Simulation confirms the possibility that massive galaxies may form by mergers, moreover, this process has an "explosive" character and is an analog of phase transition, cD-galaxies (with mass comparable to the total galaxy mass of the cluster) being formed as a new phase. What are the conditions which make this process possible? The expression for may be written as
(see, e.g., Voloshchuk 1984). Numerical solution of the Smoluchowski equation gives , , for , , respectively (if the initial distribution has a tail, may essentially depend on and be much less). Assuming where is an average density of the mass contained in galaxies and expressing the variables in astronomical units, we obtain the order of magnitude for :
Here , , is the ratio of the local density of the mass contained in galaxies to the average density of the Universe; the coefficient C in Eq. (6) is assumed . The mass of a rich cluster is , 2-7% of it is contained in galaxies (Böhringer 1995). The size (of the central part) being of the order of one megaparsec, the ratio may be several hundred to several thousand. Assuming we obtain that, for a cluster with a low velocity dispersion (), the critical time is less than the age of the Universe on condition that masses of the initial galaxies (which then merge) or more (for the case , ). A close estimate for can be obtained also for the region 6 , . On the other hand, we may consider the formation of massive galaxies and the mass function tail only if the initial mass is much less than a typical mass of a large galaxy (). For a cluster with a bigger velocity dispersion ( or more) it is much more difficult to satisfy condition (19): is large, so the kernel with should be taken; is proportional to and can be less than the age of the Universe only for very high density () and large enough initial masses (). Thus, possible dependence of on time due to cluster evolution, its space nonhomogeneity etc. can essentially influence the role of mergers, especially on small masses.
The estimate for given above is based on the assumption that dark matter belongs to the whole cluster rather than to individual galaxies. If dark matter is concentrated in galaxies 7, the ratio may increase by an order which results in the same decrease of (according to Eq. (19)).
So, the conditions necessary for the "explosive" process of mergers may be realized in many clusters.
After a cD-galaxy which contain a significant part of the total mass has formed in the cluster centre, the dynamics of the cluster is largely determined by attraction to this galaxy, and the model considered in this paper breaks down. Besides random collisions, spatial inhomogeneity and mass segregation become essential: due to dynamical friction, most massive galaxies gradually gets into the centre and are swallowed by the cD-galaxy. However, before , when there is no yet cD-galaxy in the cluster, galaxy mergers can be considered as random pairwise encounters with probability given by Eq. (7). It is also clear that mass segregation at a late stage of cluster evolution should be computed together with mass function evolution, using a spatially inhomogeneous Smoluchowski type kinetic equation, which is a much more complicated problem.
Note that galactic mergers due to dynamical friction were discussed by Hausman & Ostriker (1978). In spite of the difference in the model, the algorithm for simulation of mergers, used in their work, is analogous to the algorithm we use (and, thus, equivalent to the Smoluchowski equation), though the expression for the merger probability is different (, which also gives the "explosive" evolution).
As shown in Sects. 2-3 (see also Kontorovich et al. 1995b; Krivitsky 1995), the mass function formed by mergers with the probability given by Eq. (7) is rather steep ( for , for ; in the latter case the asymptotics seem to be non-power, so cannot be determined accurately). It is quite possible that the obtained values of the slope correspond to the steepening of the cluster galaxy luminosity function at the faint end, which was recently discovered (de Propris et al. 1995; Kashikawa et al. 1995; Bernstein et al. 1995). According to observational data analysed in these works, the effective value of the slope for faint galaxies in clusters increases up to 2-2.2 (though, as Bernstein et al. note, there is a different interpretation of these results). Possibly, this part of the luminosity function is formed by mergers and can be described by the intermediate asymptotics for the Smoluchowski equation (if the latter can be extended to small enough masses). Recent HST data shows also an excess of faint low-mass objects for field galaxies (Cowie et al. 1995).
The appearance of relatively steep intermediate asymptotics () can be easily understood from the following arguments 8. Both obtained values for the index ( for and for ) are within the range to . The mass function with corresponds to a constant mass flux 9to infinity (i.e., to cD-galaxy, in our case). However, due to nonlocality of the distribution 10(, see Vinokurov & Kats 1980) such a solution is not realized exactly in both our cases. Nonlocality leads to an essential role of interactions between low-mass and high-mass galaxies. Then the number of massive galaxies is approximately conserved, and the constant flux of their number to infinity corresponds to (Kontorovich et al. 1993). Since none of these limit cases is realized, the index is situated between these values: (), (), and is rather close to their arithmetic mean value (as we can see both from the simulation and the numerical solution of the Smoluchowski equation).
The density ratio in the above estimates was one of the most important parameters which control the possibility of effective merger process. The local value of this parameter may vary in a very wide range: from 1 (scales exceeding an average distance between massive field galaxies) to (if we take an average density of a galaxy for ). As was shown above, in clusters this parameter is large enough ( or even ) to yield an "explosive" evolution due to mergers. Local concentrations may enable analogous phenomena for field galaxies at large z (see, e.g., Komberg &Lukash 1994 ; Kontorovich 1994).
Morphological changes in cluster galaxies, which is one of the results of mergers, may be related to the change of the angular momentum distribution (cf. Toomre 1977). The possibility of dependence between Hubble's morphological type and an effective angular momentum has been discussed in the literature (see Fig. 1 in Polyachenko et al. 1971) and confirmed by an independent analysis of observational data (Kontorovich & Khodjachikh 1993; Kontorovich et al. 1995a). However, this dependence needs special consideration which is beyond the scope of this work.
The above consideration of cluster evolution takes into account only galaxy mergers in a spatially homogeneous model. It allows to obtain the "explosive" evolution, the steep part of the mass function, cD-galaxies, rapid evolution of galaxy morphological types, and a mean value of the dimensionless angular momentum which does not depend on the details of the initial distribution. In the same time, this approach has obvious limitations. The "explosive" evolution does not produce Schechter's mass function with . It is possible that the effective merger probability U changes at a late stage of cluster evolution (when massive galaxies have formed) in such a way that becomes less than one and the "explosive" process slows down 11, which leads to a flatter . Another possibility is that this part of the mass function may not be formed only by mergers.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998