Osipkov-Merritt model

Osipkov–Merritt models (named for Leonid Osipkov and David Merritt) are mathematical representations of spherical stellar systems (galaxies, star clusters, globular clusters etc.). The Osipkov-Merritt formula generates a one-parameter family of phase-space distribution functions that reproduce a specified density profile (representing stars) in a specified gravitational potential (in which the stars move). The density and potential need not be self-consistently related. A free parameter adjusts the degree of velocity anisotropy, from isotropic to completely motions. The method is a generalization of Eddington's formula for constructing isotropic spherical models.

The method was derived independently by its two eponymous discoverers. The latter derivation includes two additional families of models (Type IIa, b) with tangentially anisotropic motions.

According to Jeans's theorem, the phase-space density of stars f must be expressible in terms of the isolating integrals of motion, which in a spherical stellar system are the energy E and the angular momentum J. The Osipkov-Merritt ansatz is

where r_a, the "anisotropy radius", is a free parameter. This ansatz implies that f is constant on spheroids in velocity space since

where v_r, v_t are velocity components parallel and perpendicular to the radius vector r and Φ(r) is the gravitational potential.

The density ρ is the integral over velocities of f:

which can be written

or

This equation has the form of an Abel integral equation and can be inverted to give f in terms of ρ:

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