Gravitational Binding Energy

The gravitational binding energy of a system is the minimum energy that must be added to it in order to completely disperse it. When the GBE of a planet, star, or other gravitationally bound structure is exceeded, the particles of the system will not reform themselves, but instead drift off infinitely. The exact calculation depends on the type of structure, but for celestial bodies (planets and stars), the following formula is used:

U = 3GM2/(5-n)R

Where U is GBE in joules, G is the gravitational constant of 6.6743 × 10-11, M is the mass of the body in question in kilograms, R is its radius in meters, and n is its polytrope. While this formula is not perfectly accurate, it is within the acceptable margin of error.

Polytrope
In the simplest terms, a celestial body's polytrope is an index of its density distribution. As the polytropic index increases, the density is more heavily distributed toward the center of the body.

Example models by polytropic index

 * An index n=0 polytrope is often used to model rocky planets (such as Earth) and ice giants.
 * Neutron stars are well-modeled by polytropes with index ranging from n=0.5 to n=1.
 * A polytrope with index n=1.5 is a good model for stars with fully convective cores, brown dwarfs, and gas giants.
 * A polytrope with index n=3 is usually used to model main-sequence stars such as our Sun.
 * White dwarfs are best modeled by polytropes with index n=1.5 for low-mass white dwarfs and index n=3 for white dwarfs with higher mass.