The face that is not quite fixed

Nothing in this collection about the Moon is a small effect seen at a great distance; it is the one astronomical body close enough that a few degrees of anything is a landscape. The Moon is tidally locked, and the usual statement of what that means — one face permanently toward the Earth — is a statement about averages that no month actually satisfies. What is locked is the rate: the Moon turns once on its axis in exactly the time it takes to go once round. Uniform rotation, uniform orbital angle, no discrepancy.

But the orbital angle is not uniform. Nothing in a Kepler orbit is.

The east–west rocking is Kepler’s second law #

The Moon’s orbit has an eccentricity of about 0.055 — small, and enough. Near perigee it moves faster along its orbit than its mean rate; near apogee it moves slower. Its rotation has no idea about any of that and proceeds at the mean rate throughout.

The difference between where the Moon has actually got to and where a uniformly moving body would have got to is the equation of centre, the difference between the true and mean anomalies. It is exactly the quantity the site has already met one field away. That is libration in longitude, and it is the larger of the two optical terms. It shows an observer on Earth a strip beyond the eastern limb for a fortnight and a strip beyond the western limb for the next. There is a second place in this collection where exactly the same quantity appears, and the coincidence is worth stopping on. The Sun’s apparent motion along the ecliptic is non-uniform for the same reason, and the resulting term — the eccentricity component of the equation of time — is the equation of centre of the Earth’s orbit, amplitude $2e = 0.033$ radians, or 7.7 minutes of time. The same expression governs a clock error on Earth and how much of the Moon can be seen.

The north–south rocking is a tilt #

The second term has nothing to do with eccentricity. The Moon’s equator is inclined by 6.7° to the plane of its orbit, so as it goes round, the Earth stands over lunar latitudes ranging from 6.7° north to 6.7° south. That is libration in latitude, and it exposes each pole in turn.

The geometry behind that tilt is a genuinely elegant piece of celestial mechanics stated by Giovanni Domenico Cassini in 1693, from observation and before any theory existed to explain it. Cassini’s three laws say that the Moon rotates uniformly with a period equal to its orbital period; that its equator maintains a constant inclination to the ecliptic; and that its spin axis, the normal to its orbit and the normal to the ecliptic all lie in one plane, which turns with the regression of the lunar nodes. The third is the surprising one, and it means the whole configuration precesses as a rigid arrangement with an 18.6-year period.

The consequence for the picture is that the two librations run on different clocks. The longitude term repeats with the anomalistic month, 27.5546 days, the interval between successive perigees. The latitude term repeats with the draconic month, 27.2122 days, the interval between successive node crossings. Two periods differing by a third of a day, beating against each other with a cycle of about six years — which is why the track in the hero figure is a Lissajous figure that fills its rectangle rather than a closed loop.

And the observer contributes a third #

The two terms above are properties of the Moon. A third is a property of standing on a rotating planet.

The Earth’s radius is 6,378 kilometres and the Moon is 384,400 away, so an observer at moonrise and the same observer at moonset are looking at the Moon from vantage points separated by up to a full Earth diameter, and the direction of the Moon differs between them by up to 0.95°. That is the diurnal libration, and it is not small compared with what it adds — nearly a degree in each direction, on top of 6.3 and 6.7.

It is also the only one of the three that a single observer can see change within a night, and it is a parallax rather than a rotation. Nothing about the Moon has moved.

The remaining 41 per cent had never been seen by any human being until the Soviet probe Luna 3 returned twenty-nine photographs on 7 October 1959. What they showed was not more of the same: the near side is 31 per cent covered by dark basaltic maria and the far side is about 1 per cent, an asymmetry that is still not fully explained and that has nothing to do with the tidal lock in any direct way.

What was actually measured #

The optical librations were noticed early. Galileo describes the effect in the Dialogue of 1632 and more clearly in a letter of 1637 — he had watched features near the limb move on and off the visible disc — and Hevelius mapped the librating zones in his Selenographia of 1647. Nothing in that requires instrumentation beyond a good eye and patience: the excursions are several degrees, and a crater near the limb visibly changes its foreshortening from week to week.

What is measured now is a much smaller quantity, and it is not optical at all.

