Exact equilibria of a stellar system in a linearized tidal field

Abstract

ABSTRACT We study the motion of stars in a star cluster which revolves in a circular orbit about its parentgalaxy. The star cluster is modelled as an ellipsoid of uniform spatial density. We exhibit two2-parameter families of self-consistent equilibrium models in which the velocity at each pointis confined to a line in velocity space. We exhibit the link bet ween this problem and that of auniform rotating ellipsoidal galaxy. With minimal adaptation, Freeman’s bar models yield athird family.Key words: stellar dynamics - globular clusters: general - open clusters and associations:general 1 INTRODUCTIONIn a star cluster moving in a circular orbit about the axis of sym-metry of a galaxy, a star is subject to the self-gravity of theclusterand the tidal field of the galaxy. If studied in a rotating, acc eler-ating frame following the orbital motion of the cluster, it is alsosubject to centrifugal and Coriolis forces. If the spatial distributionof stars is uniform within an ellipsoid, and if the usual linear ap-proximation of the tidal field is adopted, the accelerations insidethe ellipsoid are all linear in the spatial coordinates. There are thentwo normal modes of oscillation in planes orthogonal to the axisof rotation. Fellhauer & Heggie (2005) studied this problem in thecase when one of the normal frequencies is imaginary. Taking forconvenience the case of an ellipsoid of revolution, they showed thatit was possible to choose the axial ratio so that it was equal to thatof the orbits in the remaining normal mode. By a very simple orbitsuperposition it was possible to construct a self-consistent model.This problem closely resembles that of constructing a modelof a uniform, rotating ellipsoidal bar (Freeman 1966a,b,c; Hunter1974, 1975). This goal has been achieved in three cases: (i) ellip-tical cylinders, in which the semi-major axis parallel to the axis ofrotation is infinite; (ii) so-called “balanced” systems, in which thecentrifugal and gravitational forces are equal along the major axis;and (iii) disks with the surface density that is obtained by projectinga uniform ellipsoid onto one plane.In this paper we set out the connection between these twoproblems, and give a more systematic treatment of the modelswhich Fellhauer & Heggie stumbled upon. In principle there arethree dimensionless parameters: two axial ratios, and one parame-ter which measures the density of the system. The requirement ofchoosing one axial ratio appropriately reduces the family to a two-parameter family. Fellhauer & Heggie chose to work with axisym-