Two scaling relations calibrated on one star

Asteroseismology has given masses and radii for hundreds of thousands of stars, which is more than every other technique combined by three orders of magnitude. It does so with two numbers extracted from an oscillation spectrum, and the extraction is robust: both numbers are frequencies, both are measured to parts in a thousand, and neither depends on the star’s distance, its brightness, or its spectrum.

The step that is not robust is what happens next. Two numbers become a mass and a radius through two relations whose exponents are exact and whose constants are fitted to a single star.

The situation is unusual in observational astronomy and worth naming for what it is: a technique whose measurement precision exceeds its accuracy by two orders of magnitude, with the entire gap sitting in two constants. The interior read from a comb of frequencies is a genuinely direct probe; what this essay is about is the last step, in which that probe’s output is compressed into two numbers and multiplied by a constant taken from somewhere else.

Where the two relations come from #

The large separation. The high-order acoustic modes of a star are nearly evenly spaced in frequency, and the spacing $\Delta\nu$ is the inverse of twice the sound travel time from the centre to the surface. Since the sound speed goes as the square root of the temperature, and the temperature profile is set by hydrostatic equilibrium, the travel time scales as the square root of the mean density. So

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

That relation is not an approximation in any deep sense; it is an asymptotic result that follows from the wave equation, and its accuracy for a real star is limited by how far the star departs from the homology the derivation assumes.

The frequency of maximum power. Oscillations are excited by convection and damped by it, and the envelope of the excited modes peaks at a frequency $\nu_{\rm max}$. The scaling here is on much weaker ground: it is argued that $\nu_{\rm max}$ tracks the acoustic cut-off frequency of the atmosphere, which is proportional to $g/\sqrt{T_{\rm eff}}$, so

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

That argument is plausible, it is supported empirically across five orders of magnitude in $\nu_{\rm max}$, and it has no derivation of the same standing as the first.

Solving the pair gives $M \propto \nu_{\rm max}^3 \Delta\nu^{-4} T_{\rm eff}^{3/2}$ and $R \propto \nu_{\rm max}\Delta\nu^{-2}T_{\rm eff}^{1/2}$. The fourth power in the mass is where the trouble is.

The asymmetry between the two relations is worth dwelling on because it decides where the effort goes. The first is asymptotic theory: it follows from the wave equation for high-order acoustic modes, its derivation makes assumptions that can be checked, and its departures from exactness can be computed from a model. The second is a scaling argument with an empirical constant. So the two ingredients of every seismic mass have completely different epistemic status, and the weaker one enters the mass with a cube.

There is a useful way to see why the exponents come out as they do. The two observables are, in effect, a mean density and a surface gravity: the first relation is a density and the second is $g/\sqrt{T}$. A density is $M/R^3$ and a gravity is $M/R^2$, so extracting $M$ and $R$ from the pair means solving two power laws whose exponents differ by one in $R$ and not at all in $M$. That near-parallelism is exactly the ill-conditioning of the previous essay’s crossing angle, in a different setting: two constraints that respond similarly to one variable determine it poorly, and the amplification factors of four and two are the numerical statement of how similarly.

The exponents amplify, and they amplify differently #

The propagation is exact and it is worth having in front of one.

A fractional error $f$ in $\Delta\nu$ gives $-2f$ in the radius and $-4f$ in the mass. A fractional error in $\nu_{\rm max}$ gives $+f$ in the radius and $+3f$ in the mass. A fractional error in the temperature gives $f/2$ and $3f/2$.

So a per cent error in the large separation is a four per cent error in the mass, and — since a main-sequence lifetime scales as roughly $M^{-2.5}$ — a ten per cent error in an age. The relation is a lever with a long arm.

The consequence for practice is that the radius is the trustworthy output and the mass is not. Seismic radii agree with interferometric radii to a per cent or two; seismic masses agree with dynamical masses to five or ten per cent; seismic ages are quoted with uncertainties of twenty per cent and are probably worse.

There is a corollary worth acting on. Because the radius is the better-determined of the two outputs, any argument that can be made with a radius should be made with a radius rather than with a mass. A planet’s radius measured from a transit depth needs the star’s radius and not its mass; a surface gravity needs both but combines them in a way that partly cancels; a mean density needs neither separately. Reformulating a question so that it asks for the well-determined combination is worth more than any improvement in the data, and it is available surprisingly often.

