A rotation locked to the orbit, but not one to one

Until 1965 Mercury was believed to keep one face to the Sun. The reasoning was sound as far as it went: Mercury is close to the Sun, tides are strong there, tides despin a body until it turns once per orbit, and the Moon — which is the nearby worked example — does exactly that. Photometric observations, made at the limit of what visual astronomy could do, appeared to agree.

Radar disagreed. Bouncing a pulse off Mercury and measuring the Doppler spread of the echo — the same trick by which a spacecraft’s position is measured from a frequency — — the approaching limb returns a blueshifted signal and the receding limb a redshifted one — gave a rotation period of 59 days against an orbital period of 88. The ratio is three to two.

What the radar actually measured #

The observation deserves its own paragraph, because it is an unusually clean example of an instrument reading a quantity nobody had thought to ask it for.

A radar pulse sent at a rotating sphere returns spread in two ways. It is spread in time, because the near point of the disc returns first and the limb returns later; and it is spread in frequency, because the approaching limb is blueshifted and the receding limb redshifted. The frequency spread is proportional to the rotation speed times the sine of the angle between the rotation axis and the line of sight, and the time spread is fixed by the planet’s radius, which is known. So one echo, resolved in both, gives a rotation rate directly.

The measured spread was about a third of what a synchronous rotation would give. Nothing in the reduction depended on identifying a surface feature and watching it move round, which is what the earlier visual work had done and what had failed: the drawings had been made at intervals close to a multiple of 88 days, so a planet turning in 59 days was seen with almost the same face presented each time.

That is a general lesson about a sampled measurement rather than a fact about Mercury. Observing a periodic phenomenon at a cadence commensurate with its period returns an alias, and the alias is a consistent answer rather than a noisy one — which is why the wrong period was believed for seventy years and was believed confidently.

Why a circular orbit has only one lock #

A body that is not perfectly round has a long axis, and the tidal field of its primary pulls that long axis towards the line joining the two bodies. If the body is turning, the axis is repeatedly dragged towards that line and repeatedly carried past it, and the couple averages to zero — unless the turning is commensurate with the orbit.

On a circular orbit the primary’s direction advances uniformly, so the only rotation rate at which the long axis holds a fixed relationship to it is exactly the orbital rate. Every other rate lets the axis drift through all orientations, and the average couple vanishes. One lock.

On an eccentric orbit the primary’s direction does not advance uniformly. It sweeps quickly near periastron and slowly near apastron, and its angular velocity at periastron exceeds the orbital mean by a large factor. A body rotating faster than the mean can still return to the same orientation relative to the primary once per orbit, or once per two orbits, at a set of half-integer multiples of the mean motion — and at each of them a non-zero average couple survives.

The strength of each is a Hansen coefficient, a Fourier amplitude of the two-body motion:

$$H(1,e) = 1 - \tfrac{5}{2}e^2 + \tfrac{13}{16}e^4,\qquad H(3/2,e) = \tfrac{7}{2}e - \tfrac{123}{16}e^3 + \cdots$$

$H(3/2,e)$ is proportional to $e$ at leading order, which is the algebraic form of “no eccentricity, no 3:2 resonance”.

The pendulum #

Near any of these resonances the dynamics reduce to one equation. Let $\gamma$ be the angle between the body’s long axis and the direction that would hold it resonant. Then

$$\ddot\gamma = -\frac{3}{2}\,\frac{B-A}{C}\,n^2\,H(p,e)\,\sin 2\gamma,$$

which is a pendulum. It has a stable equilibrium, a libration around it, an unstable equilibrium, and a separatrix dividing libration from circulation.

The libration frequency follows immediately: $\omega_{\rm lib} = n\sqrt{3\,(B-A)/C\;H(p,e)}$. For Mercury, with $(B-A)/C = 2.03\times10^{-4}$ from spacecraft gravity measurements, that gives a free libration period of about twelve years — a genuine prediction, since every quantity in it is measured elsewhere.

The permanent bulge the whole thing needs #

There is one ingredient in the pendulum equation that is easy to pass over and is not automatic: the body has to be permanently non-round.

A perfectly fluid body raised into a tidal bulge has its bulge always pointing at the primary, by construction, and such a bulge exerts no restoring couple on the rotation — the bulge is a response rather than a feature, and it moves through the body as the body turns. What the resonance grips is a frozen asymmetry: a difference between the two equatorial moments of inertia that belongs to the solid body and turns with it.

