Astronomy & Celestial Mechanics
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Stars

Every note turns back at its own depth

A sound wave heading into a star is refracted by the rising sound speed and turns around before it reaches the middle. Where it turns depends on which mode it is — so a list of frequencies is not one average of the interior but thousands of differently weighted ones, and that is a solvable problem.

Assumes Asteroseismology and Opacity.

16 min read 8 figures Measuring the unreachableLight is the only messenger

The scaling relations treat a star’s oscillation spectrum as two numbers: a mean spacing and a peak frequency, which invert to a mass and a radius. That is a great deal to get out of a power spectrum, and it throws almost everything away. The Sun has some three thousand individually measured mode frequencies, each good to parts in a million, and compressing them to two numbers is a considerable loss.

What the rest of them are good for is a profile — the sound speed as a function of depth, with an error bar at each radius — and the reason a set of frequencies can produce one is a single piece of geometry.

Where each mode turns back: ℓ = 0 through the centre, ℓ = 300 in the outer 2 per cent. Why a set of frequencies is a depth profile and a single frequency is not. An acoustic wave travelling into a star meets a rising sound speed and is refracted back; it turns where its horizontal phase speed matches the local sound speed, which happens at c(r)/r = 2πν/√(ℓ(ℓ+1)). The horizontal axis is the angular degree on a logarithmic scale and the vertical axis is the fractional radius of that turning point, drawn at 2000, 3090, 4000 microhertz. The ordering is the content. A radial mode, ℓ = 0, has no horizontal phase speed at all and passes straight through the centre. Degrees one and two turn deep in the core. By ℓ = 300 the mode is trapped in the outer 2 per cent and knows nothing about anything below. So a frequency measured to a part in ten thousand constrains an average of the interior weighted in a way the mode itself decides, and measuring thousands of modes of different degree gives thousands of differently weighted averages — which is a solvable inverse problem, and is how the base of the convection zone was located at 0.713 of the radius rather than assumed. The sound speed here is a polytrope's rather than a tabulated solar model's, so the curve is the right shape and the wrong star in its outer tenth, where the real Sun is convective and this one is not. What the picture cannot show is the frequency dependence at fixed degree, which is weaker but not negligible: a higher-frequency mode of the same degree turns slightly deeper, and the three curves separating toward the right is that effect.
Fig. 1 Where each mode turns back, against its angular degree, at three frequencies. A radial mode has no horizontal phase speed at all and passes straight through the centre. Degree one and two turn deep in the core. By degree 300 the mode is confined to the outer two per cent of the radius and knows nothing whatever about anything below it. That ordering is the entire basis of the method.

Why a wave turns

A sound wave travelling obliquely into a star meets a medium whose sound speed rises with depth, because the temperature does. The deeper part of a wavefront therefore travels faster than the shallower part, and the wavefront swings around: refraction, exactly as in an ocean or an atmosphere.

The turning point is where the wave has been bent to horizontal — where its horizontal phase speed equals the local sound speed. Writing that out,

c(r)r=2πνℓ(ℓ+1),\frac{c(r)}{r} = \frac{2\pi\nu}{\sqrt{\ell(\ell+1)}},

and the right-hand side is the whole of the mode’s identity. A high-degree mode has a large horizontal wavenumber and therefore a slow horizontal phase speed, so it matches the sound speed close to the surface and turns immediately. A low-degree mode has a fast horizontal phase speed and has to go deep before the sound speed catches up. A radial mode, with ℓ=0\ell = 0, has no horizontal propagation at all and cannot be refracted.

The upper boundary is different in kind. A mode is reflected from above not by refraction but because the density scale height becomes shorter than the wavelength near the surface: the atmosphere thins out too fast for the wave to keep going, and it bounces. So a mode is trapped in a shell — an acoustic cavity — bounded below by refraction and above by the surface, and its frequency is set by the requirement that a whole number of half-wavelengths fit into the cavity’s acoustic depth.

14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here.
Fig. 2 What that quantisation produces. Modes of successive radial order are evenly spaced in frequency, by the inverse of twice the sound travel time from surface to centre — which is why the large separation is a mean density and nothing else. The regularity is the signature of a cavity, and the small departures from it are where the structural information lives.

There is a consequence worth drawing out before the mathematics. Because the cavity’s lower boundary depends on the mode and its upper boundary does not, every mode in the star passes through the outer few per cent — and only some of them reach the middle. The information available about the Sun is therefore wildly non-uniform in depth, hugely redundant near the surface and thin at the centre. That is the opposite of what a naive picture of “sampling the interior” would suggest, and it is why the error bars on an inverted profile grow inward.

