A sphere pulls exactly like a point, and the proof is a pair of cones

Every calculation in this collection treats the Sun as a point. So does every orbit ever computed, every satellite ever flown, and every mass ever weighed by watching something go round it.

It looks like an approximation whose error nobody bothers to state. It is not an approximation. For a body whose density depends only on distance from its centre, the gravitational field outside it is exactly that of a point mass at the centre — not nearly, not to leading order, but identically, with no correction term of any size.

Newton proved it, and by his own account the proof held up the Principia for years. Without it there is no two-body problem to solve, because the Sun and the Earth are not points and the equations of motion for extended bodies do not close.

The cancellation, in one paragraph #

Take a point somewhere inside a thin uniform shell and draw a narrow double cone through it, opening in both directions. The cone cuts a small patch out of the shell in each direction.

Let the near patch be at distance $r_1$ and the far patch at $r_2$. The patches subtend the same solid angle, so their areas — and therefore their masses — are in the ratio $r_2^2/r_1^2$. Their gravitational pulls, being inverse-square, are in the ratio $r_1^2/r_2^2$. The two ratios are reciprocals, so the pulls are equal in magnitude and opposite in direction.

They cancel. And since every direction from the point can be covered by such a cone, and every cone cancels, the total force is zero.

That is the whole argument, and its economy is the point. It uses nothing but the fact that area grows as the square of distance and force falls as the square of distance, and it is exact for that reason and no other.

The second figure is worth pausing on because it defeats the intuition it is meant to correct. A point close to the shell’s wall has a great deal of shell very near it on one side and a thin distant sliver on the other, and the pull toward the near wall feels as though it must win. It does not. There is much more shell on the far side, by exactly the factor that compensates, and the compensation is not approximate.

The outside case, which is the one that matters #

The interior result is the striking one. The exterior result is the useful one, and it comes from the same integral run over a different region.

For a point outside the shell, the two patches cut by a double cone are both on the same side and their pulls add rather than cancel. Doing the integral over the whole shell — which is where Newton’s Principia spends its effort — gives a total force identical to that of the shell’s entire mass concentrated at its centre.

A solid body is a stack of nested shells, so the result extends immediately: any spherically symmetric distribution attracts an external point exactly as a point mass at its centre does. The density can vary with radius however it likes. The Sun’s central density is a hundred and fifty times that of water and its outer layers are thinner than laboratory vacuum, and it still pulls precisely like a dot.

Inside a solid body the story combines both halves. At radius $r$, every shell outside $r$ contributes nothing — the interior result — and every shell inside contributes as though its mass were at the centre. So the field depends only on the mass enclosed, which for uniform density goes as $r^3$, giving $g \propto r^3/r^2 = r$. It rises linearly to a maximum at the surface and falls as $1/r^2$ beyond.

The property being used is not “gravity” #

The theorem is often filed under gravity, and filing it there hides how narrow the requirement actually is.

Nothing in the cancellation argument mentions mass, or attraction, or Newton. It uses one thing: that the influence falls as the inverse square of distance — the same exponent that makes the orbit an ellipse rather than a rosette. Any field with that property has the same theorem, and the electrostatic case is the identical result — the field inside a charged conducting shell is zero, and that is why a metal box shields its contents.

The converse is the sharper statement. If the exponent were anything but exactly $-2$, the cancellation would fail, a point inside a shell would feel a net force, and a sphere would not pull like a point. That makes the shell theorem a test of the exponent, and the most precise tests of the inverse-square law ever performed are exactly this experiment done electrostatically: put a sensitive detector inside a charged shell and look for a field. None is found, and the resulting bound on any departure from an exponent of 2 is about one part in $10^{16}$.

That is a striking asymmetry. The inverse-square law of electrostatics is verified to sixteen figures by an experiment sitting on a bench. The inverse-square law of gravity is verified to a few parts in $10^{4}$ at laboratory scales, because gravity is forty orders of magnitude weaker and no shell can be built that shields it. The mathematics is shared; the experimental access is not.

What was actually measured #

The theorem is exact for a spherical body, and no body is spherical. The interesting empirical question is therefore not whether the theorem holds but how large its violations are — and they are measured extremely well, because they are what a satellite orbit is most sensitive to.

