A state the Moon had to jump to

Every amplitude in the libration of the Moon is an angle, and the essay on free libration ended by asking what the angles are measured against. For a locked body the reference is a Cassini state: an equilibrium in which the spin axis, the orbit normal and the normal to the plane the orbit regresses about all lie in one plane and turn together. Cassini described the Moon’s in 1693 from observation alone: the Moon’s equator is tilted a degree and a half to the ecliptic, on the opposite side from its orbit, and the three normals stay coplanar as the orbit’s node goes round every 18.6 years.

The geometry is so tidy that it reads like a fact about the Moon. It is a fact about a balance, and the balance has changed. The Moon has been receding from the Earth for four and a half billion years, and every quantity in the balance depends on its distance. Following the balance back shows that the Moon’s present state did not always exist in its present form, that another state existed when the Moon was close, and that the Moon must have moved from one to the other by a jump.

Two motions in balance #

The Moon’s spin axis is not free. The Moon is slightly flattened and slightly elongated, and the Earth’s gravity pulls on that bulge with a torque that tries to swing the axis towards the orbit normal. A spinning body answers a torque by precessing, so the axis precesses about the orbit normal, at a rate proportional to the bulge — the difference of moments of inertia $\beta = (C-A)/B$ — and to the Moon’s orbital rate.

The orbit normal is not fixed either. The Sun’s pull on the Earth–Moon system makes the Moon’s orbital plane regress, its node going backwards round the ecliptic once every 18.6 years, while its inclination stays near 5.145°. The orbit normal therefore circles the ecliptic pole.

A Cassini state is the arrangement in which the spin axis precesses about the orbit normal at exactly the rate needed to keep pace with the orbit normal’s own circling, so that in the frame turning with the node nothing moves. The condition is

$$\lambda \sin\theta\cos\theta = \sin(\theta - I),$$

where $\theta$ is the angle between the spin axis and the orbit normal, $I$ the orbit’s inclination to the ecliptic, and $\lambda$ the ratio of the spin axis’s precession rate to the node’s regression rate. Everything about the Moon’s interior enters through $\lambda$.

How many states there are #

The condition is an equation between two curves, and its solutions are where they cross. The regression side is fixed: a sine curve offset by the inclination. The torque side is a sine of twice the angle, scaled by $\lambda$. When $\lambda$ is small, as it is today, the torque curve is shallow and crosses the regression curve once at small angles — the Moon’s state, at 6.68° from the orbit normal, which is 5.145° plus Cassini’s 1.54° — and once more near 180°, a state in which the Moon would spin backwards. When $\lambda$ is large, the torque curve is steep enough to cross the regression curve twice more, and there are four states.

The boundary between the two cases is a definite number. For an inclination of 5.145° the extra pair of states exists when $\lambda$ exceeds 1.31; at exactly that value the two new crossings touch, and below it they are gone. This is the same bifurcation that governs the obliquity of a planet, where the ratio is of the spin-axis precession to the precession of the planet’s orbit and the Earth and Mars sit near it. For the Moon the ratio is not near the boundary today. It was, once.

The critical value is a function of the inclination alone, $\lambda_c = (\sin^{2/3} I + \cos^{2/3} I)^{3/2}$, and it is worth seeing how it behaves. For an orbit in the reference plane it is exactly one: the torque and the regression balance at a single ratio, and the extra pair of states appears the moment the torque wins. A tilted orbit raises the threshold, slowly at first and then faster, because the regression of a tilted orbit drags the spin axis harder: at the Moon’s 5.145° it is 1.31, at twenty degrees it would be about 1.7. The inclination of the Moon’s orbit therefore decides where the bifurcation lies, and since that inclination has its own unexplained history, the distance of 34 Earth radii carries the uncertainty of the Moon’s early inclination with it. A Moon whose orbit was tilted by ten degrees when it was young would have met the bifurcation a little closer in; one whose orbit lay in the reference plane, a little further out. In every case the bifurcation exists, and in every case the Moon passed through it.

A ratio that falls as the distance cubed #

Both rates in the ratio depend on the Moon’s distance, in opposite directions. The precession of a locked Moon’s spin axis is driven by the Earth’s torque on its bulge, and for a body turning once per orbit it goes as the orbital rate — as the distance to the minus three halves. The regression of the node is driven by the Sun’s perturbation of the Moon’s orbit, which grows as the square of the Sun’s orbital rate divided by the Moon’s — as the distance to the plus three halves. Their ratio falls as the distance cubed.

Today the ratio is 0.232: the spin axis precesses about four times more slowly than the node regresses. Halve the distance and the ratio rises eightfold. At 33.8 Earth radii it passes 1.31, the critical value, and closer than that the Moon had four Cassini states rather than two.

