Chapter 13
Hard X-Rays
Photons with energies in hard X-ray wavelengths (
keV) are produced by particles in a collisional plasma, mostly by collisions between relativistic electrons and thermal ions. Since we cannot place in-situ particle detectors into the solar corona, we have to obtain information on energetic particles from solar flares by remote-sensing of hard X-ray photons and radio photons. Moreover, since the Earth’s atmosphere absorbs hard X-rays, solar hard X-rays can only be detected by space-borne detec- tors. Major contributions to hard X-ray observations have been made over the last 25 years from hard X-ray spectrometers onboardSMM andCGRO, and from the hard X-ray imagersSMM/HXIS,Hinotori,Yohkoh/HXT, andRHESSI. Solar flare hard X- ray emission provides us a key diagnostic on particle acceleration and propagation processes. However, since hard X-ray emission is produced most prolifically when nonthermal electrons precipitate from collisionless coronal sites towards the highly- collisional chromosphere, most information comes from mapping of the electron pre- cipitation sites, their energy-dependent timing, their energy spectra, and their temporal correlation with flare signatures in other wavelengths. From these pieces of informa- tion, we have to work backward to reconstruct the magnetic topology in flare regions, to localize the acceleration sites with respect to magnetic reconnection diffusion re- gions, the particle trajectories of free-streaming and trapped particles, and ultimately to attempt a diagnostic on the physical processes of energization and acceleration in the first place. A major breakthrough during the last decade was the discovery of coronal above-the-loop-top sources and electron time-of-flight measurements, which both pin- point consistently the acceleration and injection of particles in reconnection regions.
At the time of writing we witness pioneering results fromRHESSI, which provides the first hard X-ray high-resolution imaging spectroscopy, the first high-resolution gamma- ray line spectroscopy, and the first imaging above energies of 100 keV.
Reviews on flare-related hard X-ray emission can be found in Emslie & Rust (1980), Dennis (1985, 1988), De Jager (1986), Dennis et al. (1987), Vilmer (1987), Dennis &
Schwartz (1989), Bai & Sturrock (1989), Brown (1991), Culhane & Jordan (1991), Hudson & Ryan (1995), Aschwanden (1999d, 2000b, 2002b), Lin (2000), and Vilmer
& MacKinnon (2003).
551
Table 13.1: Compilation of hard X-ray and gamma-ray detectors and imagers used for solar flare observations.
Spacecraft Instrument Energy Time Detector Imaging Operation
or Detector range resolution area resolution period OSO-5 CsI(Na) 20-250 keV 1.8 (0.2) s 71 cm
1969-75
OSO-7 NaI(Ti)
10-300 keV 10, (2.5) s 9.57 cm
1971-74 (Balloon) NaI(Ti)
keV 0.1 s 60 cm
1974
(Balloon) Ge
13-300 keV 0.008 s 300 cm
1981
Hinotori SXT 10-40 keV 7 s 113 cm
20” 1981-82
HXM 17-340 keV 0.125 s 57 cm
1981-82
SGR 0.21-6.7 MeV 2 s 62 cm
1981-82 ISEE-3 NaI(Tl) 12-1250 keV 0.125 s 22 cm
1978-..
SMM HXIS
3.5-30 keV 0.5-7 s 1.44 cm
8”, 32” 1980-89 HXRBS
20-260 keV 0.128 s 71 cm
1980-89
GRS
0.3-9 MeV 16, (2) s 200 cm
1980-89
CGRO BATSE 20-300 keV 0.064 s 2025 cm
1991-00
OSSE 0.05-10 MeV 2, (0.016) s 2620 cm
1991-00
Yohkoh HXT
14-93 keV 0.5 s 70 cm
5” 1991-01
WBS
2, (0.25) s 12 cm
1991-01
RHESSI HPGe
3 keV-20 MeV 2 s 90 cm
2.3” 2002-..
