A planet measured by the light it removes

A planet passing in front of its star removes some of the light. That is the entire measurement, and it is worth being exact about how little it contains: a number, repeated at intervals, describing how much dimmer a point of light became.

From that one number comes the radius of a body nobody has resolved, orbiting a star nobody has resolved either.

The depth is an area, and areas are what a silhouette takes #

A star of radius $R_\star$ presents a disc of area $\pi R_\star^2$. A planet of radius $R_p$ in front of it covers $\pi R_p^2$. If the star’s disc were uniformly bright, the fraction of light removed would be the ratio of those areas:

$$\delta = \frac{\pi R_p^2}{\pi R_\star^2} = \left(\frac{R_p}{R_\star}\right)^2.$$

Two squares, and the constant cancels. A planet a tenth of its star’s radius removes a hundredth of its light. Jupiter in front of the Sun would be 1.06 per cent; the Earth in front of the Sun is 84 parts per million.

The cancellation is worth pausing over, because it is the source of both the method’s power and its central limitation. Nothing in the expression is a length. The depth does not know how big the planet is; it knows the ratio, and it will report the same 1.05 per cent for a Jupiter crossing a Sun as for a body the size of the Earth crossing a star the size of Neptune’s orbit — if such a thing existed. Getting from the ratio to a radius requires knowing $R_\star$, and $R_\star$ is a property of a star that has never been resolved either.

So the first thing a transit measures is a comparison. This is the field’s habitual position rather than an unlucky one: a parallax is an angle, a colour is a temperature only through a model, and a brightness is a distance only if something is already known. The transit is the same bargain, made unusually cleanly.

The geometry is a chord across a disc #

Seen on the sky, the planet crosses the star along a straight line — a chord. Which chord it is gets its own symbol, the impact parameter $b$, the perpendicular distance from the star’s centre to the path, in units of the stellar radius. A central crossing has $b = 0$; a crossing that just grazes the limb has $b$ approaching 1.

The chord’s length follows from Pythagoras and nothing else: $2\sqrt{1 - b^2}$ in stellar radii. What matters is that this is a shorter path than the diameter, so a transit at high impact parameter takes less time — and the light curve records the time.

The two discs are tangent, externally or internally, at four moments. Astronomers number them first to fourth contact, in the order they happen. Between first and second the planet is partly on and partly off the disc, so the light falls; between second and third it is wholly on, so the light is steady; between third and fourth it comes off again. The flat middle and the sloped shoulders in the light curve are those intervals, and the ratio of their durations is a measurement in its own right, taken up on the next rung of this ladder.

The same geometry, seen from inside #

There is nothing exotic about any of this. It is the geometry of an eclipse with the observer moved.

A solar eclipse is a transit of the Moon across the Sun, observed from inside the shadow. The Moon’s angular radius happens to be close to the Sun’s, so the “depth” is essentially unity and the phenomenon is spectacular. A transit of Venus, watched from Earth, is a transit in exactly the sense meant here: Venus covers 0.1 per cent of the solar disc and the Sun dims by that much, which nobody notices without an instrument. The transits of Venus in 1761 and 1769 were the largest coordinated scientific expedition of the eighteenth century, mounted to measure the timings of exactly these contacts from widely separated places on Earth, and thereby the scale of the solar system. The method was Halley’s, the observations were fought for at appalling cost, and the arithmetic was defeated by an effect nobody had anticipated: at second and third contact the planet’s dark disc appears to cling to the limb by a black ligament, so the moment of tangency cannot be timed to better than tens of seconds. The same contacts, measured on a distant star with a photometer instead of an eye, are what the whole of this field now rests on.

The range of depths is the field’s first hard fact #

Because the depth is a square, the range across the plausible planets is enormous.

A hot Jupiter is a one per cent signal. Ground-based photometry through the atmosphere reaches a few parts in a thousand on a good night, which is why the first transiting planet was found from the ground and the hundredth was not. An Earth-sized planet around a Sun-like star is 84 ppm, roughly the precision of a photometer looking at a star for six hours from above the atmosphere, and that number — not any question of principle — is the reason space telescopes were built for this.

The escape from the problem is to change the denominator. The depth is a ratio, so the same planet in front of a smaller star gives a deeper signal. An Earth crossing a red dwarf of a fifth the Sun’s radius removes twenty-five times as much light: 0.2 per cent, an easy measurement. That is not a trick; it is the reason the small rocky planets in the current census orbit small cool stars, and it is a selection effect before it is a discovery about planet formation.

