Newtonian gravity models the attraction between masses as a force acting through a gravitational field.

Universal gravitation

For two point masses $m_1$ and $m_2$ separated by distance $r$, the magnitude of the gravitational force is

$$F=G\frac{m_1m_2}{r^2},$$

where $G$ is the gravitational constant.

The force acts along the line joining the masses and is attractive.

Gravitational field

Instead of describing every interaction pair separately, define the gravitational field $\mathbf g$ as force per unit test mass:

$$\mathbf g=\frac{\mathbf F}{m}.$$

For a point mass $M$ at the origin,

$$\mathbf g(\mathbf r)=-\frac{GM}{r^3}\mathbf r.$$

The negative sign indicates that the field points toward the source mass.

Superposition

In Newtonian gravity, fields from several source masses add vectorially:

$$\mathbf g=\sum_i \mathbf g_i.$$

For a continuous mass distribution the sum becomes an integral over density.

Gravitational potential

Because the Newtonian gravitational field is conservative, it can be written in terms of a scalar potential $\Phi$:

$$\mathbf g=-\nabla\Phi.$$

For a point mass,

$$\Phi=-\frac{GM}{r}.$$

Potential energy for a test mass is $U=m\Phi$.

Scope of the model

Newtonian gravity is extremely effective when gravitational fields are weak and speeds are small compared with the speed of light. It treats gravity as a force evolving with a universal time.

General relativity replaces that framework with a geometric description compatible with relativity, but the Newtonian model remains an important approximation and provides the limiting behavior that the relativistic theory must reproduce.