A second consequence is that the method has a natural resolution scale, set by the local wavelength. A mode of frequency 3,000 microhertz in a region where the sound speed is 200 kilometres a second has a wavelength of about seventy thousand kilometres — a tenth of the solar radius. Nothing thinner than that can be resolved directly, and features that are thinner are detected instead by the periodic signature they leave in the frequencies, which is the trick described below.

From frequencies to a profile

Each frequency is an integral of the sound speed over the mode’s cavity, with a weighting that depends on where the mode spends its time. Perturb the star’s structure slightly and each frequency shifts by the corresponding weighted integral of the perturbation:

δνiνi=∫0RKi(r) δcc(r) dr+…,\frac{\delta\nu_i}{\nu_i} = \int_0^R K_i(r)\,\frac{\delta c}{c}(r)\,dr + \dots,

where KiK_i is the mode’s kernel — computable from a reference model, and different for every mode.

That is a linear inverse problem, and the question is whether it is well posed. It is not, in general: there are infinitely many δc/c\delta c/c profiles consistent with any finite set of measured frequencies, because a profile has infinitely many degrees of freedom and the data do not.

What makes the problem tractable is that the kernels are different from each other in a structured way. The standard technique constructs, for a chosen target depth, a linear combination of the measured frequencies whose combined kernel is as close to a spike at that depth as the available modes allow. Applying the same combination to the data returns the sound speed there. The width of the achieved spike is the resolution, and it is quoted with the answer, which is the honest part of the method: the result is not “the sound speed at 0.5 R” but “an average of the sound speed over a region 0.05 R wide centred at 0.5 R”.

The same comb folded at 135.1 μHz — three ridges and their curvature. Frequency against frequency modulo Δν, for 14 radial orders of the Sun. Folding at 135.1 μHz stacks the orders into three near-vertical ridges, one for each degree, and that is what makes Δν a fact about the data rather than a fitted parameter: get it wrong and the ridges lean. What is left over is the curvature — the ℓ = 0 ridge wanders 6.7 per cent of Δν across the drawn range, which is the departure from the asymptotic relation and the part of the spectrum that knows about the star's outer layers. The small separation is here too and here it is visible: 9.00 μHz between the ℓ = 0 and ℓ = 2 ridges, 39 pixels on this axis against 39 — the same quantity that is three pixels wide in the unfolded spectrum. A constant offset of 27 μHz has been subtracted before folding so that no ridge wraps round the edge.
Fig. 3 The diagnostic that has to work before any inversion can. Folding the spectrum at the large separation lines the modes up into ridges by degree, and the identification of every mode — which ℓ, which radial order — has to be correct, because a misidentified mode contributes the wrong kernel. The curvature of the ridges is itself structural information: it comes from the gradient of the sound speed near the surface.

The construction is done in one of two ways, and both are worth naming because the choice is a real one. Optimally localised averaging builds the spike directly, trading resolution against error amplification through a single free parameter; regularised least squares fits a smooth profile and penalises curvature, with the penalty as the free parameter. They are formally equivalent under a change of variable and they fail differently in practice, so the standard is to run both and publish the pair. Where they disagree, the disagreement is the honest error.

What it found

The convection zone’s base is at 0.713±0.0010.713 \pm 0.001 of the solar radius. That number was not put into a model; it was read off an inversion, and it is one of the most precisely known interior quantities in astrophysics.

The signature is a discontinuity in the gradient of the sound speed. Below the boundary the star is radiative and its temperature gradient is set by opacity; above it the star is convective and stirred nearly to the adiabatic gradient. The two do not match smoothly, and a kink in the sound speed produces a small oscillatory signature in the frequencies themselves — a periodic component in ν\nu whose period is set by the acoustic depth of the kink. Fitting that periodicity locates the boundary without any inversion at all, which is a useful independent check.

The same technique locates the second ionisation zone of helium, about two per cent of the radius below the surface, where the ionisation changes the adiabatic exponent and produces another kink. The amplitude of that signature measures the helium abundance of the convection zone — 0.2485 by mass — and that is a measurement of an element whose lines are not visible in the solar spectrum at all, because helium’s transitions require temperatures the photosphere does not reach. The Sun’s helium content is known seismically and not spectroscopically — an odd position for the second most abundant element in the universe, and a reminder that the strength of a line says as much about the conditions as about the abundance.

