A term too small to keep that a denominator kept

In the expansion that all planetary theory rests on, the terms are arranged in a hierarchy. A term has a size set by the disturbing planet’s mass and a power of the eccentricities and inclinations, and with eccentricities of a few hundredths, each extra power makes it twenty times smaller. D’Alembert’s rule says which powers belong with which angles, so the hierarchy can be read off before anything has been computed. A theory is built by writing down the terms to a certain order and stopping. For a long time this looked as if it would succeed.

The problem is that the size of a term in the disturbing function is not the size of its effect. A term of a certain amplitude acts on the planet’s motion with an angle that turns at some rate, and its effect accumulates through a time integral, so it is divided by that rate. A slow rate means a long time pushing in the same direction, and a large response. The hierarchy applies to the function being integrated. The orbit is the integral.

For most terms it hardly matters. But there is one that matters more than any other in the Solar System, and it was not a theoretical curiosity. It was the largest known failure of Newton’s law.

A discrepancy that looked like a runaway #

Since Kepler’s Rudolphine Tables in 1627, it had been noticed that the mean motions of Jupiter and Saturn could not be held steady. In 1676 Edmond Halley compared ancient positions with modern ones and reported that Jupiter’s motion had sped up since Ptolemy, and Saturn’s had slowed. That he corrected the tables with a term growing as the square of the time tells how seriously he took it as a real, accumulating effect.

The issue was alarming because of what it implied. Jupiter speeding up means Jupiter moving inward. Saturn slowing means Saturn moving outward. Extrapolated, Jupiter would fall into the Sun in the far future, and Saturn would drift away. Euler, Lagrange and others spent decades on it, and each prize essay of the Paris Academy in the 1740s and 1750s failed to find the cause. By the 1770s Lagrange had shown that first-order terms could not change the mean distances of the planets secularly, and yet the observed drift would not go away. Some, including Euler at first, suspected a resistance of the ether or a departure from the inverse-square law.

Pierre-Simon Laplace solved it in 1785. He showed that the effect was periodic, with a very long period, and came from a term that everyone had discarded. An integration lets it be seen without the algebra.

Between about AD 150 and the seventeenth century, Jupiter’s residual rises and Saturn’s falls. That is what Halley saw. It reverses about a century later, as the integration shows, and within two more centuries it has crossed zero. A century’s observations could not have told a slow oscillation from a steady acceleration, since it is only a small part of a long cycle. The curve shows why the problem was so difficult for tables based on modern observations alone and why comparisons with ancient records seemed to demand a secular term. The amplitudes from this simple integration, about 20 and 50 arcminutes, are close to the values of 20 and 48 in the planetary theories.

One gain and one loss #

Before finding what makes this happen, the integration can test one consequence that has nothing to do with how the term was derived. The Sun, Jupiter and Saturn make a closed system with a total energy. The two planets can exchange energy with each other, because the term in question acts in both directions, while the sum of what they hold is almost constant.

Keeping the sum of $-Gm/2a$ fixed, a change in Jupiter’s semi-major axis requires a change of the opposite sign in Saturn’s. Since $n\propto a^{-3/2}$, the mean-motion changes are also opposite. And a longitude residual is the integral of a mean-motion change, so the two residuals must be exactly anti-correlated with the ratio

$$\frac{\delta\lambda_S}{\delta\lambda_J} = -\frac{m_J\sqrt{a_J}}{m_S\sqrt{a_S}}.$$

The law is what ended the alarm. A planet in a closed system cannot gain mean motion unless the other pays, and when Laplace found both sides of the account balancing in the ratio demanded, the observed drift could no longer be considered a loss from the system. Halley’s gain and loss are a single exchange, from one planet to the other. It removes the fear of the unwinding in the same way that the invariance of major axes was to remove it for all the secular terms, by Laplace and Lagrange in the same years.

There is also a small, satisfying surprise in that sign. Giving energy to Jupiter, making its orbit less tightly bound, makes it larger and slower. Gaining energy makes an orbiting body fall behind. A planet with extra energy has more semi-major axis and less speed, which is the same paradox that makes a spacecraft burn backward in order to catch up. So when Jupiter’s longitude is rising, Jupiter is losing energy to Saturn and moving inward.

Five against two #

The term is the one with the angle

$$\psi = 2\lambda_J - 5\lambda_S + \ldots$$

Because $P_S/P_J\approx 2.483$, two of Saturn’s years are nearly five of Jupiter’s, and so this particular combination of mean longitudes changes very slowly. Its rate $2n_J-5n_S$ is about one part in 73 of Jupiter’s own mean motion, which is where the 870-year period comes from.

