__ = _ . 17
Ri—l Rt ( )

Equation (14) expressed in terms of ﬁnite differences deﬁnes R,, and n, is given
by equation (16). Figure (I) shows the response of the grid to rapid non-thermal
ﬂare heating. Each line follows the position of an individual grid point as the run
progresses. Note how the grid follows the evolution of the atmosphere, responding

quickly to the large changes in temperature near the height where ﬂare heating is

15

0.20

0.15—

 

Time (s)
o
8

0.05

0.00

Figure 1.

 

1400 1200 1000 800
Height (km)

An example of grid motion during the onset of a strong ﬂare.

 

600

at its peak (z=1.1 Mm). The concentration of points moving upward through the
chromosphere toward the transition region (the region of high grid density just below

z=1.4 Mm) shows how the grid responds to a velocity wave.

2.4 Non-Linear Advection

The second term in equations (2)-(5) describes the advection of mass, momentum,
energy, and population density respectively. At each cell interface j, the amount of

a quantity q,- transported through the interface in a time (it is given by
F:<q,-><v,-—u,->6t, (18)

where the averages denote time averages, vj refers to the ﬂuid velocity at the interface,
and u,- refers to the velocity of the interface itselfas a result of grid motion. Care must
be taken when determining interface values of the cell-centered, advected quantities.
In principle, these values may differ greatly from one another, as is the case at a
shock front. In this situation, the use of simple arithmetic averaging to determine
the interface values will invariably lead to excess numerical diffusion over time; the
averaging procedure will eventually wash out the sharp differences. To mitigate these
effects, the interface value of an advected quantity is determined using the technique

of van Leer (1977). Speciﬁcally, q, is determined by upwind linear interpolation,

1 1
(13' = (2 — 31') ((11+1/2 “ §de+1/2)

1 1
: (5 + 8]) (qj—l/Z ‘— 6'de_1/2) - (19)

H

17

s,- = sign(1/2, 2),) is a switching factor equal to 1/2 if v, is zero or positive, and -1/2
if v,- is negative. Recall that 2 increases with decreasingj (a convention that becomes
convenient when the optical depth scale is introduced), thus qj_1/2 is located on the
upwind side of 2),. The switch 83- thereby chooses the upwind value of q,- to calculate
the interface value. The second order correction to the interface value, dq,_1/2, is

taken to be the harmonic mean of qu and qu_1,

2A%A%4
qu + ACE—1

 

dqj-I/z = (20)

if AqJ-qu_1 > 0, and zero otherwise. qu is deﬁned as the difference qj+1/2 — qj_1/2.
This formulation ensures monotonicity in the interpolation scheme, and prevents
numeric rippling as quantities are advected. Once the interface value of the physical

variable is found, it is time—centered as

< q, >2 0q}‘+1+(l— 0)q;-‘. (21)

Possible values of 6 lie between 0.5 and 1. The choice of 0.5 would seem ideal, but
unfortunately, this simple arithmetic time averaging results in numeric instability.

For all of the simulations, the stable value of 0 = 0.55 is used.

2.5 The Transfer Equation

The amount of energy transported by radiation in a frequency bandwidth dz/ across

an element of area dA with normal 11 into a solid angle d9 at a position r in an

18

inﬁnitesimally small time interval dt is given by

6E:I(r,n,1/,t)f'-ndAdtd1/dQ. (22)

The term 1(r, n,1/, t) is deﬁned to be the speciﬁc intensity, and provides the funda-
mental, macroscopic description of the radiation ﬁeld.

As radiation passes through matter, energy may be removed from the ﬁeld, and
transferred to the material. The amount of energy removed from a beam of radiation
in the frequency interval [1/,1/ + do], passing through matter of area dA normal to
the beam and thickness ds in direction 11, in solid angle d9 depends on the density

of interacting particles and their atomic cross-section,

(SE = x(r,n,1/,t) I(r,n,1/,t)dA ds dt d1/ d9. (23)

x(r,n,1/,t) is the opacity, expressed in units of (length) ‘1, and describes the total
extinction per centimeter of path length. It is proportional to the density of particles
interacting with the radiation and their cross-section.

The amount of energy emitted by particles in a frequency band [1/,1/ + du], in
the direction n into solid angle d9, from material of cross-section dA and path length

ds, in a time interval dt is

(SE = 1)(r,n,1/,t) dA ds dt d1/ d0. (24)

77 is the emission coefﬁcient. It has the units of energy per unit volume, per unit

19

time, per unit solid angle, per unit frequency.
With these macroscopic deﬁnitions, the transport of radiation energy through a

material can be described by a time—independent Boltzmann type equation,
(11 - V) I(r, n, V) = 17(r, n, u) — x(r, n, u) I(r,n,1/) . (25)

This equation simply says that the difference between the amount of energy that
emerges and the amount that is incident on a volume of material in a time interval
must correspond to the energy emitted by the material within the volume minus the
energy lost through absorption. In other words, photons do not spontaneously decay,
and must be either added to the radiation ﬁeld, or taken away by physical interaction
with a material medium. Assuming a plane-parallel geometry, the transfer equation

further simpliﬁes to its standard form,

81,,
t‘ a “ =11!“ — XVII... (26)
Z

 

where ,u is the direction cosine of the normal to the atmosphere. The shorthand
1w is used so that the notation is less cumbersome, and means simply that [(2) is
a function of frequency and direction. The height dependence of I (and the other
variables) is implicit in the notation.

In general, a spectral line corresponding to an atomic transition is not sharp.
Finite lifetimes of excited states and collisions with nearby atoms all contribute to

a spread in frequency that can be described by a normalized absorption proﬁle, (my.

20

Similarly, emission corresponding to an atomic transition has a spread in frequency
which can be described by a normalized emission proﬁle, div“. If it is assumed that
there is no correlation between the frequencies of absorbed and emitted photons, then
each is independently distributed in frequency across the absorption proﬁle; that is,
211,,“ : dun. This assumption is referred to as complete redistribution.

The absorption and emission coefficients can be written in such a way as to con-

tain the microscopic properties of the material. If complete redistribution is assumed,

then the absorption and emission coefﬁcients can be expressed as

X1211 = 011(711 — may) + Xuc
2hV3

771m = 7a,,G,,-n, + me (27)
(Mihalas (1978)). For bound-bound transitions, 0,, : B,j¢,,,,(h1/,j/47r) is the angle
and frequency dependent absorption cross—section for a transition from level i toj and
8,, is the Einstein coefﬁcient of radiative excitation. C3,, is the ratio of the statistical
weights of the two levels, g,/gj. For bound—free transitions, 0,, is the photoionization
cross-section, and Gij E ne<I>(T)e”“’/”T. Here, (I)(T) denotes the Saha ionization
equation in the form given by equation (5—14) in Mihalas (1978). The quantities
Xuc and 77% refer to any additional known, ﬁxed contributions to the opacity and
emissivity resulting from overlapping continuum transitions.
Since the absorption and emission coefﬁcients depend on population densities
of atoms at different energy levels, the determination of the radiation ﬁeld requires

the solution to the level population equations (equation (5)). Similarly, the solution

to the level population equations requires knowledge of the radiation ﬁeld via the
transition rates, 13,-, = 12,-, + C,,. C,, is the collisional transition rate for a transition

from state i to state j, and R,, is the corresponding radiative transition rate given

 

by
if. fo°°‘,‘,—-’;a,-.-G -(I.,,.+ 2’15) Cindy if i>j
12,-,- = (28)
1L1, 00° 71—20,,[Wd/1d1/ if i<j

(Carlsson (1986)). The extinction proﬁle, 925W, is in general a function of temperature
and ﬂuid velocity (taken to be a Voigt proﬁle in the simulations), and thus further
couples the transition probabilities to the momentum equation (equation (3)) through
its dependence on the ﬂuid velocity, 2)

Simply stated, the radiative transition rates ultimately depend on the number of
available photons at any given depth, angle, and frequency as well as the population
densities of the atoms that interact with the photons. In addition, a large fraction
of the radiation ﬁeld may be comprised of photons that were emitted from distant
regions. Thus, the equation of transfer is not only coupled to the rate equations, it
is both non-local and non-linear.

The method of complete linearization is well suited for ﬁnding simultaneous
solutions to such coupled, non-linear, non-local systems of equations. Given an ap-
proximate solution vector to the system of conservation equations (x of equation
(10)) along with an accurate determination of the deviation from the true solution
(the error term f,(x) of equation 10) the Newton—Raphson procedure requires only
an efﬁcient means of determining the Jacobian (8f,/0;r, of equation (10)) that gives

the sensitivity of the errors to variations in the variables.

22

In the case of the level population equations, the current error can be directly
evaluated, given approximate values of the level populations, atmospheric velocities
and a corresponding Feautrier solution for the intensity. However, an approximate
solution to the equation of transfer can be used to calculate the population correc-
tions (in,- (the components of the perturbation 6x of equation (10)), dramatically
reducing the amount of computer time necessary to obtain a result. As long as the
approximation is not too far aﬁeld from the true solution, the process will still yield
corrections that drive the values of the population densities toward the ﬁnal solution.

Linearization of the transfer equation is required, and proceeds along the lines
of Scharmer & Carlsson (1985). Perturbations of IV“, 11.4,, and X141 of the form

n+1 _ Tl n . . . .
1W — [W + 61W, are substituted into equation (26) to obtain

a TI 71 T1. 11 n n
115;:61“, = 677,,“ — [Md/xv“ — qu61uu' (29)
In this case, the superscript n refers to the current Newton-Raphson iteration, and

does not label a discrete timestep. The second order term involving 613,15qu is

neglected. The above equation can be recast in terms of the optical depth along the

 

 

 

line of sight of the current iteration, (17':# = ‘Xiiu dz, as
a (It: 6, I}
6 3,, = _ I“ + 13,, 1:1,. +613“. (30)
373,. x5, Xi.“

The optical depth scale has a simple physical interpretation. Recall that the opacity

X1211 has dimensions of cm '1, and hence I/qu can be interpreted as the photon mean

23

free path; that is, the average distance that a photon at a particular frequency 1/ can
expect to travel before interacting with an atom. Then Tm, is the number of mean
free paths along the line of sight at a particular frequency. The negative sign in the
deﬁnition of the optical depth scale is introduced in order that it may increase as an
observer looks inward into the atmosphere.

The source function is deﬁned as the ratio of the emission and absorption coef-

ﬁcients. Its corresponding linear perturbation is taken to be

6,11 6,11
'qu _ 1,, xi.” (3,)

n V# ,n ’
X1111 All“

653,, E

 

so that equation (30) can be expressed in the standard form

5‘
07'"

W

 

51;}, = 513,, — 653,, . (32)

The solution of this equation can be expressed in terms of an “approximate” linear

integral operator, AZ“, that acts upon the source function:

61” = mass, . (33)

1111

The expressions for the absorption and emission coefﬁcients (equation (27)) can be
linearized and substituted into equation (31). The perturbations for bound-free tran—

sitions (dropping the n superscript for brevity) can be cast in the form

6b,, = 6 (on-Cm, — 71,,ne<I>(T)e’—”V/kT)—f Kim)

24

= 0,6 (6n, - (I)(T)€_hV/kT(nj6ne '1' 726671,“)

— n,ned(<I>(T)e—h”/“T)) + time (34)

C2

2}. 3 .
67],“; = (i (laicnjneq)(T)e—hV/AT + 77W)

2], 3
: la“. ((P(T)€_hV/kT(nj6ne ‘1' 728671,)

C2

+ 71.,n86(<l>(T)e-h"/”T)) + 577% . (35)

Similarly, for bound-bound transitions, the perturbations can be written as

Iii/,- , g,-
6“, = 6 (8,,4—7r’ow (n,- — n,—) + X110)

.7

h i' i i
: Bij4—irj— (6(01,” (71,“ — 71,2) + 05,,” (6711‘ — g‘6ﬂj)) + JXVC (36)

J J

 

6771/11 :

: ———sz? (72,562,“ '1' ¢Uu6nj) + anuc - (37)

Thus, the linear perturbation 65,,“ is written entirely in terms of quantum mechanical
atomic data and the perturbations in level populations, electron densities, tempera-
ture, and ﬂuid velocities (through draw). The intensity perturbation 61,4, of equation
(33) can then be expressed as a linear integral operator acting on the coefficients 672,,
6n,, cine, (iv, and (ST.

This closes the system. With the speciﬁcation of the integral operator A“ de-
scribed below, the frequency dependent equation of transport is eliminated from the
matrix described by equation (10). Only one equation per atomic level remains,

though each is now coupled spatially to the population densities of other states.

