Astronomy: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 Plummer 3-dimensional density profile is given by
- [math]\displaystyle{ \rho_P(r) = \frac{3M_0}{4\pi a^3} \left(1 + \frac{r^2}{a^2}\right)^{-\frac{5}{2}}, }[/math]
where [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \sigma_P^2(r) = \frac{G M_0}{6\sqrt{r^2 + a^2}}. }[/math]
The isotropic distribution function reads
- [math]\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}, }[/math]
if [math]\displaystyle{ E \lt 0 }[/math], and [math]\displaystyle{ f(\vec{x}, \vec{v}) = 0 }[/math] otherwise, where [math]\displaystyle{ E(\vec{x}, \vec{v}) = \frac12 v^2 + \Phi_P(r) }[/math] is the specific energy.
Properties
The mass enclosed within radius [math]\displaystyle{ r }[/math] is given by
- [math]\displaystyle{ M(\lt 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.[2]
Core radius [math]\displaystyle{ r_c }[/math], where the surface density drops to half its central value, is at [math]\displaystyle{ r_c = a \sqrt{\sqrt{2} - 1} \approx 0.64 a }[/math].
Half-mass radius is [math]\displaystyle{ r_h = \left(\frac{1}{0.5^{2/3}} - 1\right)^{-0.5} a \approx 1.3 a. }[/math]
Virial radius is [math]\displaystyle{ r_V = \frac{16}{3 \pi} a \approx 1.7 a }[/math].
The 2D surface density is:
[math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ M(R_{1/2})=M_0/2 }[/math].
For the Plummer profile: [math]\displaystyle{ R_{1/2}=a }[/math].
The escape velocity at any point is
- [math]\displaystyle{ 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]\displaystyle{ E = \frac{1}{2} v^2 + \Phi(r) }[/math] and specific angular momentum [math]\displaystyle{ L = |\vec{r} \times \vec{v}| }[/math] are given by the positive roots of the cubic equation
- [math]\displaystyle{ 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]\displaystyle{ R = \sqrt{r^2 + a^2} }[/math], so that [math]\displaystyle{ r = \sqrt{R^2 - a^2} }[/math]. This equation has three real roots for [math]\displaystyle{ R }[/math]: two positive and one negative, given that [math]\displaystyle{ L \lt L_c(E) }[/math], where [math]\displaystyle{ L_c(E) }[/math] is the specific angular momentum for a circular orbit for the same energy. Here [math]\displaystyle{ L_c }[/math] can be calculated from single real root of the discriminant of the cubic equation, which is itself another cubic equation
- [math]\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, }[/math]
where underlined parameters are dimensionless in Henon units defined as [math]\displaystyle{ \underline{E} = E r_V / (G M_0) }[/math], [math]\displaystyle{ \underline{L}_c = L_c / \sqrt{G M r_V} }[/math], and [math]\displaystyle{ \underline{a} = a / r_V = 3 \pi/16 }[/math].
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 ([math]\displaystyle{ \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.[3]
References
- ↑ Plummer, H. C. (1911), On the problem of distribution in globular star clusters, Mon. Not. R. Astron. Soc. 71, 460.
- ↑ Dejonghe, H. (1987), A completely analytical family of anisotropic Plummer models. Mon. Not. R. Astron. Soc. 224, 13.
- ↑ 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.
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