Master Thesis Quantitative Analysis of Nonthermal ......Master Thesis Quantitative Analysis of...

Master Thesis

Quantitative Analysis of Nonthermal

Components in Solar Hard X-ray Flares

Observed with the Yohkoh Satellite

「ようこう」衛星で観測された硬X線

太陽フレアにおける非熱的成分の定量解析

Takashi Minoshima

簑島敬

Department of Earth and Planetary Science

Graduate School of Science, The University of Tokyo

Superviser : Takaaki Yokoyama

January, 2005

Abstract

The observed X-ray emission within a deka-keV energy range (called “Hard X-ray

(HXR)“) associated with a solar flare is thought to be evidence that deka-keV

nonthermal electrons are efficiently produced in solar flares. However, the particle

acceleration process in solar flares is still unknown. The analysis of solar hard

X-ray flares will be useful for discussing the particle acceleration process in solar

flares. In this thesis, we discuss characteristics of nonthermal components in solar

flares by analyzing solar hard X-ray flares observed with Yohkoh quantitatively

and statistically.

To discuss characteristics of nonthermal components in solar flares quanti-

tatively, the determination of the lower energy cutoff (Ec) in the spectrum of

nonthermal electrons is very important. However, it is not easy to derive it from

the observed HXR spectrum. Consequently, we try to derive it by assuming the

energy balance between nonthermal components and thermal components in the

impulsive phase. We apply this method to seven impulsive flare events observed

with Yohkoh, and we successfully estimate physical variables in flares such as the

Ec.

The values of the derived Ec are ranging in 20 - 45 keV. The validity of these

indirectly derived Ec is roughly provided from the HXR spectral analysis. We

suggest positive correlation between the nonthermal electron rate in the impul-

sive phase and the number density of the SXR emitting flare plasma in the pre-

impulsive phase. Positive correlation between the derived Ec and the spatial scale

of the flare is also suggested. It is the first time to our knowledge that such rela-

tionships are quantitatively shown. We expect that our work will be of benefit to

the understanding of the particle acceleration process in solar flares.

i

Contents

Abstract i

1 General Introduction 1

2 Instrumentation 9

2.1 Hard X-ray Telescope (HXT) . . . . . . . . . . . . . . . . . . 9

2.1.1 Collimator (HXT-C) . . . . . . . . . . . . . . . . . . . 12

2.1.2 Detector Assembly (HXT-S) . . . . . . . . . . . . . . . 12

2.1.3 Electronics Unit (HXT-E) . . . . . . . . . . . . . . . . 12

2.2 Soft X-ray Telescope (SXT) . . . . . . . . . . . . . . . . . . . 13

2.3 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Hard X-ray Emission Models in a Solar Flare 15

3.1 Nonthermal Bremsstrahlung . . . . . . . . . . . . . . . . . . . 16

3.1.1 Thin-Target Emission Model . . . . . . . . . . . . . . . 17

3.1.2 Thick-Target Emission Model . . . . . . . . . . . . . . 18

3.1.3 Thick-Target Emission by the Power-Law Electrons

with a Lower Energy Cutoff . . . . . . . . . . . . . . . 19

3.2 Thermal Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . 22

3.3 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . 24

4 Quantitative Analysis of Nonthermal Components in Solar

Hard X-ray Flares 25

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

iii

iv Contents

4.2 Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Total Energy of Nonthermal Electrons . . . . . . . . . 27

4.2.2 SXR Emitting Plasma Energy . . . . . . . . . . . . . . 27

4.2.3 Energy Ratio . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Observational Data . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.5 Analysis Example (2001/04/12 X2.0 Flare) . . . . . . . . . . . 30

4.6 Results of the Statistical Analysis . . . . . . . . . . . . . . . . 35

5 Summary & Discussion 39

5.1 HXR Spectral Analysis in the Rising Phase . . . . . . . . . . . 40

5.2 Nonthermal Electron Rate . . . . . . . . . . . . . . . . . . . . 45

5.3 Lower Energy Cutoff in the Spectrum of Nonthermal Electrons 46

5.4 Event Selection: Revisited . . . . . . . . . . . . . . . . . . . . 49

5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 49

A Case Studies 53

Acknowledgement 67

List of Figures

1.1 Soft X-ray image of a solar flare observed on 1992 February

21 with Yohkoh/SXT. . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Soft X-ray cusp shaped structure observed with Yohkoh/SXT

(Tsuneta et al., 1992). . . . . . . . . . . . . . . . . . . . . . . 3

1.3 An example of plasmoid ejection observed with Yohkoh/SXT

(Ohyama & Shibata, 1998). . . . . . . . . . . . . . . . . . . . 3

1.4 Loop top hard X-ray source observed in the 1992 January 13

flare. Background color is soft X-ray image observed with

Yohkoh/SXT. Contour image is hard X-ray image observed

with Yohkoh/HXT. White thick line represents the solar limb

(Masuda et al., 1994). . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Cartoon of the so-called CHSKP model. Solid lines show the

magnetic field lines. Red arrows show the directions of plasma

flow. Reconnected magnetic field lines form the flaring loops

with a cusp shaped structure (red regions). Along the re-

connected lines, nonthermal particles and thermal conduction

propagate into the chromosphere, and generate “Hα flare rib-

bons” (blue regions). . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Time profile of a solar flare (Kane, 1974). . . . . . . . . . . . . 7

1.7 Energy spectra of HXR emission observed in the 1980 June 27

flare (Lin et al., 1981). . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Yohkoh image. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

v

vi List of Figures

2.2 Schematic drawing of the HXT instrument (Kosugi et al.,

1991). It consists of three major sections: the collimator

(HXT-C), the detector assembly (HXT-S), and the electron-

ics unit (HXT-E). The aspect system (HXA) is installed along

the central axis of HXT-C and HXT-S. . . . . . . . . . . . . . 11

3.1 Calculated HXR spectra emitted by the power-law electrons

with a lower energy cutoff in both “a sharp cutoff” (red aster-

isk) and “a saturation” (blue plus sign) cases. No significant

difference between them can be seen. . . . . . . . . . . . . . . 21

3.2 Calculated HXR spectrum emitted by the power-law electrons

with a sharp lower energy cutoff (asterisk) and the fitted dou-

ble power-law spectrum (red line). Fitting parameters are

εb = 28.2 keV, γ1 = 2.27, and γ2 = 3.98, respectively. . . . . . 22

4.1 HXR time profiles observed with Yohkoh/HXT in the 2001

April 12 flare. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Hard X-ray image (contour) taken with the HXT M2-band,

overlaid on a soft X-ray image in the 2001 April 12 flare.

Hard X-ray image is synthesized by the MEM. The photon

count accumulation interval for hard X-ray image is 10:16:32 -

10:17:48 UT (impulsive phase). Contour levels are 70, 35, 12.5

% of the maximum brightness. Soft X-ray image is taken with

SXT (Be119) at 10:17:11 UT. Heliographic grids are shown by

dashed lines in 2◦ increments. . . . . . . . . . . . . . . . . . . 32

4.3 Time profiles of the M2-band count rate (red line) and the

estimated thermal energy of the SXR emitting flare plasma

(blue asterisk) in the 2001 April 12 flare. . . . . . . . . . . . . 33

4.4 Plot of |Enonth(Ec)−∆Eth|/∆Eth as a function of Ec. . . . . . 34

4.5 Plot of the derived Ec vs. M2-band peak count rate. When

the derived Ec is greater(lower) than 30 keV, called “relatively

high(low)“. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

List of Figures vii

4.6 Plot of the estimated nonthermal electron rate in the impulsive

phase vs. the volume emission measure (left panel), the num-

ber density (right panel) of the SXR emitting flare plasma in

the pre-impulsive phase. Correlation coefficients are 0.94 and

0.97, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 38


April 6 flare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Plot of the single power-law photon index obtained from the

M1-band and L-band count data (horizontal axis) vs. that

obtained from the M2-band and M1-band count data (vertical

axis) in the rising phase of the 2001 April 6 flare. . . . . . . . 42


June 2 flare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Plot of the single power-law photon index obtained from the

M1-band and L-band count data (horizontal axis) vs. that

obtained from the M2-band and M1-band count data (vertical

axis) in the rising phase of the 2000 June 2 flare. . . . . . . . . 44

5.5 Plot of the estimated nonthermal electron rate in the impulsive

phase vs. the number density of the SXR emitting flare plasma

in the pre-impulsive phase (same as the right panel of Figure

4.6). Correlation coefficient is 0.97. Solid line is the power-law

fit by the least square method. . . . . . . . . . . . . . . . . . . 46

5.6 Hard X-ray image taken with the HXT M2-band in the 2001

August 25 flare. The two HXR sources are specified by eye

(the boxed areas). . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.7 Plot of the derived Ec vs. the calculated footpoint distance.

