Analytical relations concerning the collapse time in hierarchically clustered cosmological models

By means of numerical methods, we solve the equations of motion for the collapse of a shell of baryonic matter, made of galaxies and substructure falling into the central regions of a cluster of galaxies, taking into account the effect of the dynamical friction. The parameters on which the dynamical friction mainly depends are: the peaks' height, the number of peaks inside a protocluster multiplied by the correlation function evaluated at the origin, the filtering radius and the nucleus radius of the protocluster of galaxies. We show how the collapse time (Tau) of the shell depends on these parameters. We give a formula that links the dynamical friction coefficient (Eta) o the parameters mentioned above and an analytic relation between the collapse time and (Eta). Finally, we obtain an analytical relation between (Tau) and the mean overdensity (mean Delta) within the shell. All the analytical relations that we find are in excellent agreement with the numerical integration.