The sheet that cannot keep Kepler's orbits

There is hardly a science museum without one. A large disc of stretched fabric or rubber, sagging under a heavy ball at its centre, and a slot for visitors to roll marbles round it. The marbles circle the dip, spiral inward as friction takes their energy, and fall into the middle, and a sign explains that this is how gravity works: mass curves space, and planets follow the curve. As a picture of general relativity the rubber sheet has well-known problems. As a picture of Newtonian orbits it has a simpler one, which a few minutes with a marble and a protractor reveal. The orbits are not ellipses.

The orbit that does not come back to itself described Bertrand’s theorem: of all the central forces that fall off as a power of distance, only two give orbits that close for every starting condition, the inverse square of gravity and the linear force of a spring. Any other force law makes orbits that precess, their closest approaches creeping round, tracing rosettes. The stretched sheet does not give an inverse-square force. Its marbles are orbiting under a different law of gravity, and it shows.

What shape a stretched sheet takes #

A membrane under uniform tension $T$, pulled down at its centre by a weight $W$, takes a shape governed by Laplace’s equation in two dimensions. Away from the load the surface has no force on it but tension, and the vertical component of tension around any circle must carry the whole weight: $2\pi r\,T\,z'(r) = W$. So the slope is

$$z'(r) = \frac{W}{2\pi T\,r},$$

falling as $1/r$, and the surface itself is logarithmic, $z \propto \ln r$. A marble on that surface feels a horizontal force proportional to the slope, $g z'(r)$, which falls as $1/r$, not $1/r^2$.

The logarithm is not an accident of rubber. It is the potential of a point mass in two dimensions. The number of dimensions a planet can orbit in followed Gauss’s law into other dimensions: field lines spreading from a point through $d$ dimensions thin out as $1/r^{d-1}$, so gravity in two dimensions falls as $1/r$ and its potential is the logarithm. A membrane is a two-dimensional object obeying the two-dimensional Laplace equation, and it reproduces two-dimensional gravity exactly. The sheet is a faithful model of gravity — in Flatland.

Rosettes on the sheet #

The consequence is visible in a single orbit.

The orbit was computed from the full equation of motion for a ball sliding on the curved surface, which includes the effect of the surface’s slope on the ball’s inertia: moving in or out on a sloping surface also means moving up or down, so the radial motion carries extra kinetic energy, a factor $1 + z'^2$. With no friction, the ball would trace this rosette forever, its closest approaches advancing by about 130° each time. A planet under Newton’s gravity, released the same way, would trace a fixed ellipse with its closest approaches 180° apart, one per orbit.

For a nearly circular orbit under a force falling as $r^n$, the angle between the closest and farthest points is $\pi/\sqrt{n+3}$. For the inverse square, $n = -2$, that is 180°: one closest approach per orbit, a closed ellipse. For the sheet’s $1/r$, $n = -1$, it is $180°/\sqrt{2}$, 127°. That formula is the content of the precession calculation in the one extra force that only turns the orbit, and it is exactly what the integration finds for gentle orbits on the sheet.

Three force laws compared #

The angle between closest approaches is a sensitive fingerprint of the force law, and a protractor laid on a long-exposure photograph of a marble would read it.

The cone, whose slope is the same everywhere, gives a constant inward force, and its orbits have closest approaches about 104° apart, more than three per circuit. The sheet gives about 127°. Only the funnel, whose slope falls as $1/r^2$, gives close to 180°. So a careful observer with a marble could tell the three surfaces apart by watching a single orbit, without measuring their shapes at all — exactly as astronomers inferred the inverse-square law from the fact that planetary orbits close.

The funnel’s curve rises steeply for eccentric orbits, and that is a real feature, not a numerical artefact. A funnel shaped as $-1/r$ steepens without limit towards its centre, its slope growing as $1/r^2$, and an orbit that dives towards the throat enters a region where the slope is large and the factor $1 + z'^2$ in the ball’s radial inertia departs strongly from one. The inverse-square law is then reproduced for the horizontal force but not for the ball’s motion, which is no longer that of a particle in a plane. Only orbits that stay where the funnel is shallow close properly.

