Astron. Astrophys. 334, 381-387 (1998)

2. The collapse time

DG97 showed how the expansion parameter depends on the dynamical friction, solving Eq. (7) by means of a numerical method but not taking into account the parameters on which depends.
Here we examine how the dynamical friction coefficient varies according to the parameters and how the collapse time depends on them. We consider Eq. (9) and the functions and , the former given by Chandrasekhar (1943), the latter (for clustered system) given by the so-called Holtsmark law (Holtsmark 1919):

where is the probability for a test particle of experiencing a force in the range , N is the total number of particles, is a typical particle mass and is a typical distance among the particles.
According to K80:

with:

The original expression for given by K80 has been modified by AA92 to take into account clustering, and turns out to be given by:

where , , , and are given in AA92 respectively by the Eqs. (21), (29), (31), (32) and (36).
Here, we want to remind that is a linear function of the correlation function and that the general expression for , adopting the notation of AA92 is given by:

where masses are measured in units of solar mass and distances in Mpc, so that k will be measured in unit of (Mpc) , with the average mass of the substructure. Since we have for all the cases that we consider in this paper, we can adopt the asymptotic expansion of demonstrated in the appendix of AA92:

(see AA92 for details). does not depend on , but only on (that is calculated at the origin).
We solve Eq. (9) and the other equations related for an outskirts shell of baryonic matter with inside the spherical regions (protocluster), for different values of , , and (where ), the latter quantity being better defined in Sect. 4.
After having determined solving numerically Eq. (9), we get as a function of , , and solving Eq. (7). We perform these calculations for different set of values of , , and inside the

following intervals:

The results that we have obtained are shown in Figs. 1-4. Before commenting upon the figures, we want to remark that the dependence of on is qualitatively shown in Fig. 5 by AC94. We observe that for the collapse time in the presence of dynamical friction is always larger than in the unperturbed case but the magnitude of the deviation is negligible for larger , whilst for the deviations increase steeply with lower . Then, having considered , the estimate we get for in Sect. 4 must be considered as a lower limit.
In Fig. 1 we show the collapse time in the presence of dynamical friction, versus the peaks' height, for different values of . In this picture, we show how grows for larger values of and for larger values of . Similarly, in Fig. 2 we note how increases for larger values of and of . The slope of the curves confirm our prediction on the behaviour of the collapse of a shell of baryonic matter falling into the central regions of a cluster of galaxies in the presence of dynamical friction: the dynamical friction slows down the collapse (as DG97 had already shown) and the effect, as we are showing in Figs. 1 and 2, increases as , , grow.
Here we want to remind you that we are considering only the peaks of the local density field with central height larger than a critical threshold . This latter quantity is chosen to be the threshold at which () where is the typical size of the peaks and is the average peak separation (see also Bardeen et al. 1986).

 Fig. 1. Collapse time of a shell of matter made of galaxies and substructure when dynamical friction is taken into account, versus We assume a nucleus radius of Mpc and a filtering radius Mpc. Open circles: ; filled circles: .

 Fig. 2. Collapse time of a shell of matter made of galaxies and substructure when dynamical friction is taken into account, versus We assume a nucleus radius of Mpc and a fixed correlation . Open circles: Mpc; filled circles: Mpc; crosses: Mpc.

 Fig. 3. Collapse time versus . We assume a filtering radius Mpc. and a total number of peaks of substructure . Open circles: ; crosses: ; filled circles: .

 Fig. 4. Collapse time of a shell of matter made of galaxies and substructure when dynamical friction is taken into account, versus We assume a filtering radius Mpc and a peaks' height . Open circles: ; filled circles: .

In Figs. 3 and 4 we show how the collapse time varies with the nucleus radius of the protocluster . Note how grows as decreases: the smaller the nucleus of the protocluster, the larger the time of collapse in the presence of dynamical friction; besides we show how this effect increases for larger values of both and .

© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998