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

**4.2. Survival Times**

The dynamical mass of a system of galaxies may be written as
*M* = *V*^{2} *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* = (*R*^{3} /
*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} ~ 0.01*H*_{0}^{-1} (where
*H*_{0}^{-1} ~ 2 × 10^{10} 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* ~ *H*_{0}^{-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* ~ *H*_{0}^{-1}, then it follows that
_{id} = 3;
and because
*t*_{df} ~ 0.03*H*_{0}^{-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.