Diagnostics of energetic electrons with anisotropic distributions in solar flares

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 90 • , 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.

The non-Maxwellian continuum in the X-ray, UV, and radio range

“…However, the n-distribution is a parametric distribution that describes well only the bulk of the distribution function during flares . In particular, the ndistributions do not contain the high-energy tail observed commonly during flares (e.g., Asai et al 2009;Veronig et al 2010;Zharkova et al 2010;Guo et al 2011). This high-energy tail then dominates the bremsstrahlung spectrum at the X-ray wavelengths and corresponding photon energies of tens of keV.…”

“…In solar physics, on which we focus here, departures from the equilibrium Maxwellian distribution occur every time particles are accelerated, e.g., in flares, where high-energy tails Appendix A is available in electronic form at http://www.aanda.org are ubiquitous (Petkaki & MacKinnon 2007Zharkova & Gordovskyy 2006;Veronig et al 2010;Zharkova et al 2010Zharkova et al , 2011Fletcher et al 2011;Holman et al 2011). Apart from that, departures from Maxwellian distributions can occur at low plasma densities if there are strong gradients of temperature and density, i.e., in the transition region (e.g., Owocki & Scudder 1983;Ljepojevic & MacNiece 1988;Scudder 1992;Pinfield et al 1999;, in conditions where the mean energy can change (Collier 2004), i.e., during plasma heating.…”

“…There, the power-law distributions were usually assumed to be responsible for the observed emission processes, such as bremsstrahlung in the hard X-ray range (e.g., Brown 1971;Lin & Hudson 1971;Holman et al 2003;Krucker & Lin 2008;Asai et al 2009;Warmuth et al 2009;Veronig et al 2010;Kurt et al 2010;Zharkova et al 2010;Guo et al 2011;Kontar et al 2011) or gyrosynchrotron emission in the radio range (e.g., Dulk 1985;Zhou & Karlický 1994;Kuznetsov et al 2011). Yasnov & Karlický (2009) presented an inverse method to determine the characteristics of the electron power-law tail distribution in radio bursts.…”

The ionization equilibrium and flare line spectra for the electron distribution with a power-law tail

“…X-ray observations with, e.g., RHESSI (Lin et al 2002) show that the flare electron distribution is composed of a plasma bulk, which is usually considered to be Maxwellian, and a high-energy tail, approximated with a power-law distribution (e.g., Brown 1971;Lin & Hudson 1971;Holman et al 2003;Brown et al 2008;Krucker & Lin 2008;Asai et al 2009;Warmuth et al 2009;Veronig et al 2010;Kurt et al 2010;Zharkova et al 2010;Guo et al 2011). The ionization equilibrium and synthetic spectra for the Maxwellian electron distribution are widely known (e.g., CHIANTI, Landi et al 2006;Dere et al 2009), but there are only few papers on the influence of the high-energy tail on the ionization equilibrium and the line spectra.…”

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

“…However, existing models of particle acceleration and precipitation considering not only collisional but other energyloss mechanisms unavoidably report anisotropic particle distributions which vary significantly with precipitation depth (for example, see distribution functions in Zharkova et al 2010a, hereafter Paper I). Fleishman & Melnikov (2003a, 2003b have shown that the pitch-angle anisotropy of accelerated particles, even if it is unable to support a coherent wave amplification, can change significantly the emission intensity (up to orders of magnitude) and polarization (the sign of circular polarization).…”