A correction that grows as the fifth power of the frequency

A Sun-like star rings at thousands of frequencies at once, each a standing sound wave trapped between the surface and a turning point deep inside, and the comb of frequencies it produces is the most precise measurement anyone makes of a star. Frequencies of the strongest modes are known to a tenth of a microhertz out of three thousand, thirty parts per million, from a few years of space photometry. Fitting them with a model of the star’s interior gives its mass, radius and age to a few per cent, and its core’s state with a precision no other method approaches.

That fitting has a problem that is almost embarrassing in its simplicity: the models get the top of the star wrong. The outermost few hundred kilometres of a Sun-like star are a layer of turbulent convection where the gas rises and falls at kilometres a second, and a one-dimensional model replaces that turbulence with a single parameter, a length nobody derived, fitted to one star — the distance a convective blob travels before it mixes. The parameter is tuned to give the right radius, and it cannot give the right structure. The sound waves feel the error.

The layer where the star boils #

The layer in question is the top of the part of a star that boils, and it is unlike the rest of the convection zone. Deep down, convection is efficient: a tiny excess of temperature over the adiabatic gradient carries the whole luminosity, and the precise model of the turbulence hardly matters, because any reasonable model gives a gradient indistinguishable from the adiabatic one. In the last few hundred kilometres below the photosphere the density falls so fast that convection becomes inefficient. The gas must be much hotter than its surroundings to carry the energy, the rising elements move at nearly the speed of sound, and the temperature gradient departs from the adiabatic one by an amount that depends entirely on how the turbulence is modelled.

The Sun’s own oscillations were what first showed it. When helioseismology matured in the 1980s and 1990s, inversions of the solar frequencies recovered the sound speed through the interior to a part in a thousand, matching the models everywhere except in the outermost per cent of the radius, where the discrepancy rose steeply toward the surface. The models’ interiors were right and their skins were wrong, and the frequencies carried the skin’s error in exactly the high-frequency modes that reach it. The surface effect was discovered as the one place where the standard solar model failed a test it passed everywhere else.

Ridges that bend together #

An échelle diagram is the natural way to look at a comb. Dividing each frequency by the comb’s spacing, the large separation, and plotting the remainder against the frequency itself stacks successive orders on top of one another, so that modes of each degree form a nearly vertical ridge. A model whose structure is right everywhere puts its ridges exactly on the star’s. The opening figure shows what happens when the model’s outer layers are wrong in the way real models are. At low frequency the model’s ridges and the star’s coincide. With increasing frequency the star’s modes fall behind, by an amount that grows steeply, and all three ridges — radial, dipole, quadrupole — bend together.

That they bend together is the key fact. Modes of different degree penetrate to different depths, so an error deep inside the star would shift the ridges by different amounts; that is how each note’s turning point lets the frequencies map the interior. An error confined to the outermost layers shifts every mode by an amount that depends on how much of its energy sits near the surface, which for the low-degree modes observed in distant stars is set by the frequency and hardly at all by the degree. The surface effect is a function of frequency alone.

An offset that is nothing low and microhertz high #

The offset’s form has been measured on the Sun, where both the frequencies and the interior are known well enough to isolate it.

The empirical form is a power law in frequency, proposed in 2008 from the solar comparison: the offset is proportional to the frequency divided by the frequency of maximum power, raised to an exponent near five. At low frequency, where a mode’s upper turning point lies deep below the surface and the wave barely reaches the layers the models get wrong, the offset is a third of a microhertz. At high frequency, where the wave reaches nearly to the photosphere, it is twenty. Since a frequency is measured to a tenth of a microhertz, the offset is two hundred times the measurement error at the top of the observed range and larger than it almost everywhere.

The exponent is not derived; it is fitted. Its physical origin is understood in outline — near-surface turbulence adds pressure that a one-dimensional model omits, which raises the surface and lengthens the acoustic cavity, lowering the high frequencies most — and three-dimensional simulations of surface convection, grafted onto one-dimensional interiors, reproduce most of the solar offset. But the simulations exist for a grid of stars, not for every star fitted, and the power law is applied everywhere as a correction whose shape is borrowed from the Sun.

