Astronomy & Celestial Mechanics
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Stars

A surface that slowed because the star grew

A star leaving the main sequence expands by a factor of a hundred. Conserve its angular momentum and its equator slows by four orders of magnitude, which is why every red giant is a slow rotator. One or two per cent are not, and a giant turning at fifty kilometres a second has to have been given the angular momentum by something else.

Assumes Stellar evolution, Internal rotation and Magnetic braking.

16 min read 8 figures Scales that defeat intuitionMeasuring the unreachable

When a star’s helium core contracts, its envelope expands — by a factor of sixty for a solar-mass star, and more for a heavier one. The luminosity, the surface temperature and the density all change enormously, and the star leaves the place it had been sitting rather than travelling. So does the rotation, and it changes for the simplest possible reason.

A hundredfold expansion is a four-order slowdown. Equatorial rotation speed against radius for three stars leaving the main sequence at 2, 10, 100 km/s, on the single assumption that nothing exerts a torque. Both axes are logarithmic. The specific angular momentum is held fixed along each track and checked at four radii rather than asserted, so the speed falls as the reciprocal of the radius and then faster, as the moment of inertia coefficient slides from 0.073 on the main sequence to 0.02 in a centrally condensed giant envelope. A star at 100 km/s crosses the 8 km/s line — the usual boundary for calling a giant a rapid rotator — at 45.6 solar radii, and everything larger is slower. That is why the observed giants are almost all under two kilometres a second, and why the one or two per cent that are not cannot be explained by anything the star did on its own: the angular momentum has to have been delivered, by a swallowed companion or a merger. The figure assumes no mass loss, which for the largest radii drawn is the weakest of its assumptions.
Fig. 1 Equatorial speed against radius for three stars leaving the main sequence at 2, 10 and 100 kilometres a second, on the single assumption that nothing exerts a torque. Both axes are logarithmic. The specific angular momentum is held fixed and checked at four radii rather than asserted, so the speed falls as the reciprocal of the radius and then faster, as the moment of inertia coefficient slides from 0.073 to 0.020 in a centrally condensed giant envelope. Everything past a hundred solar radii is below a kilometre a second.

The arithmetic

Angular momentum is J=k2MR2ΩJ = k^2 M R^2 \Omega, and the equatorial speed is v=ΩRv = \Omega R. So

v=Jk2MR,v = \frac{J}{k^2 M R},

and at fixed JJ, MM and k2k^2 the speed is inversely proportional to the radius. A hundredfold expansion is a hundredfold slowdown.

But k2k^2 does not stay fixed. On the main sequence the Sun’s is 0.073. A red giant’s envelope is enormously extended and its mass is concentrated towards the bottom of it, so the coefficient falls — to around 0.02 or below. That makes the slowdown faster than the reciprocal of the radius, and a hundredfold expansion becomes closer to a four-order change in surface speed.

A lever 12 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 12 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 3·10⁻¹¹ of its angular momentum in the same year — a ratio of 1,315, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 3.3·10¹⁰ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.
Fig. 2 The other way a star loses spin, on the same axes. A magnetised wind carries angular momentum away at a lever arm set by where the field can still enforce corotation — twelve stellar radii here — so the torque is far larger than the mass loss alone would suggest: a wind carrying a hundred-millionth of a solar mass a year removes a substantial fraction of the Sun’s angular momentum over its lifetime. Expansion redistributes spin inside the star; this removes it from the star altogether.

What is measured, and what it costs to measure it

The observable is a line width, and at these speeds it is a small one.