Correlation coefficient is 0.8. Solid line is the linear fit by the

least square method. . . . . . . . . . . . . . . . . . . . . . . . 48

A.1 HXR time profiles observed with Yohkoh/HXT in the 1997

November 6 flare. . . . . . . . . . . . . . . . . . . . . . . . . . 57

viii List of Figures

A.2 Time profiles of the M2-band count rate (red line) and the


(blue asterisk) in the 1997 November 6 flare. . . . . . . . . . . 57


August 18 flare. . . . . . . . . . . . . . . . . . . . . . . . . . . 58



(blue asterisk) in the 1998 August 18 flare. . . . . . . . . . . . 58



(blue asterisk) in the 2000 June 2 flare. . . . . . . . . . . . . . 59


November 24 flare. . . . . . . . . . . . . . . . . . . . . . . . . 60



(blue asterisk) in the 2000 November 24 flare. . . . . . . . . . 60



(blue asterisk) in the 2001 April 6 flare. . . . . . . . . . . . . . 61


August 25 flare. . . . . . . . . . . . . . . . . . . . . . . . . . . 62



(blue asterisk) in the 2001 August 25 flare. . . . . . . . . . . . 62

List of Tables

4.1 Selected seven impulsive flares observed with Yohkoh. . . . . . 30

4.2 Results of the statistical analysis. . . . . . . . . . . . . . . . . 36

5.1 Physical variables discussed in Section 5.1through Section 5.3. 49

ix

Chapter 1

General Introduction

The solar atmosphere is full of dynamic plasma phenomena. Recent obser-

vations have revealed that the solar atmosphere is much dynamic than we

thought, even if the Sun is “quiet”. Above all, a solar flare (Figure 1.1)

is the largest exploding phenomenon in our solar system. Its spatial scale

reaches 105 km, which is 10 times larger than the radius of the Earth. Its

energy reaches 1032 erg, which is 100 million times larger than the energy

released by a hydrogen bomb. Once the flare occurs, a huge amount of en-

ergy is released in a short time, and solar plasma is strongly heated and

accelerated. Electromagnetic waves over a very wide range of wavelengths

from X-rays (occasionally from γ-rays) to radio waves are emitted during the

flare. Almost all flares are accompanied by a huge amount of mass ejection,

the so-called coronal mass ejections (CMEs), into interplanetary space. It

affects the Earth’s magnetosphere, yields geomagnetic storm or aurora.

Classically, solar flares were merely recognized as sudden exploding and

brightening phenomena on the photosphere. But recent notable progress of

observational instruments such as Yohkoh (Ogawara et al., 1991), SOHO1,

and TRACE2 has made it possible to study the physics of solar flares in

detail. The magnetic reconnection model is widely accepted to explain the

1http://sohowww.nascom.nasa.gov/2http://vestige.lmsal.com/TRACE/

1

2 1. General Introduction

Figure 1.1: Soft X-ray image of a solar flare observed on 1992 February 21

with Yohkoh/SXT.

mechanism of a solar flare. The magnetic reconnection, topological change of

a configuration of antiparallel magnetic field lines in the presence of plasma, is

a fundamental physical process and is thought to be a highly efficient engine

to convert magnetic energy into thermal and kinetic energies of plasma flows

and particles. Observed phenomena associated with solar flares, such as cusp

shaped structure (Tsuneta et al., 1992, Figure 1.2), plasmoid ejections (e.g.,

Ohyama & Shibata, 1998, Figure 1.3), loop-top hard X-ray sources (Masuda

et al., 1994, Figure 1.4), reconnection inflow (Yokoyama et al., 2001), and so

on, morphologically confirm the standard magnetic reconnection model, the

so-called CSHKP model (Carmichael, 1964; Sturrock, 1966; Hirayama, 1974;

Kopp & Pneuman, 1976, Figure 1.5).

3

Figure 1.2: Soft X-ray cusp shaped structure observed with Yohkoh/SXT

(Tsuneta et al., 1992).

Figure 1.3: An example of plasmoid ejection observed with Yohkoh/SXT

(Ohyama & Shibata, 1998).


Figure 1.4: Loop top hard X-ray source observed in the 1992 January 13 flare.

Background color is soft X-ray image observed with Yohkoh/SXT. Contour

image is hard X-ray image observed with Yohkoh/HXT. White thick line

represents the solar limb (Masuda et al., 1994).

Solar flares are often observed in X-ray range. The interesting is that

X-ray emissions within a few times keV energy range (the so-called “Soft

X-ray (SXR)“) and within a deka-keV energy range (the so-called “Hard

X-ray (HXR)“) show quite different characteristics. Light curve of SXR

often shows “gradual” rise and decay, whereas that of HXR often shows

“impulsive” rise and decay (Figure 1.6). Energy spectra of SXR and HXR

also show different characteristic. Energy spectrum of SXR is well-fitted by

a (single temperature) Maxwellian, whereas that of HXR is well-fitted by a

(single or double) power-law function (e.g., Lin et al., 1981, Figure 1.7). The

observed power-law energy spectrum within HXR energy range indicates the

existence of deka-keV nonthermal particles in solar flares.

We can also see the spatial difference between SXR and HXR emission

sites. In SXR observation, the brilliant loop-like structure (a flare loop) is

5

Figure 1.5: Cartoon of the so-called CHSKP model. Solid lines show the

magnetic field lines. Red arrows show the directions of plasma flow. Re-

connected magnetic field lines form the flaring loops with a cusp shaped

structure (red regions). Along the reconnected lines, nonthermal particles

and thermal conduction propagate into the chromosphere, and generate “Hα

flare ribbons” (blue regions).

often seen. In HXR observation, on the other hand, we can often see the

discrete HXR sources located at the footpoints of a flare loop (See Figure

1.4 ). Sakao (1994) concluded that the high energy nonthermal electrons

precipitation into the dense chromosphere along a flare loop produce HXR

radiation via bremsstrahlung.

As mentioned above, it is the observational fact that high energy nonther-

mal particles are efficiently produced in solar flares. However, the particle

acceleration process is still unknown. The presence of a loop-top HXR source,

which is one of the most important findings made by Yohkoh, implies that the

acceleration site is located above a flare loop(s). But further observational

results and analyses are needed to confirm this proposition. Some theoretical


models about the particle acceleration mechanism in solar flares have been

discussed, but none of them are conclusive. Thus the particle acceleration

process in solar flares is one of the most important scientific issues in solar

physics.

In this thesis, we discuss characteristics of nonthermal components in so-

lar flares by analyzing hard X-ray flares observed with Yohkoh quantitatively

and statistically. We successfully estimate physical variables in flares such

as the lower energy cutoff in the spectrum of nonthermal electrons. Fur-

thermore, we examine relationships of physical variables of flare nonthermal

components. We expect that this work will be of benefit to the understanding

of the particle acceleration process in solar flares.

The plan of this thesis is as follows. In Chapter 2 we briefly review

instrumentation used for our analysis. In Chapter 3 we review hard X-ray

emission models applied to the observed hard X-ray emission. In Chapter 4

we detail our analysis. In Chapter 5 we present summary and discussion.

7

Figure 1.6: Time profile of a solar flare (Kane, 1974).


Figure 1.7: Energy spectra of HXR emission observed in the 1980 June 27

flare (Lin et al., 1981).

Chapter 2

Instrumentation

In this Chapter, we briefly review instrumentation used for our analysis. We

carry out the analysis by using the observational data of the Yohkoh satellite

(Ogawara et al., 1991, Figure 2.1) and the GOES satellite. The Yohkoh

satellite is a project of the Institute of Space and Astronautical Sciences,

Japan (ISAS1) and an international collaboration with the US and UK. It

was launched on 1991 August 30, and operated for more than a decade until

2001 December 14. Yohkoh carried four scientific instruments: the Hard X-

ray Telescope (HXT; Kosugi et al., 1991), the Soft X-ray Telescope (SXT;

Tsuneta et al., 1991), the Wide Band Spectrometer (WBS), and the Bragg

Crystal Spectrometer (BCS). The HXT and SXT were so designed as to

obtain simultaneous hard and soft X-ray flare images with high spatial and

temporal resolution. These instruments are powerful imagers to study the

high energy physics of solar flares.

2.1 Hard X-ray Telescope (HXT)

The hard X-ray telescope (HXT) is a Fourier-synthesis type imager with

64 bi-grid modulation subcollimators (SC’s). Each SC measures a spatially

modulated photon count. A set of photon count data from the 64 SC’s

1Now, Japan Aerospace Exploration Agency (JAXA)

9

10 2. Instrumentation

Figure 2.1: Yohkoh image.

can be converted into an image by using image-synthesis procedures such as

Maximum Entropy Method (MEM).

The HXT records a set of 64 photon counts when the flare mode is trig-

gered. The main capabilities of HXT are as follows:

i) Simultaneous imaging in four energy bands, namely, the L-band (13.9 -

22.7 keV), M1-band (22.7 - 32.7 keV), M2-band (32.7 - 52.7 keV), and H-

band (52.7 - 92.8 keV);

ii) Angular resolution of ∼ 5′′ with a wide field of view covering the whole

Sun (∼ 35′ by 35′);

iii) Basic temporal resolution of 0.5 s; and

iv) High sensitivity with an effective area of ∼ 60 cm2.

Instrumentally HXT consists of three major sections, i.e., the collimator

(HXT-C), the detector assembly (HXT-S), and the electronics unit (HXT-

E) (Figure 2.2). Outlines of HXT-C, HXT-S, and HXT-E are described in

Section 2.1.1 through 2.1.3, respectively.

2.1. Hard X-ray Telescope (HXT) 11

Figure 2.2: Schematic drawing of the HXT instrument (Kosugi et al., 1991).

