A day that has to be read from a shape

The length of a day seems the easiest thing to know about a planet. On Mars or Mercury it is found by watching a mark go round. A giant planet offers no mark. Its visible layer is cloud, driven by winds of hundreds of metres a second, and a spot at one latitude will return in a period that differs by many minutes from a spot at another. The number wanted is the rotation of the interior, the deep bulk of the planet, and no telescope can see that.

Yet the number is needed. The planet’s flattening, the harmonics of its gravity and the interpretation of its winds all take the rotation rate as an input. The core weighed by its gravitational field, reached through those harmonics, depends on the period of rotation as well as on what is in the planet. And the size of the winds depends directly on it, because a wind speed is a speed relative to the frame that rotates with the interior. Getting Saturn’s day wrong by six minutes means getting the equatorial jet wrong by nearly a hundred metres a second.

That is close to what happened. The earlier story of the field ended by noting that the radio period quoted for Saturn for thirty years was later found to have measured something else. What took its place is worth telling in its own right, because it inverts the usual logic: a quantity that was supposed to be an input to the study of the interior turned out to need that study in order to be known.

A clock needs a tilt #

The way to time a planet without a surface is to find something rotating rigidly with the interior and visible from outside. A magnetic field generated deep inside is the natural candidate. If the field’s axis is tilted to the spin, then the field’s orientation at any fixed outside point changes once per rotation, and so do the radio emissions tied to it. Jupiter’s decametric radio bursts give a period of 9 hours, 55 minutes and 29.7 seconds, one of the best-determined numbers in planetary science, and they have kept it steadily for decades.

Saturn does not cooperate. Its dipole is aligned with its spin to better than a hundredth of a degree, perhaps exactly, a remarkable fact on its own and one that dynamo theory does not comfortably predict. An axisymmetric field does not change when rotated about its own axis, and so it provides no periodic signal to read. A dipole’s symmetry is a problem for the clock in the same way as a uniform sphere would be a problem for anyone hoping to see it spin.

When the Voyager spacecraft passed in 1980 and 1981, they detected a radio emission from the auroral regions whose brightness rose and fell with a period of 10 hours, 39 minutes and 22 seconds. Since the same kind of signal gave an accurate period for Jupiter, this was adopted as Saturn’s rotation. It went into every table and every model. It was only when Cassini arrived in 2004 that the signal was seen to have changed. The period had lengthened by about six minutes in twenty-odd years. A planet with Saturn’s angular momentum cannot possibly have changed its rotation by that amount in that time, and the energy involved would be vast. So the signal could not be a clock.

The radio period is now known to be a property of the magnetosphere. It is different in the northern and southern hemispheres and it varies with the seasons. It seems to be driven by currents in the upper atmosphere rather than by the deep interior. No one had been measuring it wrongly, but the measurement had been read as something it was not, and all that had followed from the assumption had to be redone.

Forty years of answers #

The published values since then form a small history of the problem.

The picture has two striking features. One is that the inferences cluster at least five minutes below the old radio period, a difference with real consequences. The other is that they still disagree among themselves by about two minutes, and the most recent seismological value moved by more than a minute from the first. Saturn’s day is known better than it was, but not with the precision of Jupiter’s. Compared with the other things planetary scientists can determine about Saturn, including gravity harmonics measured to a part in a hundred thousand, it is curiously imprecise.

Each method has its own logic. To understand them, it is necessary to see what spin does to a fluid body.

What spin does to a shape #

A rotating fluid flattens, because it is held up partly by the centrifugal effect at the equator. How much depends on two things. One is how strong the rotation is compared with gravity, measured by the dimensionless rotation parameter $q = \omega^2 a^3/GM$, which for Saturn comes to about 0.158, the largest of any planet. The other is how the mass is concentrated, since a centrally condensed body has less of itself out at the equator to be flung outwards. The Darwin–Radau relation links the two with the moment of inertia:

$$\frac{C}{Ma^2} = \frac{2}{3}\left[1 - \frac{2}{5}\sqrt{\frac{5q}{2f} - 1}\,\right].$$

Given a moment of inertia, then, the flattening $f$ is a function of the period. It is nearly direct, because $q$ goes as the inverse square of the period.

