A&A 446, 1157-1163 (2006)
E. P. Kontar1 - A. L. MacKinnon2 - R. A. Schwartz3 - J. C. Brown1
1 - Department of Physics and Astronomy, University of Glasgow, G12 8QQ, UK
2 - DACE, University of Glasgow, G3 6NH, UK
3 - SSAI, Science Systems & Applications, Inc., Code 612.1, NASA/GSFC, Greenbelt, MD 20771, USA
Received 21 June 2005 / Accepted 30 September 2005
The observed hard X-ray (HXR) flux spectrum from solar flares is a combination of primary bremsstrahlung photons with a spectrally modified component from photospheric Compton backscatter of downward primary emission. The latter can be significant, distorting or hiding the true features of the primary spectrum which are key diagnostics for acceleration and propagation of high energy electrons and of their energy budget. For the first time in solar physics, we use a Green's function approach to the backscatter spectral deconvolution problem, constructing a Green's matrix including photoelectric absorption. This approach allows spectrum-independent extraction of the primary spectrum for several HXR flares observed by the Ramaty High Energy Solar Spectroscopic Imager (RHESSI). We show that the observed and primary spectra differ very substantially for flares with hard spectra close to the disk centre. We show in particular that the energy dependent photon spectral index is very different for and for and that inferred mean source electron spectra differ greatly. Even for a forward fitting of a parametric to the data, a clear low-energy cutoff required to fit essentially disappears when the fit is to - i.e. when albedo correction is included. The self-consistent correction for backscattered photons is thus shown to be crucial in determining the energy spectra of flare accelerated electrons, and hence their total number and energy.
Key words: Sun: flares - X-rays: general - methods: data analysis - techniques: spectroscopic - scattering
Comptonisation of prescribed primary HXRs can be straightforwardly studied using Monte-Carlo simulations (e.g. Bai & Ramaty 1978). This technique is ideal for obtaining the reflected, and hence the total observed, photon spectrum if one knows or assumes the form of the primary X-ray spectrum (Alexander & Brown 2002). However, the primary spectrum is unknown and unlikely to be an exact power-law, as is usually assumed. Even with a power-law distribution of primary electrons a number of physical processes exist which will cause the resultant photon distribution to deviate from pure power-law form e.g. effects of nonuniform ionization (Brown 1973; Kontar et al. 2003), beam return current losses (Zharkova & Gordovskyy 2005) and others. Results obtained from Monte Carlo simulation for power-law primary photon spectra cannot be straightforwardly applied to such situations. More sophisticated parametric modelling of the primary photon spectrum, with more theoretical uncertainties and free parameters, would become necessary. Therefore, an approach independent of the primary spectrum is required.
Understanding and modelling backscatter has become even more important with the advent of high quality X-ray spectra from the Ramaty High Energy Solar Spectroscopic Imager (RHESSI) with spectral resolution as high as keV, in combination with uncertainties as low as a few percent (for strong flares) allowing detailed spectral analysis over a broad X-ray range (Lin et al. 2002). A correct treatment of Compton backscatter by a cold medium has also been important for understanding spectra from cosmic X-ray sources (reviewed e.g. by Svensson 1996). This led to construction (Magdziarz & Zdziarski 1995) of an analytic (piecewise continuous), angle dependent, Green's function for the probability that a photon, of initial energy and drawn from a distribution isotropic in the downward hemisphere, incident on a plane-parallel atmosphere, is re-emitted in direction ( ) with energy . This was obtained by fitting the results of a large set of Monte Carlo calculations, and its form is guided, particularly in various limits, by analytical results such as those of Illarionov et al. (1979), Lightman et al. (1981), White et al. (1988). Representing as it does the results of Monte Carlo calculations, no assumptions of single scattering or on the form of primary spectrum are involved. The only major assumptions in the Green's function formulation are those already mentioned: plane geometry of the cold matter - this is well satisfied for the photospheric conditions and downward isotropy of the primary photon distribution.
Combined with a treatment of absorption (Morrison & McCammon 1983) with standard solar photosphere element abundances (Anders & Grevesse 1989), the Green's function formulation of Compton backscatter provides us with a powerful tool for interpreting flare hard X-ray spectra. Firstly, it allows us to calculate easily, without time consuming Monte Carlo simulations, the total observable photon spectrum for any specified primary spectrum, not just the power-law cases treated by Bai & Ramaty (1978). More importantly, because the Green's function may be used to express the observed spectrum in terms of any primary spectrum, we may use regularized inversion techniques (Craig & Brown 1986; Kontar et al. 2004, and references therein) to deconvolve the primary photon spectrum from observations. Features of this primary spectrum arise only from features of the HXR source electron distribution, not of the energy-dependence of backscattering, so it may be safely inverted (Kontar et al. 2004) to yield a meaningful source-averaged electron distribution in the sense of (Brown et al. 2003). Although the Green's function makes such model-independent inversion possible, this step is carried out here for the first time.