Hydrogen is half ionised by 9,555 K. Ionisation fractions from the Saha equation at an electron pressure of 20 N/m², against temperature. Each species' stages sum to one at every point, which is checked rather than assumed. Hydrogen crosses half-ionised at 9,555 K and is 99% ionised by 12,527; calcium, whose first ionisation potential is 6.113 eV against hydrogen's 13.598, is already 99% singly ionised at 6,000 K, where hydrogen is 100.00% neutral. Those two curves are the reason the same photosphere shows strong Ca II and weak Balmer at one temperature and the reverse at another — with the abundances fixed and only the temperature moving. The partition functions are held at their ground-state weights, which is the usual first approximation and shifts these curves by a few hundred kelvin rather than by their shape.
Fig. 4 The ionisation that leaves the signature. As a species passes through partial ionisation, energy that would have gone into raising the temperature goes into ionising instead, so the adiabatic exponent dips. That dip is a localised feature at a known depth, and it acts as a marker: the frequencies carry a small periodic term whose amplitude measures how much of the gas is doing it, which is to say the abundance.

The rotation profile, which is the same machinery

Everything above concerns the spherically symmetric structure and uses mode frequencies. Rotation breaks the symmetry and appears in the splittings — the small frequency differences between the components of a multiplet — and the inversion is formally identical, with a different kernel.

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.
Fig. 5 The result. The convection zone rotates differentially in latitude and the radiative interior rotates as a solid body, with a shear layer between them. It is worth stressing that the shape of this curve is an output rather than an assumption: a parameterised fit could have been imposed, and instead the profile was constructed depth by depth from combinations of splittings, each with its own resolution kernel and error bar. What the result means is a separate argument; that it is a measurement rather than a model is this one.

A third result is worth listing with the two above because it is the one that closed a controversy. The inverted sound speed matched the model built on the then-current abundances to about a part in a thousand throughout the radiative interior — which, at the time it was obtained, settled the question of whether the shortfall in solar neutrinos could be a problem with the solar model. It could not: the central temperature the sound speed implied was within half a per cent of the model’s, and the neutrino flux goes as a very high power of that temperature, so no admissible change to the Sun could produce the deficit. The resolution had to be in the neutrinos, and it was.

That is the strongest single use the technique has been put to, and it is a good illustration of what an inversion is for. It did not measure the central temperature directly — no mode resolves the innermost fifth well — but it constrained the sound speed everywhere outside it tightly enough that no model with a cooler core could be made to fit.

Where the method runs out

The resolution is not uniform, and it degrades in two places.

The deep core. Only the lowest degrees reach inside about 0.2 solar radii, and there are not many of them — perhaps a few dozen with useful precision. So the inner fifth of the Sun is measured with kernels a substantial fraction of the radius wide, and claims about the core’s rotation in particular have been made and withdrawn several times. Gravity modes, which are trapped in the radiative interior and would resolve it beautifully, have amplitudes at the surface of perhaps a millimetre per second and have never been unambiguously detected.

The surface. The outermost few hundred kilometres is where every mode has its largest amplitude and where the physics is worst — the same region in which the mixing-length parameter is a fitted stand-in for a turbulent flow: convection is vigorous, the fluid is turbulent, the mixing-length description is a crude stand-in, and none of the standard modelling assumptions holds. All modes are affected in a way that depends almost entirely on frequency rather than on degree — the “surface term” — and the standard treatment is to fit it away with a smooth function of frequency. That works, and it means that any structural signature confined to the outer per cent is indistinguishable from the correction being applied.

Which gradient is steeper, and where. The temperature gradient radiation would need in order to carry the whole luminosity, against the gradient a rising blob of gas follows, through an n = 3 polytrope with a Kramers opacity and an energy generation going as ρT^4. Wherever the first exceeds the second the layer is unstable and convects. This model puts the outer boundary at 89 per cent of the radius; helioseismology measures the base of the Sun's convection zone at 71.3 per cent, and the gap between the two numbers is the price of a polytrope with one opacity law rather than a solar model. The curve also rises above the adiabatic value inside 4 per cent of the radius, which is a convective core — it appears when the energy generation is concentrated enough, and it is the structure a star burning by the CNO cycle has.
Fig. 6 Where the trouble lives. The radiative and adiabatic gradients cross at the base of the convection zone — which the inversion locates precisely — and near the surface the actual gradient departs from both, because convection there is inefficient and neither limit applies. That region is a few hundred kilometres thick, contains a negligible fraction of the star, and is responsible for most of the residual disagreement between measured and modelled frequencies.

What it cost to measure

The Sun’s surface moves by a few tens of centimetres a second in each mode, which is a Doppler shift of parts in 10910^{9}. The mode frequencies have to be determined to parts in 10610^{6}, which requires a time series years long with no gaps — because a gap puts sidelobes into the power spectrum at the frequency of whatever caused it, and a daily gap puts them 11.57 microhertz away, which is close enough to real structure to be confused with it.

