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

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

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

$\Phi_P(r) = -\frac{G M}{\sqrt{r^2+a^2}}\,,$

where G is Newton's gravitational constant.

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.^[2]

Core radius $r_c$, where the surface density drops to half its central value, is at $r_c =a\sqrt{\sqrt{2}-1}\approx0.64a$.

Half-mass radius is $r_h \approx 1.3 a$

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

See also The Art of Computational Science^[3]

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.^[4]

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. P.Hut and J.Makino. The Art of Computational Science
4. 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.