The arrow that says which way the orbit points

The orbit that does not come back to itself establishes Bertrand’s result: of all the central force laws there are, exactly two produce orbits that close, and one of them is the inverse square. That is a statement about what happens and not about why.

The why is that the inverse square has a conserved quantity nothing else has.

What is missing from energy and angular momentum #

Start with what the two familiar constants do not say.

The energy fixes the semi-major axis. The angular momentum, together with the energy, fixes the eccentricity. Between them they determine the size and shape of the orbit completely — and they say nothing whatever about which way it points. Rotate an ellipse about the focus and neither constant changes.

For a general central force that freedom is real: the orbit precesses, and its orientation is a function of time rather than a constant. For the inverse square it is not, and something must be keeping it fixed.

That something is a vector, discovered and rediscovered so often that its name is a committee. It is built from the position and the velocity at any one instant, and the figure constructs it at five separate points on the orbit and gets the same arrow every time.

The construction is short enough to give: take the velocity, cross it with the angular momentum, and subtract the gravitational parameter times the unit vector pointing outward. Differentiating that with respect to time and using the equation of motion gives zero — but only if the force is exactly inverse-square, because the cancellation between the two terms depends on the outward unit vector’s rate of change matching the acceleration’s.

It points along the major axis and its length is the eccentricity. Those two facts are exactly the missing information: a direction and a shape parameter, which with the energy complete the description of the orbit.

That last clause is the whole of Bertrand’s theorem seen from the other side. A force falling as anything but the inverse square leaves a residue when the derivative is taken, the vector is no longer constant, and the orbit’s axis is free to move — which is precisely the precession the earlier rung measured off integrated paths.

The length is the eccentricity, checked #

The identification of the vector’s length with the eccentricity is worth checking rather than asserting, because it is the part that makes the quantity useful rather than merely conserved.

One of the two numbers is local: take the position and velocity at an instant, form the vector, measure it. The other is global: follow the whole orbit, find its nearest and furthest approach, take the ratio. Nothing about the first calculation looks at more than a moment and nothing about the second is available from one.

They agree, and the figure integrates five separate orbits to show it. That is the practical value of a conserved quantity — it turns a question about a whole trajectory into an arithmetic operation on one instant’s data.

The same trick is what makes orbit determination possible from a short arc of observations. Two positions and the time between them give a velocity; a velocity and a position give the vector; and the vector gives the shape and the orientation of the whole orbit without ever seeing the rest of it.

What breaks it #

The conservation is fragile in a specific and informative way: it survives no change to the force law at all.

Adding a term that falls as the inverse cube — small enough that the orbit is still very nearly an ellipse — makes the vector turn. It does not become noisy or oscillate within each orbit; it rotates steadily, which is what a nearly-conserved quantity does when the thing conserving it is nearly present.

The rate can be measured two ways. The vector’s angle can be tracked directly, or the direction of successive perihelion passages can be read off the integrated path, and the figure does both and requires them to agree. They do, which confirms that the vector’s rotation is the orbit’s precession and not a separate phenomenon.

A conserved quantity that stops being conserved becomes an observable. That is worth stating as a general point: the useful thing about the vector is not that it is constant for Kepler, but that its rate of change is the cleanest measure of how far a real force law departs from the inverse square.

A gentler orbit, and a longer run #

Repeating at gentler numbers is worth doing because the claim is a first-order one, and a first-order claim should get better as the perturbation shrinks.

It does. The vector’s angle is closer to a straight line, the departure from constancy within each orbit is smaller, and the two independent measurements of the rate — the vector’s own rotation and the movement of successive perihelia — agree more closely.

The eccentricity matters too, and in a direction worth noticing. A nearly circular orbit precesses more slowly for the same perturbation, because the extra inverse-cube term is felt most strongly near perihelion and a circular orbit has no perihelion to speak of. That dependence is in the relativistic formula as the factor $1/(1-e^2)$, and it is the reason the asteroid Icarus, which is much further from the Sun than Mercury, precesses almost as fast: its eccentricity of 0.83 more than compensates for its distance.

