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

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...

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Bibliographic Details
Published inarXiv.org
Main Authors Cainã de Oliveira, Maia, Leonardo Paulo, de Souza, Vitor
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 23.12.2023
Subjects
Anisotropy
Augers
Charged particles
Cosmic rays
Dipoles
Homogeneity
Isotropy
Liouville equations
Liouville theorem
Magnetic fields
Quadrupoles
Space density
Spherical harmonics
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Summary: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.
ISSN:2331-8422