The orbit that does not come back to itself

Planetary orbits are ellipses that repeat. It is easy to take that as the natural behaviour of a body going round another and to treat the ellipse as a consequence of the orbit being bounded. It is not. Under almost every central force a bounded orbit runs between a nearest and a furthest radius for ever without ever coming back to where it started, tracing a rosette that eventually visits every point of the annulus between the two.

The claim that only two force laws close every bounded orbit is Bertrand’s theorem, proved in 1873, and the interesting part is what it takes to state it correctly.

What a bounded orbit is guaranteed #

Angular momentum is conserved for any central force, so the motion stays in a plane and the radial motion decouples: the radius obeys a one-dimensional equation in an effective potential that adds the centrifugal term to the real one.

The radial motion becomes a one-dimensional problem, and a particle with energy below the top of that well oscillates between two turning points for ever. That is what a bounded orbit is guaranteed: the radius is periodic. What is not guaranteed is that the angle swept between two successive closest approaches is a rational fraction of a turn — and closure needs both.

So a bounded orbit is guaranteed to be periodic in radius. What is not guaranteed is that it should be periodic in angle at the same time, and closure requires both periods to be commensurate.

The relevant quantity is the apsidal angle: the angle swept between one closest approach and the next furthest one, which is half a radial period. An orbit closes when that angle is a rational multiple of $\pi$, and repeats after however many circuits the denominator requires.

For a nearly circular orbit the angle can be got in one line. Perturb a circular orbit and the radial oscillation has frequency $\omega_r$ while the angular motion has frequency $\omega_\theta$, and their ratio is $\sqrt{3 + \mathrm{d}\ln F/\mathrm{d}\ln r}$. For $F \propto r^n$ that gives an apsidal angle of $\pi/\sqrt{3+n}$: $\pi$ at $n = -2$, $\pi/2$ at $n = 1$, and something irrational nearly everywhere else.

The half of the theorem that is easy to miss #

That is not yet Bertrand’s theorem, and the gap is where the content is.

The formula above holds for orbits that are nearly circular. A force law could give a rational apsidal angle at one particular eccentricity by coincidence, producing one closed orbit; the theorem is about laws for which every bounded orbit closes, and that requires the angle to be independent of the amplitude.

That flatness is the theorem. A closed orbit at one amplitude and not at another is an accident of initial conditions rather than a property of the force, and the two exponents that keep their angle are the two for which the orbit’s shape is genuinely a consequence of the law.

Bertrand’s proof works by expanding the apsidal angle in the eccentricity and requiring every term beyond the first to vanish. The first condition fixes the exponent to make $\pi/\sqrt{3+n}$ rational; the next kills all but two of the survivors; the third finishes the job. It is not a deep argument, and its shape is worth noticing: an infinite family of conditions, of which the first admits many candidates and the rest eliminate all but two.

Measuring the angle rather than deriving it #

The figures here obtain the apsidal angle by integrating an orbit in Cartesian coordinates and finding the turning points of the radius on the resulting path, rather than by evaluating the quadrature that gives it. That is deliberate, and it is worth saying why, because the quadrature is not hard:

$$\Phi = \int_{r_{\min}}^{r_{\max}} \frac{L\,\mathrm{d}r}{r^2\sqrt{2m\left(E - V_{\text{eff}}(r)\right)}}.$$

The trouble is that the integrand is singular at both limits, which are precisely the two turning points — the square root vanishes there — so evaluating it numerically requires a substitution chosen to remove the singularity, and the substitution differs from case to case. More importantly, a figure that plots that integral is drawing the formula it is supposed to be testing.

Integrating instead produces a path, and the angle is then read off the path the way an observer would read it off an orbit: find successive closest approaches, take the difference of their polar angles. Everything the figure claims is a property of the drawn curve. The near-circular closed form $\pi/\sqrt{3+n}$ then has a job — it appears as a dashed comparison, and it disagrees with the measurement if the integration is wrong.

