Three causes with three shapes in one curve

Eclipsing binaries are the only stars whose masses are known directly, and the reason is that they present two independent geometrical observables. This essay is about a third one that arrives for free and is usually thrown away.

An eclipsing binary produces a sharp, repeatable event every orbit, and the time of that event can be measured to seconds. Over a century of observations that is a clock with ten million ticks and a precision of a part in a hundred million, and it is the most sensitive instrument in stellar astronomy for anything that changes an orbit slowly.

What such a clock produces is a residual: the observed time of each eclipse minus the time a constant period predicts. The plot of that residual against cycle number — the O−C diagram — is where a century of binary-star physics has been read.

The reading is not straightforward, and the reason is that several physically unrelated processes all produce residuals of a few thousandths of a day over decades. Telling them apart is the whole of the technique, and it is done entirely by shape.

Why the shapes are different #

The separation works because each cause enters the timing through a different mechanism, and mechanisms have functional forms.

A wrong period gives a straight line. If the assumed period is too short by $\delta P$, each successive eclipse is late by a further $\delta P$, so the residual accumulates linearly in cycle number. This is not physics, it is a bookkeeping error, and it is removed by refitting the ephemeris.

A changing period gives a parabola. If the period itself is drifting at a rate $dP/dE$ per cycle, the accumulated residual is the integral of a linear function, which is quadratic. Mass transfer between the components changes the period; so does angular momentum lost to a magnetised wind; so does the shrinking of the orbit by gravitational radiation. All three produce parabolas, differing only in sign and coefficient.

A third body gives a sinusoid. A distant companion moves the eclipsing pair about the system’s centre of mass, so the light from the eclipses has a varying distance to travel. The residual is the light-travel time across the pair’s orbit about the barycentre — a sinusoid whose period is the third body’s period and whose amplitude is the projected orbit size divided by the speed of light.

Apsidal motion gives an antiphase pair. If the eclipsing orbit is eccentric, the two eclipses are not half a period apart, and how far from half depends on the orientation of the orbit. As the apsides precess, the primary minimum moves one way and the secondary the other. That is the one signature that changes sign between the two eclipses, and it is the reason both have to be timed.

There is a fifth entry that is really the absence of one: nothing in this list produces a step. A discontinuity in the residual — the eclipses suddenly a minute late and staying there — has no dynamical cause at all on the relevant timescales, and when one appears it is a data problem: a change of observer, of filter, of timing convention, or an error in converting between time scales. Recognising that a step is never physics is one of the more useful pieces of pattern recognition the technique offers.

What the amplitudes are worth #

Each shape carries a physical quantity, and the quantities are of quite different kinds.

The parabola’s curvature is $dP/dE$, and dividing by the period gives a fractional rate of change. For a semi-detached system transferring mass the rate is set by the mass-transfer rate and the mass ratio, so the timing measures a flow of material nobody can see. For a detached system it measures angular momentum loss, and it is one of the few direct handles on magnetic braking.

The sinusoid’s amplitude is a length divided by $c$: the projected semi-major axis of the eclipsing pair’s orbit about the barycentre. Combined with the period it gives a mass function for the third body, in exactly the way a radial-velocity curve does — and with the same limitation, that the inclination is unknown and the mass is a lower bound.

The apsidal term’s period is the precession period, and that is the most interesting of the three, because the precession rate depends on how centrally condensed the two stars are. A point mass causes no precession; an extended star has a quadrupole that does. So an O−C diagram measures a structural property of a star’s interior, from nothing but the times of its eclipses.

It is worth putting the amplitudes side by side. A third body of a tenth of a solar mass in a twenty-year orbit gives a light-travel amplitude of a few thousandths of a day — a few minutes. A mass-transfer rate of $10^{-8}$ solar masses a year in a close binary gives a period change that accumulates to a similar amount over fifty years. An eccentricity of 0.1 with a fifty-year apsidal period gives a few thousandths of a day as well. The three are the same size for entirely unrelated reasons, which is precisely why the shapes have to do the work: nothing about the magnitude of a residual says which cause produced it.

