The interior read from a comb of frequencies

The account of why the biggest stars die first contains one sentence saying that two seismic observables scale as a star’s mean density and as its surface gravity, and that the pair therefore gives a mass and a radius. That sentence is a promissory note, and this is the rung that pays it.

The spectrum below is what the note is written on: not a light curve and not one oscillation, but a comb of some forty resonances of the Sun, each a standing sound wave filling the whole body of the star, all ringing at once and together moving the surface by a fraction of a metre per second.

Two numbers, and what each is a proxy for #

Almost everything in the spectrum can be discarded; two summary numbers carry the argument.

The first is the large separation $\Delta\nu$, the spacing between consecutive radial modes — 135.1 μHz for the Sun. The resonances are the frequencies that fit a whole number of half-wavelengths into a sound wave’s round trip through the star, so $\Delta\nu$ is the reciprocal of that crossing time, and a crossing time divided into a size is a mean density:

$$\Delta\nu \propto \sqrt{\bar\rho} \;\propto\; \sqrt{M/R^3}.$$

The second is $\nu_{\max}$, the frequency of maximum power — the centre of the envelope, 3,090 μHz, where the modes are excited hardest. The envelope tracks the atmosphere’s acoustic cutoff frequency, which for an isothermal layer is the sound speed over the pressure scale height, and that ratio is a surface gravity divided by the square root of a temperature:

$$\nu_{\max} \propto \frac{g}{\sqrt{T_{\rm eff}}} \;\propto\; \frac{M}{R^2 \sqrt{T_{\rm eff}}}.$$

One observable is a mean density and the other is a surface gravity, which is to say one is $M/R^3$ and the other is $M/R^2$ — two independent combinations of the same two unknowns.

Why the comb is nearly evenly spaced #

Even spacing is not obvious. A star is a cavity of appalling complexity, its sound speed varying by two orders of magnitude between centre and photosphere, its modes carrying three quantum numbers rather than one.

What rescues the arithmetic is that at high radial order the asymptotic form of the wave equation makes frequency linear in order:

$$\nu(n,\ell) \;=\; \Delta\nu\left[\,n + \frac{\ell}{2} + \varepsilon + \frac{\alpha}{2}(n - n_{\max})^2\right] \;-\; \ell(\ell+1)\,D_0.$$

The bracket is the comb: $n$ counts radial nodes, $\ell$ is the surface degree, $\varepsilon$ a dimensionless offset near 1.45 for the Sun set by the outermost layers, $\alpha$ a small curvature term. Set $\alpha$ and $D_0$ to zero and the spectrum is exactly periodic in $n$ with period $\Delta\nu$, the odd degrees falling halfway between the even.

The two relations invert, and that is the method #

Two equations in two unknowns, both read off one spectrum, and the algebra closes. Writing every quantity as a ratio to the Sun’s,

$$\frac{R}{R_\odot} = \frac{(\nu_{\max}/\nu_{\max,\odot})\,(T/T_\odot)^{1/2}}{(\Delta\nu/\Delta\nu_\odot)^{2}}, \qquad \frac{M}{M_\odot} = \frac{(\nu_{\max}/\nu_{\max,\odot})^{3}\,(T/T_\odot)^{3/2}}{(\Delta\nu/\Delta\nu_\odot)^{4}}.$$

A temperature is needed, and comes from the colours. Everything else on the right is a frequency. There is no evolutionary track, no opacity table, no mixing length and no distance.

The exponents in that inversion are the least glamorous and most consequential fact. A fractional error in $\Delta\nu$ arrives doubled in the radius and quadrupled in the mass; an error in $\nu_{\max}$ arrives at the first power in the radius and the third in the mass. The same spectrum therefore yields a radius good to a few per cent and a mass good to nearer ten, and no improvement in the frequencies changes the ratio between those two figures.

The fold that turns a spacing into a fact #

A spacing quoted off a spectrum invites the objection that it was fitted. The answer is a change of coordinates, not a better fit.

Cut the frequency axis into strips of width $\Delta\nu$ and stack them. If the guessed spacing is right, every radial order lands at the same horizontal position and the comb collapses into vertical ridges, one per degree; if it is wrong, each successive order is displaced a little further and the ridges lean. The lean shows up long before the error would be noticeable in the unfolded spectrum, which is what makes $\Delta\nu$ a property of the data.

Nine microhertz, and an age #

The last term in the asymptotic relation, $-\ell(\ell+1)D_0$, separates degrees of the same parity. Its observable is the small separation $\delta\nu_{02}$, the gap between an $\ell = 0$ mode and the $\ell = 2$ mode that would otherwise sit on it: 9.00 μHz for the Sun, 6.7 per cent of $\Delta\nu$ and three parts in a thousand of $\nu_{\max}$.