Four rotations, their edges, and the speed below which there is no measurement. Residual intensity against distance from line centre in velocity, for a line at 500 nm broadened by rotation alone at v sin i = 5, 20, 50, 150 km/s. Each profile is exactly zero beyond Δv = v sin i — Δλ_max = λ v sin i / c, which is 8.34e-3 nm at 5 km/s, 3.34e-2 nm at 20 km/s, 8.34e-2 nm at 50 km/s, 2.50e-1 nm at 150 km/s — so the edge is the measurement, and reading each drawn edge back returns the rotation that produced it to better than 0.5%. Two limits are drawn and they are different limits. The first is the spectrograph: at R = 100000 one resolution element is 3.00 km/s, and a rotation narrower than that is not sampled at all — this generator refuses to draw it. The second is physical: everything that is not rotation — Fe's thermal width at 6000 K, 0 km/s of microturbulence, 3 km/s of macroturbulence and the instrument — adds in quadrature to a half width of 3.12 km/s, and the rotational half width only exceeds it above v sin i = 4.00 km/s. Below that the shape belongs to the other mechanisms and v sin i is not recoverable, however good the spectrum.
Fig. 3 The rotation kernel and its limits. Rotation broadens a line into a distinctive shape with a hard edge, quite unlike a thermal or collisional profile, and its width is the equatorial speed times the sine of the inclination. Below a few kilometres a second the rotational broadening is smaller than the other contributions and the measurement becomes a difference of comparable quantities — which is exactly the regime almost every red giant is in.

There is a second measurement route that does not use a line width at all. A giant with starspots modulates its own brightness as it turns, so a long photometric record delivers a rotation period directly — and a period plus a radius from the seismic scaling relations gives an equatorial speed with no projection ambiguity in it. That route only works for stars with spots, which for giants means the magnetically active minority, so it samples the fast tail rather than the bulk. The two techniques are therefore complementary in exactly the awkward way: the one that reaches the slow majority delivers only an upper limit, and the one that delivers a real number only sees the anomalies.

That floor matters. A giant rotating at 0.5 km/s produces a broadening far below the star’s own macroturbulence, and the reported value is then an upper limit rather than a measurement. So the distribution of giant rotation speeds is well determined only above a few kilometres a second, and the bulk of the population is unresolved at the bottom of it — which is fine for this argument, since the bulk being below the floor is the prediction.

A lever 30 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 30 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 1.9·10⁻¹⁰ of its angular momentum in the same year — a ratio of 8,219, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 5.3·10⁹ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.
Fig. 4 The same wind with a longer lever, which is what a younger, faster-rotating, more strongly magnetised star has. The torque goes as the square of the Alfvén radius, so thirty radii against twelve is a factor of six — and since the field strength itself grows with rotation, the braking is self-limiting in a way that makes the final period nearly independent of the initial one. That convergence is what makes rotation usable as an age, and it is the same convergence the giant panel above shows by a completely different route.

The fast tail

Their evolutionary state is fixed by where they sit relative to a cluster’s bend, and surveys of thousands of giants find that one to two per cent rotate faster than about eight kilometres a second, and a handful faster than fifty. Those cannot have got there by evolving.

The arithmetic forbids it. A giant at fifty kilometres a second and a hundred solar radii carries a specific angular momentum that on the main sequence would have corresponded to an equatorial speed of thousands of kilometres a second, well beyond break-up. There is no main-sequence progenitor that could have supplied it.

So the angular momentum was delivered. The candidates are all binary.

Engulfment of a companion. As the star expands, anything orbiting inside the growing radius is swallowed, and it deposits its orbital angular momentum in the envelope. A Jupiter-mass planet at a tenth of an astronomical unit carries enough to spin a giant’s envelope to several kilometres a second; a low-mass star carries far more.

Tidal spin-up without engulfment. A close companion raises a tide, and if the giant’s rotation is slower than the orbit — before the star fills its lobe and the flow narrows its own channel —, the tide transfers angular momentum from the orbit to the star. The giant’s enormous radius makes the tide extremely effective — the rate goes as a high power of the radius over the separation — so a companion that was tidally irrelevant on the main sequence becomes decisive on the giant branch.

A merger. Two stars that coalesce produce a single object with the orbital angular momentum of the pair, which is very large. Such objects are expected to be rapid rotators and to have anomalous surface compositions.

