Annu. Rev. Astron. Astrophys. 1991. 29: 239-274 Copyright © 1991 by Annual Reviews. All rights reserved

2. EQUILIBRIUM MODELS

The dynamical time scale in an elliptical galaxy ranges from less than 10⁵ years in the central regions to more than 10⁹ years in the outskirts. With the possible exception of the inner nucleus, the two-body relaxation time is everywhere much larger than a Hubble time. Thus, an elliptical galaxy is a collisionless stellar system. The time scale for differential precession between neighboring orbits is usually of the order of 5-10 dynamical times, and becomes comparable to a Hubble time beyond a few optical radii. As a result, the inner regions of elliptical galaxies are likely to be in equilibrium, in agreement with their regular and smooth optical appearance, but the outer regions probably have not experienced sufficient phase-mixing to have reached equilibrium.

The structure and dynamics of a collisionless stellar system are determined completely by the phase-space distribution function f(r, v, t), which gives the distribution of the stars in the system over position r and velocity v as a function of time t. The distribution function must be non-negative, and satisfy the collisionless Boltzmann equation. In an equilibrium model ðf / ðt = 0. Integration of f over all velocities gives the density distribution (r) of the system. By Jeans' theorem, f depends on the phase-space coordinates r and v only through the isolating integrals of motion admitted by the gravitational potential of the system (179, 220, but see 40). Any collisionless dynamical model can be scaled in mass, radius and central velocity dispersion, but only two of these three parameters can be chosen freely, because of the constraint provided by the virial theorem (48).