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Astron. Astrophys. 327, 921-929 (1997)

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3. Comparison of simulation results with solution
of the Smoluchowski equation

Integrating the generalized Smoluchowski equation (1) over momenta, one can obtain the ordinary Smoluchowski equation which describe the evolution of the galaxy mass function:


Solving this equation is another independent way to find  [FORMULA]. In this section we compare the results obtained by Monte Carlo simulation with the obtained earlier (see Kontorovich et al. 1995b and Krivitsky 1995) results of direct numerical solving the Smoluchowski equation with kernels [FORMULA] and [FORMULA].

For numerical solution of Eq. (17) and analysis of the obtained results we used methods described in Krivitsky 1995; Kontorovich et al. 1995b. Fig. 7b shows a plot of the obtained mass function for [FORMULA]. One can see that an intermediate asymptotics, close to a power law  [FORMULA], is formed in the region [FORMULA]. The exponent [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA] (in the latter case [FORMULA] is defined worse, see below). The value 1.9 differs from [FORMULA] given by Cavaliere et al. (1992) but, in our opinion, agrees rather well with the plot in their work. The value of [FORMULA] for the two cases are 5, respectively, 0.26 and 0.1 (in the same units as in Fig. 7) for the initial distribution [FORMULA]. The time dependence of the number of galaxies, obtained from the solution of the Smoluchowski equation, is shown in Fig. 8. The moment displayed in Fig. 1b ([FORMULA]) approximately corresponds to [FORMULA] (whereas [FORMULA]).

[FIGURE] Fig. 7a and b. a  Numerical solution of the Smoluchowski equation in the small mass region ([FORMULA], M  is measured in units  [FORMULA], time in units [FORMULA], [FORMULA]  in units  [FORMULA]). 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 [FORMULA] (Fig. 7a) the intermediate asymptotics is not as close to a power law as for [FORMULA]. An effective slope [FORMULA] in this case is 2-3. The value of [FORMULA] is [FORMULA], that is the phase transition is very fast. However, in this case the time dependence of the distribution moments is non-power and [FORMULA] cannot be determined accurately (Kontorovich et al. 1995b; Krivitsky 1995).

[FIGURE] Fig. 8. Time dependence of the number of galaxies, obtained from the solution of the Smoluchowski equation (upper time scale for [FORMULA], lower time scale for [FORMULA]). Though the total number of galaxies decreases, at any t the number of low-mass galaxies which have never merged is much more than the number of massive tail galaxies.

In numerical solving Eq. (17), a finite limit mass [FORMULA] was introduced: the integral from 0 to [FORMULA] in the right-hand part was replaced by the integral from 0 to  [FORMULA]. 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 [FORMULA] and [FORMULA]. In the case of kernel (7) which belongs to the class [FORMULA], existence of [FORMULA] and its value essentially influence the solution, in particular, the number of galaxies [FORMULA] and distribution moments [FORMULA] as functions of time, the value of [FORMULA] etc. Moreover, van Dongen (1987) showed that the limit [FORMULA] does not exist at all for [FORMULA] (this is the case for Eq. (7)). The influence of [FORMULA] increases as [FORMULA] becomes farther from 1: for [FORMULA], especially if [FORMULA], this influence is moderate. The farther [FORMULA] moves from 1, the greater the difference between the intermediate asymptotics and the power law becomes and the worse defined [FORMULA] and [FORMULA] are; this is the case for [FORMULA].

As known (see van Dongen & Ernst 1988; Voloshchuk 1984; Krivitsky 1995; Kontorovich et al. 1995b), in many cases the solution for [FORMULA] is self-similar: [FORMULA] for [FORMULA], where [FORMULA] is a time-independent function, [FORMULA] describes a "front" moving to greater masses. The numerical solution shows that [FORMULA] for [FORMULA] is closer to the self-similar form for lower  [FORMULA]. The mass function for [FORMULA] is not self-similar (nonlocal case, see discussion in Kontorovich et al. 1995b). However, for [FORMULA] the shape of the curve becomes nearly constant (Fig. 7a): a different self-similarity appears, because mergers with the cD-galaxy 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 ([FORMULA]), 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 [FORMULA]. So we can make a conjecture that [FORMULA] for [FORMULA] depends only on the value of the limit mass and does not depend on its nature.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998