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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‐law functions of the energy  E  and angular momentum  L , i.e.   F ∝ L  − s   (− E )  q   . On the one hand, calculation of the growth rates in the framework of linear stability theory and  N ‐body simulations shows 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  &lt;  s  crit  . On the other hand, Palmer &amp; Papaloizou showed that the instability remains 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.  γ∝ exp(− s  * / s ) . In fact, for systems with a finite lifetime this means stability.

Notes on the stability threshold for radially anisotropic polytropes