Spin parameter and scale-free density perturbations in hierarchical clustering

The present attempt aims to predict the dependence of the spin parameter, , the angular momentum,J, and the typical radius,a _vir, on the mass,M, which have been found inN-body simulations of expanding density perturbations in hierarchical clustering, when virialization is attained. We show that M ⁰ for systems with same adimensional density distribution and velocity distribution, and in particular for scale-free density perturbations in hierarchical clustering. In the special case of ellipsoidal perturbations, it is also found:J M ^7/4,a _vir M ^1/2. All these results turn out to be in close agreement withN-body simulations, despite the simple model adopted. Expanding and virialized perturbations are modelled, respectively, by homogeneous and heterogeneous, similar ellipsoids which allow flat rotation curves far from the centre. Both energy and angular momentum maintain from a given time on, lying between the beginning of strong decoupling from the Hubble flow and the occurrence of maximum volume. Then the whole set of virialized ellipsoidal configurations with same energy and angular momentum are derived, and the dependence of the spin parameter on the anisotropy parameter, _pec is investigated. Turning our attention to the formation of galaxies, we derive the total mass as a function of the collapse factor, using the empirical anticorrelation between dark to visible mass ratio within the optical radius of disk galaxies and their luminous masses. Observational data related to a sample of elliptical galaxies provide evidence that the contraction in these bodies occurred in proportion to the square root of the ratio of total to luminous mass. On the contrary, it is deduced that dissipation of angular momentum in elliptical galaxies occurred more or less at the same rate. If both shape and anisotropy parameter are preserved during the collapse, typical axis rations ₂₁ = 0.98, ₃₁ = 0.69, are found to correspond to a moderate anisotropy, _pec 0.27, with a small dependence on the spin parameter in the range allowed.