The true formal solution to the transfer equation can be expressed in terms of

the classical A operator. For outgoing rays (11 > 0),
+ + 00 —(t,, — T )
AWISWI : [Wt :17. SW6 H W dtuu i (38)
VH
and for incoming rays (11 < 0),
- - —(r —t )
WI] [0 .. .. (39)

If the source function varies slowly with respect to the attenuation factor, then a
majority of the intensity will originate over a small range in optical depth. The
operator AW can then be approximated using the Eddington-Barbier relations of
Scharmer (1981) and Scharmer & Carlsson (1985). In such an approximation, the
source function is represented as a polynomial expansion in optical depth, and only
the linear dependence is retained. As such, the integral of equations (38) and (39)
can be represented by one point quadrature sums of the form

Aiilsuul z wquWUjIL) , (40)

1111

where the functional dependence on line-of—sight optical depth 7",,“ is implicitly un-
derstood. The assumption that the source function depends linearly on the optical

depth, SW, = c1 + c271,,“ yields values for the speciﬁc intensities of

I; = c1+c2(1+r,,,,)

26

Ill—p = C1 —' 62(1+ T1,“) + (C2 — C1)8_TV“ , (41)

a result of the integration of equations (38) and (39). A choice of quadrature weights

wjy = 1
wit” 2 1 — €_T"“ (42)
and functions
T+ = T“, + 1
_ _ Tug _
T — ——1 _ 64.” 1 (43)

will reproduce equations (41) from the formalism of equation (40). With the ap-
proximate AZ” operator now deﬁned, the perturbations of the population densities
are fully determined, and in principle, it is now possible to iterate the system of
equations (‘2) - (5) to a solution.

However, certain computational issues remain. The solution is intimately con-
nected to the physics of radiation transport; namely, the great variation in the opacity
(and thus the photon mean free path) over the frequency range of a spectral line. In
the line wings, the mean free path can be quite large, and in the line core, it can be
very small. A photon emitted at wing frequencies may travel great distances through
the atmosphere without interacting with another atom, while a core photon will likely

be absorbed before traveling very far from its point of origin. Non-local transport,

then, occurs most efficiently in the wings, and contributions from the core photons
play a secondary role. Contributions from the line core in regions of large optical
depth analytically cancel. If these terms are not removed from the expressions, they
can yield numerical residuals that can lead to inaccuracy in the computed results,
particularly if single precision arithmetic is used. To avoid this problem, Scharmer &
Carlsson (1985) precondition both the error terms and the linearized level population

equations by deﬁning a new operator,

6A2“ = AL, — 1 . (44)

Use of this operator removes the core photons, and vastly improves the numeri-
cal stability of the Newton—Raphson iterations by analytically cancelling the large,
lowest—order terms. Additional computational simpliﬁcations include setting 10;” = 0
when TV“ < 0.1, and using the diffusion approximation 5,,“ z BL,” (ie. LTE) for
TV“ > 10. Again, it is important to remember that all of these approximations are
used only to expedite the calculation of the coefﬁcients of the correction terms of the
level population equations, and are independent of the calculation of the error terms

on which the accuracy of the non-LTE solution depends.

28

Chapter 3

THE INITIAL ATMOSPHERE

3.1 Atomic Data

An important part of the model atmosphere calculation, as described in chapter (1),
is the non-LTE treatment of the radiation ﬁeld. Atoms important to the chromo—
spheric energy balance are treated in non-LTE: a six-level hydrogen atom, a six level
singly ionized calcium atom, a nine level helium atom and a four level singly ion-
ized magnesium atom. The atomic states, and the line and continuum transitions
calculated in detail are listed in Tables (1) - (6). Data such as oscillator strengths,

statistical weights, and atomic cross-sections were graciously provided by Carlsson

 

 

 

(1997).
Table 1. Atomic Parameters for Hydrogen ([H/H]=12.0, A=1.01)
Ion Energy (eV) Designation
H I 0.0 n=1
H I 10.19 n=2
H I 12.08 n=3
H I 12.74 n=4
H I 13.05 n=5
H II 13.60 00

 

29

 

 

Table 2. Atomic Parameters for Helium ([He/H]=11.0, A=4.00)
Ion Energy (eV) Designation
He I 0.000 132 ISO
He I 19.819 18 28 351
He I 20.615 13 23 150
He I 20.963 13 2p 3P:
He I 21.217 13 2p lPf
He II 24.601 13 251/2
He II 65.414 23 251/2
He 11 65.415 2p 2P;
He III 79.019 03 150

 

 

 

 

Table 3. Atomic Parameters for Calcium ([Ca/H]=6.36, A=40.08)
Ion Energy (eV) Designation
Ca 11 0.000 23
Ca II 1.692 2d3
Ca 11 1.699 2(15
Ca II 3.123 2p1
Ca II 3.151 2p3
Ca III 11.876 Ca III

 

30

Table 4. Atomic Parameters for Magnesium ([Mg/H]=7.58, A=24.12)

 

 

 

Ion Energy (eV) Designation
Mg II 0.000 2p6 35 251/2
Mg 11 4.426 2p6 4p 2 {72
Mg II 4.434 2p6 4p 2 30/2
Mg III 15.035 Mg III

 

The atomic collision strengths had to be extended to high energies, since during
the course of a strong flare, significant amounts of material can be heated to temper—
atures exceeding 107 K early on in the impulsive phase. Such data is relatively scarce,
thus it was necessary to use several different sources to compile the necessary colli-
sion strengths for Helium at high temperatures. Speciﬁcally, the collision strengths
of Lanzafame et al. (1993) for He I were used for atmospheric temperatures below a:
40,000 K; above that temperature, the high energy collisional data of Badnell (1984)
were used. He II collision strengths were obtained from Aggarwal (1992).

Other atomic species are included in the calculation as background continua in
LTE using the Uppsala opacity package of Gustafsson (1973). Complete redistribu-
tion is assumed for all lines, although for the Lyman transitions, the effects of partial
redistribution are mimicked by truncating the proﬁles at ten Doppler widths (see
Milkey & Mihalas (1973)). Optically thin radiative cooling due to brehmsstrahlung
and coronal abundances of carbon, oxygen, neon, and iron are not calculated in
detail, but their contributions to the radiative cooling rate, A(T), are included in

the equation of internal energy conservation (equation (4)) via an additional heating

31

Table 5.

b-b Transitions Calculated in Detail.

 

 

 

 

Atom Agj (A) Transition Atom Aij (A) Transition

H I 1215.70 Lyoz Ca II 8662.16 2d3 <———+ 2pl

H 1 1025.75 Lyﬁ Ca 11 8498.01 2d3 <—> 2p3

H 1 972.56 Ly7 Ca 11 8542.05 2d5 <-——> 2p3

H I 949.77 Ly6 aHe I 625.58 ls2 180 <-—> ls 25 3S1

H I 6562.96 Ha aHe I 601.42 ls2 180 <——> 1328 1So

H I 4861.50 H/3 He I 10830.29 ls 2s 331 <—) ls 2p 3P:

H 1 4340.62 H7 He I 584.35 Is2 1So (——-> Is 2p lP‘l’

H I 18752.27 Pa He 1 20580.82 ls 28 1S0 +——> ls 2p lP‘l’

H I 12818.86 Pﬂ He 11 303.79 ls 2S1/2 +—+ 23 2S”;

H I 40513.47 Ba He II 303.78 ls 281/2 (——+ 2p 2P;

Ca 11 3968.46 23 +—> 2p1 Mg 11 2800.41 2p6 33 281/2 {—9 2p6 4p 2P?”
Ca 11 3933.65 28 <——> 2p3 Mg 11 2795.59 2p6 38 2S1/2 +—-) 2p6 4p 2P;,,

 

aForbidden transition

term, chin = nenhA(T). The atomic absorption coefficients used in specifying A(T)

are those of Hansteen (1997).

3.2 The Pre-ﬂare Model

Two basic methods can be used to model the solar chromosphere. They have been
dubbed by Ricchiazzi & Canﬁeld (1983) as the “semiempirical” approach and the
“synthetic” approach. Semiempirical models ignore energy transport in the atmo—
sphere in favor of a ﬁxed temperature and density stratiﬁcation which reproduces
observed optical emission from static solutions to the equations of statistical equi-
librium and radiative transport. Synthetic models differ in that the atmosphere and

optical emission are produced as a result of a self-consistent solution of the basic

32

Table 6.

b—f Transitions Calculated in Detail.

 

 

 

Atom Age (A) Initial State Atom /\,-c (A) Initial State
H I 911.12 n=1 He 11 503.98 ls2 lSo
H I 3635.67 n=2 He II 2592.02 ls 23 3S1
H I 8151.31 n=3 He 11 3109.80 1828180
H I 14419.07 1124 He II 3407.63 182p 3P:
H I 22386.68 n=5 He II 3663.37 ls 2p 1P?
Ca 11 1044.00 28 He II 227.84 ls 231/2
Ca II 1218.40 2d5 He 11 911.36 2p 2P3
Ca II 1421.04 2p3 Mg II 1168.65 2p6 4p 2P‘f/2
Mg II 1169.49 2p6 4p 2P1?”

 

 

equations of hydrodynamics and radiation transport. Properly done, the latter will
result in an atmospheric structure that is as close to the solar chromosphere as the
assumptions implicit in the model will allow. Synthetic models have the added ad-
vantage that it is possible to analyze the energetics of the atmosphere, and thus
develop observational diagnostics that may aid in the interpretation of observations.
The disadvantage, of course, is they are far more difficult to generate. The synthetic
method is the method of choice if one wishes to understand the processes of energy
transport, heating, and radiation during a flare. This approach is used throughout
the course of this work.

The ﬁrst step is to produce a pre-flare atmosphere which serves as a starting
point for the flare simulations. The use of an existing semi-empirical state as the ini—
tial atmosphere would have the undesired effect of introducing non-physical dynamic

effects that do not directly result from the influx of flare energy. Instead, a synthet-

33

ically generated pre-flare state in hydrostatic and energetic equilibrium is required.
As described in Chapter (2), in order for the implicit Newton-Raphson scheme to
successfully converge upon a solution, it is necessary to have as the initial guess, an
atmosphere that is reasonably close to the true non—LTE solution. With a solution
vector of thirty variables, a depth grid of 191 points, a maximum frequency grid of
101 points over 5 angles, and an adaptive grid, generating a starting atmosphere that
will not immediately diverge is a non—trivial matter. To generate the desired pre-
flare state in hydrostatic, energetic, and radiative equilibrium, a multi-step process
is required.

The temperature and density stratiﬁcation of the initial state of Carlsson & Stein
(1997) is used as a starting approximation for the chromosphere. A quiet corona and
transition region from Hansteen (1997) is attached to the chromosphere, and this
atmosphere is then used as a starting approximation for input into a static version
of the radiation hydrodynamic code that treats only one, six level hydrogen atom in
LTE. In practice, any reasonable starting approximation for the atmosphere, could
be used at this stage, eg. the standard VAL3C chromospheric model of Vernazza et
al. (1981)) with a coronal structure calculated using a loop scaling law.

The temperature structure is held ﬁxed, and the electron densities and the LTE
populations of hydrogen are iterated to convergence. The next task is to move the grid
points into regions where high resolution is desired or necessary (such as the transition
region). The grid weights are slowly increased on each variable individually, until the
transition region is properly resolved and a sufﬁcient number of points appear in the

lower atmosphere.

34

One by one, the other atoms are added to the atmosphere, and the LTE solutions
found. Modest grid weights are used for each of the ground state population densities,
as the grid will need to adjust to rapid changes in these variables during the dynamic
runs. Once a ﬁxed temperature LTE solution is found for the atmosphere with all
four model atoms, and the grid is in place, that atmosphere is used as a starting
approximation to determine the non-LT E populations of hydrogen. Non—linearities
in the non-LTE system of equations will lead to convergence problems at this step, if
the LTE starting approximation is too far from the true solution. The convergence
properties are enhanced using the method of collisional-radiative switching described
in Hummer & Voels (1988), and implemented in the static version of the code of
Carlsson & Stein (1997). Once the atmosphere has converged to a solution, that
model is used as the starting approximation to ﬁnd the non-LTE populations densities
of helium, calcium and magnesium.

The static, non-LTE, ﬁxed temperature solution is then used as a starting ap—
proximation to solve the full set of radiation hydrodynamic equations. A dynamic
run is initiated where the coronal apex temperature is ﬁxed at 106 K at the upper
transmitting boundary, and an assumed amount of non—radiative quiescent heating
is applied over several grid zones near the bottom boundary to fix the gradient of
the temperature at the base of the photosphere. All variables are allowed to change,
and no external driving mechanism or source of heating is provided (except over
the several zones at the lower boundary). The atmosphere then relaxes to a state
of hydrostatic and energetic equilibrium. To test the dynamic stability of the ﬁnal

pre—flare atmosphere (hereafter referred to as PFl), it was allowed to evolve without

35

a source of heating for three hours of solar time. At the end of this simulation, the
average velocity ﬁeld did not exceed 25 cm sec '1, which is well below the atmospheric
velocities experienced during ﬂares.

