Kinetic Theory of Collisionless Self-Gravitating Gases: Post-Newtonian
  Polytropes
  In this paper we study the kinetic theory of many-particle astrophysical
systems and we present a consistent version of the collisionless Boltzmann
equation in the 1PN approximation. We argue that the equation presented by
Rezania and Sobouti in A&A 354 1110 (2000) is not the correct expression to
describe the evolution of a collisionless self-gravitating gas. One of the
reasons that account for the previous statement is that the energy of a
free-falling test particle, obeying the 1PN equations of motion for static
gravitational fields, is not a static solution of the mentioned equation. The
same statement holds for the angular momentum, in the case of spherical
systems. We provide the necessary corrections and obtain an equation that is
consistent with the corresponding equations of motion and the 1PN conserved
quantities. We suggest some potential relevance for the study of high density
astrophysical systems and as an application we construct the corrected version
of the post-Newtonian polytropes.