One numerical detail is worth recording because it produced a wrong answer that looked entirely reasonable. Apsides are turning points, so the step at which one is detected has the step’s own resolution, and an error of one step in the apsis position is an error of a step in the angle. Interpolating a parabola through the three samples around the turning point removes it, and without that the measured angle for the inverse square came out at 179.6° rather than 180° — close enough to look like a rounding issue and far enough to be visible on a plot of angle against eccentricity, which is exactly the plot the theorem is read from.

Why those two, and what they have in common #

Both closing laws have an extra conserved quantity that no other central force does, and the closure is a consequence of it — which is a symmetry handing over a conservation law in the case where the symmetry is not one anybody expected.

For the inverse square there is the Laplace–Runge–Lenz vector, which points from the focus toward perihelion and has a fixed length proportional to the eccentricity. It is conserved, so perihelion does not move, so the ellipse does not turn. For the linear force there is a conserved tensor doing the same job for the ellipse centred on the origin.

An orbit’s failure to close is exactly the precession of that vector, and every perturbation that breaks the pure inverse square makes it precess — a direction that returns to itself slightly late, which is the same word doing quite different work from a spinning top’s. So the question “does the orbit close” and the question “is there another conserved quantity” are the same question, and the second is the more useful phrasing because conserved quantities can be looked for systematically.

A quantity spreading in three dimensions and conserved in transit dilutes as the inverse square, which is why that exponent is not one among many. It is the exponent a conserved flux in three dimensions must have — so the two force laws Bertrand’s theorem picks out are the one geometry hands over and the one a spring gives, and there is no third candidate waiting to be found.

A line source falls as the first power and a plane not at all, so the exponent depends on the shape of the source rather than on the law. That is worth having beside the theorem because it shows what is special about the inverse square: it is the point-source case in three dimensions, and Bertrand’s result is about central forces, which is to say about point sources.

The lever between the exponent and the observation #

Bertrand’s theorem has a practical corollary. Since only the exact inverse square closes, any departure from it shows up as a rotation of the line of apsides, and the size of the rotation is a measurement of the departure.

The lever is enormous. A departure of one part in a hundred gives 6,483 arcseconds of apse motion per orbit; the innermost planet, at four hundred orbits a century, would show 720 degrees of it in that time. Run the arithmetic backwards on the actually observed unexplained residual — 43 arcseconds per century, which is a tenth of an arcsecond per orbit — and the corresponding departure is $1.6\times10^{-7}$. That residual is not a departure from the inverse square at all in the end, but a term free fall cannot remove reappearing in the equation of motion.

That is why the residual mattered so much. It is not that the discrepancy was large but that the quantity is measured with a lever of ten million, and any modification of the force law large enough to explain it would have destroyed the agreement everywhere else. The explanation that survived is not a change to the exponent at all: general relativity adds a term falling as the inverse fourth power, which is negligible except very close in.

The relativistic effective potential has an additional term with a different power of the radius, so it is not a small correction to an exponent — it is a new term, and it leaves the orbit unclosed by an amount that does not vanish as the orbit is made larger. That is why Mercury’s precession is a test of the form of the theory rather than of a parameter in it — the same reason the bending of starlight settled the question and a more accurate measurement of the gravitational constant could not.

The other things that break closure, and why they are easier #

Mercury’s 43 arcseconds are the residue after everything else has been subtracted, and the everything else is much larger. The total observed precession is about 5,600 arcseconds per century, of which about 5,025 is the precession of the coordinate frame itself and about 530 is the pull of the other planets.

Both of those break closure without touching the exponent. A planet perturbed by other planets is not in a central force at all, so Bertrand’s theorem does not apply and no closure is expected. What is striking is that the disagreement is small enough that the residual could be identified in the first place: the solar system is dominated by one mass so completely that the two-body problem is a good approximation to parts in $10^3$, and a subject where the leading approximation is that good is one in which small residuals are informative.

A circular orbit is the special case everything is measured against: the radius does not oscillate at all, so the question of an apsidal angle does not arise. Every statement about closure is a statement about orbits that are nearly circular or not circular, and the circular case is the degenerate one where the theorem has nothing to say.