It consists of three major sections: the collimator (HXT-C), the detector

assembly (HXT-S), and the electronics unit (HXT-E). The aspect system

(HXA) is installed along the central axis of HXT-C and HXT-S.


2.1.1 Collimator (HXT-C)

The collimator (HXT-C) is the X-ray optics part of the instrument. It is

a metering tube (417 mm × 376 mm × 1400 mm) with X-ray grid plates

at both ends. Each grid plate is an assembly of 64 subcollimator grids,

which provides the 64 modulation patterns necessary for image-synthesis.

The aspect system (HXA) optics, which include lenses with filters on the

front grid plate and fiducial marks on the rear plate, is equipped at the

center of the X-ray optics. It provides the optical axis direction information

of the X-ray optics with respect to the solar disk.

2.1.2 Detector Assembly (HXT-S)

The detector assembly is composed of 64 detector modules, eight high-voltage

power supply units, and two one-dimensional CCD arrays for aspect optics.

A detector module mainly consists of a NaI(T1) scintillation crystal and a

photomultiplier tube with a high-voltage bleeder string and a pre-amplifier

installed. Eight detector modules are packed together to form a detector unit

to which one DC-DC converter supplies high voltage. Eight detector units

are tied together to form the detector assembly.

2.1.3 Electronics Unit (HXT-E)

The electronics unit (HXT-E) processes the signals from HXT-S. It converts

hard X-ray pulse-height analogue signals from the individual subcollimators

into digital signals and counts the incident photon number after discriminat-

ing the photon energy into four energy bands. All digitalized photon count

data are sent to an onboard data processor (DP; Ogawara et al., 1991) every

0.5 s. HXT-E also processes the HXA signals. In addition, HXT-E controls

power/mode of the whole HXT instrument.

For more details, see Kosugi et al. (1991). We use the HXT data to

estimate physical variables of nonthermal components in solar flares (Chapter

2.2. Soft X-ray Telescope (SXT) 13

4).

2.2 Soft X-ray Telescope (SXT)

The soft X-ray telescope (SXT) is a grazing-incidence reflection telescope

which forms X-ray images in the 0.25 - 4.0 keV range on a CCD detector

(1024 × 1024 pixels). A filter wheel assembly located in front of the CCD

detector provides the capability of energy discrimination for plasma temper-

ature and emission measure diagnostics. A 119 µm beryllium filter (Be119)

and a 11.6 µm Al filter (thick Al) are sensitive to the highest energies. A

rotating shutter, also located in front of the CCD detector, provides the ca-

pability to optimize exposure time in response to the solar activity. Both are

automatically controlled by an onboard data processor. The highest spatial

resolution is ∼ 2.5′′/pixel in full-resolution images in the partial frame image

mode (64 × 64 pixels). Temporal resolution is usually up to 2 s.

We use the SXT data to estimate the volume of flare region (Chapter 4).

2.3 Data Set

In this thesis, we use the observational data of Yohkoh/HXT, Yohkoh/SXT,

and GOES. The HXT and SXT data are obtained from PLAIN Center

DARTS2. Flare events suitable for our analysis are selected by surveying

the whole flares detected with HXT, which are listed in “The YOHKOH

HXT/SXT Flare Catalogue” (Sato et al., 2003). We select seven impulsive

flare events (see Section 4.4 for more details).

The HXT data is a set of photon count data from the 64 SC’s every

0.5 s and in four energy bands, as mentioned in Section 2.1. We apply

background subtraction and, if necessary, photon counts correction of the

HXT data “saturated” due to scalar overflow and/or detector deadtime to

the data used for our analysis.

2http://www.darts.isas.ac.jp/index.html


We use the SXT data in full-resolution images in the partial frame image

mode (flare mode). Filter Be119 is selected because this is suitable to an-

alyze the high-temperature phenomena such as solar flares. The SXT data

used for our analysis are correctly calibrated by using standard calibrating

procedure. When the SXT images contain a saturated pixel(s), these images

are eliminated from our analysis.

GOES is the broadband soft X-ray telescope operated by NOAA. It ob-

serves whole Sun X-ray fluxes in the 0.5 - 4.0 and 1.0 - 8.0 A wavelength

bands (no spatial resolution except for GOES-12 SXI). We use the GOES

data for plasma temperature and emission measure diagnostics by taking the

ratio of emission observed with these two bands. Note that all amount of X-

ray fluxes detected with GOES is assumed to be emitted by the flare thermal

plasma which SXT (Be119) observes simultaneously.

Chapter 3

Hard X-ray Emission Models in

a Solar Flare

In this Chapter, we review hard X-ray (HXR) emission models applied to the

observed HXR emission in a solar flare. A significant amount of HXRs, not

only soft X-rays (SXRs), is emitted during a flare. Main characteristics of the

observed HXRs, which are quite different from those of the observed SXRs,

are (1) light curve which shows impulsive fluctuation in the early phase of

a flare (the so-called “impulsive phase”); and (2) energy spectrum which is

well-fitted by a power-law function. These characteristics of the observed

HXRs indicate that a large amount of nonthermal particles is impulsively

produced in the impulsive phase. Moreover, it is generally accepted that a

significant amount of flare energies is released during the impulsive phase.

This means that nonthermal particles play an important role in flare energet-

ics. Therefore, the study of flare nonthermal particles is important to discuss

not only the particle acceleration process in solar flares but also the nature

of flares themselves. To do this, HXRs will be a powerful mean.

HXR emission in a solar flare is a continuum emission. It is widely be-

lieved that interaction of beams of nonthermal electrons with ambient so-

lar plasma results in HXR emission via bremsstrahlung, i.e., nonthermal

bremsstrahlung. We first review nonthermal bremsstrahlung in Section 3.1.

15

16 3. Hard X-ray Emission Models in a Solar Flare

3.1 Nonthermal Bremsstrahlung

Two major models of nonthermal bremsstrahlung are the thin-target and

thick-target emission models proposed by Brown (1971). In the thin-target

case it is assumed that nonthermal electrons continue to emit HXRs without

significant modification to their distribution function. In the thick-target

case, on the other hand, it is assumed that nonthermal electrons are im-

mediately stopped (thermalized) by Coulomb collisions with dense ambient

plasma. A thin-target scenario would be applicable to nonthermal electrons

injected outward through the corona, or trapped into low-density corona. A

thick-target scenario is applicable to nonthermal electrons precipitating into

the dense chromosphere.

HXR emission in a solar flare is in the deka-keV energy range. In such a

energy range, cross section for the bremsstrahlung is expressed as a conve-

nient formula. It is direction-integrated, nonrelativistic, Bethe-Heitler cross

section (Jackson, 1962):

σB(ε, E) =8α

3r20

mec2

εEln

1 + (1− ε/E)1/2

1− (1− ε/E)1/2

=κBH

εEln

1 + (1− ε/E)1/2

1− (1− ε/E)1/2(3.1)

=7.9× 10−25

εEln

1 + (1− ε/E)1/2

1− (1− ε/E)1/2[cm2keV−1].

Here α = e2/(hc) is the fine structure constant, r0 = e2/(mec2) is the classical

electron radius, κBH = (8α/3)r20mec

2 is the constant in the Bethe-Heitler

cross section, ε is photon energy in units of keV, E is electron energy in units

of keV, respectively. The number of photons emitted by bremsstrahlung per

unit time per unit energy per unit volume is expressed as follows:

1

ε

dW

dεdV dt= npneve(E)σB(ε, E). (3.2)

Here np is the target density, ne is the source electron density, and ve(E) is

the velocity of electrons, respectively. Equation (3.2) can be rewritten as

1

ε

dW

dεdt=

∫

V

dV · (npneveσB). (3.3)

3.1. Nonthermal Bremsstrahlung 17

This equation expresses the number of HXR photons emitted from the whole

bremsstrahlung sites per unit time per unit energy. The HXR spectrum

I(ε) (photons/cm2/sec/keV) observed at the orbit of the Earth is simply

written as

I(ε) =1

4πR2

(1

ε

dW

dεdt

)

=1

4πR2

∫

V

dV · (npneveσB) [photons/cm2/sec/keV], (3.4)

where R = 1AU.

3.1.1 Thin-Target Emission Model

Let us consider the thin-target case. In this case, it is assumed that the

distribution of source electrons is not modified throughout HXR emission.

Thus, for a power-law spectrum of source electrons:

F (E) = AE−δ [electrons/sec/keV], (3.5)

right hand side of equation (3.3) becomes∫

V

dV · (npneveσB) = A∆N

∫ ∞

ε

E−δσB(ε, E)dE, (3.6)

where ∆N =∫source

np(s)ds is the column density of the source observed.

Thus, the HXR spectrum (eq. (3.4)) in the thin-target case is written as

Ithin(ε) =∆NAκBH

4πR2ε−1

∫ ∞

ε

E−(δ+1) ln1 + (1− ε/E)1/2

1− (1− ε/E)1/2dE, (3.7)

by using equation (3.1). This equation can be simplified by evaluating the

integral by parts (see, e.g., Tandberg-Hanssen & Emslie, 1988):

Ithin(ε) =∆NAκBH

4πR2

B(δ, 1/2)

δε−(δ+1) (3.8)

∝ ε−(δ+1) [photons/cm2/sec/keV].