A perturbation that acts in one part of the orbit is amplified by whatever concentrates the orbit there. That is why eccentric orbits are the sensitive probes of any departure from the inverse square, and why the tests of gravity that matter most use binary pulsars on eccentric orbits rather than circular ones.

Where the symmetry comes from #

By the argument a symmetry hands over, a conservation law comes from a symmetry, and the question is which one.

Energy comes from time-translation and angular momentum from rotation, and both are symmetries of the space the orbit lives in. There is no obvious further symmetry of three-dimensional space left over, and for a long time the vector was regarded as an accident.

It is not. The Kepler problem has a symmetry that is not a symmetry of space but of the space of states: bound orbits of a given energy are permuted by a group of rotations in four dimensions rather than three, and the three extra generators are the components of this vector. The orbit’s orientation is fixed because a rotation that would change it is not a symmetry of the problem, which is the same sentence as saying the vector is conserved.

The quantum version is the one with visible consequences. The four-dimensional symmetry is why hydrogen’s energy levels depend only on the principal quantum number and not on the angular momentum — the accidental degeneracy is not accidental, and it is removed by exactly the same departures from the inverse square that make the classical orbit precess. A screened Coulomb potential in a multi-electron atom lifts it, and the order the shells fill is what is left once it is lifted.

Forty-three arcseconds #

The historical case is the sharpest example of the previous two sections put together.

Mercury’s perihelion moves at about 574 arcseconds a century. Most of that is the pull of the other planets, calculable in Newtonian mechanics and calculated with great labour in the nineteenth century, and it accounts for all but about 43. Those 43 were the problem: a residual rotation of a vector that should not rotate.

General relativity supplies a correction to the effective potential that behaves as an inverse cube — the same shape as the perturbation in the drift figures, arriving from a theory rather than by hand — and its coefficient is fixed — there is nothing to adjust. The resulting precession is $6\pi GM/c^2a(1-e^2)$ per orbit, and for Mercury it comes to 43 arcseconds a century.

The strength of the case is that the same formula does the other planets too. Venus at 8.6, Earth at 3.8, Mars at 1.35, and the asteroid Icarus at 10 — none of them famous, all of them measured, all of them reproduced by a formula that was fixed by Mercury and then not adjusted. A theory that fitted one number would be worth little; one that predicts five is a different kind of claim.

The other things this vector does #

Once the vector is in hand, several results that are usually derived separately fall out of it.

The shape of the orbit. Take the dot product of the vector with the position and rearrange: the result is the polar equation of a conic, with the eccentricity and the orientation already in place. That is a derivation of Kepler’s first law in two lines with no differential equation solved, and it is the cleanest one there is.

The scattering angle. For an unbound orbit — the case the lens with no focal length needs for light rather than for matter — the same vector still exists and still points along the axis of the hyperbola, so the angle between the incoming and outgoing asymptotes follows from its direction without integrating anything. Rutherford’s formula comes out of it directly.

The quantum spectrum. Applying the same algebra to the operators rather than the classical quantities gives hydrogen’s energy levels without solving the radial equation at all — Pauli did it in 1926, before Schrödinger’s version, and it is still the shortest route to the answer.

And the perturbation theory. Because the vector’s rate of change is a small quantity when the perturbation is small, averaging it over an orbit gives the secular evolution of the orbit’s shape and orientation directly. That is how the long-term behaviour of a planetary system is computed, and the quantities being propagated are this vector and the angular momentum rather than positions and velocities.

A conserved quantity is a change of variables as much as a fact. Its real value is that it replaces a question about a trajectory with a question about a constant, and each of the four results above is that replacement made in a different setting.

What else moves a perihelion #

The forty-three arcseconds are a residual, and it is worth naming the things they are a residual of, because the subtraction is where the difficulty lay.

The other planets — whose mutual disturbance is the same secular problem the distance that forgets the moon treats on a different scale — contribute about 530 arcseconds a century, Venus and Jupiter dominating. Computing that was decades of work by Le Verrier and later Newcomb, and it is a secular perturbation calculation of exactly the kind the previous section describes — averaging the disturbing force over the orbits and propagating the result.