Superimposed on the geometric librations are physical librations: real oscillations of the Moon’s rotation, driven by the Earth’s torque on its permanent bulges. Their amplitude is of order a hundred arcseconds — a thousandth of the optical terms — and they are measured by bouncing laser pulses off the retroreflectors left by Apollo 11, 14 and 15 and by the Lunokhod rovers, timing the return to a few millimetres over a 384,400-kilometre round trip.

Those millimetres carry the interior. A rigid Moon would librate physically in one specific way; a Moon with a fluid layer inside it dissipates energy and lags, and the lag shows up as a small phase shift in the librations at particular frequencies. The analysis of five decades of ranging finds exactly such a dissipation, and it is the primary evidence that the Moon has a fluid outer core, some 330 to 350 kilometres in radius. A body’s interior, inferred from the timing of a reflection off its surface.

Nine per cent, seen edge on #

The extra 9 per cent that libration exposes is not nine per cent of a usable map. It is a thin annulus around the limb, and every part of it is seen at a grazing angle.

The foreshortening is severe and computable. Ground near the limb is tilted almost edge-on to the line of sight, so a square kilometre of it covers a small fraction of the pixels a square kilometre at the centre of the disc covers, and the resolution along one axis collapses while the other is unaffected. A crater at 85° from the sub-Earth point is compressed by a factor of about eleven in the radial direction. That is why the pre-Luna 3 maps of the librating zones are so much worse than the maps of the centre, and why several features in them were later found not to exist.

It also inverts the usual relationship between area and information. The librating annulus is the region most often mapped and least well known, and the only fix is to look from somewhere else — which is what an orbiting spacecraft does, and why the modern lunar coordinate frame comes from Lunar Reconnaissance Orbiter’s altimetry rather than from any Earth-based survey however long.

There is one place where the grazing view is an advantage rather than a handicap. Looking along the surface rather than down onto it is how a profile is measured: the limb’s silhouette against the sky gives the height of the terrain there directly, and successive librations rotate different terrain into that silhouette. Nineteenth-century observers built limb-profile charts from exactly this, and those charts were used well into the twentieth century to predict the exact timing of a total solar eclipse — because the last beads of sunlight shine through valleys on the lunar limb, and which valleys depends on the libration on the day.

Where the lock came from #

None of this explains why the rotation is synchronous in the first place, and the answer belongs to another anchor. The timescale for that process goes as the sixth power of the distance and inversely as the square of the satellite’s mass, which is why the Moon is locked, Pluto and Charon are locked to each other, most inner satellites of the giant planets are locked, and the Earth is not locked to the Moon — it is on its way, and the day is lengthening by about 1.8 milliseconds per century, which lunar laser ranging also measures.

What libration is worth, apart from geography #

Three uses, and none of them is about seeing more of the Moon.

It fixes the coordinate system. Selenographic longitude and latitude have to be tied to something, and what they are tied to is the mean sub-Earth point — the origin of the hero figure’s plot. Defining that origin requires separating the mean direction from the librating one, which requires the theory above. Every lunar map, every landing-site coordinate and every mascon in the gravity field is referred to a frame that libration defines.

It is a lever on the interior. The physical librations depend on the Moon’s moments of inertia, which depend on how its mass is distributed with depth. A body that is uniform, a body with a dense core and a body with a thick crust librate differently under the same torque, and the differences are measurable. The moment-of-inertia factor that comes out, 0.3931, is well below the 0.4 of a uniform sphere and is one of the strongest constraints on lunar structure.

And it complicates photometry. A crater near the limb is seen at a different angle each month, which changes its apparent brightness through the way lunar soil scatters light. Any long-term photometric monitoring of a lunar feature has to remove the libration first, and the correction is larger than most of the effects being looked for.

The same three uses transfer wholesale to every other synchronous rotator. The tidally locked planets of an M dwarf librate in longitude by exactly the same $2e$ if their orbits are not circular, and the resulting stirring of the terminator region is one of the few things about their surfaces that can be predicted from an orbit alone.

What the picture cannot show #

A fixed eccentricity, when the real one varies. The track in the hero figure is computed with $e = 0.0549$, the mean, which gives a longitude libration of ±6.3°. Solar perturbation carries the Moon’s eccentricity between about 0.026 and 0.077 over a few months, so the extreme excursions are larger than the mean ones — up to about 8.2°. The coverage figure uses the extreme value and the track figure the mean, and the difference between them is roughly three per cent of the Moon’s surface.

