Self-similar collapse with non-radial motions

We discuss the evolved mass profile near the center of an initial spherical density perturbation, delta proportional to M-epsilon, of collision-less particles with non-radial motions. W consider a scheme in which a particle moves on a radial orbit until it reaches its turnaround radius, r(*). At turnaround the particle acquires an angular momentum L = L root GM(*)r(*) per unit mass, where M-* is the mass interior to r(*). In this scheme, the mass profile is M proportional to r(3/(1+3 epsilon)) for all epsilon > 0, in the region r/r(t) << L, where r(t) is the current turnaround radius. If L << 1 then the profile in the region L << r/r(t) << is M proportional to r for epsilon < 2/3. We also present a model for the growth of dark matter halos and use it to study their evolved density profiles. In this model, halos are spherical and form by quiescent accretion of matter in clumps, called satellites. The halo mass as a function of redshift is given by the mass of the most massive progenitor, and is determined from Monte-Carlo realizations of the merger-history tree. Inside the halo, satellites move under the action of the gravitational force of the halo and a dynamical friction drag force. The associated equation of motion is solved numerically. The energy lost to dynamical friction is transferred to the halo in the form of kinetic energy. As they sink into the halo, satellites continually lose matter as a result of tidal stripping. The stripped matter moves inside the for mass scales where the effective spectral index of the initial density field is less than -1, the model predicts a profile which can only approximately be matched by the NFW one parameter family of curves. For scale-free power-spectra with initial slope n, the density profile within about 1% of the virial radius is rho proportional to r(-beta) with 3(3 + n)/(5 + n) <= beta <= 3(3 + n)/(4 + n).