Exact Analytical Solutions of Continuity Equation for Electron Beams Precipitating in Ohmic and Mixed Energy Losses
Solar and Stellar Flares: observation, theory and simulations
R. R. Dobranskis and V. V. Zharkova
V. V. Zharkova (Northumbria University)
Northumbria University
In this paper we extend the approach presented in Dobranskis & Zharkova (2014a, b) by updating the analytical solutions of continuity equation (CE) for pure Ohmic losses and developing a method for joint analytical solutions to account simultaneously for collisional and Ohmic losses. The exact solutions of CE for electron density of the beams precipitating in Ohmic losses are found at different precipitation depths for precipitating and "returning" electrons. These solutions are combined iteratively with the analytical solutions for pure collisions for the calculation of beam electron differential density for mixed (Ohmic and collisional) energy losses (MEL) and the HXR intensity for relativistic cross-section. The outcomes are compared with the numerical results found from the Fokker-Planck (FP) solutions for beams with the same parameters and the energy loss mechanisms (collisional and Ohmic losses). The HXR intensity distribution produced by the analytical MEL solution reveals a close resemblance to the results found from the numerical FP solution, while the MEL simulation calculation time is up to 30 times faster than the numerical FP. The proposed MEL approach is to be implemented in IDL code, which can be incorporated into the RHESSI SolarSoftWare and utilized at the initial investigation stage of the spectral fitting providing good preliminary electron beam parameters and an opportunity to account for both collisional and Ohmic losses simultaneously without lengthy numerical Fokker-Planck simulations.
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