Annu. Rev. Astron. Astrophys. 1979. 17: 135-87 Copyright © 1979 by Annual Reviews. All rights reserved

6.1 Crossing Times

It has been obvious for a long while that the motions of galaxies in groups and clusters, if simply interpreted, imply the existence of a substantial amount of mass in addition to that traditionally associated with individual galaxies (Burbidge & Burbidge 1975). However, this conclusion depends entirely on the assumption that the groups are at least bound, if not in virial equilibrium. It is therefore necessary at the outset to decide on the nature of small groups: bound or unbound density enhancements? A useful concept here is the crossing time (Field & Saslaw 1971). If crossing times are short compared to the Hubble time, the groups must be bound; otherwise they would have dispersed long ago. It has sometimes been argued that groups may be unbound and ``exploding'' (e.g. Ambartsumian 1961), but the fact that the great majority of galaxies are group members makes this hypothesis unattractive as a general explanation.

One may choose various definitions of the velocity and radius in defining the crossing time, and these choices make significant and systematic differences in the final values. For example, one may use as R the mean harmonic radius, R_VT, of the group (Limber 1959). R_VT is the characteristic radius used in the virial theorem. Taking V_VT equal to the square root of the mass-weighted rms space velocity with respect to the center of mass, one has t_VT = R_VT / V_VT for the virial crossing time. Using this definition of t, Turner & Sargent (1974) concluded that a large number of de Vaucouleurs' groups have long crossing times and are therefore not bound. However, Jackson (1975) demonstrated that t_VT is a poor estimator of the crossing time, yielding values that are systematically too large. He suggested instead a moment-of-inertia crossing time based on the moment-of-inertia radius. Rood & Dickel (1978a) have used the linear crossing time

(9)

where <r> is the average projected radial distance of group members from the center of mass and <V> is the average of the absolute value of the radial velocities with respect to the center of mass. Gott & Turner (1977) have adopted a similar definition of t_L. With these new definitions of crossing time, all three studies conclude that virtually all the groups identified by Sandage, Tammann, and de Vaucouleurs (STV) and Turner and Gott (TG) have crossing times significantly less than H₀^-1.

There are a number of effects which combine to bias these crossing times to systematically small values, notably the inclusion of nonmembers and the existence of binaries and subclusters, both of which increase the mean projected velocity. Even after all reasonable nonmembers are removed, however, the crossing times are still short, and it seems safe to assume that the great majority of these groups are bound density enhancements after all.

However, it is not clear that these groups have had time to reach virial equilibrium. Gott & Turner (1977) conclude that the median TG group is just now entering the virialized regime. Their calculation rests on necessarily rough dynamical estimates, however, which might not apply accurately to groups with only a few members. Given all the uncertainties, it seems possible that many of the looser groups have not virialized. We must therefore keep in mind that in some cases masses may be as much as a factor of two smaller than those obtained from the virial theorem.