The three are ordered by how much they deliver, and the ordering is steep. A Jupiter at a tenth of an astronomical unit carries about ten to the forty-eight erg seconds; a brown dwarf at the same distance carries fifty times more; a low-mass stellar companion at half an astronomical unit carries a thousand times more still. A giant’s envelope needs something in the middle of that range to be spun to a few kilometres a second, and the top of the range would overspin it past break-up. So the observed distribution of rapid rotators — mostly modest, with a rare extreme tail — is roughly what a distribution of engulfed companion masses would produce, which is the argument for the interpretation as a class rather than object by object.

The independent evidence

The engulfment hypothesis makes predictions beyond the rotation, and two of them are checkable.

The first is lithium. Lithium is destroyed in stellar interiors and is largely absent from the surfaces of ordinary giants, whose deep convective envelopes have dredged up processed material. A swallowed planet delivers material that has never been inside a star, and it is lithium-rich. So an engulfment should leave a giant that is both rapidly rotating and lithium-enhanced — and the rapid rotators are indeed enhanced in lithium far more often than ordinary giants are.

The second is the core. Engulfment adds angular momentum to the envelope and not to the core, so an engulfed giant should have an anomalously large ratio of envelope to core rotation. That is measurable.

One line, and a rotation rate at each end. Rotational splitting of mixed dipole modes in a red giant, against the fraction ζ of each mode's inertia that sits in the helium core. Every point is one multiplet; the scatter is a 0.011-nanohertz measurement error and is seeded so the drawing is reproducible. Because a mixed mode is a gravity wave in the core and a pressure wave in the envelope at once, and ζ says in what proportion, the splitting is a straight line in ζ whose value at ζ = 1 is the core's rotation and at ζ = 0 the envelope's. Fitting that line to the drawn points — rather than drawing the line the points were made from — returns a core period of 20.0 days and an envelope period of 165 days, against the 20 and 165 they were built from. The contrast is 8.3, and that is the number that does not fit: the core of a red giant has contracted by a factor of ten and the envelope has expanded by a hundred, so angular momentum conservation alone predicts a contrast of many hundreds. Something is coupling the two, and no mechanism proposed so far transports enough. What the figure cannot show is where between the two the transport happens, because ζ is a weighting and not a depth.
Fig. 5 The measurement that separates the two. In a red giant each oscillation mode is partly a gravity wave in the core and partly a pressure wave in the envelope, so its rotational splitting is a weighted average of the two rates with a weighting that shifts from mode to mode. Fitting the line returns both. An ordinary giant has a core turning about ten times faster than its envelope; a giant that has swallowed something should have a much smaller contrast, because the envelope has been given angular momentum the core never received.

Where on the branch the star is

“Red giant” names several distinct evolutionary states with very different radii, and the rotation argument is sensitive to which one a star is in.

A star climbing the giant branch for the first time has a degenerate helium core and a hydrogen-burning shell, and its radius grows steadily as the core grows. At the tip it ignites helium, and the ignition rearranges the interior: the core expands, the shell moves outwards and cools, and the envelope contracts by a large factor. The star settles onto the clump — the red giant equivalent of a main sequence, where it burns helium in the core for a hundred million years at a radius of about ten solar radii rather than a hundred.

Then it climbs again, on the asymptotic branch, to radii larger than before.

The rotation follows all of that. A star on the clump has contracted from the tip, so conservation alone should have spun its envelope back up by the ratio of the radii — a factor of a hundred in the extreme case, which would make every clump star a rapid rotator. They are not: clump stars are slow, like everything else on the branch.

The resolution is that the contraction is not adiabatic in the relevant sense. The envelope is losing angular momentum throughout — to a wind, and to the coupling with the slowly rotating interior — and the helium flash itself is a violent enough rearrangement that the assumption of a fixed specific angular momentum per shell is not obviously safe. What the observations establish is that the spin-up does not happen, and the accounting for why is not settled.