References:
) Frost (1969), Frost et al. (1971);
) Datlowe et al. (1974); Harrington et al. (1972);
) Hurley & Duprat (1977);
) Lin et al. (1981);
) Makishima (1982), Takakura et al. (1983a), Enome (1983), Tsuneta (1984);
) Anderson et al. (1978);
) Van Beek et al. (1980);
) Orwig et al. (1980);
) Forrest et al. (1980);
) Fishman et al. (1989);
!
) Kurfess et al. (1998);
"
) Kosugi et al. (1991);
) Yoshimori et al. (1991);
) Lin et al. (1998);
13.1 Hard X-ray Instruments
In Table 13.1 we provide a compilation of hard X-ray detectors and imagers that made major contributions to the study of solar flares. Most of them are space-based instru- ments, attached to solar-dedicated or all-sky astrophysical missions, and a few instru- ments have also been flown on balloon flights. In the following we describe in more detail four instruments that collected most of the flare observations.
13.1.1 SMM - HXRBS, GRS, HXIS
TheSolar Maximum Mission (SMM)was the first solar flare dedicated mission, lasting a full decade, from 1980-Feb-4 to 1989-Dec-2. The scientific highlights of the SMM mission are reviewed in the monograph of Strong et al. (1999). The instrument suite contained three hard X-ray instruments, theHard X-Ray Burst Spectrometer (HXRBS) (Orwig et al. 1980), theGamma Ray Spectrometer (GRS)(Forrest et al. 1980), and the Hard X-Ray Imaging Spectrometer (HXIS)(Van Beek et al. 1980). SMM was in an orbit with an initial altitude of 524 km and an inclination of#%$'&(
.
13.1. HARD X-RAY INSTRUMENTS 553
Figure 13.1:Cross-sectional views of theHard X-Ray Burst Spectrometer (HXRBS) onboard theSolar Maximum Mission (SMM) spacecraft (Orwig et al. 1980).
HXRBSprovided high-time resolution (0.128 s) histories of hard X-ray time pro- files in 16 channels in the energy range of # # keV. TheHXRBSinstrument consisted of a central CsI(Na) detector surrounded by a CsI active collimator element (Fig. 13.1). The CsI crystal was viewed by four photomultipliers, operated in anticoin- cidence. The duty cycle of the instrument was about 50% andHXRBSrecorded over 12,000 solar flare events (see catalog by Dennis et al. 1991).
GRSobserved at higher energies, in the range of 0.3-9 MeV, and recorded some 270 -ray flares at keV.HXISwas the first instrument to image hard X-ray flares, with a resolution of 32”, and in a fine FOV with 8”-pixels. Because theHXIS energy range was 3.5-30 keV, the images are dominated by thermal emission.
13.1.2 Yohkoh - HXT
Shortly after the SMM-decade, the first hard X-ray imager at energies keV was launched onboard the Yohkohspacecraft, called the Hard X-Ray Telescope (HXT) (Kosugi et al. 1991), while the context images in soft X-rays were recorded with the Soft X-Ray Telescope (SXT). TheYohkohmission was a solar flare dedicated mission, and both of the instruments switched to higher time and spatial resolution in flare mode.
HXTis a Fourier-synthesis imager with 64 collimator detectors, each one with a bi- grid collimator in front (Fig. 13.2), providing measurements of the sine and cosine of 32 independent spatial Fourier components of the source. Images can be reconstructed
Figure 13.2: Arrangement of the 64 subcollimators ofYohkoh/HXT which record 64 one- dimensional Fourier components. The cosine and sine Fourier element pairs are shown with solid and dashed linestyles (Kosugi et al. 1991).
from these measurements using algorithms such as the Maximum Entropy Method (MEM)andCLEAN.HXTwas able to produce images in 4 energy bands, 15-24 keV, 24-35 keV, 35-57 keV, and 57-100 keV, with an angular resolution of $ and a time resolution of 0.5 s. The lifetime ofYohkohextended over a full solar cycle, andHXTrecorded a total of 3,112 flare events during the period from 1991-Oct-1 to 2001-Dec-14, documented in theYohkoh HXT/SXT Flare Catalogue(Sato et al. 2003).
