Updated analytical solutions of continuity equation for electron beams precipitation – I. Pure collisional and pure ohmic energy losses

We present updated analytical solutions of continuity equations for power-law beam electrons precipitating in (a) purely collisional losses and (b) purely ohmic losses. The solutions of continuity equation (CE) normalized on electron density presented in Dobranskis & Zharkova are found by method of characteristics eliminating a mistake in the density characteristic pointed out by Emslie et al. The corrected electron beam differential densities (DD) for collisions are shown to have energy spectra with the index of −(γ + 1)/2, coinciding with the one derived from the inverse problem solution by Brown, while being lower by 1/2 than the index of −γ/2 obtained from CE for electron flux. This leads to a decrease of the index of mean electron spectra from −(γ − 2.5) (CE for flux) to −(γ − 2.0) (CE for electron density). The similar method is applied to CE for electrons precipitating in electric field induced by the beam itself. For the first time, the electron energy spectra are calculated for both constant and variable electric fields by using CE for electron density. We derive electron DD for precipitating electrons (moving towards the photosphere, μ = +1) and ‘returning’ electrons (moving towards the corona, μ = −1). The indices of DD energy spectra are reduced from −γ − 1 (CE for flux) to −γ (CE for electron density). While the index of mean electron spectra is increased by 0.5, from −γ + 0.5 (CE for flux) to −γ + 1(CE for electron density). Hard X-ray intensities are also calculated for relativistic cross-section for the updated differential spectra revealing closer resemblance to numerical Fokker–Planck (FP) solutions.