The third thing that is conserved

A bound orbit in a central force is a curve in a plane, and a curve in a plane needs three things said about it before it is pinned down: how big, what shape, and which way round. The first two have famous answers. The total energy fixes the semi-major axis and nothing else; the angular momentum, with the energy, fixes the eccentricity. Neither of them says anything at all about the direction of perihelion.

That is not a gap in the bookkeeping. It is a real freedom: rotate an orbit about its focus and the energy and angular momentum are unchanged, because both are scalars and the force is central. So if the orbit is to close — if the body is to come back to the same perihelion revolution after revolution — something must be holding that direction, and it has to be a conserved quantity nobody has named yet.

Counting what is available #

For motion in a plane there are four numbers of state — two positions, two velocities — and the trajectory is fixed by them. Conservation laws cut that down. Energy takes one; angular momentum takes another. Two conserved quantities leave a two-parameter family, and a two-parameter family of curves through a given point is not a closed curve: it is a curve that fills a region.

That is exactly what a general central force gives. The radial motion oscillates between a minimum and a maximum radius with one period, the angle advances steadily with another, and unless those two periods are commensurate the path never repeats. It fills an annulus.

The angle between one perihelion and the following aphelion is called the apsidal angle. If it is $\pi$, the orbit closes after one revolution: the ellipse. If it is a rational multiple of $\pi$, the orbit closes after several. If it is anything else, the orbit never closes at all, and the picture is a rosette.

What a different force law does #

The cleanest way to see this is to keep everything about the initial conditions fixed and change only the exponent in the force. Write $F \propto r^{-n}$ and integrate — not in time, but in the angle, where the orbit equation is

$$\frac{d^2u}{d\theta^2} + u = \frac{k\,u^{\,n-2}}{L^2},\qquad u = \frac{1}{r}.$$

At $n = 2$ that is the equation of a harmonic oscillator in $\theta$ with period $2\pi$ — which is precisely the statement that the ellipse closes, and the reason the inverse square is special is visible in one line.

Which force laws close every bound orbit is a separate question, and it is not this essay’s. It is a classification — Bertrand’s — and the answer is that exactly two laws qualify, the inverse square and the linear spring. What is taken from it here is one fact and no argument: the inverse square is in the list, and this essay is about what puts it there. The mechanism is a conserved quantity, and a conserved quantity is a different kind of object from a theorem about exponents.

The vector, and what it is made of #

The conserved quantity behind the inverse square’s closure is the Laplace–Runge–Lenz vector, and it has been discovered and forgotten about six times — by Hermann and Bernoulli in 1710, by Laplace in 1799, by Hamilton, by Gibbs, by Runge, by Lenz, and finally by Pauli, who used it to solve the hydrogen atom before Schrödinger’s equation was available. It is

$$\mathbf{A} = \mathbf{v}\times\mathbf{L} - GM\,\hat{\mathbf{r}},$$

a vector in the orbital plane, and the demonstration that it is constant takes three lines of differentiation and works only because the force goes as $r^{-2}$: any other exponent leaves a term over.

Two facts about it do all the work. Its magnitude is $GMe$, so it is the eccentricity, written as a vector. And its direction is towards perihelion, which is the missing third piece of information. Conserving it conserves the orientation, and an orbit whose orientation is conserved is an orbit that closes.

The surprise: a symmetry that is not in the picture #

The uncomfortable question is why an extra conservation law should exist. Noether’s theorem attaches a conserved quantity to a continuous symmetry, and energy and angular momentum come from the obvious ones: time translation and rotation in space. There is no third obvious symmetry of an inverse-square problem lying about.

There is a third symmetry, and it is not a symmetry of space. In 1935 Fock showed that the bound inverse-square problem, transformed appropriately, becomes free motion on the surface of a four-dimensional sphere; the hidden symmetry is the rotation group of that sphere, and the Laplace–Runge–Lenz vector is what the extra three rotations conserve. Nothing in the orbital plane rotates. The symmetry is in a space constructed from the momenta, and it exists only for the inverse square.