The figure sets out the difficulty. A change of about six minutes in the period moves the flattening by a little under two thousandths. A shift in the moment of inertia of just two hundredths, which is well within the range of interior models, moves it about four times as far. Interior structure and period are thoroughly entangled, and a shape measurement alone cannot pull them apart. The flattening itself is known from radio occultations of spacecraft and from the stellar occultations by the planet. Even that carries an uncertainty, because its winds distort the shape in their own right. A jet of 400 metres a second holds up a bulge of its own, and the measured shape mixes the bulge due to the deep rotation with the part due to the wind.

Exact gravity, inexact interior #

The gravity field is a better measure of the planet’s response to rotation than the shape is, because it is measured to much higher precision and depends on the mass distribution rather than on a visible surface. Saturn’s second harmonic $J_2$ responds linearly to the rotation parameter, $J_2 = \Lambda q$, with a coefficient $\Lambda$ set by the interior. The field felt by a satellite in orbit has been read from Cassini’s Grand Finale orbits, to better than a part in a hundred thousand. So the left side of that equation is known to an extraordinary degree. The problem is on the right.

The sensitivity is the point. Because the period goes as the square root of the coefficient, a one per cent uncertainty in the interior’s response is three minutes in the length of the day. Interior models that fit all the harmonics do so with a wide range of cores, helium gradients and layered compositions, and they can differ in $\Lambda$ by amounts of that order. The period can therefore be read only once the interior is known, and the interior only once the period is.

There are ways of cutting into that circle. John Anderson and Gerald Schubert, in 2007, took the planet’s measured shape and the condition that the wind field should be as small as possible, in a sense they defined through the dynamic heights required. Peter Read and collaborators in 2009 used a related condition on the potential vorticity. Ravit Helled and others in 2015 minimised the difference from the observed shape over the whole family of periods compatible with the harmonics. All these found a period around 10 hours and 32 to 34 minutes. Each one has introduced a further assumption about the winds, and they all reach similar answers, but none measures rotation directly. It is a strong clue when several different assumptions converge, though it does not amount to proof.

What the choice of frame does to the wind #

The way the period enters the winds is simple, and it says something about what a wind is. A cloud tracked at a given latitude has a definite speed in space, and that is what the images measure. The wind is the difference between that and the rotation of the deep interior beneath it.

In the old radio frame nearly all of Saturn’s zonal flow was to the east, unlike Jupiter’s, which has alternating eastward and westward jets more or less in balance. That contrast was a puzzle for many years, and meteorological models tried to explain why the two gas giants differed so greatly. Much of the contrast vanishes with a shorter period. In the newer frame, the eastward jets are weaker and westward flow appears at the latitudes where the old frame showed none. A substantial piece of apparent planetary meteorology was thus a mistake about a reference frame, and the difference between these two planets turned out to be smaller than had been thought.

That is more than a matter of bookkeeping, since a wind’s depth can be measured. The harmonics $J_6$ to $J_{10}$ that Cassini measured show a signature of winds extending to a depth of about 9,000 kilometres. The inference depends on the winds’ profile in the deep frame, since the odd and high even harmonics respond to the flow and the depth of the flow is what they measure. A shifted frame therefore affects the conclusion, and any analysis needs a consistent period running all through it.

A probe that never went in #

The most direct evidence came from a place no one had expected to be a seismometer. Saturn’s rings are a sensitive detector of periodic forces, and oscillations in the planet slightly perturb its gravity, generating spiral density waves in the C ring wherever a pattern’s speed matches a resonance in the ring particles. Mark Marley and Carolyn Porco proposed this in the early 1990s, and Matthew Hedman and Philip Nicholson measured the first such waves in 2013 in Cassini stellar occultations. These data give a pattern speed for each wave, along with its number of arms, $m$.

For a prograde fundamental mode, the pattern seen from outside moves at the rotation rate plus the mode’s own frequency divided by the number of arms:

$$\Omega_p = \Omega + \frac{\sigma_m}{m}.$$

So a mode’s measured pattern speed adds the rotation and structure together, but the two appear differently.

This is the same trick that gives the interior rotation of the Sun from its frequency splittings: a rotation adds in one way, and structure in another, and the measured pattern across many modes can be separated into the two. For Saturn, the method’s leverage is the many modes measured, each with its own $m$. Christopher Mankovich and his collaborators, in 2019, fitted a family of interior models to the patterns of the waves and to the gravity, and found a period of 10 hours, 33 minutes and 38 seconds, with an uncertainty of about a minute and a half. In 2023 a more complete treatment of the rotation’s effect on the modes, including differential rotation in the outer layers, yielded about 634.7 minutes, a shift of over a minute from the earlier value. So the method is capable of giving Saturn’s day, and it is still being refined.

