A wind that takes no mass and all the spin

A star has two obvious ways of getting rid of something: it can radiate it, or it can blow it away. Energy leaves by the first route in enormous quantities and nothing is left behind to show for it. Angular momentum cannot leave by that route at all — light carries essentially none — so the second route is the only one there is, and the accounting becomes interesting the moment the amount of material involved is written down.

The Sun’s wind removes roughly two parts in a hundred million million of its mass each year. Over four and a half billion years that is about a hundredth of a per cent, which is a number so small that the Sun’s structure has never noticed it. Over the same interval the Sun’s rotation has slowed by something like a factor of ten. One wind, two accounts, and they disagree by three orders of magnitude.

The lever, and why it squares #

The mechanism is a lever, and the length of the lever is the whole of it.

Consider one gram of gas leaving the surface with no field to hold it. It carries away the angular momentum it happened to have at the surface, which is $\Omega R^2$ per unit mass. That is the naive account, and under it a star that loses a hundredth of a per cent of its mass loses a hundredth of a per cent of its spin, in the same proportion, and rotation would be essentially permanent.

Now add a magnetic field. Near the star the field’s energy density is much larger than the wind’s kinetic energy density, so the field wins the argument about where the gas may go: the gas is channelled along field lines that are anchored in the layer of the star that boils, and it is dragged around with the star as it climbs. That corotation is not free — it is the field doing work — but it continues until the wind has picked up enough speed that its kinetic energy density matches the magnetic one. That radius is the Alfvén radius, and past it the field is carried by the flow rather than the other way round.

So the gram of gas is released at $r_A$, not at $R$, and it carries $\Omega r_A^2$ per unit mass. The loss rate becomes

$$\frac{dJ}{dt} = -\frac{2}{3}\,\dot{M}\,\Omega\,r_A^2,$$

with the two-thirds coming from averaging over a sphere rather than a single equatorial streamline. Set against the star’s own angular momentum $J = k^2 M R^2 \Omega$, the ratio of fractional rates is

$$\frac{\dot J / J}{\dot M / M} = \frac{2}{3}\frac{1}{k^2}\left(\frac{r_A}{R}\right)^2,$$

which contains no physics beyond the geometry and the star’s internal mass distribution. For the Sun, with $r_A/R$ somewhere between ten and fifty and $k^2 = 0.073$, that ratio is between about a thousand and twenty thousand.

The star’s own concentration matters here and is easy to miss. A uniform sphere would have $k^2 = 0.4$; the Sun, with most of its mass well inside half its radius, has 0.073. A centrally condensed star has less angular momentum to lose in the first place, so the same wind removes a larger fraction of it. Concentration is a multiplier on the brake.

What is actually measured, and what is inferred #

The honest position is that only one of the three numbers in that ratio is measured well.

The mass loss rate is measured, in the solar case, by particle detectors on spacecraft that have flown through the wind. That is about as direct as astronomy gets, and it is why the Sun anchors the whole subject: for every other star the mass loss has to be inferred from an emission line, from the astrospheric absorption a wind carves in the interstellar hydrogen ahead of it, or from a model.

The rotation is measured directly too, from spots crossing the disc or from the Doppler shift of light from the approaching and receding limbs.

The Alfvén radius is the number that is not measured. It is computed from a wind model, and the model needs the surface field strength, the field’s geometry, and the wind’s acceleration profile. The first of these can now be measured by Zeeman techniques on other stars, the second is partly imaged and largely assumed, and the third is a theory. So the quantity that enters squared is the one least under control, which is why published braking laws for other stars differ from each other by factors of several.

The law that follows, and the exponent that does not depend on the details #

Something useful survives that mess. Suppose only that the Alfvén radius grows with the field, and that the field grows with rotation — which is what a dynamo driven by rotation and convection does. Then the braking torque grows faster than linearly in the rotation rate, and the standard result is

$$\frac{d\Omega}{dt} \propto -\Omega^3.$$

That has an exact solution, and the solution has a property that is worth more than its accuracy:

$$\Omega(t) = \frac{\Omega_0}{\sqrt{1 + 2K\Omega_0^2 t}} \;\longrightarrow\; \frac{1}{\sqrt{2Kt}}.$$

The limit contains no $\Omega_0$. A star that was born spinning very fast brakes very hard and one that was born slow brakes gently, and the two converge onto the same curve. The initial condition is not merely diluted; it is erased.