A separation measured short #

The first consequence reaches the simplest quantity in asteroseismology, the large separation, which with the frequency of maximum power gives a star’s mass and radius through two scaling relations.

The offset is larger at higher frequency, so it compresses the spacing between successive orders, and a separation fitted as the slope of frequency against order comes out short. The bias is set by the offset’s slope near the maximum, which is the same however many orders surround it, so widening the fit to more orders barely changes it — the bias is not noise, and more data do not average it away. Seven or eight tenths of a per cent is a large error for a measurement whose statistical precision is a few hundredths of a per cent.

The large separation is usually measured not as a fitted slope but from the power spectrum directly, as the spacing that best folds the comb onto itself, weighted toward the strongest modes near the maximum. That weighting puts the measurement where the offset’s slope is moderate, and it is one reason the scaling relations work as well as they do. But it ties the measured separation to the frequency of maximum power, which changes from star to star, and so ties the bias to the star’s surface gravity — a systematic error that varies smoothly across a sample, which is the hardest kind to notice.

The scaling relations raise the separation to powers. The mass goes as its inverse fourth power and the radius as its inverse square, so seven or eight tenths of a per cent in the separation becomes three per cent in the mass and one and a half in the radius. For the Sun, on which the relations are calibrated, the error cancels: the Sun’s own frequencies carry the same offset, and the calibration absorbs it. For a star whose outer layers differ from the Sun’s — hotter, cooler, more evolved, richer or poorer in metals — the offset differs, and the cancellation is partial. That is one of the reasons the scaling relations need empirical corrections that depend on the star’s temperature and evolutionary state.

A second layer of the problem in the giants #

The effect matters most for the stars where it is hardest to calibrate. Red giants, whose oscillations have been measured by the tens of thousands, are where asteroseismology delivers ages for the Galaxy’s stellar populations, and their ages come mainly from their masses, which come from the scaling relations. A three per cent mass error for a giant is a ten per cent age error, because a giant’s age is set by how long it spent on the main sequence, which goes as roughly the inverse third power of the mass.

A giant’s outer convection zone is deep and its surface gravity low, and its near-surface turbulence is stronger relative to its structure than the Sun’s. The offset is correspondingly different in size and shape, and no giant is known well enough to calibrate it directly. The empirical corrections to the scaling relations for giants are therefore fitted against stars with independently known masses — giants in eclipsing binaries and in clusters whose turn-off ages are known — and those calibrators are few. The relations calibrated on one star are being recalibrated on a few dozen, and the surface effect is part of what the recalibration absorbs.

A correction with the right shape, or not #

Detailed modelling fits individual frequencies rather than a separation, and corrects them for the surface effect by fitting the power law’s amplitude along with the star’s parameters. The last figure but one asks what that correction achieves.

When the offset really is the power law, fitting its amplitude removes the bias completely. It costs precision: the uncertainty of the fitted separation doubles, because the correction and the separation both move the highest frequencies most and the fit cannot fully tell them apart. That is the honest price of a free correction, and it is a small one.

When the offset has a different shape — here a two-term form of the kind proposed later on physical grounds, a combination of a cubic and an inverse term, equal to the power law at the maximum — the power-law correction removes only a sixth of the bias, and leaves a misfit larger than the noise allows. The fit is wrong and says so, but it does not say in which direction to move. The correction is only as good as its shape, and the shape is the part of the surface physics that cannot be computed from first principles for an arbitrary star. Different studies use different forms, and their fitted ages for the same stars differ by amounts comparable to their quoted errors.

Ratios the surface cannot reach #

There is a way out, proposed in 2003, and it is the reason core ages from asteroseismology are trusted at all. Instead of fitting frequencies, fit combinations of frequencies that the surface does not affect.

The small separation between a radial mode and the quadrupole mode of the next lower order is the difference between two frequencies only a few microhertz apart. Both lie near the surface by the same amount, so the offset shifts them almost equally, and their difference barely changes. Divided by the large separation, which the offset compresses by a fraction that the small separation shares, the ratio cancels the surface effect to first order. The figure shows the result: where the raw spacing changes by two per cent, the ratio changes by five thousandths of a per cent.

