Osipkov–Merritt model

A free parameter adjusts the degree of velocity anisotropy, from isotropic to completely radial motions.

[2][3] 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 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: Following a derivation similar to the one above, the velocity dispersions in an Osipkov–Merritt model satisfy The motions are nearly radial (

This is a desirable feature, since stellar systems that form via gravitational collapse have isotropic cores and radially-anisotropic envelopes.

This is a consequence of the fact that spherical mass models can not always be reproduced by purely radial orbits.

[3] In his 1985 paper, Merritt defined two additional families of models ("Type II") that have isotropic cores and tangentially anisotropic envelopes.

Both families assume In Type IIa models, the orbits become completely circular at r=ra and remain so at all larger radii.

In Type IIb models, stars beyond ra move on orbits of various eccentricities, although the motion is always biased toward circular.