# Two bodies of finite size treated as two point masses in Newtonian gravity

When discussing gravitation between two bodies of finite size, for instance Earth around the Sun, we suppose the mass of Earth and the Sun to be perfectly localized at the center of each body. Is this a "useful" "good" "approximation" or the "absolute truth"?

• This is the shell theorem: en.wikipedia.org/wiki/Shell_theorem
– user4552
Commented Oct 11, 2014 at 20:04
• @BenCrowell Ah, so it has a name! I'll edit my question to reflect the correct terminology :) Also, wikipedia's first derivation is really unnecessarily tedious.
– Danu
Commented Oct 11, 2014 at 20:10
• Commented Oct 11, 2014 at 20:30

This is certainly not the absolute truth, since every human on this planet is a living witness to the fact that Earth is not pointlike: It has finite volume.

However, there's a very good mathematical reason why we can consider the sun and Earth as pointlike when doing calculations in celestial mechanics: It's because the gravitational field due to a spherical body like the sun, when seen from outside of the sun, is identical to that of a pointlike gravitational charge with the same mass as the sun. This is called the shell theorem, and it is a consequence of Gauss' law for gravity, and a very similar result holds in the context of electromagnetism (where the name Gauss' law is most famous). This is an elementary exercise that many undergraduates are asked to derive in their first E&M course.

The relevant equations are:

$$\vec \nabla \cdot {\rm \vec g}= -4\pi G\rho(\vec r)\ ({\rm gravity})\hspace{1cm}\vec\nabla\cdot \vec E =\frac{\rho(\vec r)}{\epsilon_0}\ ({\rm electrostatics})$$

One can immediately see that they are essentially telling you the same thing, up to some proportionality constants, so a similar result is expected to hold for both.

There is one big caveat: Planets and stars are of course not perfectly spherically symmetric, so we're really dealing with an approximation after all. In fact, as pointed out in the comments, it is practically impossible to achieve perfect spherical symmetry, so our result should not be expected to hold exactly in reality - even though it's often extraordinarity close. However, this is not due to the fact that the bodies are not pointlike, but rather due to their lack of symmetry.

All finite-sized objects act like point masses at sufficiently large distances. A finite-sized three dimensional object of mass $M$ has a unique minimal bounding sphere with some radius $r$ and center $c$. For any distance $R>r$ from the center, the magnitude of the gravitational force on a small test body of mass $m$ is bounded by $\frac {GMm}{(R+r)^2} \le F \le \frac {GMm}{(R-r)^2}$, and the force is directed toward the center object, plus or minus $\arcsin\left(\frac r R\right)$. As $R$ grows ever larger, this converges to the gravitational force exerted by a point mass of mass $M$ located at $c$.

An object with a spherical mass distribution (density is a function only of distance from the center of the object) looks like a point mass everywhere outside the object. Other objects don't look quite like point masses at close distances. For example, a rotating large object will have shape that is more or less that of oblate spheroid (i.e., it will have an equatorial bulge). At close distances, that equatorial bulge will result in departures from a point mass model. These departures can be significant and important. For example, the Earth's equatorial bulge is what enables us to have satellites in sun-synchronous orbits.

This equatorial bulge is but the first of an infinite number of terms in the spherical harmonic expansion of an object's gravitational field. Using spherical harmonics is a commonly used approach to modeling a non-spherical object. That there are, in theory, an infinite number of terms presents a challenge with regard to modeling. Typically, a spherical harmonics approximation is truncated to some degree and order. For objects of extreme interest (e.g., the Earth), that degree and order can be rather high. The GRACE gravity model of the Earth is complete to degree and order 360. A formerly highly classified US agency, the National Geospatial Intelligence Agency, takes this even further. They have developed a 2190x2159 spherical harmonics gravity model of the Earth.

Even that 2190x2159 is a bit coarse compared to items of interest to a geophysicist. Academia churns out a lot more geophysicists than are needed by academia, and much of that surplus finds work in the petrochemical or mining industries. Local deviations from the gravitational acceleration suggested by an oblate spheroid are key indicators that there's oil, gold, or some other precious resource hiding underground. These deviations can also be of academic interest. For example, gravitational anomalies form one of the key pieces of evidence that the North American continent almost split in two 1.1 billion years ago at the Midcontinent Rift System.

Further afield, the Moon too does not look anything like a point mass at close distances. For example, the Apollo 16 mission released a small object, PFS-2, in orbit about the Moon, the intent of which was to orbit the Moon and measure charged particles and the Moon's magnetic field. PFS-2 was intended to be in a more or less circular low lunar orbit about the Moon. That's not what happened. Instead, something bizarre happened. After 35 days of weird changes to its orbit, PFS-2 crashed into the Moon.

A truncated spherical harmonics approximation works quite nicely for large bodies because the higher order terms quickly tend to zero for large bodies. Only a handful of terms are needed to describe a gas giant or a star. Significantly more terms are needed to accurately describe an Earth-sized body, and even more terms to describe a Moon-sized body. Ever smaller objects exhibit ever more deviations from sphericality. At some point, the spherical harmonics approach just doesn't work that well. An alternate approach used for small solar system bodies is to use a 3D picture of the object. Computer graphics models tend to use polyhedra to describe the 3D shape of an object. Starting with Barnett, "Theoretical modeling of the magnetic and gravitational fields of an arbitrarily shaped three-dimensional body," Geophysics 41.6 (1976): 1353-1364, these polyhedral gravity models turn out to be rather useful for describe the gravity of a small (asteroid-sized) body.

Point masses are merely an approximation of the motion of an extended body by the motion of its center of mass. It's a good approximation for small, mostly homogeneous (at least spherically layered) bodies, e.g. planets moving around the sun. But even for the orbital motion of the Moon around the Earth it's not a good approximation anymore.