Annu. Rev. Astron. Astrophys. 1988. 26: 245-294 Copyright © 1988 by Annual Reviews. All rights reserved

4.2. Survival Times

The dynamical mass of a system of galaxies may be written as M = V² R / G, where V is an effective rms velocity of the galaxies about the system's barycenter, R is an effective radius (related to the harmonic radius) of the system, and G is the Newtonian gravitational constant. The equation for dynamical mass can be recast as an equation for dynamical time, i.e. t_dyn = R/V = (R³ / GM)^1/2 = (G )^-1/2, where is an effective density. The time scale for any specified effect of the gravitational interactions of galaxies in a system is simply t_dyn multiplied by a dimensionless parameter. For example, a derivation based on fundamentals of galactic dynamics contained in the recent textbook by Binney & Tremaine (27a) implies, in crude approximation that the time scale to slow the motion of a galaxy appreciably by dynamical friction is simply t_df ~ Nt_dyn, where N is an effective number of galaxies in the system. It follows from this line of reasoning that if derived dynamical masses lead to anomalous observed effects, so too should derived dynamical friction time scales.

In 1975, Tremaine et al. (199) showed that the dynamical friction time scale for globular clusters passing near the center of M31 is much smaller than the age of M31. Thus many of the globular clusters should have spiraled into the nucleus of M31 and their disruption products should now contribute to the mass of the nucleus. The actual fractional contribution is uncertain pending further kinematic observations of the nucleus (198b).

The formula for t_df adopted by Tremaine et al. is a logical consequence of Newtonian mechanics and follows from the classical derivation by Chandrasekhar (36a) for a test star that experiences random Newtonian gravitational encounters as it moves in an infinite homogeneous field of stars. This formula is consistent with results of N-body simulations by Barnes (21) and an analytical formalism developed by Tremaine & Weinberg (200) for a test star that rotates or revolves through a spherical system. The latter study indicates that frictional effects arise entirely from near-resonant stars.

In 1977, Hickson et al. (90) pointed out that t_df ~ 0.01H₀^-1 (where H₀^-1 ~ 2 × 10¹⁰ yr is the Hubble time scale) for compact groups of galaxies, which measures the survival time of the group members against coalescence by means of dynamical friction. N-body simulations of compact groups by Barnes (21) confirm this time scale and depict the mergers graphically. Let = N_product / N_source, where N_product is the number of galaxies that are merger products of compact groups in a volume in space containing N_source compact groups. Then we can define a dynamical friction time scale anomaly index by x = _dyn / _id - 1. The parameter _dyn is the value obtained if N_product is inferred from the application of the formula for t_df, and _id is the value obtained from the observationally identified merger products. For example, if the merger products of compact groups with properties as defined by Hickson (89) are identified with the galaxies having the same range of luminosity in the general luminosity function (a conservative upper limit), and if the formation of compact groups and their coalescence into galaxies (modeled as immutable end products) are steady-state processes over the time t ~ H₀^-1, then _dyn ~ 100 and _id ~ 10, so that x ~ 10. [Here the larger estimate of x presented in (214) has been decreased by a factor of 10 with the inclusion of new data and more refined analysis by P. Hickson (private communication, 1987).]

Some of the uncertainties in an analysis for compact groups are not present in an analysis for another class of extragalactic objects-the Abell clusters of Bautz-Morgan type I, I-II (defined to contain a supergiant, most luminous galaxy) as identified by Leir & van den Bergh (112a). These clusters are easy to recognize because the most luminous member, whether a single or binary supergiant galaxy, is much more luminous than any other cluster member. It follows from the theory of dynamical friction that when a binary supergiant galaxy merges, it becomes a single supergiant galaxy. A major advantage of this type of sample is that nearly all of the selection functions are the same for the sources (binary supergiant galaxies) and the products (single supergiant galaxies). It is observed that 25% of BM I I-II clusters contain binary supergiant galaxies (160). If each single supergiant galaxy is the merger product of a binary supergiant galaxy (a conservative upper limit) and if the formation of binary supergiant galaxies and their coalescence into single supergiant galaxies (modeled as immutable end products) are steady-state processes over the time t ~ H₀^-1, then it follows that _id = 3; and because t_df ~ 0.03H₀^-1 for the merger of a binary supergiant galaxy (160, 211b, d, g), then _dyn ~ 30, so that x ~ 10. By reestimating values of parameters, the severity of the anomaly could be reduced (198a), but its total elimination appears to require that single supergiant galaxies are mutable, i.e. they often transform into binary supergiant galaxies through the capture of a luminous cluster member. But visual inspection of BM I, I-II clusters suggests that a single supergiant galaxy is immutable in the sense that its capture of the next most luminous cluster member would result in a system identified as a single supergiant galaxy with a satellite, not a binary supergiant galaxy [i.e. the difference in luminosity between a single supergiant galaxy and the next most luminous cluster member is large compared with the difference in luminosity of the components of a binary supergiant galaxy (160)]. This important observational result should be checked both visually by independent observers and photometrically by accurate modern techniques.

A class of Abell clusters has been discovered that contains numerous binary galaxies (188). Research is needed to determine the detailed dynamical structure of these binary galaxies and to identify the structural characteristics or dynamical processes that permit this class of clusters to exist in the presence of the dynamical friction that tends to transform binary galaxies into single galaxies.