Fast Spectral Fitting of Hard X-Ray Bremsstrahlung from Truncated Power-Law Electron Spectra

Hard X-Ray bremsstrahlung continuum spectra, such as from solar flares, are commonly described in terms of power-law fits, either to the photon spectra themselves or to the electron spectra responsible for them. In applications various approximate relations between electron and photon spectral indices are often used for energies both above and below electron low-energy cutoffs. We examine the form of the exact relationships in various situations, and for various cross-sections, showing that empirical relations sometimes used can be highly misleading and consider how to improve fitting procedures. We obtain expressions for photon spectra from single, double and truncated power-law electron spectra for a variety of cross-sections and for the thin and thick target models and simple analytic expressions for the Bethe-Heitler cases. We show that above a low-energy cutoff the Kramers and Bethe-Heitler results match reasonably well with results for exact cross-sections up to energies around 100 keV; that below the low-energy cutoff, Kramers and other constant spectral index forms commonly used are very poor approximations to accurate results; but that our analytical forms are a very good match. Analytical forms of the Bethe-Heitler photon spectra from general power-law electron spectra are an excellent match to exact results for both thin and thick targets and they enable much faster spectral fitting than evaluation of the full spectral integrations.