Mercury’s is $(B-A)/C = 2.03\times10^{-4}$, measured from the spacecraft-mapped gravity field.

It is a small number and it is large enough: the libration period it gives is twelve years rather than infinity, and a body with no permanent asymmetry at all would have no spin–orbit resonance available to it whatever its eccentricity.

That requirement has a consequence for which bodies can be in such states. A body that has been molten throughout, or one whose interior relaxes on a timescale shorter than the libration period, cannot hold the asymmetry and cannot be captured. So a non-synchronous spin–orbit lock is itself evidence that a body has a rigid outer shell — which for Mercury was consistent with everything else and for a body like Europa would be a strong constraint.

Capture #

A body being tidally despun approaches the family of resonances from above, slowing steadily. It meets the 2:1 first, then the 3:2, then the 1:1. At each it either passes through or is caught, and the probability of capture is not one.

The mechanism is the one that governs orbital resonance capture: a slowly changing pendulum conserves the area its trajectory encloses, so a growing separatrix keeps what is inside it while a trajectory approaching from outside is caught only if it happens to arrive at the right phase. The fraction that does is the ratio of two areas.

For a simple tide — one whose lag is a constant angle, independent of frequency — the capture probability into the 3:2 at Mercury’s eccentricity works out at only about seven per cent. That is uncomfortably small for something that is observed to have happened.

The resolution proposed, and now generally accepted, is that Mercury has a liquid outer core, and that friction between that core and the mantle supplies a second, larger torque. With core–mantle friction included the capture probability rises above half. So Mercury’s rotation state is evidence for a liquid core, arrived at long before any measurement of one.

The measurement that confirmed the core #

That prediction was tested directly, and the test is one of the neatest measurements in planetary science.

A body in the 3:2 resonance does not rotate uniformly. The same varying torque that maintains the resonance also forces a small oscillation in the rotation at the orbital period — a forced libration, with an amplitude inversely proportional to the moment of inertia that has to be rocked. If the whole planet rocks, the amplitude is small. If only the mantle rocks, because a liquid core does not follow it, the amplitude is larger by the ratio of the two moments. Combining that with the planet’s obliquity, its gravitational flattening and its second-degree gravity coefficient gives the full set of interior parameters. That is the four-observable, two-unknown argument that turns a rotation state into a core radius, and it is the same reasoning that says whether a body’s heavy material sank.

The other resonances, and why they are empty #

If the 3:2 is populated because it was reachable, what about the 2:1, which the planet passed first?

Its strength at Mercury’s eccentricity is 0.33 against the 3:2’s 0.66, so its capture probability is correspondingly lower — and a passage through it that fails simply continues the despinning. There is also a long-term complication: Mercury’s eccentricity is not constant. Secular perturbations from the other planets vary it between about 0.1 and 0.3 over hundreds of thousands of years, and the resonance strengths vary with it, so the capture probabilities are themselves time-dependent. Integrations that include this find that Mercury may have been captured and released several times before settling.

The half-integer members of the family are worth a note of their own, because they are usually left out of the list and there is a reason for it. The resonant angle for a spin rate p times the mean motion involves the quantity 2p, so p may take half-integer values — 3:2 is p = 3/2 — and the strengths fall steeply with how far p sits from unity, roughly as the eccentricity raised to the power |2p − 2|. At Mercury’s eccentricity that is a factor of five between neighbouring members, so the 5:2 and everything above it are negligible before the question of capture arises. The family is short not because the others do not exist but because their widths are smaller than the rate at which the tide sweeps a body past them.

The 1:1 is the exception at the other end. Its strength does not vanish as the eccentricity goes to zero — a synchronous lock needs only the permanent bulge — so it is the one resonance that is always available and always the deepest. A body that survives every higher member ends there, which is why synchronous rotation is the common outcome in the solar system and Mercury’s state is the notable one.

The planet that did neither #

Venus is the counterexample, and its rotation is stranger than Mercury’s in a way that shows what the tidal argument leaves out.

It turns once every 243 days, retrograde, so its day is longer than its year and its sense of rotation is opposite to its orbit. The eccentricity is 0.0068 — nearly circular — so the family of resonances this essay is about has essentially only one member available, the synchronous one, and Venus is nowhere near it.