That gap dates the star, because $D_0$ is an integral of the gradient of the sound speed, weighted towards the deep interior. Quadrupole modes penetrate slightly less far than radial ones, so their difference is sensitive to a small central region and almost nothing else — and that is the region whose composition is changing, as hydrogen gives way to helium and the sound-speed gradient steepens. Nine microhertz is a measurement of how much hydrogen is left.

Plotting $\delta\nu_{02}$ against $\Delta\nu$ therefore gives a diagram whose model grid is a grid of masses and ages, and among main-sequence stars of similar $\Delta\nu$ the small separation falls as the star ages. The ages come out good to 10 or 20 per cent for a solar-type star — remarkable for a single field star with no cluster around it and no turn-off to read.

One honesty is owed. The ratio $\delta\nu_{02}/\Delta\nu$ rises from 0.067 for the Sun to 0.125 for the red giant below, opposite to the trend just described — but $\Delta\nu$ has itself fallen by two orders of magnitude between those stars, and a fraction of a collapsing denominator is not a clock. The diagnostic is the pair, not the quotient.

What is actually measured: twenty centimetres a second #

Every frequency above rests on an amplitude that is absurdly small. A solar radial mode moves the photosphere at about 20 cm s⁻¹ — a Doppler shift of seven parts in ten thousand million — and in brightness it is a few parts per million. Nothing about the Sun is faint, so the difficulty is not photons: the signal sits four or five orders of magnitude below the star’s own variability, and a single measurement of it is meaningless.

What rescues it is that the signal is coherent and the noise is not. A mode’s power piles up in one bin of a Fourier transform while the noise spreads across all of them, so the noise per frequency bin falls as the square root of the number of measurements. The arithmetic runs on stated inputs: 30 parts per million per half-hour cadence, some 70,000 cadences in four uninterrupted years, a per-bin noise near a tenth of a part per million — against a mode amplitude of a few. The detection is bought with time.

On the Sun the record is full-disc velocity, from ground networks since the 1970s and from space since 1996; those series fix the solar reference values these figures use. On other stars it is photometry: Kepler stared at one field for four years and produced usable combs for some hundreds of solar-type dwarfs and of order sixteen thousand red giants — two orders of magnitude more stars than the whole catalogue whose masses are measured any other way.

The dates explain the delay. Leighton, Noyes and Simon reported a patchy five-minute velocity oscillation of the solar surface in 1962 without knowing what it was; it was proposed as trapped standing waves around 1970, Deubner resolved the predicted ridges in 1975, and the evenly spaced peaks came out of full-disc velocity work at the end of that decade. Reaching a second star took twenty years more: the first convincing solar-type comb elsewhere came from a few nights on α Centauri A in 2001.

The only other place this works #

A mass and a radius in physical units, with no stellar model in the chain, is not something astronomy hands out. Double-lined eclipsing binaries are the one other source of it, and the comparison is worth making term by term.

A binary needs two stars, two visible spectra and an orbit within a degree or two of edge-on; it delivers masses and radii to 0.2 per cent from Kepler’s third law and geometry. A comb needs one star and a long time series, and delivers a radius to a few per cent and a mass to nearer ten. So the binary is far the more precise, on a few hundred systems selected by a severe geometric accident; the comb is the cruder, on tens of thousands selected by nothing but brightness.

The two methods meet on the few red giants that happen to sit in eclipsing binaries, and that meeting is the only external check the scaling relations have. It was not reassuring at first: the earliest such comparisons found seismic masses larger than the dynamical ones by of order fifteen per cent, which sent the $\Delta\nu$ relation back for a correction rather than the binaries. That is the right direction of deference, and it exists only because two methods answer one question.

One mode instead of thousands #

A classical Cepheid is the same physics with the mode count set to one: its period is the crossing time itself, while the Sun’s crossing time appears only in a spacing. A Cepheid pulsates because a helium ionisation layer acts as a valve and pumps one mode to enormous amplitude, making the star a self-excited oscillator on a limit cycle. The Sun has no such valve: its modes are driven by the turbulence of its own convection zone, a broadband source, so every mode the cavity supports is excited a little and none is excited much. Huge amplitude in one mode buys a period anyone can measure and nothing else; tiny amplitude in a thousand modes buys the interior.

The structure reaches beyond stars: the microwave background’s acoustic peaks are a standing sound wave in a different cavity, and their spacing is likewise a size divided by a sound speed. A comb is what a bounded medium makes, and a spacing read off one is a crossing time.