The PFl state is shown in Figure (2) along with its pre—ﬂare distribution of grid
points. Figures (3) - (6) compare this synthetic atmosphere to the semiempirical
quiet VAL3C atmosphere of Vernazza et al. (1981), and to the semiempirical active
MFlME atmosphere of Metcalf (1990) (used as the pre-ﬂare atmosphere of Hawley
86 Fisher ( 1994)). The PFl state more closely resembles the VAL3C atmosphere,
but there are clear differences. PF 1 includes the corona, has a transition region
located at a smaller value of column mass, and has a somewhat different photospheric
temperature structure near the bottom boundary. The Ca II K proﬁle is rather
similar to VAL3C, but the Ha proﬁle differs somewhat in the wings. This difference
results from both the choice of the heating proﬁle used near the lower boundary,
and the fact that effects of line blanketing are neglected in the simulations. The
disparity could be mitigated by an iterative procedure that would determine the type
of boundary heating necessary to force the lower photosphere into the conﬁguration
of VAL3C, but such a process would require re-relaxing the atmosphere many times.
This would be costly in terms of CPU hours, and is unnecessary in any event. Slight
alterations in the structure of the pre—ﬂare atmosphere deep in the photosphere have
little effect on the atmospheric evolution once impulsive ﬂare heating has begun. As
demonstrated in the following chapters, it is the structure and energetics of the ﬂare
heated chromosphere and transition region that are ultimately responsible for the

excess optical emission observed during ﬂares.

36

 

6.0 I I I I I I I I I I I I I

I I T I

.01 .01
O U1
l

P
01

Log Temperature (K)
I I I I l I I I I I I I f I
I I I I I I I I I I I I I I I I I I I I

5‘
C
[III
1

 

 

3.5 i 1 i i i 1 i 1 i 1 i 1 i 1
-6 -4 -2 0
Log Column Moss (g cm”)

 

Figure 2. The temperature stratiﬁcation of the PF 1 pre—ﬂare atmosphere. Shown is
the logarithm of the temperature (K) as a function of the logarithm of the column
mass (g cm ‘2). The cross-hatches denote the preﬂare position of the grid points.

37

 

 

 

 

 

 

 

 

I I I I I III I I I 1I II II IITI I I I I I I I III I I I I I III I I I I I I I I I I I I I III I IIITIIT
i I _
4,4 — I. : PF1 _
u— I '''''' _ VALSC —1
l— I _ _ — _ MFIME —I
r— I .1
A I
x 4.2 f— '—
v |
a) I l I
L. _ I .1
3
4— __ | -
E I
m 4 O — 1 —
O. _ _
Q) - _
I—
U)
I— \ —1
3.6 _ —
11 I I I I I IJII I I I I I I I I I III I I II I III I II I I I I I I I I I I I I I I III I I I I II I I I I I I
-6 —5 -4 -3 -2 -1 0

Log Column Mass (9 cm'z)

Figure 3. The temperature stratiﬁcation as a function of log column mass (g cm ‘2)

for the three model atmospheres PFl, VAL3C, and MFIME.

38

 

   

 

 

 

 

13 II I I I I I I I I I I I I I I I I I I I I I I I I I I I ITI I II ﬁT I I I I I I I I I I I I I I I I I I I I II VI Iﬂ

_ 'l/ 1

- —l

h -1

I— d

A - n
T I— .1
E ‘2 T r “—
0 " _i
v — I —
>N )— l —4
t "‘ I —1
(I) _ _i
c — I q
Q) l- I _.
Q 11 T 1 i
C - ..
O _ a
L >— ..
*6 : Z
.2 i- q
I—L’ ’- l .4
8’ ‘0 T l i
_I _. I PFI .4
: g ------ — VALSC I

: ! — — — - MF1ME I

l— l —4

>— 0 -1

9 rlgLI I I I LJ LII I I I I I I I I I I I I I I I I I I I I I I I I I I I I II I LI I III I I I I I I I I I I I I I J_I_

Log Column Moss (g cm”)

Figure 4. The electron density stratiﬁcation (cm ’3) as a function of log column

mass (g cm ‘2) for the three model atmospheres PFl, VAL3C, and MFIME.

39

 

— I ' —— PF1 —
2 O ; I: II ------ — VALSC T
‘ ' II — — — - MF1ME

1.5- II —

3
3
C
2;:
C
O
U
(D
.C
+—
O
+— _
(D
>
I;
2
Q)
m
>
I:
V)
C
Q)
+—
C

0.5 — x \ _.

1.0._ I

 

 

 

 

 

 

-1.5 -1.0 -O.5 0.0 0.5 1.0 1.5
Wavelength from Line Center (Angstroms)

Figure 5. The Ca 11 K resonance line proﬁles for the three model atmospheres PFl,
VAL3C, and MFIME. Intensity is relative to the continuum, and the wavelength
from line center is expressed in Angstroms.

40

 

_ -—————— PF1 .
------ — VALSC _
————MHME _

.0 .o .o
38 O") (I)
1

Intensity Relative to the Continuum

.0
N

 

 

 

0.0 I I I I I I J I I I I I 41 I 4 I I I I I I

-O.4 -O.2 0.0 0.2 0.4
Wavelength from Line Center (Angstroms)

 

Figure 6. The Ha line core for the three model atmospheres PFl, VAL3C, and
MFlME. Intensity is relative to the continuum, and the wavelength from line center
is expressed in Angstroms.

41

3.3 Accelerated Electrons

The energy deposition rate in a plasma due to thick target heating from a ﬂux of accel-
erated, non—thermal electrons was originally calculated by Emslie (1978). Following
Hawley & Fisher (1994), a similar, but more generalized formulation is implemented;
one that takes into account the depth dependence of the hydrogen ionization frac-
tion. The electron energy distribution is assumed to follow a power law of the form
(E/EC)_‘S for E > EC. A low energy cutoff of the electron energy spectrum, EC, is
necessary so that the total energy deposited by the beam is ﬁnite. Following Fisher
et al. (1985a) and Hawley & Fisher (1994) the low energy cutoff is taken to be 20
keV, in accordance with soft X-ray observations. The parameter 6 is the electron
spectral index, which for the case of thick target scattering is related to the hard
X—ray spectrum by I(E) oc E‘(‘5"1) (Brown (1971)). A decrease in 6 increases the
ratio of high energy to low energy electrons in the energy spectrum, and thus corre-
sponds to a “harder” beam. Variation between observed values of the spectral index
(4 < (I < 6) has only a modest effect on the evolution of the atmosphere (Mariska et
al. (1989)). A spectral index of 6 = 5 is assumed in the models.

In cgs units, the resulting heating rate per hydrogen nucleus as a function of

hydrogenic column depth, N, is

 

4 a: ~5/2
QM) = Kama-mac. [g g] 5 (N (M) (45)

Emslie (1978), where e is the fundamental unit of electric charge, and 7 = :rA +

42

(1 — :r)A’. A and A’ are the Coulomb logarithms defined by Emslie (1978), and :1:
is the hydrogen ionization fraction as a function of column depth. The values used
for A and A’ are those of Ricchiazzi (1982): A = 65.1 + 1.5lnE — 0.5ln 72),, and
A' = 25.1 + In E (the same values used in Hawley & Fisher (1994)). The electrons are
injected at the coronal apex, and they propagate along magnetic ﬁeld lines (assumed
vertical and invariant at all heights), thus the pitch angle, #0, is unity. .77 is the time-
dependent non—thermal electron energy ﬂux that enters the magnetically conﬁned
loop at its apex. The quantity BJ;C((5/2,1/3) is the incomplete beta function for
rc = N/NC, where NC = IL0E3/67re47 is the maximum column depth that electrons
of cutoff energy EC can penetrate. N‘(N) refers to the ionized column depth, given
by N"(N) = fON 7(N’)/A dN', and N; is the cutoff column depth for electrons in an
ionized plasma (a: = 1, 'y = A). Details on the derivations of these expressions can be
found in Emslie (1978), Emslie (1981), and Hawley & Fisher (1994). The resultant
non-thermal heating rate is included in the equation of internal energy (equation (4))

as a source of external heating.

3.4 Soft X-ray Irradiation

At a given height in the atmosphere, the amount of heating due to soft X-ray ir-
radiation depends in general on the non—local contributions of all other regions of
the atmosphere that are sufﬁciently energetic. To produce X-rays, we use the op-
tically thin, thermal emission spectra of Raymond & Smith (1977) along with the
temperature and electron density stratiﬁcation of the atmosphere to determine the
volume emissivity (cu) at each layer. Following Hawley & Fisher (1992), we sum

43

the emissivities into discrete wavelength and temperature bins ranging from 1-250 A
and 1-40 million Kelvin, and at each height in the atmosphere determine the emitted
power per wavelength band. Regions contributing to the thermal X-ray emissivity
are assumed optically thin, thus their emergent intensity is simply the integral of the

volume emissivity over the path of the emitting region

1
If 2 [7/ SV(TU)(IT,, = —i/cu(z)dz. (46)

5V, a function of monochromatic optical depth TV, is deﬁned in section (2.5) of chapter
(2). If emissive material is close to an optically thick atmospheric layer (r S d/x/8,
see Figure(7)), then it is assumed that the atmosphere is uniformly illuminated and

the incident ﬂux is calculated using the plane parallel approximation
1 —1
Fu(r,,) = 27r/ glue—U" — ”rd/“(11¢ — 27r/ plue”(T" — t”)/”d,u. (47)
0 0

The X-ray emission of the lower temperature, optically thick layers is assumed negli-
gible, thus the speciﬁc intensity (1,, of equation (47)) reduces to the aggregate contri-
butions of the optically thin emitting regions attenuated by the optical path length.
The plane parallel contribution to the X-ray ﬂux at a given optical depth can then

be written as

00 Tu

€,,(t,,)E2(tu — TV) d1. — 271'] (”(1”)E2m — t.) dtu, (48)

O

FAT.) = 27r/

Tu

where E; is the second exponential integral.

44

In order not to overestimate the soft X-ray heating available from a loop geome—
try, we follow Hawley & Fisher (1994) and use the Gan & Fang (1990) approximation
which treats distant (r > d/x/8 in Figure (7)) emitting regions as point sources. This
point source contribution to the incident soft X-ray ﬂux as a function of optical depth
is given by

FU(T,,) : [:0 Ewe-(ti, - WWI/city _ A” W847} — tu)/#,dty .
(49)
The quantities r, z, z’, and p’ are shown in Figure (7). The distance from an atmo-
spheric layer to an emissive region, r, can be expressed in terms of one half the total

length of the loop, L = 7rR/2, and the separation along the loop, [2’ — 2|, by

r: ﬂsin—W—Iz'—z|. (50)

7r 4L

The direction cosine in terms of these fundamental quantities is given by

,u' = cosgb :- cos ﬁk’ — z). (51)

At the loop footpoint, these expressions reduce to those of Hawley & Fisher (1994).
The total X-ray ﬂux at any atmospheric layer, then, is the sum of the plane-parallel
and point source contributions of equations (48) and (49). Since the average scale of
the depth grid in the simulations exceeds the thermalization scale length of Henoux

& Nakagawa (1977), we assume that the energy of the photoionized electrons is

45

 

 

 

 

 

 

Figure 7. Loop geometry used in the calculation of soft X-ray heating. Symbols are
deﬁned in the text.

46

transformed entirely into heat within each atmospheric layer. The volumetric heating
rate for elementj is computed as in Hawley & Fisher (1992),

Fu
Q,- = / 72.j(hl/ — Xj)0'qu—dl/ , (52)
u ,I/

with the notable difference that neutral and singly ionized helium abundances are
now calculated in detail. For each element j, X, is the ionization potential, 72]- is the
number density, and 0],, is the collisional cross section. The hydrogen photoionization
and recombination rates are calculated in the same manner as in Hawley & Fisher
(1992) and are included in the population equation (equation (5)). The total ﬂare
heating at each level of the atmosphere is a sum of the thermal contribution from
soft X-ray irradiation and the non-thermal contribution of the accelerated electrons.
The ﬂare heating is included in the equation of internal energy (equation (4)) as a
source of additional heating (Q), ensuring detailed energy balance at all depths in

the atmosphere and at all times during the ﬂare.

47

Chapter 4

FLARE ENERGETICS AND DYNAMICS

To investigate the properties of the lower atmosphere during the impulsive phase of
a solar ﬂare, three levels of non-thermal electron energy ﬂux are considered: $2109,
1010, and 1011 ergs cm "2 s ‘1 (hereafter called runs F9, F10, and F11 respectively).
These levels correspond to weak (F9), moderate (F10), and strong (F11) ﬂare heating.
The quiescent atmosphere is heated for seventy seconds in the F9 and F 10 runs, and
for a shorter four second burst in the F11 run, which is computationally much more
problematic.