Newton’s protractor #

Reading a force law from the angle between closest approaches is not a museum trick; it is how the inverse-square law was first tested. In Book I of the Principia, Newton worked out the precession of nearly circular orbits under power-law forces, the same $\pi/\sqrt{n+3}$ that the integration reproduces here, and turned it round: measure how fast an orbit’s closest approach moves, and the exponent of the force follows. The planets’ perihelia hardly moved at all, which pinned the exponent to $-2$ with remarkable precision for the time. The Moon was a problem. Its closest approach to the Earth advances by about three degrees every month, which by the formula would mean a force falling as $r^{-2.016}$, a small but definite departure from the inverse square.

Newton suspected the Sun’s pull on the Moon was responsible, and the matter was settled only in 1749, when Alexis Clairaut carried the perturbation calculation to second order and found that the Sun’s tidal pull accounts for the whole advance. The inverse-square law survived, and the episode left behind the method this essay uses: an orbit’s precession is a sensitive gauge of whether the force is exactly inverse-square, and any surplus has to be explained by some other force. Mercury’s surplus of 43 seconds of arc a century, after all the planets’ perturbations were subtracted, was explained the same way by general relativity, as the orbit special relativity cannot close and its neighbours followed.

A marble on a museum sheet makes Newton’s measurement in a few seconds, with an error of tens of degrees rather than seconds of arc, and gets an answer Newton would have recognised at once: the force is not inverse-square, and the exponent is about $-1$.

Stable rosettes and the speed circle #

That the sheet’s orbits are rosettes rather than spirals is itself a statement about the force. The effective-potential construction that the orbit that cannot be made smaller uses to find the innermost stable orbit of a black hole decides it here too: a circular orbit is stable — a slightly disturbed orbit oscillates about the circle instead of spiralling away — only if the force falls more slowly than $1/r^3$. The sheet’s $1/r$ satisfies that easily, so a marble nudged off a circular orbit wobbles in and out about it, which is the rosette. In four spatial dimensions, where gravity would fall as $1/r^3$, a membrane analogue would show no stable orbits at all; every marble would spiral in or fly off. The essay on the number of dimensions a planet can orbit in drew that conclusion for planets, and the sheet, by being one dimension short, lands on the stable side with room to spare.

The inverse square has one more fingerprint that the sheet lacks. The velocity that always lies on a circle showed that under an inverse-square force the tip of the velocity vector, drawn from a fixed origin, traces an exact circle as the body goes round its orbit — Hamilton’s hodograph. On the sheet, the velocity’s tip traces a closed curve only when the orbit is circular; for a rosette it traces a rosette of its own. The exact circle in velocity space is one more consequence of the hidden symmetry that keeps inverse-square orbits closed, and the sheet, missing the symmetry, misses all of its consequences together.

Coin funnels #

The funnel does exist, in a different exhibit: the coin funnel, or “gravity well”, into which visitors roll a coin and watch it circle, faster and faster, before it drops through a hole in the middle. Its surface is a hyperboloid, $z \propto -1/r$, chosen precisely so that the coin’s orbits resemble planetary ones.

On a shallow part of the funnel the orbit is very nearly an ellipse, advancing by only a few degrees per circuit, and the coin’s speed increases as it moves inward exactly as a planet’s does, sweeping out equal areas in equal times. The steepness of the funnel’s throat spoils this as the coin descends, and friction makes the coin spiral in, which a planet does not. But the funnel is the right shape and the sheet is the wrong one, and the museums that have both are showing two different laws of gravity side by side under one explanation.

There is one more subtlety. A rolling coin or marble is not a sliding point. Some of its kinetic energy is in its spin, two-fifths of its translational energy for a solid sphere, and its effective inertia is larger than its mass by a factor $7/5$. That slows every orbit by $\sqrt{7/5}$ but, for a shallow surface, leaves every shape unchanged: rosettes stay rosettes with the same angles, and ellipses stay ellipses. A rolling ball also has gyroscopic effects on a curved surface, coupling its spin to its path, but they are small for gentle slopes.

Kepler’s third law, and two others #

The second clear difference is in timing.

For a circular orbit, the inward force supplies the centripetal acceleration: $v^2/r = g z'(r)$. On the sheet, $z' \propto 1/r$, so $v^2$ is the same at every radius. Every circular orbit on a stretched sheet has the same speed, and its period is simply proportional to its circumference. On the cone, $v^2 \propto r$ and the period goes as $\sqrt{r}$. Only on the funnel does $v^2 \propto 1/r$, and the period grows as $r^{3/2}$, Kepler’s third law.