D’Alembert’s rule says the coefficient of a term with this angle has to have at least a power of the eccentricities or inclinations equal to the difference in the integers, $|5-2|=3$. It is a third-order term. With eccentricities of 0.05, the coefficient is $e^3\approx10^{-4}$ of a leading term, which was why no one thought to compute it. Laplace noticed something that had escaped others: that the effect on the longitude is the second integral of a force, and the term’s rate appears twice in the denominator.

$$\delta\lambda \sim \frac{m'}{M}\,C\,e^{3}\,\frac{3n^2}{(2n_J-5n_S)^2}.$$

The squared denominator is enormous. With a ratio near 73, and the factor of three in front, that is a multiplier of about $16{,}000$, and it takes this term from a coefficient of order $10^{-8}$ to many arcminutes in the longitude.

The integration makes this checkable in a direct way. If the argument is correct, the amplitude must follow the denominator when nothing else is changed. So the experiment is to change the one thing that cannot be changed in the sky, the period ratio, and integrate all over again.

The experiment is a fine example of what a computer can add to an old argument. Laplace’s derivation was a triumph of analysis; the expansion had to be carried to the third order by hand, with hundreds of separate coefficients, and an error anywhere would have ruined it. A modern integration is able to test the central claim as a controlled experiment, with an imaginary Saturn, in a few seconds. It gets the slope, without any of the expansion, because the small divisor is in the physics and not in the mathematics. Had the real ratio been 2.40 instead of 2.48, the effect would have been some three arcminutes, hidden among others, and the crisis would probably not have occurred.

Where the ordering went wrong #

An instructive way to see the failure is to set out all the near-commensurabilities of the pair, as the eighteenth-century analysts might have done had they thought of it.

These are order-of-magnitude estimates with their coefficients set to one, so the top of the figure does not rank the terms in detail. They do place the shape of the problem. In the lower line the size falls smoothly with the order of the term, as the rule intends. In the upper line a spike stands at one rational number, for no reason in the theory other than a fact about these two planets’ periods. The rational number has no special dynamical status. It merely happens to be close.

So the trouble with the hierarchy is that it treats the terms by their form and leaves out the one number, the frequency, that depends on the actual system. The frequencies are a function of the planets’ periods, and they are a kind of number that behaves erratically: the near-integer ratios are dense on the line, so that for any pair of periods, some combination of small integers lies close, and as the order increases, ever closer ones appear. This is the seed of the small-divisor problem, and it means that no expansion in powers of the eccentricity can be made uniformly accurate.

A few years after this, the same type of term was found to be responsible for the gaps in the asteroid belt that bear Kirkwood’s name, and for the existence of commensurabilities among satellites of Saturn and Jupiter.

A number that mattered for the whole century #

It is hard now to appreciate what Laplace’s result was for its time. The great inequality had been the biggest obvious failure of Newtonian gravitation, nearly a hundred years after the Principia. When the term was included, with the secular variations it also implies, Laplace found that the tables could be brought to within about two arcminutes of the observations going back to ancient times. The Bureau des Longitudes in Paris adopted his theory for the planets. It was the final step in the program of explaining every known motion in the Solar System by the single law.

It also established a mood. From it came Laplace’s view that the system of the world is stable and that its motions are entirely computable by a sufficiently capable intelligence; and in particular the remark about having no need of a hypothesis, which refers to the corrections that Newton thought God might have to apply. The reassurance rested on an argument that was not as rigorous as was believed. Poincaré showed a century later that the series do not converge in the usual sense, for a reason connected with small divisors of exactly this type, and chaos on billion-year scales has now been shown in the inner planets. But the great inequality itself is real, and periodic, and the argument for it has survived without change.

What the integration leaves out #

Three bodies in a plane. The integration ignores Uranus and Neptune, and it drops the inclinations. The period of the inequality in full theories is about 880 years, and it slowly varies; the integration’s value of 870 comes from starting elements chosen to give the observed period ratio. Starting exactly from the osculating semi-major axis of Saturn at the year 2000 gives a noticeably different beat, because an osculating element at a single date is itself swinging by more than the difference that matters. Saturn’s semi-major axis was set to 9.58 astronomical units so as to give the observed ratio 2.483, and that fitting has to be stated.

The smoothing hides much. The curves average over 40 years to remove the terms at the two planets’ synodic period and near it. Those terms are present, and they are also large, a few arcminutes at the periods of 20 and 60 years. Some residual ripple remains visible in the first figure, and with almost no smoothing it dominates.

A chart of this kind makes it clear that the smooth curve is not what any observer measured.

