The Plummer model or Plummer sphere is a density law that was first used by H. C. Plummer to fit observations of globular clusters.<ref>Plummer, H. C. (1911), On the problem of distribution in globular star clusters, Mon. Not. R. Astron. Soc. 71, 460.</ref> It is now often used as toy model in N-body simulations of stellar systems.
Description of the modelEdit
The Plummer 3-dimensional density profile is given by <math display="block">\rho_P(r) = \frac{3M_0}{4\pi a^3} \left(1 + \frac{r^2}{a^2}\right)^{-{5}/{2}},</math> where <math>M_0</math> 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 <math display="block">\Phi_P(r) = -\frac{G M_0}{\sqrt{r^2 + a^2}},</math> where G is Newton's gravitational constant. The velocity dispersion is <math display="block">\sigma_P^2(r) = \frac{G M_0}{6\sqrt{r^2 + a^2}}.</math>
The isotropic distribution function reads <math display="block">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},</math> if <math>E < 0</math>, and <math>f(\vec{x}, \vec{v}) = 0</math> otherwise, where <math display="inline">E(\vec{x}, \vec{v}) = \frac{1}{2} v^2 + \Phi_P(r)</math> is the specific energy.
PropertiesEdit
The mass enclosed within radius <math>r</math> is given by <math display="block">M(<r) = 4\pi\int_0^r r'^2 \rho_P(r') \,dr' = M_0 \frac{r^3}{(r^2 + a^2)^{3/2}}.</math>
Many other properties of the Plummer model are described in Herwig Dejonghe's comprehensive article.<ref>Dejonghe, H. (1987), A completely analytical family of anisotropic Plummer models. Mon. Not. R. Astron. Soc. 224, 13.</ref>
Core radius <math>r_c</math>, where the surface density drops to half its central value, is at <math display="inline">r_c = a \sqrt{\sqrt{2} - 1} \approx 0.64 a</math>.
Half-mass radius is <math>r_h = \left(\frac{1}{0.5^{2/3}} - 1\right)^{-0.5} a \approx 1.3 a.</math>
Virial radius is <math>r_V = \frac{16}{3 \pi} a \approx 1.7 a</math>.
The 2D surface density is: <math display="block"> \Sigma(R) = \int_{-\infty}^{\infty}\rho(r(z))dz=2\int_{0}^{\infty}\frac{3a^2M_0dz}{4\pi(a^2+z^2+R^2)^{5/2}} = \frac{M_0a^2}{\pi(a^2+R^2)^2},</math> and hence the 2D projected mass profile is: <math display="block">M(R)=2\pi\int_{0}^{R}\Sigma(R')\, R'dR'=M_0\frac{R^2}{a^2+R^2}.</math>
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: <math>M(R_{1/2}) = M_0/2</math>.
For the Plummer profile: <math>R_{1/2} = a</math>.
The escape velocity at any point is <math display="block">v_{\rm esc}(r)=\sqrt{-2\Phi(r)}=\sqrt{12}\,\sigma(r) ,</math>
For bound orbits, the radial turning points of the orbit is characterized by specific energy <math display="inline">E = \frac{1}{2} v^2 + \Phi(r)</math> and specific angular momentum <math>L = |\vec{r} \times \vec{v}|</math> are given by the positive roots of the cubic equation <math display="block">R^3 + \frac{GM_0}{E} R^2 - \left(\frac{L^2}{2E} + a^2\right) R - \frac{GM_0a^2}{E} = 0,</math> where <math>R = \sqrt{r^2 + a^2}</math>, so that <math>r = \sqrt{R^2 - a^2}</math>. This equation has three real roots for <math>R</math>: two positive and one negative, given that <math>L < L_c(E)</math>, where <math>L_c(E)</math> is the specific angular momentum for a circular orbit for the same energy. Here <math>L_c</math> can be calculated from single real root of the discriminant of the cubic equation, which is itself another cubic equation <math display="block">\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,</math> where underlined parameters are dimensionless in Henon units defined as <math>\underline{E} = E r_V / (G M_0)</math>, <math>\underline{L}_c = L_c / \sqrt{G M r_V}</math>, and <math>\underline{a} = a / r_V = 3 \pi/16</math>.
ApplicationsEdit
The Plummer model comes closest to representing the observed density profiles of star clustersTemplate:Citation needed, although the rapid falloff of the density at large radii (<math>\rho\rightarrow r^{-5}</math>) 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.<ref>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.</ref>