A transit late by the width of an orbit

A planet that transits its star is a clock, and a second planet pulling on it makes the clock run fast and slow. The wander has a shape, the shape has a period, and fitting it weighs a planet nobody has seen. Every timing variation of that kind has the same cause: a force acting on the planet that transits.

There is a second cause, and it needs no force on the planet at all. A transit is an event at the star, and the observer learns of it by light that has crossed the distance from the star. If the star is itself moving towards and away from the observer — carried round the centre of mass it shares with a distant companion — the distance changes and so does the arrival time. The planet keeps perfect time. The messenger does not.

The effect has a name older than astronomy’s knowledge of any planet beyond the Sun, because it is how light was first shown to take time to arrive.

The first transit-timing residual #

Io circles Jupiter every 42.5 hours and disappears into Jupiter’s shadow once each orbit, so its eclipses are a clock anyone with a telescope and a pendulum can read. In the 1670s Ole Rømer, working at the Paris Observatory, noticed that the clock did not keep time. Eclipses observed while the Earth was moving towards Jupiter came progressively earlier than a uniform table predicted, and eclipses observed while it was moving away came progressively later. The residuals did not depend on where Io was in its orbit, or on where Jupiter was in its own. They depended on where the Earth was.

The explanation Rømer gave, in 1676, was that light takes time to cross the Earth’s orbit. He predicted that an eclipse in November of that year would come some ten minutes later than the tables said, and it did. It was Christiaan Huygens who turned the delay into a speed, taking Rømer’s estimate of about twenty-two minutes for light to cross the diameter of the orbit, which is too long by a third, together with a solar distance that was itself poorly known. The number that came out was about seventy per cent of the modern value. The existence of a finite speed was the result; a good value had to wait for James Bradley’s aberration of starlight half a century later.

The error in Rømer’s twenty-two minutes is instructive rather than embarrassing. Io’s eclipses are not instantaneous: the moon fades into the shadow over minutes, and whether an observer timed its disappearance or its reappearance depended on which side of Jupiter the shadow fell, which itself changes with the Earth’s position. A systematic that follows the Earth round its orbit is exactly the signal being measured, so it could not be averaged away, only modelled. The same is true of the stellar version below: anything that changes with the companion’s orbital phase — spots on the star, a changing blend of light — imitates the delay.

Read as an exoplanet astronomer would read it, Rømer’s measurement is a transit-timing residual: an O−C diagram whose wander has a period set by something other than the clock, and whose amplitude is a length divided by the speed of light. The modern version turns it round. The speed of light is known exactly, so the delay measures the length.

A star carried round its own barycentre #

Suppose a star with a transiting planet has a companion a long way out — another star, a brown dwarf, a giant planet on a decades-long orbit. Neither body of a bound pair is still: the host star orbits the pair’s centre of mass on an ellipse scaled down by the mass ratio,

$$a_\star = a\,\frac{m_c}{M_\star + m_c},$$

and the component of that motion along the line of sight changes the distance its light has to travel. The transit times acquire a delay equal to that line-of-sight position divided by the speed of light. For an orbit of eccentricity $e$, argument of periastron ω and inclination $i$, the delay swings by

$$\Delta t = \frac{a_\star \sin i}{c}\sqrt{1 - e^2\cos^2\omega}$$

either side of its mean, a formula written down by John Irwin in 1952 for eclipsing binaries, whose eclipses were the first stellar clocks in which the effect was deliberately sought. Two stars in orbit about each other are themselves a clock, and a third star moving the pair round a wider orbit shifts every eclipse together; Algol’s own eclipses carry a light-time term with the 680-day period of its third star.

The curve is not a sinusoid. An eccentric companion carries the star quickly through periastron and slowly through apoastron, and the delay spends most of its time near one extreme and swings through the other in a hurry. The asymmetry is the eccentricity, the phase of the asymmetry is ω, and both are recovered from the shape alone — the same inversion that turns the velocity curve of a star into an orbit, applied to position rather than to speed. A delay is the integral of a radial velocity divided by $c$, so the two measurements are one orbit read at two derivatives.

What is not recovered is the inclination on its own. The delay contains $a_\star \sin i$, and exactly as with a velocity curve, the mass that comes out is a minimum mass unless something else supplies the angle. For a companion that is also resolved in images, its motion across the sky supplies it; for one that is not, the minimum is all there is.