A mean longitude is a derived quantity. Nobody observes it. The observable is a position on the sky, from which the mean element is obtained by removing the short-period perturbations and the equation of centre, using a theory. The eighteenth-century observations came with the theories of their day. The history is therefore not as clean as it appears, since Halley’s reduction of a position had its own errors, of about the same size as his result in some cases.

The historical markers are a guide. Ptolemy’s positions have an accuracy of perhaps ten minutes of arc, and some were adjusted. The fit that Halley made was to a handful of points over 1,500 years. The figure shows the phase, not the quality, of the evidence.

The same beat in a cooler key #

The other main long-period term of the same kind is a useful control. Uranus and Neptune have period ratio of about 1.96, close to 2:1. A second-order term should be strong, by the same logic. And it is: there is a known inequality with a period of about 4,200 years, whose amplitude is about a degree in Uranus’s longitude and which caused trouble in working out Uranus’s motion after its discovery. It was not recognised at first. In the 1840s it was a part of the residuals that led Adams and Le Verrier to predict Neptune. Their predictions used an assumption that Neptune lay further out than it does, based on Bode’s rule, and the position they gave was good only for the few decades either side of the discovery.

The Earth and Venus have another such relation: their eight-to-thirteen near-commensurability gives a small but known term in the Earth’s longitude, of a period of about 240 years, discovered by George Airy in 1832 and said to be the most laborious calculation of its time. A comparatively tiny inequality, of less than three arcseconds, it was nonetheless a test of the accuracy of the solar tables. It is a fifth-order term, whose coefficient would be expected to be utterly negligible.

There is even a connection with something on the ground. Some of the long cycles in eclipse timing have the same arithmetic, in that a slow beat between nearly commensurate periods gives a long repetition, though eclipses need no force at all. In that case the near-commensurability merely repeats a geometry. In the planetary case it repeatedly pushes in the same direction, and it is the addition of force to the same clock that gives the large response.

Why nobody looked there first #

Why did it take so long to think of the term? In retrospect it looks obvious. The periods of Jupiter and Saturn are known; it is simple arithmetic that two of one are nearly five of the other; and Kepler himself had remarked on the ratio. The reason is partly technical. Before Lagrange, theories of planetary motion were not written in a form where a term’s frequency could be seen by inspection. The disturbing function was expanded in series for particular positions, and terms were kept by size. The rule of d’Alembert connecting the eccentricity power with the angle had not been stated in a way that would direct attention to such a term. Each mathematician who worked on the problem computed what the ordering told them to compute, and it never reached the third order.

Part of the reason is that, at first, the discrepancy seemed to have been explained in other ways. Euler’s prize work on it introduced the method of variation of elements, the single most useful analytic tool in the subject, and gave rise to the equations of how forces move elements. Lagrange’s treatment produced the first proper secular theory. So the failure to find the term had positive results; those developments, in turn, made it possible to see that a term of the third order might matter. And the same procedure was then available for the bound on how far the planets’ orbits can wander, which is the other half of what Laplace thought he had proved.

Laplace’s own route is suggestive. He had shown in 1773 that first-order perturbations cannot produce a secular change in mean distance. If that was so, the observed drift had to be a long-period effect, and any long-period effect required a small denominator. He then searched the list of small integer combinations of the two mean motions for one that was slow. So the derivation came after a clear statement of what was logically possible, and the expansion of the hard term was justified by a theorem about the easy ones. It is a good description of how a large calculation should be used.

Still open: whether a term this large is a coincidence #

Why should Jupiter and Saturn lie so close to a resonance at all? Their ratio of 2.48 is suspiciously near but not at 5:2. Models of the early Solar System, particularly the family of Nice models, place these two planets originally nearer together, probably in a 3:2 resonance, and have them migrate apart through the 2:1 ratio by scattering of a large disc of planetesimals. In that case the present ratio is simply where the planets stopped when the disc ran out, and its closeness to 5:2 would have no deeper significance. A slightly larger amount of disc would have given a different ratio, different terms, a smaller or larger inequality, and possibly no crisis in celestial mechanics.

It may be that the large inequality is only an accident of when the migration ended. If so, it is a curiosity in the same way that the coincidence of the Sun’s and Moon’s angular sizes is, in that a basic understanding of the science was shaped by an accident that needed no explanation. Whether the migration left other traces that fix the final ratio, in the structure of the Kuiper belt or the asteroid belt’s inclination, is a live question. The present term may eventually offer a test: an integrator can already run backward to see how recently it was this size, but only as far back as chaos permits the orbits to be calculated, which is not to the epoch when the question would be answered.