In each case, the atmosphere goes through two distinct phases. The first phase
is one of gentle evaporation, its general characteristics shown in Figures (8) - (12)
for the intermediate F10 run. This stage is characterized by the formation of one,
or possibly two, temperature “plateaus” in the upper chromosphere that result from
the direct deposition of energy by the non-thermal ﬂare electrons. These plateaus are
hot regions high in the atmosphere (but below the corona) where the temperature
gradient is near a local minimum. The second phase is one of explosive evaporation
(shown in Figures (13) - (16) for the F10 run). This stage is characterized by the
formation of large compression waves and shocks which propel mass upward into the

transition region and corona and downward toward the temperature minimum region.

48

5.0

4 . 8
4 . 6
A
5
a)
‘5 4.4
E
a: .
D- 4 2 .1“ II|| ‘
. l\ Ill m .
E \ ‘\\\\\ '
p.— . I”, "II'III
4 , O 0' 1'
. 8 .
3 z. 1,, ,, 7’ 1111
’/ l’l/ /// / l / "
.. :11 l
'2. 9 /”’///,’;’/o////’////"/////// . Ill .

,4 mm
@011/751/1/0/Ul/ 1 I' I "I, "ﬂu" "l
/ / l’WI”’3lf "'""”'"’”””"llu
%“// l)

[fl/I’ll, ' \
\\\\\\\\\\\\\\\\(,
\\\\\\\\\\\\

II ’/ I

/ ///
W’sz'ai’t
“Ml/”I, |
9’! ’Iql

’I
l/I’l

     

 

Q ."=>
(3 .Q x—x.\Q‘“

Figure 8. The temperature structure of the F10 chromosphere during the ﬁrst two
seconds of ﬂare heating. Shown is the log of the temperature (K) as a function of
height (Mm) and time (s). A temperature plateau at T w 104 K between z 1.0 and
1.5 Mm is well developed after m 0.23. Two plateaus (one at T z 10‘1 K and one at
T m 105 K are well developed after 2.08. The motion of the grid is also apparent.

49

-3

Log Gas Density (9 cm )

 

 

 

 

 

0.0 0.5 1 _° 1 .5
Ho‘gh‘t (Mm)

Figure 9. The density structure of the F10 chromosphere during the ﬁrst two seconds
of ﬂare heating. Shown is the log of the gas density (g cm ‘3) as a function of height
(Mm, 1 Mm = 106 meters) and time (s). The lack of large peaks near the point of
maximum ﬂare heating is indicative of the gentle phase.

50

U!

 

o
v
>
z: 1 3
m
c
a:
O
c
_._
8
n.
a: II”
I1,”Iz,”Ir, ’0'
I I l
’1/ ’I/ ’I/ ’Ix”//”/. H
040/00,] ’10 ’0’ ’0 ’I; '
I, ’l ’I/ ’0 //I ’11"
441mm, 1,, III/o,
”/,”//l’//”/ ”I '1:
I I I I/ ’l
”rz,”//”’//”II'
1 O a r,,//,, I”, 1,, .
’I/ ’1’, 01M. n
I I”. 0%10/1'1‘
\ .1 - '
-\ .Q
,_
22’
Q)
E:—
.P

 

.6
ivwva

Q .6
A
Q '0 <3 -7- Q ' ““9“"

Figure 10. The proton density stratiﬁcation of the F10 chromosphere during the
ﬁrst 1.2 seconds of ﬂare heating. Shown is the log of the proton density (cm '3) as a
function of height (Mm) and time (s).

51

a
z"
I
m
s
E
3:,
E 4
U
.0.
O
>

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Figure 11. The velocity structure of the F10 chromosphere during the ﬁrst 1.2
seconds of ﬂare heating. Shown is the gas velocity (km s") as a function of height
(Mm) and time (s). The modest ﬂows are characteristic of the gentle phase.

52

           

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53

After 0.2 seconds of ﬂare heating in the F10 run, the atmosphere has developed
a plateau at 104 K between % 1.0 and 1.5 Mm (shown in Figure (8)). At this
temperature, most (but not all) of the energy deposited by the non-thermal electrons
is radiated away, and the atmosphere is in a state of near equilibrium. This is not
a true equilibrium state however, particularly near the point where ﬂare heating
is at its maximum (z 1.1 Mm). There, the atmosphere continues to be heated,
and increased numbers of thermal and non—thermal collisional ionizations elevate the
number density of protons between R 0.9 and 1.2 Mm (see Figure (10)). These
ionizations are sufﬁcient in number to effect a change in the stopping depth of cutoff
energy electrons. Thus, even though there is no signiﬁcant material ﬂow in this region
(as shown in Figures (9) and (11)), the heating proﬁle of the accelerated electrons
(shown in Figure (12)) begins to change, and the point of maximum energy deposition
slowly moves upward toward the corona.

After 3 0.3 seconds, radiative losses are no longer able to balance a majority of
the non-thermal energy inﬂux, and the atmosphere goes through another time of rapid
heating. By s 0.7 seconds, optically thin metal losses are able to effectively mitigate
the inﬂux of ﬂare energy, and the atmosphere returns to a state of near-equilibrium.
A second temperature plateau is formed between z 1.1 and 1.5 Mm (see Figure
(8)), and is bounded below by a locally hot region where the ﬂare heating is at its
maximum. As before, the density structure is only modestly altered (see Figure (9)),
but the changes in the hydrogen ionization fraction affect the heating proﬁle of the
non-thermal electrons (shown in Figure ( 12)). The proﬁle spreads out and becomes

less concentrated, and the stopping depth of cutoff energy electrons continues a slow,

54

steady upward motion. This stage persists for m 25 seconds. During this time,
chromospheric material continues its gradual evaporation, and the transition region
slowly moves toward the corona.

After approximately 25 seconds, the radiative losses are no longer able to balance
the continued deposition of ﬂare energy, and the atmosphere enters the explosive
phase. The bulk of the energy of the ﬂare electrons is no longer radiated away, and
is thus able to do work in the atmosphere. Chromospheric material is rapidly heated
to coronal temperatures (shown in Figure (13)), and large amounts of material are
propelled upward into the transition region and corona (the “explosive evaporation”
described by Fisher et al. (1985a)), and downward toward the temperature minimum
region (the “chromospheric condensation” described by Fisher et al. (1985b)) as
shown in Figures (14) and (15). As large amounts of material move toward the
corona, many of the electrons thermalize before they reach the chromosphere, directly
heating the evaporated plasma and continuing to force this material higher up into
the loop. As the run progresses, the ﬂare heating proﬁle broadens, and develops
peaks that correspond to local density maxima in the atmosphere (see Figure (16)).

In the remainder of this chapter, the energetics and dynamics of the F9 and
F 11 runs are analyzed in detail with special attention paid to the regions where ﬂare
emission originates. The F10 run is an intermediate case with properties of both the
F9 and F11 runs; thus the above discussion should provide a general framework for
understanding the more detailed analysis each of the limiting cases. The F9 run is in
a state of gentle evaporation through the entirety of the ﬂare. The F11 run evolves

in the same manner as F10, but does so much more quickly. In chapter (5), the

55

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Figure 13. The F10 atmosphere during the time of explosive evaporation. Shown is
the log of the temperature (K) as a function of height (Mm) and time (3). Note how
the grid moves in response to the temperature and density structure. The narrow
local temperature minimum marks the condensation, and is located at z z 1.3 Mm
after 50 seconds. A chromospheric temperature plateau located between 2. 0.7 and
1.1 Mm persists through the explosive phase.

56

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57

the log of the gas density (g cm ‘3) as a function of height (Mm) and time (s). The
upward moving density peak is the evaporated material, and the downward moving

peak marks the chromospheric condensation.

 

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Figure 15. The velocity structure of the F10 atmosphere during the time of explo—

sive evaporation. Shown is the gas velocity (km S“) as a function of height (Mm)

and time (8). Negative velocities denote downward moving plasma (ie. toward the
58

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59

resulting line and continuum radiation is described for all three simulations.

4.1 The Weak Flare

The case of the weakly heated atmosphere (run F9) is considered ﬁrst. As shown
in the ﬁrst frame of Figure (17), the accelerated electrons penetrate rapidly into the
upper chromosphere where a majority thermalize in a narrow region centered at a
height of 1.07 Mm (1 Mm = 106 m = 1 megameter, height is measured relative to
the point where T5000 : 1 ). This location corresponds to the maximum penetration
depth of cutoff energy electrons, where the bulk of the electron ﬂux is thermalized.
There, the temperature rapidly rises, and a local maximum appears at the point
where the ﬂare heating is strongest (shown in the ﬁrst frame of Figure (18)). A
modest peak in the gas pressure develops and instigates mass ﬂows both upward
toward the transition region, and downward toward the photosphere. As is evident
by twenty seconds after ﬂare onset (panel four of Figure (17)), increased neutral
hydrogen density in the upper chromosphere has reduced slightly the depth to which
the cutoff energy electrons can penetrate, and caused the heating proﬁle of the beam
to broaden, and its maximum to move toward the transition region.

When the accelerated electrons are stopped and thermalize, the local temper-
ature increases, collisionally populating the high energy bound states of atoms in
the region. Additionally, non—thermal Coulomb collisions increase the local electron
density by their direct ionization of neutral hydrogen and helium. This leads to a
signiﬁcant increase in the number of recombinations and de—excitations in the region
where ﬂare energy deposition is highest. The bound-bound de-excitations reduce

60

 

 

 

 

 

 

 

 

 

Figure 17. The F9 atmosphere. The solid line represents the log of the electron
density as a function of height, 2 (Mm), at time t (log n.3 , [nelzcm ‘3). The dot-
ted line denotes the pre—ﬂare electron density, and the dashed line is the log of the

non-thermal electron heating, scaled so that it will ﬁt on the plot (log Qe—+12.5,
[Qe—]=ergs cm ‘3 s ’1).

61

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Figure 18. The F9 atmosphere. The solid line represents the log of the temperature,
T (K), as a function of height, 2 (Mm), at time t. The dotted line denotes the pre—ﬂare

temperature structure.

the internal energy of the atoms, and transfer that energy to the radiation ﬁeld by
emitting photons. Similarly, the recombinations strengthen the radiation ﬁeld by
reducing the thermal energy of a local ﬂuid parcel. It is the total number of these
transitions that determines the overall level of radiative cooling at a given height in
the atmosphere.

After the initial seconds, energy deposited by the accelerated electrons is nearly
balanced by elevated levels of Balmer and Paschen recombination, Hﬂ and H7 emis-
sion, and the h and k resonance transitions of Mg II. Recombinations from the He
II ground state to excited energy levels of He I also strongly contribute to the in—
creased amount of radiative cooling in the region where ﬂare heating is strongest.
This strengthening of the radiation ﬁeld at deeper levels leads to slight increases
in the number of photon absorptions higher up in the chromosphere, providing an
additional source of heating in these layers as the ﬂare progresses.

The top panel of Figure (19) shows the temperature and density stratiﬁcation
of hydrogen, the proton density, and the normalized beam heating proﬁle of the F9
atmosphere two seconds after ﬂare heating has begun; the bottom panel shows the
state of the atmosphere after sixty seconds. After a full minute of continual non-
thermal heating, the gentle evaporation of chromospheric material into the loop has
increased the number density of hydrogen atoms between 1.4 and 3.8 Mm. At the
same time, the non—thermal collisions have elevated the hydrogen ionization fraction.
Thus, the accelerated electrons from the corona thermalize over a much wider region
as the ﬂare progresses. Increased numbers of thermal collisions local to the beam

heated area further populate the n=4 and n=5 energy states of hydrogen, causing

63

elevated levels of HF and H7 emission. These additional sources of cooling, along
with recombinations from He II to excited states of He I and resonance transitions
of Mg II, reinforce the recombination losses from the elevated electron density in the
area, and a local temperature minimum is formed at 1.28 Mm just below the point
of maximum ﬂare heating. A similar process is responsible for the creation after two
seconds of the temperature minimum at 1.05 Mm shown in the top panel of Figure
(19))-

Moderating the total cooling rate at 1.3 Mm are photoionizations from the N =2
level of He II and ground state hydrogen, ionizations from Ca II to Ca III, and Lya
absorptions at line core frequencies. However, in the 250 km wide region imme-
diately above 1.3 Mm, where the temperature is much higher, recombinations and
de-excitations at the above frequencies are the dominant processes. These transitions
become the primary source of radiative cooling in this region, mitigating the broad
inﬂux of energy from the ﬂare electrons. This alteration in the energy balance is
responsible for the abrupt change in the temperature gradient just above the point of
maximum ﬂare heating. Optically thin cooling from metal transitions is the dominant
cooling mechanism at the temperature peak, and prevents the non-thermal electrons
from heating the evaporated plasma above a 40,000 K.

