Astron. Astrophys. 327, 921929 (1997)
2. Joint mass and angular momentum distribution:
Monte Carlo simulation
2.1. Generalized Smoluchowski equation
and merger probability
The kinetic equation which describes
(the generalized Smoluchowski equation) is
where
etc. The function
(the kernel of the equation) is a characteristic
of the probability for the merger
. Taking into account mass and momentum
conservation laws, W can be rewritten as
(without orbital momentum). If we take into consideration the
orbital momentum
, the second
function should be replaced by a function of a
finite width
. The function U can be calculated as
, where
is the merger crosssection, v is
the relative velocity at infinity, the bar means an average over
velocities.
An exact computation of collision dynamics and determining the
merger crosssection is a very complicated problem. Nevertheless, main
features of this process are known, both from analytical consideration
and numerical experiments (Roos & Norman 1979; Aarseth & Fall
1980; Farouki & Shapiro 1982; Farouki et al. 1983;
Chatterjee 1992) and enables one to formulate conditions necessary
for a merger: the galaxies must pass at a small distance (interaction
is especially intense if the outer parts overlap) and the relative
velocity must be small enough. Namely, we shall assume below that a
merger occurs provided that (i) the minimal distance between the
two galaxies is less than the sum of their radii
and (ii) the relative velocity at infinity
is less than some limit value which is of the order of the escape
velocity
, i.e.,
,
. Taking into account gravitational focusing, we
may derive from the former condition that the impact parameter
. Thus, the merger crosssection is
We shall assume that the galaxy peculiar velocity distribution is
Gaussian
^{3}with the root mean
square velocity
(obviously, the relative velocity distribution
(at infinity) in this case is also Gaussian, but the root mean square
velocity is
). So,
After integration we obtain:
If the density of a galaxy
does not depend on its mass, then the radius
R and mass M are related as
. For FaberJackson and TullyFisher laws (), using the virial theorem () and the massluminosity relation of the form
, we obtain
. Below we shall take
Asymptotical behaviour of Eq. (5) is
the coefficients
,
. Here mass
corresponds to
:
where
. We shall assume
in the further consideration.
Note that the difference of U in the small mass region from
that in Cavaliere et al. 1991 is caused by the velocity
restriction for mergers in Eq. (3). For
collisions without mergers are more probable
than mergers (). The latter, however, gives the "explosive"
evolution in the small mass region too. Effects related to collisions
without mergers may be important (see footnote 1), however, this
question is beyond the scope of this paper.
As
is, in general, a function of time, the
coefficients
and
and mass
depends on time too (see, e.g., Kontorovich
et al. 1992). Below we shall assume them to be constant,
neglecting changes of velocities and masses both due to capture of new
members or evaporation of galaxies from clusters and groups, and
mergers itself, or other reasons.
In the region
galaxy peculiar velocities are much less than
stellar velocities in the galaxy; the relationship for
is inverse. In this paper we consider both the
asymptotical regions
and
and the intermediate case where
is within the range of simulated masses.
Numerical experiments of galaxy merging (Farouki & Shapiro 1982) show that the merger probability U depends both on masses and
momenta, reaching a maximum when
,
and
have the same direction. Nevertheless, this
dependence is less essential than the dependence on masses (,
for
, which leads to the "explosive" evolution). In
this paper we shall use the simple model (3) which does not take into
account the dependence of the merger probability on the mutual
orientation of angular momenta.
It is known from the Smoluchowski equation theory (Ernst 1986;
Voloshchuk 1984) that behaviour of the solution essentially depends
both on the homogeneity power u in the mass dependence of
U and on the asymptotics of U for very different masses
which can be characterized by exponents
:
For the case of galaxy mergers (Eq. (7)), obviously,
and
()or
(). If
then an analog of phase transition takes place
in the system. For an initial distribution localized in the small mass