Why the constant is not one #

The proportionalities above become equations by inserting the solar values, which amounts to assuming that a star with the Sun’s mean density has the Sun’s large separation. That is true to the extent that the star is homologous to the Sun, and it is less true the further the star is from being a solar analogue.

Two departures matter.

The mass distribution. A red giant has an enormously centrally condensed structure, with a dense degenerate core and a vast tenuous envelope. Its mean density is the same quantity as a dwarf’s mean density and its sound-travel time is not related to it in the same way, so the constant in the first relation shifts — by a few per cent for a giant, in a way that depends on its evolutionary state.

The surface. The outer layers of a star are where the model is worst: convection is inefficient there, the treatment is one-dimensional, and the frequencies computed from a model are systematically too high by several microhertz. That is the surface effect, it affects all modes in a way that depends on frequency, and it biases the fitted $\Delta\nu$.

Both are corrected by comparison against theoretical models, which reintroduces the model dependence the technique was prized for avoiding. Corrections of two to four per cent in $\Delta\nu$ are typical for giants, and the four in the exponent turns that into ten to sixteen per cent in mass.

There is a third departure that is smaller and instructive: the relations are usually written with a temperature in them, and the temperature comes from spectroscopy or photometry with its own systematic. A hundred kelvin — which is about the accuracy of a spectroscopic temperature — is 1.7 per cent, which enters the mass with a power of three halves, giving 2.6 per cent. So a technique advertised as distance-independent and model-independent carries a spectroscopic systematic in its mass at the level of a few per cent, and there is no version of the relations that avoids it.

It is worth being explicit about what “calibrated on the Sun” means operationally, because it sounds like a minor convention. The solar values of the two observables are known to better than a tenth of a per cent, so the calibration introduces no measurement error at all. What it introduces is an assumption of homology: the claim that a star of a different mass, composition and evolutionary state relates its observables to its mass and radius by the same constants the Sun does. That assumption is testable only by finding stars with independent masses, and the number of such stars with detected oscillations is a few dozen. So the calibration rests on one star for its value and on a few dozen for its transferability, against a catalogue of hundreds of thousands.

What was actually measured #

The relations are tested wherever a mass or radius is available independently, and the tests are the reason the size of the problem is known.

Eclipsing binaries with oscillating components. A handful of systems have a red giant showing oscillations in a binary with a measurable dynamical mass. The seismic masses come out systematically high by about five per cent before correction and agree to within a few per cent after the model-based corrections are applied — which is a validation of the corrections rather than of the raw relations.

Interferometric radii. For bright nearby dwarfs the angular diameter is measurable directly, and combined with a parallax it gives a radius with no seismology in it. Seismic radii agree to about a per cent, which is the strongest evidence that the first relation is sound.

Cluster members. Giants in a cluster share an age and a distance, so their seismic masses should agree with each other and with the cluster’s turn-off mass. They do, to within the scatter, and the comparison constrains the mass scale to a few per cent.

And the distance test. A seismic radius plus an effective temperature gives a luminosity, which with an apparent magnitude gives a distance — a route entirely independent of parallax. Comparing that against measured parallaxes tests the radius scale, and it is one of the ways the parallax zero point itself has been checked.

Where the picture stops #

Three limits stand out, and the second is the one being worked on.

The second relation has no derivation. The scaling for $\nu_{\rm max}$ rests on an argument about the acoustic cut-off that is dimensionally right and quantitatively unestablished. It works empirically over an enormous range, which is evidence that something is right about it, and there is no theory that says the constant should be the same for a giant as for a dwarf.

Individual frequencies are better than the scalings. Fitting the actual mode frequencies against a stellar model — rather than compressing them into two numbers — uses far more information and gives masses and ages several times more precise. It also requires a model, so it trades one dependence for another, and it is only possible for stars with high enough signal-to-noise to resolve individual modes.

And the ages are the weakest output and the most wanted. Galactic archaeology needs ages for large samples, seismology is the only technique that can supply them in bulk, and the propagation above means that a ten per cent age is close to the best that scaling relations can do — which is not good enough to distinguish the formation epochs of the Galaxy’s components.

