A lever set by the dipole and not the field

The Sun has lost about a ten-thousandth of its mass to its own wind over its whole life and most of its rotation to the same wind, and the whole of that asymmetry is one geometric factor: the gas leaving the surface is forced by the magnetic field to keep turning with the star until it is far out, at the Alfvén radius, where the wind’s own momentum finally beats the field. A gram of gas released there carries the angular momentum of a point a dozen solar radii from the axis, and the torque on the star goes as the square of that distance.

The essay that set this up noted, almost in passing, that the distance depends on how the field is arranged — that a field which closes over near the surface holds the wind for a shorter way than one that reaches out — and that this is the least satisfactory part of the subject. It also matters most. The clock that rotation provides runs on the torque, the torque runs on the square of the lever, and the lever runs on the geometry. What follows makes that dependence quantitative, from the scalings that magnetised wind simulations give, and follows it to the conclusion the scalings force: the field that brakes a star is not the field that is easiest to measure.

How far each kind of field can hold a wind #

A star’s surface field can be written as a sum of multipoles: a dipole, like a bar magnet, with field lines leaving one hemisphere and returning to the other; a quadrupole, with four regions of alternating polarity; an octupole, with six; and so on. Close to the surface every order can be equally strong. Far away they are not, because a field of order $l$ falls with distance as $r^{-(l+2)}$: the dipole as the inverse cube, the quadrupole as the inverse fourth power, the octupole as the inverse fifth.

The Alfvén radius is where the field’s energy density falls to the wind’s kinetic energy density, so a field that falls faster reaches that point sooner. Simulations of magnetised winds, run for each order separately, give the Alfvén radius as a power of a single combination of the star’s properties, the wind magnetisation

$$\Upsilon = \frac{B_\star^2 R_\star^2}{\dot M\, v_\text{esc}},$$

the surface field squared over the wind’s momentum flux. Adam Finley and Sean Matt fitted the results in 2017 and 2018 as $R_A/R_\star = K_l\,\Upsilon^{m_l}$, with exponents of 0.229 for a dipole, 0.134 for a quadrupole and 0.087 for an octupole — close to the $1/(2l+2)$ a simple balance of energy densities gives.

The magnetisation is chosen here so that a dipole holds the wind to twelve stellar radii, about where the Sun’s Alfvén surface is usually put. At that magnetisation a quadrupole of the same surface strength holds the wind to 5.7 radii and an octupole to 3.9. The difference grows with the magnetisation, because the dipole’s lever rises most steeply with it: a more strongly magnetised star gains a great deal more lever from a dipole than from any higher order.

Why the exponents are a quarter, a sixth and an eighth #

The fitted exponents are close to what a one-line balance gives, and the balance shows where the geometry enters. At the Alfvén radius the magnetic energy density equals the wind’s kinetic energy density, $B^2/8\pi \approx \rho v^2/2$. The wind carries a fixed mass flux outward, $\dot M = 4\pi r^2 \rho v$, so $\rho v^2 \propto \dot M v/r^2$. A field of order $l$ has $B = B_\star (R_\star/r)^{l+2}$. Setting the two equal,

$$B_\star^2 \left(\frac{R_\star}{r}\right)^{2l+4} \propto \frac{\dot M v}{r^2} \quad\Longrightarrow\quad \left(\frac{r}{R_\star}\right)^{2l+2} \propto \frac{B_\star^2 R_\star^2}{\dot M v} = \Upsilon,$$

so the Alfvén radius grows as $\Upsilon^{1/(2l+2)}$: a quarter for a dipole, a sixth for a quadrupole, an eighth for an octupole. The simulations give slightly smaller exponents because the wind accelerates outward rather than moving at one speed, but the ordering and nearly the values are this arithmetic. Each higher order adds two to the power at which the field falls, and that buys two in the denominator of the exponent: the lever of a complicated field grows only weakly however strong the field is made.

The torque goes as the square #

The torque is the mass-loss rate times the star’s rotation rate times the square of the Alfvén radius, so a shorter lever costs twice.

At the Sun’s magnetisation a quadrupole brakes at 22 per cent of the rate of a dipole with the same surface strength, and an octupole at 11 per cent. Nothing changed between the three cases except the arrangement of the field on the surface: the same total field strength, the same wind, the same star. Rearranged from one large loop into four smaller ones, the field brakes the star at a fifth of the rate.

That is the quantitative form of the earlier statement that geometry matters more than strength. A factor of two in the field strength, at fixed geometry, changes the torque by a factor of about $2^{0.92} = 1.9$, because the Alfvén radius goes as $B^{2m}$ and the torque as its square; the change from dipole to quadrupole at fixed strength changes it by a factor of almost five. The magnetism of the star could double and the braking fall at the same time, if the doubling came with a change of arrangement.

