Two radii, from a light curve alone

Almost every stellar radius ever quoted is an inference. A spectrum gives a temperature; a distance and a brightness give a luminosity; and the Stefan–Boltzmann relation turns the two into a radius — but only after the temperature has been extracted from a model atmosphere and the luminosity from a bolometric correction, each of which was itself calibrated on something.

There is one route that does not work this way, and it is worth being precise about what makes it different. A pair of stars whose orbit happens to be seen edge-on eclipses, and the geometry of the eclipse is read directly off the light curve. What comes out is not a radius but a fraction: how large each star is compared with the separation between them. The separation then comes from the two velocity curves and Kepler’s third law, in physical units, and the two multiply.

No model of a stellar interior appears anywhere in that chain. Nor does a distance, nor a bolometric correction, nor an assumed temperature scale.

That is a short list of absences and it is the whole reason a few hundred obscure variable stars underwrite the mass and radius scale of the entire subject. Everything else — the radii on an evolutionary track, the calibration of a spectral type, the size assumed for the host of a transiting planet — is anchored, directly or through two intermediaries, to these systems.

What each feature of the curve fixes #

An eclipse is a geometry problem, and the light curve encodes the geometry in a way that is over-determined enough to be solved.

The two depths. During the primary eclipse the smaller, hotter star is hidden; during the secondary the cooler one is. The ratio of the depths is therefore the ratio of the two surface brightnesses, and it involves no distance and no absolute calibration whatever. Turning that ratio into a ratio of temperatures needs only the shape of the spectrum in the observing band.

The durations. How long an eclipse lasts, in units of the orbital period, is a statement about how large the stars are compared with their separation. Two eclipses give two constraints; two stars have two fractional radii; the system is exactly determined, given the inclination.

The shape between contacts. A total eclipse has a flat bottom, because for a while the smaller star is entirely hidden and nothing changes. A partial one does not. That single qualitative difference removes the worst degeneracy in the problem — between the inclination and the sum of the radii — and it is the reason totally eclipsing systems are worth far more than partial ones.

The degeneracy is worth spelling out, because it is the same one that afflicts every eclipse problem. A shallow eclipse can mean a small companion crossing the centre of the disc or a large one grazing the edge, and in a partial eclipse those two produce nearly the same curve. What separates them is the duration of ingress compared with the duration of the whole eclipse: a central crossing has a short ingress and a long flat middle, a grazing one has almost nothing but ingress and egress. Measuring that ratio well needs cadence, which is why the systems solved to a per cent are the ones that have been watched at a few seconds’ sampling rather than a few minutes’.

The spacing of the minima. In a circular orbit the secondary eclipse falls exactly half a period after the primary. In an eccentric one it does not, and the displacement measures the projected eccentricity — while the ratio of the two eclipse durations measures a different combination of the same quantities. Both come free with the light curve, and neither needs a spectrum.

What the light curve cannot do alone #

The fractional radii are dimensionless. Nothing in the light curve says how far apart the stars are, and therefore nothing in it says how large they are.

The somewhere else is the spectroscopy. Both stars are bright enough to show lines, so both radial-velocity curves can be measured, and a double-lined spectroscopic binary gives the two velocity semi-amplitudes. Kepler’s third law then gives $M_1 + M_2$ and the semi-major axis directly, in solar masses and kilometres — subject to the factor of $\sin i$, which the light curve has already supplied.

That is the whole of it. Masses come from the two velocity curves, the separation comes with them, the fractional radii come from the light curve, and the products are two radii in metres.

The complications that are not small #

Three effects distort the light curve enough that ignoring them produces wrong radii rather than imprecise ones, and each of them changes the shape of the curve in a way that a fit will happily absorb into the radii if it is not told otherwise. Two are about the stars being close together and one is about their surfaces not being uniformly bright. None of the three is a correction at the level of the last decimal place; all three move the answer by more than the formal error bars on a good light curve.

Reflection. Each star heats the hemisphere of the other that faces it, so the system is brighter near the phases where the heated face is visible. In a close pair this is a smooth few-per-cent modulation between eclipses, and neglecting it biases the eclipse depths.

Distortion. Stars close enough to eclipse frequently are close enough to be tidally elongated, and an ellipsoidal star presents a varying cross-section through the orbit. The signature is a modulation at half the orbital period, and once it is large the notion of “the radius” of the star starts to require a definition.

That last is why the useful systems are detached — both stars well inside their Roche lobes, nearly spherical, and not exchanging material. The near-contact systems are astrophysically fascinating and are useless as rulers.

Why the answers agree, and what it means when they do not #

The system is over-determined, and that is the property worth exploiting.

