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.^[1] 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

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

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

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

where G is Newton's gravitational constant.

Properties

The mass enclosed within radius $r$ is given by

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

Many other properties of the Plummer model are described in Herwig Dejonghe's comprehensive paper.^[2]

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

Half-mass radius is ${\displaystyle r_{h}\approx 1.3a}$

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

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 (${\displaystyle \rho \rightarrow r^{-5}}$) 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.^[3]

References

1. Plummer, H. C. (1911), On the problem of distribution in globular star clusters, Mon. Not. R. Astron. Soc. 71, 460
2. Dejonghe, H. (1987), A completely analytical family of anisotropic Plummer models. Mon. Not. R. Astron. Soc. 224, 13
3. Aarseth, S. J., Henon, M. and Wielen, R. (1974), A comparison of numerical methods for the study of star cluster dynamics. Astronomy and Astrophysics 37 183.