Only two force laws let an orbit come back

An orbit has two clocks in it and they are not the same clock. One counts the time from periapsis back to periapsis — the radial period, the time for the body to fall in, turn, and climb out again. The other counts the time to go once round in angle. For a planet the two agree exactly, and because they agree the path shuts on itself and the same ellipse is retraced for ever.

Nothing requires them to agree. The well angular momentum builds has a bottom, and a body displaced from the bottom of a well oscillates radially while it also revolves; whether the two rhythms are commensurate is a question about the shape of the well, and the shape of the well is a question about the force. Change the exponent by a hundredth and the two clocks drift apart, the path never repeats, and what was an ellipse becomes a rosette that fills an annulus.

The remarkable part is how few exponents let the clocks agree. Not a sparse set, not a family with parameters: two.

The angle that has to be measured #

The quantity to compute is the angle the body sweeps while its radius goes from its smallest value to its largest. Call it the apsidal angle. If it is $180°$ the far end of the orbit is directly opposite the near end, the next half-lap retraces the first by symmetry, and the path is closed after one revolution.

Getting it is a matter of dividing one rate by another. The angular rate is fixed by the conservation law that equal areas express, $\dot\theta = L/r^2$. The radial rate comes from the energy, $\dot r = \sqrt{2(E - U_{\rm eff}(r))}$. Their ratio is $d\theta/dr$, and integrating it between the turning points gives

$$\Phi = \int_{r_-}^{r_+} \frac{L/r^2}{\sqrt{2\left(E - U_{\rm eff}(r)\right)}}\,dr .$$

That integral is the whole subject, and it has an unpleasant feature: at both ends the square root vanishes, because a turning point is by definition where the radial speed is zero. The integrand goes to infinity at each limit, integrably but not gently, and no evenly spaced rule in $r$ survives it.

The standard escape is a change of variable that makes the singularities cancel. Write the radius as the midpoint of the two turning points plus a sine, $r = A + B\sin\theta$ with $A$ the mean and $B$ the half-difference. Then $dr$ carries a factor of $\cos\theta$, and near each end the vanishing square root also behaves like $\cos\theta$, so the two cancel and what is left is a smooth function of $\theta$ over half a turn. Every apsidal angle in this essay is that integral, evaluated at the midpoints of four thousand equal steps, which never lands on either end.

The other half of the setup is less obvious and matters more. To compare force laws fairly, every orbit drawn has to be the same orbit in some sense, and the natural sense is that it spans the same pair of radii. So the energy and angular momentum are solved for from the turning points rather than chosen: demanding that the radial speed vanish at both $r_-$ and $r_+$ is two equations in the two unknowns, and it fixes them. An orbit that runs from 0.6 to 1.4 under one force law is compared with an orbit that runs from 0.6 to 1.4 under another, and nothing else about them is held fixed, because nothing else can be.

Why the easy answer is not the answer #

There is a closed form, and it is the reason this question looks settled until it is examined.

Take an orbit only slightly eccentric. The body sits near the bottom of the well and executes small radial oscillations about the circular radius. A well is quadratic near its minimum, so the oscillation is simple harmonic, and its frequency follows from the second derivative of the effective potential there. Dividing that by the angular frequency gives

$$\Phi_{\rm circ} = \frac{\pi}{\sqrt{3+p}} ,$$

for a force $F \propto r^{p}$. It is three lines of algebra and it contains a great deal. At $p = -2$ it gives $\pi$, which is the closed ellipse. At $p = +1$ it gives $\pi/2$: a quarter turn from periapsis to apoapsis, so four of them make a full revolution and the orbit closes after two radial periods. As $p$ falls toward $-3$ the denominator vanishes and the angle diverges, which is the statement that the circular orbit is losing its stability — below $-3$ there is no minimum in the effective potential at all, only a monotone slide, and a body perturbed inward keeps going.

The temptation is to stop there. If $\Phi_{\rm circ}/\pi$ is a rational number the orbit closes, and rational numbers are dense, so closure looks common: $p = -2.25$ gives $\Phi = 120°$, three of which make a full turn, and the path shuts after three radial periods.

It does — for that one orbit. The trap is that $\Phi_{\rm circ}$ is the limit of the apsidal angle as the eccentricity goes to zero, and a force law that closes only in that limit closes only the orbits that are already circles.

The distinction is worth stating plainly because it is the difference between a curiosity and a theorem. A force law closes a single orbit if its apsidal angle for that orbit is a rational multiple of $\pi$; it closes every bound orbit only if its apsidal angle is the same rational multiple for all of them at once. The second is an enormously stronger demand, and Bertrand proved in 1873 that exactly two power laws meet it.

The proof is an expansion of the apsidal angle in the eccentricity. The first correction beyond the near-circular value involves the third and fourth derivatives of the potential, and setting it to zero for all orbits forces the potential into one of two forms. Continuing to the next order confirms there is nothing else. The result is short to state and was not easy to find, and it is one of the few theorems in classical mechanics whose content is an absence.

