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

$\rho _{P}(r)={\bigg (}{\frac {3M}{4\pi a^{3}}}{\bigg )}{\bigg (}1+{\frac {r^{2}}{a^{2}}}{\bigg )}^{-{\frac {5}{2}}}\,,$

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

$\Phi _{P}(r)=-{\frac {GM}{\sqrt {r^{2}+a^{2}}}}\,,$

where G is Newton's gravitational constant. The velocity dispersion is

$\sigma _{P}^{2}(r)={\frac {GM}{6{\sqrt {r^{2}+a^{2}}}}}\,.$

The distribution function is

$f({\vec {x}},{\vec {v}})={\frac {24{\sqrt {2}}}{7\pi ^{3}}}{\frac {Na^{2}}{G^{5}M^{5}}}(-E({\vec {x}},{\vec {v}}))^{7/2}\,,$

if $E<0$ and $f({\vec {x}},{\vec {v}})=0$ otherwise, where $E({\vec {x}},{\vec {v}})={\frac {1}{2}}v^{2}+\Phi _{P}(r)$ is the specific energy.

Properties

The mass enclosed within radius $r$ is given by

$M(<r)=4\pi \int _{0}^{r}r^{2}\rho _{P}(r)dr=M{r^{3} \over \left(r^{2}+a^{2}\right)^{3/2}}$.

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

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

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

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

The radial turning points of an orbit characterized by specific energy $E={\frac {1}{2}}v^{2}+\Phi (r)$ and specific angular momentum $L=|{\vec {r}}\times {\vec {v}}|$ are given by the positive roots of the cubic equation

$R^{3}+{\frac {GM}{E}}R^{2}-\left({\frac {L^{2}}{2E}}+a^{2}\right)R-{\frac {GMa^{2}}{E}}=0$.

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 ${\underline {E}}=Er_{V}/(GM)$, ${\underline {L}}_{c}=L_{c}/{\sqrt {G\,M\,r_{V}}}$, 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.