Electron acceleration with improved Stochastic Differential Equation method: cutoff shape of electron distribution in test-particle limit

TL;DR

This paper introduces an advanced stochastic differential equation method incorporating skew Brownian motion to simulate electron acceleration at astrophysical shocks, analyzing how the cutoff shape varies with diffusion parameters in different limiting cases.

Contribution

The paper develops a novel SDE-based simulation method with shock crossing asymmetry and applies it to study electron cutoff shapes under various physical limits.

Findings

Reproduces previous age-limited cutoff results with a≈2β.

Roughly reproduces analytical expectation a≈β+1 in cooling-limited cases.

Fits numerical results well in escape-limited cases, with some deviations from previous formulas.

Abstract

We develop a method of stochastic differential equation to simulate electron acceleration at astrophysical shocks. Our method is based on It\^{o}'s stochastic differential equations coupled with a particle splitting, employing a skew Brownian motion where an asymmetric shock crossing probability is considered. Using this code, we perform simulations of electron acceleration at stationary plane parallel shock with various parameter sets, and studied how the cutoff shape, which is characterized by cutoff shape parameter $a$, changes with the momentum dependence of the diffusion coefficient $\beta$. In the age-limited cases, we reproduce previous results of other authors, $a\approx2\beta$. In the cooling-limited cases, the analytical expectation $a\approx\beta+1$ is roughly reproduced although we recognize deviations to some extent. In the case of escape-limited acceleration, numerical result fits analytical stationary solution well, but deviates from the previous asymptotic analytical formula $a\approx\beta$.