Here B(a, b) is the standard beta function. Under the thin-target assumption,

we find that if the source electron spectrum is a power-law, then the resultant


HXR spectrum is also a power-law Ithin(ε) = aε−γ, and the spectral index

of the HXR spectrum is larger than that of the source electron spectrum

(γ = δ + 1).

3.1.2 Thick-Target Emission Model

Let us consider the thick-target case. In this case, it is assumed that non-

thermal electrons immediately lose their energy via Coulomb collisions with

dense ambient plasma. Thus, we first summarize expressions for Coulomb

collisions. The energy loss rate by Coulomb collisions is expresses as follows:

dE

dt= −σE(E)npve(E)E. (3.9)

Here σE(E) is the cross section for the Coulomb energy loss and has the

following form:

σE(E) =2πe4 ln Λ

E2=

K

E2, (3.10)

where ln Λ is the Coulomb logarithm which is treated as almost constant.

Suppose that an electron with initial energy E0 collides with dense target

plasma and emits photons by bremsstrahlung until it is thermalized. The

number of photons per unit energy ν(ε, E0), at an energy of ε, emitted in

above situation is expressed as

ν(ε, E0) =

∫ end(E=ε)

start(E=E0)

npσB(ε, E)dl

=

∫ t1(E=ε)

t0(E=E0)

npσB(ε, E(t))ve(t)dt

=

∫ E0

ε

σB(ε, E)

EσE(E)dE, (3.11)

where we use equation (3.9). By using this equation, right hand side of

equation (3.3) becomes∫

V

dV · (npneveσB) =

∫

S

dS

∫

L

dl · (npσBneve)

=

∫

S

dS

∫ E0

ε

dEσB(ε, E)

EσE(E)neve. (3.12)


Thus, for a power-law spectrum of injected electrons:

F (E0) = AE−δ0 [electrons/sec/keV], (3.13)

the HXR spectrum (eq. (3.4)) in the thick-target case is written as

Ithick(ε) =1

4πR2

∫ E0=∞

E0=ε

F (E0)

∫ E=E0

E=ε

σB(ε, E)

EσE(E)dEdE0

=AκBH

4πR2Kε−1

∫ E0=∞

E0=ε

E−δ0

∫ E=E0

E=ε

ln1 + (1− ε/E)1/2

1− (1− ε/E)1/2dEdE0,

(3.14)

by using equations (3.1) and (3.10). This equation can be simplified by re-

versing the order of integration considering the area of integration in (E, E0)

space (see also, e.g., Tandberg-Hanssen & Emslie, 1988):

Ithick(ε) =AκBH

4πR2K

B(δ − 2, 1/2)

(δ − 1)(δ − 2)ε−(δ−1) (3.15)

∝ ε−(δ−1) [photons/cm2/sec/keV].

Also under the thick-target assumption, we find that if the electron injection

spectrum is a power-law, then the resultant HXR spectrum is also a power-

law Ithick(ε) = aε−γ. The spectral index of the HXR spectrum is smaller than

that of the electron injection spectrum (γ = δ − 1).

To summarize, we can numerically derive the electron injection spectrum

F (E0) = AE−δ0 from the observed power-law HXR spectrum Ithick(ε) = aε−γ

under the thick-target assumption as follows (Hudson et al., 1978):

A = 3.28× 1033γ2(γ − 1)2B

(γ − 1

2,3

2

)· a,

δ = γ + 1.

(3.16)

3.1.3 Thick-Target Emission by the Power-Law Elec-

trons with a Lower Energy Cutoff

In both the thin-target and thick-target emission models reviewed in Sections

3.1.1 and 3.1.2 respectively, a power-law spectrum of HXR emitting electrons


extends to lower energies indefinitely (cf. eq. (3.5) and (3.13)), that is, the

number of nonthermal electrons becomes infinity. Therefore, the spectrum

of nonthermal electrons must have a lower energy cutoff (Ec) to keep the

number of nonthermal electrons within a reasonable range. It seems to be

known that when the spectrum of nonthermal electrons has the lower energy

cutoff, the resultant HXR spectrum becomes flat toward the lower energy

and the broken energy of HXR spectrum (εb) is smaller than Ec (e.g., Nitta

et al., 1990). Gan et al. (2001b) studied this issue quantitatively.

In this Section, we present the “modified” thick-target emission model

based on Gan et al. (2001b). The HXR spectrum in the thick-target case is

written as (cf. eq. (3.14))

Ithick(ε) = I0ε−1

∫ E0=∞

E0=ε

F (E0)

∫ E=E0

E=ε

ln1 + (1− ε/E)1/2

1− (1− ε/E)1/2dEdE0, (3.17)

where I0 = κBH/(4πR2K). Here F (E0) will be given as a power-law with a

lower energy cutoff of the form:

F (E0) =

{AE−δ

0 , if E0 ≥ Ec,

C, if E0 < Ec,(3.18)

instead of equation (3.13). Here we consider C = 0 (“a sharp cutoff” case)

or C = AE−δc (“a saturation” case). Putting equation (3.18) into equation

(3.17), we get the modified thick-target emission form:

if ε ≥ Ec,

Ithick(ε) = AI0ε−1

∫ E0=∞

E0=ε

E−δ0

∫ E=E0

E=ε

ln1 + (1− ε/E)1/2

1− (1− ε/E)1/2dEdE0,

(3.19)

otherwise,

Ithick(ε) = CI0ε−1

∫ E0=Ec

E0=ε

∫ E=E0

E=ε

ln1 + (1− ε/E)1/2

1− (1− ε/E)1/2dEdE0

+ AI0ε−1

∫ E0=∞

E0=Ec

E−δ0

∫ E=E0

E=ε

ln1 + (1− ε/E)1/2

1− (1− ε/E)1/2dEdE0.

(3.20)


Figure 3.1: Calculated HXR spectra emitted by the power-law electrons

with a lower energy cutoff in both “a sharp cutoff” (red asterisk) and “a

saturation” (blue plus sign) cases. No significant difference between them

can be seen.

Obviously, equation (3.19) is same as equation (3.14). This means that

equation (3.16) is applicable to the HXR spectrum with energy above Ec. We

numerically calculate equations (3.19) and (3.20) in both “a sharp cutoff”

and “a saturation” cases. The resultant HXR spectra in case of Ec = 40 keV

and δ = 5 are shown in Figure 3.1.

These calculated HXR spectra can be well-fitted by a double power-law

of the form:

I(ε) =

{a1ε

−γ1 , if ε < εb,

a2ε−γ2 , if ε ≥ εb,

(3.21)

where a2 = a1εγ2−γ1

b . The condition γ1 < γ2 is said to “break down” (Dulk

et al., 1992). Figure 3.2 shows the calculated HXR spectrum in case of

Ec = 40 keV, δ = 5, and “a sharp cutoff”, and the fitted double power-law


Figure 3.2: Calculated HXR spectrum emitted by the power-law electrons

with a sharp lower energy cutoff (asterisk) and the fitted double power-law

spectrum (red line). Fitting parameters are εb = 28.2 keV, γ1 = 2.27, and

γ2 = 3.98, respectively.

spectrum by using equation (3.21). Gan et al. (2001b) studied a theoretical

relationship between γ1 and γ2, as well as the relationship between εb/Ec and

γ2, to establish how to determine the lower energy cutoff from the observed

HXR spectrum. We do not show details of the theoretical results of Gan

et al. (2001b), but note the rough relationship of the following form:

εb = (0.6− 0.8)Ec. (3.22)

3.2 Thermal Bremsstrahlung

It is well-known that the observed HXR spectrum is often characterized by

a combination of exponential-like and power-law rather than a single power-

law (e.g., Lin et al., 1981). It is also well-known that the observed SXR

3.2. Thermal Bremsstrahlung 23

spectrum is well-fitted by a single temperature Maxwellian (although several

line emissions contaminate the SXR continuum spectrum). Such exponential-

like spectrum is interpreted as thermal bremsstrahlung from the (super) hot

plasma. In this Section, we briefly summarize thermal bremsstrahlung.

Let us consider a volume V , containing a uniform density ne of hot elec-

trons with an isothermal Maxwellian energy distribution, viz.

f(E) = 2πne

(1

πkT

)3/2

E1/2e−E/kT [electrons/cm3/erg], (3.23)

where k is the Boltzmann’s constant. The bremsstrahlung produced by in-

teraction of these electrons with ambient plasma is written as

Ithermal(ε) =npV

4πR2

∫ ∞

ε

f(E)ve(E)σB(ε, E)dE

=1

4πR2

(8

πme

)1/2npneV

(kT )3/2

κBH

ε

×∫ ∞

ε

exp(−E/kT ) ln1 + (1− ε/E)1/2

1− (1− ε/E)1/2dE (3.24)

by using equation (3.4). Integrating by parts, we find

Ithermal(ε) =1

4πR2

(8κ2

BH

πmek

)1/2npneV

εT 1/2exp(−ε/kT )g(ε/kT ), (3.25)

where

g(a) =

∫ ∞

0

e−ax

√x(1 + x)

dx (3.26)

is a slowly varying function of order unity. We may neglect this factor. Thus,

equation (3.25) is written as

Ithermal(ε) =D

4πR2

EM

εT 1/2exp(−ε/kT ) [photons/cm2/sec/keV], (3.27)

where D =√

(8k2BH/πmek) = 5.7 × 10−12 cm3s−1K1/2, and EM = npneV '

n2eV is the volume emission measure of the source.

