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.