The distance is reached early. The Moon formed close to the Earth — within a few tens of Earth radii, on any account of its formation — and receded fastest at first, when the tides it raised on a fast-spinning Earth were strongest. The usual reconstructions put it past 34 Earth radii within a few hundred million years of its formation. Whatever happened at the bifurcation happened during the period of the Moon’s youth when its interior was still hot, its crust still forming, and its shape still being set.

The state that disappeared #

The hero figure follows each state through the distance. The state the Moon occupies today, state 2, continues inward to ever larger obliquities: 15° at 40 Earth radii, 45° at the bifurcation, 80° at 20. State 1, which exists only inside the bifurcation, has a small obliquity — the spin axis near the orbit normal — growing from about a degree at 20 Earth radii to 21° at 33.8. State 4, between them, is unstable.

A Moon that formed close to the Earth, with its spin axis nearly perpendicular to its orbit, would have settled into state 1, the low-obliquity state, because that is where tidal dissipation drives a spin axis that starts near the orbit normal. As the Moon receded, state 1 drifted to larger obliquity and state 4 approached it; at 33.8 Earth radii they met and both vanished. The Moon’s spin axis, which had been sitting at an equilibrium, found itself at a point where there was no equilibrium at all, and it was carried round, by the same precession that had held it, to the only stable state remaining — state 2, at about 45°. That is the jump the green line in the hero figure marks, from an obliquity of 21° on one side of the orbit normal to 45° on the other.

The argument was made by William Ward in 1975, using the same balance with the numbers of the time. It does not depend on details. It requires only that the Moon formed inside 34 Earth radii, that its shape was roughly what it is now, and that tidal dissipation had brought its spin axis to the stable state available — and each of those is generally accepted.

Following an equilibrium that moves #

A spin axis does not have to sit exactly at a Cassini state; it can circle one, precessing about it at a frequency set by the same balance. Tidal dissipation inside the Moon damps that free circling, slowly, and draws the axis onto the state. Once there, the axis stays there even as the state itself moves, provided the state moves slowly compared with the period of the free circling — a few decades for the Moon — which the Moon’s recession, a few centimetres a year now and perhaps metres a year at first, always allowed.

That following is what makes the jump inevitable rather than possible. A Moon that was in state 1 when young stayed in state 1 as the state drifted outward with the recession, the axis tracking the equilibrium like a marble in a bowl that is slowly being tilted. The tracking is an adiabatic invariance of the same kind that preserves other slowly-changing quantities in celestial mechanics: the area enclosed by the axis’s free circling stays constant while the bowl changes shape. What adiabatic following cannot survive is the bowl disappearing. When states 1 and 4 merge, the bowl around state 1 shrinks to nothing, the invariance breaks, and the axis is released onto whatever path the new geometry offers — which carries it round to state 2.

The tilt Cassini described is a reading of the ratio #

Since the jump, the Moon has been moving outward along state 2, and its obliquity has been falling. The tilt Cassini described is the obliquity minus the orbit’s inclination, and for a weak torque it is nearly $I\lambda/(1 - \lambda)$: proportional to the ratio itself. At 40 Earth radii the Moon’s axis was 14.5° from the ecliptic pole. Today it is 1.54°. When the Moon reaches 70 Earth radii it will be 0.88°, and in the limit of a Moon so far away that the Earth’s torque no longer mattered, the spin axis would stay fixed in space while the orbit regressed around it, and the tilt would be zero.

That is the sense in which the tilt measures the Moon. The ratio $\lambda$ contains the bulge, and once the orbital rates are known, the tilt gives the bulge.

The measured tilt of 1.5427° gives $\beta = 6.24 \times 10^{-4}$. Laser ranging to the reflectors the Apollo astronauts left on the Moon measures the physical librations directly and gives $6.31 \times 10^{-4}$. The two agree to about a per cent, and the remaining difference is the part of the torque the simple condition leaves out — the Moon’s second, smaller moment difference, which also contributes, and terms of higher order in the eccentricity and inclination of the orbit. The full theory used with the ranging data includes them and agrees with the tilt to far better than a per cent. It is the same measurement that weighed Mercury’s core, with the Moon’s numbers: a body locked in a Cassini state carries its moments of inertia in the angle of its axis.

Why Mercury sits in the other state #

Mercury is also in a Cassini state, and it is the other one. Its spin axis is tilted only two arcminutes from its orbit normal, in state 1 — the low-obliquity state the Moon lost. The difference is the ratio. Mercury’s orbit precesses very slowly, its node regressing once in about three hundred thousand years under the planets’ perturbations, while its spin axis precesses relatively fast under the Sun’s strong torque on its bulge; its $\lambda$ is far above the critical value, and it has all four states available, of which it has settled into the one nearest its orbit normal.

