          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 