13.1.3 CGRO - BATSE
TheBurst and Transient Source Experiment (BATSE)on theCompton Gamma Ray Observatoryis the most sensitive all-sky hard X-ray and -ray detector system ever flown. It consists of 8large-area detectors (LADs), with an area of 2025 cm
each, placed at the eight corners of the spacecraft (see Fig. 1.3). In addition,BATSEis also equipped withspectroscopy detectors (SD), which consist of NaI(Tl) scintillators with a front area of 127 cm
, cover a broad energy range of 15 keV 110 MeV and have 7.2% energy resolution. BATSE was operated in different energy and time binning modes, triggering automatically to a higher rate after a burst trigger. Weak solar flare events were recorded with a low time resolution of 1.024 s in four energy channels (25- 50, 50-100, 200-300, 300 keV), while burst trigger events were recorded with 16 ms and 64 ms time resolution in 16 energy channels. CGROoperated during the period from 1991-Apr-5 to 2000-Jun-4, andBATSE recorded a total of 8021 burst triggers with the following identification of events: 2704 astrophysical gamma ray bursts, 1192 solar flares, 1717 magnetospheric events, 78 terrestrial gamma flashes, 2003 transient sources, and 185 soft gamma repeaters.
13.1. HARD X-RAY INSTRUMENTS 555
Figure 13.3:Layout of the RHESSI telescope that was mounted on a rotating spacecraft. The telescope (left) contains a set of 9 front grids and 9 identical rear grids which together modulate the incoming hard X-ray photons. The mounting of the 9 grids is shown on the right: The grid pitch (slit and slat) increases by a factor of
from grid 1 to 9, so that each one modulates a particular angular Fourier period. The modulated throughput is detected by 9 cooled germanium detectors, one behind each of the rear grids (left), (Hurford et al. 2002).
13.1.4 RHESSI
TheReuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI), launched on 2002-Feb-5, is a Fourier imager of the class ofrotation-modulated collimators (RMC), which spins with a period of s.RHESSIuses 9 collimators, each one made up of a pair of widely separated grids. Each grid is a planar array of equally-spaced, X-ray- opaque slats separated by transparent slits (Fig. 13.3). The slits of each pair of grids are parallel to each other and their pitches are identical, so that the transmission through the grid pair depends on the direction of the incident X-rays. Different Fourier com- ponents are measured at different rotation angles and with grids of different pitches.
For RHESSI, the grid pitches range from p=34 m to 2.75 mm in steps of . This provides angular resolutions spaced logarithmically from 2.3” to 180”. The images are reconstructed with algorithms likeClean,Maximum Entropy Method (MEM),Maxi- mum Entropy Method Visibilities (MEMVis),Pixon, andForward-Fitting. RHESSI is the first telescope to provide hard X-ray imaging at such high angular resolution.
Another prime capability ofRHESSIis its high energy resolution, thanks to the cooled germanium detectors, i.e., keV FWHM at 3 keV, increasing to keV at 5 MeV. This allows the many gamma ray lines with typical FWHM of 2-100 keV in the 1-10 MeV range to be resolved for the first time. Instrumental descriptions of the RHESSI instrument can be found in Lin et al. (1998, 2002), the imaging concept is described in Hurford et al. (2002), and the spectrometer in Smith et al. (2002).
2θ b
v
e
-+Ze
Figure 13.4: Elastic scattering of an electron ( ) off a positively charged ion ( ). The electron moves with velocity v on a path with impact parameter and is deflected by an angle of , with
v
, according to the Rutherford formula. Electromagnetic radiation (bremsstrahlung) is emitted as a consequence of the acceleration of the particle during the swing-by around the ion.