And a fourth, and it is about the population the technique reaches. Oscillations are detectable when their amplitude exceeds the photometric noise, and the amplitude scales steeply with luminosity — so seismology works easily on red giants, with difficulty on solar-type dwarfs, and not at all on anything hotter than about the Sun’s temperature, where convective envelopes are too thin to excite modes. The sample is therefore selected by evolutionary state, which correlates with mass and age, and any population statistic built from it inherits that selection. What a survey could have seen has to be modelled before what it found means anything, and seismic samples have an unusually sharp and unusually well-understood selection boundary.

Why a calibrated relation is a different object from a derived one #

The general point is one this collection meets repeatedly, and it is worth stating cleanly.

A relation with derived exponents and a fitted constant behaves quite differently from one that is derived throughout. Its differential use is excellent: comparing two stars, the constant cancels and the exponents do the work, so relative masses and relative radii are far better determined than absolute ones. Its absolute use inherits everything about the calibrating object, and there is only one calibrating object.

That is the same structure as a Coulomb logarithm calibrated against simulations and as a mixing length calibrated on the Sun: a piece of physics whose form is understood and whose normalisation is imported. Such a relation is enormously useful and it is not a measurement in the sense a parallax is, and the distinction matters most when the result is being compared against a prediction that shares the same calibration.

The escape, where one is available, is to find a second calibrator of a different kind. For the seismic relations that is what the eclipsing binaries and the interferometric radii provide, and the programme of finding more such benchmarks is the main line of work in the field — not better photometry, of which there is already enough, but a handful of stars measured a completely different way.

One remaining observation about why the technique is nevertheless transformative. Before seismology, a field star’s mass was not measurable at all — it was inferred from a position in a colour–magnitude diagram, which requires a distance and a model and gives a factor. Seismology gives it to five or ten per cent, for hundreds of thousands of stars, from photometry alone. That a five per cent mass is a poor measurement by the standards of an eclipsing binary is true and beside the point: the only stars whose masses were known numbered a few hundred, and they were not a representative sample of anything. A worse measurement of a much larger and fairer sample is a different kind of instrument, and most of what has been learned about the Galaxy’s stellar populations in the last fifteen years came from it.

A final observation on how the field has responded, because the response is instructive. Rather than trying to derive the second relation, the effort has gone into building a set of benchmark stars — objects with masses and radii from eclipsing binaries, interferometry, or cluster membership, and with detected oscillations — against which the relations are calibrated empirically as a function of evolutionary state. That converts a theoretical problem into an observational programme, it works, and it changes the character of the technique: the relations become an interpolation across a calibrated grid rather than a piece of physics. The distance ladder has the same architecture, and it has the same vulnerability — a systematic in the benchmarks propagates to everything, undiluted, and cannot be found from within.

One further consequence is worth carrying, because it decides how such masses should be used in a population study. The calibration is a single multiplicative constant applied to every star, so an error in it moves every seismic mass in the same direction by the same fraction — it is a shared systematic rather than a random one. A sample of ten thousand seismic masses therefore has a statistical precision far better than its accuracy, and averaging does not help at all. That matters most for the quantities derived by comparing a sample against a model: a galactic age distribution built from seismic ages inherits the calibration’s error as a rigid shift of the whole distribution, which looks like a real feature and is not. The remedy is the usual one and it is being pursued — anchor the constant on stars whose masses are known by other means, which for this purpose means eclipsing binaries and interferometrically resolved stars, and there are a few dozen of them.

End on what makes the relations worth using despite all of this. They require two numbers read off a power spectrum and nothing else — no distance, no spectrum, no model of the star’s interior — and they deliver a mass and a radius for any star with a long enough light curve. Nothing else in stellar astrophysics does that, which is why an approximate constant fitted on one star has been applied to hundreds of thousands.

Where the ladder goes next #

The rung directly above is the surface effect: what it is physically, why one-dimensional models get the outer layers wrong, and how the empirical corrections are constructed. The one above that is what replaces the scalings — fitting individual mode frequencies against models, and what that costs in model dependence for what it buys in precision.