A baseline shorter than the orbit #

The opening example had ninety years of transits, more than two orbits of the companion. No transit survey is that long. The first space photometry of transiting planets is less than two decades old, and a companion on a forty-year orbit shows a survey only a fraction of its delay.

The period that absorbed the delay is not an error in the ordinary sense. It is the Doppler shift of a clock carried at the star’s line-of-sight speed. A clock approaching at velocity $v$ has its ticks compressed by $v/c$, and 1.140 seconds out of a 4.2-day period is a fractional shift of $3.1\times10^{-6}$, which is a speed of 0.94 kilometres a second — the star’s velocity towards the observer, averaged over the twelve years. No timing measurement over a short baseline can separate a planet’s true period from that shift, because a steady velocity changes nothing else about the signal.

A clock can never measure its own constant velocity; it can only measure its acceleration. That is the principle a timing series over a partial orbit rests on, and it has a useful consequence. Every planet in the same system is carried at the same speed, so every period is shifted by the same factor, and period ratios survive untouched. A pair’s distance from resonance, which is what the size of its gravitational timing variations depends on, is immune to the companion.

It is also how spacecraft are navigated. A position measured from a frequency is a Doppler shift integrated over a tracking pass, and a transmitter whose clock rate is uncertain cannot be distinguished, over a short pass, from a transmitter moving at a slightly different speed; tracking stations solve for both by exploiting the acceleration the Earth’s rotation imposes on the line of sight each day. The exoplanet version has no such daily lever, only the slow curvature of the companion’s orbit.

The same arithmetic applies to the observer’s own motion. Before any exoplanet transit time is compared with another, it is corrected for the Earth’s position in its orbit — Rømer’s sixteen minutes, removed — and for the Sun’s motion about the solar system’s barycentre, a few seconds dominated by Jupiter. The corrected times are expressed in a time scale defined at the barycentre rather than at the telescope, and a timing residual left over after that correction is a statement about the target’s barycentre, not about the Earth’s. The ephemerides that supply the Earth’s position are themselves fitted to observations corrected for the time light took to arrive, so the correction for light travel is present at both ends of every measurement in this essay.

What a companion has to be #

The size of the delay depends on two things a companion might have, and the combination rules out most of what the word “planet” suggests.

The delay is a lever: the star’s orbit is the companion’s orbit scaled by the mass ratio, so the delay rises with separation. The orbital period rises faster, as the three-halves power, and a survey sees a full cycle only while the period is shorter than its baseline. Between those two constraints lies a narrow window. A Jupiter at 5 AU, the Sun’s own companion, delays the Sun’s clock by 2.5 seconds over twelve years, below the timing precision of any transit. A companion of stellar mass at 10 AU delays it by minutes and completes its orbit within a survey’s life.

The Sun is a useful test of the arithmetic, seen from outside. An observer on a planet of another star, timing the Earth’s transits across the Sun, would find them delayed by Jupiter’s pull on the Sun by ±2.5 seconds over twelve years, and by Saturn’s by a further ±1.4 seconds over twenty-nine and a half. The two terms beat against each other, since their periods are nearly in the ratio of five to two, and the Sun’s own path about the barycentre loops between about one and two solar radii from it. None of that would be visible in the Earth’s transits, each of which lasts thirteen hours and could be timed to perhaps a minute. The solar system’s giant planets, which dominate its dynamics, would be invisible to this method from any distance; what the method sees is companions a hundred times heavier.

So the objects found this way around transiting-planet hosts are stars and brown dwarfs. That is not a disappointment. A distant stellar companion shapes a planetary system’s dynamics — it can drive the slow exchange of eccentricity and inclination that delivers hot Jupiters to their stars — and a timing series that finds the companion finds the thing that may have put the transiting planet where it is.

A brown dwarf closer in is the marginal case.

The same effect has found planets, just not around stars with transiting planets. A pulsar is a clock far better than any transit, keeping time to microseconds, and the first planets known outside the solar system were found in 1992 as periodic delays in the pulses of PSR B1257+12 — the pulsar carried round the barycentre of its planets, the smallest of them about twice the mass of the Moon. That was a Rømer delay measured on a clock read against a model of everything between it and the telescope, at a precision that makes a planet’s lever arm visible. Eclipse timings of close binary stars have been used the same way, and there the history is a caution: several planets proposed around evolved binaries from their eclipse timing predicted future eclipses that then arrived at the wrong times, and the proposed systems turned out to be dynamically unstable within a few thousand years. A light-time model with enough companions will fit any wander, and the test of such a fit is the next eclipse rather than the quality of the last.