13.2 Bremsstrahlung
The most important radiation mechanism that produces a continuum of emission in hard X-ray wavelengths is bremsstrahlung, which results from emission of photons when electrons are elastically scattered in the electric Coulomb field of ambient ions (Fig. 13.4). We distinguish three different situations: (1)Thermal bremsstrahlungre- sults when the colliding electrons have the same temperature as the ambient plasma ( 2.3), (2)Thick-target bremsstrahlungoccurs when the incident electrons have first been accelerated to a much higher (non-thermal) energy (in a collisionless plasma) and then become collisionally stopped when they hit a thermal plasma, and (3)Thin- target bremsstrahlungoccurs when electrons are continuously accelerated in a colli- sional plasma and the X-ray spectrum is nearly unchanged from the acceleration or injection spectrum.
13.2.1 Bremsstrahlung Cross-Sections
Elastic scattering of a single electron ( ) off an ion with charge ( ) is quanti- fied by thedifferential scattering cross section "!$#%& (' with theRutherford formula (Eq. 2.3.5). The derivation of thisdifferential scattering cross sectionfor bremsstrahlung in Coulomb collisions can be found in standard textbooks on classical electrodynamics (e.g., Jackson 1962;15.2). Theradiation cross-section)+*,.-0/2143, which specifies how much energy is radiated in bremsstrahlung photons at frequency15/ &&&6/21 7 (1 by an in- cident electron with velocity- is calculated by an integration over all scattering angles
8
or impact parameters9 within the possible range of9:<;6=?>@9A>B9C:D
(Eq. 2.3.7;
Fig. 2.5). In the classical derivation of bremsstrahlung, the upper limit is estimated by the maximum momentum transfer 9:D FE #CG E #IHKJ2- , and the lower limit is set by the collision time. The integral of this radiation cross-section is also called the Coulomb IntegralLNMO,QPR3 (orGaunt factorS0,Q15/ TU3, see definition in Eq. 2.3.8). The clas- sical derivation is non-relativistic, and conservation of momentum and photon energy
13.2. BREMSSTRAHLUNG 557
are not considered; it is thus only valid for thermal bremsstrahlung (2.3).
For non-thermal bremsstrahlung, however, conservation of energy and momentum must be considered, which is (for weakly relativistic electrons),
E 1 / , &#'&
3
9 E
,
3
,
3 / , &#'&# 3
where E G
%%#IH J and E G
% #&H J are the kinetic energies of the electrons before
and after the collision,
1 and E
14% are the energy and momentum of the photon,
and9 is the momentum transferred to the scattering center. The ratio of the maximum to the minimum momentum transfer is given by the kinematic limits,GR G 3 and,G G 3,
9 :<D
9 :R;6=
E
G+ G
G G E
E , 3
E ,
143
1 & , &#'&
3
which leads to the non-relativistic (or weakly-relativistic)Bethe-Heitler cross-section (Jackson 1962; Eq. 15.29)
) *,.-0/153
E (
H
L6M
,
153
1 & , &#'&
3
The numerical factor E corresponds to the quantum-mechanical result in theBorn approximation, first calculated by Bethe & Heitler in 1934. Inserting the fine structure constant E
%
E %
, the classical electron radius &J E
%&HKJ
E
#'&$
!
"
cm, the kinetic energy of an electron E H J
, 3 ,
%%# 3H J
, considering only collisions between electrons (
E
/ H
E H J ) and protons ( E ), and using the Born approximation ( E ), the cross-section reads,
) * ,.-0/2143
E (
J
H J
L6M
,
143
1
,#"%$
'&)(+*-,/.
3 &
, &#'& 3
The radiation cross-section) *I,.-0/153 (per frequency unit 0 ) can be converted into a photon cross-section! , / 3, per photon energy E
1 E 0 , which is defined by (e.g., Lang 1980, p.43),
! , / 3 E ) *I,.-0/153
0 E
) * ,.-0/2143
, &#'&("3
which yields the commonly used form of the Bethe-Heitler cross-section (e.g., Brown 1971; Hudson et al. 1978)
! , / 3 !
