Plummer model explained

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 Plummer 3-dimensional density profile is given by$$\backslash rho\_P(r)\; =\; \backslash frac\; \backslash left(1\; +\; \backslash frac\backslash right)^,$$where

M _{0} is the total mass of the cluster, and a is the Plummer radius, a scale parameter that sets the size of the cluster core. The corresponding potential is$$\backslash Phi\_P(r)\; =\; -\backslash frac,$$where G is Newton's gravitational constant. The velocity dispersion is$$\backslash sigma\_P^2(r)\; =\; \backslash frac.$$

The isotropic distribution function reads$$f(\backslash vec,\; \backslash vec)\; =\; \backslash frac\; \backslash frac\; (-E(\backslash vec,\; \backslash vec))^,$$if

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

Properties

The mass enclosed within radius

r is given by

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

Core radius

r _{c} , where the surface density drops to half its central value, is at $r\_c\; =\; a\; \backslash sqrt\; \backslash approx\; 0.64\; a$.

Half-mass radius is

r _{h} = \left( (1)/(0 . 5 ^{2/3}) - 1 \right) ^{-0.5} a ≈ 1 . 3 a.

Virial radius is

r _{V} = (16)/(3 \pi) a ≈ 1 . 7 a .

The 2D surface density is:$$\backslash Sigma(R)\; =\; \backslash int\_^\backslash rho(r(z))dz=2\backslash int\_^\backslash frac\; =\; \backslash frac,$$and hence the 2D projected mass profile is:$$M(R)=2\backslash pi\backslash int\_^\backslash Sigma(R\text{'})\backslash ,\; R\text{'}dR\text{'}=M\_0\backslash frac.$$

In astronomy, it is convenient to define 2D half-mass radius which is the radius where the 2D projected mass profile is half of the total mass:

M(R _{1/2} ) = M _{0/2} .

For the Plummer profile:

R _{1/2} = a .

The escape velocity at any point is$$v\_(r)=\backslash sqrt=\backslash sqrt\backslash ,\backslash sigma(r),$$

L = |\vec{r} x \vec{v}| are given by the positive roots of the cubic equation$$R^3\; +\; \backslash frac\; R^2\; -\; \backslash left(\backslash frac\; +\; a^2\backslash right)\; R\; -\; \backslash frac\; =\; 0,$$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$$\backslash underline\backslash ,\; \backslash underline\_c^3\; +\; \backslash left(6\; \backslash underline^2\; \backslash underline^2\; +\; \backslash frac\backslash right)\backslash underline\_c^2\; +\; \backslash left(12\; \backslash underline^3\; \backslash underline^4\; +\; 20\; \backslash underline\; \backslash underline^2\; \backslash right)\; \backslash underline\_c\; +\; \backslash left(8\; \backslash underline^4\; \backslash underline^6\; -\; 16\; \backslash underline^2\; \backslash underline^4\; +\; 8\; \backslash underline^2\backslash right)\; =\; 0,$$where underlined parameters are dimensionless in Henon units defined as \underline{E} = E r _{V} / (G M _{0)} , \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 → 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]

Notes and 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.