*Astron. Astrophys. 334, 381-387 (1998)*
## 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 R form
preferentially around the local maxima of the primordial density
field, once it is smoothed on the filtering scale
. 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 , given by:
where is the mean background density,
*z* is the redshift and is the mean
overdensity within the nonlinear region. After the *turn around*
epoch, the fluctuations start to recollapse when their overdensity
defined by
reaches the value .
The evolution of the density fluctuations is described by the power
spectrum, given by:
with:
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 , 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:
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 is the
collapse time of a shell of baryonic matter in the absence of
dynamical friction (GG72), one can write:
where is the critical density at a time
and is the *average
density* inside at .
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 , can be written in the form:
where is the overdensity within
, is the coefficient of
dynamical friction and is the expansion
parameter of the shell (see GG72 Eq. 6), that can
be written as:
Supposing that there is no correlation among random force and their
derivatives, we have:
(K80), where is the average
*"duration"* of a random force impulse of magnitude *F*,
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
depends on the dynamical friction coefficient and how
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
depends on the peaks' height
, on the parameter 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 and on the nucleus radius of the
protocluster . 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:
In Sect. 4 we give a semi-analytical relation that links the
dynamical friction coefficient with the parameters on which it
depends:
Linking these two relations we also find the dependence of the
dimensionless collapse time on the parameters
used:
In Sect. 5 we give a semi-analytical relation between
and and
:
Finally, in Sect. 6 we summarize our results and comment on their
possible implications.
© European Southern Observatory (ESO) 1998
Online publication: May 15, 1998
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