A duration that measures an eccentricity

A transit light curve contains more than a depth. It contains a duration, and a duration is a speed: the planet crossed a chord of known length in a measured time, so its transverse velocity is known.

For a circular orbit that velocity follows from the period and the orbital radius, and the orbital radius follows from the period and the star’s mass. Rearranged, the duration gives the star’s mean density — a stellar property, measured from a planet’s shadow, with no spectroscopy in it at all.

That is a remarkable enough result on its own, and it is worth pausing on how unusual the route is: a property of a star, inferred from the shadow of an object a thousandth of its size, using nothing but a clock. What makes it more useful is what happens when the orbit is not circular.

A planet is measured by the light it removes, and the amount removed gives a radius ratio. This essay is about the other number the same light curve contains, and about the fact that it is a measurement of the star until something else measures the star, at which point it becomes a measurement of the orbit.

Where the factor comes from #

Kepler’s second law says the planet sweeps equal areas in equal times, so its speed varies around the orbit as the inverse of its distance from the star. A transit happens at one particular point on the orbit — the one where the planet passes in front — and the speed there is set by the distance there.

The distance at transit depends on the eccentricity and on where the orbit’s perihelion points relative to the line of sight, which is the argument of periapsis $\omega$. Writing it out,

$$\frac{T}{T_{\rm circ}} = \frac{\sqrt{1-e^2}}{1 + e\sin\omega},$$

which is a factor of one when the orbit is circular and can be anything from $\sqrt{(1-e)/(1+e)}$ to its reciprocal otherwise.

Two properties of that expression matter. The extremes are exact reciprocals, so the circular duration is the geometric mean of the longest and shortest possible transits at a given eccentricity. And the effect is first order in $e$, not second — a modest eccentricity of 0.2 changes the duration by up to twenty per cent, which is enormous compared with the precision a duration is measured to.

It is worth writing the density relation out once, because it is short and it shows where the assumption enters. Combining Kepler’s third law with the geometry of a central transit gives

$$\rho_\star \approx \frac{3\pi}{GP^2}\left(\frac{a}{R_\star}\right)^3,$$

and $a/R_\star$ comes from the transit’s duration and depth. Everything on the right is measured from the light curve and the period; nothing about the star’s spectrum, distance or brightness appears. That is why a transit is a stellar density measurement, and it is the reason space photometry produced better densities for a hundred thousand stars than spectroscopy ever had. The eccentricity enters as a correction to the geometry, and it is exactly the correction this essay is about.

Turning it into a measurement #

The duration anomaly is a measurement of $e\sin\omega$, and that requires knowing what the circular duration would have been — which requires the stellar density.

Three routes supply it independently of the transit.

Asteroseismology. The large frequency separation of a star’s oscillations is proportional to the square root of its mean density, so a light curve long enough to resolve the oscillations gives the density to a per cent or better. That is the best route and it applies only to bright stars.

A parallax and a spectrum. A parallax gives the luminosity, an effective temperature gives the radius through the Stefan–Boltzmann law, and a mass from a spectrum or an evolutionary model gives the density. That is available for essentially every transiting planet host now, at a precision of some ten per cent.

A stellar model. Fitting the spectroscopic temperature, gravity and metallicity to an evolutionary track gives a density with a model attached, at similar precision and with correlated systematics.

Which of the three is used decides what the measurement is worth, and the ordering is instructive. Asteroseismology gives one per cent and applies to a few hundred systems; parallax plus spectroscopy gives ten per cent and applies to all of them; a model gives ten per cent with correlated errors and applies to all of them. Since the eccentricity’s uncertainty is inherited almost entirely from the density, the method delivers eccentricities good to a few thousandths for the seismic subsample and to a few hundredths for everything else — the same asymmetry the seismic scaling relations produce, where a small subsample measured superbly calibrates a large one measured adequately.

There is also a fourth route that deserves mention because it inverts the logic and is now common. In a system with several transiting planets, all the transits must be consistent with one stellar density. That is an over-determined system: several durations, one density, and one eccentricity per planet. Solving it gives the density and the eccentricities together, with no external stellar measurement at all, and the consistency requirement is what makes it work. It is the same manoeuvre as requiring two stars in a binary to have one age — the information comes from the shared parameter rather than from the precision of any single measurement.