A rotating body bulges at its equator. The Earth’s equatorial radius exceeds its polar radius by 21.4 kilometres, a flattening of one part in 298. That departure from sphericity means the external field is not quite a point mass’s, and the leading correction is a term conventionally called $J_2$:

$$J_2 = 1.0826 \times 10^{-3}.$$

That number is known to eight significant figures, and it is known that well because its effects on satellite orbits are enormous rather than subtle. The bulge exerts a torque on an inclined orbit, causing the node and the argument of periapsis to drift — for a low Earth orbit at 51°, the orbital plane rotates by about 5 degrees per day. A satellite whose position is predicted with a point-mass Earth is wrong by hundreds of kilometres within a week — and the effect is used deliberately by Sun-synchronous orbits.

The measurement runs the other way round, and this is what makes the field so productive. Tracking satellites precisely enough gives the coefficients of the Earth’s gravity field, and those coefficients are a map of how mass is distributed inside a planet nobody can look into. Modern models carry the expansion to degree 2190 — a spatial resolution of about 9 kilometres — and are derived from the GRACE and GOCE missions, which measured the field by flying two spacecraft in formation and watching the distance between them change as they passed over mass anomalies.

The most consequential product of that work is not a static map. It is the changes: GRACE measured the field repeatedly and the differences track where water has moved. Ice loss from Greenland and Antarctica, groundwater depletion in northern India, the redistribution after a large earthquake — all of them are measured as a change in a gravitational coefficient, which is a change in how far a body departs from pulling like a point.

Where the theorem breaks, and by how much #

The failures are worth ordering by size, because the ordering is not what intuition suggests.

Non-sphericity. The largest, and it is $10^{-3}$ for the Earth, $10^{-2}$ for Saturn, and negligible for the Sun, whose flattening is about $10^{-5}$. This is the reason planetary satellites need correction terms and heliocentric orbits do not.

Proximity. All the corrections fall off faster than the main term — $J_2$’s contribution goes as $1/r^4$ rather than $1/r^2$ — so they matter close in and vanish quickly. A geostationary satellite feels the bulge a hundred times less than a low one does, which is one reason transfers to it are planned with a point-mass model and flown with a corrected one.

Being inside. The theorem says nothing about the field within a body’s own radius except through the enclosed mass, and that is a real limitation rather than an academic one: it is why the tidal field exists at all, and why a galaxy’s rotation curve is a measurement of its mass distribution rather than of its total mass. Nothing else. Composition, internal structure, rotation and temperature are all invisible from outside as long as the density depends on radius alone. A hollow sphere and a solid one of the same mass and radius produce identical external fields, and no external measurement can tell them apart. That is a genuine and permanent limit on what gravity can reveal.

The tunnel, and what a body deep inside feels #

The interior result has a consequence that reads as a party trick and is a serious tool at other scales.

Drop something down a straight tunnel bored through the centre of a uniform planet. Above it, at every moment, is a shell of material that pulls on it not at all. Below it is a sphere of enclosed mass, which pulls as a point at the centre. The enclosed mass goes as $r^3$ and the field as $r^3/r^2$, so the restoring force is proportional to the displacement — which is Hooke’s law, and the motion is simple harmonic.

The period follows in one line: $2\pi\sqrt{R^3/GM}$, which for the Earth is 84.4 minutes. The object falls to the far side, arrives with zero speed, and comes back. It is not a coincidence that the same 84.4 minutes is the orbital period of a satellite skimming the surface — both are $2\pi\sqrt{R^3/GM}$, because a grazing orbit and a diametral tunnel are the same oscillation seen along different axes. A body dropped through the Earth and a body in the lowest possible orbit stay level with each other the whole way.

The serious version is what happens to a star inside a galaxy. It feels only the mass interior to its orbit, exactly as the tunnel object does, so measuring how fast stars move at each radius measures how much mass lies inside that radius. Do it for a spiral galaxy and the enclosed mass keeps growing far beyond where the light stops — which is the observation that made dark matter unavoidable. The whole argument rests on the interior half of the shell theorem: without it, the outer stars would feel the whole galaxy and the measurement would say nothing about where the mass sits.

The generalisation: it is Gauss’s law, and it explains the exponent #

The cone argument is Newton’s and it is elegant. It is also, in modern terms, a special case of something more general, and the more general statement explains why the inverse square is the exponent that makes it work.

