A fluid that turns as one piece

The Sun’s surface does not turn at one rate. Sunspots near the equator come round in about twenty-five days and spots at forty degrees of latitude take twenty-eight, and this has been known since the 1630s, when it was worked out from drawings. It is a genuine measurement of a rotation and not of anything else, which is more than can be said for most quantities in this collection: a spot is a feature on the surface, and watching it come round is an angle divided by a time.

That much is unsurprising. A rotating fluid with convection in it will not turn rigidly, because a rising parcel carries its angular momentum upward and arrives moving too fast or too slow for the level it has reached. What was expected was that the interior would do more of the same — the same reasoning that makes the outer third of the Sun boil also predicts what boiling does to rotation — that the differential rotation would deepen, and that the surfaces of constant angular velocity would be cylinders aligned with the rotation axis, which is what a rotating, convecting, barotropic fluid is obliged to produce.

Neither expectation survived the measurement. Inside the convection zone the surfaces are very nearly radial rather than cylindrical, so the latitude dependence persists all the way down with hardly any change. And beneath about seven-tenths of the radius it stops entirely: the radiative interior turns as one solid piece.

Two questions that sound like one #

The distinction matters because it separates two questions that sound like one. How fast does the Sun rotate has an answer to three figures. How is that rotation distributed had no answer at all until the 1980s, and could not have one, because there was no observable inside the star to carry it. Rotation is not visible in a brightness, or a spectrum, or a mass; it is visible in the frequencies of sound.

What a splitting is #

A non-rotating star’s oscillation modes come in degenerate multiplets. A mode of angular degree $\ell$ has $2\ell+1$ patterns on the sphere that differ only in how they are oriented, and in a spherically symmetric star they must all have the same frequency, because nothing distinguishes one orientation from another.

Rotation distinguishes them. A pattern that travels in the direction of rotation is carried along and appears at a higher frequency; one travelling against it appears lower. The degeneracy splits, and to first order the spacing between adjacent components is exactly the angular velocity — averaged over the star, but with a weighting that the mode itself sets.

That weighting is what makes the technique a probe of depth rather than a single number. Each mode occupies its own cavity: it is refracted back before reaching the centre, and where it turns depends on its degree. So a low-degree mode averages the rotation over nearly the whole star, and a high-degree one over the outer skin, and the difference between their splittings is the rotation of the part in between. This is the same structure of argument as the scaling relations that give a mass and a radius from two numbers, taken one level further: there the whole spectrum was compressed into two global quantities, and here the individual frequencies are kept and their differences are the data.

The latitude dependence comes from the other index. Within a multiplet the components differ in how many nodal lines they have running through the poles, and a component with many of them is confined near the equator while one with few spans the whole sphere. So a multiplet’s pattern of splittings — not merely its overall spacing — resolves the rotation in latitude as well as in radius, and the two-dimensional map that results is what shows the surfaces to be radial rather than cylindrical.

The inversion is not a fit to a parameterised profile. It is a construction: for a given target depth, a combination of the measured splittings is found whose combined weighting is as close to a spike at that depth as the available modes allow, and the same combination applied to the data returns the rotation there. The width of the achieved spike is the resolution, and it is quoted alongside the answer — which is the reason nobody claims to know the rotation of the inner fifth of the Sun.

What was actually measured, and what it cost #

The Sun’s surface moves by a few tens of centimetres per second in each mode. Detecting that is a Doppler measurement at the level of parts in $10^{9}$ of the speed of light, and the mode frequencies must be measured to parts in $10^{6}$ — which means a time series many years long, without gaps, because a gap in a time series puts sidelobes into the spectrum that are indistinguishable from real modes.

Two solutions were built and both are still running. The first is a network of six identical instruments spaced around the Earth’s longitudes, so that the Sun never sets on the observation. The second is a spacecraft at the first Lagrange point, in permanent sunlight. Between them the solar oscillation spectrum has been recorded almost continuously since the mid-1990s, and the rotation profile is a fit to more than a hundred thousand individual splittings.

