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The '''Plummer model''' or '''Plummer sphere''' is a density law that was first used by [[Henry_Crozier_Keating_Plummer|H. C. Plummer]] to fit observations of [[globular cluster]]s.<ref>Plummer, H. C. (1911), [http://adsabs.harvard.edu/abs/1911MNRAS..71..460P On the problem of distribution in globular star clusters], ''Mon. Not. R. Astron. Soc.'' '''71''', 460</ref> It is now often used as [[toy model]] in [[N-body simulation]]s of stellar systems.
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== Description of the model ==
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[[Image:Plummer_rho.png|thumb|right|220px|The density law of a Plummer model]]
 
The Plummer 3-dimensional density profile is given by
 
:<math>\rho_P(r) = \bigg(\frac{3M}{4\pi a^3}\bigg)\bigg(1+\frac{r^2}{a^2}\bigg)^{-\frac{5}{2}}\,,</math>
 
where ''M'' is the total mass of the cluster, and ''a'' is the '''Plummer radius''', a scale parameter which sets the size of the cluster core. The corresponding potential is
 
:<math> \Phi_P(r) = -\frac{G M}{\sqrt{r^2+a^2}}\,,</math>
 
where ''G'' is [[Isaac Newton|Newton]]'s [[gravitational constant]].
 
== Properties ==
 
The mass enclosed within radius <math>r</math> is given by
:<math>M(<r) = 4\pi\int_0^r r^2 \rho_P(r) dr = M{r^3\over\left(r^2+a^2\right)^{3/2}}</math>.
 
Many other properties of the Plummer model are described in [[Herwig Dejonghe]]'s comprehensive paper.<ref>Dejonghe, H. (1987), [http://adsabs.harvard.edu/abs/1987MNRAS.224...13D A completely analytical family of anisotropic Plummer models]. ''Mon. Not. R. Astron. Soc.'' '''224''', 13 </ref>
 
Core radius <math>r_c</math>, where the surface density drops to half its central value, is at <math>r_c = \frac{a}{\sqrt{2}} \approx 0.7 a</math>.
 
[[Half-mass radius]] is <math>r_h \approx 1.3 a</math>
 
[[Virial radius]] is <math>r_V = \frac{16}{3 \pi} a \approx 1.7 a</math>
 
See also '''The Art of Computational Science'''<ref>P.Hut and J.Makino. The Art of Computational Science [http://www.artcompsci.org/kali/vol/plummer/ch09.html]</ref>
 
== Applications ==
The Plummer model comes closest to representing the observed density profiles of [[star clusters]], although the rapid falloff of the density at large radii (<math>\rho\rightarrow r^{-5}</math>) is not a good description of these systems.
 
The behavior of the density near the center does not match observations of elliptical galaxies, which typically exhibit a diverging central density.
 
The ease with which the Plummer sphere can be realized as a [[Monte Carlo method|Monte-Carlo model]] has made it a favorite choice of [[N-body simulation|N-body experimenters]], in spite of the model's lack of realism.<ref>Aarseth, S. J., Henon, M. and Wielen, R. (1974), [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1974A%26A....37..183A A comparison of numerical methods for the study of star cluster dynamics.] ''Astronomy and Astrophysics'' '''37''' 183.</ref>
 
==References==
{{reflist}}
 
 
{{Physics-stub}}
[[Category:Astrophysics]]

Latest revision as of 23:29, 3 December 2014

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