On the instability of weakly radially anisotropic star clusters

We discuss contradictions existing in the literature in the problem on the stability of collisionless spherical stellar systems, which are the simplest anisotropic generalization of the well-known polytropic models. On the one hand, calculations of the growth rates within the framework of a linear stability theory and N -body simulations suggest that these systems should become stable when the parameter s characterizing the degree of anisotropy of the stellar velocity distribution becomes lower than some critical value s crit > 0. On the other hand, according to Palmer and Papaloizou, the growth rate should be nonzero up to the isotropic limit s = 0. Using our method of determining the eigenmodes of stellar systems, we show that even though the mode growth rates in weakly radially anisotropic systems of this type are nonzero, they are exponentially small, i.e., decrease as γ ∝ exp(− a/s ) when s → 0. For slightly radially anisotropic systems with a finite lifetime, this actually implies stability.