A pendulum’s level sets close because the motion is periodic in its single coordinate, and one coordinate cannot fail to close. That comparison locates the difficulty exactly: closure is a question about two periods being commensurate, so it can only arise once there are two, and a system with one degree of freedom is closed for a reason that has nothing to do with its force law.

What closure buys, and what it cost when it went #

It is worth asking why the closure of orbits mattered historically, given that no observation ever showed an orbit closing exactly.

The answer is that closure is what makes an orbit describable by a small number of fixed parameters. An ellipse with a fixed orientation is six numbers, and it stays six numbers for ever; a rosette needs a seventh that grows without bound, and any table of it goes out of date. Every ephemeris before the nineteenth century was built on the assumption that the elements were constants, and the whole apparatus of celestial mechanics — the osculating elements, the variation of parameters, the secular and periodic terms — is machinery for treating slowly changing elements as almost-constants.

So the discovery that the elements do change was not a small correction to a calculation but a change in what kind of object an orbit is. Le Verrier’s careful accounting of Mercury’s precession in 1859 was the extreme case: he found 38 arcseconds a century unaccounted for, later refined to 43, and the size of the anomaly was one part in $10^5$ of the total precession — which is a measure of how good the almost-constant description had become.

The proposals for explaining it are a useful catalogue of what a residual invites. An unseen planet inside Mercury’s orbit, which was searched for and named before it was found not to exist. A slight oblateness of the Sun, which would have produced an extra term but is measured to be far too small. And a modification of the exponent, which the lever above rules out immediately: the departure needed is $10^{-7}$, and the same departure would have thrown the outer planets off by amounts long since excluded. Only the fourth proposal — an additional term of a different power, arising from a different theory of gravity altogether — fitted the constraint that the effect be large at Mercury and invisible everywhere else.

The other closing law, where it is actually found #

The linear force appears in the figures as a mathematical companion to the inverse square, and it is easy to read it as one. It is not: it is the exact force inside a uniform sphere, and the orbits it produces are realised in a place worth describing.

Apply Gauss’s law inside a body of uniform density and only the mass within the current radius counts. That mass grows as $r^3$ and the force it exerts falls as $r^{-2}$, so the product is proportional to $r$: the gravitational field inside a uniform sphere rises linearly from zero at the centre to its surface value. That is $F \propto -r$, the second of Bertrand’s two laws, exactly.

So a body moving in a cavity inside a uniform planet traces an ellipse centred on the planet’s centre, and it closes. Every orbit closes, at every eccentricity, which is what the flat line in the amplitude figure says.

The period of that motion is the striking part, because it contains nothing about the orbit. For a harmonic force the period is independent of amplitude, so every path inside the sphere — a wide ellipse, a narrow one, a straight line through the centre — takes the same time, and that time depends only on the density:

$$T = \sqrt{\frac{3\pi}{G\rho}}.$$

For the Earth’s mean density it is 84.4 minutes.

Two consequences follow that are usually met as curiosities and are really the same fact. A tunnel bored straight through a uniform Earth — along any chord, not only through the centre — would let a dropped object oscillate from one end to the other and back in 84.4 minutes, because the component of a linear force along a straight line is itself linear. And a satellite in a circular orbit skimming the surface takes 84.4 minutes too, because at the surface the interior solution and the exterior one meet. The tunnel and the orbit are the same ellipse, one squashed flat and one opened out.

That number recurs in navigation. A pendulum tuned to an 84-minute period has the property that accelerating its support does not deflect it from the local vertical, which is what a Schuler-tuned inertial platform exploits — and the period is 84 minutes for the same reason.

The real Earth is not uniform. Its core is four times denser than its mantle, so the field does not rise linearly and the orbits inside do not close. Computing the fall through the centre with a measured density profile gives about 38 minutes rather than 42, which is a substantial correction and a nice illustration of how much of this argument depends on the word uniform.

