Ion-neutral collisions and the ion sound wave

Ion-neutral collisions can be taken into account in the (linearized) ion kinetic equation by the Boltzmann collision operator K. Gross and Jackson (1959) have shown that K can be approximated convergently by simple operators KN (N=0, 1, 2... infinity ), KN preserving the N first eigenvalues of K and approximating the others by a single constant lambda . This approximation scheme is discussed precisely for the computation of the complex wavenumber k( omega ) of the ion sound wave. The eigenvalues of K are evaluated for the elastic ion-neutral interaction, the best choice of lambda is determined, and the convergence of the approximations is considered. It is shown that N=3 leads to an error smaller than 2 per cent. This allows one to describe the properties of the ion sound wave (phase velocity and damping) at frequencies omega about the ion neutral collision frequency, where simpler models fail and where the transition from collisional to Landau damping occurs.