AbstractAn exact solution to the energy-dependent Milne problem for isotropic scattering has been obtained using a simple separable scattering kernel. The extrapolation distance and angular distribution at the surface of the half-space have been calculated using the free-gas scattering cross sections. A further calculation for a very narrow kernel shows that the extrapolation distance is insensitive to the inelastic part of the scattering kernel, but depends mainly on the energy dependence of the mean free path. The results have been compared with numerical work obtained from the THERMOS code and thus provide a measure of the accuracy of THERMOS for this type of problem. The results also give physically reasonable bounds on the extrapolation distance and angular distributions.