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. It is now often used as toy model in N-body simulations of stellar systems.

Description of the model

File:Plummer rho.png

The density law of a Plummer model

The Plummer 3-dimensional density profile is given by $${\rho }_P(r)=\frac{3M_0}{4\pi a^3}{(1+\frac{r^2}{a^2})}^{-5/2},$$ 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 $${\mathrm{\Phi }}_P(r)=-\frac{G{M}_0}{\sqrt{r^2+a^2}},$$ where G is Newton's gravitational constant. The velocity dispersion is $${\sigma }_P^2(r)=\frac{G{M}_0}{6\sqrt{r^2+a^2}}.$$

The isotropic distribution function reads $$f(\overrightarrow{x},\overrightarrow{v})=\frac{24\sqrt{2}}{7{\pi }^3}\frac{a^2}{G^5{M}_0^4}(-E(\overrightarrow{x},\overrightarrow{v}){)}^{7/2},$$ if $E<0$, and $f(\overrightarrow{x},\overrightarrow{v})=0$ otherwise, where $E(\overrightarrow{x},\overrightarrow{v})=\frac{1}{2}{v}^2+{\mathrm{\Phi }}_P(r)$ is the specific energy.

Properties

The mass enclosed within radius $r$ is given by $$M(<r)=4\pi \underset{0}{\overset{r}{\int }}r{\text{'}}^2{\rho }_P(r^{\prime })d{r}^{\prime }=M_0\frac{r^3}{(r^2+a^2{)}^{3/2}}.$$

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

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

Half-mass radius is $r_h={(\frac{1}{0.5^{2/3}}-1)}^{-0.5}a\approx 1.3a.$

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

The 2D surface density is: $$\mathrm{\Sigma }(R)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\rho (r(z))dz=2\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{3a^2{M}_{0}dz}{4\pi (a^2+z^2+R^2{)}^{5/2}}=\frac{M_0{a}^2}{\pi (a^2+R^2{)}^2},$$ and hence the 2D projected mass profile is: $$M(R)=2\pi \underset{0}{\overset{R}{\int }}\mathrm{\Sigma }(R^{\prime })R^{\prime }d{R}^{\prime }=M_0\frac{R^2}{a^2+R^2}.$$

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: $M(R_{1/2})=M_0/2$.

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

The escape velocity at any point is $$v_{\mathrm{e}\mathrm{s}\mathrm{c}}(r)=\sqrt{-2\mathrm{\Phi }(r)}=\sqrt{12}\sigma (r),$$

For bound orbits, the radial turning points of the orbit is characterized by specific energy $E=\frac{1}{2}{v}^2+\mathrm{\Phi }(r)$ and specific angular momentum $L=|\overrightarrow{r}\times \overrightarrow{v}|$ are given by the positive roots of the cubic equation $$R^3+\frac{G{M}_0}{E}{R}^2-(\frac{L^2}{2E}+a^2)R-\frac{G{M}_0{a}^2}{E}=0,$$ 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 $$\underset{\_}{E}{\underset{\_}{L}}_c^3+(6{\underset{\_}{E}}^2{\underset{\_}{a}}^2+\frac{1}{2}){\underset{\_}{L}}_c^2+(12{\underset{\_}{E}}^3{\underset{\_}{a}}^4+20\underset{\_}{E}{\underset{\_}{a}}^2){\underset{\_}{L}}_c+(8{\underset{\_}{E}}^4{\underset{\_}{a}}^6-16{\underset{\_}{E}}^2{\underset{\_}{a}}^4+8{\underset{\_}{a}}^2)=0,$$ where underlined parameters are dimensionless in Henon units defined as $\underset{\_}{E}=E{r}_V/(G{M}_0)$, ${\underset{\_}{L}}_c=L_c/\sqrt{GM{r}_V}$, and $\underset{\_}{a}=a/r_V=3\pi /16$.

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 ($\rho \to r^{-5}$) 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.