Sections 2, 3 describe the Green's function formulation and the incorporation of absorption respectively. Section 4 describes how to recover the primary spectrum from data, and Sect. 5 applies this to real RHESSI data. In Sect. 6 we discuss applications and implications of our work.
As discussed in Sect. 1, downward-emitted photons will be either absorbed or scattered, and some scattered toward the observer, adding to the total flux detected. The backscattered photons will be emitted from an extended region of the photosphere, out to the horizon distance but with greatest intensity in an area of extent comparable to the altitude of the primary source (Brown et al. 1975; Bai & Ramaty 1978; Schmahl & Hurford 2004). The Green's function implemented here ignores spatial structure, however, simply summing over all photons that emerge in the correct direction. This makes it ideal for dealing with spatially unresolved spectra.
While the Green's function (Magdziarz & Zdziarski 1995) was obtained for use in studying cosmic X-ray sources, it applies to any plane-stratified slab of "cold'' material (i.e. the scattering electrons are non-relativistic). Scattering takes place only on electrons, whether free or atomic. To account for elements heavier than hydrogen the Compton cross-section is multiplied by an effective mean atomic number z=1.2. The detailed density structure of the medium is irrelevant (Tomblin 1972).
Absorption, on the other hand, does depend strongly on chemical composition, and the best estimate of photospheric abundances should be included when adding absorption to the Green's function treatment. We use standard abundances given by Anders & Grevesse (1989). The heavy elements Fe/Ni play the most important role in the range from 10 keV up to 30 keV, while lighter elements contribute below 10 keV (Morrison & McCammon 1983).
The total observed spectrum
is given by the sum of
i.e. using (1)
In practice measurements yield discrete quantities and the
integral Eq. (2) is presented in matrix form
|Figure 1: Green's functions including Compton scattering and photoelectric absorption for three primary photon energies keV and ( ).|
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The shape of the Green's function depends on the maximum energy of a primary photon. For primary photons with low energies keV, the Green's function has a rather simple structure close to a Dirac's delta function (Fig. 1) showing that backscattering is dominated by the first scattering, while the contribution from higher orders of scattering is small. In the range below keV photoelectric absorption dominates scattering and Compton scattering can be treated as diffuse monochromatic radiation transfer. The Green's function then can be replaced by (Chandrasekhar 1960, where the form of is given), , , are the total Compton and absorption cross-sections respectively. gives the single-scattering albedo modified by absorption and completely controls the influence of absorption. Application to the solar case simply requires evaluation of with solar photospheric abundances. In this energy range below 10 keV the total Compton scattering cross-section can be approximated by the Thompson cross-section . The absorption cross-section is based on the element abundances from Anders & Grevesse (1989). We use the coefficients of a piecewise polynomial fit to the numerical results given by Morrison & McCammon (1983) in the range below 10 keV and extrapolate them as above 10 keV. Due to rapid decrease of the absorption cross-section around 12 keV. The generated Green's matrices (4) have uncertainty around 6% (due to the approximations used) and 1 keV energy resolution.
|Figure 2: Upper panel: primary (solid line), reflected (dotted line) and total (dashed line) photon spectra calculated assuming a primary spectrum , and using the Green's function for a X-ray source at heliocentric angle . The lower panel shows the reflectivity, defined as the ratio of reflected to primary fluxes . The reflectivity taken from Bai & Ramaty (1978) is shown with diamonds. Two absorption edges of Fe at 7.1 keV and Ni at 8.3 keV are seen in the reflected component.|
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The absorption is photo-electric in nature and below 10 keV has a complex structure with multiple absorption edges due to elements heavier than hydrogen. In consequence it is also strongly influenced by elemental abundances. Figure 2 shows the two absorption edges of Fe at 7.1 keV and Ni at 8.3 keV in the reflected component. The spectrum shows features in agreement with the study of the reflection spectra in intergalactic nuclei by Ross & Fabian (1993). Here we ignored the fluorescence of the lines which is beyond the scope of our paper.