Watching the clocks separate #

The quadrature produces a number; a number is easier to believe when the path is drawn from the same integral that measured it.

The second of those is what a non-closing orbit looks like when the departure is not small, and the eye reads it as a different kind of motion. It is not. It is the same two clocks running at rates whose ratio is irrational, and the ellipse of the first figure is the exceptional case rather than the normal one.

The centred ellipse deserves a sentence of its own. A body on a spring moves as two independent harmonic oscillators, one along each of two perpendicular axes, at the same frequency and with whatever relative phase the initial conditions supply. The sum of two perpendicular sinusoids of equal frequency is an ellipse about the origin, always, and the source of the force sits at the middle. So both closing laws give ellipses and the ellipses are not the same ellipse: one has the attractor at a focus and the other at the centre, and no continuous change of the exponent carries one into the other, because everything between them fails to close.

Newton got there first and got the number wrong #

The apsidal angle was a test of the force law a hundred and eighty years before Bertrand classified the laws, and the episode is worth recounting because it shows the measurement being used in the direction it is actually useful.

In the first book of the Principia Newton works out what happens when an inverse-cube term is added to any central force: the orbit keeps its shape in a rotating frame, so a closed orbit becomes a precessing one at a rate the added term fixes exactly. He then runs the argument backwards. The Moon’s line of apsides makes a complete circuit in 8.85 years, which is a precession of about three degrees a lunar month, and inverting his own result turns that into an effective exponent for the force holding the Moon.

The answer came out near $-2.016$ rather than $-2$, and Newton reported it as a confirmation with a discrepancy he attributed to the Sun’s perturbation. He was right about the cause and wrong about the size: the solar perturbation accounts for the whole of the Moon’s apsidal motion, but the first-order calculation he could do gives only half of it. The missing half occupied Clairaut, d’Alembert and Euler for a decade in the 1740s, and was briefly taken as evidence that the inverse-square law needed an extra term. Clairaut carried the expansion to second order in 1749, found the other half, and withdrew the objection.

The whole of that argument rests on the fact this essay is about. An apsidal motion is a measurement of the force law only because the exact inverse square has none, and a disagreement about a precession is therefore a disagreement about gravity rather than about an orbit. The eighteenth century’s near-miss and the nineteenth’s genuine residual at Mercury are the same measurement made twice, with the difference that the second one survived every subtraction.

The modern bound from the same observable is tight. Lunar laser ranging determines the Moon’s distance to a centimetre and its apsidal motion correspondingly well, and after every Newtonian term has been removed the residual constrains a departure in the exponent to a few parts in $10^{11}$. The measurement Newton made with a calendar is still the measurement, six orders of magnitude better.

The two cases are not obscure #

It would be a tidier story if the two exceptions were mathematical accidents with no physical tenants. They are the two most common force laws in the subject.

The inverse square is gravity outside any spherical body, which the shell theorem guarantees exactly rather than approximately. Everything this collection says about planets, moons, binary stars and spacecraft lives there.

The linear law is gravity inside a uniform sphere. Enclosed mass grows as the cube of the radius, the inverse-square attraction of that mass falls as the square, and the product is proportional to $r$. So a body dropped down a shaft through an idealised planet executes simple harmonic motion, which is why the tunnel takes the same time whichever pair of points it joins, and a body given some sideways motion in the same shaft traces a closed centred ellipse.

The only two force laws in the continuum that close every orbit are the field outside a sphere of matter and the field inside one. Nothing in Bertrand’s proof knows about spheres. The theorem is about the exponents 1 and $-2$, and it happens that those are the two exponents that a lump of matter actually produces, on either side of its own surface.

That is the kind of coincidence worth being suspicious of, and the suspicion does not survive inspection: the two exponents are related by nothing except that $1$ and $-2$ both give $3+p$ equal to a perfect square of a rational, which is what the closed form needs. There is no deeper connection, and the collection is entitled to find the arrangement fortunate rather than meaningful.

There is also a direction to the failure, and it is easy to lose. Between $p = -3$ and $p = -2$ the denominator $\sqrt{3+p}$ is less than one, the apsidal angle exceeds $180°$, and the apsides advance — the orbit precesses the way the body goes round. Between $p = -2$ and $p = +1$ the angle is under $180°$ and the apsides regress. So the inverse square is not merely a point where closure happens; it is the boundary between two qualitatively different behaviours, and a measured precession has a sign that says which side of it the force is on. Mercury’s residual advances, which excludes a shallower-than-inverse-square attraction outright and was noticed as a constraint long before anyone could explain the number.

What a small departure does #

Between the two exceptions the whole continuum fails, and most real systems sit slightly off one of them rather than far from both. The useful question is therefore not whether an orbit closes but how fast its apsides turn when it nearly does.

Expand the apsidal angle about $p = -2$. A small excess $\delta$ in the exponent shifts $\Phi_{\rm circ}$ by $-\pi\delta/2$ to first order, which means the apsides advance or regress by $\pi\delta$ per revolution — a precession rate directly proportional to the departure from the inverse square, with no threshold beneath which it vanishes.

