A rocking nobody started

Four numbers that weigh a planet’s core measured Mercury’s forced libration — the rocking a body does in response to the uneven torque an eccentric orbit applies — and turned its amplitude into a statement about how much of the planet’s moment of inertia is following the mantle.

Every part of that measurement is a response to something external. The forcing has the orbit’s period, its amplitude is set by the torque, and the body contributes only the moment of inertia being swung.

A locked body also oscillates at a period of its own, and that oscillation has no external cause at all.

The pendulum, and its period #

A body locked to its orbit sits in a potential minimum: its long axis points at its primary, and displacing it from that direction produces a restoring torque proportional to the displacement and to the difference between the two equatorial moments of inertia.

That is a pendulum, and its angular frequency is

$$\omega_{\rm free} = n\sqrt{\frac{3(B-A)}{C}}\,,$$

with $n$ the orbital mean motion and $(B-A)/C$ the body’s triaxiality — the fractional difference between its two equatorial moments, divided by the polar one. For a body in a three-to-two resonance rather than one-to-one the expression carries an extra factor depending on the eccentricity.

Two things about that follow immediately.

The period is set by the body. The orbital period enters as a scale, and everything else is a property of the shape. Nothing about the torque, the primary’s mass or the orbit’s eccentricity appears, so measuring the free period is measuring $(B-A)/C$ directly.

And it is long. Triaxialities are of order $10^{-4}$ to $10^{-3}$, so the square root is a few per cent and the free period is tens to hundreds of orbits — years for a moon, years for a planet.

Why it should not be there #

A pendulum with dissipation rings down. The tides that keep a body locked in the first place dissipate energy, and the amplitude of any free oscillation decays on a timescale of about $Q$ of its own periods, with $Q$ the tidal quality factor — a number that is meaningless without the period it was measured at and which for rocky bodies is of order tens to hundreds and which a heat flow depends on to a power nobody can compute.

A free period of three years and a $Q$ of a hundred gives a damping time of about fifty years. Even a $Q$ of a thousand gives five hundred.

Against the age of the solar system those are nothing. Whatever free libration a body was left with when it settled into its lock has been gone for four and a half billion years, and anything oscillating now was started since.

So a free libration is not a property of a body; it is an event in its recent past, and the quantity it measures is not a shape but a disturbance. That inverts the earlier measurement, where every number was a static property.

Two oscillations in one spectrum #

What a rotation measurement delivers is a time series of orientation, and what an analysis does with it is take a spectrum. Both librations appear in it, at different frequencies, and separating them is the whole of the method.

The forced terms are at the orbital frequency and its harmonics, plus the frequencies of every periodicity in the orbit — the perturbations from the Sun and the planets put lines at their own frequencies too, and for the Moon there are dozens of them with amplitudes above a milliarcsecond. Every one of those lines has a predicted frequency and a predicted amplitude, both computable from the orbit and the body’s moments.

The free terms are at frequencies nothing in the orbit produces. That is the identification: a line at 2.9 years in the lunar rotation spectrum corresponds to no combination of orbital frequencies, so it is the body’s own.

The forced spectrum is a prediction and the free lines are a residual, which is an unusually clean division. It also means the free libration’s detectability depends on how well the forced terms are modelled: an error in a predicted forced amplitude leaves a residual at a forced frequency, and if that frequency happens to lie near a free one the two are hard to separate.

For the Moon the free longitude period of 2.9 years sits comfortably clear of any strong forced line, which is why it was identified early and has survived every revision of the model. The latitude modes at 74.6 and 80.1 years are closer to the periods of several long-term orbital terms, and their amplitudes have moved more.

What the Moon’s requires #

The Moon’s free libration is detected, and it has been for decades, because lunar laser ranging measures the orientation of the lunar surface to a few milliarcseconds.

Three free modes exist and all three are seen: a longitude libration with a period of 2.9 years and an amplitude of about 1.8 arcseconds, and two latitude modes at 74.6 and 80.1 years with amplitudes of tens of arcseconds. Each is a different oscillation of the same pendulum in a different degree of freedom.

All three should have damped long ago, and there is no agreed explanation for any of them. The candidates are worth listing because each is a different kind of statement about the Moon.

