A perigee that moved twice as fast as Newton

The Moon’s orbit is an ellipse whose long axis turns, once in about 8.85 years. This has been known in round terms since Hipparchus. It is the largest single thing the Sun does to the Moon that is not periodic, and if gravity is an inverse square and the Sun’s tide is known, its rate is a clean calculation. Newton made it in the Principia and got half the right answer, and said so: in Book I, having worked out the rate from the Sun’s force, he remarks that the apse of the Moon is about twice as swift. Three of the strongest mathematicians of the next generation, Clairaut, d’Alembert and Euler, then repeated the calculation independently. They all found the same: half.

The event that followed is one of the best stories in celestial mechanics, because of what the failure turned out to be. There was no error in the mathematics. Three separate people did not make the same mistake. The law was good. The trouble was in an assumption so reasonable that nobody questioned it: if a parameter is small, the first term in its expansion is what matters.

The first term, and the factor of two #

Take the Moon’s orbital rate $n$ and the Sun’s angular rate round the sky $n'$. Their ratio $m=n'/n$ is about 0.0748, one over 13.37 months a year. The Sun’s tide, the difference between its pull on the Moon and on the Earth, stretches the Moon’s orbit along the Sun’s line and squeezes it across, by a force proportional to $n'^2 r$, which is a small fraction, $m^2$, of the Earth’s pull.

Averaged over a month and a year, that tide has one simple effect: its mean is a small inward radial force, proportional to distance, of $-\tfrac12 n'^2 r$. A central force that is not exactly inverse-square turns an ellipse’s axis, and the amount follows from a short calculation. It comes to

$$\frac{\dot\varpi}{n} = \frac{3}{4}m^2 .$$

With $m=0.0748$ this is 0.0042 of a turn for each month: a perigee period of about 18 years. That is the classic result. The measured value is 8.85 years. There is nothing subtle about the size of the discrepancy, a factor of two in a quantity that had been measured for centuries with an accuracy of better than one part in a thousand. A later scholium of Newton’s claims a rate close to the observed 40.7° a year, by a calculation that has never been reconstructed with any confidence; he evidently knew what number he wanted and could not derive it openly.

A direct integration makes it possible to see what is going on without going through the series. An orbit is integrated under just the Earth’s pull and a tide from a Sun moving on a circle, nothing else.

The result is unusually clean. The integration needs no lunar theory and uses just two forces, yet it reproduces the observed perigee period to about a per cent. That should settle at once any doubt about the cause. It also shows the structure of the discrepancy. At small values of $m$, for a satellite whose orbit is much faster than the Sun’s motion, the first-order law is close. As $m$ grows the true rate rises away from it, and by the Moon’s $m$ the true value is twice as big.

So no force was missing. What was missing was a term in $m$.

Clairaut’s crisis #

It is worth pausing on how serious this seemed to those involved. In November 1747, Alexis Clairaut read a paper to the Paris Academy that said the first-order calculation could not be made to agree with the Moon, and hence that the law of gravity must have a second term, varying as an inverse fourth power of the distance, which would matter at the Moon’s distance and be small at the planets’. It is easy to see what a shock this was, since the Newtonian synthesis was then only just being established in France, and Clairaut was among its leading supporters.

Buffon replied at once, without mathematics. He argued that a physical law should be simple and that a law with two terms was a sum of causes, not one; it would be a poor law of nature that depended on some arbitrary distance. It is a shaky argument, and Clairaut told him so. It later turned out to be right.

To make this concrete, ask what exponent would be required to produce the Moon’s perigee rate from the force law alone. By Newton’s own theorem of revolving orbits, a pure central force as $r^{-q}$ has a near-circular orbit whose apsidal angle per radial cycle corresponds to $q=3-1/(1+\dot\varpi/n)^2$.

That number, 2 and 4/243, is what Newton reports in the Principia as the exponent if the whole of the apsidal motion were due to the law. He treats it as showing that the actual law is close to the square. He knew the Sun must account for at least part of it, but his calculation gave only half. The difference to be explained then corresponds to an exponent higher by 0.0085. Note the last consequence stated in the caption. Gravity falls off with distance from the Earth’s surface to the Moon, and Newton’s “Moon test” compares the two. With the actual exponent of 2, the test works to within a per cent or so, given the measurements of the Earth’s radius from Picard. With 2.0085, a discrepancy of 3.5 per cent should appear. That could almost have been used to resolve the matter in advance, though Clairaut’s proposed fourth-power term was designed to be negligible at the Earth’s surface scale and would have gone the other way.

A series that does not know it is small #

In May 1749, Clairaut announced to the Academy that he had found the problem, and that the law was fine. Going to the next order in the perturbation calculation gave a term that almost brought the theoretical rate into agreement with observation. He had been computing to first order in the Sun’s force and neglecting the corrections to the orbit, including those that the Sun’s force itself had made. When those were fed back, the result nearly doubled. Euler checked it, was astonished and asked to be shown the method. D’Alembert confirmed the calculation independently.