Two solutions were built. One is a network of six identical instruments spread around the Earth’s longitudes so that the Sun never sets on the observation. The other is a spacecraft at the first Lagrange point, where the Sun never sets at all. Both have been running since the mid-1990s, and the mode frequencies they produce are the input to everything above.

For other stars the equivalent is photometric rather than Doppler, and the amplitudes are parts per million in brightness. Space photometry made it possible, and the yield has been tens of thousands of red giants — where the modes are at lower frequency and higher amplitude — against a few hundred solar-type dwarfs. The same photometry, from the same instruments, was collected to find planets by the light they remove, and the seismology is a by-product of a transit survey: two entirely different sciences extracted from one set of brightness measurements, and the second of them turns out to give the stellar radii the first one needs.

The plane the two relations make, and how badly it is conditioned in mass. The large separation against the frequency of maximum power, both logarithmic, for four stars whose masses, radii and temperatures are stated and whose frequencies are computed from them — the Sun at 3,090 and 135 μHz; a subgiant at 671 and 39.0 μHz; a red-clump star at 34 and 4.06 μHz; a red giant at 4.43 and 0.862 μHz. Over the plane are lines of constant radius, at 1, 3, 10, 30 solar radii, and of constant mass, at 0.8 and 2. The two families are not at right angles and that is the finding: the constant-mass lines run at a slope of about 0.77 and the constant-radius lines at 0.5, so a factor of 2.5 in mass moves a star only 0.11 decades across the plane. Mass enters the inversion at the quarter power of the observables and radius at the first, which is why an asteroseismic radius is good to a few per cent and an asteroseismic mass to something nearer ten. Every point here was put on the plane by the relations and then read back off it: solving the two equations for mass and radius returns the stated values to 7.8e-16.
Fig. 7 Why giants dominate the sample. Their oscillation frequencies are tens of microhertz rather than thousands, so a four-year photometric time series resolves them easily, and their amplitudes are hundreds of times larger. What is lost is degree: only ℓ = 0, 1 and 2 are visible in disc-integrated light, because higher degrees cancel when the whole disc is summed into one number. So a distant star gives three kernels where the Sun gives three hundred, and the inversion that resolves the Sun’s interior is not available anywhere else.
Two stars at the same point of every other diagram, 4.1 times apart in one. Above: the gravity-mode period spacing against the large frequency separation, for 90 shell-burning giants and 55 core-burning ones. These are the same stars in every other measurement. They have the same luminosity, the same temperature, the same colour, the same surface gravity and — inside the band drawn — the same Δν, which means the same mean density; the scaling relations of the rung below return the same mass and the same radius for both. What separates them is a quantity that comes from nowhere near the surface: ΔΠ₁ is set by the buoyancy frequency integrated across the core, and a core that has ignited helium is expanded and convective, so its integral is smaller and its period spacing larger. The two sequences do not touch — 61 seconds against 248 — and the gap sorts a catalogue of tens of thousands of giants into stars burning hydrogen in a shell and stars burning helium in a core, by a Fourier transform of a light curve. Below: what that looks like in the spectrum itself, drawn over two radial orders. The number of mixed ℓ = 1 modes between consecutive radial modes is Δν/(ΔΠ₁ν²), so the shell burner has about 12 of them and the core burner about 3: the star with the larger period spacing has the sparser spectrum. The picture cannot show what the same modes are also used for and cannot settle — the splitting of each mixed mode gives the rotation rate of the core separately from the envelope, and the cores come out spinning some ten times faster than the surface and a hundred times slower than any model of angular-momentum transport predicts.
Fig. 8 What the two-number summary cannot do, and what the rest of the spectrum can. Δν and ν_max give a mass and a radius through relations good to a few per cent, and they use two numbers out of three thousand — but a star climbing the giant branch and a star burning helium in its core sit at the same point of that plane, at the same luminosity, the same radius and very nearly the same mass. The period spacing of their mixed modes differs by a factor of four, because one has a convective core and the other does not. That is a fact about the interior recovered from the part of the spectrum a two-number summary throws away, and it is the shortest argument for doing the inversion at all.

What a single bad frequency does

An inversion is a weighted sum of measured frequencies, and the weights are not small and not all positive. That has a consequence for error propagation which is the central practical difficulty of the method and which the resolution kernels do not by themselves reveal.

To build a narrow spike at a chosen depth, the combination has to arrange for the broad kernels of many modes to cancel almost everywhere and add only in one place. Cancellation of that kind requires large coefficients of both signs, and large coefficients multiply the measurement errors as well as the signal. So the narrower the spike demanded, the larger the error on the answer — not because the data got worse, but because the combination that localises them amplifies their noise.

