Astron. Astrophys. 327, 921929 (1997) 3. Comparison of simulation results with solution

Fig. 7a and b. a Numerical solution of the Smoluchowski equation in the small mass region (, M is measured in units , time in units , in units ). b Solution in the large mass region, for comparison (the same case as in Cavaliere et al. (1992), but in a wider mass range and somewhat more precise). 
In the case (Fig. 7a) the intermediate asymptotics is not as close to a power law as for . An effective slope in this case is 23. The value of is , that is the phase transition is very fast. However, in this case the time dependence of the distribution moments is nonpower and cannot be determined accurately (Kontorovich et al. 1995b; Krivitsky 1995).
Fig. 8. Time dependence of the number of galaxies, obtained from the solution of the Smoluchowski equation (upper time scale for , lower time scale for ). Though the total number of galaxies decreases, at any t the number of lowmass galaxies which have never merged is much more than the number of massive tail galaxies. 
In numerical solving Eq. (17), a finite limit mass was introduced: the integral from 0 to in the righthand part was replaced by the integral from 0 to . Physically, such a substitution corresponds to a sink at large masses. As it was shown by Krivitsky 1995, Kontorovich et al. 1995b, consequences of this replacement are different for kernels with and . In the case of kernel (7) which belongs to the class , existence of and its value essentially influence the solution, in particular, the number of galaxies and distribution moments as functions of time, the value of etc. Moreover, van Dongen (1987) showed that the limit does not exist at all for (this is the case for Eq. (7)). The influence of increases as becomes farther from 1: for , especially if , this influence is moderate. The farther moves from 1, the greater the difference between the intermediate asymptotics and the power law becomes and the worse defined and are; this is the case for .
As known (see van Dongen & Ernst 1988; Voloshchuk 1984; Krivitsky 1995; Kontorovich et al. 1995b), in many cases the solution for is selfsimilar: for , where is a timeindependent function, describes a "front" moving to greater masses. The numerical solution shows that for is closer to the selfsimilar form for lower . The mass function for is not selfsimilar (nonlocal case, see discussion in Kontorovich et al. 1995b). However, for the shape of the curve becomes nearly constant (Fig. 7a): a different selfsimilarity appears, because mergers with the cDgalaxy dominate and the dependence of U on the smaller mass vanishes.
Both in direct solution of the Smoluchowski equation and in numerical simulation of mergers, there exists a finite limit mass: the mass of the sink in the former case, the total mass of the system in the latter one. However, the problems which are solved in this section and in Sect. 2 are not equivalent. Nevertheless, simulation shows that, in spite of the essential influence of the limit mass (), the mass function obtained by simulation of mergers in Sect. 2 and the one obtained as a direct solution of the Smoluchowski equation have good agreement in the region . So we can make a conjecture that for depends only on the value of the limit mass and does not depend on its nature.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998
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