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

The Plummer 3-dimensional density profile is given by

: $\rho_P(r) = \frac \left(1 + \frac\right)^,$

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

: $\Phi_P(r) = -\frac,$

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

: $\sigma_P^2(r) = \frac.$

The distribution function is

: $f(\vec, \vec) = \frac \frac (-E(\vec, \vec))^,$

if $E < 0$, and $f(\vec, \vec) = 0$ otherwise, where $E(\vec, \vec) = \frac12 v^2 + \Phi_P(r)$ is the specific energy.

Properties

The mass enclosed within radius $r$ is given by
: M(

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 $r_c = a \sqrt \approx 0.64 a$.

Half-mass radius is $r_h = \left(\frac - 1\right)^ a \approx 1.3 a.$

Virial radius is $r_V = \frac a \approx 1.7 a$.

The 2D surface density is:

$\Sigma(R)=\int_^\rho(r(z))dz=2\int_^\frac=\frac$,

and hence the 2D projected mass profile is:

$M(R)=2\pi\int_^\Sigma(R')\, R'dR'=M_0\frac$.

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: $M(R_)=M_0/2$.

For the Plummer profile: $R_=a$.

The escape velocity at any point is
:$v_(r)=\sqrt=\sqrt\,\sigma(r) ,$

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

:$R^3 + \frac R^2 - \left(\frac + a^2\right) R - \frac = 0,$

where $R = \sqrt$, so that $r = \sqrt$. 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

:$\underline\, \underline_c^3 + \left(6 \underline^2 \underline^2 + \frac\right)\underline_c^2 + \left(12 \underline^3 \underline^4 + 20 \underline \underline^2 \right) \underline_c + \left(8 \underline^4 \underline^6 - 16 \underline^2 \underline^4 + 8 \underline^2\right) = 0,$

where underlined parameters are dimensionless in Henon units defined as $\underline = E r_V / (G M_0)$, $\underline_c = L_c / \sqrt$, and $\underline = a / r_V = 3 \pi/16$.

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 ($\rho\rightarrow r^$) 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.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.

References

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Astrophysics