The only stars whose masses are known

Mass decides everything about a star — its luminosity, its temperature, its lifetime, and how it ends. That sentence is repeated in every account of stellar astrophysics, and it raises a question that is answered much less often: how does anybody know a star’s mass?

Not by looking at it. A star’s spectrum gives a temperature and a surface gravity; its brightness and distance give a luminosity. Every route from those to a mass runs through a model of stellar structure, which is to say through the theory that the mass was supposed to be testing.

There is exactly one class of object where the chain contains no model at all.

Two spectra give a ratio; the eclipse gives the rest #

The chain is short enough to state completely, which is precisely what makes it valuable.

A spectroscopic orbit of a double-lined pair gives the period $P$, the eccentricity, and the two velocity semi-amplitudes $K_1$ and $K_2$. From those alone,

$$\frac{M_1}{M_2} = \frac{K_2}{K_1}, \qquad (M_1 + M_2)\sin^3 i = \frac{P(K_1+K_2)^3}{2\pi G}.$$

The ratio is exact and needs nothing further. The sum is entangled with the inclination, which a spectrograph cannot see — the same $\sin i$ ambiguity that leaves an exoplanet’s mass a lower bound.

An eclipse removes it. If the system eclipses, the orbit is nearly edge-on, and the shape of the eclipse fixes the inclination precisely: how long each eclipse lasts, whether it is flat-bottomed or V-shaped, and how the two durations compare all depend on how far off centre the smaller disc passes. Solving that geometry gives $i$ to a few hundredths of a degree in a good system.

That is the whole method: two spectra and one light curve give two masses and two radii, in physical units, from Kepler’s laws and geometry. No stellar atmosphere, no evolutionary track, no colour–temperature calibration, no distance. The best-observed systems reach 0.2 per cent in mass and radius, which is a precision unmatched by any other stellar measurement.

What decides whether the solution is any good #

Not every eclipsing binary delivers this, and the conditions are restrictive.

Both spectra must be visible. If one star is much fainter, only one set of lines is measurable, and only the mass function comes out — a lower bound rather than a mass. That excludes most pairs with a large mass ratio, which is a bias worth remembering.

The stars must be detached. If either fills its own Roche lobe, it is distorted into a teardrop, its light curve is continuously varying rather than flat between eclipses, and the radius that comes out is a fitted parameter of a deformation model rather than a radius. Detached systems only.

And the eclipse must be well sampled. The inclination is inferred from the shape of the ingress and egress, which for a good system last a few hours out of a period of days.

What the method cannot deliver #

It gives masses and radii. It does not give temperatures, luminosities or distances, and each of those requires something extra.

Temperature comes from the spectrum or the colours, through a calibration built on other stars. The light curve gives only the ratio of the two surface brightnesses, from the ratio of the eclipse depths, so one temperature must come from outside and the second follows.

Luminosity then follows from $L = 4\pi R^2\sigma T^4$ with the measured radius — which is the one place a stellar luminosity is computed from a measured radius rather than from a distance and a flux.

Distance can be recovered, and this is the elegant part: with a radius in kilometres and a temperature, the flux at the star’s surface is known; the flux received is measured; the ratio is the inverse square of the distance. That is an eclipsing binary distance, and it is a geometric method that works far beyond parallax range. Distances to the Large Magellanic Cloud good to one per cent have been obtained this way, and that number is one of the anchors of the whole distance ladder.

What it calibrates #

The reason a few hundred systems matter so much is that everything else in stellar astronomy is checked against them.

The same is true one level up. A stellar evolution model predicts, for a given mass and composition and age, a radius and a luminosity. Testing it requires a star whose mass and radius are known independently — and a binary provides two such stars, of the same age and the same composition, which is a far stronger test than one star would be. A single system with two well-measured components tests an isochrone at two points at once.

What was actually measured: Algol, and a paradox #

Algol — $\beta$ Persei — was the first, and its history contains the method’s most instructive surprise.

John Goodricke measured its period in 1783 as 2 days 20 hours 49 minutes, from naked-eye estimates of its brightness, and proposed that a dark body was passing in front of it. He was 19 and he was right, a century before anyone could check. The spectroscopic confirmation came from Hermann Vogel in 1889, who found the expected radial-velocity variation.

