The two main arenas of astrophysical dynamics are celestial mechanics and stellar dynamics. The former deals with the motion of few bodies in a near-Kepler potential; the latter with the motion of many bodies in a non-Kepler potential. I would like to discuss the hybrid problem of many bodies in a near-Kepler potential, which is relevant to a number of astrophysical systems, including protoplanetary disks and the centers of galaxies containing massive black holes.

Consider a spherical system of radius r, containing N bodies of mass m orbiting in the gravitational field of a body of mass M ≫ N m. Assume that the orbits have moderate eccentricities and random orientations and imagine taking a time exposure of the system over several orbits. Each body is then smeared into an approximate Kepler ellipse, which precesses slowly on a timescale tprec. Each ellipse exerts a force on other bodies at comparable radius, f ~ Gm/r2. The mean force from all the ellipses is Fm ~ N f. The mean force determines the precession time through the relation L ~ rFmtprec, where L ~ (GMr)1/2 is the specific angular momentum of an orbit. Thus tprec ~ (M/Nm)tcr, where tcr ~ (r3/GM)1/2 is the crossing time.