We describe a Neumann iteration method for solution of the electron transport equation with the independent variables of energy, pitch angle, and altitude. The method has been developed to study electron energy deposition in the high-latitude thermosphere and mesosphere. It is fast and can compute numerical point response function solutions of the electron transport equation. Inelastic cross sections input to the model are empirically based but are constrained by theoretical consistency conditions. Angular elastic cross sections are also empirically based. Thus, the transport solutions obtained represent a test of compatibility between various sets of cross sections and energy deposition measurements. Energy coupling of the electron transport equation is treated using an efficient, discrete energy loss method that allows the use of a widely separated energy grid at high energy. However, there are limitations on the bin width that are due to physical considerations. This method allows separation of the electron transport equation into a system of single-energy equations coupled in energy. The angular and spatial parts of each single-energy equation can be solved by Neumann iteration. We employ an angular grid rather than an expansion of the angular dependence in terms of spherical harmonics.

To obtain accurate phase function integrals at a low computational cost, we have developed a numerical quadrature based on the use of analytic phase function forms. We have also developed a quadrature for the spatial integration based upon the known analytic behavior expected from the solution of the transport equation. As an application of the model, we make the first published comparison between a direct solution-as opposed to a Monte Carlo method-using discrete energy loss from detailed atomic cross sections and monoenergetic, monodirectional electron beam injection experiments. Results obtained from the model for the practical range are within 10% of the observed range for electrons in N2 over the energies studied from 300 eV to 12 keV. The scaled energy deposition profiles computed by the model tend toward shape invariance at higher energies when plotted as a function of the fraction of the practical range as also seen in empirical studies. The computed shape is generally in good agreement with empirical determinations, being best at the highest incident energies. ¿American Geophysical Union 1987