The conservation of energy requires that the total energy gain or loss at any
point in the atmosphere must be due to the sum total of heat ﬂow, radiative losses,
viscous dissipation, and the work done by gravity, pressure, and the ﬂare electrons.
The atmospheric energy balance after one minute of ﬂare heating is shown in Figure

(20). In both the top and bottom panels, the cross-hatches denote the total gain in

64

Log Temperature (K)

Log Temperature (K)

 

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Height (Mm)

Figure 19. The F9 atmosphere at 28 (top panel) and 608 (bottom panel). The
temperature (solid line), total hydrogen number density (cross-hatches), and proton
density (dashed line) are shown in relation to the position of the normalized heating
proﬁle of the non-thermal electrons (dotted line). The densities are expressed per
cubic centimeter.

65

 

 

 

   

 

 

 

 

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Figure 20. Energy balance for the F9 run after 60 seconds of ﬂare heating. Shown
in the bottom panel is the non—thermal electron heating (solid line), the radia—
tive heating due to optically thick transitions calculated in detail (dashed line), and
radiative heating due to optically thin metal cooling (dotted-dashed line). In both
panels, the (cross-hatches) indicate the total gain in material energy. The top panel
shows the amount of non-thermal energy that is not radiated away (solid line), the
work done by pressure (dashed line), the work done by gravity (dotted line), and
the conductive heating (thrice dotted, once dashed line). The conductive heating is
non-negligible only in the coronal transition region, and to a much lesser extent at
1.7Mm. The viscous dissipation is everywhere negligible. Note the difference in scale
between the top and bottom panels.

66

material energy (the net atmospheric heating rate) per unit mass, as expressed by
the Lagrangian derivative of the internal and kinetic energy of the ﬂuid. The bottom
panel shows just how effectively the optically thick (dashed line) and optically thin
(dotted-dashed line) radiative cooling balance the high level of external ﬂare heating
(solid line). However, the balance is not exact; the radiative losses do not entirely
counteract the energy deposited by the accelerated electrons. This excess is indicated
by the solid line in the top panel of Figure (20). Although there is a net gain in
energy throughout the region, and the work done by the pressure (also shown in the
top panel of Figure (20)) in moving and compressing material is non-negligible, the

vast majority of the ﬂare energy in this case has been radiated away.

4.2 The Strong Flare

A signiﬁcantly higher level of thick target heating, corresponding to a non-thermal
energy ﬂux of .75'21011 ergs cm ‘2 s ‘1 (the F11 run), causes the atmosphere to respond
in a distinctly different manner (see Figures (21) and (22)). Initially, a majority of
the non-thermal electrons penetrate to a depth of 1.0 Mm, where they thermalize and
heat the atmosphere to z 104 K within a few milliseconds. As was seen in the F9 run,
modest pressure gradients develop in response to this rapid temperature increase, and
material begins to ﬂow away from the point of maximum energy deposition. As the
density increases, the cutoff column depth moves upward, and by 0.02 seconds the
beam begins to deposit much of its energy at 1.05 Mm (shown in the ﬁrst panel of
Figure (21)) creating the weak, local temperature maximum at that height evident
in the ﬁrst panel of Figure (22).

67

Directly below this point, sustained ﬂare heating has increased the number of
thermal collisions, populating the excited states of atoms in the region. At the same
time, direct non—thermal ionizations of hydrogen by the accelerated electrons have
increased the local density of electrons. Thus, as in the F9 run, the amount of
radiative cooling in this region due to de-excitations and recombinations increases,
causing the dip seen in the temperature proﬁle at 1.0 Mm. There is a second local
temperature maximum at 0.95 Mm, well below the point where the ﬂare electrons are
currently depositing most of their energy. Although the direct inﬂux of ﬂare energy
at this height is insufﬁcient to sustain this temperature peak, it persists due to the
high number of photoionizations that are driven by the recombination radiation from
above.

This early stage of atmospheric evolution is characterized by the formation of a
chromospheric plateau, referring to the broad region at z 104 K where the temper—
ature gradient is at a local minimum. This plateau is bounded from below by the
photoionization peak, and from above by the coronal transition region. The local
temperature minimum close to the lower boundary of the plateau behaves like the
chromospheric condensations of Fisher et al. (1985a), but it is not, strictly speak—
ing, the same phenomenon. The temperature minimum occurs without a substantial
increase in the local gas density, and without the downward motion and steep ve-
locity gradients usually associated with a fully developed condensation. The optical
line emission formed in this region is not signiﬁcantly Doppler shifted, and the pro-
ﬁles remain essentially symmetric about their nominal central wavelengths. As time

goes on, there is an increase in the temperature at the height where ﬂare heating is

68

 

 

 

 

 

 

 

 

 

Figure 21. The F11 atmosphere. The solid line represents the log of the electron
density as a function of height, 2 (Mm), at time t (log nc , [nc]=cm ‘3). The dot-
ted line denotes the pre—ﬂare electron density, and the dashed line is the log of the
non-thermal electron heating, scaled so that it will ﬁt on the plot (log Qc—+10.5,
[Qe—]=ergs cm ’3 3‘1).

69

 

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temperature structure.

70

strongest; a point which itself is not static, and slowly progresses upward towards
the transition region. After only 0.04 seconds, the optically thick transitions are no
longer able to mitigate the inﬂux of beam energy. Mass ﬂows increase, the point of
maximum ﬂare heating moves higher, and the temperature rapidly rises.

At a temperature of approximately 105 K, optically thin metal cooling is able to
balance the non-thermal power input of the ﬂare electrons, and a near equilibrium
state is formed. Mass ﬂows subside, and the point of maximum ﬂare heating settles
at m 1.1 Mm. However, this is not a true equilibrium state; particularly near the
coronal transition region (z 1.4 Mm) and at the point where ﬂare energy deposition
is at its peak. Figure (23) shows the energetics of the atmosphere after 0.5 seconds
of ﬂare heating. At 1.1 Mm, the energy input of the ﬂare electrons is almost entirely
radiated away; thus, the local temperature maximum is heated rather slowly. Near
the coronal transition region, the radiative loss rates are much reduced since the gas
density is signiﬁcantly lower than at 1.1 Mm. Flare electrons that thermalize in this
region are able to do more work in the atmosphere, instigating material ﬂows, and
rapidly heating the plasma. Similarly, in a narrow photoionization dominated region
at m 0.9 Mm, the energy deposited by the accelerated electrons is able to effectively
heat the atmosphere. Although coronal temperatures high in the loop exceed 106
K, there is not yet a sufficient amount material at that temperature to produce a
noticeable amount of soft X-ray heating in the chromosphere.

The atmospheric temperature structure is now characterized by two separate
plateaus at z 104 K and 105 K (shown in Figure (22) at 0.2s and 0.5S). As before,

the lack of large velocity gradients at the formation depth of emission lines keep the

71

 

 

 

 

 

 

 

 

 

b 400

'9')

To»

U, 200
9

m

is o

i’

o -200
a.

7m :
'm 500 :—
W Z
9 _
Q) ..
2 O

O :
g -500 j
o _
a r—

1.00 1.10 1.20 1.30 1.40 1.50
Height (Mm)

Figure 23. Energy balance for the F11 run after 0.5 seconds of ﬂare heating.
Shown in the bottom panel is the non-thermal electron heating (solid line), the
radiative heating due to optically thick transitions calculated in detail (dashed line),
and radiative heating due to optically thin metal cooling (dotted-dashed line). In
both panels, the (cross—hatches) indicate the total gain in material energy. The
top panel shows the amount of non-thermal energy that is not radiated away (solid
line), the work done by pressure (dashed line), and the viscous work (dotted line).
The conductive heating, buoyancy work, and soft X-ray heating are negligible by
comparison. Note the difference in scale between the top and bottom panels.

observed proﬁles generally unshifted and symmetric. These structures are relatively
long lasting; in the energetic F11 run this pseudo—equilibrium state persists for 0.6
seconds; in the weak F9 run it lasts throughout the entire minute-long event, as
shown in Figure (18).

In the F11 run, the slow heating of the chromosphere continues, until the at-
mosphere is no longer able to effectively radiate away the non-thermal energy input.
This occurs after 0.6 seconds of ﬂare heating. At that moment, a signiﬁcant portion
of the energy from the non-thermal electrons is no longer being radiated away, and
it is able to do work in the atmosphere. Material is forced away from the point of
maximum ﬂare heating (1.13 Mm) forming a shock front that rapidly moves upward
into the corona, and a narrow compression wave that propagates downward into the
chromosphere (see Figure (22) from 1.03 to 4.03). Within the downward moving
wave, high densities induce increased radiative cooling, and a local temperature min-
imum appears. This is the chromospheric condensation of Fisher et al. (1985a), and
the process just described is referred to as “explosive” evaporation by Fisher et al.
(1985b). Figure (24) shows an expanded view of the chromospheric condensation and
plateau that have developed after four seconds of strong ﬂare heating. In front of
the downward moving condensation is a narrow region at 0.97 Mm that is heated in
part by increased photoionizations stemming from recombination radiation produced
within the condensation itself. Above the condensation, a new transition region has
formed at R: 1.0 Mm, a location that is 400 km deeper than the position of the
original, pre—ﬂare transition region.

It is important to note that each distinct phase of atmospheric evolution has its

73

 

SOTTTIITTITTTTTIITTTTYTTTTTIITIIIIII[ITIIITITTTIIII

(II

.p

3
Log Electron Density (cm-3)

 

Log Temperature (K)

N

 

Lllllll/lllllllllllllllilLLLllllllllllli

 

 

ﬂ
A

3.51L1L1111111L41_LlllllllLLlJl1ll11111111111]1111111

0.60 0.70 0.80 0.90 1.00
Height (Mm)

 

Figure 24. The temperature structure (solid line) and the electron density stratiﬁ—
cation (dashed line) after 4.03 of non-thermal heating in run F11.

74

own deﬁnite observational signature. During the gentle phase, the absence of large
velocity gradients in the chromosphere leads to emission lines that are generally sym-
metric and non-distorted, reﬂecting the state of the atmosphere at the depth of their
formation. If the non-thermal energy input of the ﬂare is sufﬁcient, the atmosphere
will progress to the explosive phase. Much of the added optical emission during this
time originates in the high density condensation (and fast moving evaporation). The
emergent intensity reﬂects the material motion and steep velocity gradients of these
regions, thus observed line proﬁles are often highly Doppler shifted and asymmetric
about their centers. The next chapter focuses on the details of the optical radiation

formed at representative times during each phase.

75

Chapter 5

OPTICAL EMISSION

5.1 Continuum emission

Figures (25), (26), and (27) show the ﬂare continuum emission at the early and late
stages of the F11, F10, and F9 runs respectively. Figures (28), (29), and (30) Show
the time evolution (light curve) of the continuum for each run at 5000 A and in
the wings of Ha. It is clear that in each case, ﬂare onset is characterized by an
initial reduction in continuum intensity. This is a new and unexpected result that
requires some discussion. The case of intermediate ﬂare heating, the F10 run, is
considered in detail since the heating is strong enough to produce noticeable effects
in the continuum emission, and the timescale is long enough to be observable.

In Figures (31) and (32) the same plots are shown for the electron density, non-
thermal heating, and temperature proﬁles for F10, as were given previously for F9
and F 11. As accelerated electrons thermalize in a geometrically thin region in the
chromosphere near 1.1 Mm (as shown at 0.23 in Figure (31)), they liberate large
numbers of electrons from ground state neutral hydrogen atoms, thereby increasing
the local ionization fraction. As the accelerated electrons thermalize, the temperature

quickly rises (see Figure (32)), and the number of collisional excitations increase. This

76

Normalized lrradiance

Figure 25.

1.00
0.90
0.80
0.70
0.60
0.50

0.40: -'

0.023

 

 

Tﬁ

 

 

AALLAJ

LAlLl

 

 

0.4 0.6 0.8
Wavelength [pm]

recombination radiation.

KJ

K]

Normalized lrradiance

0.90:
0.80 r
0.70
0.60

0.50 .

0.40

Normalized Balmer and Paschen continua for the F11 run. The dotted
line is the continuum emission of the pre—ﬂare atmosphere. The early impulsive phase
is dominated by hydrogen photoionization, and the later impulsive phase by hydrogen

 

 

 

 

 

Wavelength [,um]

Normalized lrradiance

Figure 26.

1.00:
0.90:
0.80
0.70
0.60
0.50:

0.40:

 

 

 

 

 

0.4 0.6 0.8 1.0
Wavelength [pm]

Normalized lrradiance

1.00;
0.90
0.80
0.70
0.60
0.50

0.40'

 

 

 

 

 

0.4 0.6 0.8
Wavelength [pm]

1.0

Normalized Balmer and Paschen continua for the F10 run. The dotted
line is the continuum emission of the pre—flare atmosphere.