An impact. A large enough strike would excite the oscillation, and the damping time gives the required recency — within the last few thousand years for the longitude mode. Impacts of that size are rare enough that this requires an unlikely coincidence.

A core–mantle interaction. The Moon has a small fluid core, and turbulent coupling at the boundary between it and the solid mantle can transfer energy into the libration rather than out of it. This is the favoured explanation for the latitude modes and it requires a core that is still liquid, which is consistent with what the forced libration and the seismic data say.

And a resonance. If some periodic forcing in the lunar orbit happens to lie near a free period, it can pump the oscillation continuously. Several candidate forcings exist at roughly the right frequencies and none is a convincing match.

That three explanations exist and none is established is the honest state of it. What the detection establishes without argument is that something has been putting energy into the Moon’s rotation recently, and the amount is small but not zero.

What Mercury’s absence bounds #

The same search has been made for Mercury and has found nothing.

The radar and spacecraft measurements of Mercury’s rotation are consistent with the forced libration alone, and the residuals bound any free libration at well under an arcsecond. Since the damping time is of order thousands of years even at a generous $Q$, that is not surprising — the surprising case is the Moon.

The bound is nonetheless worth something. An upper limit on a free libration is an upper limit on how much energy has been put into the planet’s rotation in the last few thousand years, from any source: impacts, internal activity, or coupling to a core that is known to be liquid.

That last one is the interesting constraint. If turbulent core–mantle coupling excites the Moon’s free libration, the same process should excite Mercury’s — and Mercury’s core is far larger relative to the planet. The non-detection therefore bounds the vigour of whatever is happening at Mercury’s core–mantle boundary, which is a statement about a region three thousand kilometres down obtained from a rocking measured across a hundred million kilometres of space — the same chain the moment-of-inertia factor runs down, one step further.

What was actually measured #

For the Moon: the round-trip travel time of laser pulses to retroreflector arrays left on the surface, measured to a few millimetres over fifty years.

The orientation of the Moon comes out of that as a by-product. A range to a point on a rotating body depends on where that point is, so a long series of ranges to several points constrains the rotation — and the free modes appear as periodic residuals at frequencies no forced term produces, which is why they are identifiable at amplitudes far below the forced libration’s.

Three things about the measurement are worth stating.

It is a fit, not an observation. The lunar orientation comes from a solution that simultaneously fits the Moon’s orbit, the Earth’s rotation, the stations’ positions, the retroreflectors’ positions, several relativistic terms and the lunar interior model. A free libration amplitude is a parameter of that solution, and its error depends on correlations with everything else in it.

The signal is small and the baseline is long. An amplitude of 1.8 arcseconds is about fifteen metres at the lunar surface, against a range measured to millimetres — so the detection is not marginal in signal-to-noise. What makes it hard is separating a 2.9-year periodicity from everything else with a period near that.

And it has improved because the model has. Several of the free-mode amplitudes moved substantially as the solution’s treatment of the lunar core and of tidal dissipation was refined, which is the normal behaviour of a parameter that is correlated with the thing being modelled.

For Mercury: radar speckle tracking from the Earth and altimetry and gravity from a spacecraft in orbit, combined into a rotation model whose residuals bound the free term.

An amplitude that is a rate #

The most useful thing about a damped oscillator with an unknown excitation is that its amplitude is a rate rather than a state, and it is worth doing the arithmetic because it turns an angle into a power.

The energy in a free libration is the moment of inertia times the square of the angular velocity of the oscillation — for the Moon, with an amplitude of 1.8 arcseconds at a period of 2.9 years, that comes to a few times $10^{12}$ joules.

If the oscillation is in a steady state, that energy is being supplied as fast as it is dissipated, which is once per damping time. At a damping time of decades the required power is of order $10^{3}$ watts.

A thousand watts is a laughably small number by any planetary standard — the Moon’s own radiogenic heat output is of order $10^{12}$ — and that is exactly why the observation is difficult to use. The free libration is a detector of an energy input so small that almost anything could supply it, which makes the detection robust and the interpretation nearly unconstrained.

The upper limit for Mercury is correspondingly weak in absolute terms and correspondingly strong as a statement about rates. What it excludes is not a process of a given power but a process that has deposited more than a certain energy into the rotation within the last few damping times, which for a planet whose interior is known to be partly liquid is a genuine constraint on how vigorously that liquid is moving.