The evolutionary state has to be known before a rotation rate means anything, and it is known — from the seismic quantities that separate a shell-burning star from a core-burning one at the same temperature and luminosity. Without that separation the giant rotation distribution is a mixture of populations that behave differently.

Why the slowdown matters beyond the star itself

A slow surface has consequences that reach well outside the envelope, and two of them are worth setting out because they turn a kinematic fact into an evolutionary one.

The first is the dynamo. A convective envelope generates a field only if it is rotating: the shear and the Coriolis force are what organise the convection into something that can wind a field. A giant that has slowed to a fraction of a kilometre a second has a Rossby number — the rotation period divided by the convective turnover time — that is very large, and its dynamo is correspondingly feeble. Ordinary red giants are magnetically quiet, and the ones that are not are, again, the rapid rotators. Magnetic activity in a giant is therefore a second, independent flag for the same anomaly the rotation flags, and the two agree object by object.

The second is the shape of the star. A rotating star is oblate and gravity-darkened, and both effects scale with the ratio of the rotation rate to the break-up rate. A giant’s break-up speed is small, because break-up goes as the square root of mass over radius and the radius is enormous — a hundred-solar-radius giant breaks up at about forty kilometres a second rather than the four hundred of a main-sequence star. So the fifty-kilometre-a-second rapid rotators are not merely unusual; they are at or above their own break-up limit, which is one of the reasons the reported extreme cases are treated with suspicion and re-measured.

A star 1.10 times wider than it is tall, and 6 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.7 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.095 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.511, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.674 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.104 at the theoretical exponent 0.25 and 1.078 at the 0.19 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.06 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.
Fig. 6 What being near that limit would do. A star at seventy per cent of its critical angular velocity is measurably oblate and its equator is measurably cooler than its poles, so its apparent temperature depends on the observer’s position. For a giant this would corrupt the temperature that the seismic scaling relations need in order to return a mass and a radius — which is the same measurement the rotation is being read from. A genuinely near-critical giant would therefore be a star whose fundamental parameters are all mutually entangled.

The braking that stops

There is one more reason a giant is slow, and it works in the same direction.

A cool giant has a deep convective envelope, so it has a dynamo, so it has a magnetic field and a wind — which means it is being magnetically braked as well as being slowed by expansion. A star does not keep all the mass it was born with, and a giant’s mass loss rate is far higher than a main-sequence star’s, and its radius is far larger, so the lever arm is longer in absolute terms.

Four different beginnings and one ending. Rotation period against age for four stars born turning at 0.3, 1, 3, 8 days, integrated under a braking law that goes as the cube of the rotation rate below a saturation period of 3.4 days and linearly above it. Both axes are logarithmic. The tracks span a factor of 25.7 at ten million years, 1.93 at six hundred million, and 1.09 at the age of the Sun — the initial condition is not merely diluted, it is erased, because the solution Ω = Ω₀(1 + 2KΩ₀²t)^(−1/2) tends to (2Kt)^(−1/2) with no Ω₀ left in it. The late slope measured off the drawn curve is 0.500 against the one half the law demands. The single constant K is fixed by one requirement, that the attractor pass through the Sun at 25.4 days and 4.57 billion years, and the track drawn for the slowest starter arrives at 26.6 days. What the figure cannot show is the saturated branch's physical cause: above a few days' rotation the dynamo stops responding to faster rotation, and that plateau is measured rather than derived.
Fig. 7 The braking law as it applies on the main sequence, where it is calibrated. On the giant branch the same mechanism operates with a much larger radius and a much larger mass loss rate, so it removes angular momentum quickly — but it also has far less time to act, since the giant branch lasts a per cent or so of a star’s life. Which of expansion and braking dominates the observed slowness has been argued both ways, and the answer differs between the sub-giant branch, where there is time, and the tip, where there is not.

The two effects are hard to separate observationally because they predict the same thing for the bulk of the population. They differ for the fast tail: braking would eventually remove delivered angular momentum too, so an engulfment signature should decay, and the fraction of rapid rotators should fall with evolutionary state along the giant branch. It does.

