Stellar dynamics around black holes in galactic nuclei

We classify orbits of stars that are bound to central black holes in galact ic nuclei. The stars move under the combined gravitational influences of th e black hole and the central star cluster. Within the sphere of influence o f the black hole, the orbital periods of the stars are much shorter than th e periods of precession. We average over the orbital motion and end up with a simpler problem and an extra integral of motion: the product of the blac k hole mass and the semimajor axis of the orbit. Thus the block hole enforc es some degree of regularity in its neighbourhood. Well within the sphere o f influence, (i) planar, as well as three-dimensional, axisymmetric configu rations - both of which could be lopsided - are integrable, (ii) fully thre e-dimensional clusters with no spatial symmetry whatsoever must have semi-r egular dynamics with two integrals of motion. Similar considerations apply to stellar orbits when the black hole grows adiabatically. We introduce a f amily of planar, non-axisymmetric potential perturbations, and study the or bital structure for the harmonic case in some detail. In the centred potent ials there are essentially two main families of orbits: the familiar loops and lenses, which were discussed by Sridhar and Touma. We study the effect of lopsidedness, and identify a family of loop orbits, the orientation of w hich reinforces the lopsidedness. This is an encouraging sign for the const ruction of self-consistent models of eccentric discs around black holes, su ch as in M31 and NGC 4486B.