Two planets late together, and two late in turn #

A timing wander on its own does not say whether it is a delay in the light or a pull on the planet. In a system with two transiting planets the distinction is immediate, and it is the test that makes the effect usable.

The light-travel delay belongs to the star, so it shifts every planet’s transits by the same amount at the same moment, and two planets’ residuals lie on top of each other. The gravitational variation of a near-resonant pair is an exchange: the pair trades orbital energy back and forth, one planet’s period lengthening while the other’s shortens, and their residuals move in opposition. Correlated wander is light; anti-correlated wander is gravity. With a single transiting planet the test is unavailable, and the shape of the curve over a full cycle — a Keplerian orbit’s asymmetry against a resonant sinusoid’s symmetry — is all that is left to decide on.

That test extends to anything else that shares the star. Radial velocities of the host show the companion’s pull directly, and a light-travel delay of $\Delta t$ corresponds to a velocity semi-amplitude of about $2\pi c\,\Delta t / P_c$ for a circular orbit: 16 minutes over 38 years is a kilometre a second, easily measured. A gravitational timing variation from a nearby planet corresponds to a velocity signal at that planet’s period instead. The two measurements of the same system therefore disagree about period in the one case and agree in the other.

What was actually measured #

The cleanest case so far is a hot Jupiter whose clock appeared to be running fast. WASP-4b orbits its star every 1.34 days, and more than a decade of transit times showed its period shortening by about ten milliseconds a year — the signature expected if tides were draining its orbit into the star, which would have made it the second planet known to be spiralling in. Velocities of the star, gathered over the same years, showed the star itself accelerating towards the observer at a few centimetres a second per day. An acceleration of that size shifts a clock’s apparent period at exactly the rate the transits showed, because of the Doppler argument above, and it left no room for orbital decay at the claimed rate: the star was being pulled by a companion further out, and the planet’s orbit was not measurably shrinking at all.

WASP-12b is the counterexample, and the pair is what makes the method credible. Its transits also run early, faster, and its star’s velocities show no comparable acceleration, so its period change is attributed to genuine decay. Two planets with the same symptom received opposite diagnoses, and in each case the deciding measurement was the star’s velocity, not the transits.

The precision that matters is not the precision of one transit time but of the whole series. A delay with a semi-amplitude of one minute, sampled by a hundred transits each timed to a minute, is detected at many standard deviations; the same minute sampled by five transits is noise. Rømer had the same advantage: Io eclipses every 42.5 hours, and a delay built up over months was accumulated over a hundred eclipses.

What the delay leaves out #

The planet’s own orbit is held fixed. The figures treat the companion as moving the star and nothing else. A companion also perturbs the planet directly, through its tidal field, and at the separations drawn that effect is a small fraction of the light-travel delay — but it grows with the companion’s mass and falls with the cube of its distance, so it is not negligible for a close companion.

The companion’s orbit is Keplerian. Over decades the star’s orbit about the barycentre is itself perturbed by anything else in the system, and a fit that assumes a single Keplerian companion will mistake an extra body for a changing eccentricity.

The delays are geometric only. At the precision of pulsar timing the gravitational delay of the light as it climbs out of the companion’s potential, and relativistic corrections to the clock itself, enter as well; at the precision of transit timing they do not.

Still open: how a quadratic trend is split between an orbit shrinking and a star accelerating #

A transit-timing series over a few years that curves away from a straight line can mean that the planet’s period is really changing — tides draining its orbit — or that its star is accelerating along the line of sight under a companion’s pull. Both produce a parabola in the same diagram. The velocities of the star separate them in principle, since a decaying orbit moves no star; in practice the acceleration needed to imitate orbital decay is a few metres a second per decade, near the limit of velocity measurements sustained that long, and the stars most worth asking are faint. For the handful of hot Jupiters whose transit times are curving, which half of the explanation holds is decided by instruments that have not yet been pointed at them for long enough.