78

Normalized lrradiance

Figure 27.

1.00:
0.90
0.80
0.70:;
0.60

0.50:

0.40:

 

 

 

Normalized lrradiance

 

 

0.6 0.8 1.0

Wavelength [pm]

ﬂux in this case are barely noticeable.

1.00

0.90

0.80

0.70

0.60

0.50.

0.40:

 

VI'V

vlvrvvvvvrvlvvvvvvvv

"VY‘V'V'V

 

 
   
  

AA

ALLLLLALLLLLAAILAALLAA

 

IAIALLLA‘AAA‘ A

 

AAAAAAJLJIAAAAL‘

 

0.6

Wavelength [,um]

1.0

Normalized Balmer and Paschen continua for the F9 run. The dotted line
is the continuum emission of the pre-ﬁare atmosphere. The changes in continuum

 

 

 

Normalized Intensity

 

 

 

 

 

 

 

 

 

 

.Z‘ _
.5 _
C _
2 f
.E I
U _‘
Q) —i
.5 _
3 I
E —_
° :
Z _:

0.0 0.5 1 .0 1 .5 2.0

Time [sec]
Figure 28. Continuum intensity (normalized between the preﬂare value and the

maximum value at 2.05) at 5000 A (solid line) for the F11 run. The normalized Ha
line wing intensities are shown at +2.5 A (dashed line) and -2.5 A (dotted-dashed
line). For this case, the lag between the 5000 A continuum brightening and the Ha
wing brightening is R: 0.1 seconds.

80

,>_~ 1'07 __ _______————————-'-','-_’:-;:-'_'-.;—.V:VE
-— C / \ ________________________________ n
2 _ 2 r - ~ ................. 1
>— / ' —

.9.) 0.5 t-/ /./ l/_’—_————-——-——=
s :4 . :
'o _ ................................................................................................... _‘
Q) 0 0 _
.5 _
'6 _ _
E '0-5 ;— —_
L ,_ .—
29 * _
_1.0 '_' gm 1 . . A . . . . . . 1 . . - . . - . - A I - A . - z - - . - - . . . . . . .—‘

O 1 2 3 4 5

Time [sec]

>‘ 1.0 ,. "ﬁ-Tﬂfv" :.:____.—.._ ____________________ J
.6 .I' 1
§ 0.5 1
C ‘l
— A
‘0 ................................................................................................... L
Q) 0 O _4
.t' _
6 _
E -O.5 _ '__
L ._ _.
20 ~— ..
_1o0 _ A A A A I A A A A 1 A A A A l A - A A 1 + —"

O 5 1O 15 20 25

Time [sec]
Figure 29. Continuum intensity (normalized between the preﬂare value and the

 

 

 
   

 

 

 

 

  

 

 

 

maximum value at 25.05) at 5000 A (solid line) for the F10 run. The normalized
Ha line wing intensities are shown at +2.5 A (dashed line) and -2.5 A (dotted-dashed
line). For this case, the lag between the 5000 A continuum brightening and the Ha
wing brightening is as 1.08.

81

 

'o

IIITITTITI

9
U:

llllillllh

I
.0
U‘I

 

 

Normalized Intensity
C
o

iLLl 111111 1

I
'o

 

 

 

ﬁvﬁ’v'vvffﬁ‘rvvvlvv vvvvvv

—l
I

{
I.
.4
1

4

4

1

4

4

4

1

1
d
<

.

4

4

4

1

4

.0
01

I
.0
U1

Normalized Intensity
C
o

 

 

I
'o

 

 

Time [sec]

Figure 30. Continuum intensity (normalized between the preﬂare value and the
maximum value at 60.05) at 5000 A (solid line) for the F9 run. The normalized Ha
line wing intensities are shown at +2.5 A (dashed line) and -2.5 A (dotted-dashed
line). For this case, the lag between the 5000 A continuum brightening and the Ha
wing brightening is m 60.08.

 

 

 

 

 

 

 

 

Figure 31. The F10 atmosphere. The solid line represents the log of the electron
density as a function of height, 2 (Mm), at time t (log ne , [nc]=cm ‘3). The dot-
ted line denotes the pre—ﬂare electron density, and the dashed line is the log of the
non-thermal electron heating, scaled so that it will ﬁt on the plot (log Qe-+11.5,
[Qe—]=ergs cm ‘3 8‘1).

83

 

            
          

 

 

 

 

 

 

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1 2 3 4 ‘l 2 3 4

l\)
(24
.p.
l\)
(A
.p.

Figure 32. The F10 atmosphere. The solid line represents the log of the temperature,
T (K), as a function of height, 2 (Mm), at time t. The dotted line denotes the pre—ﬂare
temperature structure.

84

 

 

 

 

 

T Y I I Y I I 1 T I I I T r
0.6 h — 1 —> oo " —
- .......... 2 —> oo ..... d
_ _..__ 3 —> 00 d
_ ...... — 4 —> oo , _ 4
A .' Z
._ _____ _ 5 _> oo . .
U r—— ' 4 —4
a: 0.4 I
U , .
C F -
I _ _
.9
0?. - . _ . s
5 x x 2
0.2 — // \\ _
I .' / \ 1
C ’- ' / ‘_ \ : -
\ . z] / T,.\Ti\. \\
>— ' / / ‘~ \ -4
‘— ,-/_, 7;.- \ .\_ \ 3
~ 1"" ‘\_\‘\_ \2 _
o . o J" "if? 1 ‘3
— -H\- a
’- 1 1 l l l 1 1 l l l l L l l L d
0.8 1.0 1.2 1.4 1.6
Height (Mm)

Figure 33. The fraction of the total number of hydrogen atoms undergoing pho-
toionizations after 0.2 seconds of flare heating in run F10.

results in a sharp rise in the population densities of excited states of hydrogen which
leads to the increase in the number of photoionizations shown in Figure (33).
Photons from the photosphere that normally would escape and be seen as con-
tinuum radiation are now being absorbed by photoionizations in the upper chromo—
sphere. This is the cause of the observed reduction in continuum intensity. Figure
(34) shows that after 0.2 seconds of ﬂare heating, there are net population gains in all
of the excited states of hydrogen. The n=2 energy state is the most abundant of the
excited states in the chromospheric plateau, followed by progressively smaller densi-
ties in the more energetic levels. Conversely, ground state hydrogen populations are
reduced at the point of maximum ﬂare heating (z 1.1 Mm) In large part, this is due
to rapid collisional ionizations from interactions with the high energy non-thermal
electrons. Thus, there is little direct photoionization from the ground state, and
no noticeable dimming in the Lyman continuum. The large overpopulations in the
excited states lead to a signiﬁcant reduction in the Balmer continuum, followed by
progressively smaller reductions in the Paschen, Brackett, and higher order continua.
It should be noted that non-thermal collisional excitations and ionizations of
the excited states of hydrogen are neglected in the computation, since the detailed
simulations of Ricchiazzi & Canﬁeld (1983) show that the dominant non-thermal
contribution to the collisional rates comes from ground state ionization of hydrogen,
and that thermal collisional transitions between the upper levels of hydrogen domi-
nate the non—thermal contributions at temperatures above % 4000 K. If there were
signiﬁcant increases in non—thermal ionizations from excited levels, the amount of

continuum dimming would be reduced.

86

 

 

+ I I I I I I I I I I I I I I I I I I 44- I I I T I I I I I I I I I
+++
1 5 r W t—Os ‘ ‘ +++++

Log Number Density (cm—3)

 

 

 

 

 

 

 

++ /W V
" .\. ....... __ """"" \ ....... / ' \9/
10- \ ........... / ~ - “10.2:
* i .(7’
C
Q)
Q
. , E
‘ 3
Z
05
O
_l
O O
irriiiirliirlirrilll trillinlrlrlrlriirri
0.6 0.8 1.0 1.2 0.6 0.8 1.0 1.2
Height (Mm) Height (Mm)

Figure 34. Number Densities per unit volume of the hydrogen ground state (cross-
hatches), n=2 energy level (solid line), n=3 energy level (dotted line), n=4 energy
level (dashed line), n=5 energy level (dotted-dashed line), and the proton density
(thrice dotted, once dashed line). The left panel shows the pre-ﬂare density stratiﬁ-
cation, and right panel shows the density proﬁles after 0.25 of run F10.

87

There is a “turnover” point that occurs one second after ﬂare onset in the F10
run (see ﬁgure (29)) where the continuum intensity ceases to be less than, and begins
to be stronger than the pre—ﬂare value. This corresponds to the time when the total
number of recombinations equals the number of photoionizations in the plateau.
Although the hydrogen ionization fraction increases immediately after ﬂare onset,
recombinations do not immediately dominate the bound-free transition rates. The
time delay between the onset of the ﬂare and the point at which recombination
radiation becomes dominant can be understood by examining the ratio of the total

number of recombinations to photoionizations,

n R -
i: c c: . 53
as niRic ( )

 

Here, nc refers to the number density of ions and n,- the number density of atoms in

state i. The rates of radiative recombination and photoionization, RC,- and Ric, are

 

given by
Ric = 471' foo a,c(h1/)_1J,,d1/ (54)
and
. * oo _ 3
RC,- : 47r (331—) f 21—0 WW + Ju e-hV/dez/ , (55)
nc yo ht/ c2

where die is the photoionization cross—section at frequency V, JV is the mean intensity,
and V0 is the cutoff frequency for a bound-free transition from level i. The asterisk
superscript denotes LTE values of the density (details on the derivation of these

equations can be found in Mihalas (1978)). The rate of recombination, then, has

88

 

 

 

 

 

 

 

 

 

P-_——

AAAAAAAAAAAAAAAAA

A A

 

0.8 1.0 1.2 1.4 0.8 1.0 1.2 1.4
Height (Mm) Height (Mm)

Figure 35. The F10 atmosphere after 0.4 seconds of ﬂare heating near the time of
maximum Paschen dimming. The top left panel shows the ratio of the proton density
no to the number density of the n=3 energy state of hydrogen (log TLC/713). The top
right panel shows the ratio of the recombination rate to the photoionization rate
from the n=3 excited state (log Reg/R33) The bottom left panel shows the ratio of
the number of recombinations to the number of photoionizations (log ncch/ngRgc),
and the bottom right panel shows the ion density (log np, solid line) and the number
density of hydrogen in the n=3 energy state (log n3, dashed line) per unit volume.

89

 

 

 

 

 

 

 

 

 

 

 

0.8 1.0 1.2 1.4 0.8 1.0 1.2 1.4
Height (Mm) Height (Mm)

Figure 36. The same as Figure (35), but after one second of ﬂare heating in run
F10. This corresponds to the onset of continuum brightening.

90

a local component proportional to the Planck function, and a non—local component
proportional to the value of the radiation ﬁeld. The photoionization rate depends
only upon the latter.

The ratio 4b, (of equation (53)) and the contributions to this ratio are plotted
in Figures (35) and (36) for two representative times during the ﬂare. Figure (35)
shows the state of the atmosphere after 0.4 seconds of ﬂare heating; a time when
the number of photoionizations dominate the number of recombinations in the upper
chromosphere. Figure (36) shows the atmosphere after one second of heating when
recombination radiation starts to dominate, and Paschen brightening begins. It is
clear from the ﬁrst panels of Figures (35) and (36) that the population ratio nC/ng
is the determining factor in whether the broad emitting region between 1.1 and 1.4
Mm is dominated by photoionizations, and if so, for how long.

The time delay can be qualitatively understood as follows. The elevated level
populations of high energy bound states of hydrogen (shown in the bottom right
panel of Figure (35) for the n=3 excited state of hydrogen) are the result of increased
thermal collisions due to the inﬂux of ﬂare energy from the corona. However, this
population excess is short lived. The high number of photoionizations liberate elec-
trons into the continuum, and quickly depopulate the bound states of the hydrogen
atom. The number of photoionizations then fall just as increases in the electron
density cause a rapid rise in the total number of recombinations. The net result is a
switchover between the dominance of photoionization absorption and recombination
emission in the plateau.

The emission in the near wings of Ho (a more noticeable effect in the stronger

91

F11 run) also originates in the plateau, and begins almost immediately after the
onset of non-thermal heating, responding rapidly to the elevated population density
in the region. Since there are no great disparities between the population levels of
the n=2 and 1123 energy states of hydrogen during the early stages of the ﬂare, the
radiative balance is controlled by the strength of the non-local radiation ﬁeld and
the local value of the Planck function. The time lag that occurs between Ha wing
emission and continuum brightening is almost entirely controlled by the amount of
time it takes for recombination radiation to dominate the plateau. In general, the
stronger the event (ie. the larger the non-thermal electron energy ﬂux), the less
time it takes for this to occur, and the lag between the continuum and line wing
brightening decreases. Although a speciﬁc ﬂare event is not being modeled here, this
process may be responsible for the delay between Ha wing brightening and 5000 A
continuum brightening that has been observed on these time scales by Neidig et al.
(1993) in the white light ﬂare of March 7, 1989.

