Self-consistent dynamical models with a finite extent -- III. Truncated power-law spheres

TL;DR

This paper develops a family of finite-extent, self-consistent dynamical models for spherical systems with power-law density profiles, providing explicit formulas and conditions for their physical viability based on orbital structure and density slope.

Contribution

It introduces a new class of analytical, finite-extent dynamical models with power-law density profiles supported by tangential Cuddeford orbital structures, including explicit conditions and formulas.

Findings

Truncated power-law spheres are supported if and only if γ ≥ 2β₀.

Derived explicit expressions for velocity dispersion profiles.

Established the density slope-anisotropy inequality as a necessary and sufficient condition.

Abstract

Fully analytical dynamical models usually have an infinite extent, while real star clusters, galaxies, and dark matter haloes have a finite extent. The standard method for generating dynamical models with a finite extent consists of taking a model with an infinite extent and applying a truncation in binding energy. This method, however, cannot be used to generate models with a pre-set analytical mass density profile. We investigate the self-consistency and dynamical properties of a family of power-law spheres with a general tangential Cuddeford (TC) orbital structure. By varying the density power-law slope $\gamma$ and the central anisotropy $\beta_0$, these models cover a wide parameter space in density and anisotropy profiles. We explicitly calculate the phase-space distribution function for various parameter combinations, and interpret our results in terms of the energy distribution of bound orbits. We find that truncated power-law spheres can be supported by a TC orbital structure if and only if $\gamma \geqslant 2\beta_0$, which means that the central density slope-anisotropy inequality is both a sufficient and a necessary condition for this family. We provide closed expressions for structural and dynamical properties such as the radial and tangential velocity dispersion profiles, which can be compared against more complex numerical modelling results. This work significantly adds to the available suite of self-consistent dynamical models with a finite extent and an analytical description.