A rectangle, when the real region is not one. Drawing longitude against latitude makes the librating zone look like a box. On the sphere it is not: near the limb the exposed strip narrows toward the poles because the geometry of a cap is not the geometry of a rectangle, which is why the coverage figure’s boundaries are curves rather than straight lines.

And the physical librations, at all. They are three orders of magnitude below the optical ones and would not be a visible line width on any of these plots. Every figure here is the geometry of a perfectly rigid Moon on a Keplerian orbit, and everything currently interesting about lunar rotation is in the residual.

The handful of photons that come back #

The physical librations are measured by laser ranging, and the photon budget of that measurement is worth setting out because it is one of the more extreme in observational astronomy.

A ranging station fires a short pulse containing something like ten to the nineteen photons. The beam diverges — diffraction alone, through a metre-class telescope, spreads it to a couple of kilometres by the time it reaches the Moon, and atmospheric turbulence makes it several times worse.

A retroreflector array is a fraction of a square metre. So the fraction of the outgoing pulse that strikes it is of order one in a hundred million.

A corner-cube reflector returns light back along the direction it came from, which is what makes the measurement possible at all — but diffraction at the corner cube spreads the return beam too, to several kilometres at the Earth, so the fraction that lands on the receiving telescope is another one in a hundred million.

Multiply the two, allow for the atmosphere twice and for the detector’s efficiency, and the expected return is of order one photon per shot, or fewer. A ranging session fires for minutes and accumulates a few hundred returns, and each one has to be distinguished from the sky background by arriving in a time window a few nanoseconds wide, predicted in advance from the current best model of the Moon’s orbit and rotation.

The measurement is therefore self-referential in a mild way: finding the return requires a prediction good to a few nanoseconds, and the prediction comes from the model the returns are used to improve.

A handful of photons per minute, timed to a fraction of a nanosecond, accumulated for five decades, is what a fluid lunar core was inferred from.

The pole that also has to be defined #

The essay has said that libration fixes the origin of the lunar coordinate system. It fixes the origin and it leaves a choice about the axes, and the choice matters at the level real missions work at.

One natural frame is defined by the Moon’s mean orientation: the axis pointing at the mean sub-Earth direction and the pole at the mean rotation axis. That is the mean-Earth frame, and it is what a map drawn from Earth-based observation naturally uses, because both of its reference directions are things an Earth-based observer can locate.

The other natural frame is defined by the Moon’s own mass distribution: the three principal axes of its inertia tensor. That is the principal-axis frame, and it is what a dynamical model uses, because the equations of rotation are simplest in it.

The two do not coincide. The Moon’s principal axes are tilted relative to its mean orientation by a few hundredths of a degree, which at the lunar surface is very nearly a kilometre.

A kilometre is not a subtlety for a lander. Coordinates published in one frame and used in the other put a target that distance away, and the discrepancy has caused real confusion — early datasets were sometimes published without saying which frame they used, and the offset is small enough to be missed and large enough to matter.

The modern convention is to work in the principal-axis frame internally, because that is where the dynamics live, and to publish in the mean-Earth frame, because that is what maps use — with the transformation between them supplied by the same libration model that measures the interior.

A frame is defined by the same oscillation that measures the core, and getting the definition wrong is a kilometre of error in a quantity nobody thought was uncertain.

One implication is worth stating because it inverts the usual reading. Libration is normally described as a bonus — the extra nine per cent — and it is more useful as a constraint. The amplitude of each rocking is a function of quantities that are otherwise hard to reach: the free libration in longitude depends on the Moon’s own moments of inertia, and its damping depends on whether there is a fluid core to dissipate energy against. The rocking is a measurement of the interior and not only of the geography, which is why lunar laser ranging tracks the libration angles to milliarcseconds rather than merely mapping what they reveal.

One more reading covers the coverage at the Moon’s mean eccentricity rather than its extreme.

Where the ladder goes next #

The rung above is the lock itself as a dynamical problem: why a body settles into a 1:1 spin–orbit resonance at all, why Mercury settled into 3:2 instead, and what decides which. The rung beside it is the use of these measurements as geophysics — the Moon’s moments of inertia, its Love numbers and its core, all extracted from how a nominally locked body fails to be quite locked.