Note that Bai & Ramaty (1978) and Zhang & Huang (2004) assumed a different absorption approximation from us based on earlier photoelectric absorption cross-sections by Fireman (1974) and photospheric abundances by Withbroe (1971). However, we should note that the difference is rather modest accounting for around 30% of reflectivity at low energies and becoming smaller at higher energies, where absorption is less important (Fig. 2). The reflectivity, and thus the contribution of Compton back-scattered photons into observed spectra is spectrum-dependent (Fig. 3). Therefore, the albedo spectrum depends on the shape of the primary spectrum (Fig. 3), - previous studies (Bai & Ramaty 1978; Alexander & Brown 2002) considered only the results for prescribed power-law primary spectra. This difference is important when one tries to infer a primary spectrum from an observed one.
We verified that we can reproduce the forward results of Bai & Ramaty (1978) by numerically evaluating Eq. (1) for the particular case and the parameters described in Fig. 2. Consistency with previous treatments being established in this way, we now turn to the inverse problem - deduction of the primary spectrum from observations of the total spectrum.
|Figure 3: Spectrum dependency of the reflected flux. Upper panel: primary (dashed line), reflected (solid line) and total (dotted line) photon spectra calculated for a modelled flare spectrum for keV, for keV. The lower panel shows the reflectivity for a X-ray source at heliocentric angle , defined as the ratio of reflected to primary fluxes (solid line). The reflectivity assuming power-law for a primary source with spectral index 3 (dotted line) and 2 (dashed line).|
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In the absence of data noise we could immediately invert
Eq. (2) to obtain :
For a given instrument such as RHESSI, the count flux
is the linear composition of instrument response matrix
and the photon flux
coming from the
Sun (Schwartz et al. 2002)
This approach has already been incorporated into the SPEX software (Schwartz et al. 2002) including the object oriented version (OSPEX). The software is publicly available as a part of Solar SoftWare (SSW).
|Figure 4: Temporal variation in four energy channels (4 seconds cadence) of the count rates summed over seven front RHESSI detectors for the September 17, 2002 solar flare. The vertical lines show the accumulation interval selected 5:50:48-5:51:36 UT for further spectral analysis.|
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We consider the spectrum around flare maximum (5:50:48-5:51:36 UT) when we have enough counts over the range 3 keV up to >100 keV. We used 7 out of 9 front segments, excluding detectors 2 and 7 due to their low energy resolution at the time of observation (Smith et al. 2002). The time interval is shown in Fig. 4.
After background subtraction, spectral results are shown in Fig. 5 for both the total observed spectrum, and the primary spectrum determined via the method of the previous section and using (isotropic emission). We note how removal of backscattered photons, as expected, reduces the primary flux and steepens the spectrum in the 20-50 keV photon energy range and slightly flattens the spectrum at energies above the albedo maximum.
|Figure 5: Spectrum of the September 17, 2002 solar flare in the time interval 5:50:48-5:51:36 UT. The solid line shows the observed spectrum; the dash line is the primary spectrum (corrected for Compton back-scattering).|
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The albedo changes the spectral indices of photon spectra, both of
power-law fits and of the local energy dependent spectral index
Figure 6 shows the energy dependent spectral index for our flare spectrum. Both spectra (with and without albedo correction) increase toward the ends of the energy interval. At low energies this is probably due to a thermal component while at higher energies it is likely connected with softening of a nonthermal electron spectrum. The spectral index of the observed data shows a clear minimum around 15 keV, while the spectrum corrected for albedo has a rather extended minimum in the broad range 15-30 keV. The albedo correction increases the minimum value of from 2.0 to 2.6. The opposite effect can be seen at higher energies: the spectral index at 40 keV decreases from 3.5 to 3.2, albedo correction making the primary spectrum harder. Qualitatively similar results have been obtained for forward modelled spectra (Bai & Ramaty 1978; Zhang & Huang 2004).
|Figure 6: Energy dependent photon spectral index of the September 17, 2002 solar flare. The solid line shows the spectral index of the observed spectrum; the dash line is the spectral index for the primary spectrum (corrected for Compton back-scattering). The confidence intervals represent the range of solutions found by allowing the incident photon spectrum to range randomly within the estimated (instrument + shot noise) errors.|
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The primary photon flux is directly connected with the mean source
electron (so-called thin target) spectrum (Brown 1971; Brown
et al. 2003)
Solution of Eq. (9) for is a somewhat unstable inverse problem (Craig & Brown 1986; Piana et al. 2003) and so results are strongly affected when the observed is used instead of the albedo corrected required for correct solution of Eq. (9). is important in the study of acceleration mechanisms and of the flare electron energy budget. Of particular interest are data such as from the flare discussed above which exhibit very flat regions which may correspond to with a low energy cut-off in the mean electron spectrum which would have major implications for electron acceleration and propagation, and for the energy budget (Kontar & Brown 2005).
