On spherically-symmetric accretion by a collisionless polytrope
An isolated, spherically-symmetric, self-gravitating, collisionless system is always a polytrope when it reaches equilibrium (Nakamura 2000). This strongly suggests as a corollary, however, that the same polytrope dominates its precursor states, since the dynamical equations for its constituents can be time-reversed. Moreover this assumption, which precludes a polytrope from ever accreting 100% of the mass from an infalling shell, as a subsequent state will eventually be a polytrope, is confirmed now by our finding that a collisionless polytrope cannot accrete 100% of an infalling shell while simultaneously guaranteeing that the entropy of the Universe as a whole increases. These strictures are only evaded by the shedding of some mass. A polytrope must lose mass to gain mass. We deduce from the time-reversible property of a collisionless polytrope that the scalar sum, P, over constituent momenta in its rest frame is an independent state variable that is conserved with respect to its surface radius through interactions between a polytrope and an infalling shell. This new constraint, together with conservation of energy, enables us (i) to show that an isolated polytrope is indeed stable against spherically-symmetric mass-loss, which is the essential content of our initial assumption; (ii) to calculate both the velocity and the fraction of infall-mass returned to infinity, provided the "accretion law" between the change in mass and surface radius is specified. Numerical results confirm a frequent empirical finding (Livio 2000) that the velocity of a mass outflow is of the same order of magnitude as the escape velocity from the system.