Statistical mechanics of collisionless orbits. IV. Distribution of
  angular momentum
  It has been shown in previous work that DARKexp, which is a theoretically
derived, maximum entropy, one shape parameter model for isotropic collisionless
systems, provides very good fits to simulated and observed dark-matter halos.
Specifically, it fits the energy distribution, N(E), and the density profiles,
including the central cusp. Here, we extend DARKexp N(E) to include the
distribution in angular momentum, L^2, for spherically symmetric systems.
First, we argue, based on theoretical, semi-analytical, and simulation results,
that while dark-matter halos are relaxed in energy, they are not nearly as
relaxed in angular momentum, which precludes using maximum entropy to uniquely
derive N(E,L^2). Instead, we require that when integrating N(E,L^2) over
squared angular momenta one retrieves the DARKexp N(E). Starting with a general
expression for N(E,L^2) we show how the distribution of particles in L^2 is
related to the shape of the velocity distribution function, VDF, and velocity
anisotropy profile, \beta(r). We then demonstrate that astrophysically
realistic halos, as judged by the VDF shape and \beta(r), must have linear or
convex distributions in L^2, for each separate energy bin. The distribution in
energy of the most bound particles must be nearly flat, and become more tilted
in favor of radial orbits for less bound particles. These results are
consistent with numerical simulations and represent an important step towards
deriving the full distribution function for spherically symmetric dark-matter
halos.
