Notes on the stability threshold for radially anisotropic polytrope
  We discuss some contradictions found in the literature concerning the problem
of stability of collisionless spherical stellar systems which are the simplest
anisotropic generalization of the well-known polytrope models. Their
distribution function $F(E,L)$ is a product of power-low functions of the
energy $E$ and the angular momentum $L$, i.e. $F\propto L^{-s}(-E)^q$. On the
one hand, calculation of the growth rates in the framework of linear stability
theory and N-body simulations show that these systems become stable when the
parameter $s$ characterizing the velocity anisotropy of the stellar
distribution is lower than some finite threshold value, $s<s_\textrm{crit}$. On
the other hand Palmer & Papaloizou (1987) showed that the instability remained
up to the isotropic limit $s=0$.
  Using our method of determining the eigenmodes for stellar systems, we show
that the growth rates in weakly radially-anisotropic systems are indeed
positive, but decrease exponentially as the parameter $s$ approaches zero, i.e.
$\gamma\propto \exp(-s_{\ast}/s)$. In fact, for the systems with finite
lifetime this means stability.
