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

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2. The collapse time

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

[EQUATION]

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

[EQUATION]

with:

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

following intervals:

[EQUATION]

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 [FORMULA] on [FORMULA] is qualitatively shown in Fig. 5 by AC94. We observe that for [FORMULA] 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 [FORMULA], whilst for [FORMULA] the deviations increase steeply with lower [FORMULA]. Then, having considered [FORMULA], the estimate we get for [FORMULA] 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 [FORMULA]. In this picture, we show how [FORMULA] grows for larger values of [FORMULA] and for larger values of [FORMULA]. Similarly, in Fig. 2 we note how [FORMULA] increases for larger values of [FORMULA] and of [FORMULA]. 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 [FORMULA], [FORMULA], [FORMULA] grow.
Here we want to remind you that we are considering only the peaks of the local density field with central height [FORMULA] larger than a critical threshold [FORMULA]. This latter quantity is chosen to be the threshold at which [FORMULA] ([FORMULA]) [FORMULA] [FORMULA] where [FORMULA] is the typical size of the peaks and [FORMULA] is the average peak separation (see also Bardeen et al. 1986).


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

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

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

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

In Figs. 3 and 4 we show how the collapse time varies with the nucleus radius of the protocluster [FORMULA]. Note how [FORMULA] grows as [FORMULA] 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 [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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