We used standard SPEX software (Schwartz et al. 2002) to fit to the observed X-ray counts an approximate parametric comprising an isothermal plus broken power-law form with low energy cut off. The minimum fit of the observed spectra gives emission measure cm-3, temperature kT=1.36 keV, spectral indexes , with break energy EB=54 keV and a low energy cut-off at 18 keV very clearly visible above the isothermal part (Fig. 7).
Next we carried out the same model fit but using the instrument response matrix corrected for albedo as described above - i.e. we repeat the process for instead of I. The result shows the same tendency as we saw in the analysis of energy dependent photon spectral index. The primary photon spectrum is softer at low energies and harder at high energies. Now we find a slightly bigger emission measure cm-3 and slightly smaller temperature kT=1.29 keV. The spectral index of energetic electrons becomes softer at low energies and harder at high energies with the break energy at higher energies EB=59 keV. Most importantly, in contrast with the fit to the observed spectrum, the primary spectrum fit yields a much lower low energy cut-off, around 10 keV, not clearly visible above the large thermal component there (Fig. 7). Therefore, we cannot conclude that the true derived from the primary photon spectrum has a well defined low-energy cut-off.
|Figure 7: Mean electron flux spectra of the September, 2002 solar flare. The solid/dash lines show the spectrum without and with albedo correction.|
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|Figure 8: Inverted mean electron flux of the August 20, 2002 solar flare for the time interval 08:25:20-08:25:40 UT. The dash line shows the spectra with albedo correction. The confidence intervals represent the range of solutions found by allowing the incident photon spectrum to range randomly within the estimated (instrument + shot noise) errors.|
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Figure 8 gives a further example from the flare of August 20, 2002 (Kasparova et al. 2005). We have obtained regularized using the method of Kontar et al, 2005 starting from the total observed spectrum and the albedo-corrected primary spectrum . Note that a local maximum in at keV, difficult to accommodate in conventional thick-target models (Kontar & Brown 2005), is no longer required when albedo is properly accounted for.
We have shown how the Compton backscatter Green's function of Magdziarz & Zdziarski (1995) may be employed to deduce primary hard X-ray spectra from observations. We have applied this procedure to RHESSI data, particularly from a couple of flares with hard photon spectra. Without a treatment of albedo, spectral hardening found in some flares at photon energies of keV appears to require a local minimum in the mean fast electron distribution (Piana et al. 2003; Kasparova et al. 2005). These local minima are particularly interesting, since they might, if steep, (Kontar & Brown 2005) be inconsistent with the very widely used collision-dominated thick-target model for X-ray production (Brown 1971; Lin & Hudson 1976). Here we have seen that a complete treatment of albedo removes much of the spectral hardening in this photon energy range, potentially restoring the viability of the collisional thick target.
The major assumption here is the isotropy of the downward directed radiation. At the relevant photon energies here 10-100 keV, the intrinsic bremsstrahlung cross-section polar diagram has a characteristic width of about (see figure in Massone et al. 2004). Since the emitting electron angular distribution will be broadened by pitch-angle scattering (Leach & Petrosian 1981; MacKinnon & Craig 1991), the resulting hard X-ray flux might be fairly close to downward isotropic.
Inclusion of the albedo effect reduces the number of energetic
electrons required for the production of the observed spectra. The
total flux of energetic electrons
It should be also stressed here that these are rather conservative (lower limit) estimates of the albedo correction, assuming an isotropic primary X-ray source. In fact, if the electrons are strongly downward directed, the back-scattered photons could produce a several times larger contribution to the observed spectrum. This would have a major effect and clearly one must consider the albedo to get a realistic idea of the flare electron spectrum and energy budget. The capacity to account completely for the effects of albedo also restores some optimism over discussing, in terms of X-ray spectra, issues such as the lowest energies at which electron acceleration operates (see Zhang & Huang 2004), though electron transport close to the thermal speed complicates the interpretation of the X-ray spectrum there (Galloway et al. 2005).
We are grateful to Gordon Holman and Jana Kasparova for valuable discussions and to Kim Tolbert for adding our code to OSPEX package. EPK, ALM, and JCB acknowledge the financial support of a PPARC Rolling Grant. RAS is supported by contract NAS5 - 01160.