Plummer model

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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[edit]

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.


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]


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]


  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.