A planet tipped in one blow would have its moons going backwards

Most planets carry a tilt modest enough to give seasons and little else. Uranus has an obliquity of 98 degrees, so its axis lies almost in the plane of its orbit. For about two decades at a stretch, one pole faces the Sun and the other sits in darkness. Obliquity is not a fixed property, and it can be changed in two ways: by a slow resonance, or by being hit. For Saturn the slow route has a serious claim. For Uranus the standard account since Viktor Safronov’s work in the 1960s has been a collision with something of about Earth’s mass, late in the formation of the planets.

The moons were taken as a straightforward supporting clue. The five large moons of Uranus and all its rings orbit in the planet’s tilted equator, as Voyager 2 confirmed in 1986, and they go round in the same direction as the planet spins. A planet knocked over by an impact would be expected to rebuild its moons from debris around its new equator, and so the moons would naturally lie there and move prograde. For a long time that seemed sufficient.

It is not, and the reason is a fairly simple piece of mechanics which was pointed out by Alessandro Morbidelli and collaborators in 2012. A disc that falls onto a new equator does not forget which way it was going, so the moons’ sense of revolution is a fossil record of what the tilt was before the last blow. Working through that yields a claim that is both surprising and precise: a single impact on an initially upright Uranus should have left retrograde moons.

What a tilted ring does #

Suppose a planet suddenly acquires a new spin direction, with a ring of material left behind in its former orbit, inclined to the new equator. The new equatorial bulge exerts a torque on the ring, and the ring’s plane precesses about the spin axis at a rate given by the same oblateness that moves a satellite’s node:

$$\dot\Omega = -\frac{3}{2}J_2\left(\frac{R}{a}\right)^2 n\cos i.$$

The rate falls steeply with distance, as the seven-halves power of the radius.

So rings at different distances are out of step within a short time on any geological reckoning. A tilted, flat disc becomes a twisted one, and soon the rings cross each other’s paths. Crossing orbits means collisions, and collisions in a disc dissipate energy but conserve total angular momentum. The motions that cancel out are the ones that differ from one ring to the next. Eventually only a thin flat disc in the plane of symmetry remains, namely the planet’s equator.

That collapse explains why moons lie in the equator even if the material that formed them was left somewhere else. It is the same process that made the Moon’s tilt so puzzling in the opposite direction, and it has an interesting consequence for the sense of motion.

The sign that survives #

When a disc collapses, the part of its angular momentum along the spin axis cannot change, as the planet’s axisymmetric bulge exerts no torque in that direction. The part perpendicular to the spin is averaged away by precession, because each ring’s perpendicular component points in a different direction once the rings are out of step. So what is left is the component along the axis.

So the sign of the moons’ revolution is set by whether the tilt of the disc to the new spin axis is under or over ninety degrees, and that is all. Take a Uranus that started upright, with a ring system in its equator. Knock it to 98 degrees in one blow. The old disc now lies 98 degrees from the new spin, and its angular momentum along the new axis is negative. When it collapses, it will make a disc in the new equator going backwards. Moons formed from it would be retrograde, and that is not what is observed.

The same holds for debris from the impact itself, in a slightly less direct way. An impact that delivers the angular momentum to tip the planet also launches material in orbit, and much of that shares the sense of the impactor’s approach, so it tends to be prograde with the new spin. However, the question then becomes what happens to the pre-existing satellite system, since gas giants forming with a disc naturally acquire moons, and these are scattered by the blow into crossing orbits and ground up. They provide a large share of the mass that eventually forms the outer satellites. That was the argument: if the debris of the old system dominates at large distances, its sense of rotation has to be accounted for.

A last blow that cannot be the only one #

The escape is straightforward, and it is the striking part. Nothing in the argument forbids an impact that delivers less than ninety degrees. A planet that was already tilted before the last blow can arrive at 98 with a final turn of under 90 degrees.

This is a very small requirement geometrically, as the prior tilt need only be more than eight degrees, if the last blow turned the axis in just the right plane. In general the required prior tilt depends on the direction of the turn. With an arbitrary direction, a substantial tilt is more likely to give a final turn below 90 degrees, and so Morbidelli and his co-authors concluded that Uranus had to have had a tilt comparable to Neptune’s, about 30 degrees, before it was finally knocked to 98. Neptune thus becomes a clue to Uranus’s early history, a sibling planet which seems to show what a moderately struck ice giant looks like.