A small signal with a short memory is a sensitive instrument for recent events and a poor one for large ones, and knowing which of the two a measurement is before quoting it is most of using it correctly.

Where the model stops #

The damping estimate is a one-line argument. Treating a free libration as a lightly damped harmonic oscillator with a decay time of $Q$ periods is the standard approximation, and the dissipation in a real body is frequency-dependent in a way that a single $Q$ does not capture — which is what the quality factor’s own caveat is about. At the free libration’s frequency, which is very low, the dissipation is likely stronger than at the tidal frequency where $Q$ is usually quoted, which shortens the damping time further.

Excitation and damping are not separate. The core–mantle coupling that is a candidate excitation is also a dissipation mechanism, and what is observed is the equilibrium between them. A steady-state amplitude set by a balance is a different kind of measurement from a decaying one, and which of the two the Moon’s modes are is not settled.

The three lunar modes are treated here as one phenomenon. They are not: the longitude mode and the two latitude modes have different restoring torques, different periods, and probably different excitations, and lumping them costs the argument some of its force.

And the figures use a single triaxiality per body. A real body’s moments are known to a few per cent at best, and for the icy satellites they come from a hydrostatic assumption rather than from a measurement — so the free periods drawn for them are predictions rather than data.

What excites one and not the other #

Setting the Moon’s detection beside Mercury’s non-detection is more useful than either alone, because the two bodies differ in the ways that matter and the comparison narrows the candidates.

Impacts. Mercury is struck far more often than the Moon — it sits deeper in the Sun’s well, where impactors are faster and more numerous — so an impact-driven excitation should be more evident there, not less. The comparison argues against impacts.

Core–mantle coupling. Mercury’s core is proportionally enormous and the Moon’s is small, so this mechanism should also favour Mercury. That the Moon shows the effect and Mercury does not is awkward for it, unless the coupling depends on something other than the core’s size — on the boundary’s topography, or on whether the core is convecting, or on the presence of a solid inner core.

Resonance with an orbital term. This is the one the comparison does not argue against, because it depends on an accident: whether some periodicity in the body’s orbit happens to lie near its own free period. The Moon’s orbit is perturbed by the Earth and the Sun and is rich in long-period terms; Mercury’s is perturbed by the other planets and is not rich in the same way.

So the comparison favours the explanation that is a coincidence, which is unsatisfying and is what the evidence says. It is also testable: a resonant excitation predicts a specific frequency and a specific phase relationship, and the lunar ranging solution is now long enough to look for both.

The generalisation #

The shape worth carrying is that a system’s own frequency and a system’s forced response measure different things, and that the difference is whether the answer is a property or a history.

A forced response is a statement about a body’s structure and about the forcing, and it persists as long as the forcing does. A free response is a statement about the body’s structure and about when it was last disturbed, and it decays. A measurement at a system’s own frequency is a clock on the last event, and the clock’s rate is the dissipation.

That is why looking for free oscillations is a standard way of asking whether something is still active. The same argument is made about a star’s oscillation modes, about a star’s interior read from its own comb of frequencies, about a planet’s normal modes after a large quake, and about the Earth’s own free oscillations, which ring for days after a great earthquake and are how the deep interior is sounded.

The second reading is about upper limits. A non-detection of a free libration is a bound on an energy input, and an energy input is a rate rather than a state — so a null result here constrains a process rather than a quantity, which is an unusually useful thing for a null result to do. The value of a non-detection depends on how fast the thing being looked for decays, and a fast-decaying signal turns an absence into a tight constraint on the present rather than a weak one on the past.

Still open: the plane a libration is measured against #

What comes next is the thing the figures here have taken for granted. Every amplitude quoted is an angle, and an angle needs a reference — for a locked body the reference is a Cassini state, an equilibrium in which the spin axis, the orbit normal and a fixed normal stay coplanar while all three precess together.

That state is itself a solution of the same rotational dynamics, it has its own free oscillation about it, and the obliquity it fixes is what weighs a whole planet’s moment of inertia. Whether a body is in the state it appears to be in, and which of the four Cassini states it settled into, is a question about a body’s rotational history that the present rotation only partly answers.