Hemispherical transport equation: modeling of quasiparallel collisionless shocks

Abstract

Using a so-called hemispherical model we derive a general transport equation for cosmic ray and thermal particles scattered in pitch angle by magnetic inhomogeneities in a moving collisionless plasma. The weak scattering through 90 degrees results in isotropic particle distributions in each hemisphere. The consideration is not limited by small anisotropies and by the condition that particle velocities are higher than characteristic flow velocity differences. For high velocities and small anisotropies the standard cosmic ray transport equation is recovered. We apply the equations derived for investigation of injection and acceleration of particles by collisionless shocks.

Numerical simulations of diffusive shock acceleration in SNRs

“…This high injection efficiency results in the significant shock modification already at early stages of SNR expan- sion while the thermal sub-shock compression ratio is close to 2.5 during the simulation. This is in agreement with the radioobservations of young extragalactic SNRs [23] and with the modeling of collisionless shocks [24]. Similar values of the injection efficiency were found in hybrid modeling (see e.g.…”

The influence of the Alfvénic drift on the shape of cosmic ray spectra in SNRs

“…This dependence of the injection efficiency on the shock velocity results in the significant shock modification already at early stages of SNR expansion. This is in agreement with the observations of young extragalactic SNRs [13] and with the modeling of collisionless shocks [14].…”

Solving the 90° Scattering Problem in Isotropic Turbulence

“…Already in the years after QLT was developed, it had been realized (e.g., Fisk 1974;Bieber et al 1988) that, in isotropic turbulence, the vanishing pitch-angle scattering at pitch angles close to 90 causes an infinitely large Њ parallel mean free path. The invalidity of QLT for describing scattering at such pitch angles is usually called the 90 problem Њ (e.g., Sakai & Kato 1984;Felice & Kulsrud 2001;Shalchi 2005;Ragot 2006;Zirakashvili 2007;Yan & Lazarian 2004. Several nonlinear theories have been developed in the 1970s (e.g., Völk 1973Völk , 1975Jones et al 1973Jones et al , 1978Owens 1974;Goldstein 1976) to solve this problem.…”