After fifty seconds of ﬂare heating, the energy deposition of the beam is less
localized, and material has moved away from the initial height of ﬂare energy depo-
sition. At this time, the hydrogen ionization fraction is approximately two orders
of magnitude greater than at 0.2 seconds. The resulting increase in electron density
throughout the plateau, and the elevated gas density within the chromospheric con-
densation both contribute to the signiﬁcant level of hydrogen recombination radiation
emanating from high in the atmosphere. The maximum brightening now occurs in
the Lyman continuum, followed by diminishing increases in Balmer, Paschen, and

higher order continua. The location of the emergent “white light” ﬂare is the broad,

92

t=50.05

 

 

 

Wavelength (nm)
Log Temperature (K)

 

 

 

 

1.0 1.2 1.4 1.6
Height (Mm)

Figure 37. The grey scale background contour denotes the continuum emergent
intensity contribution function plotted against height (Mm) and wavelength (nm) for
the F10 run 505 after the onset of non—thermal heating. Also shown is the temperature
stratiﬁcation of the atmosphere (log T (K)). The chromospheric plateau is the region
between z 0.7 Mm and 1.15 Mm, and the condensation is located between R 1.3 and
1.4 Mm.

93

high temperature chromospheric plateau, and the narrow, high density chromospheric
condensation, as shown in Figure (37). The source of the white light ﬂare is hydrogen

recombination radiation.

5.2 Line Emission

During ﬂares, mass motions and strong velocity gradients are associated with chro-
mospheric evaporations and condensations as material is accelerated through atmo-
spheric plasma. Mass motions local to an emitting region will Doppler shift the
discrete energies at which atomic species can either emit or absorb photons. The
peak opacity is therefore shifted by an amount corresponding to the local gas veloc-
ity (taken to be positive outward, in the direction of the corona), and consequently
the emission or absorption proﬁles are shifted. Strong velocity gradients distort the
normally symmetric extinction proﬁles; where they are negative, atoms lying increas-
ingly farther above a given atmospheric layer absorb radiation at progressively higher
frequencies, effectively blue-shifting the extinction curve. Similarly, positive velocity
gradients redshift the extinction proﬁle. The stronger the velocity gradient in an
emitting region, the more distorted an emission or absorption line can become.
Figures (38) and (39) show the time evolution of the Ca 11 K lines in response to
the moderate and strong ﬂare heating of runs F10 and F11. In each case, the early
phase of the ﬂare is characterized by relatively symmetric, non-distorted proﬁles
reﬂecting the pseudo-equilibrium state of the atmosphere at that time. The emission
in the Ca II K proﬁle after only 0.2 seconds of heating in the F10 run resembles the
emission from the semi-empirical preﬂare state of Metcalf(1990) shown in Figure (5).

94

 

wirTﬁrvvv'vvvavr IvvvvtvvvrTvv vv'vvvvrvvvv'rv vv'v‘lvv'vvverT
- 4 n I

8”t=o.2s § Tt=1.o § 1’ t=5.0s § ‘,.’t=3o.0s

 

 

 

 

 

 

 

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

     

  
 
 

I
\ ,——

 

 

 

 

 

O
.I.AAALL+A_ALAA AAlAL

-O.5 0.0 0.5 -O.5 0.0 0.5

 

Figure 38. The time evolution of the Ca 11 K line proﬁle for run F10. The vertical
axis is normalized intensity with respect to the continuum, and the horizontal axis
denotes the number of Angstroms from line center.

95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

oooooooooooooooooooooooooooooo

        

 

7.“. i".
0.0

  

 

 

   

n
l
.1 ..:T

0.....i .
-O.5 0.0 0.5

-O.5 0.0 0.5

4 l

-O.5

 

 

 

 

ALL.

0.5

 

Figure 39. The time evolution of the Ca II K line proﬁle for run F11. The vertical
axis is normalized intensity with respect to the continuum, and the horizontal axis
denotes the number of Angstroms from line center.

96

This stage of the ﬂare precedes the explosive phase (where steep velocity gradients in
the atmosphere distort the line proﬁles and cause distinct asymmetries as material
evaporates into the corona, and pushes downward into the chromosphere).

In order to determine the depth of formation of line radiation, the prescription of
Carlsson & Stein (1997) is followed, and the formal solution of the transfer equation

for emergent intensity (equation (26)) is cast in the form

1 21 _ u
IS = — Sune TV/l‘X—dz

[1 20 Tu

, (56)

Note that the integrand represents the fraction of the emergent intensity emanating
from height 2. This intensity contribution function is separated into the product of
three physically meaningful terms: Xu/Tu, Tue—TV/lf, and 5V. Xv is the monochro-
matic opacity per unit volume, which is proportional to the product of the density
of emitting particles and their collisional cross-sections. The ratio XU/Tu is quite
sensitive to mass motions in the atmosphere, and becomes large where there are
many emitting particles at small optical depth. Tue—TV/lt is an attenuation term
that peaks near TV 2 1. The source function 5,, is the ratio of the emissivity to the
opacity of the atmosphere. Complete redistribution is assumed; thus the line source
function is independent of frequency across the line core.

Figure (40) shows how, after .50 seconds of ﬂare heating in the F10 run, the Ca
11 K resonance line reﬂects the dynamic state of a radiatively cooled, dense conden-
sation moving downward into upward moving hot chromospheric material. Behind

the compression front, the contribution to the absorption (and thus the value of the

97

 

Height (Mm)

Height (Mm)

 

40 20 0 -20 -40 40 20 0 -20 -40
Au (km/s) Av (km/s)

Figure 40. The components of the intensity contribution function for Ca 11 K after
50 seconds of ﬂare heating in run F10. See text for details.

98

optical depth) is strongest on the blue side of the proﬁle, thus XU/Tu is strongest in
red. In front of the wave, the opposite is the case, as optical depth becomes large on
the red side of the proﬁle. The source function increases along with increased density
in the condensation, but is still strongly decoupled from the Planck function. The
contribution to the emergent intensity C,, a product of the three terms 5V, Tue—TVA”,
and XV/TV, is shown in the lower right panel of Figure (40) along with the Ca II K
line proﬁle. The strong “central” peak emanates from a broad region of heated gas
between 0.8 and 1.1 Mm near to optical depth unity where densities are high enough
to couple the radiation to the thermal pool. However, the red line broadening is a
direct result of the increased number of emitters in the relatively optically thin down-
ward moving condensation front. The smaller, blue peak originates between 1.05 and
1.15 Mm where the atmosphere is near to LTE and the source function is strong.
Since the extinction proﬁle is considerably steeper on the blue side of line center,
and since there is a reduction in Xu/Tu along the blue side of 73,21, the blue peak is
less prominent than the central feature. It should be noted that features extending
far into the wings of the lines should be interpreted cautiously, since this analysis
assumes complete redistribution over all transitions.

After four seconds of strong ﬂare heating in the F11 run, a chromospheric con-
densation just below 1 Mm is well developed. The large red peak in the Ca 11 K
line shown in the last panel of Figure (39) (and the redshift in the Ha line shown in
the last panel of Figure (43)) mirrors the 40 km s‘1 downward ﬂuid velocity within
the geometrically thin condensation. Elevated densities of Ca 11 in the condensa-

tion collisionally couple the radiation ﬁeld to local conditions, thus the strength of

99

Height (Mm)
O 7‘ 7‘
to o o
u o u:

.0
40
o

0.85 *

1.05

0.95

Height (Mm)

0.90 _ ._ log 8,

0.85 ‘

 

60 4O 20 0 ~20 -40 -60 60 40 20 O -20 -4O -60
Au (km/s) Au (km/s)

Figure 41. The components of the intensity contribution function for Ca 11 K after
4 seconds of ﬂare heating in run F11. See text for details.

100

this emission peak reﬂects the local gas temperatures within the condensation. Fig-
ure (41) shows how the emission core originates below the condensation at the top
of the chromospheric temperature plateau located at 0.9 Mm. At this height, the
source function is again coupled to the Planck function; however, the core remains
bright even though the plateau is substantially cooler than the condensation. This
is because contributions to the core intensity occur over a signiﬁcant portion of the
chromospheric plateau, owing in part to the steep proﬁle of the extinction curve, and
in part to elevated emission through the region.

The temporal evolution of the H0 emission line is shown in Figures (42) and
(43) for the F10 and F11 runs respectively. In both cases, one second of ﬂare heating
has yet to produce velocity gradients near regions that signiﬁcantly contribute to
the emission, therefore the proﬁles remain symmetric. However, in the case of the
strong ﬂare (run F11), a greater number of energetic electrons are able to penetrate
deeply into the chromosphere. The resulting shifts in the density structure move line
center optical depth TV 2 1 slightly deeper in the atmosphere; just below the initial
thermalization point of the cutoff energy electrons, in a region where the source
function has become strong and thermal depopulation of emitters less pronounced.
Thus, while the line is nearly twice as strong as in the case of the moderately strong
ﬂare, the central reversal is signiﬁcantly weaker. The development of the blue bright
point in the near wing of Ha at 1.5 seconds in the F11 run marks the onset of explosive
evaporation, and the redshift marks the downward mass ﬂows in the condensation.

Unlike calcium, the Ho radiation ﬁeld is signiﬁcantly de—coupled from local con-

ditions throughout the condensation. In particular, the line source function falls off

101

 

 

 

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v v v v v v

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t=70.0s §

 

 

 

 

Figure 42. The time evolution of the Ho line proﬁle for run F10. The vertical axis is
normalized intensity with respect to the continuum, and the horizontal axis denotes
the number of Angstroms from line center.

102

 

t=o.02s§ " t=0.25

 

 

 

 

 

 

 

 

Figure 43. The time evolution of the Ha line proﬁle for run F11. The vertical axis is
normalized intensity with respect to the continuum, and the horizontal axis denotes
the number of Angstroms from line center.

103

_ mix—T.) .
’~"' ’

.w s»
O U!

l"
01

Height (Mm)

Height (Mm)

 

200 100 O -100 -200 200 100 0 -100 -200
AV (km/s) Au (km/s)

Figure 44. The components of the intensity contribution function for Ha after 50
seconds of ﬂare heating in run F10. See text for details.

104

immediately below the interface between the condensation and the lower transition
region where the contribution to the emergent intensity is highest. Optical depth
unity occurs lower in the condensation, thus XV/TV dominates the intensity contribu—
tion. Fluid velocities in the evaporating plasma (at 1.4 Mm) exceed 130 km/s, and
as a result, the bright peak in the blue wing of Ha (seen in the ﬁnal frame in Figure
(43)) is shifted nearly 2.25 A away from line center. This emission stems from a
preponderance of emitting material in a region of small optical depth coupled with
an increase in the value of the source function towards the lower boundary of the
evaporation region.

The Ha proﬁle of Figure (44), formed at the same time as the calcium proﬁle
of Figure (40), reﬂects both the dynamic state of the condensation and the explosive
evaporation. Again, the downward moving gas in the condensation blue shifts the
absorption proﬁle causing an increase in XV/Tu in red. This increases emission on
the red side of the core, however the effect is less dramatic than the shift in the
Ca II K proﬁle since the Ha core is signiﬁcantly more broad. The more noticeable
effect near line center is due to the upward moving material immediately preceding
the condensation front. In this region, opacity increases blueward, and we see a
brightening in the blue peak close to line center.

The central reversal in the Ha core at this stage of the ﬂare is a consequence of the
sharp decline in the value of the source function at the upper edge of the condensation
near line center optical depth one. The central reversal is much stronger shortly after
ﬂare onset, since the thermalization of high energy electrons from the corona causes

additional collisional depopulation of emitting species near optical depth unity. The

105

bright feature in the blue wing originates in the fast moving evaporating plasma high
in the loop. This emission stems from a preponderance of emitting atoms in a region
of small optical depth coupled with a modest increase in the value of the source

function at the lower boundary of the evaporating region.

5.3 Soft X—ray heating and the Gradual Phase

In the F9 run, material in the upper chromosphere is not heated past 106 K, and thus
soft X-ray irradiation does not contribute to the energy balance in the atmosphere
either in the impulsive phase, or in the gradual phase (when the heating from the
non—thermal electrons comes to an end). However, in runs F10 and F11, greater
amounts of material are heated to higher temperatures in the upper atmosphere,
and the contribution to the energy balance from thermal soft X—ray radiation is
no longer negligible. During the impulsive phase of the F10 and F11 runs, this
contribution is, at its maximum, R: 3 orders of magnitude less than the energy input
provided by the non—thermal electrons, and as such has little effect on the dynamic
or emission characteristics of the atmosphere. However, after the ﬂux of accelerated
electrons subsides, the X-rays provide the primary source of heating throughout the
chromospheric plateau.