Most planets never do this at all #

The transit requires an alignment, and the alignment is unlikely.

For a circular orbit, a transit happens when the line of sight lies within one stellar radius of the orbital plane at the planet’s distance — which, averaged over random orientations, has probability exactly $R_\star/a$. Everything follows from that expression. Kepler’s third law turns the period into a distance, so the probability falls as $P^{-2/3}$.

The consequences are severe and entirely quantitative. A survey that finds one transiting hot Jupiter has looked past about nine systems that contain one and do not show it. A survey that finds one Earth analogue has looked past two hundred. Every occurrence rate quoted in this field is a count divided by a probability, and dividing by a small number honestly is most of the work.

There is a compensation worth stating alongside, because it is easy to present the geometry as pure loss. A transiting planet is enormously more informative than a non-transiting one. The alignment that makes the transit possible also fixes the inclination near 90°, which is precisely the quantity a radial-velocity orbit cannot supply; and the same alignment puts the planet’s atmosphere in front of the star once per orbit and behind it once per orbit, which is where every measurement of an exoplanet’s air comes from.

What was actually measured #

HD 209458 b, 1999. The planet had already been found by the wobble of its star. Two teams — one led by David Charbonneau, one by Gregory Henry — pointed small telescopes at the star at the times the radial-velocity orbit predicted a transit might fall, and found a dip of 1.7 per cent lasting about three hours. The telescope Charbonneau’s group used had a 10 cm aperture. That is the whole apparatus: a lens the size of a saucer, a photometer, and a prediction good enough to say when to look.

The importance was not the planet, which was already known about. It was that the depth gave a radius, the radial velocity gave a mass, and the two together gave a density of about 0.37 g/cm³ — establishing, for the first time by measurement rather than by inference from a spectrum, that the object was a gas giant and not a small star or a stellar artefact. The debate about whether the radial-velocity signals were really planets ended with that light curve.

Kepler, 2009–2013. A 0.95 m telescope staring at one 115-square-degree field, measuring the brightness of about 150,000 stars every thirty minutes for four years, at a precision of a few tens of parts per million for the brightest. It found over 2,600 confirmed planets. Its successor TESS trades depth for sky coverage, surveying almost the whole sky in 27-day segments and finding the bright, nearby systems that can afterwards be studied in detail.

The radius that is not measured. Every one of those planetary radii is a stellar radius multiplied by a measured ratio, so the catalogue’s accuracy is the accuracy of the stellar radii. This turned out to matter enormously. Before Gaia’s parallaxes, the radii of Kepler’s host stars were typically uncertain by 20–40 per cent, and the planetary radii inherited it in full. The California–Kepler Survey re-derived the host parameters with high-resolution spectroscopy, and Gaia then fixed the distances; the planetary radii moved, and the resulting distribution turned out to have structure in it that nobody had been able to see — a gap where planets ought to be. A systematic error in the denominator had been smearing it out.

The exposure that is longer than the ingress #

There is a systematic in the depth that has nothing to do with the star and everything to do with the shutter, and it is worth setting out because it caught a whole survey.

A photometer integrates for some interval and reports one number per interval. If that interval is short compared with the features of the light curve, the samples trace the curve. If it is not, each sample is an average of the curve over the interval, and the reported curve is the true one convolved with a box of the exposure’s length.

For a transit the sharpest features are the ingress and egress, which for a hot Jupiter last of order twenty minutes and for a planet on a wide orbit rather longer. An exposure of thirty minutes therefore smears them substantially: the corners round off, the shoulders lengthen, and the flat bottom shortens.

The consequences run in a specific direction. A smeared light curve looks like one with a longer ingress and a shorter flat section, which is what a higher impact parameter produces — so a fit that ignores the integration returns a grazing geometry for a planet that crossed the middle of the disc. Since the depth of a grazing transit is not the square of the radius ratio, the recovered radius is biased too.

The repair is straightforward and is now standard: the model is evaluated at a fine time grid and averaged over the exposure before being compared with the data, so the model is smeared in the same way the data are. What it costs is computation, since the model has to be evaluated many times per data point, and what it buys is that the parameters recovered are the true ones rather than the smeared ones.

