1 Introduction and formulation of the integral equations

We consider the analysis of solar flare hard X-rays (HXR) in the energy range from around ten to a few hundred keV emitted by collisional bremsstrahlung of high energy electrons on ions of the ambient plasma. We assume that the electron distribution function is isotropic and the hydrogen dominated plasma is fully ionized (cf. discussion of the latter by Brown et al. 1998). It is well-known that HXRs carry detailed information about flare electrons and, in particular, the shape of the spectra are related to electron acceleration and energy transport. Careful consideration of techniques for optimal recovery of such information is particularly timely with NASA's planned launch of the HESSI flare mission, the first to carry a high resolution HXR spectrometer. Through the 1960s and 70s only low spectral HXR resolution ($\approx \! 20-30 \%$) was available and without any spatial resolution. The 1980s and 90s saw the first HXR imaging (SMM, Hinotori and YohKoh missions) and the separate advent of Ge detector applications yielding HXR resolution to $\approx \!1\%$. The latter allowed the first possibility of genuine HXR spectral inversion while the former gave the first indications of the HXR source structure. HESSI is the first instrument to combine both these functions and will enable new insight into the spectral distibution of source electrons as functions of position and time, facts of major interest to flare modelling. To obtain the maximum benefit from these data will require the development of optimal methods for deconvolving the observed photon spectra as well as the spatial information obtained by the Fourier imaging technique involved.

In the case of an optically thin source of collisional bremsstrahlung, the total rate of photon emission $g(\epsilon)$ per unit photon energy $\epsilon$ is given by  

$$g(\epsilon)=\int_V n({\vec{r}}) \int_{\epsilon}^{\infty} F(E,{\vec{r}}) Q(\epsilon,E) {\rm d}E {\rm d}{\vec{r}}$$ (1)

where V is the volume of the emitting region, $n({\vec{r}})$ is the density of the ions in the plasma, $F(E,{\vec{r}})$ is the local electron flux spectrum, E is the electron energy and $Q(\epsilon,E)$ is the bremsstrahlung cross-section. We note that g represents the output at the sun while actual measurements are of the output at the earth but since only the shape of the spectrum and the relative errors are considered here, multiplicative constants are irrelevant.

In a non-thermal atmosphere, Eq. (1) can be simplified according to two different models. If the electron lifetimes inside the source region are long compared to the observation time, then $F(E,{\vec{r}})$ is not modified by energy loss and reflects the instantaneous flux spectrum of high-energy electrons. In this non-thermal thin-target model (Brown 1971) by defining the average ion density  

$$\overline{n}= \frac{1}{V} \int_V n({\vec{r}}) {\rm d}{\vec{r}}$$ (2)

and the average electron distribution function  

$$f(E)=\frac{1}{\overline{n} V} \int_{V} n({\vec{r}}) F(E,{\vec{r}}) {\rm d}{\vec{r}}$$ (3)

equation (1) can be simplified to  

$$g(\epsilon)=\int_{\epsilon}^{\infty} f(E) Q(\epsilon,E) {\rm d}E .$$ (4)

If the lifetimes of the electrons in the source region are short compared to the observation time, then electrons have to be continuously injected at a rate $F_0(E_0)~{\rm s}^{-1}$ per unit injection energy E₀. Within this non-thermal thick-target framework the photon flux is given by (Brown 1971)  

$$g(\epsilon) = \int_{\epsilon}^{\infty} F_0(E_0) \int_{\epsilon}^{E_0} \frac{Q(\epsilon,E)}{R(E)} {\rm d}E {\rm d}E_0$$ (5)

where R(E) represents the average collisional energy loss rate of an electron of energy E per unit plasma column density. Equation (5) can be written in the form of a Volterra integral equation of the first kind  

$$g(\epsilon) = \int_{\epsilon}^{\infty} \frac{Q(\epsilon,E)} {R(E)} G(E) {\rm d}E$$ (6)

by defining  

$$G(E)=\int_{E}^{\infty} F_0(E_0) {\rm d}E_0.$$ (7)

Both these models involve linear integral equations of the first kind, where the data are represented by the continuous bremsstrahlung spectrum and the solutions are energetic electron distribution functions - the electron average distribution function f(E) or the injection rate F₀(E₀). To find these functions from Eqs. (4)-(7) an explicit form for the bremsstrahlung cross-section $Q(\epsilon,E)$ must be chosen. For typical applications ($\epsilon\simeq 10-100$ keV) non-relativistic expressions are often adequate. The simplest form is given by Kramers' formula (Koch & Motz 1959)  

$$Q(\epsilon,E)=\frac{Q_0}{\epsilon E}$$ (8)

where for hydrogen  

$$Q_0 = \frac{8}{3} \frac{r_{0}^{2}}{137} m c^2$$ (9)

with r₀ the classical radius and m the rest mass of the electron respectively. A more accurate approximation is given by the Bethe-Heitler formula  

$$Q(\epsilon,E)=\frac{Q_0}{\epsilon E} \log\frac{1+\sqrt{1-\frac{\epsilon}{E}}}{1-\sqrt{1-\frac{ \epsilon}{E}}}.$$ (10)

However relativistic and semi-relativistic corrections have also been introduced (Gould 1980; Haug 1997) and, in general, the sensitivity of the solutions of Eqs. (4) or (5) to variations in the cross-section is an interesting and important problem (see results below).

Inversion of Eq. (4), with the Bethe-Heitler cross-section only, has been addressed in (Piana 1994). In that paper the ill-posed nature of the problem of determining f(E) from knowledge of $g(\epsilon)$ was discussed and Tikhonov regularized versions of the solution were computed by using the singular system of the integral operator in the case of discrete data. Applications to noisy simulated and real spectra were also considered. In the present paper we want to generalize the singular value decomposition (SVD) approach to the case of non-thermal bremsstrahlung for both thick and thin targets and for both forms of Q above. Our aim is to provide a quantitative basis, in terms of SVD analysis, for evaluation of the differences resulting from the two cross-section approximations, and between the thick- and thin-target cases, in respect of the ill-posedness of the problems, i.e., how accurately the source functions can be recovered from noisy data in each case. In Sect. 4 we compare these results with the analogous ones for the more ill-posed problem of thermal bremsstrahlung inversion. In a forthcoming paper, devoted to applications, we will use the SVD analysis to invert a real HXR spectrum emitted during a solar flare.