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

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.