L6M 21
1
,3"%$
'4 &65
3 , &#'&
3
! E $
77
J H J
E &
$98
,#"%$
'4 &65
3 & , &#'&$"3
A simpler form can sometimes be used, neglecting the logarithmic term, which is the Kramers cross-section,
! , / 3 ! & , &#'&:"3
TheBethe-Heitler andKramerscross-sections are only applicable to non-relativistic or mildly relativistic electrons. For higher energies, a full-relativistic cross-section has been derived (Elwert 1939; Koch & Motz 1959), which was applied in form of a multiplicative Elwert factor by Holt & Cline (1968). An expansion of the fully- relativistic cross-section up to 6-th order of the momentumG E % #&H J is given in Haug (1997). Comparisons show that the relative error using the non-relativistic cross- section can exceed 10% already for mildly-relativistic energies of keV (Haug 1997).
13.2.2 Thick-Target Bremsstrahlung
The total X-ray emission from an emitting volume is proportional to the number of collisions between electrons and ions (mainly protons), i.e., to the product of their densities integrated over the volume,OJ , 3 . The bremsstrahlung cross-section
! , / 3 has the unit of a target area per photon energy, (cm
keV
), see Eq. 13.2.7), so the product of the cross-section! , / 3 with the velocity-0, 3 of the incident elec- trons corresponds to a target volume per time unit. To obtain the total number of emitted photons (of a given photon energy E 1 ) we have also to integrate over all contributions from electrons with energies higher than the photon energy, i.e., ,
O
E
! , / 3 -0, 3
OJ&, 3
,
4
&65
3 / ,
&# & 3
which has the unit of photons per time and energy, (s
keV
). Assuming a uniform target density over the volume, so that is constant and defining E & , the mean photon count rate at Earth distance, E AU, has to be scaled by a factor of
%
, yielding an observed hard X-ray intensity! , 3 of
! , 3 E
" #$
E
! , / 3 -0, 3% J , 3 / ,
&# &
3
which has the units of photons per detector area, time, and energy. If we plug in the non-relativistic electron velocity-0, 3 E& # %&H J and the Bethe-Heitler cross-section
! , / 3 (Eq. 13.2.7) into the integral of the hard X-ray spectrum! , 3 (Eq. 13.2.11), we obtain
!,
3 E
"
!
L6M 1 1 #
H J
'
OJ&, 3 & ,
&# & # 3
This equation implicitly defines theinstantaneous nonthermal electron spectrum J&, 3 that is present in the hard X-ray emitting source. For thick-target emission, however, the electron injection spectrum ( J , 3 that is externally injected into the hard X-ray emitting source is different from J&, 3. The transformation from the injection spectrum
(IJ , 3 into the source spectrum J , 3 is defined by the energy loss process. If the energy losses are purely Coulomb collisional, we have the following energy loss function,
E *)
, 3-0, 3 / ) E P & ,
&# & 3
13.2. BREMSSTRAHLUNG 559
The total number of photons emitted from an electron with initial energy at photon energy during braking, as long as , is
10,
/ 3 E
! , / 3, 3 -$, 3 "/ , &#'&
3
which can be written as an energy integral by substituting, 3 -$, 3 " E , %
) 3
from Eq.(13.2.13),
10,
/ 3 E
! , / 3 ) & , &#'&
3
Since the electrons are decelerated until they are at rest in the target, the ambient plasma density is not relevant for the emitted photon flux1 , / 3 (Eq. 13.2.15), which is thus independent of . The total photon emission rate from the region !, 3 can then be expressed in terms of theelectron injection spectrum( J , 3 incident into the X-ray emitting region per second (with eq. 13.2.15),
! , 3 E
" #$
E
"
1 , / 3 (J ,
3
& , &#'&
( 3
E
(IJ , 3
! , / 3 ) & , &#'&
3
This equation implicitly defines theinjection spectrum( J , 3 by the observed photon spectrum!, 3 .
