Self-similar spherical collapse with non-radial motions

We derive the asymptotic mass profile near the collapse centre of an initial spherical density perturbation, delta proportional toM(-epsilon), 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 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) much less than L, where r(t) is the current turnaround radius. If L much less than 1 then the profile in the region L much less than r/r(t) much less than 1 is M proportional to r for epsilon < 2/3, and remains M proportional to r(3/(1 + 3 epsilon)) for epsilon greater than or equal to 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)r(n) with k(t) varying adiabatically.