  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 dF/dJ < 0 at constant energy,
all spherical systems are secularly stable. If dF/dJ > 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,
non-rotating 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.