What appears is a power series in $m$. Worked out in full centuries later by Delaunay and Hill, it starts

$$\frac{\dot\varpi}{n} = \frac34 m^2 + \frac{225}{32}m^3 + \frac{4071}{128}m^4 + \frac{265493}{2048}m^5 + \ldots$$

Look at the coefficients. They grow by factors of about 9 and 4.5 and 4, so the true expansion parameter is not $m$ at all. The ratio of the second term to the first is $\tfrac{75}{8}m$, about $9.4m$, and at $m=0.0748$ that is 0.70.

Hence a small quantity’s powers do not necessarily fall quickly. The true speed of convergence is fixed by the size of the coefficients, and they are large here for a reason that a good physical explanation should be able to give. It took more than a century to get to the bottom of it, and the whole expansion it belongs to gave no warning. George Hill in 1877 and 1878 showed how to expand around a much better starting point than the Keplerian ellipse. Instead of starting from an ellipse perturbed by the Sun, he used a periodic orbit in the rotating frame that already includes the Sun’s effect, the variation orbit; then the apsidal motion comes as the eigenvalue of an infinite determinant. Hill’s result gives the rate to many figures from just a few terms. It is the root of the theory of linear stability of periodic orbits.

What the perigee actually does #

The slow series suggests something else: the rate is not a steady, uniform turning. Reading the direction of the perigee from one passage to the next reveals a very different motion from the smooth line that the word “rate” suggests.

So the true perigee moves forward and backward by over thirty degrees. In the orbit-averaged equations, the instantaneous rate depends strongly on the angle $\psi$ between the perigee and the Sun:

$$\frac{\dot\varpi}{n} = \frac34 m^2\left[1+\frac52\cos2\psi\right].$$

It is positive and large when the perigee points along the Sun’s line, and negative when at right angles to it. The Sun’s tide is not the steady force which the mean calculation pretends. The same term also changes the Moon’s eccentricity, and that combination is the evection, the largest periodic inequality in the Moon’s motion after the equation of the centre, known since Ptolemy’s work.

Because that rate has a cosine factor, its plain average over $\psi$ is exactly the first-order value. That was the error’s hiding place. The angle $\psi$ does not change evenly. It changes as the difference between the Sun’s motion and the perigee’s. When the perigee is moving forward, in the same direction as the Sun, the angle changes more slowly, and when the perigee is moving back, it changes faster. So the Moon’s orbit spends a greater part of its time at the forward rates.

This is a real part of the second-order effect, though only a part. The closed form gives the time-weighted rate as $m-\sqrt{(m-A)^2-B^2}$, with $A=\tfrac34 m^2$ and $B=\tfrac{15}{8}m^2$. Expanded, it gives a term $\tfrac{225}{128}m^3$, one quarter of the true coefficient. The other three quarters come from what the monthly average has thrown away: within each month, the Sun’s force reshapes the Moon’s orbit (this is the variation, discovered by Tycho), and that distorted orbit then responds to the tide differently from a Kepler ellipse. So the physical account can be taken only part of the way without calculation. The rest has no simple picture, and that is what the coefficient’s size is telling.

A moral with a measurable consequence #

The successful calculation had consequences far beyond the Moon. When Clairaut’s result was published, the law of gravitation had just survived its most severe test, and at the very time that the French Academy’s support of Newtonian theory was being established. Buffon’s metaphysical objection now looked prophetic. In 1752 Clairaut won the St Petersburg Academy’s prize with the theory. Then in 1758 he used the same method to compute the delay in the return of Halley’s comet from the action of Jupiter and Saturn, predicting perihelion in mid-April 1759; it came on 13 March, within his stated margin of a month.

It also shows that a measure of how well a theory works is confounded by the order at which the theory is truncated. The Moon has been a warning since then. The more a theory’s parameter looks small, the more it is tempting to stop too early. The same thing recurred in another form: Adams in 1853 found that Laplace’s theory of the Moon’s secular acceleration had neglected a term that halved it, in the same kind of calculation, and Delaunay confirmed it. The remainder of the acceleration turned out to be from tides in the Earth’s oceans, which nobody had suspected. So the second-order term again removed a false agreement and pointed to new physics.

A consistency check of the same kind is available for an artificial satellite. A satellite of the Earth, at the height of a GPS satellite, has $m$ near 0.0014, and $\tfrac34 m^2$ is quite accurate. There the solar part of the node’s and perigee’s rates is computed with just the first term and it is good to a few per cent, so for artificial satellites the factor of two simply never appears. Only for a body as far out as the Moon does the true rate depart. There is thus something unusual in the fact that the failure occurred at all for the first body ever studied.

