Gauss' Law For Gravity - Poisson's Equation and Gravitational Potential

Poisson's Equation and Gravitational Potential

Since the gravitational field has zero curl (equivalently, gravity is a conservative force) as mentioned above, it can be written as the gradient of a scalar potential, called the gravitational potential:

Then the differential form of Gauss's law for gravity becomes Poisson's equation:

This provides an alternate means of calculating the gravitational potential and gravitational field. Although computing g via Poisson's equation is mathematically equivalent to computing g directly from Gauss's law, one or the other approach may be an easier computation in a given situation.

In radially symmetric systems, the gravitational potential is a function of only one variable (namely, ), and Poisson's equation becomes (see Del in cylindrical and spherical coordinates):

while the gravitational field is:

When solving the equation it should be taken into account that in the case of finite densities ∂ϕ/∂r has to be continuous at boundaries (discontinuities of the density), and zero for r = 0.