Count the constraints: two depths, two durations, two ingress shapes, the spacing and the ratio of durations. Count the unknowns: two fractional radii, the inclination, the ratio of surface brightnesses, the eccentricity and the orientation. The counts are close, and once the velocity curves are added — two amplitudes, and the same eccentricity and orientation measured a second and completely independent way — the problem is comfortably over-determined.

So the eccentricity derived from the eclipse spacing can be checked against the eccentricity derived from the shape of the velocity curves, and they have no reason to agree unless the model is right. When they disagree, something in the picture is wrong: a third body, a spot distorting one eclipse, or a distorted star. That internal check is largely absent from single-star work, where a temperature and a luminosity give a radius and there is nothing to test it against.

The mass ratio is read off the two velocity amplitudes and nothing else, which means it is worth seeing what a different ratio looks like in the same system.

That division is the whole reason an eclipsing double-lined binary is worth so much more than either half of it. Spectroscopy alone gives $M \sin^3 i$ and cannot separate a heavy pair seen obliquely from a light pair seen edge-on. Photometry alone gives fractional radii and an inclination and no scale at all. Put together, each supplies exactly what the other lacks, and the result is a mass and a radius in kilograms and kilometres with no step that assumes anything about how a star works.

There are of order a hundred systems in the Galaxy where both halves have been done well enough for the masses and radii to be good to one or two per cent, and they are the calibration sample for everything else in stellar astrophysics. Every relation between mass and luminosity, every isochrone, every asteroseismic scaling law is checked against them, and when one of those relations is revised it is these systems that decide whether the revision was an improvement. A hundred systems is not many for a calibration sample, and the number grows slowly: an eclipsing binary bright enough for high-resolution spectroscopy of both components is a rare object, and the photometry has to span years to pin the period well enough for the radii to be worth having.

What the measurement is for #

A few hundred systems are known well enough to give masses and radii to better than three per cent, and their value is out of all proportion to their number, because everything else in stellar astronomy is calibrated against them.

Two eclipsing stars and a distance #

There is a second use for the same systems, and it is the reason they appear in the extragalactic literature.

Once the radii are known in kilometres and the surface brightnesses are known from the spectra, the total luminosity of each star follows — and comparing that with the observed flux gives a distance, with no rungs underneath it. Two orbits of one pair, and a distance falls out is the version of this argument for a resolved visual pair; the eclipsing version needs no resolution at all and therefore works much further away.

The surface-brightness step is the weak one, and it is worth naming. It uses an empirical relation between a star’s colour and its surface brightness, calibrated on nearby stars with measured angular diameters. So the method is not quite free of calibration — but the calibration is interferometric angular diameters, which are themselves geometric, and the chain has one link where the older ladders had five.

The systems that are also clocks #

An eccentric eclipsing binary does something else useful: its apsidal line precesses, and the precession shows up as a slow drift in the timing of the secondary eclipse relative to the primary.

The rate depends on how centrally concentrated the two stars are, because a centrally concentrated star responds to its companion’s tide less than a uniform one. So a timing measurement made over decades constrains the internal density profile — a quantity that otherwise reaches the outside world only through the frequencies of a star’s own oscillations. Two independent probes of stellar interiors, and both of them are timing measurements.

The general relativistic contribution to the same precession is present too, and in the tightest systems it is comparable with the classical term. Separating the two requires knowing the stars well enough to predict the classical part, which the light curve and velocity curves supply — so an eclipsing binary can be a test of gravity in the same way Mercury’s orbit was, with the advantage that its companion is a star rather than a planet and the effect is correspondingly larger.

How many there are, and why so few #

The Kepler and TESS missions between them have catalogued tens of thousands of eclipsing binaries. The number solved to the three-per-cent standard that makes a system useful as a benchmark is a few hundred.

The gap is entirely in the spectroscopy. A light curve is cheap: a survey satellite delivers one for every eclipsing system in its field, for free, at cadence. The two velocity curves are not. Both stars have to be bright enough for their lines to be separated, which means a magnitude limit far brighter than the photometric one; the lines have to be resolvable, which fails when the components are similar and the orbit is wide; and a full orbit has to be covered, which for a long-period system means years of scheduled time on a spectrograph.

So the population that is photometrically enormous and spectroscopically tiny has the shape one would expect: nearly all the benchmarks are bright, nearby, short-period systems of comparable and unremarkable masses. The regimes where stellar models are least secure — the very low masses, the very massive stars, the metal-poor ones — are exactly where the benchmarks are thinnest.

The clock the eclipses keep #

A light curve gives radii, and the times of the eclipses give something the shapes cannot.

Eclipses recur on a strictly periodic schedule, so a table of observed minima against cycle number should be a straight line. Departures from it are measurable to seconds over decades, and each kind of departure has a cause.

