A model of the interaction between incident precipitation and H2 atmospheres is described. The local degraded primary and secondary electron density distributions are calculated by using the continuous slowing down approximation. The altitude distribution of the ionization rate and various H And EUV H2 emissions are calculated for four different incident electron spectra. A total EUV H2 emission efficiency of 10.6 kR/incident erg cm-2 s-1 is obtained for a pure H2 atmosphere. Comparison with the Voyager Jupiter observations indicates that an incident energy flux of about 8 erg cm-2 s-1 was present at the time of the encounter if the emission is located in an H2-dominated region. The local thermospheric heating rate was about 4 ergs cm-2 s-1 for Jupiter and of the order of 0.1 erg cm-2 s-1 for Saturn. A globally averaged atomic hydrogen production rate of ~1¿1010 atoms/cm2 s-1 is induced by the Jovian auroral electron precipitation, largely exceeding the solar EUV dissociation rate.