The solid-body tide raised by the Sun should have despun Venus and left it synchronous long ago. What appears to have prevented that is the atmosphere.

A thick atmosphere heated on the day side and cooled on the night side develops a pressure bulge, and because the heating has a thermal lag the bulge does not sit at the subsolar point. It is a tide raised by heat rather than by gravity, and it exerts a torque on the solid planet through the surface. For Venus, whose atmosphere is ninety times the Earth’s by mass, that thermal torque is comparable with the gravitational one — and it acts in the opposite sense, because a thermal bulge leads where a gravitational bulge lags.

The equilibrium is therefore a balance between two tides of different physical origin, and the rotation rate at which they cancel is not synchronous and need not even be prograde. Venus’s present state is consistent with such a balance, which is the standard account and is not a complete one: it does not by itself explain how the planet came to be retrograde in the first place, and the two candidate histories — a chaotic evolution of the spin axis, or a large early impact — are still argued.

Mercury’s rotation is decided by its interior and Venus’s by its atmosphere, which is an unhelpfully large difference between two planets of nearly the same size. It also means that a spin state is not a reliable interior diagnostic for any body with substantial air, which removes the technique from exactly the planets whose interiors are hardest to reach.

What it means for the surface #

A 3:2 rotation with a large eccentricity produces a solar day of 176 Earth days — two orbits — and a peculiar illumination pattern. Two opposite longitudes are at the subsolar point at perihelion on alternate orbits, and those are the hottest places on the planet; the two longitudes ninety degrees away have the Sun overhead at aphelion and are substantially cooler. There is a measurable longitudinal asymmetry in surface properties matching that pattern.

The tilt that comes with the lock #

There is a companion state to the spin–orbit resonance that is usually described separately and is really the same physics applied to a different angle.

A body’s spin axis precesses under the torque its primary exerts on its equatorial bulge, and its orbit’s plane precesses under the perturbations of everything else. If the two precession rates come into step, the spin axis can be captured into a state in which it stays at a fixed angle to the orbit normal and the two precess together — a Cassini state, named for the observation of it in the Moon three centuries before the theory.

For Mercury the relevant Cassini state has an obliquity of about two arcminutes: the spin axis is very nearly perpendicular to the orbit and not exactly so, and the small offset is maintained rather than being a leftover. That is what makes it a measurement.

The size of the offset depends on the body’s moment of inertia — a more concentrated body precesses differently — so measuring the obliquity to a few per cent measures the polar moment of inertia to a few per cent. It is one of the four observables that the interior determination described above uses, and it is the one that would be least accessible if the body were not in a resonance at all.

There is a further consequence for the libration measurement. The forced libration in longitude and the obliquity respond to different combinations of the moments of inertia, so having both is what separates the mantle’s moment from the whole planet’s. A body in a Cassini state with a measurable libration is therefore a body whose interior is over-determined, and that is why the case for Mercury’s liquid core is regarded as closed while the equivalent case for several icy moons is not.

It is worth adding what the same argument says about the bodies where it fails. An icy moon with a subsurface ocean has a mantle decoupled from its core in the same way Mercury’s is, so the libration amplitude is a measurement of the shell’s moment of inertia — and that is how the oceans of Enceladus and Europa have been argued for. The difficulty is that the libration also depends on the shell’s rigidity and on how well it is coupled to the ocean beneath, neither of which is measured, so the same observation that closes the case for a metallic core leaves an icy one open.

A resonance is a constraint, and a constrained system reveals more than a free one, which is the reason a planet in an unusual rotation state is worth more as an instrument than one in the ordinary one.

One more eccentricity shows where the three-to-two state stops being available at all.

Where the ladder goes #

The nearest rung is the synchronous case done properly: the Moon’s rotation is locked but not rigidly, and its own librations — free, forced, and geometric — expose about 59 per cent of its surface from the Earth.

Further out, the same machinery applies to bodies whose eccentricity is maintained rather than damped. A moon held eccentric by a resonance with another moon is trapped in a state where the tide never finishes, and the energy it cannot get rid of comes out as heat — which is why a moon can be molten with no radioactive source. Whether a body ends up in a non-synchronous state, a synchronous one, or an eternally heated one is decided by the same coefficients drawn in this essay’s first figure.