## 3. Shell collapse timeOne of the consequences of the angular momentum acquisition by a mass shell of a proto-cluster is the delay of the collapse of the proto-structure. As shown by Barrow & Silk (1981) and Szalay & Silk (1983) the gravitational interaction of the irregular mass distribution of proto-cluster with the neighbouring proto-structures gives rise to non-radial motions, within the protocluster, which are expected to slow the rate of growth of the density contrast and to delay or suppress the collapse. According to Davis & Peebles (1977) the kinetic energy of the resulting non-radial motions at the epoch of maximum expansion increases so much to oppose the recollapse of the proto-structure. Numerical N-body simulations by Villumsen & Davis (1986) showed a tendency to reproduce this so-called previrialization effect. In a more recent paper by Peebles (1990) the slowdown in the growth of density fluctuations and the collapse suppression after the epoch of the maximum expansion were re-obtained using a numerical action method. In the central regions of a density peak () the velocity dispersion attains nearly the same value (Antonuccio-Delogu & Colafrancesco 1995) while at larger radii () the radial component is lower than the tangential component. This means that motions in the outer regions are predominantly non-radial and in these regions the fate of the infalling material could be influenced by the amount of tangential velocity relative to the radial one. This can be shown writing the equation of motion of a spherically symmetric mass distribution with density (Peebles 1993): where and are, respectively, the mean radial and tangential streaming velocity. Eq. (13) shows that high tangential velocity dispersion may alter the infall pattern. The expected delay in the collapse of a perturbation may be calculated solving the equation for the radial acceleration (Peebles 1993): where is the angular momentum and the acceleration. Writing the proper radius of a shell in terms of the expansion parameter, : remembering that and that , where is the Hubble constant and assuming that no shell crossing occurs so that the total mass inside each shell remains constant, that is: or integrating the equation once more: where Eq. (14) or equivalently Eq. (18) may be solved using the initial conditions: , and using the function found in Sect. 2 to obtain the time of collapse, . In Figs. 2 - 4 we compare the results for the time of collapse, , for with the time of collapse of the classical GG spherical model:
As shown the presence of non-radial motions produces an increase in the time of collapse of a spherical shell. The collapse delay is larger for a low value of and becomes negligible for . This result is in agreement with the angular momentum-density anticorrelation effect: density peaks having low value of acquire a larger angular momentum than high peaks and consequently the collapse is more delayed with respect to high peaks. Given we also calculated the total mass gravitationally bound to the final non-linear configuration. There are at least two criteria to establish the bound region to a perturbation : a statistical one (Ryden 1988b), and a dynamical one (Hoffman & Shaham 1985). The dynamical criterion, that we have used, assumes that the binding radius is given by the condition that a mass shell collapse in a time, , smaller than the age of the universe : We calculated the time of collapse of GG spherical model,
, using the density profiles given in Eq. (5)
for and then we repeated the calculation
taking into account non-radial motions obtaining
. Then we calculated the binding radius,
, for a GG model solving
for In Fig. 5 we compare the peak mass obtained from GG model, using Hoffman & Shaham's (1985) criterion, with that obtained from the model taking into account non-radial motions.
As shown for high values of (), the two models give the same result for the mass while for the effect of non-radial motions produces less bound mass with respect to GG model. Before concluding this section we want to discuss the applicability of the spherical model and consequently the use of spherical shells in our model. The question of the applicablity of the idealized spherical collapse model and that of the secondary infall model (SIM) to realistic systems and initial conditions is almost as old as the model itself (GG; Gunn 1977; Filmore & Goldreich 1984; Quinn et al. 1986; Zurek et al. 1988; Quinn & Zurek 1988; Warren et al. 1991; Crone et al. 1994). In a recent paper Zaroubi et al. (1996) investigate the applicability of the quoted model to realistic systems and initial conditions by comparing the results obtained using the spherical model and SIM with that of a set of simulations performed with different N-body codes (Treecode and a "monopole term" code). They introduced a numerical model to trace the evolution of density peaks under the assumption of aspherically symmetric force and realistic initial conditions instead of the spherically symmetric force and initial conditions as assumed in the SIM. They obtained a good agreement between the SIM and the simulations. © European Southern Observatory (ESO) 1998 Online publication: August 6, 1998 |