That is a remarkable inference to come out of a sign. Nothing about the present rotation of Uranus or the present orbits of its moons contains any information about the size of a former impact. Yet the moons’ prograde motion constrains the sequence. It rules out one specific history, and that history happens to be the simplest one.

How the odds change with two blows #

The next question is how common each sequence would be. Treat each impact as adding a spin vector in a random direction to whatever the planet had, and follow what emerges.

The picture is intuitive. A single large blow can certainly produce a tilt near ninety degrees, but it does so with the wrong sign. Two smaller blows can do the same job with the correct sign, in a modest but appreciable share of trials. A sequence of many smaller impacts would give a broader spread of final tilts. Since late-stage accretion of the giant planets is thought to have involved a few very massive bodies, a pair of large late impacts is not at all an implausible history. Ice giants in simulated growth histories often collide with several bodies of Earth mass and above.

How much angular momentum this requires gives a further sense of scale.

This is a useful sanity check. The bodies involved are not unreasonable by the standards of late planet formation, nor are they far outside what the isolation mass for the outer solar system suggests. It also means the planet’s spin itself could have been substantially reset, since a body of this mass carries as much angular momentum as the planet has. There is no reason to think that Uranus’s present rotation of about seventeen hours is primordial. The rotation and the tilt may both be a legacy of the last big blows.

The boundary of the argument #

The account assumes the planet’s bulge is in charge of the debris, and this has a limit. Beyond a certain distance the Sun’s tide dominates over the equatorial bulge in determining a satellite’s plane, as the theory of the Laplace plane explains. An orbit inside that transition precesses about the planet’s equator. An orbit outside it precesses about the planet’s orbital plane.

Uranus’s regular moons reach out only to 23 radii, comfortably inside that radius. In fact the problem is at the outer moons’ scale, where the precession period is thousands of years and the collisional damping is much slower. Simulations of the collapse find that an outer disc would not precess fast enough with today’s $J_2$ to randomise before it reforms into moons. Morbidelli’s group proposed that a compact disc of debris from the impact itself, close in and massive, raised the effective oblateness temporarily, as the dashed curve in the first figure shows, by a factor large enough to speed things up. The disc then spreads and is lost back onto the planet, and the moons remain.

That proposed compact disc has proved to be a problem. Julien Salmon and Robin Canup, in 2022, followed the evolution of such debris in more detail and concluded that the outer debris had to start from a broader distribution to reach Oberon’s distance, and that the pre-impact obliquity would then be even more narrowly constrained. Shigeru Ida and colleagues pointed out in 2020 that the debris from a collision with an ice giant would be largely water vapour, which behaves very differently from a rocky disc as it cools. Raluca Rufu and Canup also found it may be challenging to explain the current system from an impact alone. So the qualitative argument stands, but its quantitative details remain a live area.

The alternative route #

There is another class of explanation that does not involve an impact at all, and it fits the sign automatically, because a slow tilt carries the moons with it. If a planet’s obliquity grows slowly, moons near the planet stay in its equatorial plane adiabatically, just as Saturn’s do as it precesses, and they keep their sense of rotation with it. So they would stay prograde at any tilt, with no need for an extra blow.

The way to tilt Uranus slowly is through a spin-orbit resonance, in which the planet’s axis precession matches a frequency of the orbital plane’s motion. It works for Saturn, in the sense that its present tilt can be matched by such a capture. For Uranus, the difficulty is that its spin precession is very slow, and to bring it into resonance needs a large additional source of torque. Gwenaël Boué and Jacques Laskar in 2010 proposed a temporary massive moon. Others have suggested a dense, extended disc during formation. Going all the way to 98 degrees in a single adiabatic sweep has proved hard, because the resonance typically stops well short, and these models need a long time. The two routes can also be combined, and it has been argued that a resonance could produce the first few tens of degrees, and a smaller impact the remainder. That combination would satisfy the moons’ constraint, as it would supply the prior tilt required.

