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Cosmic-Ray Proton to Electron Ratios

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}.