In a mixed field only the dipole reaches far enough #

Real stellar fields are not pure multipoles. They are mixtures, and the question that decides how the scalings apply is what a mixture does.

The answer, which Finley and Matt found in their mixed-field simulations and fitted as a broken power law, is that the orders do not add. Close to the star the quadrupole may dominate, but it falls faster, and at some distance the dipole overtakes it. If that crossover is inside the Alfvén radius — and for any substantial dipole it is — the wind at the Alfvén radius sees only the dipole, and the torque is exactly what the dipole component would give on its own. The quadrupole, however strong at the surface, contributes nothing to the lever.

At the Sun’s magnetisation that holds whenever the dipole is more than about a quarter of the surface field strength. Only when the dipole is weaker than that does the quadrupole reach the Alfvén radius first and set a floor under the torque. At higher magnetisation the threshold falls further: a strongly magnetised star braked by a field that is 85 per cent quadrupole by strength is still braked by its 15 per cent dipole and nothing else.

So the torque depends on the dipole, not on the field. A star’s surface can be covered in strong small-scale structure — spots, active regions, loops of every size — and none of it brakes the star. The braking is done by the faint, global, bar-magnet component that underlies it, and a measurement that does not isolate that component does not measure the torque.

Spots do not brake #

The strongest magnetic fields on the Sun’s surface are in sunspots, where the field reaches three thousand gauss and is strong enough to darken the gas it threads. They contribute almost nothing to the Sun’s braking. A sunspot pair is a small loop: two patches of opposite polarity a few tens of thousands of kilometres apart, connected by field lines that arch up and come back down. Seen from a distance, it is a high-order multipole, and its field has fallen to nothing long before the solar wind reaches the Alfvén surface.

What the wind feels is the open field — the field lines that do not come back, rooted mostly in the coronal holes where the gas pressure opens the corona — and those are the large-scale, low-order part of the field, dominated near solar minimum by the polar dipole. A star covered in spots and active regions can have an enormous total field and a modest open flux, and it brakes according to the second. That is the same conclusion as the mixed-field figure, reached from the Sun’s own surface: strength concentrated in small structures is invisible to the wind.

The field each measurement sees #

Stellar fields are measured in two ways, and they see different things.

The first uses the splitting of spectral lines into polarised components. In unpolarised light the splitting broadens the lines, and the broadening measures the total field strength averaged over the surface — every spot and every loop, whatever its polarity, since the broadening does not care about sign. The second follows the circular polarisation of the lines as the star rotates and reconstructs a map of the surface field: Zeeman–Doppler imaging. Because opposite polarities cancel in polarised light, small structures of mixed sign vanish from the map, and what it recovers is the large-scale field — typically a tenth to a fifth of the total that broadening measures. The dipole is a part of that.

Since the torque goes as the field strength to the power 0.92, almost in proportion, the differences between these measurements carry straight into the torque. In the example drawn, with a large-scale field 15 per cent of the total and a dipole 8 per cent, a torque computed from the total field as though it were a dipole is ten times too large, and one computed from the large-scale map without separating the dipole is nearly twice too large.

That is not a hypothetical correction. Studies that computed torques from Zeeman–Doppler maps of dozens of Sun-like stars, using these scalings, found them broadly consistent with the torques the stars’ observed spin-down requires, within the large uncertainty in their mass-loss rates — which is possible only if the dipole component is the one carrying the torque. A torque from the total field would overshoot the observed spin-down by an order of magnitude.

The difficulty is that the dipole is the hardest part to measure precisely. A Zeeman–Doppler map needs the star to rotate fast enough that its surface features are resolved in Doppler shift, which nearly excludes slowly rotating, old stars like the Sun — exactly the stars whose braking is in question. For those, the dipole is inferred from the rotation rate and the magnetic activity rather than measured.

The young end, where the brake is already as strong as it gets #

The same geometry sets the other end of a star’s history. A young star rotating once a day or faster has a strong dynamo, and its field strength stops rising with rotation rate below a Rossby number of about a tenth: the activity saturates. Zeeman–Doppler maps of such stars, which rotate fast enough to be mapped well, show strong large-scale fields that are often, though not always, dominated by a dipole, sometimes with a large toroidal component wrapped around the rotation axis that a polarised map can see and a wind cannot use.