The rapid rotators that are not anomalies

There is a class of fast-rotating giant that requires no companion at all, and separating it from the engulfment cases is the main difficulty in interpreting the tail.

A star more massive than about two solar masses does not have a convective envelope on the main sequence, so it has no magnetic brake and arrives at the giant branch turning fast — hundreds of kilometres a second rather than a few. Expansion still slows it by the factor this essay computes, and it starts from a hundred times higher, so it ends up at a few kilometres a second by ordinary evolution.

That means a giant at five kilometres a second is either a low-mass star that swallowed something or an intermediate-mass star that did not need to. The two are told apart by mass, and the mass comes from the seismic scaling relations — which is another reason the asteroseismic samples are the ones the statistics are drawn from.

There is a second and subtler contaminant at the same place in the distribution. A star with a close companion that has not been engulfed can be tidally synchronised: if the orbital period is short enough, the tide forces the giant’s rotation towards the orbital rate, and a giant synchronised to a ten-day orbit is turning at several kilometres a second without anything having been swallowed. Those systems are identifiable by their radial velocities, and removing them is routine — but only for the ones bright enough for repeated spectroscopy.

The one-to-two-per-cent figure quoted for the anomalous tail is therefore a figure after two removals, and how complete either removal is depends on whether the star was in a sample that had the ancillary data. The raw fraction of fast rotators is several times higher.

There is a third population worth mentioning because it is the one that would falsify the whole reading. A giant that has not swallowed anything and has no companion and is nevertheless rotating fast would require the slowdown argument itself to be wrong. None has been convincingly identified — every well-characterised fast rotator has turned out to have a companion, an anomalous composition, or a mass high enough to explain it — and the absence of that population is the strongest evidence the argument has.

It is an absence rather than a detection, so it is only as good as the follow-up. The systems that have been checked are the bright ones, and a fast rotator too faint for radial-velocity monitoring is a fast rotator whose companion has not been looked for.

What the picture cannot show

The tracks in the opening figure assume no mass loss, and for a star near the tip of the giant branch that is the weakest assumption in the essay. A star losing a substantial fraction of its envelope loses angular momentum with it, and on a lever arm that is at least the stellar radius — so the real tracks bend further down than the drawn ones at large radii, by an amount that depends on a mass loss rate known to a factor of several.

They also treat the star as rotating as one body. It does not: the core and the envelope have different rates, and the coefficient used to convert between angular momentum and surface speed is an average over a structure that is changing. The measured core-to-envelope contrast of about ten says the approximation is wrong by a factor of that order somewhere inside, and right at the surface, which is where the observable is.

There is a third omission, and it is the one that limits the whole comparison rather than any single track. A projected rotation speed is not a rotation speed: what a line width gives is the equatorial velocity multiplied by the sine of the inclination, and the inclination of a single giant is unknown. For a population that is fixable — the average of the sine over randomly oriented axes is known, so a distribution of measured values can be deprojected into a distribution of true ones — but for one star it is not, and a star measured at twenty kilometres a second may be turning at twenty or at fifty. Every statement above about a fast tail is therefore a statement about the tail of a deprojected distribution, and the deprojection assumes the axes are randomly oriented, which is exactly what would fail if the fast rotators had been spun up by a companion in a preferred plane.

The way round it is asteroseismic. The relative amplitudes of the components a rotating star’s oscillation modes split into depend on the angle between the rotation axis and the line of sight, so a star with detected oscillations gives its own inclination — and then the line width gives a genuine equatorial speed. That is available for a few thousand giants and for none of the brightest ones, which are the stars the extreme rotation claims are made about.

One more expansion factor brackets the range a real giant covers.

