The auroral downward field-aligned current is mainly carried by electrons accelerated up from the ionosphere into the magnetosphere along magnetic field lines. Current densities are typically of the order of a few ¿ Am-2, and the associated electrons are accelerated to energies of several hundred eV up to a few keV. This downward current has been modeled by Temerin and Carlson (1998) using an electron fluid. This paper extends that model by describing the electron populations via distribution functions and modeling all of the F region. We assume a given ion density profile, and invoke quasi-neutrality to solve for the potential along the field line. Several important locations and quantities emerge from this model: the ionospheric trapping point, below which the ionospheric population is trapped by an ambipolar electric field; the location of maximum E$parallel$, of the order of a few mVm-1, which lies earthward of the B/n peak; the acceleration region, located around the B/n peak, which normally extends between altitudes of 500 and 3000 km; and the total potential increase along the field line, of the order of a few hundred V up to several kV. The B/n peak is found to be the central factor determining the altitude and magnitude of the accelerating potential required. Indeed, the total potential drop is found to depend solely on the equilibrium properties in the immediate vicinity of the B/n peak.