Plummer model: Difference between revisions

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The '''Plummer model''' or '''Plummer's modelsphere''' is a density profilelaw and potential pair. Itthat was first used by [[H. C. Plummer (1911)]] to fit the 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 a stellar distribution[[toy model]] in simulations[[N-body simulation]]s of stellar systems.

== Description of the model ==

[[Image:Plummer_rho.png|thumb|right|220px|The density law of a Plummer model]]

The Plummer density profile for Plummer model 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 densitysize of the cluster core. The corresponding potential is then

:<math> \Phi_P(r) = -\frac{G M}{\sqrt{r^2+a^2}}\,,</math>

where ''G'' is [[Isaac Newton|Newton]]'s [[gravitational constant]].

== Properties ==

The Plummer density distribution can be used as an analytical [[toy model]] either for globular clusters or galaxies. The total mass of the model, given by

:<math>M = \int 4\pi r^2 \rho_P(r) dr</math>

is finite. It is decrescent at large distances from the center, <math>\rho_P \sim r^{-5}</math> (<math>r \gg a</math>), and is constant near the center, <math>\rho_P \sim \mathrm{constant}</math> (<math>r \ll a</math>).

The mass enclosed within radius <math>r</math> is given by

The behaviour near the center does not match observations of elliptical galaxies, which show a density divergence all the way to the [[Hubble Space Telescope]] resolution. This makes the Plummer density law a rather poor description of them.

:<math>M(<r) = \int 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>

== 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 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}}

===General resources===

* NASA Astrophysics Data System (http://adswww.harvard.edu/) has a collection of past articles, from all major astrophysics journals and many conference proceedings.

===Books===

* Binney, James; Tremaine, Scott (1987). ''Galactic Dynamics'', Princeton University Press, Princeton, New Jersey.

* Heggie, Douglas; Hut, Piet (2003). ''The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics'', Cambridge University Press.

===Articles===

* Aarseth, S. J.; Henon, M.; Wielen, R. (1974). A comparison of numerical methods for the study of star cluster dynamics. ''Astronomy and Astrophysics'' '''37''' 183. [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1974A%26A....37..183A NASA ADS] (This article has an appendix on how to create a Plummer model)

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