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

$${\displaystyle \rho _{P}(r)={\frac {3M_{0}}{4\pi a^{3}}}\left(1+{\frac {r^{2}}{a^{2}}}\right)^{-{5}/{2}},}$$

 

where ${\displaystyle M_{0}}$  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

$${\displaystyle \Phi _{P}(r)=-{\frac {GM_{0}}{\sqrt {r^{2}+a^{2}}}},}$$

 

where G is Newton's gravitational constant. The velocity dispersion is

$${\displaystyle \sigma _{P}^{2}(r)={\frac {GM_{0}}{6{\sqrt {r^{2}+a^{2}}}}}.}$$

 

The isotropic distribution function reads

$${\displaystyle f({\vec {x}},{\vec {v}})={\frac {24{\sqrt {2}}}{7\pi ^{3}}}{\frac {a^{2}}{G^{5}M_{0}^{4}}}(-E({\vec {x}},{\vec {v}}))^{7/2},}$$

 

if ${\displaystyle E<0}$ , and ${\displaystyle f({\vec {x}},{\vec {v}})=0}$  otherwise, where ${\textstyle E({\vec {x}},{\vec {v}})={\frac {1}{2}}v^{2}+\Phi _{P}(r)}$  is the specific energy.

Properties

The mass enclosed within radius ${\displaystyle r}$  is given by

$${\displaystyle M(<r)=4\pi \int _{0}^{r}r'^{2}\rho _{P}(r')\,dr'=M_{0}{\frac {r^{3}}{(r^{2}+a^{2})^{3/2}}}.}$$

 

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

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

Half-mass radius is ${\displaystyle r_{h}=\left({\frac {1}{0.5^{2/3}}}-1\right)^{-0.5}a\approx 1.3a.}$ 

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

The 2D surface density is:

$${\displaystyle \Sigma (R)=\int _{-\infty }^{\infty }\rho (r(z))dz=2\int _{0}^{\infty }{\frac {3a^{2}M_{0}dz}{4\pi (a^{2}+z^{2}+R^{2})^{5/2}}}={\frac {M_{0}a^{2}}{\pi (a^{2}+R^{2})^{2}}},}$$

 

and hence the 2D projected mass profile is:

$${\displaystyle M(R)=2\pi \int _{0}^{R}\Sigma (R')\,R'dR'=M_{0}{\frac {R^{2}}{a^{2}+R^{2}}}.}$$

 

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: ${\displaystyle M(R_{1/2})=M_{0}/2}$ .

For the Plummer profile: ${\displaystyle R_{1/2}=a}$ .

The escape velocity at any point is

$${\displaystyle v_{\rm {esc}}(r)={\sqrt {-2\Phi (r)}}={\sqrt {12}}\,\sigma (r),}$$

 

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

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

 

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. Here ${\displaystyle L_{c}}$  can be calculated from single real root of the discriminant of the cubic equation, which is itself 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 ${\displaystyle {\underline {E}}=Er_{V}/(GM_{0})}$ , ${\displaystyle {\underline {L}}_{c}=L_{c}/{\sqrt {GMr_{V}}}}$ , 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.