Plummer model

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.

Description of the model

The density law of a Plummer model

The Plummer 3-dimensional density profile is given by $$\rho _{P}(r)={\frac {3M_{0}}{4\pi a^{3}}}\left(1+{\frac {r^{2}}{a^{2}}}\right)^{-{5}/{2}},$$ 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.

Properties

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 ${\underline {a}}=a/r_{V}=3\pi /16$.

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 ($\rho \rightarrow r^{-5}$) 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.