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 .