That linearity is why the measurement is worth making. The forty-three arcseconds a century left over in Mercury’s motion are a departure of this kind, though not a change in the exponent: general relativity adds a third term to the effective potential rather than altering the second, and the third term is negative and steep. The observable is the same, and so is the logic — an orbit that did not shut was the evidence, and it was evidence because for an inverse square it had to.

That last point is the honest limitation of the drawing this whole argument is built on, and it is worth being blunt about. The effective potential answers where a body can go with complete authority. It answers how the body’s angle advances only through an integral that has to be done, and the two closing exponents are invisible in the curve.

What actually closes, and what has been checked #

The theorem is about idealised power laws, and no physical system is one. It is fair to ask what the observational content is.

The strongest statement is negative and it is about the solar system. Planetary apsides do turn: Mercury’s line of apsides advances by 574 arcseconds a century, of which all but 43 comes from the pull of the other planets. The departures from a pure inverse-square problem are therefore large and known, and what makes the residual interesting is that everything else was subtracted first. Bertrand’s theorem is what licenses treating an unexplained precession as evidence of new physics, because under the exact inverse square there is no precession to explain away.

The laboratory version is sharper. Torsion-balance experiments look for departures from the inverse square at short range by comparing the force between masses at different separations, and they constrain a fractional change in the exponent to better than a part in $10^{10}$ over laboratory distances. Planetary ranging does the same over astronomical ones. Neither experiment is described as a test of Bertrand’s theorem, but the quantity being bounded is exactly the $\delta$ whose linear consequence is a precession.

And there is a case where the exponent is genuinely different and the rosettes are the observed behaviour. A star orbiting inside a galaxy feels the enclosed mass rather than a point mass, and in the region where the rotation curve is flat the enclosed mass grows in proportion to the radius, so the force falls as $1/r$ — the logarithmic potential, $p = -1$.

The measurement that reaches this is not of one star’s rosette, which nobody will watch for a hundred million years. It is the ratio of the radial to the angular frequency in the solar neighbourhood, read off the velocity distribution of nearby stars, and it comes out near $1.4$ rather than the $1$ a closed orbit would need. The rosette is inferred from a statistic rather than watched, which is the ordinary situation in this field and worth saying out loud.

Where the theorem stops #

It is about power laws, and gravity in a real system is not one. A potential that is inverse-square at one radius and logarithmic at another has an apsidal angle that varies from orbit to orbit, and Bertrand’s classification simply does not apply to it. What survives is the local statement: at each radius the effective exponent sets the local precession rate.

It is Newtonian. Adding the relativistic $-GML^2/c^2r^3$ term to the effective potential produces an apsidal angle greater than $180°$ for every bound orbit, so no relativistic orbit closes, including the ones around the Sun. The ellipse of the first figure is an approximation that happens to be superb, and the precession it fails to contain is how the failure was found.

And it says nothing about stability. A closed orbit and an unclosed one are equally stable or unstable; closure is about commensurability, stability about the sign of a second derivative. Both change at $p = -3$, where the well disappears, and at every other exponent they are unrelated properties — which is worth holding onto, because “the orbit does not close” has the sound of an instability and is not one.

Why an absence is the interesting result #

A theorem that classifies is usually valued for what it admits. This one is valued for what it excludes, and the pattern is worth naming because it recurs throughout this subject.

The strength of the result is not that the inverse square closes orbits — that was known from the conic sections and is visible in the conserved vector that points at periapsis, whose very existence is the closure written as a constant of the motion. The strength is that nothing else does. Before Bertrand, an unexplained precession could be attributed to a slightly different exponent, and the attribution was not obviously wrong. After Bertrand, any exponent other than $-2$ predicts a precession for every orbit with a magnitude tied to the departure, so one clean measurement of one planet bounds the exponent for all of them.

That is the same structure as the over-determination arguments elsewhere in this collection, arrived at from the other direction. A cluster weighed three ways is convincing because three independent routes agree; this is convincing because one route excludes a continuum. Both replace a fit with a test, and a test is what a measurement has to be able to fail.

There is a last consequence worth drawing out, about what the extra conserved quantity is doing. A general central force conserves energy and angular momentum, which between them reduce the problem to the radial one these essays are built on and no further. The inverse square conserves something more, and so does the spring — a vector in the first case and a tensor in the second — and those extra constants are precisely what pins the orbit’s orientation and stops the apsides moving. Bertrand’s theorem and the existence of that extra constant are two statements of the same fact, and the second is the one that generalises: look for a closed orbit and one finds a conservation law that was not asked for.

Still open: what the next question is #

The well has been treated throughout as a fixed curve with a body sliding along it, and the next question is what happens when the curve itself moves — when the mass at the centre changes, or the angular momentum is slowly taken away. The energy is then not conserved and neither is the shape of the orbit, and what does survive is neither of the two quantities this essay has been dividing by. It is the area enclosed in the radial phase plane, it is conserved only if the change is slow, and how slow is slow enough is the whole of the argument.