What it cannot separate #

The method has two structural limitations, and both are geometric.

It measures $e\sin\omega$, not $e$. A planet whose perihelion points across the line of sight — $\omega = 0$ or $180°$ — transits at the semi-latus rectum, where its speed is the circular value, and shows no duration anomaly however eccentric it is. So a null result is not a circular orbit; it is an orbit whose eccentricity vector is perpendicular to the line of sight.

The impact parameter enters too. A transit’s duration depends on the chord’s length as well as on the speed, and the chord depends on how centrally the planet crosses. A high impact parameter shortens the transit exactly as a perihelion passage does, and disentangling them requires the light curve’s shape — the ingress duration relative to the total, which is sensitive to the impact parameter and not to the speed.

When the duration changes #

Everything so far treats the duration as a fixed property of the system. It is not, and the way it drifts is a third measurement hiding in the same light curve.

The duration depends on the impact parameter, and the impact parameter depends on the orbit’s orientation relative to the line of sight. Anything that turns the orbital plane therefore changes the duration slowly over many transits. Two things do: a companion whose gravity precesses the node, and an oblate star whose quadrupole does the same. Neither shifts the transit times appreciably; both change how long the transits last.

The effect is small and it accumulates. A nodal precession of a thousandth of a degree per year moves the impact parameter by a few thousandths, which changes the duration by seconds — undetectable in one transit and unmistakable in a decade of them. A handful of systems now show measured duration drifts of a minute or two per decade, and in each case the interpretation is a plane that is slowly turning.

The measurement is worth having because it constrains things nothing else reaches. A duration drift from an oblate star measures that star’s quadrupole moment and the angle between its spin and the orbit at once — the same two quantities an obliquity measurement gets from a line profile, obtained from photometry over years instead of from spectroscopy in one night. A drift from a companion measures the companion’s mass and inclination in a regime where it may be producing no other signal at all.

There is a sharper version for systems seen at high impact parameter. Near the grazing limit the duration depends steeply on the impact parameter, so a small change in the plane produces a large change in the transit — and in a few cases the planet has precessed out of transit entirely, or into one, within the span of the observations. A planet that stops transiting is an awkward thing to publish and it is an extremely precise measurement of a precession rate.

And it separates the two blind spots. The duration anomaly measures one component of the eccentricity vector and the secondary eclipse’s timing measures the other; a duration drift measures neither and instead constrains the orbital plane, which is the quantity both of the first two assume is fixed. Three quantities from one photometric series, distinguished by whether they appear in the duration, in the timing, or in the derivative of the duration. That is a good deal of orbital information from a signal that is a dip of a per cent.

What was actually measured #

The technique was proposed in the 1980s and became practical when space photometry produced tens of thousands of transits with durations measured to minutes.

The population’s eccentricity distribution. Applying the method statistically to a large sample gives a distribution of $e$ without measuring any individual orbit well. The answer for small planets in multi-planet systems is that eccentricities are small — of order 0.03 — and for single transiting planets rather larger, which is one of the observational statements underpinning the idea that compact multiple systems are dynamically cold.

Individual eccentric planets. For a handful of systems with asteroseismic densities and long-period transiting planets, the eccentricity comes out to a few hundredths and agrees with radial-velocity measurements where both exist. That agreement is the validation of the method.

And the circularisation boundary. Because the method works on planets too faint for radial velocities, it extends the measurement of where tidal circularisation has taken hold to a much larger sample, and confirms the sharp transition at a few days that the tidal theory predicts.

Where the picture stops #

The picture stops in three places, and the second is the one that biases populations.

The stellar density is the limiting term. At a ten per cent density the eccentricity’s uncertainty is of order 0.05 for a favourable orientation, which is not competitive with radial velocities for a bright star and is the only option for a faint one. Improving it means improving the star, not the photometry.

