Relaxation dynamics of hot protons in a thermal bath of atomic hydrogen

We present a rigorous kinetic theory formulation of the relaxation of hot protons (${\mathrm{H}}^{+}$) in a bath of thermal atomic hydrogen (H). We apply the (well-known) quantum-mechanical scattering theory to (${\mathrm{H}}^{+}$,H) collisions and calculate the differential elastic cross section as a function of collision energy and scattering angle. This calculation includes the effects of both direct and charge-exchange scattering. We then solve the time-dependent Boltzmann equation numerically for the ${\mathrm{H}}^{+}$ distribution function with an initial δ-function distribution. We also consider two approximate models for the collision dynamics, each based on the assumption that charge exchange dominates the relaxation and that no momentum is transferred in a collision (the linear-trajectory approximation). The first model uses the Rapp-Francis [J. Chem. Phys. 37, 2631 (1962)] energy-dependent cross section in the exact kernel which defines the Boltzmann collision operator. The second model uses a hard-sphere cross section in an approximate collision kernel. We compare the relaxation behavior calculated with the approximate formulations with the exact solution. We also calculate the mobility of ${\mathrm{H}}^{+}$ in H and compare the exact and approximate formulations. This study has applications to processes in astrophysics and aeronomy such as the nonthermal escape of H from planetary atmospheres.