Black holes and Galactic density cusps -- I. Radial orbit cusps and bulges

TL;DR

This paper investigates the distribution functions resulting from self-similar radial infall of collisionless matter, exploring steady and time-dependent cases, and discusses implications for galaxy formation and central mass embedding.

Contribution

It introduces a detailed analysis of distribution functions during radial infall, including steady and time-dependent scenarios, and provides methods to embed central masses in these models.

Findings

Steady state systems are described by a universal distribution function.

Logarithmic potential yields Gaussian distribution for inverse-square density.

Central masses can be embedded via iteration, with corrections for different cases.

Abstract

In this paper, we study the distribution functions that arise naturally during self-similar radial infall of collisionless matter. Such matter may be thought of either as stars or as dark matter particles. If a rigorous steady state is assumed, then the system is infinite and is described by a universal distribution function given the self-similar index. The steady logarithmic potential case is exceptional and yields the familiar Gaussian for an infinite system with an inverse-square density profile. We show subsequently that for time-dependent radial self-similar infall, the logarithmic case is accurately described by the Fridmann and Polyachenko distribution function. The system in this case is finite but growing. We are able to embed a central mass in the universal steady distribution only by iteration, except in the case of massless particles. The iteration yields logarithmic corrections to the massless particle case and requires a `renormalization' of the central mass. A central spherical mass may be accurately embedded in the Fridmann and Polyachenko growing distribution however. Some speculation is given concerning the importance of radial collisionless infall in actual galaxy formation.