region, a slowly decreasing distribution tail is formed; in a finite
time the tail reaches infinite masses (in an idealized case). The
second moment of the distribution becomes infinite at
. This phenomenon may, in principle, lead to
fast formation of massive galaxies and quasars by mergers (Kontorovich
et al. 1992), and also to formation of cDgalaxies in groups and
clusters (Cavaliere et al. 1991).
2.2. Numerical simulation of mergers: methods
Direct analytical or numerical solving the generalized Smoluchowski
equation with orbital momentum is a very complicated problem due to
the great number of variables. This difficulty can be avoided by using
numerical Monte Carlo simulation of merging process. In this section
we present such simulation. A finite system consisting of N
galaxies (referred to as "cluster" below, though it may also be a
group) was considered. Pairs of these galaxies merged (with
probability, proportional to
) until the number of the galaxies reduced to
some
. For each merger mass and angular momentum
conservation laws
,
were fulfilled. Distribution of the angular
momenta (intrinsic
and orbital
) over directions was taken isotropic. It was
assumed that the merger probability U depends only on masses
according to Eqs. (5) and (7) and does not depend on momenta. The
absolute value of the orbital momentum was computed in accordance with
Eq. (3), namely,
, v being a random number distributed in
the range 0 to
with probability density
, and the impact parameter
being a random number distributed in the range
0 to
with probability density
. The initial galaxy mass distribution was
chosen exponential
(the results are independent of the exact
expression for initial distribution, only the fact that it is
localized in the region of small masses
and decreases rapidly for large ones is
essential). Instead of the momentum S, it is convenient to
introduce a dimensionless momentum
, similar to Peebles' parameter
(see also Doroshkevich 1967). Dimensionless
momenta
of the initial galaxies were distributed
uniformly in the range 0 to 1 in our simulation (as for
, the exact form of the distribution is
unessential). Computations were carried out for the following
parameters:
,
,
.
In conclusion we describe the procedure of simulation.
 N initial galaxies are simulated, with mass and angular
momentum distributed according to
.
 Two random integer numbers, i and j,
distributed uniformly in the range
(n is the current number of
galaxies) and satisfying the condition
, are simulated.
 Galaxies number i and j merge with probability
. The mass and momentum of the new galaxy are
calculated as described above. With probability
the galaxies do not merge, and jump to
item 2 is executed. Here
depends on time.
 Items 23 are executed until the number of galaxies n
becomes equal
.
The algorithm used in our simulation was somewhat different from
the simplified scheme given above. The reason was that the merger
probability
is very small for the majority of galaxies and
so the simulation time is very large. In actual simulation the
algorithm was modified as follows:
 the probability to choose the i th galaxy in
item 2 was
instead of
, where
is the number of initial galaxies which have
subsequently merged into the i th galaxy;
 to compensate this change, the function
was used instead of
in item 3.
Obviously, these modifications do not influence the result of the
simulation. In the same time, the number of cycles reduces because the
average value of
is closer to 1 than the average value of
.
2.3. Numerical simulation of mergers: results
After some time, an analog of phase transition takes place in the
system of merging galaxies, similarly to what occurred in the work by
Cavaliere et al. (1991). The system divides into two phases: a
giant galaxy which contains a major part of the total mass and many
small galaxies (in the cases
and
this transition is more evident than for
). These giant galaxies formed by mergers can
probably be identified as cDgalaxies in the centres of groups or
clusters (cf. Hausman & Ostriker 1978; Cavaliere et al. 1991
). The majority of lowmass galaxies are those which have never merged
(Fig. 1). Such behaviour is related to strong dependence of the merger
probability on masses ( or
) which, in turn, results in a steep mass
function ( for
,
for
, see below).

