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On self‐similar collapsing solutions to the kinetic equations of stellar dynamics

We discuss the application of the kinetic equations of stellar dynamics to the self‐similar collapse of a spherical cluster of stars. We extend to the Balescu‐Guernsey‐Lenard collision kernel a number of results previously established for the Landau kernel, such as the fact that the scaling properties of the equations determine completely the time‐asymptotic radial density profile of solutions that collapse in finite time (they must approach an inverse‐cube power law as collapse is completed). However, we further prove that no such solutions actually exist that either are bounded or have inverse‐power asymptotics near the center of the cluster, with the possible exception of solutions whose total energy is exactly zero. We also study the invariance properties of the orbit‐averaged Kuzmin‐Hènon‐Poisson equations under similarity transformations, and we show that in this case the exponent α in the radial density profile is not restricted to α=3 but can also take values in the Lynden‐Bell‐Eggleton range 2<α≤2.5. Thus, in the orbit‐averaged case the similarity analysis per se does not support the notion of a single, “universal” power law for the density of stars during the late stages of core collapse.