Collisional and return current heating functions for beam-heated models of solar flares

In beam-heated models of solar flares, the bulk of the energy deposited in the flare atmosphere resides in the low-energy end of the electron spectrum. Since the shape of the spectrum at low energy is not well determined observationally, various forms of low-energy cut-off have been assumed in theoretical modelling. Certain results of such modelling may depend strongly on the assumed spectrum. We derive the heating distributions for various spectra, both for collisional energy loss and for Ohmic dissipation of the return current, and show that none of the spectra are fully satisfactory, according to the criteria that for both collisional and Ohmic heating, the heating rate should be bounded, continuous, and smooth, and have a tractable functional form. A simple form of electron spectrum is suggested, which satisfies these criteria.