Forty-three arcseconds, after everything else

The number everybody remembers is 43. The number actually observed is 5,600, and the distance between those two figures is the whole methodological interest of the case.

The orbit that does not close #

An orbit under an exact inverse-square attraction closes: the body returns to the same point of space after one revolution, and the ellipse’s long axis stays put. That is a special property of the $1/r^2$ force and of essentially nothing else. Under any other central force the radial oscillation and the angular circulation have different periods, and the apsidal line — the line joining the nearest and furthest points — turns.

So a moving perihelion is a diagnostic, in the same way a mass sum extracted from a period is: both convert a departure from an idealised two-body relation into a physical quantity. It says the force is not exactly inverse-square, or that something else is pulling, or that the coordinate system is not fixed. All three are true for Mercury, in that order of decreasing size.

Subtracting the coordinate system #

The largest term is an artefact of how positions are recorded.

Which coordinate a position is quoted in is a question with consequences, and here it dominates the answer. Longitudes in the solar system are measured from the vernal equinox, the intersection of the Earth’s equator with the ecliptic. That intersection moves, because the Earth’s spin axis precesses under the torque the Sun and Moon exert on its equatorial bulge, completing a circuit in about 25,800 years. In arcseconds per century that is 5,025.6, and it applies to every longitude in the sky, Mercury’s perihelion included. That is the first lesson of the account and it generalises well past this case. A residual can only be trusted to the accuracy of the largest thing subtracted from it. The 43 arcseconds exist as a number only because the precession constant was known to a few parts in ten thousand by the 1850s, from a century and a half of meridian observations of stars, and not because anybody measured Mercury better.

Subtracting the planets #

What is left, 574 arcseconds a century in an inertial frame, is mostly the other planets, and computing their contribution is the hardest part of the calculation. Each planet’s perturbation on Mercury’s apsidal line is a secular term in the perturbation theory of the elements, and the terms are not small compared with each other:

Venus contributes 277.9 arcseconds a century, Jupiter 153.6, the Earth 90.0, Saturn, Uranus and Neptune together 7.3, and Mars 2.5. The sum is 531.5. Venus’s contribution alone is six times the residual, and Jupiter’s is three and a half times it, so both must be right to better than one per cent before the leftover means anything.

Le Verrier did this in 1859 and found a discrepancy of 38 arcseconds a century. Simon Newcomb redid it in 1882 with better planetary masses and better observations and got 43, and his figure stood essentially unchanged until spacecraft arrived. It is worth registering how thin the margin is: the entire anomaly is smaller than the correction for Mars plus Saturn plus a decent error in Venus’s mass.

The wrong answers, and why they were reasonable #

The residual was known for fifty-six years before it was explained, and the explanations offered in between are not foolish. Each is a plausible modification, and each was killed by a measurement.

A planet inside Mercury’s orbit. Le Verrier had already found Neptune by exactly this reasoning, so proposing Vulcan was not a leap but a method. Required: a body or belt of the right mass at the right distance. Searched for during solar eclipses and in transits across the Sun for decades; never found, and the modern limits are far below what would be needed.

Solar oblateness. A Sun with an equatorial bulge has a quadrupole moment $J_2$, and a quadrupole produces apsidal precession. To supply 43 arcseconds a century requires $J_2 \approx 2\times10^{-5}$, which corresponds to an oblateness of a few parts in ten thousand — large enough to have been visible in the solar disc. Helioseismology now measures the Sun’s $J_2$ at about $2.2\times10^{-7}$, which contributes some 0.03 arcseconds a century. The bar for it is in the hero figure, and it is the smallest one there.

A different exponent. Asaph Hall proposed in 1894 that the force falls as $r^{-2.00000016}$, chosen to give exactly the observed advance. This is the most instructive failure: it fits Mercury by construction and then makes a prediction about the Moon, whose apsidal motion is measured, and the prediction is wrong. A modification tuned to one residual has to survive every other orbit in the solar system, and a one-parameter fit has nowhere to hide.

Two residuals, and what separated them #

Le Verrier is the author of both the most celebrated inversion of a residual in the subject and the least successful, and he used the same method for each. It is worth being precise about what distinguished them, because the difference is not luck.

