The double-power approach to spherically symmetric astrophysical systems
  In this paper, we present two simple approaches for deriving anisotropic
distribution functions for a wide range of spherical models. The first method
involves multiplying and dividing a basic augmented density with polynomials in
$r$ and constructing more complex augmented densities in the process, from
which we obtain the double-power distribution functions. This procedure is
applied to a specific case of the Veltmann models that is known to closely
approximate the Navarro-Frenk-White (NFW) profile, and also to the Plummer and
Hernquist profiles (in the appendix). The second part of the paper is concerned
with obtaining hypervirial distribution functions, i.e. distribution functions
that satisfy the local virial theorem, for several well-known models. In order
to construct the hypervirial augmented densities and the corresponding
distribution functions, we start with an appropriate ansatz for the former and
proceed to determine the coefficients appearing in that ansatz by expanding the
potential--density pair as a series, around $r=0$ and $r=\infty$. By doing so,
we obtain hypervirial distribution functions, valid in these two limits, that
can generate the potential--density pairs of these models to an arbitrarily
high degree of accuracy. This procedure is explicitly carried out for the
H\'enon isochrone, Jaffe, Dehnen and NFW models and the accuracy of this
procedure is established. Finally, we derive some universal properties for
these hypervirial distribution functions, involving the asymptotic behaviour of
the anisotropy parameter and its relation to the density slope in this regime.
In particular, we show that the cusp slope--central anisotropy inequality is
saturated.
