Analytical treatment for the development of electromagnetic cascades in intense magnetic fields

TL;DR

This paper develops an analytical model for electromagnetic cascades in intense magnetic fields, revealing a characteristic photon spectrum with a sharp cutoff and a broken power-law shape, validated by Monte Carlo simulations.

Contribution

It introduces a simple analytical function to describe the saturated cascade photon spectrum, dependent only on the product of cascade scale and magnetic field strength.

Findings

Photon spectrum exhibits a sharp cutoff due to magnetic absorption.

Spectral energy distribution follows a broken power-law with indices 0.5 and 0.125.

Analytical fit matches simulation results with over 96% accuracy.

Abstract

In a strong magnetic field, a high-energy photon can be absorbed and then produce an electron-positron pair. The produced electron/positron will in turn radiate a high-energy photon via synchrotron radiation, which then initiates a cascade. We built a one-dimensional Monte-Carlo code to study the development of the cascade especially after it reaches the saturated status, when almost all the energy of the primary particles transfers to the photons. The photon spectrum in this status has a cut-off due to the absorption by magnetic fields, which is much sharper than the exponential one. Below the cut-off, the spectral energy distribution (SED) manifest itself as a broken power-law with a spectral index of $0.5$ and $0.125$, respectively, below and above the broken energy. The SED can be fitted by a simple analytical function, which is solely determined by the product of the cascade scale $R$ and the magnetic field perpendicular to the motion of the particle B_{\perp}, with an accuracy better than 96\%. The similarity of the spectrum to that from the cascade in an isotropic black-body photon field is also studied.