Uranus’s residual was in the shape of a perturbation. A body further out produces a very particular signature: a term whose amplitude and phase are tied together, whose period is set by the synodic period of the two planets, and which changes sign as the perturber is overtaken. Le Verrier could fit the residual with a body’s mass, distance and longitude and check that the fit had the right functional form, not merely the right size. It did, and Neptune was one degree from where he said.

Mercury’s residual was secular. A steady advance carries almost no information about its cause: an interior planet, a belt of small bodies, a solar bulge, a modified exponent and a relativistic correction all produce a constant rate, and the observation is a single number. One number cannot choose among five one-parameter hypotheses. Every one of them could be fitted, and four of them were.

What broke the deadlock was not a better measurement of Mercury. It was that each hypothesis made a second prediction somewhere else — Vulcan should be visible at an eclipse, an oblate Sun should show a flattened disc, a modified exponent should move the Moon’s perigee — and each of those second predictions failed. General relativity’s second predictions, the deflection of starlight and the gravitational redshift, did not.

The methodological rule this leaves is worth carrying: a secular residual is a weak constraint and a periodic one is a strong constraint, because the second has a shape and the first has only a size. It is the same reason a transit timing residual weighs a planet while a steady drift in a period could be almost anything.

The number general relativity computes #

Einstein’s field equations give, for a test body orbiting a mass $M$, an apsidal advance per orbit of

$$\Delta\varpi = \frac{6\pi GM}{a c^2 (1-e^2)},$$

with no adjustable parameter of any kind. Everything in it is measured elsewhere.

Two features of that formula are worth pausing on, because they explain why Mercury and not another planet.

It is largest for the innermost orbit, and Mercury is innermost by a factor of two. And it is amplified by eccentricity, and Mercury’s eccentricity of 0.2056 is by a wide margin the largest among the planets.

What is actually being measured now #

The modern determination does not use meridian-circle observations of Mercury at all. It uses ranging: radar echoes from Mercury’s surface from 1966 onwards, then the Doppler and range tracking of the MESSENGER spacecraft in orbit from 2011, which fixes Mercury’s position along its orbit to a few metres rather than to the tenths of an arcsecond an optical position gives.

The number that comes out is not “the perihelion advance” as an isolated quantity. It is a parameter in a global fit: hundreds of thousands of range and Doppler measurements to many spacecraft and planets are fitted simultaneously for the planetary ephemeris, the masses of the asteroids, the solar quadrupole moment, and the post-Newtonian parameters. The relativistic advance appears as the combination $\tfrac13(2 + 2\gamma - \beta)$ multiplying the classical expression, where $\gamma$ and $\beta$ are 1 in general relativity, and the fits constrain that combination to 1 within a few parts in a thousand.

This is a different kind of measurement from Newcomb’s and it is worth saying how. Newcomb had one observable and subtracted known terms from it. The modern analysis has no step at which 43 arcseconds is isolated; the residual is never formed, because everything is estimated at once. The advantage is that the correlations between the solar quadrupole and the relativistic term — which affect the orbit in nearly the same way, and are the largest remaining ambiguity — are handled rather than assumed away. The cost is that no single number in the output is the observation.

And there is a system where the same effect is enormous. The binary pulsar PSR B1913+16 has a periastron advance of 4.226598 degrees per year: some 35,000 times Mercury’s rate of 43 arcseconds a century, because both stars are neutron stars and the orbit is about thirty times smaller than Mercury’s — 1.95 million kilometres against 57.9 — with an eccentricity of 0.617. Measured to nine significant figures from pulse timing, it is used not as a test but as a tool — because the relativistic advance depends on the total mass, it delivers the mass sum of the two stars from timing alone.

The other two predictions, and their different fates #

The residual described here was known for fifty-six years before it was explained, which means that when the explanation arrived it explained something already on the books. That is a retrodiction, and its evidential weight has been argued about ever since.

The case against giving it much weight is the standard one: a theory constructed by someone who knew the number could have been arranged to produce it. The case for giving it a great deal is that it could not, and the reason is the formula. The advance comes out of the field equations with no free parameter — the mass, the semi-major axis and the eccentricity are all measured elsewhere — so there was nothing to adjust. A theory that produced 30 arcseconds a century, or 60, would have been wrong and could not have been repaired.