Fig. 1ac. Masses M and dimensionless angular momenta
of simulated galaxies (): a initial, b and c formed by mergers (each dot represents one galaxy). A distribution tail, independent of the initial conditions, is formed due to mergers. The righthand part of the figure () corresponds to cDgalaxies. For
(c) separation of galaxies into the two phases, normal and cD, can be seen better than for
(b). Each diagram shows
galaxies (b and c  100 clusters of
galaxies each);
,
.


Fig. 2. Masses M and dimensionless angular momenta
of simulated galaxies for the case
.
,
, 100 clusters ( galaxies).

The mass function formed by mergers in the case
(i.e.,
) is shown in Fig. 3. In the region
it is close to a power law
,
. The rise near the righthand end corresponds to
cDgalaxies the masses of which are comparable to the total masses of
their clusters. In the case
(i.e.,
), and in the intermediate case of a finite
the mass function is steeper.

Fig. 3. Mass function obtained by Monte Carlo simulation (). The values of parameters are the same as in Fig. 1. Mass is given in units
,
is normalized to unity. The part of the plot near
corresponds to cDgalaxies.


Fig. 4. Mass function obtained by Monte Carlo simulation for different
(). The average slope changes with
.

The obtained momentum distribution at fixed mass in the
asymptotical region of large masses is close to the normal
distribution (Fig. 5a). Thus, the distribution tail may be represented
as

Fig. 5a and b. Comparison of the angular momentum cumulative distribution at a fixed mass with the normal distribution (Eq. (10)) for
. a The region of the distribution tail () for
. The distribution is close to the normal one. b cDgalaxies for
. The distribution differs from the normal one (the significance level in the KolmogorovSmirnov test is
).

where
is the mass function,
is the average square value of the
momentum S for given mass M. cDgalaxies in
the cases
and
make an exception (Fig. 5b).
KolmogorovSmirnov test shows evidently that their momentum
distribution differs from the normal one.
For simulation shows that, irrespectively of the
initial momentum, the root mean square value of the dimensionless
momentum
becomes constant at large masses:
that is
(Figs. 6a and 1). The distribution at small masses depends on
.
For
the dimensionless momentum decreases with mass
(Figs. 6b and 2).

Fig. 6a and b. Dependence of the root mean square dimensionless momentum
on mass. a
. In the lefthand part of the plot the main contribution is given by small galaxies which have never merged,
being determined by the initial distribution for them. For large galaxies momenta are determined by mergers and do not depend on the initial conditions,
. b The same for different
().
decreases with M.

The fact that
for
at large masses has a simple qualitative
explanation. Consider the change of the mass and momentum due to
mergers. As the mass function decreases rapidly (i.e., the number of
small galaxies is very large) and
, it is natural to suppose that the main
contribution to the change of the mass M and momentum S
of a given massive galaxy is associated with accretion of small
galaxies
(as, e.g., in Kontorovich et al. 1992).
However, in this case the situation is different. The rate of changing
M and S due to mergers with low mass galaxies () can be expressed as
since
,
for
. In our case
, the slope of the mass function
. Therefore, the main contribution to
integral (14) is given by large galaxies (); contribution of large and small galaxies in
Eq. (13) are of the same order.
Next we consider successive mergers of galaxies with equal large
masses M and momenta
. The new galaxy has a mass
and momentum
^{4}
. Thus, the new value of the dimensionless
momentum
. As a result of successive mergers,
tends to the equilibrium value
(). So, the result
is a consequence of the fact that the main
contribution to the change of mass and momentum is given by mergers
between comparable mass galaxies. The obtained value for
is somewhat different from Eq. (11) due to
simplifying assumptions made in the derivation. Note, that for the
isotropic momentum distribution without allowance for the orbital
momentum
,
; for the anisotropic distribution (Kats &
Kontorovich 1990, 1991)
,
.
In the case
the main contribution to Eqs. (13), (14) is
given by small masses
(due to large
) and we can expect decreasing of
as
. This can be demonstrated as follows. If
integral
converge at infinity then it is possible to
replace the upper limit of integrals (13), (14) by infinity. Then
Therefore,
An analysis of observational data (Kontorovich & Khodjachikh
1993; Kontorovich et al. 1995a) confirms that
, the coefficient k being rather close
to
, which is in accordance with Eq. (12).
Note that allowance for dependence of the merger probability on
momenta may give an increase of
: numerical experiments show that a merger is
more probable if all momenta have the same direction (Farouki &
Shapiro 1982). On the other hand,
is sensitive to the exact form of the merger
crosssection (in particular, to the value of
in Eq. (3)).
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998
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