The implicit equations for theinstantaneous source spectrum J , 3 (Eq. 13.2.12) and theelectron injection spectrum( J , 3 (Eq. 13.2.17) can be transformed intoAbelian integral equations and the analytical solution was calculated by Brown (1971). He assumed a powerlaw function for the observed hard X-ray spectrum ! , 3 (i.e., left- hand-side of Eqs. 13.2.12 and 13.2.17),
! , 3 E ! , 3
,M "%$
4
&)5
3/ , &#'&
$ 3
where is a reference energy, above which the integrated photon flux is ! (pho- tons cm
s
keV
), and is the powerlaw slope (not to be confused with the Lorentz factor). The parameters and of the hard X-ray spectrum can be time- dependent. The total number of photons above a lower cutoff energy is the integral of Eq. (13.2.18),
!,
3 E
! , 3 E !
, M "%$
3 & , &#'&
: 3
Brown (1971) solved the inversion of Eqs. (13.2.12) and (13.2.17) and found the fol- lowinginstantaneous nonthermal electron spectrum J&, 3, present in the X-ray emit- ting region, with the associatedelectron injection spectrum("J , 3,
OJ , 3 E &( 8 , 3
# / # !
,& L&
"
(
M 4 &65
3 /
( J , 3 E # &( $8
, 3
# / # !
,& L&
"
(
M 4 &65
3 /
,
&# &# 3
with (cm
) the mean electron or proton density density in the emitting volume, [keV] the lower cutoff energy in the spectrum,! (photons cm
s
keV
) the total X-ray photon flux at energies , and
,GO/I3 is theBeta function,
,G /I3
E
,
3
/ ,
&# &# # 3
which is calculated in Hudson et al. (1978) for a relevant range of spectral slopes and is combined in the auxiliary function9&,03,
9&, 3 E , 3
# / # &# & ,
&# &# 3
So the powerlaw slope of the electron injection spectrum ( E ) is steeper than that ( ) of the photon spectrum in the thick-target model. With this notation we can write the electron injection spectrum as
(IJ , 3 E
#'&( $ 8
!
,
3 9&,03
!
,& L& " ( M 4
&)5
3 &
,
&# &# 3
The total number of electrons above a cutoff energy is then
,
3 E
(IJ&, 3
E # &( $8
!
9I, 3 , 3
!
,& L& " (
M 3 &
,
&# &# 3
The power in nonthermal electrons above some cutoff energy is
,
3 E
(IJ&, 3 E # &( $98
9&,03!
,4 &65
3 &
,
&# &#%("3
or a factor of (keV/erg)= &( 8
smaller in cgs-units,
,
3 E
( J , 3 E & 8
9&,03!
,
&)(+*
3 &
,
&# &# 3
Solar flares have typical photon count rates in the range of! E
(photons s
cm
) at energies of 20 keV and slopes of . Thus, for E E # keV, and using9&,03 &#
(Eq. 13.2.23), we estimate with Eq. (13.2.27) a nonthermal power of , # keV) 8
8
!
erg s
. Integrating this power over typical flare durations ofND * J
s yields a range of E , #
keV)8ND * J 8
8
[erg] for flare energies. A frequency distribution of total nonthermal flare energies in electrons ( # keV) that covers this range has been determined in Crosby et al. (1993), see Fig. 9.27.