The waves carry a further surprise. Some of the observed waves lie close together in pattern speed and do not correspond to a single fundamental mode per $m$. The splitting appears to arise from coupling with gravity modes confined to a stably stratified layer deep in the planet, and it has been read as a sign of a diffuse core, spread over a large fraction of the radius. The rings thus show both the day and the core at once, from outside, as helioseismology’s frequency comb does for the Sun.

The surprise in a measured quantity #

There are two lessons in this episode that apply beyond Saturn. One is that the distinction between a measurement and what it measures can hide for decades. The radio period was perfectly genuine and precisely determined, it was repeatable and its source was understood in general terms. Yet the inference from Jupiter to Saturn, from one planet whose field is tilted to another whose field is not, was never in fact justified. Measured quantities are much less easily questioned than modelled ones.

The other is the shape of the problem. Science normally proceeds with a measured input and a modelled output, and this one has neither: the rotation, the interior and the winds are determined together, in a joint fit. That is a less satisfying kind of answer, as it does not admit of a single decisive experiment. It also means a change in any one part reaches all the others. An improved model for the deep layer changes the mode frequencies, which changes the period, which changes the winds, which changes the harmonics that were used to fit the layer. Saturn has become a system of equations, with its day among the unknowns.

The same doubt, further out #

The episode raises an obvious question about the two ice giants. Their fields are strongly tilted, so a radio clock should in principle work well, and the periods in the tables, about 17 hours 14 minutes for Uranus and 16 hours 7 minutes for Neptune, come from radio emission recorded by Voyager 2. But each planet was visited once, for a few days, in the 1980s, and a period from one short flyby cannot be checked for the drift that Saturn showed.

Ravit Helled, John Anderson and Gerald Schubert applied the wind-minimising method to these two in 2010, and they arrived at periods different from the Voyager values by more than half an hour, in opposite directions: Uranus shorter, Neptune longer. That result is contested and rests on the same kind of assumption as the Saturn work. For Uranus a separate line has since opened. Auroral features tracked over a decade of ultraviolet imaging with a space telescope give a period close to the Voyager one, and more accurate, so its radio clock seems to have been sound. That is what a tilted field should provide, and it makes Saturn look like the exception that it is.

The difference between the two kinds of planet is the difference this whole account turns on. A clock can be read only where the symmetry is broken. A field tilted by fifty-nine degrees gives a loud signal once per rotation, and a field tilted by less than a hundredth of a degree gives nothing of the kind, and whatever periodic signal is found near it must have some other origin.

What the picture cannot show #

These figures treat Saturn’s rotation as one number, though it need not be. Differential rotation in the deep interior would mean that a single period is itself an average, and the most recent seismological analyses, finding differences among modes, allow for this. The Darwin–Radau relation is a first-order approximation, reasonably good for the Earth but growing inaccurate at Saturn’s rotation rates, where higher-order terms in $q$ matter and the relation can be off by more than the differences discussed here. The approximation serves in the figure only to show the entanglement of period and moment.

The mode-frequency formula in the ring figure is schematic, chosen to display the separation between a uniform offset and a dependence on $m$. It is not a calculation of Saturn’s real modes, which need a model of its interior with corrections for rotation at high order. The wind profile in the figure is a drawn sketch with a few jets, and only the difference between its two curves is computed. Nor does anything here prove that the seismic period is the correct one. The distinction between it and the others is in the agreement of several independent methods to within about a minute or two, and that is an argument, not a measurement.

Still open: whether Saturn has one rotation at all #

If the deep interior rotates differentially, then there is no single day to find, and the disagreement among estimates would be a physical result. The gravity harmonics seen by Cassini suggest that the winds decay by about 9,000 kilometres down, and below that the planet should be close to rigid, since a magnetic field and conducting metallic hydrogen brake differences in rotation. But the layer the ring waves probe has been suggested to be stably stratified, and such a layer can in principle rotate at a different rate from what lies around it. The further step is to detect modes that reach different depths, each with its own effective rotation, as was done for the Sun, and the tachocline that came out of that work was a surprise. Whether Saturn has anything similar is a question whose answer may come from waves in the rings alone, since no probe going to Saturn in the foreseeable future will be able to measure that inside.