Relativistic van Allen belts in magnetospheres of pulsars and white dwarfs

Abstract

We consider dynamics and multi-frequency emission patterns of relativistic van Allen belts - particles trapped in the magnetosphere of neutron stars and white dwarths. We account for synchrotron radiative losses and effects of relativistic beaming of radiation. The system is non-Hamiltonian (non-energy conserving): this results in a wide non-scalable variety of spectral and temporal behaviors. There are three types of trapped particles' trajectories: (i) oscillating (particles experience multiple bounces between magnetic bottles); (ii) precipitating (particles fall onto the star with finite transverse momentum); (iii) freezing (particles lose their transverse motion before falling onto the star). The separation between regimes (i) and (ii) depends both on the ratio of the bounce time to cooling time at magnetic equator $\tau_{ 0} $, $\eta _0 = R_0/( c \tau_0) \leq 1$, as well as the initial pitch angle $\alpha_0$; regimes (i) and (ii) are separated at $ \alpha_{0, crit} \sim \eta_0^{3/10}$. Resulting emission patterns show large variety: single or double peaked, and/or flat hat with sharp walls. Multi-frequency profiles - in optical and X-ray bands - can be used to get information about physical (magnetic field strength, injection point) and geometrical (dipolar angle and the line of sight) properties.