13.2. BREMSSTRAHLUNG 561
In some work, the photon spectrum (Eq. 13.2.18) is specified with the variable , which gives the photon flux at 1 keV and relates to! in Eq. (13.2.18) by,
E ! , 3
/ , &#'&# $ 3
The observed hard X-ray photon spectrum!, 3 observed at Earth (Eq. 13.2.18), the thick-target electron injection spectrum( JI, 3 (Eq. 13.2.24), and the total power in non- thermal electrons above some cutoff energy , i.e., , ), Eq. (13.2.27), are then
! , 3 E
, M 7")$
4 &65
3 / , &#'&# : 3
( J , 3 E # &( $98
9I, 3
,& L&
"
(
M 4 &65
3/ , &#'&
3
,
3 E & 8
9I, 3
, 3 ,
3
,
&6( * 3 & , &#'&
3
13.2.3 Thin-Target Bremsstrahlung
For the thin-target case, the electrons are continuously accelerated and the hard X-ray spectrum is nearly identical with the acceleration or injection spectrum. The thin- target electron injection spectrum( J , 3 is given in Lin & Hudson (1976) and Hudson et al. (1978),
( J , 3 E & 8
,03
' ,& L& " ( M 4 &)5
3/ , &#'&
#3
and the function
,03 is defined in terms of theBeta function,
, 3 E , 3
, / 3 , &3
& , &#'&
3
We see that a given photon spectrum, e.g., with a slope of E , implies a flatter electron injection spectrum in the thin-target case ( E
E
#'& ) than in the thick-
target case ( E UE ). Hard X-ray sources observed in chromospheric heights are generally interpreted in terms of thick-target bremsstrahlung. In occulted flares, how- ever, when the hard X-ray sources at the supposed chromospheric flare loop footpoints are blocked, extended sources of hard X-ray and gamma-ray emission are sometimes observed over a considerable coronal height range, which are likely to be produced by thin-target bremsstrahlung (e.g., Datlowe & Lin 1973; Barat et al. 1994; Trottet et al. 1996). The fact that coronal hard X-ray sources are generally much weaker than chromospheric footpoint sources implies that thick-target bremsstrahlung is generally dominant (in non-occulted flares).
1 10 100 Photon energy ε [keV]
100 102 104 106 108
Photon flux I[photons cm-2 s-1 keV-1]
Large flare T=30 MK log(n)=11 log(EM)=49 d=1000 s
εc=19.5 keV Small flare T=10 MK log(n)=10 log(EM)=46 d= 100 s
εc= 4.7 keV
Figure 13.5:Left: Theoretical hard X-ray spectrum consisting of a thermal and a nonthermal (powerlaw) component with equal energy content above the cutoff energy. The parameters are chosen for a large flare with
MK,
cm
,
cm
,
, and duration
s; and for a small flare with
MK,
cm
, 7
cm
,
, and duration
s;Right: Synthetic spectra between 4 and 10 keV that show the 6.7 keV Fe/Ni feature that becomes progressively prominent for increasing flare temperatures. The spectra have been calculated with the CHIANTI code, with coronal abundances of Fe and Ni, and smoothed with a gaussian filter with a width of FWHM=0.8 keV.
Spectra are given in 1 MK intervals from 8 to 33 MK. Fluxes are those at the mean solar distance and for a flare with volume emission measure
cm
(Phillips 2004).
13.3 Hard X-ray Spectra
13.3.1 Thermal-Nonthermal Spectra
Soft X-ray measurements show that the flare plasma has typical electron temperatures in the range ofT J MK (see compilation in Table 9.4; e.g., Pallavicini et al. 1977; Metcalf & Fisher 1996; Reale et al. 1997; Garcia 1998; Sterling et al. 1997;
Nitta & Yaji 1997). This temperature range corresponds to electron energies of E
T J &:
#'&( keV (Appendix E). In the thick-target bremsstrahlung model (13.2.2),
nonthermal populations of electrons and ions accelerated in the corona precipitate to the chromosphere, heat up the plasma at the flare loop footpoints, and cause an over- pressure that drives upflows of heated plasma into the flare loops seen in soft X-rays. In this so-calledchromospheric evaporationprocess we expect that the energy of the pre- cipitating nonthermal electrons has to exceed the thermal energy of the heated plasma that is produced as a consequence. Let us compute such a combined thermal-plus- nonthermal hard X-ray spectrum in order to understand the energy ranges in which the two components dominate, the cutoff energy that separates them, and the overall spectral shape.
