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[[Image:Osipkov-Merritt-Models.jpg|thumb|200px|Osipkov-Merritt distribution functions, derived from galaxy models obeying [[Jaffe's law]] in the density. The isotropic model, <math>f=f(E)</math>, is plotted with the heavy line.]]


'''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 function]]s 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 [[wikt:radial|radial]] motions.  The method is a generalization of [[Eddington's formula]]<ref>[[Arthur Eddington|Eddington, A.]] (1916), [http://adsabs.harvard.edu/abs/1916MNRAS..76..572E  The distribution of stars in globular clusters,] ''Mon. Not. R. Astron. Soc.'', '''76''', 572</ref>  for constructing isotropic spherical models.


The method was derived independently by its two eponymous discoverers.<ref>Osipkov, L. P. (1979), [http://adsabs.harvard.edu/abs/1979PAZh....5...77O  Spherical systems of gravitating bodies with an ellipsoidal velocity distribution,] ''Pis'ma v Astron. Zhur.'', '''5''', 77</ref><ref name="Merritt">[[David Merritt|Merritt, D.]] (1985), [http://adsabs.harvard.edu/abs/1985AJ.....90.1027M  Spherical stellar systems with spheroidal velocity distributions,] ''Astron. J.'', '''90''', 1027</ref> The latter derivation includes two additional families of models (Type IIa, b) with tangentially anisotropic motions.
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==Derivation==
 
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
 
:<math>f = f(Q) = f(E+J^2/2r_a^2)</math>
 
where ''r<sub>a</sub>'', the "anisotropy radius", is a free parameter. This ''ansatz'' implies that ''f'' is constant on spheroids in velocity space since
:<math>
2Q = v_r^2 + (1+r^2/r_a^2)v_t^2 + 2\Phi(r)
</math>
where ''v''<sub>r</sub>, ''v''<sub>t</sub> 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'':
:<math>
\rho(r) = 2\pi\int\int f(E,J) v_t dv_t dv_r
</math>
which can be written
:<math>
\rho(r) = {2\pi\over r^2} \int_\Phi^0 dQ f(Q) \int_0^{2r^2(Q-\Phi)/(1+r^2/r_a^2)} dJ^2\left[2(Q-\Phi)-(J^2/r^2)(1+r^2/r_a^2)\right]^{-1/2}
</math>
 
or
 
:<math>
\rho(r) = {4\pi\over 1+r^2/r_a^2} \int_\Phi^0 dQ \sqrt{2(Q-\Phi)}f(Q).
</math>
 
This equation has the form of an [[Abel integral equation]] and can be inverted to give ''f'' in terms of ''ρ'':
 
:<math>
 
f(Q) = {\sqrt{2}\over 4\pi^2} {d\over dQ} \int_Q^0 {d\Phi\over\sqrt{\Phi-Q}} {d\rho^'\over d\Phi},\ \ \ \ \ \rho^'(\Phi) = \left[1+r(\Phi)^2/r_a^2\right]\rho\left[r(\Phi)\right].
</math>
 
==Properties==
 
Following a derivation similar to the one above, the velocity dispersions in an Osipkov–Merritt model satisfy
 
:<math>
{\sigma_r^2\over\sigma_t^2} = 1 + {r^2\over r_a^2}.
</math>
 
The motions are nearly radial (<math>\sigma_r\gg\sigma_t</math>) for <math>r\gg r_a</math> and nearly isotropic (<math>\sigma_r\approx\sigma_t</math>) for <math>r\ll r_a</math>. This is a desirable feature, since stellar systems that form via [[gravitational collapse]] have isotropic cores and radially-anisotropic envelopes.<ref>van Albada, T. (1983), [http://adsabs.harvard.edu/abs/1982MNRAS.201..939V  Dissipationless galaxy formation and the R to the 1/4-power law,] ''Mon. Not. R. Astron. Soc.'', '''201''', 939</ref>  
 
If ''r<sub>a</sub>'' 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<sub>a</sub>'' that generate negative ''f'''s are unphysical. This result can be used to constrain the maximum degree of anisotropy of spherical galaxy models.<ref name=Merritt/>
 
In his 1985 paper, Merritt defined two additional families of models ("Type II") that have isotropic cores and tangentially anisotropic envelopes. Both families assume
 
:<math>f = f(E-J^2/2r_a^2)</math>.
 
In Type IIa models, the orbits become completely circular at ''r=r<sub>a</sub>'' and remain so at all larger radii.
In Type IIb models, stars beyond ''r<sub>a</sub>'' 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<sub>a</sub>''.
 
Carollo ''et al.'' (1995)<ref>Carollo, C. M. ''et al.'' (1995), [http://adsabs.harvard.edu/abs/1995MNRAS.276.1131C  Velocity profiles of Osipkov-Merritt models,] ''Mon. Not. R. Astron. Soc.'', '''276''', 1131</ref> derive many observable properties of Type I Osipkov–Merritt models.
 
==Applications==
 
Typical applications of Osipkov–Merritt models include:
 
-- modelling of [[star cluster]]s,<ref>Lupton, R.'' et al.'' (1989), [http://adsabs.harvard.edu/abs/1989ApJ...347..201L  The internal velocity dispersions of three young star clusters in the Large Magellanic Cloud,] ''Astrophys. J.'', '''347''', 201</ref> [[galaxies]],<ref>Nolthenius, R. and [[Holland Ford|Ford, H.]] (1987), [http://adsabs.harvard.edu/abs/1986ApJ...305..600N  The mass and halo dispersion profile of M32,] ''Astrophys. J.'', '''305''', 600</ref> [[dark matter halo]]s <ref>Sotnikova, N. Ya. and Rodionov, S. A. (2008), [http://adsabs.harvard.edu/abs/2008arXiv0809.3946S  Anisotropic Models of Dark Halos,] ''Astron. Lett.'', '''34''', 664-674</ref> and [[galaxy cluster]]s;<ref>Lokas, E. and Mamon, G. (2001), [http://adsabs.harvard.edu/abs/2001MNRAS.321..155L  Properties of spherical galaxies and clusters with an NFW density profile,] ''Mon. Not. R. Astron. Soc.'', '''321''', 155</ref>
 
-- constructing anisotropic galaxy models for studies of dynamical [[Instability#Instabilities of stellar systems|instabilities]].<ref>May, A. and [[James Binney|Binney, J.]] (1986), [http://adsabs.harvard.edu/abs/1986MNRAS.221P..13M  Testing the stability of stellar systems,] ''Mon. Not. R. Astron. Soc.'', '''221''', 13</ref><ref>Saha, P. (1991), [http://adsabs.harvard.edu/abs/1991MNRAS.248..494S  Unstable modes of a spherical stellar system,] ''Mon. Not. R. Astron. Soc.'', '''248''', 494</ref>
 
==See also==
 
* [[Stellar dynamics]]
 
== References ==
 
{{reflist|2}}
 
{{DEFAULTSORT:Osipkov-Merritt model}}
[[Category:Astrophysics]]

Revision as of 00:14, 11 February 2014


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