The Effect of the Electric Field Induced by Precipitating Electron Beams on Hard X-Ray Photon and Mean Electron Spectra

The effect of a self-induced electric field is investigated analytically and numerically on differential and mean electron spectra produced by beam electrons during their precipitation into a flaring atmosphere as well as on the emitted hard X-ray (HXR) photon spectra. The induced electric field is found to be a constant in upper atmospheric layers and to fall sharply in the deeper atmosphere from some ``turning point'' occurring either in the corona (for intense and softer beams) or in the chromosphere (for weaker and harder beams). The stronger and softer the beam, the higher the electric field before the turning point and the steeper its decrease after it. Analytical solutions are presented for the electric fields, which are constant or decreasing with depth, and the characteristic ``electric'' stopping depths are compared with the ``collisional'' ones. A constant electric field is found to decelerate precipitating electrons and to significantly reduce their number in the upper atmospheric depth, resulting in their differential spectra flattening at lower energies (<100 keV). While a decreasing electric field slows down the electron deceleration, allowing them to precipitate into deeper atmospheric layers than for a constant electric field, the joint effect of electric and collisional energy losses increases the energy losses by lower energy electrons compared to pure collisions and results in maxima at energies of 40-80 keV in the differential electron spectra. This, in turn, leads to the maxima in the mean source electron spectra and to the ``double power law'' HXR photon spectra (with flattening at lower energies) similar to those reported from the RHESSI observations. The more intense and soft the beams are, the stronger is the lower energy flattening and the higher is the ``break'' energy where the flattening occurs.