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

Abstract

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.

Keywords:

• Stellar dynamics
• Globular cluster evolution
• Similarity solutions

Acknowledgments

This work was supported by NSF Grant DMS‐0318532 and by PSC‐CUNY Award 66557‐00 35. The author is grateful to M. Kiessling for many fruitful discussions, and to both anonymous reviewers for their helpful comments.

Notes

¹In order to determine how Q _BGL ( v ,  w ) in (3) transforms under scaling transformation, recall the well‐known formula for the composition of the Dirac distribution with a function, in order to move all scaling factors in the argument of δ( k ·( v − w )) to the denominator.

²The total momentum 𝒫 can always be set equal to zero by an appropriate coordinate boost.

³One should be careful not to confuse the two concepts of singularity formation and core collapse, where the latter is defined as a dramatic accumulation of mass at the center of the cluster accompanied by the formation of a stationary halo in finite time. Clearly, the two phenomena can occur simultaneously, but also separately and without each other.