To summarize, we can deduce the volume emission measure and the tem-

perature of the thermal bremsstrahlung source from the observed exponential-

like X-ray spectrum.


3.3 Summary of This Chapter

In this Chapter, we reviewed several HXR emission models applied to the

observed HXR emission in a solar flare. In Section 3.1, we reviewed nonther-

mal bremsstrahlung. Two major models of nonthermal bremsstrahlung are

the thin-target and thick-target emission models.

In the thin-target emission model reviewed in Section 3.1.1, we find that if

the source electron spectrum is a power-law F (E) = AE−δ, then the resultant

HXR spectrum is also a power-law Ithin(ε) = aε−γ, and the spectral index

of the HXR spectrum is larger than that of the source electron spectrum

(γ = δ + 1).

In the thick-target emission model reviewed in Section 3.1.2, we find that

if the electron injection spectrum is a power-law F (E0) = AE−δ0 , then the

resultant HXR spectrum is also a power-law Ithick(ε) = aε−γ, and the spectral

index of the HXR spectrum is smaller than that of the electron injection

spectrum (γ = δ − 1). Using equation (3.16), we can numerically derive the

electron injection spectrum from the observed power-law HXR spectrum.

In Section 3.1.3, we presented the modified thick-target emission model by

considering the lower energy cutoff in the spectrum of nonthermal electrons.

When the spectrum of nonthermal electrons has the lower energy cutoff,

the resultant HXR spectrum becomes flat toward the lower energy and the

broken energy of HXR spectrum is smaller than the lower energy cutoff in

the spectrum of nonthermal electrons. We show that, in the modified thick-

target emission model, the numerical relationship between (A, δ) and (a, γ)

(eq. (3.16)) is applicable to the HXR spectrum with energy above Ec. Rough

relationship between εb and Ec is given by equation (3.22).

In Section 3.2, we reviewed thermal bremsstrahlung. We find that the

volume emission measure and the temperature of the thermal bremsstrahlung

source can be deduced from the observed exponential-like X-ray spectrum,

such as the SXR spectrum.

Chapter 4

Quantitative Analysis of

Nonthermal Components in

Solar Hard X-ray Flares

4.1 Introduction

Many observations have revealed that a significant amount of nonthermal

particles is produced in a solar flare and nonthermal particles play an im-

portant role in flare energetics. However, the particle acceleration process is

still unknown. Therefore, the study of flare nonthermal particles is very im-

portant. Because it is widely believed that the flare hard X-rays are emitted

by nonthermal electrons, the analysis of solar hard X-ray flares can provide

information about nonthermal electrons in solar flares, and it may provide

an explanation about the acceleration process in solar flares. Aim of this

thesis is to discuss characteristics of nonthermal components in solar flares

by analyzing hard X-ray flares quantitatively and statistically.

To discuss characteristics of nonthermal components in solar flares quan-

titatively, the determination of the lower energy cutoff (Ec) in the spectrum

of nonthermal electrons is very important. It determines the total number

and energy of nonthermal electrons, and it may be related to the acceleration

25

26 4. Quantitative Analysis of Nonthermal Components in Solar Hard X-ray Flares

mechanism in solar flares. Because HXR spectrum becomes flat toward the

lower energy and the broken energy of HXR spectrum (εb) is smaller than

Ec when the spectrum of nonthermal electrons has the lower energy cutoff,

to estimate the broken energy of the observed HXR spectrum must be a key

clue to deduce the lower energy cutoff in the spectrum of nonthermal elec-

trons. However, it is not easy to obtain it from the observed HXR spectrum

because of contamination due to thermal emission at the lower energy. For

a long time, one assumes the Ec to be 20 keV or 30 keV without any justifi-

cation. Quantitative determination of the lower energy cutoff is necessary to

discuss characteristics of nonthermal components. Consequently, we try to

derive the lower energy cutoff in the spectrum of nonthermal electrons with

some assumptions.

4.2 Analysis Method

It is well-known that the variation of the time integral of the microwave

intensity Imicro often closely matched the variation of the SXR intensity (ISXR)

in solar flares, the so-called “Neupert effect” (Neupert, 1968):

Imicro ∝ dISXR

dt. (4.1)

Because the variation of the HXR intensity (IHXR) is similar to that of the

microwave intensity, the concept of the Neupert effect can be extended to

include the relationship between the SXR and the HXR intensity, viz.

IHXR ∝ dISXR

dt. (4.2)

Since the SXR intensity is thought to represent the total energy of the SXR

emitting flare plasma and the HXR intensity would be thought to represent

the energy release rate of nonthermal electrons, the concept of the Neupert

effect will be additionally extended as follows:∫

dEnonth

dtdt = α∆ESXR. (4.3)

4.2. Analysis Method 27

Here dEnonth/dt is the energy rate of nonthermal electrons, ∆ESXR is the

energy build-up of the SXR emitting flare plasma during the impulsive phase,

and α represents the ratio between the total energy of nonthermal electrons

and the energy build-up of the SXR emitting flare plasma, respectively. In

the Section 4.2.1 through 4.2.3, we will show how to derive the lower energy

cutoff in the spectrum of nonthermal electrons from equation (4.3).

4.2.1 Total Energy of Nonthermal Electrons

Adopting the thick-target emission model, we can numerically derive the

spectrum of nonthermal electrons F (E) = AE−δ from the observed HXR

spectrum and equation (3.16). A and δ are obtained as a function of time,

namely, A(t) and δ(t). The energy rate of nonthermal electrons is given as

follows:

dEnonth

dt=

∫ ∞

Ec

EA(t)E−δ(t)dE

=A(t)

δ(t)− 2E−δ(t)+2

c . (4.4)

In general, Ec must be a function of time. But we assume that it is time-

independent. The total energy of nonthermal electrons released during the

impulsive phase (left hand side of eq. (4.3)) is given as follows:∫

dEnonth

dtdt =

∫A(t)

δ(t)− 2E−δ(t)+2

c dt ≡ Enonth(Ec). (4.5)

Because both A(t) and δ(t) are given, equation (4.5) is a function of Ec.

4.2.2 SXR Emitting Plasma Energy

Based on Wu et al. (1986), we evaluate the energy of the SXR emitting flare

plasma. ESXR is expressed as follows:

ESXR = Eth + Econvect + Etr + Erad + Econduct, (4.6)

where Eth is the thermal energy, Econvect is the convection energy, Etr is the

turbulent energy, Erad is the time integral of the radiation loss rate, and


Econduct is the time integral of the thermal conduction loss rate of the SXR

emitting flare plasma, respectively. In general, Econvect and Etr are 2-3 orders

of magnitude lower than Eth. Typical time scale of the radiation loss rate

of the SXR emitting flare plasma (T ∼ 107 K) is an order of magnitude

longer than the impulsive duration (∼ 100 sec). Therefore, Econvect, Etr,

and Erad are generally negligible compared with Eth during the impulsive

phase. Evaluation of Econduct, on the other hand, is difficult because it is

very sensitive to temperature. Consequently, we will neglect this term in

this thesis. We simply evaluate ESXR as follows:

ESXR ' Eth = (3nekT )V = 4.14× 10−16EM1/2V 1/2T, (4.7)

where EM is the volume emission measure, V is the volume, and T is the

temperature of the SXR emitting flare plasma, respectively. Since all of these

variables are estimated from the SXR observational data, ∆ESXR ' ∆Eth is

calculable.

4.2.3 Energy Ratio

The ratio between the total energy of nonthermal electrons and the energy

build-up of the SXR emitting flare plasma is important quantity in flare en-

ergetics. In our analysis, however, α is assumed to be unity. The assumption

of α = 1 means that the energy build-up of the SXR emitting flare plasma

completely results from the thermalization of nonthermal electrons.

Hence, we can numerically derive the lower energy cutoff in the spectrum

of nonthermal electrons by solving the following equation:

Enonth(Ec)−∆Eth = 0. (4.8)

Using the derived Ec, we can estimate several physical variables of nonther-

mal electrons.

4.3. Observational Data 29

4.3 Observational Data

In this thesis, we use the observational data of Yohkoh/HXT, Yohkoh/SXT,

and GOES (see Section 2.3).

The Yohkoh/HXT observational data is applied to obtain the incident

HXR spectrum. Taking the ratio of the measured photon counts in two

adjacent energy bands, we can obtain the incident HXR spectral parameters,

a(t) and γ(t), with a single power-law assumption. The M2-band (32.7 - 52.7

keV) count data and the H-band (52.7 - 92.8 keV) count data are mainly

applied to obtain them in this thesis.

The Yohkoh/SXT (Be119) observational data is applied to estimate the

volume of the SXR emitting flare plasma. Because SXT is the 2-dimensional

soft X-ray imager, we estimate the volume of the SXR emitting flare plasma

from the SXT (Be119) observational data with simple geometric uncertainty

as follows:

V = (0.5− 1.0)× S3/2, (4.9)

where S is the area observed with SXT (Be119). The area is calculated from

the SXT pixels at which the data number exceeds 3 percent of the maximum

one in each image.

