Astron. Astrophys. 327, 921-929 (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 cross-section, v  is the relative velocity at infinity, the bar means an average over velocities.

An exact computation of collision dynamics and determining the merger cross-section 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 cross-section is

We shall assume that the galaxy peculiar velocity distribution is Gaussian ³with 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 Faber-Jackson and Tully-Fisher laws (), using the virial theorem () and the mass-luminosity 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 cD-galaxies 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.

1. N initial galaxies are simulated, with mass and angular momentum distributed according to .
2. 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.
3. 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.
4. Items 2-3 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 cD-galaxies in the centres of groups or clusters (cf. Hausman & Ostriker 1978; Cavaliere et al. 1991 ). The majority of low-mass 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. 1a-c. 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 right-hand part of the figure () corresponds to cD-galaxies. 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 right-hand end corresponds to cD-galaxies 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 cD-galaxies.

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  cD-galaxies for . The distribution differs from the normal one (the significance level in the Kolmogorov-Smirnov test is  ).

where is the mass function, is the average square value of the momentum S for given mass M. cD-galaxies in the cases and make an exception (Fig. 5b). Kolmogorov-Smirnov 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 left-hand 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 cross-section (in particular, to the value of in Eq. (3)).

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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