Self-similar spherical collapse with non-radial motions

We derive the asymptotic mass profile near the collapse centre of an initial spherical density perturbation, δ∝M−ε, of collisionless 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∝r3/(1+3ε) for all ε>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≪1 is M∝r for ε<2/3, and remains M∝r3/(1+3ε) for ε≥2/3. The derivation relies on a general property of non-radial orbits which is that the ratio of the pericentre to apocentre is constant in a force field k(t)rn with k(t) varying adiabatically.