Just as in electrostatics we can expand in a Taylor series the electrostatic potential as the infinite sum of the contributions of a monopole, a dipole, a quadrupole, etc., could we apply the same reasoning to the gravitational field? More importantly, even if we could, would this be of any interest? I'm assuming it wouldn't, since a mass distribution will never cancel out its "gravitational monopole". For instance, if we have a proton and an electron, then the system's net charge is null, but this would never be the case with ordinary matter, since mass would simply add up. Therefore, we would always have the first term of the multipole expansion dominating over the rest. Hence, the question that follows quite naturally is: is there any case in which a multipole expansion of the gravitational potential is relevant?
2 Answers
Yes, the multipole expansion of the gravitational potential is used all the time in astrophysics. Because the source of Newtonian gravitational field is the mass density, which is always positive, there can never be a gravitational dipole moment in a body's center of mass frame. The lowest correction to the gravitational potential of an approximately spherical celestial body must then come from its quadrupole moment.
Since rotating bodies (like the Earth or the sun) are pretty consistently oblate, they can have significant quadrupole moments. The magnitudes of the forces due to the quadrupole terms in the expansion are usually much smaller than the monopolar forces, even at quite close ranges, as correctly noted in the question. However, they can nonetheless have important observable effects, because of the way they perturb the gravitational interactions between bodies. The oblateness of the Earth is about $f=\frac{1}{300}$, which is large enough that it needs to be accounted for when doing precision of calculations of lunar or satellite trajectories. (The tidal torques exerted on the Earth's oblate bulge by the sun and moon are also what is responsible for the precession of the Earth's axis over a 25,700-year period.)
I am just adding something to the excellent answer by @Buzz. Due to the absence of negative masses, all the multipole moments depend on the position of the origin of the coordinate system. Still, in addition to their usefulness when dealing with the gravitational forces of approximately spherical objects, the multipole expansion of clusters of stars or galaxies is routinely exploited in fast algorithms for solving the gravitational N-body problem (see this Wikipedia page for an example).
