The ionization created in the Earth's upper atmosphere by an electron beam emitted from a low altitude spacecraft is described by two coupled nonlinear differential electron transport equations for the up-goin (along a magnetic field line) and down-going differential energy flux. The equations are solved numerically, using the mas spectrometer and incoherent scatter data (MSIS) atmospheric model and the international reference ionosphere (IRI) model, yelding estimates of the differential electron energy flux density at the spacecraft location. At altitudes below 200--250 km and for beam energies around 1 keV, it is shown that secondary electrons supply a significant contribution to the return current to the spacecraft and thereby reduce the spacecraft potential. Our numerical results are in good agreement with observations from the Charge 2 sounding rocket experiment. ¿ American Geophysical Union 1990