The honest limits are worth stating with the result. Below about 0.2 solar radii there are very few modes with any sensitivity — only the lowest degrees reach there, and there are not many of them — so the rotation of the core is measured with large error bars and has been the subject of repeated claims of detection and retraction. The safe statement is that the deep interior rotates within about ten per cent of the rest of the radiative zone, and that anything faster than about twice the surface rate is excluded.

It is worth being explicit about how strange the result in the hero figure is, because familiarity has made it look ordinary. A gas has no rigidity. Two shells of a radiative stellar interior at different radii are not attached to each other in any mechanical sense, and there is no reason whatever for them to agree on a rotation rate to within a per cent — any more than two layers of the Earth’s atmosphere do. Whatever enforces the agreement has to reach across four-tenths of a solar radius and keep doing so for the star’s whole life.

The layer that should not be sharp #

Between the differentially rotating convection zone and the rigidly rotating interior there is a transition, and it is thin: a few per cent of the radius, which for the Sun is a few tens of thousands of kilometres. It is called the tachocline, and both of its properties are problems.

The first problem is that it is sharp. A shear layer in a fluid spreads, by viscosity if by nothing else, and over four and a half billion years even the Sun’s very small microscopic viscosity would have spread this one across a substantial fraction of the radiative zone. It has not spread. Something is holding it, and the leading candidate is a weak fossil magnetic field in the radiative interior — a field of a few gauss would be enough, and it would also explain the rigidity above. A few gauss is a very small field by stellar standards and an entirely undetectable one: it is buried under the convection zone, and the only thing that leaves the centre of a star is neutrinos, which do not care about magnetism.

The second problem is that it exists at all. A rigidly rotating radiative zone is not what an ordinary fluid does. It is what a fluid does when angular momentum can be transported efficiently over long distances, and the only mechanism available in a non-convective, non-turbulent region is magnetic: field lines threading different radii act like springs, and a spring between two shells enforces corotation. But a fossil field also has to be confined — it must not diffuse up into the convection zone, where its shear would produce far more differential rotation at the surface than is observed — and arranging that is delicate.

The connection between the magnetic cycle and this layer is the strongest argument for the tachocline’s importance and it is not a measurement. Nobody has observed a field at 0.7 solar radii. What is observed is a shear layer in the right place with the right stratification, and a cycle that needs one.

There is a competing account in which the cycle is generated in the near-surface shear layer instead — a second, much thinner region of strong shear in the outermost few per cent, visible in the hero figure as the drop just below the surface. It is favoured by some simulations and it removes the storage problem, at the cost of removing the tachocline’s explanation for the equatorward migration. The argument is live, and the observation that would settle it is a measurement of the field at depth, which nothing can currently make.

Giants, and the angular momentum that went somewhere #

The Sun is one star, and one star cannot say whether a rigid interior is general. Red giants can, and their answer is more interesting.

A giant has a helium core the size of the Earth inside an envelope the size of Mars’s orbit, and the two are coupled only by whatever transports angular momentum between them. The core got there by contracting, and contraction conserves angular momentum: a core that shrinks by a factor of thirty in radius should, left alone, spin up by a factor of nine hundred, and the full calculation from the main sequence gives factors of tens of thousands.

Measuring the core rotation of a giant is possible because a giant’s modes are mixed. An acoustic mode in the envelope can couple to a gravity mode trapped in the dense core, and the resulting hybrid carries information from both regions in one frequency. The splitting of a mixed mode is therefore a weighted average of core and envelope rotation with a weighting that shifts from mode to mode — and inverting a few dozen of them separates the two.

The measured core-to-envelope ratio is about ten. Not ten thousand. So angular momentum is being removed from the core almost as fast as contraction concentrates it, by a mechanism efficient enough to be nearly rigid and yet not quite — and no proposed mechanism produces the right number. The discrepancy is not a small residual to be absorbed by tuning: it is the difference between a core that would be spinning near break-up and one that is barely spinning at all. Magnetic torques from a field generated in the core are too strong or too weak depending on the assumed geometry; internal gravity waves are promising and hard to calibrate; and the meridional circulation that classical stellar rotation theory relies on is far too slow.

