We examine the stability of a low-mass stellar system surrounding a massive central object. Examples of such systems include the centers of galaxies or star clusters containing a massive black hole and the Oort comet cloud. If the self-gravity of the stellar system is the dominant non-Keplerian force, such systems may be subject to slowly growing (secular) lopsided instabilities. Stability to secular modes is largely determined by the dependence of the distribution function F on angular momentum J. If ?F/?J 0 at constant energy, all spherical systems are secularly stable. If ?F/?J 0, as is expected if there is a loss cone at low angular momentum, all spherical systems in which F = 0 at J = 0 (an empty loss cone) are only neutrally stable, and flattened, nonrotating systems are generally unstable. These results suggest that secular instabilities may dominate the structure and evolution of the stellar systems in the centers of galaxies.
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