Analytic calculations of the spectra of ultra-high energy cosmic ray
  nuclei. I. The case of CMB radiation
  We present a systematic study of different methods for the analytic
calculation of ultra-high energy nuclei diffuse spectra. Nuclei propagating in
the intergalactic space are photo-disintegrated and decrease their Lorentz
factor due to the interaction with cosmic microwave background and
extragalactic background light. We calculate the evolution trajectories in the
backward time, that describe how atomic mass number $A$ and Lorentz factor
$\Gamma$ change with redshift $z$. Three methods of spectra calculations are
investigated and compared: {\it (i)} trajectory method, {\it(ii)} kinetic
equation combined with trajectory calculations and {\it (iii)} coupled kinetic
equations. We believe that these three methods exhaust at least the principal
possibilities for any analytic solution of the problem. In the most
straightforward method {\it(i)} only trajectory calculations are used to
connect the observed nuclei flux with the production rate of primary
(accelerated) nuclei $A_0$. In the second method {\it (ii)} the flux (space
density) of primary nuclei, and secondary nuclei and protons are calculated
with the help of kinetic equation and trajectories are used only to determine
the generation rates of these nuclei. The third method {\it (iii)} consists in
solving the complete set of coupled kinetic equations, written starting with
primary nuclei $A_0$, then for $A_0-1$ etc down to the $A$ of interest. The
solution of the preceding equation gives the generation rate for the one which
follows. An important element of the calculations for all methods is the
systematic use of Lorentz factor instead of energy. We consider here the
interaction of nuclei only with the cosmic microwave background, this case is
particularly suitable for understanding the physical results.
