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A kinetic equation for the collisional evolution of stable, bound, selfgravitating and slowly relaxing systems is established, which is valid when thenumber of constituents is very large. It accounts for the detailed dynamics andself consistent dressing by collective gravitational interaction of thecolliding particles, for the system's inhomogeneity and for differentconstituent's masses. The evolution of the one-body distribution function isdescribed in action angle space. The collision operators are expressed in termsof the collective response function allowed by the existing distributionfunctions at any given time and involve particles in resonant motions. The setof equations which describe the coupled evolution of the distribution functionsand of the potential is derived for spherical systems. In the homogeneouslimit, which sacrifices the description of the evolution of the spatialstructure of the system, but retains the effects of collective gravitationaldressing, the kinetic equation reduces to a form similar to the Balescu-Lenardequation of plasma physics.

A Balescu-Lenard type kinetic equation fot the collisional evolution of stable self-gravitating systems