That is the trade-off the method is built around, and it is made explicit: the free parameter in the construction is a weight on error amplification against a weight on the width of the kernel, and it is varied to produce a curve of resolution against uncertainty. A point is chosen on that curve and both numbers are quoted.

The consequence for a single bad datum is worse than it would be in an ordinary fit. A frequency measured wrongly by ten standard deviations does not merely add noise; it enters every localised average that includes it, with whatever coefficient the combination assigned, and it can produce a smooth spurious feature in the inverted profile rather than an obvious outlier. Because the profile is built from overlapping combinations, one bad mode can bend a whole region.

The defences are the ordinary ones and they matter more here than usual. Modes are measured independently by more than one instrument and cross-checked. The identification of every mode is verified on the folded diagram before anything is inverted. Inversions are repeated with subsets of the data — leaving out a range of degrees, or a range of frequencies — and a feature that moves when a subset is dropped is not reported.

An inversion is not robust in the way a fit is, because the operation that gives it its resolution is the same operation that gives it its sensitivity to a single wrong number, and there is no way to have one without the other.

Why the inverse problem is worth the trouble

A reader might reasonably ask why anybody constructs kernels and localised averages instead of simply fitting a stellar model to the frequencies and reporting its interior. That is done too, and it answers a different question.

A model fit says: among the structures my code can produce, this one matches best. An inversion says: whatever the structure is, here is what the data require of it at this depth, to this precision. The first is a statement about a family of models and the second is a statement about the star, and when the model family does not contain the truth — which is exactly the situation the sound-speed disagreement below points to — only the second is any use.

The distinction has a practical edge. A model fit reports a helium abundance and a mixing length as fitted parameters, and their error bars are the error bars of a fit, which assume the model is right. An inversion reports a sound speed with a resolution kernel attached, and the kernel is a statement about what was actually constrained. The two disagree most where the model is worst, which is precisely where the answer matters.

Where the ladder goes

The next rungs of this anchor are about what happens when the inversion disagrees with the model, which it now does. The sound speed the inversion returns and the sound speed a model built on the current solar abundances predicts differ by nearly a per cent just below the convection zone — a disagreement between two ways of knowing the same star that has been open for twenty years and is the strongest current evidence that stellar opacities are wrong.

There is a rung about the other stars in the other direction too. A binary in which both components oscillate gives two seismic radii and two dynamical masses from the orbit, which is the only external check the scaling relations have ever had. The agreement is at the few-per-cent level and the residuals are systematic, which is the sort of result that generates the next decade of work.

There is also a thread running out toward other stars. The three degrees available for a distant star are not enough for an inversion, but they are enough for the kink signatures: the acoustic glitch from a helium ionisation zone is detectable in the best asteroseismic targets, and it measures a helium abundance for a star nobody can resolve. The technique that measured the Sun’s helium is now being applied where no other method exists at all.

Where this goes next

14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here. The interior read from a comb of frequencies A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age. 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. A better measurement that made the model worse: 0.9 per cent in the sound speed. The fractional difference between the Sun's sound speed as its own oscillations measure it and as a structural model predicts it, against fractional radius. Zero would be agreement. The lower curve is the model built on the solar abundances used until the mid-2000s, and it hugs the axis: a part in a thousand across most of the interior, which was for a long time the best-tested piece of stellar physics anybody had. The upper curve is the same model with the abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard, which lowered carbon, nitrogen and oxygen by around thirty per cent. The disagreement grows to 0.9 per cent, and it is not spread through the star: it peaks at 0.683 of the radius, just beneath the base of the convection zone at 0.713. The same substitution moves the model's own convection-zone base from 0.715 to 0.729, against a seismic value known to about a thousandth. What is being tested here is not really the abundances but what converts a composition into a structure, which is the opacity: the metals whose abundances fell are exactly the ones whose bound–free absorption dominates at those temperatures, and an opacity larger by some fifteen per cent near that boundary would restore the agreement. Laboratory measurements of iron at those conditions have since come in high by about that much, which is a satisfying result to have arrived at by way of a discrepancy in the sound speed of the Sun. The curves are published inversions and model differences rather than anything computed here; what the figure adds is where they peak and by how much. A better measurement that made the model worse The Sun's composition was re-measured with better atmospheres and better physics, and the carbon, nitrogen and oxygen abundances fell by about thirty per cent. The improved model then disagreed with the Sun's own oscillations by ten times as much as the model it replaced, and it still does.

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Acoustic cavityAsteroseismologyConditioningConvectionHelioseismologyInversionKernelRegularisationRotational splittingSound speedTurning point