Cosmic-Ray Proton to Electron Ratios

TL;DR

This paper derives a simple approximate formula for the proton-to-electron energy density ratio in relativistic particles, useful for characterizing cosmic-ray compositions, especially at injection conditions.

Contribution

It introduces a new approximate expression for the proton-to-electron energy density ratio based on spectral indices and mass ratios, applicable under specific conditions.

Findings

Provides a formula for U_p/U_e in terms of spectral index q and mass ratio

Relates the energy density ratio to the known number density ratio at 1 GeV

Useful for estimating cosmic-ray particle compositions

Abstract

A basic quantity in the characterization of relativistic particles is the proton-to-electron (p/e) energy density ratio. We derive a simple approximate expression suitable to estimate this quantity, U_p/U_e = (m_p/m_e)^(3-q)/2, valid when a nonthermal `gas' of these particles is electrically neutral and the particles' power-law spectral indices are equal -- e.g., at injection. This relation partners the well-known p/e number density ratio at 1 GeV, i.e. N_p/N_e = (m_p/m_e)^{(q-1)/2}.