Convergence is what makes the whole subject useful. It means that after a few hundred million years a star’s rotation is a function of its age and its mass alone, and that is an age read off a rotation period rather than a fossil of its birth.

Where the braking stops #

There is a boundary, and it is sharp enough to have a name.

Braking of this kind requires a magnetic field, the field requires a dynamo, and the dynamo requires a convective envelope — a layer where the gas overturns, so that rotation can wind and shear the field into something strong. A star of about 1.3 solar masses and above has no such layer: its envelope carries energy by radiation, and the convective zone is confined to a small core deep inside. Above that mass the wind has no strong ordered field to ride, the Alfvén radius collapses towards the surface, and the lever disappears. The observational signature is a step. Cool stars rotate slowly, hot stars rotate fast, and the transition is abrupt. Nothing about a star’s structure changes abruptly at that mass — the convective envelope thins continuously — but the braking is so steeply dependent on it that a gradual structural change produces a discontinuity in the outcome. That step is the same boundary that decides whether a star can pull a misaligned planet back into its own equatorial plane, for the same reason: it is a boundary in dissipation rather than in shape.

The regime where the brake stops responding #

The braking law above assumes the field grows with rotation, and above a threshold rotation rate it stops doing so.

The observation is direct. Measures of magnetic activity — chromospheric emission, X-ray luminosity, the fraction of the surface covered in spots — rise steeply as a star’s rotation period shortens, and then flatten. Above a rotation of roughly ten times the Sun’s, an activity indicator no longer distinguishes a fast rotator from a faster one: the dynamo has saturated.

Why it saturates is not settled. The candidate explanations are that the field reaches the strength at which its own pressure is comparable to the gas pressure in the convection zone and cannot grow further; that the surface simply runs out of room for more active regions; or that the dynamo’s own back-reaction on the convection limits it.

The consequence for the braking is what matters here. If the field is capped, the Alfvén radius is capped, and the torque no longer grows as the cube of the rotation rate but as the first power. The braking of a very fast rotator is therefore far weaker relative to its spin than the unsaturated law would give, and the convergence described above is delayed accordingly.

That shows in the tracks. Without saturation, a star born at half a day would brake to the common curve within a few million years; with it, the fastest rotators stay fast for a hundred million years and then fall onto the curve abruptly when they drop below the saturation threshold. Young clusters show exactly that structure: a fast-rotating sequence, a slow one, and a sparsely populated gap between them, with the gap moving to longer periods as the cluster ages.

The saturation is what makes the early rotational history readable, because it prevents the initial conditions from being erased immediately — and it is why the clusters between ten and a few hundred million years old are the ones the whole calibration rests on.

The field the lever is attached to #

The braking is not a steady, structureless drain. The field that sets the Alfvén radius is generated by a dynamo that is itself organised, and on the Sun it is organised into an eleven-year cycle. The field’s own organisation is watched directly in the eleven-year march of sunspots towards the equator, and it is not steady. A dipolar field opens few field lines and holds the wind out to a large radius; a complicated multipolar field closes over on itself near the surface and holds it to a small one. Since the torque goes as the square of that radius, the geometry matters more than the total field strength. This is the least satisfactory part of the subject and the part where the observations of other stars are least helpful, because a Zeeman measurement of a distant star delivers a disc-averaged line-of-sight field and the geometry has to be reconstructed.

Measuring the flux directly #

The Alfvén radius was described above as the quantity that is computed rather than measured, and for stars it remains so. For the Sun it has now been measured, by flying through the region where it sits.

A spacecraft in the solar wind measures the plasma’s velocity vector and its density, and the tangential component of the velocity — the part perpendicular to the radial direction — is exactly what carries angular momentum. Multiply by the density and the radius and integrate over the sphere, and the result is the angular momentum flux, with no model in it.

The measurement is hard because the tangential velocity is small: a few kilometres a second against a radial flow of four hundred. It is a component of a vector at the half-per-cent level, and it requires the instrument’s own pointing to be known better than that.

