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In this paper we consider simultaneous analytical solutions of continuity equations 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 differential densities at given precipitation depth. The differential densities of precipitating electrons derived from the analytical solutions for MELs 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 in 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 different initial energy fluxes and spectral indices is calculated using the MEL approach for different ratios between the differential densities of precipitating and returning electrons. The number of returning electrons can be even further enhanced by a magnetic mirroring, not considered in the present model, while dominating at lower atmospheric depths where the magnetic convergence and magnitude are the highest. 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 fitting to Reuven High Energy Solar Spectroscopic Imager observations of nine C, M and X class flares revealing a close fit to the observations and good resemblance to numerical FP solutions.

Updated analytical solutions of continuity equation for electron beams precipitation – II. Mixed energy losses