The stability of spherical stellar systems with a compact central mass

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Abstract
We consider the stability of spherical stellar systems with anisotropic distribution functions peaked or singular for radial orbits. We consider the effect of a small central point mass on the instability to non-radial modes previously found in these systems. We show that as the point mass grows in magnitude purely growing modes are converted to overstable modes before becoming stable once the point mass is larger than a critical value. We estimate numerical values for such critical masses in the case of a generalized polytropic model system.

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