Self-similar spherical collapse with non-radial motions

Abstract

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 $L=\mathcal{L}\sqrt{G{M}_{\ast }{r}_{\ast }}$ 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 $r/r_t\ll \mathcal{L}$, where r_t is the current turnaround radius. If $\mathcal{L}\ll 1$ then the profile in the region $\mathcal{L}\ll r/r_t\ll$ 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.

6 pages, 1 figure, two column MNRAS style

References in corpus (2)

• Formation of cuspy density profiles - a generic feature of collisionless gravitational collapse
• Self-similar collapse of collisional gas in an expanding Universe

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