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 escape velocity at any point is For bound orbits, the radial turning points of the orbit is characterized by specific energy formula_20 and specific angular momentum formula_21 are given by the positive roots of the cubic equation where formula_23, so that formula_24. This equation has three real roots for formula_25: two positive and one negative, given that formula_26, where formula_27 is the specific angular momentum for a circular orbit for the same energy. Here formula_28 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_30, formula_31, and formula_32. 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_33) 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.