The Moon’s orbit regresses much faster — once in 18.6 years, because the Sun’s perturbation of a satellite’s orbit is strong — and its spin axis precesses slowly, because the Earth’s torque on a small, nearly spherical body is weak. That puts the Moon’s ratio below critical, where state 1 does not exist. So the two best-measured locked bodies in the Solar System sit on opposite sides of the bifurcation, and the measurement that weighed Mercury’s core through its state-1 obliquity weighs the Moon’s flattening through its state-2 one. The Galilean satellites, whose orbits regress under Jupiter’s oblateness and the other satellites’ pulls, are expected to sit in state 1 with obliquities of a fraction of a degree, and measuring those obliquities from orbit is one of the ways their interiors will be weighed.

A jump the Moon would have felt #

A change of obliquity from 21° to 45° is not a quiet event for a body raised in tides.

A locked Moon with a tilted spin axis is flexed by the Earth’s tide twice each orbit, because the tidal bulge sweeps north and south across its surface. The heating that flexing produces grows as the square of the sine of the obliquity and, through the tide’s strength and frequency, as a high power of the orbital rate. Along the Moon’s path it was hundreds of times today’s rate while the Moon sat in state 1 near the bifurcation, simply because the Moon was close; the jump to state 2 multiplied it by four in the time the spin axis took to swing over, a few precession periods; and the heating then decayed as the Moon receded and its obliquity relaxed along state 2.

How much energy that deposited depends on how long the Moon spent near 34 Earth radii, which depends on how fast it was receding, which depends on the tidal dissipation in the early Earth — none of which is known to better than a factor of several. Estimates of the total range from a modest warming of the interior to enough to affect the crust as it formed, and the timing coincides with the period in which a moon is heated by not being allowed to relax, on any body. The figure gives the shape of the episode, not its size.

What an obliquity jump could leave behind #

An event that rotated the Moon’s spin axis by tens of degrees relative to its orbit, during the early epoch when its surface was forming, might be expected to have left marks. A change of obliquity changes where the Moon’s surface is coldest and where tidal stresses are largest, and a change of the spin axis relative to the Moon’s body — true polar wander — changes which parts of the surface face the poles.

Two lines of evidence have been read this way, neither conclusively. The distribution of ice in permanently shadowed craters near the lunar poles is offset from the present poles in a way that suggests the poles have moved by several degrees since the ice was deposited, but that movement is attributed to changes in the Moon’s interior rather than to the Cassini transition, which preceded the ice by billions of years. And the Moon’s shape — its bulge is larger than its present orbit and spin would support — is a fossil of a time when the Moon was closer and spinning faster, frozen in while the crust was still pliable. The jump’s own fossil, if it exists, is in that frozen shape, set while the Moon was near the distance where the jump happened.

What the balance leaves out #

The calculation holds the Moon’s bulge fixed at its present value. The bulge itself is a fossil of the early orbit, set when the Moon was closer and spinning faster, and if it was frozen in near the bifurcation then $\beta$ was close to its present value there — which is what the calculation assumes. If the bulge was larger earlier, the bifurcation was further out.

The orbit’s inclination to the plane it regresses about is held at 5.145°. That plane is not the ecliptic when the Moon is close: it is the Laplace plane, which lies near the Earth’s equator for a close Moon and moves to the ecliptic as the Moon recedes, and the Moon’s inclination to it has its own unexplained history. A different inclination moves the critical ratio and the obliquities, though not the existence of the bifurcation.

And the condition is the two-term relation of the simplest model of a precessing top. The full treatment includes the triaxial terms that reconcile the tilt with the ranging data, the eccentricity of the orbit, and the dissipation that makes the spin axis lead the Cassini state very slightly — by a quarter of an arcsecond today, a lead that measures the friction between the Moon’s mantle and its fluid core along with its solid tide.

Still open: what the jump did to a young Moon #

The Moon’s present Cassini state is an equilibrium it reached by elimination: the state it would naturally have occupied when young ceased to exist as it moved outward. That much follows from the balance and the distances. What does not follow is what the transition did.

The answer depends on three numbers the Moon’s early history has not yet supplied: how fast it was receding at 34 Earth radii, how dissipative its interior was then, and how large its bulge was. The ranging data, which measure the present dissipation and the present lead of the spin axis, constrain how the Moon’s interior responds now; the rocks returned from the Moon constrain its thermal history; and models of the early Earth’s tides constrain the recession. Put together, they would say whether the Cassini transition was a footnote in the Moon’s thermal history or one of its defining events — whether the tilt that measures the Moon’s flattening today also records the moment, four and a half billion years ago, when the Moon’s spin axis was forced to the other side of its orbit.