A saturated star’s torque grows only in proportion to its rotation rate, because the field no longer grows with it, and that slower dependence is why the youngest stars in a cluster keep a wide spread of periods for a hundred million years before converging. The geometry enters here too: two young stars of the same mass and rotation, one with a dipole-dominated field and one with a field broken into higher orders, brake at rates that differ by the factors in the second figure, and the spread of periods in young clusters is wider than any single braking law reproduces. A star’s field geometry, set by the details of its dynamo early on, may decide how quickly it joins the converged sequence that gyrochronology then reads — and the stars that never converge at all, the fully convective ones, are those whose dynamo and field arrangement differ most from the Sun’s.

A clock stopped by a rearrangement #

The geometry is the leading explanation for the observation that motivated this essay: old Sun-like stars spin faster than the braking law predicts, as though their braking weakened at about the Sun’s age. Whether the field could do that without weakening in strength is a quantitative question, and the scalings answer it.

The model is deliberately simple. The star’s field strength falls as its rotation slows, through the Rossby number — the rotation period divided by the time a convective cell takes to turn over — as the dynamo weakens; the torque follows the dipole’s strength to the power 0.92; and the braking is normalised so the star reaches the Sun’s 25.4 days at the Sun’s age. With the geometry fixed, the period keeps growing — in this model almost in proportion to the age — and reaches 46 days at nine billion years. If, once the Rossby number passes two, the dipole’s share of the field drops to fifteen per cent of what it was, the torque falls fivefold, the spin-down nearly stalls, and the star reaches only 32 days at nine billion years.

Nothing in that calculation changed the field’s total strength at the transition. A dynamo that reorganised itself — trading a global dipole for smaller structures of the same total strength — would stop the clock. That is what the observed weakened braking requires of the field, and Zeeman–Doppler maps of slowly rotating stars, the few that can be made, do show more complex, less dipolar fields at higher Rossby numbers, among the slower rotators. Whether the reorganisation is a sharp transition, as the asteroseismic ages of old stars suggest, or a gradual drift, and why it happens near the Sun’s own Rossby number, is what no dynamo model yet predicts.

The Sun’s own lever #

The Sun is the one star whose field geometry has been followed through time, and it changes with the eleven-year cycle. At solar minimum the large-scale field is close to a dipole, anchored in the strong polar fields; at maximum the polar fields reverse, the dipole weakens to a fraction of its minimum strength, and the field is dominated by higher orders from the active regions. The Sun’s torque therefore varies through the cycle, by more than its total field strength does, and the scalings say the variation should follow the dipole.

That makes the polar field that is kept as the cycle’s memory — the one quantity that predicts the next cycle — also the quantity that sets the Sun’s spin-down at solar minimum. The two are measured from the same magnetograms for different reasons, and the agreement of the Sun’s present torque, inferred from the solar wind’s angular momentum measured in situ by spacecraft, with what its dipole implies is one of the best tests the scalings have. The wind measurements are difficult and somewhat lower than the torque that the Sun’s spin-down history requires, which is itself a hint that the Sun’s present braking may be weaker than its average over its life.

What the scalings leave out #

The Alfvén radii come from fits to simulations that are axisymmetric, with each multipole aligned with the rotation axis, an isothermal or polytropic corona and a single wind temperature. Real fields are tilted and non-axisymmetric, and a dipole tilted far from the rotation axis brakes less than an aligned one. The mass-loss rate, which enters the magnetisation, is measured for only a handful of Sun-like stars, from the absorption their winds cause where they meet the interstellar gas; it is uncertain by a factor of several, which moves the magnetisation by the same factor and the Alfvén radius by less, because of the small exponents. The stall model ties the dipole’s share to the Rossby number by an assumption rather than a dynamo. And the solar magnetisation is set to reproduce a quoted Alfvén radius rather than computed from measured wind properties.

What does not depend on these is the structure: higher orders hold the wind for shorter distances, the torque goes as the square, and in a mixed field the dipole alone reaches the Alfvén radius.

Still open: what decides how much of the field is dipole #

A star’s dipole is the part of its field the dynamo organises on the largest scale, and why a dynamo produces more or less of it is the question underneath every statement here. Fast rotators have strong, often dipole-dominated large-scale fields; slow rotators like the Sun have weaker ones, more complex, and varying through a cycle. Simulations of convective dynamos reproduce the general trend — less dipolar fields as rotation slows — and disagree about where the change happens and how abruptly. The observations that would settle it are Zeeman–Doppler maps of slowly rotating stars, which are hard for the reason already given, and long records of stellar activity cycles, which take decades. Until then the braking law that turns rotation into age is calibrated at one end by young clusters and at the other by the Sun and by asteroseismic ages, and the field geometry that connects them is inferred from the spin-down rather than measured, which runs the argument in the opposite direction to the one a clock needs.