Plummer model

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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[edit]

The density law of a Plummer model

The Plummer 3-dimensional density profile is given by

where M is the total mass of the cluster, and a is the Plummer radius, a scale parameter which sets the size of the cluster core. The corresponding potential is

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

The distribution function is

if and otherwise, where is the specific energy.


The mass enclosed within radius is given by


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

Core radius , where the surface density drops to half its central value, is at .

Half-mass radius is

Virial radius is

The radial turning points of an orbit characterized by specific energy and specific angular momentum are given by the positive roots of the cubic equation


where so that . This equation has three real roots for , two positive and one negative given that , where is the specific angular momentum for a circular orbit for the same energy. Here can be calculated from single real root of the discriminant of the cubic equation which is itself another cubic equation

where underlined parameters are dimensionless in Henon units defined as , , and .


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 () 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]