Self-similar spherical collapse with non-radial motions

We derive the asymptotic mass profile near the collapse center of an initial spherical density perturbation, δ ∝ M, of collision-less particles with non-radial motions. We show that angular momenta introduced at the initial time do not affect the mass profile. Alternatively, we 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 √ GM∗r∗ per unit mass, where M∗ is the mass interior to r∗. In this scheme, the mass profile is M ∝ r for all ǫ > 0, in the region r/rt ≪ L, where rt is the current turnaround radius. If L ≪ 1 then the profile in the region L ≪ r/rt ≪ is M ∝ r for ǫ < 2/3. The derivation relies on a general property of non-radial orbits which is that ratio of the pericenter to apocenter is constant in a force field k(t)r with k(t) varying adiabatically.