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Plummer model

Mathematical model in astronomical systems

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Source updated November 11, 2025

The Plummer model or Plummer sphere is a density law that was first used by H. C. Plummer to fit observations of globular clusters. It is now often used as toy model in N-body simulations of stellar systems.

01Description of the model

The Plummer 3-dimensional density profile is given by $\rho _{P}(r)={\frac {3M_{0}}{4\pi a^{3}}}\left(1+{\frac {r^{2}}{a^{2}}}\right)^{-{5}/{2}}={\frac {3M_{0}a^{2}}{4\pi (a^{2}+r^{2})^{{5}/{2}}}},$ where $M_{0}$ is the total mass of the cluster, and a is the Plummer radius, a scale parameter that sets the size of the cluster core. The corresponding potential is $\Phi _{P}(r)=-{\frac {GM_{0}}{\sqrt {r^{2}+a^{2}}}},$ where G is Newton's gravitational constant. The velocity dispersion is $\sigma _{P}^{2}(r)={\frac {GM_{0}}{6{\sqrt {r^{2}+a^{2}}}}}.$

The isotropic distribution function reads $f({\vec {x}},{\vec {v}})={\frac {24{\sqrt {2}}}{7\pi ^{3}}}{\frac {a^{2}}{G^{5}M_{0}^{4}}}(-E({\vec {x}},{\vec {v}}))^{7/2},$ if $E<0$ , and $f({\vec {x}},{\vec {v}})=0$ otherwise, where ${\textstyle E({\vec {x}},{\vec {v}})={\frac {1}{2}}v^{2}+\Phi _{P}(r)$ is the specific energy.

02Properties

The mass enclosed within radius $r$ is given by $M(<r)=4\pi \int _{0}^{r}r'^{2}\rho _{P}(r')\,dr'=M_{0}{\frac {r^{3}}{(r^{2}+a^{2})^{3/2}}}.$

Many other properties of the Plummer model are described in Herwig Dejonghe's comprehensive article.

Core radius $r_{c}$ , where the surface density drops to half its central value, is at ${\textstyle r_{c}=a{\sqrt {{\sqrt {2}}-1}}\approx 0.64a$ .

Half-mass radius is $r_{h}=\left({\frac {1}{0.5^{2/3}}}-1\right)^{-0.5}a\approx 1.3a.$

Virial radius is $r_{V}={\frac {16}{3\pi }}a\approx 1.7a$ .

The 2D surface density is: $\Sigma (R)=\int _{-\infty }^{\infty }\rho (r(z))dz=2\int _{0}^{\infty }{\frac {3a^{2}M_{0}dz}{4\pi (a^{2}+z^{2}+R^{2})^{5/2}}}={\frac {M_{0}a^{2}}{\pi (a^{2}+R^{2})^{2}}},$ and hence the 2D projected mass profile is: $M(R)=2\pi \int _{0}^{R}\Sigma (R')\,R'dR'=M_{0}{\frac {R^{2}}{a^{2}+R^{2}}}.$

In astronomy, it is convenient to define 2D half-mass radius which is the radius where the 2D projected mass profile is half of the total mass: $M(R_{1/2})=M_{0}/2$ .

For the Plummer profile: $R_{1/2}=a$ .

The escape velocity at any point is $v_{\rm {esc}}(r)={\sqrt {-2\Phi (r)}}={\sqrt {12}}\,\sigma (r),$

For bound orbits, the radial turning points of the orbit is characterized by specific energy ${\textstyle E={\frac {1}{2}}v^{2}+\Phi (r)$ and specific angular momentum $L=|{\vec {r}}\times {\vec {v}}|$ are given by the positive roots of the cubic equation $R^{3}+{\frac {GM_{0}}{E}}R^{2}-\left({\frac {L^{2}}{2E}}+a^{2}\right)R-{\frac {GM_{0}a^{2}}{E}}=0,$ where $R={\sqrt {r^{2}+a^{2}}}$ , so that $r={\sqrt {R^{2}-a^{2}}}$ . This equation has three real roots for $R$ : two positive and one negative, given that $L<L_{c}(E)$ , where $L_{c}(E)$ is the specific angular momentum for a circular orbit for the same energy. Here $L_{c}$ can be calculated from single real root of the discriminant of the cubic equation, which is itself another cubic equation ${\underline {E}}\,{\underline {L}}_{c}^{3}+\left(6{\underline {E}}^{2}{\underline {a}}^{2}+{\frac {1}{2}}\right){\underline {L}}_{c}^{2}+\left(12{\underline {E}}^{3}{\underline {a}}^{4}+20{\underline {E}}{\underline {a}}^{2}\right){\underline {L}}_{c}+\left(8{\underline {E}}^{4}{\underline {a}}^{6}-16{\underline {E}}^{2}{\underline {a}}^{4}+8{\underline {a}}^{2}\right)=0,$ where underlined parameters are dimensionless in Henon units defined as ${\underline {E}}=Er_{V}/(GM_{0})$ , ${\underline {L}}_{c}=L_{c}/{\sqrt {GMr_{V}}}$ , and ${\underline {a}}=a/r_{V}=3\pi /16$ .

03Applications

The Plummer model comes closest to representing the observed density profiles of star clusters, although the rapid falloff of the density at large radii ( $\rho \rightarrow r^{-5}$ ) 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 experimenters, in spite of the model's lack of realism.

Watch videos about Plummer modelExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Plummer model”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Plummer rho.png by Unknown author, Public domain

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