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. The velocity dispersion is

${\displaystyle \sigma _{P}^{2}(r)={\frac {GM}{6{\sqrt {r^{2}+a^{2}}}}}\,.}$ 

The distribution function is

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

where ${\displaystyle E({\vec {x}},{\vec {v}})={\frac {1}{2}}v^{2}+\Phi _{P}(r)}$  is the specific energy.

Properties

The mass enclosed within radius ${\displaystyle r}$  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 ${\displaystyle r_{c}}$ , 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 ${\displaystyle r_{V}={\frac {16}{3\pi }}a\approx 1.7a}$ 

The radial turning points of an orbit characterized by specific energy ${\displaystyle E={\frac {1}{2}}v^{2}+\Phi (r)}$  and specific angular momentum ${\displaystyle L=|{\vec {r}}\times {\vec {v}}|}$  are given by the positive roots of the cubic equation

${\displaystyle R^{3}+{\frac {GM}{E}}R^{2}-\left({\frac {L^{2}}{2E}}+a^{2}\right)R-{\frac {GMa^{2}}{E}}=0}$ .

where ${\displaystyle R={\sqrt {r^{2}+a^{2}}}}$  so that ${\displaystyle r={\sqrt {R^{2}-a^{2}}}}$ . This equation has three real roots for ${\displaystyle R}$ , two positive and one negative given that ${\displaystyle L<L_{c}(E)}$ , where ${\displaystyle L_{c}(E)}$  is the specific angular momentum for a circular orbit for the same energy. This can be calculated from another cubic equation

${\displaystyle {\underline {E}}\,{\underline {L}}_{c}^{3}+\left(6{\underline {E}}^{2}{\underline {a}}^{2}+{\frac {1}{2}}\right){\underline {L}}_{c}^{2}+\left(12{\underline {E}}^{3}{\underline {a}}^{4}+20{\underline {E}}{\underline {a}}^{2}\right){\underline {L}}_{c}+\left(8{\underline {E}}^{4}{\underline {a}}^{6}-16{\underline {E}}^{2}{\underline {a}}^{4}+8{\underline {a}}^{2}\right)=0}$ 

where underlined parameters are dimensionless in Henon units defined as ${\displaystyle {\underline {E}}=Er_{V}/(GM)}$ , ${\displaystyle {\underline {L}}_{c}=L_{c}/{\sqrt {G\,M\,r_{V}}}}$ , and ${\displaystyle {\underline {a}}=a/r_{V}=3\pi /16}$ .

Applications

The Plummer model comes closest to representing the observed density profiles of star clusters^{[citation needed]}, 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.