In2.3 we defined the thermal flux spectrum 4, 3 (Eq. 2.3.13), which yields the photon number by dividing through the photon energy , i.e., the photon spectrum is
! , 3 E
, 32% . Assuming a uniform temperature throughout the flare volume and
13.3. HARD X-RAY SPECTRA 563
using the units E %
cm
,T E T J %
K,
T E &$ ( keV, and
the photon energy in units of keV, we obtain from Eq. (2.3.13) the following thermal photon spectrum,
! , 3 E # &( 8
T
'
J
&
J
&$ ( T
, M "%$
4 &)5
3 &
, & & 3
The total energy in thermal electrons is (with Eq. 9.6.1),
E J T J E & 8
!
T
,
&)(+*
3 / , & &# 3
where thevolume emission measureis defined as E
J (assuming a filling factor of unity).
On the other side we can quantify the total energy in nonthermal electrons above some cutoff energy , by multiplying the power (Eq. 13.2.31) with the flare duration
, ND * J 3,
7= 4,
3 E ,
3 8 6D * J E & 8
, 3
,
S53
, & & 3
where we used the approximation9&, 3 &#
and denote E ND * J %
s.
If we assume energy equivalence between thermal and nonthermal energies, i.e.,
E
=
, 3 with Eqs. (13.3.2-3), we find a condition for the photon flux constant , which then yields the nonthermal photon spectrum (Eq. 13.2.29),
! = , 3 E & 8
T
!
, 3
& , & & 3
The cutoff energy of the nonthermal spectrum can be defined by the intersection of the thermal with the nonthermal photon spectrum, i.e.,! , E 3 E ! = , E 3, which yields the following expression,
E &$ ( T
LNM
, 3
T '
& , & & 3
In Fig. 13.5 we plot such a theoretical thermal-plus-nonthermal photon spectrum based on the energy equivalence between both components. For a large flare (T E / E
/
E / E
, and E ) we find a nonthermal cutoff energy of E :'&
keV, and for a small flare (T E / E & / E & / E , and E ) we find a nonthermal cutoff energy of E & keV. Thus, in such a model where the nonthermal energy is fully converted into thermal energy, we expect nonthermal cutoff energies in the range of # keV. At lower energies we expect the thermal (near-exponential) spectrum to dominate, while the nonthermal (powerlaw- like) spectrum dominates at higher energies .
Spectral fitting to observed hard X-ray spectra have indeed confirmed the exis- tence of the generic thermal-plus-nonthermal model described above. Early hard X- ray detectors, such as NaI(Tl) and CsI(Na) on ISEE-3 or HXRBS/SMM did not have
Figure 13.6: Observed hard X-ray spectra from the 1980 June 27 flare, labeled with the time in s after 1614:41.87 UT. The flux of each spectrum is offset by a factor
. The dashed lines show fits of the thermal spectrum and the dotted lines fits of a thick-target bremsstrahlung spectrum produced by DC electric field acceleration in the runaway regime (Benka & Holman 1994).
sufficient spectral resolution to resolve the steep thermal spectrum at any energy, but high-resolution spectra with cooled germanium detectors (from balloon flights by Lin et al. 1981 or RHESSI) clearly reveal the detailed spectral shape of the thermal-plus- nonthermal spectrum as calculated in Fig. 13.5. The hard X-ray spectrum of the 1980 June 27 flare was observed with high spectral resolution in the 13-300 keV range and revealed the presence of asuperhot temperature componentwith a maximumT J MK and an emission measure of E # &: 8
cm
(Lin et al. 1981; Lin &
Schwartz 1987). The same flare was also fitted with a DC electric field acceleration model for the nonthermal component (Fig. 13.6; Benka & Holman 1994; Kucera et al. 1996). For such “superhot” temperatures ofT J MK, the thermal component
