4 The thermal case

For completeness and comparison we also give here (based on Piana et al. 1995) the parallel analysis of the thermal inverse problem where the emission is assumed to come from an optically thin thermal source (Craig & Brown 1976). If $n_{\rm e}({\vec{r}})$, $T({\vec{r}})$ are the electron density and temperature at the point ${\vec{r}}$ in the source region and $F(\epsilon,T({\vec{r}}))$ is the photon spectral distribution function for an isothermal source element, the photon spectrum is given by  

$$g(\epsilon)=\int_V F(\epsilon,T({\vec{r}})) n_{\rm e}^{2}({\vec{r}}) {\rm d}{\vec{r}} .$$ (56)

From this equation, once the Kramers' cross-section is adopted and via suitable changes of variable (Craig & Brown 1976) we have  

$$g(\epsilon)=\frac{1}{\epsilon} \int_{0}^{\infty} \frac{\xi(T)}{\sqrt{kT}} \mbox{exp}\left(-\frac{\epsilon}{kT}\right) {\rm d}T$$ (57)

where k is the Boltzmann constant and $\xi(T)$ is the differential emission measure defined by  

$$\xi(T)=\int_{S_{\rm T}} \frac{n_{\rm e}^{2}}{\vert\nabla T\vert} {\rm d}S_{\rm T}$$ (58)

with $S_{\rm T}$ a constant temperature surface.

Equation (57) can be easily transformed into the inverse Laplace transform problem  

$$g(\epsilon)=\frac{1}{\epsilon} \int_{0}^{\infty} \eta(y) \exp(-\epsilon y) {\rm d}y$$ (59)

with y=1/kT and  

$$\eta(y)=y^{-\frac{3}{2}} \xi\left(\frac{1}{ky}\right)~$$ (60)

(all the multiplicative constants have been put equal to one). A linear inverse problem with discrete data  

$$\vec{g} = L \eta$$ (61)

can also be written in this case, with L defined by  

$$(L \eta)(\epsilon_n)= \frac{1}{\epsilon_n} \int_{0}^{\infty} \eta(y) \exp(-\epsilon_n y) {\rm d}y ~~~~n=1,\ldots,N.$$ (62)

If we introduce the $L^2(0,\infty)$ functions  

$$\phi_n(y)= \frac{1}{\epsilon_n} \exp(-\epsilon_n y)~~~n=1,\ldots,N$$ (63)

the Gram matrix in this case is given by  

$$G_{mn}= \frac{1}{\epsilon_m \epsilon_n} \frac{1}{\epsilon_m + \epsilon_n}$$ (64)

for $n,m=1,\ldots,N$.

In Table 3 we compare the condition numbers for N=10, 25, 50, 100 geometrically sampled points (the energy range is again $\epsilon_1=10$ Kev, $\epsilon_N=100$ keV) for the thermal case and the non-thermal models (thin- and thick-target) when the Kramers cross-section is used (the Kramers' cross-section is the one adopted to derive Eq. (57)). As one can see, the Laplace inversion problem is much more unstable than the non-thermal problems.

 

Table 3: Condition numbers for the thin-target, thick-target and thermal models when the Kramers cross-section is adopted, for different numbers N of sampling points. The photon energy range is $\epsilon_1=10$ keV, $\epsilon_N=100$ keV and the sampling is geometric