 
Astron. Astrophys. 334, 381387 (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 socalled 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. 14. 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
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