The surprise arrived when the two components were characterised. Algol’s more massive star is an ordinary main-sequence B star; its less massive companion is a subgiant — evolved, expanded, past the main sequence. That is backwards. More massive stars evolve faster, by a wide margin, so in a pair born at the same time the heavier one should leave the main sequence first. The lighter one has, and this became known as the Algol paradox.

The resolution is that the mass ratio is not the one the system was born with. The subgiant was originally the more massive star; it evolved first, expanded, filled its Roche lobe, and transferred most of its envelope to its companion — which is now the heavier of the two and still on the main sequence. Mass transfer had to be invented to explain one binary, and it turned out to be the mechanism behind novae, X-ray binaries and a large fraction of type Ia supernovae.

The methodological point is worth keeping. The paradox was only visible because the masses were measured rather than inferred. Had they been estimated from the luminosities, using the mass–luminosity relation, the answer would have been consistent and wrong.

Why an eclipse is worth so much more than a transit #

The comparison with a transiting planet is instructive, because the geometry is identical and the information content is not.

A transit gives the planet’s radius as a ratio to the star’s, from the depth; it gives the orbital period; and with the transit’s shape it gives the stellar density. What it does not give is any absolute length or mass, because the star’s own radius is not measured — it is taken from a model, or from an asteroseismic analysis, or from an interpolation on a colour. Every exoplanet radius in every catalogue is a ratio multiplied by somebody’s estimate of a stellar radius, and the errors in the second are usually larger than the errors in the first.

An eclipsing binary breaks that regress because both bodies are stars with visible spectra. The spectroscopic orbit supplies the absolute scale — kilometres, from a velocity multiplied by a time — that a transit has no way to obtain. The whole difference is one measurable line spectrum.

That is also why the two fields are coupled in one direction only. Improvements in binary-star radii propagate into exoplanet radii, since the calibrations they anchor are what stellar radii are estimated from; nothing about a transit improves a binary.

The sample is small, and biased in known ways #

The catalogue of systems with masses and radii good to three per cent contains a couple of hundred entries, and it is not a random sample of stars.

It favours nearly equal masses, because both spectra must be detectable. It favours short periods, because a long-period system eclipses rarely and its orbit takes years to trace. It favours main-sequence stars, because giants in close pairs are usually interacting. And it is thin at the extremes: below about 0.3 solar masses the components are faint and the models are known to disagree with the measurements by 5 to 10 per cent in radius, and above about 20 solar masses there are very few systems at all.

That last gap matters more than it sounds. The mass–luminosity relation is calibrated where the binaries are, and extrapolated where they are not — which is precisely the regime where the Eddington limit becomes relevant and the relation is expected to change shape.

There is one route to a mass that is better than a binary orbit and works only for one kind of object. Pulsar timing in a relativistic binary measures post-Newtonian effects — the periastron advance, the Shapiro delay of the pulses passing the companion, the orbital decay — and each is an independent combination of the two masses. PSR J0740+6620’s mass is $2.08 \pm 0.07$ solar masses, obtained from the Shapiro delay alone, and it is that number rather than any spectrum that constrains what matter does at nuclear density.

What a mass is worth, at four significant figures #

It is fair to ask why anybody needs a stellar mass to 0.2 per cent when the theory being tested has larger uncertainties than that. The answer is that a few of the tests are sharp.

The clearest is the radius. For a star of given mass, composition and age, stellar structure theory predicts a radius, and the prediction is not adjustable — the mixing-length parameter shifts it, but only by a per cent or two for a solar-type star. Measured radii good to a few tenths of a per cent therefore either confirm the models or do not, and for low-mass stars they do not: measured radii of M dwarfs in binaries run 5 to 10 per cent larger than predicted, with temperatures correspondingly lower, and the discrepancy has resisted twenty years of attention. The favoured explanation is magnetic activity inhibiting convection in a star that is convective throughout, which is a proposal about the transport mechanism rather than about the mass.

The second sharp test is the age. Two stars in a binary share a birthday, so a model must fit both with one age and one composition — four measured numbers against two free parameters. Systems where one component has left the main sequence are the most demanding of all, because the evolved star’s radius changes fast and pins the age tightly.

Neither test is available anywhere else. That is the argument for the precision: it is not that a mass to four figures is useful in itself, but that the surplus precision is what converts a measurement into a refutation.

