The kinetic effects of electron beam precipitation and resulting hard X-ray intensity in solar flares^*

Cybernetics and Virtual Systems Department, University of Bradford, Bradford BD7 1DP, UK e-mail: m.gordovskyy@bradford.ac.uk

Received: 16 April 2004
Accepted: 25 October 2004

Abstract

The numerical solutions of the time-dependent kinetic Fokker-Planck equation are presented for fast electrons injected from the solar corona into a flaring atmosphere and precipitating with energy and pitch-angle diffusion into a loop with a converging magnetic field. The electrons are assumed to lose their energy in Coulomb collisions with particles of the partially ionised ambient plasma and Ohmic heating owing to the electric field induced by the precipitating beam. The electric field induced by a precipitating electron beam is found to cause a return current beam, which comes back to the source in the corona with a wide pitch angle distribution. The return current is assumed to arise from the ambient plasma and from beam electrons scattered into negative pitch-angles. Energy and pitch-angle distributions of precipitating and return current electron beams at various atmospheric depths are presented along with the precipitating beam abundances, energy fluxes and resulting hard X-ray bremsstrahlung (photon) spectra for electron beams with power law energy spectra with spectral indices of 3, 5, 7 and initial energy fluxes of 10⁸, 10¹⁰, 10¹² erg cm^-2 s^-1. Energy distributions of the return current cover energies lower than 60 keV for weaker soft beams and increase to 65 keV for moderate soft beams, or to 70–75 keV for more intense and hard beams. The maxima are at 30 keV for weaker soft beams and are shifted towards higher energies, up to 50 keV, for harder and more intense beams. 
As a result, the photon spectra of hard X-ray emission emitted from a flaring atmosphere are found to have a broken power-law (elbow-type) shape with a higher energy part retaining the spectral index  associated with the electron beam's initial index and varying slightly with the beam parameters. However, the lower energy part of the X-ray photon spectra shows a much smaller increase or even a substantial decrease of its spectral index  for more intense or harder beams. These simulated broken power-law photon spectra produced by precipitating electron beams with a single spectral index agree reasonably well with the photon energy spectra from the flares of 20 and 23 July, 2002 observed by the RHESSI payload.