Simultaneous analytical solutions of continuity equations are sought for
electron beam precipitation a) in collisional losses and b) in Ohmic losses, or mixed
energy losses (MEL), by applying the iterative method to calculate the resulting dier-
ential densities at given precipitation deph. The differential densities of precipitating
electrons derived from the analytical solutions for mixed energy losses reveal increased flattening at energies below 10-30 keV compared to a pure collisional case. This flattening becomes stronger with an increasing precipitation depth turning into a positive slope at greater precipitation depths in the chromosphere resulting in a differential density distribution with maximum that shifts towards higher energies with increase of a column depth. While the differential densities combining precipitating and returning electrons are higher at lower energies than those for a pure collisional case.
The resulting hard X-ray (HXR) emission produced by the beams with dierent initial
energy fluxes and spectral indices is calculated using the MEL approach for different
ratios between the dierential densities of precipitating and returning electrons. The proposed MEL approach provides an opportunity to account simultaneously for both collisional and ohmic losses in flaring events, which can be used for a quick spectral fitting of HXR spectra and evaluation of a fraction of returning electrons versus precipitating ones. The semi-analytical MEL approach is used for spectral fitting to RHESSI observations of nine C, M and X class flares revealing a close fit to the observations and Fokker-Planck solutions.