This is one of the clearest quantitative failures in stellar physics, and it was invisible for as long as the only measurable interior was the Sun’s, where the contrast never arises because a main-sequence star’s core has not contracted.

What the giants say about when it happens #

A single number — a core spinning ten times its envelope — is a constraint on the transport but not a diagnosis. What narrows it is following the ratio along the evolutionary sequence, and the sequence is available because a survey measured tens of thousands of giants at once.

The picture that emerges is that core rotation falls as a star ascends the giant branch. That is a stronger statement than the ratio alone. The core is contracting throughout that ascent, so conservation of angular momentum applied to the core in isolation demands that it spin up by a large factor; what is observed is that it spins down. Angular momentum is not merely being prevented from concentrating — it is being removed from the core faster than contraction supplies it.

The rate at which it is removed can then be compared against the timescale of the contraction, and the comparison says the coupling time between core and envelope is of order ten million years or less at that stage. That is short: shorter than the evolutionary timescale, which is why the ratio stays near ten rather than growing, and far shorter than anything the classical hydrodynamic transport mechanisms supply.

The subsequent stage is more informative still. When a low-mass star ignites helium in its core, the core expands and the contraction reverses; a core that had merely been coasting on its earlier angular momentum would then spin down by the expansion factor and no more. The measured core rotation rates of the resulting clump stars are lower than that, which means transport continued to remove angular momentum through the transition rather than switching off with the contraction.

So whatever does the transporting is not driven by the contraction itself. It operates at a rate set by something else — a magnetic field’s strength, or the amplitude of the internal waves the convective envelope launches downward — and it operates whether the core is shrinking or growing. That eliminates a family of proposed mechanisms in which the shear produced by contraction is what drives the transport, which is most of the family that classical rotating-star theory offers.

Why a star’s rotation is also its age #

There is a practical payoff to all of this that has nothing to do with interiors, and it turns the sky into a clock in a way no orbit does. A magnetised wind leaving a rotating star carries away angular momentum with a very long lever arm, because the field forces the escaping gas to corotate out to many stellar radii before releasing it. The result is a braking law under which rotation decays roughly as the square root of time — which means a measured rotation period is an age.

That method, gyrochronology, is now one of the few ways to date an ordinary field star, and it depends directly on the question this essay is about. If the interior and the envelope were decoupled, the surface would brake while the interior kept its angular momentum, and the surface rate would recover whenever the coupling caught up — making the period a poor clock. The observed tightness of the period–age relation for stars older than a few hundred million years is itself evidence that main-sequence interiors are rotationally coupled to their surfaces on timescales short compared with the braking.

The assumptions gyrochronology runs on are worth naming, because the same two are what make a pulsar’s characteristic age unreliable by factors of twenty: a braking law whose index is assumed rather than measured, and an initial rotation rate assumed to have been fast enough not to matter. An age derived from a braking law inherits both, whatever the star.

One more set of kernels shows which part of the star each mode actually samples.

Where the ladder goes #

The next rungs of this anchor divide by what the rotation is being read for. One direction is the dynamo: what the tachocline’s shear actually does, why the cycle is eleven years rather than eleven hundred, and why some stars stop cycling altogether. Another is transport: the giants’ missing factor of a thousand is a well-posed question with a wrong answer, and whatever fixes it will also change how much material is mixed into a burning core and therefore how long stars live.

There is also a thread that leads out of stars entirely. Angular momentum concentrated by contraction and then removed is the same problem that governs how a collapsing cloud becomes a star at all rather than a disc that never accretes, and the same problem that decides how fast a neutron star is born spinning. In each case the naive conservation argument overshoots by orders of magnitude and something has to carry the excess away, and in each case what carries it is thought to be a magnetic field whose strength nobody can measure.

And there is a methodological thread running through all of it, which this collection takes up under the disagreement between two ways of knowing the Sun. The inversions above assume a structural model — the mode cavities and the kernels are computed from one — so a sound speed that is wrong by a per cent propagates into the rotation profile. The two measurements are not independent, and the interior of the nearest star is known to a precision at which that matters.