The GOES observational data is applied to obtain the volume emission

measure and the temperature of the SXR emitting flare plasma. We assume

that all amount of X-ray fluxes detected with GOES is emitted by the flare

plasma which SXT (Be119) observes simultaneously, although GOES detects

whole Sun X-ray fluxes. With this simplification, the volume emission mea-

sure and the temperature obtained from the GOES observational data are

treated as those of the SXR emitting flare plasma.

4.4 Event Selection

In this thesis, we analyze seven impulsive flares observed with Yohkoh which

are listed in Table 4.1. These flares are selected according to the following


Table 4.1: Selected seven impulsive flares observed with Yohkoh.

M2-band peak count rate

date GOES class (cts/sec/SC)

1997/11/06 X9.4 5901

1998/08/18 X2.8 115

2000/06/02 M7.6 78

2000/11/24 X2.3 1479

2001/04/06 X5.6 992

2001/04/12 X2.0 152

2001/08/25 X5.3 3653

criteria:

(a) M2-band peak count rate exceeds 30 cts/sec/SC;

(b) Double-source structure is seen in M2-band;

(c) SXT (Be119) images without saturated pixels at the end of the HXR

burst are available; and

(d) There are both the HXT and SXT observational data in the “pre-impulsive

phase”, in which no significant enhancement of X-ray emission still occurs.

Discrimination of double-source structure is made by eye with “The YOHKOH

HXT/SXT Flare Catalogue” (Sato et al., 2003).

4.5 Analysis Example (2001/04/12 X2.0 Flare)

Let us present an analysis example. The flare commenced on 12 April,

2001 at ∼ 10:11 UT. Figure 4.1 shows the HXR time profiles observed with

Yohkoh/HXT. The GOES soft X-ray class was X2.0. Figure 4.2 shows the

X-ray images taken with HXT and SXT.

4.5. Analysis Example (2001/04/12 X2.0 Flare) 31

Figure 4.1: HXR time profiles observed with Yohkoh/HXT in the 2001 April

12 flare.


Figure 4.2: Hard X-ray image (contour) taken with the HXT M2-band, over-

laid on a soft X-ray image in the 2001 April 12 flare. Hard X-ray image is

synthesized by the MEM. The photon count accumulation interval for hard

X-ray image is 10:16:32 - 10:17:48 UT (impulsive phase). Contour levels are

70, 35, 12.5 % of the maximum brightness. Soft X-ray image is taken with

SXT (Be119) at 10:17:11 UT. Heliographic grids are shown by dashed lines

in 2◦ increments.

4.5. Analysis Example (2001/04/12 X2.0 Flare) 33

∆Eth

∆tT05 T90

Figure 4.3: Time profiles of the M2-band count rate (red line) and the es-

timated thermal energy of the SXR emitting flare plasma (blue asterisk) in

the 2001 April 12 flare.

Figure 4.3 shows the time profiles of the M2-band count rate (red line)

and the estimated thermal energy of the SXR emitting flare plasma (blue

asterisk). Black solid lines at ∼ 10:16 UT and at ∼ 10:23 UT denote “T05”

and “T90”, respectively. Here T05(T90) is the time at which the time integral

of the M2-band count rate reaches 5(90) percent of the total amount of the

measured photon counts in M2-band and is defined as the onset(end) of the

impulsive phase. The region surrounded by green lines is defined as the

pre-impulsive phase (tpre = T0 ∼ T0.05). The SXT images taken between ∼10:19 and ∼ 10:23 UT are eliminated from this analysis because they contain

a saturated pixel(s). Note that the volume of the SXR emitting flare plasma

is assumed to be V = S3/2 (see eq. (4.9)) in this example.

We first estimate the thermal energy build-up of the SXR emitting flare


Figure 4.4: Plot of |Enonth(Ec)−∆Eth|/∆Eth as a function of Ec.

plasma ∆Eth = Eth(T90)−Eth(tpre) and the impulsive duration ∆t = T90−T05. In this example, ∆Eth of 1.3 × 1031 (erg) and ∆t of 410 (sec) are

obtained. We also estimate several physical variables in the pre-impulsive

phase such as the volume emission measure, the temperature, and the number

density n = (EM/V )1/2. The total energy of nonthermal electrons released

during the impulsive phase, Enonth(Ec), is obtained by the method mentioned

in Section 4.2.1.

Next, we derive the optimal Ec by comparing Enonth(Ec) with ∆Eth (see

Figure 4.4). In this example, the optimal Ec is derived to be 25 keV.

Finally, we estimate several physical variables of nonthermal components.

The total number of nonthermal electrons in the impulsive phase is given as

N =

∫∫ ∞

Ec

A(t)E−δ(t)dEdt, (4.10)

and the nonthermal electron rate in the impulsive phase is estimated as

4.6. Results of the Statistical Analysis 35

follows:

dN

dt∼ N

∆t. (4.11)

In this example, N and dN/dt are estimated to be 2.3× 1038 (electrons) and

5.6 × 1035 (electrons/sec), respectively. The analysis results in other flares

are shown in Appendix A.

4.6 Results of the Statistical Analysis

The results of the statistical analysis are summarized in Table 4.2 and Figure

4.5.


Tab

le4.

2:R

esult

sof

the

stat

isti

calan

alysi

s.

EM

an

aT

a∆

Eth

∆t

Ec

NdN

/dt

M2-

band

peak

coun

tra

te

date

(104

8cm

−3)

(109

cm−

3)

(107

K)

(103

1er

g)(s

ec)

(keV

)(1

038)

(103

5se

c−1)

(cts

/sec

/SC

)

1997

/11/

060.

52.

2−

4.7

1.4

1.4−

2.0

158

>50

b···

···

5901

1998

/08/

181.

63.

0−

4.3

1.2

0.7−

1.0

406

31−

34c

1.0−

1.5

2.4−

3.7

115

2000

/06/

0220

18−

261.

30.

2−

0.3

5020−

22d

0.5−

0.7

9.0−

1478

2000

/11/

2432

28−

391.

21.

0−

1.3

146

23−

25d

1.7−

2.5

12−

1714

79

2001

/04/

062.

04.

3−

6.0

1.3

1.6−

2.3

374

43−

48c

1.6−

2.7

4.2−

7.3

992

2001

/04/

121.

63.

5−

5.0

1.3

0.9−

1.3

410

25−

28d

1.4−

2.3

3.5−

5.6

152

2001

/08/

255.

05.

5−

7.8

1.1

2.0−

2.8

464

38−

44c

2.0−

3.2

4.1−

6.9

3653

aV

aria

bles

inth

epr

e-im

puls

ive

phas

e.bT

hede

rive

dE

cha

san

unce

rtai

nty.

cT

hede

rive

dE

cis

“rel

ativ

ely

high

”.dT

hede

rive

dE

cis

“rel

ativ

ely

low

”.

4.6. Results of the Statistical Analysis 37

Figure 4.5: Plot of the derived Ec vs. M2-band peak count rate. When the

derived Ec is greater(lower) than 30 keV, called “relatively high(low)“.

We classify the analyzed events based on the derived Ec into “relatively

high Ec” ones (Ec > 30 keV) and “relatively low Ec” ones (Ec < 30 keV).

Since the derived Ec in the 1997 November 6 flare is high compared with the

energy range of M2-band (32.7 - 52.7 keV), it will have an uncertainty (see

Section 3.1.3). We don’t use this result for the following discussions in this

thesis.

From the results of the statistical analysis, we find a relationship between

the nonthermal electron rate in the impulsive phase and the volume emission

measure/the number density of the SXR emitting flare plasma in the pre-

impulsive phase. Figure 4.6 is the plot of the estimated nonthermal electron

rate in the impulsive phase vs. the volume emission measure (left panel),

the number density (right panel) of the SXR emitting flare plasma in the

pre-impulsive phase. Correlation coefficients are 0.94 and 0.97, respectively.

This result, indicating the positive correlation between the production rate


Figure 4.6: Plot of the estimated nonthermal electron rate in the impulsive

phase vs. the volume emission measure (left panel), the number density

(right panel) of the SXR emitting flare plasma in the pre-impulsive phase.

Correlation coefficients are 0.94 and 0.97, respectively.

of nonthermal electrons and the number density of the ambient plasma in

the pre-flare stage, is thought to be plausible. It is the first time to our

knowledge that the clear relationship between nonthermal component and

thermal component is quantitatively shown.

Further discussions will be presented in Chapter 5.

Chapter 5

Summary & Discussion

In the previous Chapter, we analyzed solar hard X-ray flares observed with

Yohkoh quantitatively and statistically. Main results are as follows:

i) The values of the derived lower energy cutoff (Ec) in the spectrum of

nonthermal electrons are ranging in 20 - 45 keV; and

ii) Positive correlation between the nonthermal electron rate in the im-

pulsive phase and the volume emission measure/the number density of

the SXR emitting flare plasma in the pre-impulsive phase is shown.

The values of the derived Ec, which are slightly higher than the usually ac-

cepted 20 keV, seem to be in reasonable range, although several authors ar-

gued that the lower energy cutoff is much higher than 20 keV (e.g., Gan et al.,

2001a). In our analysis, however, the lower energy cutoff in the spectrum of

nonthermal electrons is indirectly derived by assuming the energy balance

between nonthermal components and thermal components (eq. (4.3)). The

verification of the validity of the derived Ec is necessary to confirm our anal-

ysis results. In the next Section, we will check the validity of the derived

Ec.

