Phase-space consistency of stellar dynamical models determined by separable augmented densities

Details Phase-space consistency of stellar dynamical models determined by separable augmented densities Phase-space consistency of stellar dynamical models determined by separable augmented densities

2012-01-31

arxiv.org/abs/1202.0004v1

Mon. Not. R. Astron. Soc. 422, 652-664 (2012)

Astronomy and Astrophysics

Astrophysics

Cosmology and Extragalactic Astrophysics

accepted for the publication in MNRAS, 12 journal pages (7pp in main body + 4 appendices)

Scientific paper

10.1111/j.1365-2966.2012.20642.x

Assuming the separable augmented density, it is always possible to construct a distribution function of a spherical population with any given density and anisotropy. We consider under what conditions the distribution constructed as such is in fact non-negative everywhere in the accessible phase-space. We first generalize known necessary conditions on the augmented density using fractional calculus. The condition on the radius part R(r^2) (whose logarithmic derivative is the anisotropy parameter) is equivalent to the complete monotonicity of R(1/w)/w. The condition on the potential part on the other hand is given by its derivative up to any order not greater than (3/2-beta) being non-negative where beta is the central anisotropy parameter. We also derive a specialized inversion formula for the distribution from the separable augmented density, which leads to sufficient conditions on separable augmented densities for the non-negativity of the distribution. The last generalizes the similar condition derived earlier for the generalized Cuddeford system to arbitrary separable systems.

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