Computational difﬁculties prevented an analysis of the gradual phase during
the F11 run, as regions of explosive evaporation and strong condensations required
more grid points to resolve than were available in the atmosphere. This problem will
be addressed in future simulations, where the number of atmospheric zones will be

doubled. The F10 run did not present such difﬁculty, and was able to progress to

106

 

 

 

 

 

 

 

 

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The F10 atmosphere during the gradual phase, when the non-thermal

heating has come to an end. The solid line represents the temperature T (K) as a
function of height, 2 (Mm), at time t. The dotted line denotes the pre—ﬂare temper-

ature structure.

Figure 45.

107

 

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. ,, E m,
t=70.0s ‘" I t=71.0s : t=75.0s 3:" ) t=80.0s
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Figure 46. The F10 atmosphere during the gradual phase, when the non-thermal
heating has come to an end. The solid line represents the electron density as a
function of height, 2 (Mm), at timet (log nc , [ne]=cm '3 ). The dotted line denotes the
pre-ﬂare electron density, and the dashed line is the soft X-ray heating (log Q3+14.5 ,

[QI]=ergs cm ’3 s ‘1).

108

100 seconds. After 70 seconds, the level of non-thermal electron ﬂux was reduced to
zero over a period of one second. Figure (45) shows the evolution of the temperature
during this time, and Figure (46) shows the electron density stratiﬁcation along with
the X—ray heating proﬁle. The majority of the X-ray heating is concentrated in a
region just below the condensation in the upper plateau. Although the heating is still
concentrated locally, a greater fraction penetrates more deeply into the chromosphere
than did the heating from the non-thermal electrons during the impulsive phase.
The soft X-rays in the F10 run heat the chromospheric plateau, but are insuf-
ﬁcient to overcome the optically thick radiative cooling in the region. The radiative
losses in the plateau compensate for the additional X-ray heating providing the pri-
mary source of cooling in the plateau between z 0.6 and 1.0 Mm as shown in Figure
(47). However, the temperature remains high throughout the lower plateau, as the
p - dv (pressure) work continues to provide a source of material energy in the region.
Unlike the impulsive phase, it is the atmospheric pressure that now dominates the
energy balance in the lower plateau. The situation is different in the narrow region
just below the coronal transition region where the X-ray heating is highest (between
z 1.0 and 1.1 Mm). The increased density from the condensation provides an efficient
source of radiative cooling in the region, overwhelming the inﬂux of heating from the
soft X-rays. The radiative losses dominate the energy balance, and result in rapid
cooling, strengthening the local temperature minimum near 1.0 Mm as time goes on.
The static models of Hawley & Fisher (1992) and Hawley & Fisher (1994) ignore
the coronal X—ray ﬂux as a factor in computing the ionization equilibrium of helium.

The level populations of neutral and singly—ionized helium are thus overestimated.

109

 

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m
In
E”
Q)
is
3
3
o
o.
0.50 0.60 0.70 0.80 0.90 1.00
Height (Mm)

Figure 47. Chromospheric energy balance for the F10 run during the gradual phase.
Shown in the bottom panel is the soft X-ray heating (solid line), the radiative heat-
ing due to optically thick transitions calculated in detail (dashed line), and radiative
heating due to optically thin metal cooling (dotted-dashed line). In both panels,
the (cross-hatches) indicate the total gain in material energy. The top panel shows
the net radiative energy gain (solid line), the work done by pressure (dashed line),
and the viscous work (dotted line). The conductive heating and buoyancy work are
shown, but are negligible by comparison.

110

 

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ll1111111111llllllALLLLJllJllllll111111llllllllllllllllllllllllll

 

0.50 0.60 0.70 0.80 0.90 1.00 1.10
Height (Mm)

Figure 48. The soft X—ray heating proﬁle after 90.0 seconds of run F10 (during
the gradual phase). Shown, are the total X-ray heating (solid line), the contribution
to the X-ray heating due to hydrogen (dashed line), the contribution due to helium
(dotted-dashed line), and the contribution of other metals (dotted line). All quantities

are expressed as log Q3, where [Qr]=ergs g ’1 s ‘1.

111

This tends to reduce the amount of coronal X-ray ﬂux that reaches the lower atmo-
sphere, since large neutral helium populations are effective absorbers of soft X-rays.
Figure (48) shows the contribution of hydrogen, helium, and other metals to the
overall level of soft X-ray heating. In all regions of the lower atmosphere, it is the
helium atom that contributes most strongly. Although the total amount of X-ray
heating in the latter stages of run F10 is too little to overcome the radiative losses
in the chromosphere, stronger ﬂares (as modeled in run F11) heat large amounts of
chromospheric material to coronal temperatures more rapidly. It is in these cases that
the X-ray heating is likely to play a more signiﬁcant role, and this proper treatment
of helium will undoubtedly become a crucial contributor to the energy balance and
emission characteristics of the lower atmosphere.

Figures (49) and (50) show the time evolution of the Ho and Ca II K line proﬁles
through the early gradual phase. The immediate reduction in temperaturejust below
the coronal transition region reduces the number of hydrogen atoms in excited energy
levels, causing an immediate reduction in the core emission of H0. The third panel of
Figure (51) shows how the value of the source function mirrors the reduction in the
value of the Planck function in that region, even though the amount of emission does
not reﬂect the temperature of the surrounding plasma. The primary contribution
to the Ha core comes from this region; thus the emission line fades. The emission
in Ca II K is more persistent. As shown in Figure (52), the contribution to the
core emission at this frequency comes from a region between 1.1 and 1.2 Mm where
temperatures remain relatively high. The extinction proﬁle remains relatively steep

in the upper plateau, allowing for additional contributions to Ca II K core emission

112

over a relatively wide area.

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The time evolution of the Ha line proﬁle for run F 10 during the gradual
phase. The vertical axis is normalized intensity with respect to the continuum, and

the horizontal axis denotes the number of Angstroms from line center.

Figure 49.

114

 

 

   

 

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Figure 50. The time evolution of the Ca II K line core for run F10 during the gradual

phase. The vertical axis is normalized intensity with respect to the continuum, and
the horizontal axis denotes the number of Angstroms from line center.

115

’

Height (Mm)

\
l
I
I
I
l
I
l

V

Height (Mm)

 

0 0
Av (km/s) Av (km/s)

Figure 51. The components of the intensity contribution function for Ha during the
gradual phase after 90 seconds in run F10. See text for details.

116

'o

.0
on

Height (Mm)

53
0*:

Height (Mm)

 

-20 -40

4O 20 —20 —40 40 20

0 0
Av (km/s) Av (km/s)

Figure 52. The components of the intensity contribution function for Ca II K during
the gradual phase after 90 seconds in run F10. See text for details.

117

Chapter 6

CONCLUSIONS

6.1 Summary

Optical observations that are highly resolved spatially and temporally can provide
a detailed view of the dynamic and energetic state of the chromosphere during the
impulsive phase of a solar ﬂare. Atmospheric velocities can be inferred from detailed
analysis of shifts and distortions in the emission of lines formed at or near where the
bulk of ﬂare energy is being deposited. The results of these simulations are ultimately
aimed at a direct comparison with ﬂare observations, in an effort to understand the
transport of radiation, and the energetics of the chromosphere during the time when
energy influx from the corona is at its peak. The numerical simulations support
the hypothesis that a sudden ﬂux of non-thermal electrons is a major factor in the
appearance of optical continuum emission during strong ﬂares, and that that emission
originates throughout a broad temperature plateau formed in the upper chromosphere
immediately after ﬂare onset. Speciﬁcally, the primary conclusions of this analysis

are the following:

0 Even the strongest impulsive events can be described as having two phases,

a gentle phase characterized by a state of near equilibrium, and an explosive

118

phase characterized by large material ﬂows, and strong hydrodynamic waves
and shocks. The line emission proﬁles that occur during the gentle phase are
generally symmetric and undistorted, while those occurring during the explosive
phase are highly asymmetric and show large emission peaks and troughs. In
each case, general features in the proﬁles can be can be used to infer the dynamic

state of the atmosphere.

Recombination radiation from the chromospheric plateau and the chromospheric
condensation is the primary cause of the white light continuum brightening ob—

served during strong ﬂares.

The time lag between the brightening of the Paschen continuum and the bright-
ening of the near wings of Ha is controlled by the amount of time it takes for
electron densities in the plateau to become high enough, and the densities of
hydrogen atoms in high energy bound states to become low enough, to allow

recombination radiation to dominate the region.

During the impulsive phase, when the accelerated electrons are directly heating
the upper chromosphere, the contribution of the thermal soft X-rays to the
overall energy balance in the lower atmosphere is negligible by comparison.
After the non—thermal heating has come to an end, the X-rays are the dominant

contributor to the energy balance of the chromosphere.

119

6.2 Limitations of the Model

Although this analysis of the ﬂaring atmosphere is the most sophisticated of its
type to date, one must guard against an overinterpretation of the results. A one-
dimensional description of the solar atmosphere near the footpoint of a ﬂare loop
can bear a close resemblance to observations only if the observations are of sufficient
spatial resolution, and only if the deposition of the energy is localized. Both of
these conditions being met, a direct comparison with the observations is possible,
though several points should be kept in mind. First, this analysis assumes complete
redistribution over all transitions. The Ca II K line proﬁle will be affected to a degree
by the neglect of the effects of partial redistribution. Although much can be learned
about how the emission responds to the dynamic state of the atmosphere, the details
of the line profile itself, particularly in the wings, should be treated with caution.
Second, the details of the near wing emission of Ha in the quiet atmosphere and in
the late gradual phase will be affected by the neglect of line blanketing. This affects
the structure of the atmosphere in the upper photosphere, and overestimates the
amount of absorption in the wings of Ha, as can be seen in the comparison between
the semi—empirical VAL3C atmosphere and the pre-ﬂare state, PFl. Although this
limitation affects the details of the proﬁle, it will have little impact on the general
emission trend once significant ﬂare heating has begun since the wing brightening
originates in the upper chromosphere.

The ﬂare model atmosphere is subject to several other simplifying assumptions.

First, is assumed that the ratio of gas to magnetic pressure is low so that MHD

120

effects can be effectively ignored, and a plane-parallel, field aligned approximation
in the lower atmosphere is valid (an assumption that holds true for reasonable field
strengths (Nagai & Emslie (1984)). This one-dimensional treatment neglects the
sideways escape of radiation. Second, the X-ray and non—thermal electron heating
of the lower atmosphere is calculated using the simple, circular loop geometry of
Figure (7), rather than a more sophisticated tapered loop geometry than bears a
closer resemblance to a true magnetic structure. Finally, it is assumed that the ions

and electrons are at the same temperature.

6.3 Future Work

There are several issues that are of interest, and that could be explored with the
current set of models. Emission from the chromosphere is observed to persist long
after impulsive heating has subsided. It is possible that the contribution from thermal
soft X—rays is responsible for maintaining the high chromospheric temperatures, and
thus giving the long decay time of the line emission, but this has yet to be conclusively
shown. The F10 run did not heat large amounts of material to coronal temperatures,
thus there was only a modest amount of heating in the lower atmosphere due to X-ray
irradiation from above. There is likely to be a great deal more heating in the more
energetic runs as they progress into the gradual phase, and a comparison between
models that include the contribution of the X-rays and those that do not should yield
insight into the energetics of the atmosphere during the gradual phase. The standard
ﬂare model manifests itself empirically as a proportionality between the time integral
of the hard X-ray ﬂux and the observed thermal soft X-ray ﬂux. This correlation

121

is known as the “Neupert” effect (Neupert (1968), Dennis & Zarro (1993)). These
dynamic models can be used to test this effect for a variety of ﬂare energies.

The heating due to high energy, accelerated protons can be included in the
analysis, and the resulting optical diagnostics should be of importance in the debate
as to whether protons can produce the type and amount of observed optical emission
at the footpoints of the ﬂare loop.

The code can be modiﬁed to include the effects of partial redistribution, and this
should provide more realistic estimates of radiative cooling in the chromosphere, as
well as more realistic line proﬁles. Since the latest version of the code includes a self—
consistent treatment of the magnetic ﬁeld, the expression for the non-thermal heating
can be adjusted, and the effects of constant ﬁeld along the loop, and ambipolar
diffusion in the transition region can be explored and compared with the non-magnetic
runs.

Finally, this type of analysis can be extended to stellar ﬂares. Of particular
interest is the active M-dwarf AD-Leo. The abundance of ﬂare data makes it a

perfect candidate for further study.

122

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