Thermal bremsstrahlung hard X-ray and primary energy release in flares

Various methods are explored for obtaining regularized solutions of the severely ill-posed Laplace inversion problem involved in deriving plasma temperature (T) structure (differential emission measure xi(T)) from bremsstrahlung spectra. Inversions of simulated data show that zero-order regularization (Tikhonov regularization in L2 space) is very unsatisfactory even with weighting, while first-order regularization (Tikhonov regularization in Sobolev space) yields reasonable results. The method is applied to a high-resolution hard X-ray flare spectrum observed by Lin and Schwartz (1987) and yields a positive solution for xi(T) showing that a purely thermal interpretation is possible for that event. The form of xi(T) found has two broad features: one peaking at around 107 K and falling off steeply toward 2 x 108 K; a second spread around a peak near 4.5 x 108 K. The interpretation of such xi(T) in terms of plasma heating and conductive flux is discussed in terms of plasma heat fluxes and heating rates. For 1-D geometry, the distribution of the plasma heating rate H(T) per unit volume is inferred from xi(T) in the limits of classical diffusive conduction and of saturated heat flux, the former being relevant at T below around 5 x 107 K and the latter at much higher T. We find there exists a maximum in H(T) around 2 x 108 K, a fact which may be important for energy release theories.