 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 OsipkovMerritt formula generates a oneparameter family of phasespace 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 selfconsistently related. A free parameter adjusts the degree of velocity anisotropy, from isotropic to completely radial motions. The method is a generalization of Eddington's formula^{[1]} for constructing isotropic spherical models.
The method was derived independently by its two eponymous discoverers.^{[2]}^{[3]} The latter derivation includes two additional families of models (Type IIa, b) with tangentially anisotropic motions.
Contents
Derivation
According to Jeans's theorem, the phasespace 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 OsipkovMerritt 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:
Properties
Following a derivation similar to the one above, the velocity dispersions in an OsipkovMerritt model satisfy
The motions are nearly radial () for and nearly isotropic () for . This is a desirable feature, since stellar systems that form via gravitational collapse have isotropic cores and radiallyanisotropic envelopes.^{[4]}
If r_{a} is assigned too small a value, f may be negative for some Q. This is a consequence of the fact that spherical mass models can not always be reproduced by purely radial orbits. Since the number of stars on an orbit can not be negative, values of r_{a} that generate negative f's are unphysical. This result can be used to constrain the maximum degree of anisotropy of spherical galaxy models.^{[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 = r_{a} and remain so at all larger radii. In Type IIb models, stars beyond r_{a} move on orbits of various eccentricities, although the motion is always biased toward circular. In both families, the tangential velocity dispersion undergoes a jump as r increases past r_{a}.
Carollo et al. (1995)^{[5]} derive many observable properties of Type I OsipkovMerritt models.
Applications
Typical applications of OsipkovMerritt models include:
 modelling of star clusters, ^{[6]} galaxies,^{[7]} dark matter halos ^{[8]}and galaxy clusters; ^{[9]}
 constructing anisotropic galaxy models for studies of dynamical instabilities.^{[10]}^{[11]}
See also
References
 ^ Eddington, A. (1916), The distribution of stars in globular clusters, Mon. Not. R. Astron. Soc., 76, 572
 ^ Osipkov, L. P. (1979), Spherical systems of gravitating bodies with an ellipsoidal velocity distribution, Pis'ma v Astron. Zhur., 5, 77
 ^ ^{a} ^{b} Merritt, D. (1985), Spherical stellar systems with spheroidal velocity distributions, Astron. J., 90, 1027
 ^ van Albada, T. (1983), Dissipationless galaxy formation and the R to the 1/4power law, Mon. Not. R. Astron. Soc., 201, 939
 ^ Carollo, C. M. et al. (1995), Velocity profiles of OsipkovMerritt models, Mon. Not. R. Astron. Soc., 276, 1131
 ^ Lupton, R. et al. (1989), The internal velocity dispersions of three young star clusters in the Large Magellanic Cloud, Astrophys. J., 347, 201
 ^ Nolthenius, R. and Ford, H. (1987), The mass and halo dispersion profile of M32, Astrophys. J., 305, 600
 ^ Sotnikova, N. Ya. and Rodionov, S. A. (2008), Anisotropic Models of Dark Halos, Astron. Lett., 34, 664674
 ^ Lokas, E. and Mamon, G. (2001), Properties of spherical galaxies and clusters with an NFW density profile, Mon. Not. R. Astron. Soc., 321, 155
 ^ May, A. and Binney, J. (1986), Testing the stability of stellar systems, Mon. Not. R. Astron. Soc., 221, 13
 ^ Saha, P. (1991), Unstable modes of a spherical stellar system, Mon. Not. R. Astron. Soc., 248, 494
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