Distribution functions for a family of general-relativistic Hypervirial models in collisionless regime

TL;DR

This paper derives distribution functions for a family of relativistic hypervirial models in collisionless regimes, extending Newtonian models to general relativity and analyzing their properties and constraints.

Contribution

It provides explicit relativistic distribution functions for hypervirial models with constant anisotropy, generalizing previous Newtonian results and exploring their physical restrictions.

Findings

Distribution functions depend on energy and angular momentum as F=l^{n-2}ξ(ε).

Explicit polynomial and beta function forms for ξ(ε) are derived for odd and even n.

Constraints on model parameters ensure non-negativity of the distribution function.

Abstract

By considering the Einstein-Vlasov system for static spherically symmetric distributions of matter, we show that configurations with constant anisotropy parameter $\beta$ have, necessarily, a distribution function (DF) of the form $F=l^{-2\beta}\xi(\varepsilon)$, where $\varepsilon=E/m$ and $l=L/m$ are the relativistic energy and angular momentum per unit rest mass, respectively. We exploit this result to obtain DFs for the general relativistic extension of the Hypervirial family introduced by Nguyen and Lingam (2013), which Newtonian potential is given by $\phi(r)=-\phi_o /[1+(r/a)^{n}]^{1/n}$ ($a$ and $\phi_o$ are positive free parameters, $n=1,2,...$). Such DFs can be written in the form $F_{n}=l^{n-2}\xi_{n}(\varepsilon)$. For odd $n$, we find that $\xi_n$ is a polynomial of order $2n+1$ in $\varepsilon$, as in the case of the Hernquist model ($n=1$), for which $F_1\propto l^{-1}\left(2\varepsilon-1\right)\left(\varepsilon-1\right)^2$. For even $n$, we can write $\xi_n$ in terms of incomplete beta functions (Plummer model, $n=2$, is an example). Since we demand that $F\geq 0$ throughout the phase space, the particular form of each $\xi_n$ leads to restrictions for the values of $\phi_o$. For example, for the Hernquist model we find that $0\leq \phi_o \leq2/3$, i.e. an upper bounding value less than the one obtained for Nguyen and Lingam ($0\leq \phi_o \leq1$), based on energy conditions.