The reason it is worth a section is that the correction is invisible in the residuals. A fit that ignores the integration can match the data very well, with an entirely respectable chi-squared, and return a wrong impact parameter and a wrong radius. A systematic that does not degrade the fit is the kind that survives peer review, and this one did, in published parameters for hundreds of planets, until the long-cadence data were refitted.

There is a related and more familiar version of the same trap in the detection stage rather than the fitting one. A search algorithm looking for box-shaped dips has to choose a grid of trial durations, and a transit shorter than the finest grid spacing is recovered with reduced significance or missed entirely. Since duration falls with orbital period, the shortest-period planets are exactly the ones a coarse grid loses — and the occurrence rate at short periods is one of the numbers the surveys were built to measure. Both problems are consequences of a continuous signal being sampled by a discrete instrument and analysed on a discrete grid, and both are handled by making the model as coarse as the data rather than the other way around.

Where the picture stops #

The floor is not flat. A real stellar disc is brighter at the centre than at the limb, because a sight line to the centre reaches deeper and hotter material before it becomes opaque. So the planet blocks more light in mid-transit than at the edges, the curve sags, and the depth exceeds $(R_p/R_\star)^2$. The exact statement in this essay holds for a uniform disc and for nothing else; the correction is between a few per cent and a few tens of per cent depending on wavelength, and it is a piece of stellar physics smuggled into what looked like a geometry problem.

A depth is not a planet. Anything that removes one per cent of the light from a pixel produces the same signal: a faint eclipsing binary blended into the aperture, a background star with a deep eclipse diluted by the bright foreground, a grazing stellar companion. The follow-up work in this field is mostly the elimination of those, and “validated” rather than “confirmed” is the word used when the elimination is statistical rather than by an independent mass measurement.

The star is not a steady lamp. Spots rotating across the disc, granulation, flares and pulsation all move the brightness. A planet crossing a starspot produces a small upward bump inside the transit, since the planet is covering a patch that was darker than average — an anomaly that is a nuisance for the radius and a gift for anyone mapping the star’s surface.

A single dip is not a period. One event is a candidate; the period comes from repetition, and the repetition is what a four-year mission was for. A planet with a period longer than the observing window is invisible to the method, however large it is — which is the most severe of the biases and the one that has no fix except waiting.

The depth is a ratio of areas, and the two things that spoil it — a small planet and a blended star — are worth drawing at the sizes at which each becomes fatal.

The generalisation #

The transit is an occultation, and occultations are the oldest precision instrument in astronomy. When the Moon passes in front of a star, the star’s disappearance is instantaneous within about 0.05 seconds, which sets an upper limit on the star’s angular diameter — the technique measured stellar diameters before interferometry could. When an asteroid passes in front of a star, timings from observers strung across the shadow track give the asteroid’s outline as a set of chords, and that is how the shapes of hundreds of minor planets are known.

When Uranus occulted a star in 1977, the star blinked several times before and after the planet itself passed — and the rings of Uranus were discovered in the timings, having never been seen. Nine years later the same technique found Pluto’s atmosphere, in the softness of the star’s disappearance.

Every one of these is the same measurement as a transit: something opaque crosses something bright, and the shape of the interruption is read. What is unusual about the exoplanet case is only that the shadow cannot be resolved, so the geometry has to be recovered from a single number varying in time.

And the family of depths at a much longer period, since the geometric probability of a transit falls as the orbit widens and the surviving detections are correspondingly biased.

Where this goes next #

The depth is one of four things a light curve carries. The others are the total duration, the duration of the ingress, and the shape of the floor — and together they fix the impact parameter, the mean density of the star, and, with a little help, the orbit’s eccentricity.

Later rungs on this anchor: the four contacts and what their ratios fix. Limb darkening, and the stellar physics hidden in the floor. The stellar density recovered from the duration alone. Transit timing and duration variations. The Rossiter–McLaughlin effect, and the misaligned orbits it revealed. Grazing transits and the degeneracies they carry. Blends, false positives, and what validation means. Transit spectroscopy of the terminator. Secondary eclipses and phase curves. The transits of Venus, and the astronomical unit. Occultations by the Moon, by asteroids and by Pluto. And the transit of an exoplanet observed by another exoplanet’s star, which is not a technique but a reminder that the geometry is common and the observers are not.