The sheet’s equal speeds are a striking thing to look for. Marbles circling near the edge of a museum sheet and marbles circling close to the centre move at about the same speed, the outer ones simply taking longer to get round. In the solar system, Mercury moves at 48 kilometres per second and Neptune at 5. The sheet’s flat speed curve is, by coincidence, what galaxies show: stars far from a galaxy’s centre circle at nearly the same speed as those further in, which is part of the evidence for dark matter, since visible matter alone would make the outer stars slower. A stretched sheet is, accidentally, a better model of a galaxy’s rotation curve than of the solar system — though for an unrelated reason, since a flat rotation curve in a galaxy comes from mass spread out with distance, not from gravity being two-dimensional.

The sheet and the curvature of spacetime #

The rubber sheet is usually offered as an illustration of general relativity, in which mass curves spacetime and bodies follow the curvature. That analogy has well-known defects. The sheet’s marbles are pulled down by the Earth’s gravity, so the illustration uses gravity to explain gravity; the curvature that matters most in general relativity for slow-moving bodies is the curvature of time, not of space, which the sheet does not show at all; and the sheet’s shape is that of two-dimensional gravity anyway. A ball at rest on a curved-space surface would stay at rest; it is the warping of time, the slower ticking of clocks lower in the potential, that makes things fall.

The point can be made quantitatively. Near the Earth, the fractional difference in the rate of clocks one metre apart in height is about one part in 10¹⁶, as the clock that runs slow lower down measured, and that gradient in the rate of time is enough, by itself, to make an apple fall at 9.8 metres per second squared. The corresponding curvature of space near the Earth is of the same tiny size, but it affects a slow apple only in proportion to the square of its speed divided by that of light, about 10⁻¹⁶ for a falling apple, and is utterly negligible. A sheet that curves only space, then, would leave the apple where it is. It is the bending of time that a picture of falling has to show, and no shape of rubber can show it. None of that makes the sheet useless. It does show, correctly, that a central dip in a potential makes things orbit, that faster things orbit further out, and that orbits precess unless the force has a special form. But what it shows most precisely is the difference between force laws, which is the opposite of what it is usually used to illustrate. The arrow that says which way the orbit points described the hidden conserved quantity, the Laplace–Runge–Lenz vector, that holds an inverse-square orbit’s ellipse fixed in space. The sheet’s marbles have no such quantity, and their rosettes are what an orbit looks like when it is missing.

What the picture cannot show #

The orbits are computed for a frictionless point sliding on an ideal surface, with constant gravity. Real exhibits have friction, rolling resistance and air drag, which make every marble spiral inward within a few circuits, and the rosette pattern is visible only for the first orbit or two. A sheet’s shape is logarithmic only far from the load and close to the centre the heavy ball itself, and the sheet is distorted by the marble’s own weight, which adds a small dimple that travels with it. The figures use a slope of 0.2 at the reference radius for the shapes and orbits on the sheet and cone and 0.05 for the funnel’s orbit; steeper surfaces add larger corrections from the $1 + z'^2$ factor.

The apsidal angles quoted, 104°, 127° and 180°, are the limits for nearly circular orbits on shallow surfaces. The integrations show how they drift for more eccentric orbits; the drift depends on the surface’s steepness, and a different choice of slope would move the curves. The treatment of a rolling ball as a sliding point with extra inertia is an approximation valid for gentle slopes; on steep surfaces, rolling introduces effects of its own.

Still open: what a surface can model #

A curved surface under gravity can mimic any central force, by choosing its shape so that its slope matches the force. The limitation is the $1 + z'^2$ factor, which makes the mimicry inexact wherever the surface is steep, and the requirement of gravity to drive it. Physicists have built more elaborate analogues of gravitational systems — fluid flows in which sound behaves like light near a black hole, optical materials that bend light as curved spacetime does, ultracold atoms whose excitations feel an effective geometry — and debate continues about how much such analogues can teach about the systems they imitate, and where they mislead. The rubber sheet is the oldest analogue and the one with the clearest failure: it shows two-dimensional gravity and is used to explain three-dimensional, and the difference is written in every orbit it draws.

A stretched sheet under a central weight sags as ln r, the potential of gravity in two dimensions, so its marbles feel a force falling as 1/r, trace rosettes with closest approaches about 127° apart for gentle orbits, and circle at the same speed at every radius; a cone gives 104° and periods as $\sqrt{r}$; only a $-1/r$ funnel gives Kepler’s closed ellipses and periods as r^(3/2), and then only where it is shallow. The habit worth carrying is to ask what law an analogy actually obeys, because a model can look right and still be the answer to a different question.