39

40 5. Summary & Discussion

5.1 HXR Spectral Analysis in the Rising Phase

The observed HXR spectrum is often contaminated by thermal emission at

the lower energy, that is, the nonthermal characteristic is contaminated by

the thermal characteristic. However, the HXR spectrum in the rising phase

is probably less contaminated by thermal emission because thermal emission

in the rising phase is not so intense. Therefore, we expect that the HXR

spectrum in the rising phase can directly provide information about nonther-

mal components, that is, the broken energy of HXR spectrum (εb; see Section

3.1.3) can be seen in the rising phase. Because the broken energy of HXR

spectrum is a key clue to deduce the lower energy cutoff in the spectrum of

nonthermal electrons, the analysis of the HXR spectrum in the rising phase

will be useful for discussing the validity of the derived Ec. Low spectral

resolution of Yohkoh/HXT with only four energy bands makes it difficult

to analyze HXR spectrum in detail, but rough discussion about the broken

energy of HXR spectrum is possible. In this Section, we discuss the validity

of the derived Ec by analyzing the HXR spectrum in the rising phase.

5.1. HXR Spectral Analysis in the Rising Phase 41

Figure 5.1: HXR time profiles observed with Yohkoh/HXT in the 2001 April

6 flare.

Let us first present the analysis of the HXR spectrum in the rising phase of

the 2001 April 6 flare. This flare commenced on 6 April 2001 at ∼ 19:12 UT.

Figure 5.1 shows the HXR time profiles observed with Yohkoh/HXT. The

GOES soft X-ray class was X5.6. The derived Ec is 43 - 48 keV (“relatively

high”; see Table 4.2). Figure 5.2 shows the relationship between the single

power-law index obtained from the M1-band and L-band count data, and that

obtained from the M2-band and M1-band count data in the rising phase of

this flare. This result means that the observed HXR spectrum in the rising

phase of this flare shows the broken-down spectral form, like Figure 3.1 or

3.2. Using the measured photon counts in HXT four energy bands, we can

obtain the HXR spectral parameters with a double power-law (eq. (3.21))

assumption (double power-law fitting). From the double power-law fitting,

the broken energy of the HXR spectrum in the rising phase of this flare


Figure 5.2: Plot of the single power-law photon index obtained from the M1-

band and L-band count data (horizontal axis) vs. that obtained from the

M2-band and M1-band count data (vertical axis) in the rising phase of the

2001 April 6 flare.

is derived to be ∼ 28 keV. The observed HXR spectra in the rising phase

of other “relatively high Ec” flares show the tendency similar to this. The

values of the broken energy of the HXR spectrum in the rising phase of these

flares are derived to be ∼ 25 keV. Since the relationship between εb and Ec is

expressed by equation (3.22), the tendency that the derived Ec is “relatively

high” (Ec > 30 keV) is not inconsistent with the derived εb of 25 - 28 keV.

5.1. HXR Spectral Analysis in the Rising Phase 43

Figure 5.3: HXR time profiles observed with Yohkoh/HXT in the 2000 June

2 flare.

Next, we present the analysis of the HXR spectrum in the rising phase of

the 2000 June 2 flare. This flare commenced on 2 June 2000 at ∼ 18:52 UT.

Figure 5.3 shows the HXR time profiles observed with Yohkoh/HXT. The

GOES soft X-ray class was M7.6. The derived Ec is 20 - 22 keV (“relatively

low”; see Table 4.2). Figure 5.4 shows the relationship between the single

power-law index obtained from the M1-band and L-band count data, and

that obtained from the M2-band and M1-band count data in the rising phase

of this flare. Figure 5.4 is quite different from Figure 5.2. We can not see

the broken-down HXR spectral form in the rising phase of this flare. The

observed HXR spectra in the rising phase of other “relatively low Ec” flares

show the tendency similar to this. This result will be interpreted as follows: If

the lower energy cutoff in the spectrum of nonthermal electrons is “relatively

low” (Ec < 30 keV), then the broken energy of the resultant HXR spectrum


Figure 5.4: Plot of the single power-law photon index obtained from the M1-

band and L-band count data (horizontal axis) vs. that obtained from the

M2-band and M1-band count data (vertical axis) in the rising phase of the

2000 June 2 flare.

is estimated to be εb <∼ 20 keV from equation (3.22). Such εb is in the L-band

energy range (13.9 - 22.7 keV). Therefore, it will be hard to identify this from

the HXR spectral analysis with HXT.

The discussion in this Section is summarized as follows:

(a) The observed HXR spectra in the rising phase of the “relatively high

Ec” flares show the broken-down spectral form and the values of the broken

energy of the HXR spectrum in the rising phase of these flares are 25 - 28

keV;

5.2. Nonthermal Electron Rate 45

(b) The observed HXR spectra in the rising phase of the “relatively low Ec”

flares, on the other hand, don’t show the broken-down spectral form.

These tendencies are not inconsistent with a theoretical relationship by

Gan et al. (2001b). Thus, we conclude that the validity of the lower en-

ergy cutoff in the spectrum of nonthermal electrons derived in the previous

Chapter is roughly provided.

5.2 Nonthermal Electron Rate

In this Section, dependence of the nonthermal electron rate on the number

density of the SXR emitting flare plasma in the pre-impulsive phase is dis-

cussed. Figure 5.5 shows the relationship between the estimated nonthermal

electron rate in the impulsive phase (dN/dt) and the number density of the

SXR emitting flare plasma in the pre-impulsive phase (n). Correlation coeffi-

cient is 0.97. The power-law fit by the least square method is dN/dt ∝ n0.61.

In the 2-dimensional magnetic reconnection model, the electron injection rate

into the reconnection region is roughly evaluated as the product of the mass

flux into the reconnection region 2nvin and the area of the reconnection re-

gion Srec. By assuming the nonthermal electron rate to be proportional to

the electron injection rate into the reconnection region, it is simply evaluated

as follows:

dN

dt∝ nvinSrec

≈ nMAvASrec ∝ MABcSrecn1/2, (5.1)

where vin is the inflow velocity into the reconnection region, vA is the Alfven

velocity, MA = vin/vA is the reconnection rate, and Bc is the magnetic field

strength in the corona, respectively. Our result of the power-law fit dN/dt ∝n0.61 to the observation is similar to equation (5.1) in terms of n. Therefore,

the correlation in Figure 5.5 might indicate the magnetic reconnection process

itself. However, it is very rough discussion. Above all, the values of the

number density of the SXR emitting plasma in the pre-impulsive phase have


Figure 5.5: Plot of the estimated nonthermal electron rate in the impul-

sive phase vs. the number density of the SXR emitting flare plasma in the

pre-impulsive phase (same as the right panel of Figure 4.6). Correlation

coefficient is 0.97. Solid line is the power-law fit by the least square method.

a strong uncertainty due to the estimation of the volume of the SXR emitting

plasma in the pre-impulsive phase (n = (EM/V )1/2). It is, however, probable

that the nonthermal electron rate in the impulsive phase and the number

density of the SXR emitting plasma in the pre-impulsive phase have the

positive correlation.

5.3 Lower Energy Cutoff in the Spectrum of

Nonthermal Electrons

In this Section, dependence of the lower energy cutoff in the spectrum of

nonthermal electrons on the spatial scale of the flare is discussed. To reveal

5.3. Lower Energy Cutoff in the Spectrum of Nonthermal Electrons 47

Figure 5.6: Hard X-ray image taken with the HXT M2-band in the 2001

August 25 flare. The two HXR sources are specified by eye (the boxed

areas).

this dependence will be useful for understanding the acceleration mechanism

in solar flares. Because all the analyzed flare events show typical double-

source structure in M2-band (Section 4.4), we evaluate the spatial scale of

these flares by the HXR sources separation distance (the footpoint distance)

in HXT M2-band image at the HXR peak time.

To measure the footpoint distance, we first specify the two HXR sources

by eye (Figure 5.6) and determine the centroids of the HXR sources (xi, yi)

in image coordinate. Next, we convert the image coordinates (xi, yi) into

the heliographic coordinates (li, bi) (see Aschwanden et al., 1999b, Appendix

A, for the heliographic coordinate transformations). Finally, we calculate


Figure 5.7: Plot of the derived Ec vs. the calculated footpoint distance.

Correlation coefficient is 0.8. Solid line is the linear fit by the least square

method.

the footpoint distance d on the solar surface as follows (Aschwanden et al.,

1999a):

d = 2πR¯

√[(l1 − l2) cos b1]2 + (b1 − b2)2

360◦, (5.2)

where R¯ = 7× 105 km ≈ 1000′′ is the solar radius. Calculation results are

shown in Table 5.1.

Figure 5.7 shows the relationship between the derived Ec and the cal-

culated footpoint distance. Correlation coefficient is 0.8. The linear fit by

the least square method is Ec = 0.98d + 2.7. This result is suggestive. The

straightforward interpretation about this result is that the DC electric field

acceleration of electrons out of a thermal plasma (e.g., Holman, 1985) effi-

ciently works in the impulsive phase of a solar flare, if the footpoint distance

corresponds to the size of the acceleration region. Further analyses are needed

5.4. Event Selection: Revisited 49

to discuss this issue, on which we will continue to study.

