Revisiting the implications of Liouville's theorem to the anisotropy of cosmic rays

TL;DR

This paper develops a mathematical framework using spherical harmonics to analyze how magnetic fields influence the directional distribution of cosmic rays, revealing conditions for isotropy preservation and predicting anisotropy signals.

Contribution

It introduces a novel solution to Liouville's equation for cosmic ray propagation, linking source properties and magnetic fields to observed anisotropies.

Findings

Isotropy is preserved only with isotropic, homogeneous sources.

A transient dipole appears due to magnetic effects in continuous source models.

Predictions for the energy at which quadrupole signals should be observed.

Abstract

We present a solution to Liouville's equation for an ensemble of charged particles propagating in magnetic fields. The solution is presented using an expansion in spherical harmonics of the phase space density, allowing a direct interpretation of the distribution of arrival directions of cosmic rays. The results are found for chosen conditions of variability and source distributions. We show there are two conditions for an initially isotropic flux of particles to remain isotropic while traveling through a magnetic field: isotropy and homogeneity of the sources. In case isotropically-distributed sources inject particles continuously in time, a transient magnetic induced dipole will appear. This dipole will vanish if the system reaches a steady-state. The formalism is used to analyze the data measured by the Pierre Auger Observatory, contributing to the understanding of the dependence of the dipole amplitude with energy and predicting the energy in which the quadrupole signal should be measured.