That is the connection worth carrying away from this rung, because it is where celestial mechanics turns out to have been solving a problem in atomic physics. The same hidden symmetry is why the hydrogen atom’s energy levels depend only on the principal quantum number and not on the angular momentum — states of different shape at the same energy, exactly as ellipses of different eccentricity at the same semi-major axis have the same period. The degeneracy in a spectrum and the closure of an orbit are the same fact. Break the inverse square in an atom, by screening it with other electrons, and the levels split by angular momentum; break it in an orbit, by any of the ways below, and the perihelion moves.

Breaking it, three ways #

Nothing in the solar system is an exact inverse-square two-body problem, so nothing in it has an exactly conserved $\mathbf{A}$, and every perihelion in it moves.

The three ways it breaks in practice are worth separating, because they have different signs and different sizes.

The first is other bodies. A planet perturbed by the rest of the solar system is not in a two-body problem at all, and the time-averaged effect of the others is a slow rotation of its apsidal line. This is by far the largest term for Mercury: about 530 arcseconds per century, of which Venus supplies 277 and Jupiter 153. It is computed from the elements that do not stay constant and was known to a fraction of an arcsecond by 1860.

The second is that the primary is not a point. An oblate body’s potential contains a $J_2$ term, which is a small addition falling off as $r^{-4}$, and it turns the apsides too — read off a satellite’s node and its perigee rather than off a planet’s.

The third is that the force is not exactly Newton’s.

What the rate looks like #

The three sources add, and what they add up to is a rosette so tight that the drawing has to lie about the rate to show it at all.

What was actually measured #

A perihelion is not observed. What is observed is a sequence of positions, and a perihelion direction is a parameter fitted to them — so the statement “Mercury’s perihelion advances by 574 arcseconds per century” is the output of an orbit determination and carries whatever the determination carried.

For most of the nineteenth century those positions were meridian-circle transits: the time at which the planet crossed a wire, turned into a right ascension, with the observer’s own clock and latitude in the chain. Le Verrier’s 1859 value came from a century of such observations plus the transits of Mercury across the solar disc, which are far more precise than any transit across a wire because they are timed against the Sun’s limb rather than against a star catalogue. The 43 arcseconds he could not account for were about a tenth of the scatter in any individual determination and were believable only because there were so many of them.

The modern number comes from a different measurement entirely. Since the 1960s Mercury’s orbit has been fixed by radar ranging — the round-trip time of a pulse, which gives a distance and not a direction — and since 2011 by tracking the MESSENGER spacecraft, whose position relative to the planet is known to metres. The residual is now known to about one part in $10^4$ of itself, and it is worth noticing that the two methods have almost nothing in common: one measures angles from a moving platform against a star catalogue, the other measures times of flight. An orbit determined from directions alone and one determined from ranges alone are conditioned in completely different ways, and agreement between them is a stronger result than either.

The cleanest apsidal measurements are not in the solar system at all. In a close binary pulsar the periastron advance is degrees per year rather than arcseconds per century — four degrees a year for PSR B1913+16 — because the relativistic term rises steeply as the orbit tightens, and the timing of an orbit measured through its pulse arrival times is precise enough to see it accumulate within a single observing season. There, the failure of the Laplace–Runge–Lenz vector to be conserved is not a residual after other effects are removed. It is the signal.

The vector as a coordinate #

A conserved quantity that points somewhere and has a length is also a pair of numbers to describe an orbit with, and using it that way removes a defect that the classical elements have.

The trouble with the classical set is that two of its angles are undefined in the cases anybody cares most about. The argument of pericentre is the angle from the ascending node to the point of closest approach — and on a circular orbit there is no closest approach, so the angle is undefined and the numerical derivatives of anything computed from it blow up. The longitude of the node has the same problem for an orbit with no inclination.

Neither defect is in the physics. A nearly circular orbit is a perfectly ordinary orbit, and a satellite in one is not doing anything singular; the singularity is in the description.

The remedy is to carry the eccentricity vector itself — two components in the orbital plane, usually written $h = e\sin\varpi$ and $k = e\cos\varpi$ — instead of the eccentricity and the angle separately. At zero eccentricity both components are zero, which is a perfectly good value for a number to take, and nothing is undefined. The same trick applied to the inclination gives two more components, and the resulting equinoctial elements are non-singular everywhere except for a retrograde equatorial orbit.