Where the model stops, unevenly #

The claim above was almost no stellar model, and the qualification has three parts.

The $\Delta\nu$ relation has a physical derivation behind it, but its constant is set at the Sun; the $\nu_{\max}$ relation is weaker still, resting on a proportionality to the acoustic cutoff which has no complete theory and is calibrated on one star. So the Sun is inside the chain — a far shorter chain than an evolutionary track, and not nothing.

The near-surface layers are the second part. One-dimensional convection gets the outermost few hundred kilometres wrong, and the resulting offset — the surface term — reaches several microhertz at the solar $\nu_{\max}$. It biases $\Delta\nu$ at the per cent level, and a per cent in $\Delta\nu$ is four per cent in the mass.

The third is the conditioning visible in the scaling plane, and it is a limit rather than an error bar. Mass moves a star across that plane by the quarter power of the observables while radius moves it by the first, so the mass is squeezed into a tenth of a decade where the radius has three. No amount of extra signal fixes a geometry that unfavourable; what fixes it is a second kind of observation, which is what the binaries supply.

What the picture cannot show #

Mode widths, and heights that are draws rather than values. Every mode here is one line of zero width. A real solar mode is about 1 μHz wide, because it is damped and lives a few days, and a real peak is a Lorentzian whose centroid must be fitted. On the drawn axis 1 μHz is a third of a pixel — which is why the drawing can get away with lines and why an analysis cannot. And convective driving is random, so the envelope is smooth only in expectation.

Rotation. Every mode with $\ell > 0$ is split into $2\ell+1$ components by rotation, and the drawing gives each degree one line. Those splittings are the only measurement of a star’s internal rotation there is, so the figure drops the subject’s most spectacular result to keep the comb legible.

Mixed modes, which is the giant panel’s real deficiency. In an evolved star the $\ell = 1$ modes couple to gravity modes trapped in the dense core, and what should be one ridge becomes a forest. Those modes are how a red giant is told apart from a red-clump star of identical $\Delta\nu$, and the drawing shows a clean ridge where the real spectrum shows the interesting mess.

And the granulation background. Convection produces a rising continuum at low frequency which, in a giant, sits directly beneath the envelope. These drawings have no noise floor; in practice the floor is the star.

The splitting that measures a rotation nobody can see #

The drawings give each degree one line, and a real spectrum gives each degree several. The reason is rotation, and what it delivers is the one quantity in stellar physics that has no other route to it at all.

A mode with $\ell > 0$ is degenerate in a spherical star: the $2\ell+1$ orientations of its angular pattern have identical frequencies. Rotation breaks the symmetry, because a wave travelling with the rotation and one travelling against it complete their circuits at different rates, so the degenerate multiplet splits into components separated by roughly the rotation frequency.

That is straightforward. What makes it a measurement of the interior is that different modes sample different depths. A mode’s splitting is an average of the rotation rate along its own cavity, weighted by where it spends its time — so a set of modes with different penetration depths gives a set of differently weighted averages, and inverting them returns a profile.

For the Sun the answer is the one described elsewhere in this collection: a differentially rotating convection zone above a radiative interior that turns as a solid body, with a thin shear layer between.

For a red giant the answer is more startling and is the field’s most consequential result. The mixed modes of an evolved star penetrate to the core, so their splittings measure the core’s rotation directly — and the core turns about ten times faster than the envelope.

Ten sounds like a lot and it is far too little. A star’s core contracts enormously between the main sequence and the giant branch while its envelope expands, and if each conserved its own angular momentum the core would end up spinning hundreds or thousands of times faster than the surface. What is observed is ten.

So angular momentum is being transported outward from the core, efficiently, by something. The candidates are internal gravity waves, magnetic fields threading the interior, and instabilities driven by the shear itself, and none of them at its computed strength produces the observed coupling. Every standard stellar-evolution code underestimates the transport by one to two orders of magnitude.

A frequency splitting of a few hundred nanohertz, measured on stars too distant to resolve, has falsified the angular-momentum transport in every model of stellar interiors, and the replacement has not been found.

Where the ladder goes next #

Later rungs on this anchor: $\varepsilon$ as a diagnostic of the surface layers; mixed modes and the period spacing of the core gravity modes, which separate hydrogen-shell from helium-core burning at identical $\Delta\nu$; rotational splitting, and the finding that giant cores turn far more slowly than angular momentum conservation alone permits; and the inversions that recover a whole sound-speed profile rather than two summary numbers.

And a rung facing outward: every transiting planet’s radius is a ratio multiplied by a stellar radius, and for the best-characterised systems that stellar radius comes from a comb — which makes a planet’s size a consequence of counting sound waves in the star it crosses.