One more amplitude belongs on the list because it is the smallest one anybody has measured and the most important. General relativity contributes to the apsidal motion of a close binary in addition to the classical tidal and rotational terms, and for the tightest systems the relativistic contribution is the larger of the two. Separating them requires knowing the stars’ internal structure, which is what the measurement is for — so the analysis is a simultaneous solution rather than a subtraction, and the relativistic term is included from theory rather than fitted. That is a rare case in this collection of a relativistic correction being an input to a measurement rather than its output.

Where it goes wrong #

The technique’s reputation is mixed, and the reasons are worth stating because they are all versions of one problem: sinusoids are easy to fit and hard to believe.

A period change and a long-period sinusoid are nearly degenerate. Over a baseline shorter than the sinusoid’s period, half a cycle of a sine is indistinguishable from a parabola. Since the observing baseline is a century and the third-body periods claimed are often decades, this is the normal situation rather than an edge case.

The data are heterogeneous. A century of eclipse timings contains visual estimates by eye, photographic minima, photoelectric measurements and modern CCD photometry, with precisions spanning three orders of magnitude and systematic offsets between methods. Fitting them together requires weights, and the weights are estimated rather than measured.

Spots move the minima. A star with a large spot has an asymmetric light curve, so the fitted time of minimum is displaced. As the spot evolves and the star rotates, the displacement wanders on the timescale of the activity cycle — which is years to decades, exactly the range where third bodies are claimed.

The last of these is the leading explanation for a class of results known as the Applegate mechanism: cyclic period changes in active binaries, of the right period and amplitude to be third bodies, that are more plausibly the star’s own magnetic cycle redistributing angular momentum inside it and changing its quadrupole moment.

There is a fourth difficulty and it is statistical rather than physical: the residuals are not independent. A timing error in the reference epoch propagates into every subsequent residual, and a small error in the assumed period produces a linear trend that is then partly absorbed by whatever is being fitted. Fitting a sinusoid to residuals that have already had a linear and quadratic term removed is not the same as fitting it to the raw times, and the covariance between the terms is large. Published significances for timing detections have historically been computed as though the points were independent, which they are not — the same error as assuming white noise in a periodogram, in a different setting.

What was actually measured #

Three results, of three different kinds, and each is what one of the three shapes was worth.

Apsidal motion and stellar interiors. For about fifty eclipsing binaries the apsidal period has been measured, giving the internal structure constant $k_2$ — the same quantity a tidal Love number describes. Comparing against stellar models was a long-standing discrepancy: observed stars appeared more centrally condensed than models predicted, by amounts of tens of per cent. The discrepancy largely disappeared when relativistic apsidal motion was included and when models with convective overshoot were used, and what is left is one of the few observational constraints on internal structure that does not go through the surface.

The orbital decay of a compact binary. For a pair of neutron stars the parabola’s coefficient is the gravitational-wave luminosity, and the measured value agrees with general relativity to a fraction of a per cent — an orbit measured to be shrinking at exactly the predicted rate. That is the same technique applied to pulse arrival times rather than eclipse times, and it is the most precisely confirmed prediction in gravitational physics.

Third bodies, some of which are real. Circumbinary companions found by eclipse timing have in several cases been confirmed by radial velocities or by direct detection. In many other cases they have not, and the field’s own assessment is that a timing detection alone, without a physical check, is a hypothesis rather than a discovery.

Where the picture stops #

Three, and the second is the one that decides what is believable.

The technique measures the derivative of a slow thing. Any effect that changes the period on a timescale longer than the observing baseline is absorbed into the ephemeris and vanishes. There is no way to detect a companion with a period of five hundred years from a century of data, however precise.

Nothing in the diagram is a detection on its own. Every shape has more than one physical cause, and the causes are separated by their functional forms, which are all smooth and all fitted with few parameters. A residual with three free sinusoids in it can be fitted well by many combinations. The rule the field has converged on is that a timing signal is a candidate until something else confirms it.