A third body. If the eclipsing pair orbits a common centre with a third star, the pair is alternately nearer to and further from the observer, and the eclipses run early and late by the light-travel time across that outer orbit — up to minutes for a wide companion. The pattern is periodic at the outer orbit’s period, and fitting it gives that orbit without ever detecting the third star.

Apsidal motion. In an eccentric system the primary and secondary eclipses are not half a period apart, and if the orbit’s major axis rotates the two eclipses drift in opposite directions. The rate of that rotation is set by how centrally condensed the two stars are, plus a relativistic term — so a century of eclipse timings measures the internal structure of stars that cannot be resolved.

Mass transfer. Moving mass between the components changes the orbital period systematically, so a parabolic term in the timing diagram is a transfer rate, measured in solar masses per year without observing any material.

So the same system yields radii from the shapes and a dynamical history from the timings, and the second requires nothing but a long enough record — which is why eclipse minima observed by amateurs a century ago are still being used.

What a spot does to the shape #

The light curve is modelled as one uniform disc crossing another, and stellar surfaces are not uniform.

A cool spot is a region of the photosphere a few hundred to a couple of thousand kelvin cooler than its surroundings, covering anything from a fraction of a per cent of the disc to, on very active stars, tens of per cent. It affects the measurement in two distinct ways, and they have opposite characters.

Out of eclipse, the spots rotate in and out of view and the system’s total brightness varies at the rotation period. That modulation is a nuisance for the eclipse fit — it changes the baseline the depths are measured against — and it is a signal in its own right, giving the rotation period directly and, from its evolution, the lifetime of the spots.

During eclipse, the occulting body can pass in front of a spot and cover it. Covering a dark region raises the total light briefly, so the smooth eclipse profile acquires a small bump at the moment of crossing.

That bump is worth more than it costs. Its position within the eclipse says where on the disc the spot was, so a series of eclipses maps the surface — and if the same spot is crossed on successive eclipses, its drift measures the star’s rotation relative to the orbit. Aligned systems cross the same spots repeatedly; misaligned ones do not, which turns the presence or absence of repeated bumps into a measurement of the geometry.

So the defect in the model is also an instrument, and the ordinary situation is that a system’s activity is fitted alongside its radii rather than removed before them — which means the radii from an active star carry a systematic that a quiet one does not.

The practical rule that follows is to prefer the quietest systems for the most precise radii, and to accept that the systems most useful for measuring activity are the ones whose radii are least trustworthy.

Both facts are consequences of the same surface, and neither can be improved by observing for longer.

Where the picture stops #

Selection is severe and it is not random. A binary eclipses only if its orbit is nearly edge-on, and the probability of that goes as the sum of the radii over the separation — so the systems that eclipse are preferentially close, and close systems have been interacting. Every catalogue of eclipsing binaries is biased towards short periods, and the long-period detached systems that would test evolutionary models at the interesting ages are the ones nobody has enough of.

The temperature scale is still an assumption. The light curve gives a ratio of surface brightnesses cleanly, and the absolute temperature of either star comes from spectroscopy, which comes from model atmospheres. So a radius is model-free and a temperature is not, and any comparison with an evolutionary track is done in a plane where one axis is clean and the other is not.

A third star dilutes everything. If a fourth object falls inside the photometric aperture — a physical companion or a chance alignment — its light adds to the baseline and does not eclipse, so every eclipse is shallower than it should be and every fractional radius comes out too small. The bias is not subtle: ten per cent of third light takes about five per cent off the radii, and it takes the surface-brightness ratio with it. Space photometry makes this worse rather than better, because the pixels are large and the point spread function is wide, and it is the reason a spectroscopic search for a third set of lines is now a routine part of solving a system rather than an afterthought. The check that catches it is the same over-determination the previous section relied on: a diluted light curve fits, but it fits with a surface-brightness ratio that the spectra disagree with, and the disagreement is proportional to the contamination.

And spots ruin it. A star with a large starspot has a light curve that is not the light curve of a spherically symmetric star, and the distortion is at the per-cent level and varies over months. For active systems — which means most cool ones — the spot modelling is the dominant uncertainty in the radii, and it is the leading candidate explanation for the persistent finding that low-mass stars in binaries are several per cent larger than models predict.

Where this ladder goes next #

Later rungs on this anchor: the radius inflation problem in low-mass stars, and whether it is magnetic activity, spots or the models; eclipsing binaries containing a compact object, where the same geometry weighs a neutron star or a black hole; apsidal motion as a probe of internal structure, and its relativistic component; the use of eclipsing systems in clusters, where an independent mass and radius pin the cluster’s age and composition at once; and the extension of the surface-brightness distance method beyond the Local Group.