The results have been surprising in a specific way. The angular momentum flux measured close in is larger than the models predicted, and it is not distributed as expected: much of it is carried by the magnetic field’s own stress rather than by the particles, and the split between the two changes with distance in a way the simple picture does not contain.

There is also a structural finding. The tangential flow is not uniform — it varies between the fast wind from open field regions and the slow wind from the boundaries of closed ones — so the total is an average over a structured flow rather than a property of a spherically symmetric one.

The single number in the braking law turns out to be a compressed description of something with structure, which is the ordinary fate of a single number, and the correction is not yet large enough to change the exponent in the braking law.

What the wind removed, and what removed the rest #

It would be a mistake to conclude that magnetic braking accounts for all of the Sun’s missing rotation. It does not.

The bulk of the problem was solved during formation, when a collapsing cloud with far too much spin per unit mass had to hand most of it outwards to a disc. What the wind has done since is take a star that arrived on the main sequence turning in a few days and slow it to twenty-five, which is a factor of about ten and is the last act rather than the main one.

The distinction matters when the present rate is used to extrapolate. Integrating the measured solar torque backwards under a constant field gives a young Sun spinning implausibly fast, and the fix is the same one that the Moon’s recession needs: the present rate is not the historical average, because the quantity that sets it was different in the past.

The braking that appears to stop #

There is a difficulty at the old end of the sequence that has emerged in the last decade and that bears directly on whether a rotation period is an age.

Rotation periods measured for field stars with independent asteroseismic ages show the expected relation up to about the age of the Sun and then depart from it: stars older than that rotate faster than the braking law predicts, by an increasing margin.

Two readings are on offer. The first is that the braking genuinely weakens — that beyond some critical rotation, the field’s geometry changes to a configuration with a smaller Alfvén radius, so the lever shortens and the torque collapses. That is called weakened magnetic braking, and it has a plausible mechanism: as a star slows, the dynamo’s ability to maintain a large-scale dipole falls, and a multipolar field closes over near the surface.

The second is that the sample is contaminated. Rotation periods are measured from spot modulation, and old inactive stars have few spots — so the ones with measurable periods are the unusually active ones, which are the unusually fast rotators. That is a selection effect that produces exactly the observed departure.

Distinguishing them requires rotation periods for old stars measured without spots, which is possible for the small number with asteroseismic detections, since the rotational splitting of the oscillation modes gives a period regardless of activity. Those measurements are consistent with the weakening being real, on a sample of a few dozen.

If it is real, gyrochronology fails for exactly the stars whose ages are hardest to get any other way, and the Sun sits near the boundary — which means the Sun is either at the end of its braking or in the middle of it, and the two give different answers for the rotation of every star older than it.

The ambiguity will not be settled by adding field stars, because the selection effect scales with the sample. What settles it is an old cluster with a well-determined age and periods measured for its slow rotators, and clusters older than four billion years are both rare and distant.

The lever and the law it produces are worth reading at settings that bracket the ones the essay uses, because the exponent is supposed to be independent of them and the amplitude is not.

The same accounting somewhere else #

The best check that the argument is about accounting rather than about stars is that the identical structure appears where nothing resembles a star. The same accounting is what turns a pulsar’s slowing into a statement about its own field. A neutron star and a cool dwarf have nothing in common except that both have a magnetic field, both eject something, and neither can radiate angular momentum away. That is enough to make the arithmetic identical.

And the calibration itself at a bluer reference colour, since the relation is fitted separately for every spectral type.

Where the ladder goes #

The immediate next rung is the one the convergence makes possible: if the tracks forget their initial conditions, a rotation period becomes an age, and the clock that works by forgetting is what that costs and what it buys.

Two harder directions lead out of it. One is the geometry problem — the Alfvén radius enters squared and is computed rather than measured, so the single largest uncertainty in stellar age-dating is a magnetic field configuration nobody has resolved. The other is what happens to the star’s interior while its surface is being braked: the wind grips the outermost layer, and something has to carry the torque inwards to the rest. Whether it does so quickly enough for the star to brake as one body is a question the Sun’s own interior rotation answers, and answers in the affirmative in a way that no proposed mechanism comfortably explains.