Where the error budget actually sits #

A method with four steps has four places to lose precision, and they do not contribute equally. Following them is worth doing because the answer is counter-intuitive in one place and explains why this class of system is so much better than any other.

The mass sum goes as $(K_1+K_2)^3$, so a velocity amplitude measured to 0.3 per cent gives a mass to 0.9 per cent. The cube is unforgiving, and it is why the observational effort goes into the spectroscopy rather than into the photometry: doubling the number of velocity points buys a factor in the mass that doubling the number of light-curve points does not.

The period is free. It is measured over decades of eclipse timings and is known to seven or eight digits in any system that has been watched, so it contributes nothing to the budget.

The inclination is the surprise. The mass sum goes as $1/\sin^3 i$, which looks alarming — and it is, for a spectroscopic orbit with no eclipse, where $i$ is unknown altogether. For an eclipsing system it is the opposite. The derivative of $\sin^3 i$ vanishes at $i = 90°$, so near edge-on the mass is insensitive to the inclination to first order: an error of half a degree at 89° propagates to under two parts in ten thousand. The one quantity the spectroscopy cannot supply is the one the geometry delivers for nothing, precisely because the systems that eclipse are the systems where the dependence is flattest.

The radii are different in character. They come out as fractions of the separation, from the eclipse durations, and there the light curve carries the whole weight — with limb darkening as the systematic floor. That is why a mass can be known to 0.2 per cent and a radius to 0.5, in the same system, from the same data.

The two halves of the solution are worth redrawing at the geometry the essay’s own error budget turns on, since the inclination is what converts a spectroscopic minimum mass into a mass.

The systems that eclipse, and the ones that were never going to #

The sample’s biases listed above are about what is measurable. There is a prior bias, purely geometric, that decides which systems enter the catalogue at all.

A binary eclipses only if the line of sight lies within a narrow band about the orbital plane, and the probability of that for a randomly oriented orbit is close to $(R_1+R_2)/a$. Two solar-type stars in a ten-day orbit are separated by about twenty solar radii and have a combined radius of two, so roughly one such pair in ten eclipses. Push the period to ten years and the separation grows by a factor of thirty; the probability falls to a third of a per cent.

That single expression explains the shape of the whole field. Short-period systems are over-represented by construction, not by any observational preference — and short-period systems are the ones most likely to have interacted, which is exactly the population the method requires to be untouched. The geometry selects for the systems the physics wants excluded, and the detached, well-behaved, long-period pairs that would be the cleanest tests are the ones least likely to be oriented usefully.

Wide-field photometric surveys have changed the supply and not the bottleneck. Kepler, OGLE, ASAS-SN and TESS between them have catalogued tens of thousands of eclipsing binaries, and a light curve is now the cheap half of the measurement. What each one still needs is a spectroscopic orbit — dozens of high-resolution spectra of a star faint enough that a single exposure takes an hour on a large telescope, spread over a period long enough to cover the orbit. The catalogue of light curves has grown by two orders of magnitude and the catalogue of masses by rather less than one, and the difference is entirely telescope time on the half of the measurement that cannot be automated.

What the picture cannot show #

Limb darkening. The light curves here are computed from uniform discs. A real stellar disc is brighter at the centre than at the edge, which rounds the shoulders of an eclipse and deepens its middle. Every real solution fits limb-darkening coefficients or takes them from model atmospheres — and there, quietly, a stellar model does enter the chain, at a level of a few tenths of a per cent in the radius.

A filter. The depths drawn are bolometric — all wavelengths at once. Real photometry is in a band, and the depth ratio in the $V$ band is not the depth ratio in the infrared, because the two stars have different temperatures. Multi-band photometry uses that difference on purpose, to get the temperature ratio.

And the third body. Many close binaries have a distant third component, which shows up as a slow drift in the systemic velocity and a light contribution that dilutes both eclipse depths. Missing it biases the radii low, and a light curve alone cannot rule it out.

And the distance the same solution yields, computed from a different set of trial surface brightnesses.

Where the ladder goes next #

The rung above is the interferometric orbit: resolving a visual binary directly, which combined with a spectroscopic orbit gives the masses and the distance without any eclipse at all. The rung beside it is the population question — what the distribution of mass ratios and periods says about how binaries form, which is a statistical statement about the very sample whose biases this essay has just enumerated.