So there is now a logical structure instead of a story. A single impact from upright is excluded by the sign of the moons, unless their mass is entirely of impact origin in a prograde sense. What remains is a prior tilt, by an impact or a resonance, with a subsequent event. Each of these is a statement with testable consequences for the moons’ composition and the planet’s interior.

The other ice giant #

Neptune is the obvious comparison, and it complicates the picture in useful ways. It has about the same mass and size as Uranus and a tilt of 28 degrees, within the range the argument needs as a prior condition. If the two planets formed alike and suffered similar histories, Neptune looks like Uranus without its last blow. That reading is attractive because it asks for only one additional event. It also requires a coincidence that nobody can yet check, since an obliquity of about thirty degrees might itself have needed an impact, and two similar planets receiving their last blows at very different scales is no more than a small-number statement about random events.

The moons of the two planets do not make the comparison easier. Uranus has five large regular moons, all near its equator. Neptune has no comparable system at all. Its largest moon, Triton, goes round backwards on an inclined orbit, and the capture of so massive a body would have scrambled and largely destroyed any regular moons it found there, so the system that might have recorded Neptune’s earlier history has probably gone. What ought to be the control in this experiment is missing.

Their interiors are a further clue. Uranus emits almost no heat beyond what it receives from the Sun, while Neptune emits more than twice what it receives. One proposed explanation is that a giant impact left a composition gradient that blocks convection in Uranus’s interior, or that it was the other way about and Neptune was struck in a way that stirred it. Those proposals predict different internal layering, and a measurement of that layering would be evidence about the blow independent of the moons. So far both planets have been visited just once, by one spacecraft passing through in a matter of hours.

A fossil that is one bit long #

The surprising connection is the economy of the evidence. The sense of revolution of a moon is a single bit: forward or backward. That single bit, with a little dynamics, constrains an event more than four billion years old for which no other record exists. It does so because of conservation. A quantity conserved through a catastrophic rearrangement passes straight through it, and what comes out carries its sign.

The same logic appears elsewhere. A retrograde hot Jupiter is a record of an interaction that a flat disc could not have produced. Triton’s backward orbit shows it was captured and not formed in place, an inference made from the same single bit. In each case the direction of motion has a meaning larger than the magnitude of anything else that could be measured about the object, and it is hard to fake by any process that conserves angular momentum.

The Uranus case is unusual in one respect. For a captured moon, the sign is about the moon’s own history, whereas here it is a claim about the planet’s history, read off moons which were probably not even present when the argument’s first event occurred.

What the picture cannot show #

The figures treat the disc’s angular momentum as a simple projection, and they imagine that nothing else adds to it. In reality the impact’s own ejecta adds angular momentum of its own, in a prograde sense, and a very massive and vaporised disc behaves differently from a ring of solid moonlets. If all of the moons formed from such ejecta, with almost nothing from a pre-existing system, the sign argument would not apply as stated. The relative contribution of each is the central uncertainty, and hydrodynamic simulations disagree on it.

The turn in the steps figure is drawn as if everything happens in one plane. The random trials use one size per blow and treat each impact as adding angular momentum instantly in an isotropic direction, ignoring the fact that an impactor’s orbit has a preferred plane, namely the plane of the disc where the planets formed. Those are serious simplifications, and the numbers are meant only to show that two blows open a route that one blow closes. The Laplace-radius figure ignores the effect of existing moons on the effective $J_2$, which raises the radius somewhat. Nothing here determines the masses or the timing of the bodies that struck Uranus.

Still open: what the moons are made of #

The most direct test of the origin of the moons is their composition. Uranus’s large moons are a little denser than Saturn’s mid-sized moons, with a substantial fraction of rock, and that has to be explained by whatever disc they formed from. A vaporised impact disc rich in water from an ice giant’s mantle would have made them icier, while a primordial co-accretion disc or rocky impact debris would give a different ratio. A mission to Uranus has been the highest priority in the most recent American planetary decadal survey, and an orbiter would measure the moons’ interiors and masses for the first time at a precision comparable to what was done for the Galilean satellites. Those measures would say whether the moons are remnants of an old system ground up and rebuilt or new bodies condensed from a vaporised disc. The sign of their orbits has already said that the last blow was not the first. Their composition may say how many came before it.