The Plummer model or Plummer sphere is a density law that was first used by H. C. Plummer

Henry Crozier Keating Plummer

Henry Crozier Keating Plummer FRS was an English astronomer.-Early years and education:Born in Oxford, Plummer was the son of William Edward Plummer and nephew of the distinguished astronomer John Isaac Plummer. He gained his education at St. Edward's School and then Hertford College...

 to fit observations of globular cluster

Globular cluster

A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes and relatively high stellar densities toward their centers. The name of this category of star cluster is...

s. It is now often used as toy model

Toy model

In physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory....

 in N-body simulation

N-body simulation

An N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity . In cosmology, they are used to study processes of non-linear structure formation such as the process of forming galaxy filaments and galaxy halos from dark...

s of stellar systems.

Description of the 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

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's gravitational constant

Gravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

.

Properties

The mass enclosed within radius is given by.

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

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 () 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

Monte Carlo method

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

 has made it a favorite choice of N-body experimenters

N-body simulation

An N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity . In cosmology, they are used to study processes of non-linear structure formation such as the process of forming galaxy filaments and galaxy halos from dark...

, in spite of the model's lack of realism.