Analytic calculations of the spectra of ultra high energy cosmic ray nuclei. II. The general case of background radiation

We discuss the problem of ultra high energy nuclei propagation in extragalactic background radiations. The present paper is the continuation of the accompanying paper I where we have presented three new analytic methods to calculate the fluxes and spectra of ultra high energy cosmic ray nuclei, both primary and secondary, and secondary protons. The computation scheme in this paper is based on the analytic solution of coupled kinetic equations, which takes into account the continuous energy losses due to the expansion of the universe and pair-production, together with photo-disintegration of nuclei. This method includes in the most natural way the production of secondary nuclei in the process of photo-disintegration of the primary nuclei during their propagation through extragalactic background radiations. In paper I, in order to present the suggested analytical schemes of calculations, we have considered only the case of the cosmic microwave background radiation, in the present paper we generalize this computation to all relevant background radiations, including infra-red and visible/ultra-violet radiations, collectively referred to as extragalactic background light. The analytic solutions allow transparent physical interpretation of the obtained spectra. Extragalactic background light plays an important role at intermediate energies of ultra high energy cosmic ray nuclei. The most noticeable effect of the extragalactic background light is the low-energy tail in the spectrum of secondary nuclei.

💡 Research Summary

The paper presents a comprehensive analytic framework for calculating the propagation of ultra‑high‑energy cosmic‑ray (UHECR) nuclei through extragalactic background radiation. Building on the three analytic schemes introduced in the companion “Paper I,” the authors now solve coupled kinetic equations (CKE) that simultaneously incorporate continuous energy losses due to cosmic expansion, electron‑positron pair production, and photo‑disintegration of nuclei. The key advance is the inclusion of the full extragalactic background light (EBL)—infra‑red, optical, and ultraviolet photons—in addition to the cosmic microwave background (CMB).

The kinetic equation for a nuclear species of mass number A reads

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