Self-similar spherical collapse with non-radial motions

We derive the asymptotic mass profile near the collapse center of an initial spherical density perturbation, $\delta \propto M^{-\epsilon}$, 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 $L={\cal L} \sqrt{GM_* r_*}$ per unit mass, where M_* is the mass interior to r_*. In this scheme, the mass profile is $M\propto r^{3/(1+3\epsilon)}$ for all $\epsilon >0$, in the region $r/r_t\ll {\cal L}$, where r_t is the current turnaround radius. If ${\cal L} \ll 1$ then the profile in the region ${\cal L} \ll r/r_t \ll$ is $M\propto r$ for $\epsilon <2/3$. 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^{n} with k(t) varying adiabatically.