Stability of the stellar disks of flat galaxies. III. The effect of collisions

According to observations, the occurrence of gravitational collisions in the disk of our own Galaxy is evidenced by the encounters between stars and giant molecular clouds. In the solar vicinity the timescale for encounter between stars and clouds is around τ ∼ 10⁹ yr. Because this timescale is much smaller than the age of the Galaxy T ∼ 10¹⁰ yr, the theory of stability of the stellar disk has to be extended to include the effects of star-cloud gravitational encounters. In this paper, a method of investigating the small-amplitude oscillations and their stability of a collisional two-dimensional galactic disk is developed, through the studying of dispersion relations. A spatially inhomogeneous, differentially rotating disk is considered, with the property that the epicyclic frequency, as well as the angular velocity of rotation, greatly exceeds the frequency v_c ∼ 1/τ of binary collisions between particles. The resulting kinetic equation for the perturbed distribution function can be solved by successive approximations, neglecting the influence of star-cloud encounters on the equilibrium velocity distribution of stars in the zeroth-order approximation. Kinetic theory with the model Bhatnagar-Gross-Krook collisional integral is used, so that the analysis is extended to regions of wavelengths and eigenfrequencies ω of oscillations inaccessible by the hydrodynamic approach developed by Lynden-Bell & Pringle and others. A general dispersion relation is obtained and analyzed in the different cases of weak, ω² ≫ v_c², and strong collisions, ω² ω² ≪ v_c². Using this dispersion relation, the effects of encounters on the dispersion laws both of Jeans and gradient perturbations are considered. The influence of particle encounters on the growth of the oscillating instability, studied in the first paper of the series, is investigated also.