The field of a point source spreads over spheres. The area of a sphere grows as $r^2$, so if the total flux through every sphere is the same — which is what it means for the source to be conserved and the space to be empty — the field strength must fall as $1/r^2$. The exponent is not a fact about gravity. It is a fact about three-dimensional space.

That reformulation is Gauss’s law, and once it is available the shell theorem takes three lines. Draw a sphere through the point of interest, concentric with the mass distribution. By symmetry the field is radial and constant on that sphere. The flux through it is the field strength times the area, and it equals the enclosed mass times a constant. Divide, and the field at radius $r$ is $GM_{\text{enc}}/r^2$ — which is both halves of the theorem at once, the interior case being the one where $M_{\text{enc}}$ is zero.

The surprising consequence is dimensional. In a space with $n$ spatial dimensions the same argument gives a force falling as $r^{-(n-1)}$, and Bertrand’s theorem then says that closed, non-precessing orbits exist only for the inverse square and the linear spring. Both conditions are met only in three dimensions. In four spatial dimensions gravity would fall as $1/r^3$, and a planet’s orbit would be unstable — the smallest perturbation would send it spiralling into the star or away forever.

The stability of orbits is therefore evidence about the dimensionality of space, and the chain runs entirely through the fact that a sphere pulls like a point.

One last observation about what the theorem costs. Because the external field of a spherically symmetric body carries no information about its interior, gravity alone can never determine what a planet is made of — only how much of it there is and how the mass departs from spherical symmetry. Everything known about planetary interiors comes from combining the gravity field with something else: seismology for the Earth, moment-of-inertia arguments from the response to tides, or the magnetic field. The theorem that makes orbits computable is the same one that makes interiors invisible.

The other shape it works for #

The theorem is usually stated for spheres and it is not confined to them, and the extension is worth knowing because it is genuinely surprising.

Consider a shell bounded by two similar, concentric, similarly oriented ellipsoids — an ellipsoidal shell of uniform density, thick or thin. The field everywhere inside the cavity is exactly zero, for the same reason as in the spherical case: the double-cone construction still cuts two patches whose masses and distances stand in the compensating ratio.

That is Newton’s result too, and it is stronger than the spherical one in an unobvious way. A rotating fluid body settles into an oblate spheroid rather than a sphere, and its interior is a stack of such shells — so the interior field of a rotating planet is far more tractable than its irregular shape suggests, and the classical theory of figures is built on exactly that.

What does not extend is the exterior half. Outside an ellipsoid the field is emphatically not that of a point mass at the centre: it carries the multipole terms that make satellite orbits precess and that this essay’s last figure draws. The interior result survives the loss of spherical symmetry and the exterior one does not, which is the opposite of what a first guess suggests, since it is the exterior case that seems the more forgiving.

The asymmetry has a clean reason. The interior cancellation is local — it happens cone by cone, and each cone’s cancellation needs only that the two patches are cut from surfaces of the same family. The exterior result is a statement about a whole integral, and the integral notices the shape.

One consequence is worth stating because it is the reason the theorem is used far outside gravity. The cancellation needs only that the force fall as the inverse square of distance and act along the line joining the two bodies; it has nothing to do with mass, or attraction, or gravity in particular. So the same result holds for electrostatics, and the field inside a charged conducting shell is zero for the same reason a cavity at the centre of the Earth would be weightless. The theorem is a statement about an exponent, and the experiments that test it most precisely are electrostatic ones — which bound the exponent’s departure from two at a few parts in $10^{16}$, far tighter than any gravitational test could.

One more reading shows the interior profile of the solid case beside the shell.

The ladder from here #

Later rungs on this anchor: Newton’s original integral, done the way he did it. Gauss’s law for gravity, and the divergence form. The interior field of a non-uniform sphere. The $J_2$ term derived, and the node and apsidal drift rates it produces. Spherical harmonic expansions of a planet’s field, and what the low-order terms say about its interior. GRACE and the time-varying field. The gravitational field inside a galaxy, and rotation curves. Ehrenfest’s argument about dimensionality and stable orbits. The Cavendish experiment and the measurement of $G$. And the shell theorem in general relativity, where Birkhoff’s theorem says something stronger and stranger: the exterior field of any spherically symmetric body, even one that is pulsating violently, is exactly static.

Newton delayed publishing for nearly twenty years, and one of the reasons he gave was that he could not prove a sphere pulls like a point. Until he could, the whole apparatus rested on an assumption he was not willing to make.