Stationary and impulsive injection of electron beams in converging magnetic field

Abstract

Aims. We study time-dependent precipitation of an electron beam injected into a flaring atmosphere with a converging magnetic field by considering collisional and Ohmic losses with anisotropic scattering and pitch angle diffusion. Two injection regimes are investigated: short impulse and stationary injection. The effects of converging magnetic fields with different spatial profiles are compared and the energy deposition produced by the precipitating electrons at different depths and regimes is calculated. Methods. The time dependent Fokker-Planck equation for electron distribution in depth, energy and pitch angle was solved numerically by using the summary approximation method. Results. Steady state injection was found to be established for beam electrons 0.07-0.2 s after the injection onset depending on the initial beam parameters. Energy deposition by a stationary beam is strongly dependent on a self-induced electric field but more weakly dependent on a magnetic field convergence. Energy depositions by short electron impulses are found to be insensitive to the self-induced electric field but are strongly affected by magnetic convergence. Short beam impulses are shown to produce sharp asymmetric hard X-ray bursts on a timescale of the order of tens of milliseconds, often observed in solar flares.

Manifestations of Energetic Electrons With Anisotropic Distributions in Solar Flares. Ii. Gyrosynchrotron Microwave Emission

“…This electron beam injection is considered as the upper boundary condition in the current Fokker-Planck study of electron precipitation into the loop legs. It can be noted from our previous Fokker-Planck simulations (Siversky & Zharkova 2009a;Zharkova et al 2010a) that a substantial part of those energetic electrons can return back to the acceleration site on the top (e.g., due to reflection from the magnetic mirror at the loop footpoint or to the effect of self-induced electric field) and join the ambient plasma electrons dragged into the RCS to be accelerated inside. Since the electric field accelerating the electrons toward a footpoint should decelerate the returning particles, the RCS region acts as a barrier preventing the particles from penetrating into the opposite half of the magnetic loop and bouncing between the footpoints.…”

“…Such a state is usually achieved in about 70-200 ms after the injection onset (Siversky & Zharkova 2009a). Since we do not consider the particles bouncing between the loop footpoints and their accumulation inside the loop, this time would basically be the travel time of the particles from the top of the loop to the footpoint and back; higher energy electrons can return faster than lower energy ones unless they are thermalized by collisions.…”

“…Since we do not consider the particles bouncing between the loop footpoints and their accumulation inside the loop, this time would basically be the travel time of the particles from the top of the loop to the footpoint and back; higher energy electrons can return faster than lower energy ones unless they are thermalized by collisions. However, in the models including the self-induced electric field, some additional time is required to form a stationary return current compensating for the current of precipitating electrons (Siversky & Zharkova 2009a). Since the beams are assumed to have power-law energy distributions, the return current is mainly carried by low-energy electrons (tens of keV) whose density is highest and whose travel time defines the formation timescale of the quasi-stationary state.…”

“…We consider the models with pure collisions (C), collisions and electric field (CE), collisions and magnetic field convergence (CB), and with all the above losses (CEB) in order to investigate the relative contribution of the different processes to formation of the electron distributions and their resulting MW emission. In the simulations including the magnetic field convergence, the magnetic field (B) is assumed to increase exponentially with depth between the loop top and the characteristic depth in the transition region (Siversky & Zharkova 2009a). The background (thermal) plasma density (n 0 ) in the corona is assumed to be constant because the nearly barometric (exponential) height scale of the hot coronal plasma heated during the flares is expected to be very large-much greater than the typical loop height; the hydrodynamic simulations also show that plasma density in the corona is nearly constant or slowly increases with depth (Zharkova & Zharkov 2007).…”

“…In addition, spatial resolution of the existing MW instruments is not sufficient to obtain local values of the emission intensity and polarization, so all the parameters are being averaged over some area. However, this averaging of each parameter is strongly dependent on its depth variations as shown by Siversky & Zharkova (2009a). Therefore, one needs to apply a method that can determine electron distribution functions at different levels of a coronal magnetic loop (independently of MW observations), with a minimal number of the initial conditions.…”