A Monte Carlo model has been constructed to describe the energization and energy degradation of low-energy protons in an H2 atmosphere in the presence of parallel electric field. The energy spectrum of Ly &agr; collision events is employed to determine the volume emission rate (VER) of H Ly &agr; emissions. Numerical experiments have been performed to study the effect of initial proton energy, electric field, neutral number density, initial pitch angle, and cutoff limit on H Ly &agr; VER. The present study demonstrates that energization of solar EUV-generated low-energy protons by parallel electric fields is incapable of producing optically thin Ly &agr; emissions on Uranus through direct collision with H2. However, nonthermal H atoms, produced through acceleration of protons in parallel electric fields, play an important role in enhancing the Ly &agr; intensity through resonant scattering of solar Ly &agr; flux: resulting emissions are consistent with the broadening of the Ly &agr;-line observed on Jupiter by IUE. The Monte Carlo model is also applied to the problem of Doppler-shifted H Ly &agr; emissions from the auroral atmosphere of Jupiter. Energetic neutral H precipitation from the ring current is discussed as a possible source of optically thin H Ly &agr; emissions on outer planets. Estimates are also given for H&agr;, H&bgr;, and H2 band emissions resulting from the impact of fast H (of ring current origin) on H2. The input energies and fluxes of precipitating ring current particles are, however, too uncertain to permit definite conclusions. ¿ American Geophysical Union 1993