Gravitational Potential - Mathematical Form

Mathematical Form

The potential V at a distance x from a point mass of mass M can be defined as the work done by the gravitational field bringing a unit mass in from infinity to that point:

where G is the gravitational constant. The potential has units of energy per unit mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as r tends to infinity, it approaches zero.

The gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object. Because the potential has no angular components, its gradient is:

where x is a vector of length x pointing from the point mass toward the small body and is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law:

The potential associated with a mass distribution is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, and if the point masses are located at the points x₁, ..., x_n and have masses m₁, ..., m_n, then the potential of the distribution at the point x is:

If the mass distribution is given as a mass measure dm on three-dimensional Euclidean space R3, then the potential is the convolution of −G/|r| with dm. In good cases this equals the integral

where |x − r| is the distance between the points x and r. If there is a function ρ(r) representing the density of the distribution at r, so that dm(r)= ρ(r)dv(r), where dv(r) is the Euclidean volume element, then the gravitational potential is the volume integral

If V is a potential function coming from a continuous mass distribution ρ(r), then ρ can be recovered using the Laplace operator Δ using the formula:

This holds pointwise whenever ρ is continuous and is zero outside of a bounded set. In general, the mass measure dm can be recovered in the same way if the Laplace operator is taken in the sense of distributions. As a consequence, the gravitational potential satisfies Poisson's equation. See also Green's function for the three-variable Laplace equation and Newtonian potential.