7.1. Cross-sections and merger rates
Consider the interaction of a perturbing galaxy of mass Mp with another galaxy of mass MG. If the perturber passes by with a large velocity v, it will be a good approximation to assume that the stars in the perturbed galaxy do not move during the encounter. Using this impulsive encounter approximation (Spitzer, 1958), the velocity change of a star in the perturbed galaxy is
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(7.1) |
where v is assumed to be parallel to the y-axis, and rp is the impact parameter for the collision. The total change in the internal energy of the perturbed galaxy is
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(7.2) |
where
G2
is the mean-square radius of the perturbed galaxy. Equation
(7.2) is valid if the energy change due to first order terms in
v is zero (e.g.
Richstone, 1976;
White, 1979a).
Now, if the galaxies are identical, with orbital energy
E0 = 1/4MV2, Eq. (7.2) applies for
each galaxy, hence
(Tremaine (1981)),
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(7.3) |
where the relationship on the r.h.s. of Eq. (7.3) is approximately
valid if the galaxies possess brightness profiles which follow the de
Vaucouleurs r1/4 law with half-mass radius
rh and a peak velocity
dispersion . Equation
(7.3) illustrates that in a rich cluster of
galaxies, in which the typical encounter velocity is
v >>
, even a
grazing encounter will produce a relatively small change in the total
internal energy. Equations (7.1) also show that the stars in the outer
parts of a galaxy will be most affected by such an encounter, with the
result that some mass loss must occur. Detailed computations of this
effect have been carried out by
Richstone (1975,
1976
and references therein).
Now consider a slow interpenetrating encounter. Equation (7.3) then
states that
E /
E0
1. The entire orbital energy will be soaked up
during the encounter, and one concludes that the galaxies will merge
(Alladin, 1965;
Alladin et al., 1975).
Clearly such an
application of Eq. (7.3) violates the impulsive approximation on which
it is based, but recently several numerical studies
(Toomre, 1977;
van Albada and van
Gorkom, 1977;
Roos and Norman, 1979;
White, 1978,
1979b)
(9)
have confirmed the suspected stickiness of galaxies in low
velocity encounters. The results are actually in quite good agreement
with those obtained from the impulse approximation.
Results for the capture cross-sections from some of these numerical studies are summarized in Figure 7.1.
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Figure 7.1. Summary of numerical results on
the capture cross-section.
v(rp) is the relative velocity between two
galaxies at the pericenter
of their two-body orbit. ve(rp) is
the escape velocity at
pericenter. The half-mass radii of the two galaxies are
rh1 and
rh2. E0 is the total
orbital energy. The triangle at
v(0) = 1.16ve(rp) is from the
numerical experiments of
van Albada and van
Gorkom (1977).
The squares are from the numerical experiments of
Roos and Norman (1979)
and the filled circles are from experiments E through I
of White (1978),
all of which led to merging. The open circle is from
White's experiment D which did not merge before the calculation was
stopped. The solid line shows the capture cross-section used by
Aarseth and Fall (1980)
in their cosmological N-body experiments
[rp / 2(rh1 +
rh2)]2 +
[v(rp) /
1.16ve(rp)]2
|
Although the numerical calculations have confirmed earlier
suspicions that merging could occur, the possibility that they play a
significant role in galaxy evolution has been seriously entertained
only recently. Since the ratio of the optical half-mass radius of a
typical giant galaxy to the average intergalaxy separation is
extremely small,
rh /
4 ×
10-3, it was thought that a merger would be
an extremely rare event. However, it now seems possible that the
optical radius of a typical spiral galaxy is a very poor indicator of
the mass distribution. Dark halos may contain most of the mass and
extend out to perhaps over 100 kpc
(Section 2.2). Because the true
extent of the dark halos is uncertain it is difficult to make a
reliable estimate of the current merger rate. One argument, due to
Toomre (1977,
also Toomre and Toomre,
1972)
suggests that it is quite
high. Toomre cites eleven examples from the NGC catalogue, of galaxies
which appear to be currently in the process of merging. These pairs
exhibit strong tidal tails and in some cases the main body of the
system can hardly be separated into two distinct components. Toomre
then argues that the tails must be transient phenomena with a median
age of
5 × 108
yr. Thus if the merger rate were constant
during the past 1010 yr, one would expect
200 merger
galaxies out of a total of
4000 NGC
galaxies. Since the merger rate is likely to have been even
higher in the past, this number could even be a considerable
underestimate. It does than appear possible that mergers have been
important in determining the morphology of galaxies and it is relevant
to ask what merger remnants might look like now and also where they
might be found.
9 See also the fascinating N-body experiments of Holmberg (1941) in which gravitational forces were calculated using light bulbs and photocells. Back.