Non-radial motion and the NFW profile

The self-similar infall model (SSIM) is normally discussed in the context of radial orbits in spherical symmetry. However it is possible to retain the spherical symmetry while permitting the particles to move in Keplerian ellipses, each having the squared angular momentum peculiar to their "shell". The spherical "shell", defined for example by the particles turning at a given radius, then moves according to the radial equation of motion of a "shell" particle. The "shell" itself has no physical existence except as an ensemble of particles, but it is convenient to sometimes refer to the shells since it is they that are followed by a shell code. In this note we find the distribution of squared angular momentum as a function of radius that yields the NFW density profile for the final dark matter halo. It transpires that this distribution is amply motivated dimensionally. An effective "lambda" spin parameter is roughly constant over the shells. We also study the effects of angular momentum on the relaxation of a dark matter system using a three dimensional representation of the relaxed phase space.