The Sun’s oblateness would contribute, and this was the leading alternative explanation. A flattened Sun has a quadrupole moment whose effect on the orbit falls as an inverse cube, which is the same functional form as the relativistic term, so it can be fitted to the same number. What killed the explanation is that the required flattening is far larger than the measured one — helioseismology now fixes the Sun’s quadrupole moment well enough that its contribution is under a hundredth of an arcsecond — and that it would have got the other planets wrong.

The precession of the equinoxes contributes about 5,000 arcseconds a century to the apparent motion, and is a property of the coordinate system rather than of Mercury. It is removed before anything else is considered.

That last item is a reminder about what “measured” means here. The observed quantity is the direction of Mercury’s perihelion against the stars, and turning it into a physical precession requires removing the motion of the reference frame, the pull of every other planet, and the light-time and aberration corrections. The residual is small, and the credibility of a small residual rests entirely on the size of what has been subtracted from it — which is why the case took fifty years to be believed and why the agreement with Venus, Earth, Mars and Icarus mattered so much when it came.

Where the model stops #

The perturbation is treated as small. The vector’s rotation is steady only to first order; at larger perturbation the orbit is no longer nearly elliptical, the vector’s length varies within each orbit as well as its direction, and the two-measurement agreement in the third figure would fail.

The relativistic correction is the leading term. The full Schwarzschild orbit is not an ellipse with a rotating axis, and near a compact object the higher terms matter enormously — the precession per orbit can approach a whole turn, and the language of a slowly turning ellipse stops applying.

Only two bodies are present. Real planetary precession is dominated by the other planets, and separating the relativistic part is a subtraction of two much larger numbers. The 43 arcseconds is a residual, and its credibility rests on the Newtonian calculation being right to a part in ten.

The orbit is assumed bound and non-degenerate. A circular orbit has zero eccentricity and therefore a vector of zero length, which points nowhere; its orientation is genuinely undefined, and the whole apparatus says nothing about it. That is not a defect — a circle has no perihelion — but it means the vector’s direction becomes an increasingly poorly determined quantity as the orbit rounds, which matters for the orbit that has to shrink, where gravitational radiation circularises a binary and the axis it once had stops meaning anything.

And the classical vector has an ordering problem in quantum mechanics. Its operator version needs symmetrising before it commutes properly with the Hamiltonian, and the four-dimensional symmetry it generates applies to the bound states of the non-relativistic Coulomb problem and not, unchanged, to anything else.

What the pictures cannot show #

The first figure draws five arrows and they lie exactly on top of one another, which is the result and is also invisible. A figure whose content is that nothing happens has to state the number it measured, and the caption does what the drawing cannot.

The drift figure plots an angle against time and cannot show that the vector’s length is also very nearly constant under the perturbation — that the perturbed orbit is still an ellipse of the same shape, merely turning. That constancy is a separate fact and it is what makes “precession” the right word rather than “distortion”.

A third omission is that the vector has three components and every figure here draws two. In a plane orbit the third is zero by construction, since the vector lies in the orbital plane, and the whole of the extra information is the two numbers a plane needs. In three dimensions the vector is perpendicular to the angular momentum and the two together carry five independent numbers against the six a state needs — the sixth being where along the orbit the body is, which no conserved quantity can supply. That accounting is the cleanest statement of what the constants of a problem can and cannot do: they describe the orbit and never the position on it.

Where the ladder goes next #

The orbit-stability ladder began with the orbit that cannot be smaller, where general relativity supplies an innermost stable circular orbit that Newtonian gravity does not, and continued with the orbit that does not come back to itself and Bertrand’s theorem. This rung asks what the inverse square has that the others lack. The rungs after it: the three-body problem, where none of these constants survives; the secular perturbation theory that computes the Newtonian part of Mercury’s motion; and the precession of a gyroscope in orbit, which is a rotation of a different vector by the same geometry.

The habit worth carrying away is that a missing conserved quantity is a question worth asking. Two constants determined the orbit’s shape and left its orientation free, and noticing the gap is what leads to the third — which turned out to carry a symmetry, a spectroscopic degeneracy and a test of general relativity behind it.