Einstein himself regarded it as the strongest of the three classical tests, and by his own account the calculation gave him palpitations. The other two were genuine predictions and they fared quite differently.

The deflection of starlight passing the Sun was predicted at 1.75 arcseconds, twice the value a Newtonian treatment of light as corpuscles would give, and was measured at the eclipse of 1919. The measurement was at the edge of what the instruments could do; the plates from one of the two expeditions were of poor quality and were discounted on grounds that have been re-argued periodically ever since. The result was accepted at the time, and the modern value of the same quantity — from radio interferometry and from spacecraft ranging — is far more precise and agrees.

The gravitational redshift was the third, and it was the last to be confirmed. It is the weakest of the three in the sense that it follows from the equivalence principle alone rather than from the field equations, so a positive result tests less. It was not measured convincingly until 1959, in a laboratory, over a height of twenty-two metres.

Three predictions, one retrodicted precisely with no adjustable quantity, one measured marginally and correctly, one not confirmed for forty years — and the first is the one whose logic is the least often stated properly.

The correlation that has replaced the residual #

The modern measurement does not isolate 43 arcseconds, and the reason is not merely bookkeeping. Two of the terms in Mercury’s orbit are nearly indistinguishable from each other.

The relativistic advance and the advance caused by the Sun’s own quadrupole moment both produce a steady rotation of the apsidal line, and their dependence on the orbit’s size and shape is similar enough that a fit to Mercury alone cannot cleanly separate them. What the data determine is a combination, and quoting either one requires a value for the other from somewhere else.

For a long time the way out was to argue about the Sun’s interior. If the solar core rotated much faster than the surface, the quadrupole would be larger and part of the 43 arcseconds would be Newtonian after all. That proposal was live into the 1970s and was settled by helioseismology, which measures the rotation of the interior directly and finds it rotating as a solid body at about the surface rate — which pins the quadrupole at a value two orders of magnitude too small to matter.

So the correlation is now broken by an independent measurement rather than by the orbit, and the residual argument has been replaced by a joint fit in which the seismic value enters as a prior. The remaining ambition is to break it within the orbital data themselves, by measuring Mercury’s position well enough and long enough that the two terms’ small differences in behaviour become detectable — which requires the kind of tracking a spacecraft in orbit around the planet provides, over years.

The quantity that was once a leftover is now one parameter in a fit with a known degeneracy, and that is a fair description of how a historical anomaly matures into a measurement.

What the picture cannot show #

The size. Every drawing of a precessing orbit exaggerates, and the exaggeration here is a factor of twelve thousand. No honest picture of Mercury’s orbit shows this effect, because 0.104 arcseconds per revolution is 5 parts in $10^7$ of a turn.

The subtraction. The hero figure shows an account with a residual at the bottom, and it cannot show the uncertainty on each line. That is where the argument actually lived for half a century: Newcomb’s 43 was 43 ± 2 or so, and the case for a new theory of gravitation rested on believing the Venus term to a fraction of a per cent. A bar chart makes a subtraction look like arithmetic when it is an error budget.

The rest of the account. The hero figure stops at seven terms because those are the ones that were argued about. A modern ephemeris carries more: the perturbations of the largest asteroids, which shift Mercury’s perihelion by a few hundredths of an arcsecond a century and are the reason asteroid masses appear as free parameters in the same fit; the Lense–Thirring term from the Sun’s rotation, of order 0.002; and the tidal deformation of Mercury itself. Each is far below the residual and none is below the precision the residual is now quoted to, which is 0.04 arcseconds a century. An account is only finished relative to a tolerance.

The mechanism. The formula is quoted here and derived nowhere. Its terms come from the Schwarzschild solution, and the split of the effect between the curvature of space and the nonlinearity of the field is a question about coordinates as much as about physics — which is precisely why the modern statement is a constraint on a combination of post-Newtonian parameters rather than a claim about which half of the effect is which.

Where the ladder goes next #

Two rungs. Downward in mass and upward in field strength: the same expansion carried one order further gives the frame-dragging of a rotating mass, measured on the Earth by LAGEOS and Gravity Probe B and dominant in the accretion discs of black holes. And sideways, into the systems where the residual is no longer small — a binary pulsar whose orbit shrinks measurably each year, where what is being watched is not a correction to Newtonian gravity but the energy the orbit is radiating away.