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

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

where $E=v^2 + \Phi(r)$ 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 $r_{h}\approx 1.3a$

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

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.