# On the non-thermal electron-to-proton ratio at cosmic acceleration sites

## TL;DR

This paper reevaluates the electron-to-proton luminosity ratio in cosmic ray sources by considering different spectral indices and minimum energies, challenging previous assumptions and providing a more nuanced understanding of particle acceleration.

## Contribution

It introduces a detailed calculation of the electron-to-proton ratio without assuming equal spectral indices, considering the impact of minimum energy thresholds and spectral differences.

## Key findings

- The ratio of luminosities can vary significantly when spectral indices differ.
- Minimum energy assumptions greatly influence the electron-to-proton ratio.
- Spectral behavior differences are crucial for accurate source modeling.

## Abstract

The luminosity ratio of electrons to protons as it is produced in stochastic acceleration processes in cosmic ray sources is an important quantity relevant for several aspects of the modeling of the sources themselves. It is usually assumed to be around 1:100 in the case of Galactic sources, while a value of 1:10 is typically assumed when describing extragalactic sources. It is supported by observations that the average ratios should be close to these values. At this point, however, there is no possibility to investigate how each individual source behaves. When looking at the physics aspects, a 1:100 ratio is well supported in theory when making the following assumptions: (1) the total number of electrons and protons that is accelerated are the same; (2) the spectral index of both populations after acceleration is $a_e=a_p\approx 2.2$. In this paper, we reinvestigate these assumptions. In particular, assumption (2) is not supported by observational data of the sources and PIC simulation yield different spectral indices as well. We present the detailed calculation of the electron-to-proton ratio, dropping the assumption of equal spectral indices. We distinguish between the ratio of luminosities and the ratio of the differential spectral behavior, which becomes necessary for cases where the spectral indices of the two particle populations are not the same. We discuss the possible range of values when allowing for different spectral indices concerning the spectral behavior of electrons and protons. Additionally, it is shown that the minimum energy of the accelerated population can have a large influence on the results.