That is the practical descendant of the whole essay. The quantity that is conserved because the force is exactly inverse-square is also the quantity whose components make the best coordinates when the force is not quite inverse-square — because the components go smoothly to zero where the classical angles fall apart, and a perturbation that would be described as “the pericentre swung through 180 degrees” is described instead as the vector passing near the origin.

What it looks like when it is nearly conserved #

The essay’s three breaking mechanisms make the vector move. Where the perturbation is small and slow, the motion has a shape worth knowing, because it is the whole content of secular perturbation theory.

Average the perturbing force over both orbits — the perturbed one and the perturber’s — and what remains is a slow drift of the eccentricity vector. For a satellite perturbed by a distant companion on a fixed orbit, that drift traces a circle in the plane of the two components, at constant speed, with a centre displaced from the origin.

The displacement is the forced eccentricity: the value the orbit would settle at if it were being held, imposed by the perturber’s own eccentricity and geometry. The radius of the circle is the free eccentricity: whatever the orbit was born with, preserved in magnitude and rotating in direction.

Two consequences follow immediately, and neither is obvious from the classical elements. An orbit whose free eccentricity happens to equal its forced eccentricity passes exactly through the origin once per circuit — meaning it becomes precisely circular, briefly, and then eccentric again with the pericentre pointing the opposite way. And an orbit’s maximum eccentricity is the sum of the free and forced values, which is what has to be checked when asking whether a body will ever cross another’s orbit.

The vector’s conservation is therefore the zeroth-order statement and its slow circulation is the first-order one, and the whole of classical secular theory is the calculation of where the centre of that circle sits and how fast the drift runs.

Where the circulation is measured #

The picture above is not only a computational convenience; it is read directly off two populations.

The asteroid belt. Every asteroid’s eccentricity vector is the sum of a forced part imposed mainly by Jupiter and a free part of its own. Plotting the whole belt in the plane of the two components shows the forced centre displaced from the origin and the free eccentricities scattered around it — and a collisional family, being fragments of one parent body, appears as a tight clump, because the fragments began with nearly the same vector and drift around the same circle together.

That is how families are identified. Their members are not close together in space, since they have long since spread all the way around the belt, and they are not close in the classical elements either. They are close in the free eccentricity and free inclination, which are the parts of the description the perturbations leave alone.

The planets themselves. The solar system’s own secular solution is the same calculation with eight mutually perturbing bodies rather than one perturber, and its output is a set of frequencies at which the eccentricity vectors circulate — periods of tens to hundreds of thousands of years. The Earth’s eccentricity oscillates between about 0.005 and 0.06 as a result, and that oscillation is one of the orbital cycles that pace the ice ages.

So a quantity conserved exactly in the two-body problem becomes, in the real system, a slow clock, and its frequencies are among the few things about the solar system’s long-term behaviour that can be stated without integrating it.

The frequencies themselves are not stable over the age of the solar system, which is the one caveat that has to travel with them: the secular system is weakly chaotic, and two of its frequencies pass close enough to a resonance that the solution’s long-term behaviour cannot be extrapolated beyond a few tens of millions of years.

Which is the ordinary condition of the subject: an exact statement about two bodies, a good approximation for many, and a horizon beyond which the approximation and the system part company.

What the ladder has done #

The first rung of this anchor established that a single inverse-square force allows exactly one family of curves, the conics, and that one number chooses among them. This rung says what holds a member of that family still.

The two statements together are a strong claim about the inverse square: it is not merely one force law among many that happens to be the one gravity uses, it is one of exactly two that produce closed orbits, and it possesses a symmetry that no drawing of the orbital plane contains. Both of those are properties of the exponent 2 and of nothing else.

Where the ladder goes next #

Two rungs are visible from here. One is Bertrand’s theorem proper — not the two closing laws but the proof that there are no others, which is a statement about which functions can make an apsidal angle independent of the orbit and is more delicate than the near-circular formula suggests. The other is the quantum end: the same hidden symmetry, the same vector, and a spectrum whose accidental degeneracy is neither accidental nor about spectra.