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
where "formula_2" 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
where "G" is Newton's gravitational constant. The velocity dispersion is
The distribution function is 
if formula_6, and formula_7 otherwise, where formula_8 is the specific energy.
Properties.
The mass enclosed within radius formula_9 is given by 
Many other properties of the Plummer model are described in Herwig Dejonghe's comprehensive article.
Core radius formula_11, where the surface density drops to half its central value, is at formula_12.
Half-mass radius is formula_13
Virial radius is formula_14.
The 2D surface density is:
formula_15,
and hence the 2D projected mass profile is:
formula_16.
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: formula_17.
For the Plummer profile: formula_18.
The radial turning points of an orbit characterized by specific energy formula_19 and specific angular momentum formula_20 are given by the positive roots of the cubic equation
where formula_22, so that formula_23. This equation has three real roots for formula_24: two positive and one negative, given that formula_25, where formula_26 is the specific angular momentum for a circular orbit for the same energy. Here formula_27 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 formula_29, formula_30, and formula_31.
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 (formula_32) 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.