1999MNRAS.303..483S

We classify orbits of stars that are bound to central black holes in galactic nuclei. The stars move under the combined gravitational influences of the black hole and the central star cluster. Within the sphere of influence of the black hole, the orbital periods of the stars are much shorter than the 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 black hole mass and the semimajor axis of the orbit. Thus the black hole enforces some degree of regularity in its neighbourhood. Well within the sphere of influence, (i) planar, as well as three-dimensional, axisymmetric configurations – both of which could be lopsided – are integrable, (ii) fully three-dimensional clusters with no spatial symmetry whatsoever must have semi-regular dynamics with two integrals of motion. Similar considerations apply to stellar orbits when the black hole grows adiabatically. We introduce a family of planar, non-axisymmetric potential perturbations, and study the orbital structure for the harmonic case in some detail. In the centred potentials 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 which reinforces the lopsidedness. This is an encouraging sign for the construction of self-consistent models of eccentric discs around black holes, such as in M31 and NGC 4486B.