Table 5.1: Physical variables discussed in Section 5.1 through Section 5.3.

Ec εb d dN/dt

date (keV) (keV) (arcsec) (1035 sec−1)

1998/08/18 31− 34 ∼ 25 31.3 2.4− 3.7

2000/06/02 20− 22 · · · 18.8 9.0− 14

2000/11/24 23− 25 · · · 26.2 12− 17

2001/04/06 43− 48 ∼ 28 31.7 4.2− 7.3

2001/04/12 25− 28 · · · 26.4 3.5− 5.6

2001/08/25 38− 44 ∼ 25 42.6 4.1− 6.9

5.4 Event Selection: Revisited

In our analysis, flare events are selected according to the several criteria (see

Section 4.4). Some of them give a strict limitation against event selection.

Very impulsive flares are not selected for our analysis because of the criterion

’(c) SXT (Be119) images without saturated pixels at the end of the HXR

burst are available’. The criterion ’(d) There are both the HXT and SXT

observational data in the “pre-impulsive phase”’ also gives a strict limitation

against event selection. Because, in most flares observed with Yohkoh, the

X-ray intensity rapidly increases after the flare mode onset, this criterion

limits the number of available events. Therefore, the selected flares for our

analysis are “particular” ones. In fact, the six of the selected flares are GOES

X class flares. Our results can not be necessarily applied to other flares, such

as very impulsive flares, weak flares, and long-duration flares.

5.5 Concluding Remarks

In this thesis, we studied characteristics of nonthermal components in solar

flares by analyzing solar hard X-ray flares observed with Yohkoh quantita-


tively and statistically. We successfully estimated physical variables in solar

flares such as the lower energy cutoff in the spectrum of nonthermal electrons

with some assumptions (Chapter 4). Relationships of physical variables of

flare nonthermal components are suggested (Chapter 5). It is the first time

to our knowledge that such relationships are quantitatively shown.

Many problems of flare physics still remain. Here we mention the two

of still remaining problems. First problem is the “total number problem”

stated, e.g., by Miller et al. (1997). This problem is that the total number

of nonthermal electrons derived from the observed HXR spectrum is larger

than the total number of electrons contained in the whole flare region. In this

thesis, this problem is remained unsolved. Because the total number of non-

thermal electrons closely depend on the lower energy cutoff in the spectrum

of nonthermal electrons (eq. (4.10)), accurate determination of the lower

energy cutoff is necessary to settle this problem. The Reuven Ramaty High

Energy Solar Spectroscopic Imager1 (RHESSI), which is the newest solar hard

X-ray/gamma-ray imager, may give us an answer to this problem because

it has an ability to observe solar hard X-rays with high spectral and spatial

resolution. Second problem is about particle acceleration process. There

remain many questions about particle acceleration process, such as accelera-

tion mechanism, acceleration timing and duration, acceleration site, and so

on. We presented a rough discussion in Section 5.3. Highly spatially-resolved

spectral analysis (called “imaging spectroscopy”) is necessary to discuss the

particle acceleration process in solar flares more in detail. In addition, the-

oretical study of particle acceleration process is also necessary. None of the

proposed acceleration mechanism in solar flares, such as the DC electric field

acceleration, the stochastic acceleration, and the shock acceleration, are con-

clusive at the present time. Both theoretical and observational study must

be done to discuss not only the particle acceleration process in solar flares

but also the nature of flares themselves.

We expect that our work will be of benefit to the understanding of the

1http://hesperia.gsfc.nasa.gov/hessi/

5.5. Concluding Remarks 51

particle acceleration process in solar flares. And we believe that, in the

future, the mysteries of this attractive phenomenon are completely revealed.

Appendix A

Case Studies

Here we briefly present the results of the quantitative analysis in the selected

flare events except that presented in Section 4.5. The analysis method is

detailed in Chapter 4.

1997/11/06 X9.4 Flare

This flare commenced on 6 November, 1997 at ∼ 11:51 UT. Figure A.1

shows the HXR time profiles observed with Yohkoh/HXT. The GOES soft

X-ray class was X9.4.

Figure A.2 shows the time profiles of the M2-band count rate (red line)


asterisk). Black solid lines at ∼ 11:53 UT and at ∼ 11:55:30 UT denote

“T05” and “T90”, respectively. The region surrounded by green lines is

defined as the pre-impulsive phase (tpre = T0 ∼ T0.05). Note that the

volume of the SXR emitting flare plasma is assumed to be V = S3/2 (see eq.

(4.9)).

The estimated physical variables are as follows: ∆Eth = Eth(T90) −Eth(tpre) = 2.0×1031 (erg), ∆t = T90−T05 = 158 (sec), and Ec > 50 (keV).

Since the derived Ec is high compared with the energy range of M2-band

(32.7 - 52.7 keV), it will have an uncertainty (see Section 3.1.3). Therefore,

we don’t estimate the total number of nonthermal electrons (N) and the

53

54 A. Case Studies

nonthermal electron rate (dN/dt) in the impulsive phase.

1998/08/18 X2.8 Flare

This flare commenced on 18 August, 1998 at ∼ 8:17 UT. Figure A.3 shows

the HXR time profiles observed with Yohkoh/HXT. The GOES soft X-ray

class was X2.8.



asterisk). Black solid lines at ∼ 8:18:40 UT and at ∼ 8:25:20 UT denote


defined as the pre-impulsive phase (tpre = T0 ∼ T0.5). Note that the volume

of the SXR emitting flare plasma is assumed to be V = S3/2.

The estimated physical variables are as follows: ∆Eth = Eth(T90) −Eth(tpre) = 1.0 × 1031 (erg), ∆t = T90 − T05 = 406 (sec), Ec = 31 (keV),

N = 1.5× 1038 (electrons), and dN/dt = 3.7× 1035 (electrons/sec).

2000/06/02 M7.6 Flare

This flare commenced on 2 June, 2000 at ∼ 18:52 UT. Figure 5.3 shows


class was M7.6.



asterisk). Black solid lines at ∼ 19:22 UT and at ∼ 19:22:50 UT denote


defined as the pre-impulsive phase (tpre = T0 ∼ T0.5). Note that the volume

of the SXR emitting flare plasma is assumed to be V = S3/2.



2000/11/24 X2.3 Flare

This flare commenced on 24 November, 2000 at ∼ 14:54 UT. Figure A.6

55

shows the HXR time profiles observed with Yohkoh/HXT. The GOES soft

X-ray class was X2.3.




and “T90”, respectively. The region surrounded by green lines is defined as

the pre-impulsive phase (tpre = T0 ∼ T0.05). Note that the volume of the

SXR emitting flare plasma is assumed to be V = S3/2.



2001/04/06 X5.6 Flare

This flare commenced on 6 April, 2001 at ∼ 19:12 UT. Figure 5.1 shows


class was X5.6.



asterisk). Black solid lines at ∼ 19:14:30 UT and at ∼ 19:20 UT denote


defined as the pre-impulsive phase (tpre = T0 ∼ T0.05). Note that the

volume of the SXR emitting flare plasma is assumed to be V = S3/2.



2001/08/25 X5.3 Flare

This flare commenced on 25 August, 2001 at ∼ 16:25 UT. Figure A.9 shows


class was X5.3.



56 A. Case Studies


and “T90”, respectively. The region surrounded by green lines is defined as

the pre-impulsive phase (tpre = T0 ∼ T0.05). Note that the volume of the

SXR emitting flare plasma is assumed to be V = S3/2.



57

Figure A.1: HXR time profiles observed with Yohkoh/HXT in the 1997

November 6 flare.

T05 T90

∆Eth

∆t

Figure A.2: Time profiles of the M2-band count rate (red line) and the

estimated thermal energy of the SXR emitting flare plasma (blue asterisk)

in the 1997 November 6 flare.

58 A. Case Studies

Figure A.3: HXR time profiles observed with Yohkoh/HXT in the 1998 Au-

gust 18 flare.

T90T05

∆Eth

∆t



in the 1998 August 18 flare.

59

T05 T90

∆Eth

∆t



in the 2000 June 2 flare.

60 A. Case Studies

Figure A.6: HXR time profiles observed with Yohkoh/HXT in the 2000

November 24 flare.

T05 T90

∆Eth

∆t



in the 2000 November 24 flare.

61

T90T05

∆Eth

∆t



in the 2001 April 6 flare.

62 A. Case Studies

Figure A.9: HXR time profiles observed with Yohkoh/HXT in the 2001 Au-

gust 25 flare.

T90T05 ∆Eth

∆t



in the 2001 August 25 flare.

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Acknowledgement

The author would like to thank Prof. Yokoyama T., who is my supervi-

sor, for his kindhearted guidances, valuable advices, and fruitful discussions,

throughout this work. I am very pleased with that I have stimulating years

in his laboratory.

I would like to thank Prof. Terasawa T., and Hoshino M. for their precise

and fruitful comments. I am grateful to all members of the Yohkoh team.

Especially, I would like to thank Drs. Masuda S., and Sato J. for their helpful

advices and fruitful discussions. I also would like to thank Drs. Shibasaki K.,

Shimojo S., and Asai A. for their helpful and valuable advices. They gave

me stimulating experiences at Nobeyama Radio Observatory. Significant

discussions with students in the Solar-Terrestrial Physics Group are gratefully

acknowledged.

67

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