Diagnostics of energetic electrons with anisotropic distributions in solar flares

I. Hard X-rays bremsstrahlung emission

¹ Department of Computing and Mathematics, University of Bradford, Bradford BD7 1DP, UK e-mail: [v.v.zharkova;t.siverskyi]@brad.ac.uk
² Institute of Solar-Terrestrial Physics, Irkutsk, Lermontov str. 126a, 664033, Russia e-mail: a_kuzn@iszf.irk.ru

Received: 8 December 2008
Accepted: 12 November 2009

Abstract

Aims. The paper aims are to simulate steady-state distributions of electrons beams precipitating in collisional and Ohmic losses with pitch angle anisotropy into a flaring atmosphere with converging magnetic field and to apply these to the interpretation of HXR photon spectra, directivity and polarization observed for different photon energies and flare positions on the solar disk.

Methods. Summary approximation method is applied to a time-dependent Fokker-Planck equation by splitting the temporal derivative equally between the derivatives in depth, energy and pitch angles and finding the solutions in forward and backward directions for each variable.

Results. For softer beams, there is a noticeable flattening of the photon spectra at lower energies caused by the self-induced electric field that increases for larger viewing angles. For the models with an electric field, the HXR emission with lower energies (30 keV) becomes directed mainly upwards at upper atmospheric levels owing to the increased number of particles moving upwards, while in deeper layers it again becomes directed downwards. The polarization maximum shifts to higher energies with every precipitation depth approaching 25 keV for the models with pure collisions and 100 keV for the models with return currents. At deeper layers, the polarization decreases because of the isotropization of electrons by collisions. The maximum polarization is observed at the viewing angle of , becoming shifted to lower angles for softer beams. The integrated polarization and directivity shows a dependence on a magnetic field convergence for harder beams, while for softer beams the directivity is strongly affected by the self-induced electric field changing from a downward motion to an upward one at upper atmospheric depths.

Conclusions. The proposed precipitation model for an electron beam with wider pitch angle dispersion of 0.2 taking into account collisional and Ohmic losses allowed us to fit the double power law HXR photon spectra with a spectrum flattening at lower energies observed in the flares of 20 and 23 July 2002. The observed directivity of HXR photons of 20 keV derived for a large number of flares located from the disk center to limb is also reproduced well by the theoretical directivity calculated for an electron beam with a very narrow pitch angle dispersion of 0.02. The simulated polarization of this narrowly-directed electron beam fits up to 90% of all the available polarimetric observations carried out at various locations across the solar disk.