How much of this the integration can show #

The Sun’s orbit is a circle and in the Moon’s plane. That leaves out the solar eccentricity, which adds about one per cent to the rate through a term in $e'^2$, and the inclination of the Moon’s orbit, which enters at a similar size. The Moon’s own eccentricity enters as well, through a term of order $e^2 m^2$, and this reduces the rate slightly. The measured rate is the sum of those, and the integration’s agreement to within a per cent is about what it should be, and not a closer match that would need to be explained.

The tide is the quadrupole alone. The parallax terms, one power of the Moon’s distance over the Sun’s, are about a four-hundredth, and they are the source of a well-known smaller inequality. The figure’s integration leaves out this term, and it also leaves out the planets’ effects, which add a few parts in a thousand.

The integrator is high in order, because a low-order one misleads. A plain leapfrog at 300 steps per orbit introduces a numerical precession of its own, of a size comparable with the true rate at small $m$, so the point at $m=0.01$ came out negative when that was first tried. Fourth-order steps were needed to get the small-$m$ values right. There is a moral here too, since a low-order integrator is quite good at producing plausible wrong rates in an orbit’s perigee.

A series is not a cause. The terms of the expansion do not correspond to distinct physical mechanisms that could be separated in nature. The breakdown of the $m^3$ coefficient into lingering and short-period parts given above is a choice of arrangement.

The same expansion, on a smaller moon #

The series can also be used in reverse, for moons elsewhere. The Sun’s effect on a planetary satellite is measured by that satellite’s own $m$. Hyperion’s orbit round Saturn, Iapetus and Triton have values of various sizes. For a distant irregular moon of Jupiter, $m$ can reach 0.2 or 0.3, the term $\tfrac34 m^2$ is nowhere near adequate, and the series fails to converge at all. For those moons, the perigee can do something that it never does for the Moon, and in some cases it can get locked with the Sun’s position. An orbit whose perigee stays at a fixed angle to the Sun is a resonance of the sort that underlies Kozai–Lidov dynamics for inclined orbits, and such a lock is seen in the orbits of some irregular satellites, a state called evection resonance.

The Moon itself may have passed through it. When the Moon was closer to the Earth, about ten times closer than now, its perigee rate was dominated by the Earth’s oblateness, and it fell as the Moon moved out. There was a moment at which the perigee’s rate passed through the Sun’s annual rate, $\dot\varpi=n'$. If the Moon was captured there for a time, the resonance would have transferred angular momentum from the Earth–Moon pair to the Earth’s orbit round the Sun. That has been proposed as a way to take the fast-spinning post-impact system to its present angular momentum, in a state the Moon had to pass through. Whether it occurred depends on how long the capture lasted, and on how dissipative the Earth and Moon were at that time.

What the two-force test rules out #

It is easy to overlook how much the agreement of that integration rules out. A hundred years after Clairaut, people still had reason to worry about the law of force at the Moon’s distance. There were suggestions for a modification of gravity at short range, and for an effect of the Sun’s light. A direct integration with just two forces at a measured ratio, reaching the observed rate within a per cent, leaves very little room for a further term in the force law that acts at the Moon’s distance. The modern limits from laser ranging are far tighter. The perigee’s rate is measured to about a part in a hundred thousand, and the corresponding limit on any addition to the inverse-square law at that range is the strongest in the Solar System.

The perigee has a useful role for that reason. Any term that turns an orbit’s long axis without changing its shape is a test of the law of force, and there are few sharper. For the planets that test gave a separate result. Mercury’s 43 arcseconds were left over only after the same perturbation theory had been carried far enough for the other planets, and they stayed left over, and they needed a new theory. It is notable that the same observable gave opposite verdicts on the same law, in the two cases. In the one, an apparent failure of the law was a failure of the calculation. In the other, an apparent failure of the calculation was a failure of the law. Nothing in the size of the discrepancy told which was which, and Le Verrier, who found Mercury’s excess, reasonably assumed from the Clairaut case and from his success with Neptune that it had a Newtonian cause, in an unseen planet.

Still open: what a long series should be understood as #

The series for the rate has now been computed to very high orders by computer algebra, and its convergence radius for the Hill problem is known: the series in $m$ can be summed for values of $m$ up to the point where Hill’s family of periodic orbits ceases to be stable, at about $m\approx0.56$ of an orbital rate in some conventions. So the Moon is well inside it, and the slow convergence is not a failure of analysis as such. What is lacking is a simple statement of why the coefficients have the sizes they do.

Hill’s determinant computes the number, and it gives no physical picture of the factor of nine. A short account of why the second term should be 70 per cent of the first, as opposed to a tenth of it, would be worth having, and nothing here supplies it. The lingering argument gives a quarter of it. The rest is written in the variation orbit and its stability, and the best that can currently be said is that a Moon a thirteenth of the way to being in step with the Sun is much closer to that resonance than the number 0.0748 suggests.