Regularized solution of the solar Bremsstrahlung inverse problem: model dependence and implementation issues

Figures & data

Figure 1. The six cross-sections considered in the article: (a) Kramers cross-section (formula K in the text); (b) Bethe–Heitler cross-section (formula BH in the text); (c) Bethe-Heitler cross-section with Elwert correction for Coulomb effects (formula BHE in the text); (d) 3BN cross-section with highly relativistic effects (formula 3BN in the text); (e) Haug interpolated cross-section (formula H in the text); (f) 3BN cross-section with electron-electron interaction effects (formula 3BNee in the text).

Figure 2. Singular spectra for the six (discretized) integral kernels: (a) formulas K and BH; (b) formulas BHE, 3BN, H and 3BNee.

Figure 3. The first three singular functions for the six (discretized) integral kernels: (a) u₁(E) for formulas K and BH; (b) u₁(E) for formulas BHE, 3BN, H and 3BNee; (c) u₂(E) for formulas K and BH; (d) u₂(E) for formulas BHE, 3BN, H and 3BNee; (e) u₃(E) for formulas K and BH; (f) u₃(E) for formulas BHE, 3BN, H and 3BNee.

Figure 4. Estimate of the spectral index by means of regularized inversion: (a) input photon spectrum, a power-law with constant spectral index 5; (b) re-constructed electron spectral index obtained in the case of the six cross-sections considered in the article.

Figure 5. Re-construction of a theoretical input averaged electron flux spectrum by means of regularized inversion: (a) theoretical input electron spectrum, a broken power-law with thermal component; (b) simulated photon flux spectrum; (c) regularized solutions in the case of formulas K and BH; (d) regularized solutions in the case of formulas BHE, 3BN and H.

Table 1. Re-construction errors (%) in the restoration of Equation (61) in the case of five cross-sections considered in the article. Formula (62) has been used to assess the re-construction errors

Cross-section	err
K	20.7
BH	30.8
BHE	2.6
3BN	9.0
H	8.5

Figure 6. Application to RHESSI data: (a) count flux in the case of the solar flare of 3 November 2003. The counts detected in five increasing energy channels are plotted without background removal (solid: 6–12 keV, dotted: 12–25 keV, dashed: 25–50 keV, dash dot: 50–100 keV, dash dot dot: 100–300 keV); (b) the photon flux spectrum constructed from the counts in (a); (c) reconstructed averaged electron flux spectrum for formulas K and BH; (d) reconstructed averaged electron flux spectrum for formulas BHE, 3BN, H and 3BNee.