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

4. How the collapse time depends on the parameters

As already shown by AC94, it is possible to write down the dynamical friction coefficient as the sum of two terms:

where is the coefficient of dynamical friction of an unclustered distribution of field particles whilst takes into account the effect of clustering. We rewrite Eq. (22) as:

where, as demonstrated by A92, the ratio depends only on and whilst is given by A92:

and, for a fixed value of , depends only on and . In the previous formula, is defined in Bower (1991), whilst and are defined in AC94. Therefore, we rewrite Eq. (23) as:

where the parameters on which depends are the following:    is the peaks' height;    is the filtering radius;    is the parameter of the power-law density profile. A theoretical work (Ryden 1988) suggests that the density profile inside a protogalactic dark matter halo, before relaxation and baryonic infall, can be approximated by a power-law:

where on a protogalactic scale.    is the product where is the total number of peaks inside a protocluster and is the correlation function calculated in . AA92 have demonstrated that in the hypothesis , where is the average mass of the subpeaks, the dependence of the dynamical friction coefficient on and may be expressed as a dependence on a single parameter that we define as:

The analytical relation , that links the dynamical friction coefficient with the parameters on which it depends, can be rewritten as the product of two functions:

where

and

With a least square method, we find the best function . First, we find an analytical relation between the dynamical friction coefficient in the absence of clustering and the parameters and . We obtain:

with:

We perform a test between the values of obtained from the last equation and the values of calculated integrating Eq. (24). Our result for the range is excellent: .
We do the same job for the quantity , finding:

with:

An analogue test gives for the range .
The function is given by the product of Eqs. (31) and (32). These contain 54 terms.

However, we have also found an empirical formula with only 13 terms that represents a good approximation. We performed a test between the results obtained from the 13-term equation and the results obtained from the numerical integration. The result is . The same test performed on the 54-term equation gives . Our 13-term equation reads as:

with:

The function is given by:

that is:

where the values of are given in Sect. 2 and the function is given by Eq. (33) or by Eq. (34).

© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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