A hundredfold expansion is a four-order slowdown. Equatorial rotation speed against radius for three stars leaving the main sequence at 2, 10, 100 km/s, on the single assumption that nothing exerts a torque. Both axes are logarithmic. The specific angular momentum is held fixed along each track and checked at four radii rather than asserted, so the speed falls as the reciprocal of the radius and then faster, as the moment of inertia coefficient slides from 0.073 on the main sequence to 0.02 in a centrally condensed giant envelope. A star at 100 km/s crosses the 8 km/s line — the usual boundary for calling a giant a rapid rotator — at 45.6 solar radii, and everything larger is slower. That is why the observed giants are almost all under two kilometres a second, and why the one or two per cent that are not cannot be explained by anything the star did on its own: the angular momentum has to have been delivered, by a swallowed companion or a merger. The figure assumes no mass loss, which for the largest radii drawn is the weakest of its assumptions.
Fig. 8 The same accounting for an eightyfold expansion rather than two hundredfold. The surface speed falls in proportion to the radius at fixed angular momentum, so even this smaller expansion takes a solar-type rotator to under a kilometre a second — which is why every red giant observed rotates slowly.

Where the ladder goes

The nearest rung is the transport question this keeps running into: why a giant’s core turns only ten times faster than its envelope when conservation alone allows hundreds. Everything in this essay is about the envelope, and the envelope is the half of the problem where the answer is known.

The other direction is the use of the fast tail as a census. If a rapidly rotating, lithium-rich giant is a star that has swallowed a planet, then counting them is counting planetary systems that ended — and the rate can be compared with the number of close-in planets found around main-sequence stars of the same mass. The two numbers are in the same range, which is a satisfying closure, and both are uncertain enough that the comparison is not yet a constraint.

Where this goes next

The mirror: an envelope 62 times larger around a core 1.48 times smaller. Envelope radius and core radius against helium-core mass for a 1 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 2.0 to 122 solar radii — 0.57 astronomical units — while the degenerate core contracts from 13,848 to 9,368 kilometres, and the two are separated by a factor of 364 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5000 to 3700 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number. The star that swells because its centre shrank A helium core contracts, and the envelope around it expands by a factor of sixty. The central density rises at the same time as the radius, which is the opposite of what a self-gravitating body is expected to do, and a burning shell between the two is the whole reason it happens. The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives. A fluid that turns as one piece The Sun's surface turns faster at its equator than at its poles, and everyone expected the inside to do something similar. Below seven-tenths of the way down it does not — the radiative interior rotates as rigidly as a bell, and nothing in hydrodynamics makes a fluid do that. Three profiles of equal equivalent width, 28.6 mÅ. Three absorption profiles with the same equivalent width — 28.6 milliångström, 1.72 km/s at 500 nm, matched to better than 0.1% by root-finding over the quadrature — differing in nothing but shape. Left: the cores, on a common velocity axis. Right: the same three normalised to their own half widths, on a logarithmic depth scale. The thermal profile is a Gaussian set by Fe's mass at 6000 K, 1.34 km/s; the collisional one a Lorentzian of γ = 2.28e-3 nm; the rotational one the classical kernel of a disc turning at v sin i = 1.73 km/s, which is exactly zero beyond 1.28 half widths and is the only one of the three with an edge. Matching the areas does not match the widths: the half widths are 1.11 km/s (thermal), 1.35 km/s (rotational), 0.68 km/s (collisional), a factor of 1.98 between the widest and the narrowest. At three half widths the collisional wing is 49 times the thermal one and at five it is 1.27·10⁶ times — six decades, which is why a line's shape stays diagnostic long after its width has stopped being so. 11% of the Lorentzian's own equivalent width lies beyond the right-hand panel's edge and is not drawn anywhere. The same width for three different reasons Thermal motion, rotation and collisions each widen an absorption line, and they can be tuned to areas that agree to a part in a thousand. What separates them is the shape, and the shape carries a rotation speed from one profile and a surface gravity from another.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Angular momentumAsteroseismologyConvective envelopeCore envelope couplingLithium enrichmentMagnetic brakingMass loss rateMoment of inertiaPlanet engulfmentProjected rotation velocityRed giantStellar mergerStellar rotationTidal spin-up