The ratios keep what matters. The small separation measures the gradient of the sound speed near the core, which changes as hydrogen is burned into helium — it is a clock — and the ratio retains that sensitivity while discarding the surface. Ages fitted to ratios rather than frequencies are therefore largely independent of how the surface was modelled, which is why they are the standard for the best-observed stars. The ratios do need the quadrupole and dipole modes as well as the radial ones, and those are measured only in stars bright enough and observed long enough to resolve the closely spaced peaks, which limits the method to a few hundred of the best targets rather than the tens of thousands with detected oscillations. What the ratios cannot give is the mean density, which is what the large separation itself carries and the surface corrupts; models fitted to ratios get their density from the frequencies after all, and inherit some of the surface’s uncertainty there.

Why the Sun gets it wrong #

The outer layers of the Sun are where its energy changes how it is carried, from convection below to radiation above, through a layer a few hundred kilometres thick where the gas is neither fully convective nor in equilibrium with the light. Convective cells there rise at a few kilometres a second, overshoot into the stable photosphere, and supply a turbulent pressure comparable to a tenth of the gas pressure. A one-dimensional model has no turbulent pressure and a convective flux that switches off sharply at the surface. Both make its outer layers more compact than the Sun’s, which shortens the acoustic cavity for the modes that reach it and raises their frequencies — the direction the offset has.

Non-adiabatic effects add a second error. The models compute oscillations as adiabatic, but in the outermost layers the gas exchanges heat with the radiation field on timescales comparable to the oscillation periods, and the waves are damped and shifted. That part of the offset depends on the modes’ interaction with the convection itself, which is the least understood physics in stellar oscillations and the reason the photons on the sides of the lines and the granulation that shapes them have their own literature.

Three-dimensional simulations of surface convection, which follow the turbulence directly, reproduce most of the solar offset when their averaged structure replaces the model’s outer layers. They are expensive and exist for a grid of stellar parameters. Patching them onto one-dimensional interiors for individual stars is now done for the best-studied targets, and it removes most of the need for an empirical correction — and with it most of the question of its shape.

The same layer limits a measurement in a different field. The rising and sinking cells that put the offset into the oscillation frequencies also shift and distort the star’s spectral lines, because the hot rising gas is brighter and blueshifted and the cool sinking gas fainter and redshifted. Their changing pattern adds a jitter of tens of centimetres a second to any measurement of the star’s velocity — the metre per second that is not the star — and it is the floor that searches for Earth-like planets around Sun-like stars now reach. The surface that spoils the frequencies spoils the velocities too, and the simulations that would fix one would help with the other.

What the figures leave out #

The frequencies here come from an asymptotic formula rather than from a stellar model, and the offset is applied to all three degrees identically. Real offsets scale with each mode’s inertia, which for low-degree modes is nearly independent of degree at fixed frequency but not exactly; the residual dependence on degree is small and is what lets the most careful analyses separate surface from interior. The figures also treat the observed modes as measured with uniform precision, while in practice the modes near maximum power are measured best and those at the edges of the range worst, so the high-frequency end where the offset is largest is also the least weighted in a fit.

And the comparison uses one star, a Sun-like one. Red giants have their maximum power at much lower frequencies and their modes probe a different structure, and the offset in giants is smaller relative to their large separation but harder to calibrate, since no giant is known as well as the Sun.

Still open: the surface of every other star #

The surface effect turns the most precise measurement in stellar astrophysics into one whose accuracy is set by the physics of convection in a layer a fraction of a per cent of the star’s radius. For the Sun the effect is measured and the correction is calibrated. For every other star the correction is borrowed, and its shape — the thing that decides whether it recovers the star or merely makes the fit look better — is assumed.

The path forward is the one already taken for the best stars: replace the outer layers of each model with averages from three-dimensional simulations of the right temperature, gravity and composition, so that the offset is computed rather than fitted. Where that has been done for a handful of well-studied stars, the remaining offsets are small and their shapes consistent. Extending it to the tens of thousands of stars that space photometry has measured oscillations for, including those that missions now flying will add, is a computational problem rather than an observational one — and until it is solved, the ratios that the surface cannot reach are what the ages of stars rest on.