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

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4. How the collapse time depends on the parameters

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where the parameters on which [FORMULA] depends are the following: [FORMULA]    [FORMULA] is the peaks' height; [FORMULA]    [FORMULA] is the filtering radius; [FORMULA]    [FORMULA] 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:

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where

[EQUATION]

and

[EQUATION]

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

[EQUATION]

with:

[EQUATION]

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

[EQUATION]

with:

[EQUATION]

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

[EQUATION]

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

[EQUATION]

with:

[EQUATION]

The function [FORMULA] is given by:

[EQUATION]

that is:

[EQUATION]

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

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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