Analytic Solutions of Ultra-High Energy Cosmic Ray Nuclei Revisited

The chemical composition of ultra-high energy cosmic rays is a key question in particle astrophysics. The measured composition, inferred from the elongation rates of cosmic ray showers, looks in general very different from the initial source composition: resonant photo-disintegration in the cosmic radiation background proceeds rapidly at the highest energies and the initial composition quickly becomes lighter during propagation. For a statistical analysis of continuously improving cosmic ray data it is desirable to know the secondary spectra as precisely as possible. Here, we discuss exact analytic solutions of the evolution equation of ultra-high energy cosmic ray nuclei. We introduce a diagrammatic formalism that leads to a systematic analytic expansion of the exact solution in terms of second order effects of the propagation. We show how the first order corrections of this expansion can improve the predictions of secondary spectra in a semi-analytical treatment.