Why it is these two and not two others #

The essay has said that the two closing laws each have an extra conserved quantity, which is true and reads as a coincidence — two unrelated exponents that happen to share a property. They are not unrelated, and the relation between them is the sharper answer to the question.

Treat the orbital plane as the complex plane and map every point to its square. An orbit under the linear force becomes, after the map and a reparametrisation of time, an orbit under the inverse square. The geometry follows immediately: an ellipse centred on the origin, squared, is an ellipse with the origin at a focus. The two orbit shapes in the opening figure are the same curve seen through the map.

The correspondence extends to a whole family. Two power-law exponents are dual in this sense when

$$(n+3)(n'+3) = 4,$$

and squaring is the case that takes $n = 1$ to $n' = -2$. The apsidal angles inherit the relation: since $\Phi = \pi/\sqrt{3+n}$, dual exponents have $\Phi\Phi' = \pi^2/2$.

That last identity is the answer. For both members of a dual pair to close, both angles must divide a full turn, and their product must be $\pi^2/2$. The only way to write $\pi^2/2$ as a product of two such angles is $\pi \times \pi/2$ — a half turn and a quarter turn — and those are the inverse square and the linear force. Bertrand’s two laws are a pair rather than two separate accidents, and the pairing is what makes the list finite.

The observation is old, in the sense that Newton knew the two shapes were both ellipses and the map was found by Bohlin in 1911, and modern in the sense that Arnold made it the centrepiece of an essay arguing that Hooke’s law and Newton’s are one law in two coordinate systems. It also explains a historical oddity: Hooke, who guessed the inverse square, had spent years on the mechanics of springs, and the two problems he worked on turn out to be the same problem.

Where the model stops #

Central forces only. Everything here assumes the force points along the line joining the two bodies and depends only on the separation. Drop either and angular momentum is not conserved, the motion need not stay in a plane, and the notion of an apsidal angle stops being well defined.

Two bodies. The theorem is about one particle in a fixed central field, which describes two bodies exactly through the reduced mass and describes three bodies not at all.

No dissipation. A bounded orbit is bounded because energy is conserved. Any drag — gas, tides, gravitational radiation — makes the orbit shrink, and the question of whether it closes becomes secondary to the question of how long it lasts.

Power laws. The figures cover $F \propto r^n$, and Bertrand’s theorem covers every central force whatever, which is a stronger statement than the figures can demonstrate. What the figures show is that the closing exponents are isolated within the power-law family, and the theorem’s content is that adding all the non-power-law forces to the search finds nothing new.

What the pictures cannot show #

The rosettes are drawn for six radial oscillations because more would fill the annulus and become a black ring. What the figure therefore cannot show is the actual long-term behaviour of a non-closing orbit, which is that it comes arbitrarily close to every point between the two radii — and no finite drawing can distinguish that from a curve that would eventually close after very many circuits.

The apsidal-angle plot ends at the exponent −3 with an empty region beyond, and that emptiness is the physics rather than a limitation of the computation: there are no bounded orbits to measure there. But an empty region on a graph looks like a failure to compute, and the figure has no way to say which it is except in the caption.

And nothing here shows the conserved vector. The Laplace–Runge–Lenz vector is the reason the ellipse holds still, it points at perihelion, and it does not lie in any of the spaces these figures are drawn in — it is a property of the orbit rather than a place on it.

Where the ladder goes next #

This ladder began with the innermost orbit that can exist at all and has now asked a question about the whole family: which force laws produce orbits that repeat. The rungs beyond are the perturbation theory that turns “does not close” into “precesses at a computable rate”, the resonances that appear when two orbital periods are commensurate, and the extra conserved quantity itself, which is the modern way to state the whole subject.

The habit worth carrying away is about how special cases become invisible. Ellipses are treated as the natural shape of an orbit because the only orbits anybody sees are ellipses, and the only orbits anybody sees are ellipses because gravity happens to be one of the two laws in existence with that property. A feature shared by every example available is easy to mistake for a feature of the subject, and the test is to compute what a neighbouring case would do — which here takes one exponent, one integration, and a rosette.