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

6. Conclusions and discussion

In Sect. 2 of this work we showed in a quantitative way how the collapse time of a shell of baryonic matter made of galaxies and substructure depends on some parameters. When one of the parameters or or increases, the collapse time grows. It means that the effects of the presence of dynamical friction should be more evident in the outer regions of rich clusters of galaxies. Besides, we show how the collapse time of an infalling shell increases with decreasing values of , and becomes very large for Mpc (see Fig. 4). As a consequence, the slowing down of the collapse of an outer shell within a cluster of galaxies owing to the dynamical friction is more remarkable in the clusters with nucleus of little dimension.

In Sect. 3 of this paper we give an analytical relation that links the dimensionless collapse time with the coefficient of dynamical friction . This relation is in excellent agreement with the numerical integration of Eq. (7) for (see Fig. 5). Then, we find an analytical relation between and the parameters on which it depends for the value (we remind that the effects of the dynamical friction are negligible for ). This is Eq. (28), that can be considered as a "low order" approximation to a more realistic situation of an outer shell of a cluster of galaxies with . We also find an empirical formula, Eq. (34), that is a good approximation of Eq. (33). Moreover, the dependence of the dimensionless collapse time on , , and is shown (Eq. (35)).
Finally, we give an analytical relation that links with and . Here, we wish to stress the usefulness of an analytical relation like . This is a powerful tool to estimate the effect of the dynamical friction in the outer regions of clusters of galaxies (Gambera et al. in preparation) and to compare the observational data with the theoretical ones. It is a good method to test how important the role of the dynamical friction is in the collapse of the clusters of galaxies.

© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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