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

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1. Introduction

The problem of the formation and evolution of clusters of galaxies has been one of the crucial topics of the last years (see e.g. Ryden & Gunn 1987, Colafrancesco et al. 1989, Antonuccio-Delogu 1992, Kaiser 1993, Colafrancesco & Vittorio 1993, Croft & Efstathiou 1994, Dutta & Spergel 1994, Mosconi et al. 1994, Sutherland & Dalton 1994, Efstathiou 1994 and Colafrancesco et al. 1995). It is well known that the formation of cosmic structures is strictly related to the evolution of the density perturbations: in the present paradigm of structure formation, it is generally assumed that cosmic structures of size [FORMULA] R form preferentially around the local maxima of the primordial density field, once it is smoothed on the filtering scale [FORMULA]. These linear density fluctuations eventually evolve towards the nonlinear regime under the action of gravitational instability; they detach from the Hubble flow at turn around epoch [FORMULA], given by:

[EQUATION]

where [FORMULA] is the mean background density, z is the redshift and [FORMULA] is the mean overdensity within the nonlinear region. After the turn around epoch, the fluctuations start to recollapse when their overdensity defined by

[EQUATION]

reaches the value [FORMULA].
The evolution of the density fluctuations is described by the power spectrum, given by:

[EQUATION]

with:

[EQUATION]

Since the density field depends on the power spectrum, which in turn depends on the matter that dominates the universe, the mean characteristics of the cosmic structures depend on the assumed model. In this context the most successful model is the biased Cold Dark Matter (hereafter CDM) (see e.g. Kolb & Turner 1990; Peebles 1993; Liddle & Lyth 1993) based on a scale invariant spectrum of density fluctuations growing under gravitational instability. In such a scenario the formation of the structures occurs through a "bottom up " mechanism. A simple model that describes the collapse of a density perturbation is that by Gunn & Gott (1972, hereafter GG72). The main assumptions of this model are: (a) the symmetry of the collapse is spherical; (b) the matter distribution is uniform in that region of space where the density exceeds the background density; (c) no tidal interaction exists with the external density perturbations and (d) there is no substructure (collapsed objects having sizes less than that of main perturbation).

Point (d) is in contradiction to the predictions of CDM models. It is well known that in a CDM Universe, an abundant production of substructures during the evolution of the fluctuations is predicted.

The problem of the substructures in a CDM Universe and its consequences for structure formation have been widely studied in previous papers (see e.g. Antonuccio-Delogu 1992, hereafter A92; Antonuccio-Delogu & Atrio-Barandela 1992, hereafter AA92; Antonuccio-Delogu & Colafrancesco 1994, hereafter AC94, Del Popolo et al. 1996, Gambera 1997 and Del Popolo & Gambera 1997 hereafter DG97). Being a rather recent topic in cosmology much work still has to be done.
The presence of substructure is very important for the dynamics of collapsing shells of baryonic matter made of galaxies and substructure of [FORMULA], falling into the central regions of a cluster of galaxies. As shown by Chandrasekhar & von Neumann (1942, hereafter CVN42; 1943), in the presence of substructure it is necessary to modify the equation of motion:

[EQUATION]

since the graininess of mass distribution in the system induces dynamical friction that introduces a frictional force term.
Adopting the notation of GG72 (see also their Eqs. 6 and 8) and remembering that [FORMULA] is the collapse time of a shell of baryonic matter in the absence of dynamical friction (GG72), one can write:

[EQUATION]

where [FORMULA] is the critical density at a time [FORMULA] and [FORMULA] is the average density inside [FORMULA] at [FORMULA]. The equation of motion of a shell of baryonic matter in presence of dynamical friction (Kandrup 1980, hereafter K80; Kashlinsky 1986 and AC94), using the dimensionless time variable [FORMULA], can be written in the form:

[EQUATION]

where [FORMULA] is the overdensity within [FORMULA], [FORMULA] is the coefficient of dynamical friction and [FORMULA] is the expansion parameter of the shell (see GG72 Eq. 6), that can be written as:

[EQUATION]

Supposing that there is no correlation among random force and their derivatives, we have:

[EQUATION]

(K80), where [FORMULA] is the average "duration" of a random force impulse of magnitude F, [FORMULA] is the probability distribution of stochastic force (which for a clustered system is given in Eq. 37 of AA92).
DG97 solved Eq. (7) numerically, showing qualitatively how the expansion parameter [FORMULA] depends on the dynamical friction coefficient and how [FORMULA] changes in the presence of dynamical friction, but without undertaking a more complete study of the dependence on the parameters.
The plan of this paper is as follows. In Sect. 2 we show how [FORMULA] depends on the peaks' height [FORMULA], on the parameter [FORMULA] that we define there as the correlation function at the origin multiplied by the total number of peaks inside a protocluster, on the filtering radius [FORMULA] and on the nucleus radius of the protocluster [FORMULA]. A more detailed description of these parameters is given below. In Sect. 3 we give an analytical relation between the dynamical friction coefficient and the collapse time:

[EQUATION]

In Sect. 4 we give a semi-analytical relation that links the dynamical friction coefficient with the parameters on which it depends:

[EQUATION]

Linking these two relations we also find the dependence of the dimensionless collapse time [FORMULA] on the parameters used:

[EQUATION]

In Sect. 5 we give a semi-analytical relation between [FORMULA] and [FORMULA] and [FORMULA]:

[EQUATION]

Finally, in Sect. 6 we summarize our results and comment on their possible implications.

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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