A null is not a circle. Because the method is blind to $e\cos\omega$, a population analysis has to marginalise over orientation, and orientations are not uniformly sampled: a planet is more likely to transit if it passes close to the star, which favours transits near perihelion. So the transiting sample is biased towards the orientations that produce short transits, and correcting for that requires assuming the eccentricity distribution being measured.

And the eccentricity itself is bounded below by zero. Fitting an eccentricity from a duration anomaly runs into exactly the boundary problem that makes a fitted eccentricity refuse to be zero, with the extra complication that the constrained combination is not the modulus but one component of the vector. The right procedure is to fit in $(e\cos\omega, e\sin\omega)$ and report a posterior, which is what modern analyses do and what the early ones did not.

A fourth sits alongside them because it changes what the method measures for a particular class of system. A planet with a large radius ratio — a hot Jupiter around a small star — has a transit whose duration is affected by the planet’s own size, since the chord is traversed by a disc rather than a point. The correction is calculable and it is degenerate with the impact parameter in the same way, and for the deepest transits it is a per cent-level effect on the inferred density. That is small compared with the eccentricity signal and not small compared with an asteroseismic density, so for the best-measured systems it is a term that has to be carried.

There is one more result that the technique produced and that would have been hard to get otherwise. Applying it to the planets found around evolved stars — subgiants and low-luminosity giants, whose densities are known well from seismology — showed that their orbits are systematically more circular at a given period than those of planets around dwarfs. That is the tidal circularisation timescale doing its work over the star’s expansion, and it is a measurement of tidal dissipation in the planet’s host rather than in the planet. Nothing about it required a radial velocity, which for a giant would have been swamped by the star’s own surface motion.

Why a duration is a richer observable than a depth #

The general lesson is worth separating out, because it applies well beyond transits.

A depth is a ratio of areas and a duration is a ratio of times, and times are measured better than fluxes. A transit depth is limited by photometric precision, which is limited by the star, the atmosphere and the detector. A transit duration is limited by the timing of two events, which for a well-sampled light curve is far better. So the duration carries more information per unit of observing effort than the depth does, and it carries a different quantity.

That is the same asymmetry that runs through the whole of this collection: a frequency is measured better than an amplitude, a phase better than a level, an interval better than a flux. Whenever a measurement can be arranged so that the wanted quantity appears in a time rather than in a brightness, it should be.

The photoeccentric effect is an unusually clean example because the same light curve delivers both. Nothing extra is observed. What changed was noticing that the duration, which had been used to check the fit, was a measurement of a quantity nobody had been extracting.

A closing observation about the history, because it is a good example of an observable being reinterpreted rather than discovered. Transit durations were measured from the first transiting planet onward, and they were used as a consistency check — does the duration agree with the density expected for a star of this type. Disagreements were attributed to the stellar parameters, which were poorly known. When the stellar parameters became well known, the same disagreements became a measurement of something else. Nothing about the data changed; what changed was which quantity in the relation was the uncertain one. The same reversal happened to eclipse timings, where a residual that had been a nuisance became a detection method.

Be explicit about how much of the transiting population this reaches, because the method’s value is statistical rather than individual. Radial velocities give an eccentricity for a planet bright enough and massive enough to move its star measurably, which is a few thousand systems and is biased towards large planets around bright stars. The photoeccentric method gives one — weakly — for every transiting planet with a stellar density, which is tens of thousands and is biased only by the transit probability. So the two techniques measure the same quantity for almost disjoint samples, and the population-level statements about eccentricity in the small-planet regime come entirely from the second. The planets that were not seen is the other half of that accounting: what a survey could have detected decides what a distribution means, and here the detection criterion and the measurement are the same light curve.

Note too that the measurement costs nothing at the telescope. Every quantity in it was recorded the moment the transit was observed, and the analysis is a reinterpretation of an archive rather than a new observation — which is why the population-level results arrived within a year of the stellar densities improving, and why they will improve again whenever the stars do.

Where the ladder goes next #

The next rung is the light curve’s shape rather than its extent: how the ingress and egress durations separate the impact parameter from the speed, and what a limb-darkening law contributes to that separation. The one above that is the same measurement made on a system with several transiting planets, where the durations must all be consistent with one stellar density — an over-determined system that measures the eccentricities and checks the star at the same time.