And the timings themselves are model-dependent. A “time of minimum” is the output of a fit to a light curve, and different fitting methods — a parabola through the bottom, a bisector of chords, a full light-curve model — give times differing by more than their formal errors when the curve is asymmetric. A century of timings measured by half a dozen conventions has systematic steps in it at the boundaries.

A fourth limit is worth adding because it is the one that has grown in importance. Space photometry produces eclipse times of extraordinary precision — seconds rather than minutes — but over baselines of a few years rather than a century. A short, precise series and a long, coarse one constrain different things: the first sees short-period effects and the second sees secular ones, and combining them requires reconciling their systematic offsets, which are larger than the modern errors. The best-determined systems are those where a century of visual and photographic timings has been tied to a few years of space data through a decade of overlapping ground-based CCD photometry, and the tie is the weakest link.

Why the shapes matter more than the amplitudes #

The general lesson is one this collection meets in every field where a residual is being interpreted.

A residual with a single unexplained wiggle is uninformative, because any number of causes could produce a wiggle. A residual whose shape is diagnostic is a measurement, because the shape excludes causes. Everything valuable in eclipse timing comes from the fact that four physically distinct effects impose four mathematically distinct forms, and that one of them — the antiphase behaviour of apsidal motion — is unique to a single cause.

That is the same argument as separating a spectral line’s shape from its position and as reading the Shapiro delay’s logarithmic dependence out of a navigation residual. In each case the discriminating information is in a functional form rather than in a magnitude, and in each case the observation has to be designed to expose it — by timing both eclipses, by measuring line by line, by observing near conjunction.

The corollary is the discipline the field learned the hard way: an observation that collapses the diagnostic shape is not a cheaper version of the full one, it is a different and much weaker measurement. Timing only the primary minimum halves the observing cost and removes the ability to distinguish a third body from a turning orbit entirely.

One further observation about why the technique survives despite its poor reputation. The reason is arithmetic: a timing measurement’s sensitivity grows as the square of the baseline for a period change and linearly for a light-travel effect, while nothing about the instrument has to improve. A century-old visual estimate of an eclipse time, accurate to five minutes, still contributes usefully to a modern parabola fit, because it is a century away. Very few measurements in astronomy have that property — most old data are superseded rather than incorporated — and it is why amateur timings, plate archives and nineteenth-century visual estimates are still being mined.

It is also worth being explicit about what makes the parabola trustworthy where the sinusoids are not. A parabola has one free parameter beyond the ephemeris, it is monotonic in its second derivative, and no plausible instrumental effect produces one — a drift in timing convention gives a step or a linear trend, spot activity gives a wandering wiggle, and neither integrates to a clean quadratic. So a well-sampled parabola over a long baseline is about as robust as this kind of measurement gets, which is why the shrinking orbit of a compact binary is quoted as a precision test of gravity while a third body from the same technique is quoted as a candidate.

One practical caution belongs with all of this, and it concerns how long a baseline has to be before a shape can be claimed. A parabola, a sinusoid and a straight line are indistinguishable over a short enough interval — every smooth function is a straight line locally — so a decade of timings can be fitted equally well by all three hypotheses, and the fitted parameters will differ wildly between them. What separates them is curvature accumulated over a span comparable to the timescale of the effect, which for a third body is its orbital period and for a period change is decades. That is why the historical archive matters so much here: a photographic plate from a century ago contributes more to distinguishing the three shapes than a year of modern photometry, and several of the best-characterised systems rest on measurements nobody alive made.

One final practical note: the three shapes are separable in principle and correlated in any real fit, because a finite baseline of noisy timings does not constrain them independently. The right output is therefore a joint posterior over all three amplitudes rather than three separately fitted numbers, and the difference matters most for the smallest of them — which is usually the one somebody wants to claim.

Where the ladder goes next #

The rung above this one is the apsidal constant itself: what a precession rate says about how a star’s mass is distributed, and why the comparison against models was in disagreement for forty years. Above it again sits the same clock applied to a system with a planet rather than a star — a transit that runs late, which is the same three shapes at a thousandth of the amplitude.