Four contact points, and what they fix

The depth of a transit is one number and it says one thing. The rest of the light curve — how long the event lasts, how long the floor is flat, how steep the shoulders are — is three more numbers, and they turn out to say a great deal more than seems reasonable.

They fix which chord the planet took. And then, with the orbital period and no other information whatever, they give the mean density of the star.

Two durations, and the shape between them #

Call the interval from first to fourth contact $T_{14}$ — the whole event — and from second to third $T_{23}$, the flat part. Their difference is the time the planet spends partly on the disc, half at each end.

The two are set by two things: how long the chord is, and how fast the planet crosses it. The chord’s half-length is $\sqrt{(1+k)^2 - b^2}$ for the outer contacts and $\sqrt{(1-k)^2 - b^2}$ for the inner ones, where $k = R_p/R_\star$. So the ratio of the durations depends on $b$ and $k$ alone — the speed cancels — and since $k$ is already known from the depth, the ratio gives $b$.

Two transits of the same planet, then, can have the same depth and completely different silhouettes. That is the content of the shape: the depth says what crossed, the shape says where.

Kepler’s third law is hiding in the duration #

Here is the step that looks like sleight of hand and is not.

The planet crosses the star’s face at its orbital speed, which is $2\pi a/P$. The distance it covers is the chord, which is $2R_\star\sqrt{1-b^2}$. So

$$T_{14} \approx \frac{P}{\pi}\,\frac{R_\star}{a}\sqrt{1-b^2},$$

and everything in it is known except the ratio $a/R_\star$ — which the equation therefore delivers, from the duration, the period and the shape.

Now bring in the harmonic law. For a planet much lighter than its star,

$$\frac{a^3}{P^2} = \frac{GM_\star}{4\pi^2},$$

so

$$\left(\frac{a}{R_\star}\right)^3 = \frac{GM_\star P^2}{4\pi^2 R_\star^3} = \frac{G P^2}{3\pi}\cdot\frac{3M_\star}{4\pi R_\star^3} = \frac{GP^2\rho_\star}{3\pi}.$$

The stellar mass and radius appear only in the combination $M_\star/R_\star^3$, which is the mean density. Rearranged:

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

A transit light curve measures the mean density of the star it crosses. Not the mass, not the radius, but the ratio that Kepler’s third law happens to isolate — and it does so from a brightness curve and a clock, with no spectrum, no parallax and no stellar model. The consequence is practical rather than ornamental. A density is a strong constraint on what kind of star is being looked at: a solar-type dwarf is about 1.4 g/cm³, a red dwarf ten times that, a subgiant a tenth. It is also the quantity a star’s own structure is organised around — the balance between a pressure gradient and a weight is a statement about mass over radius cubed before it is a statement about anything else — so the number a light curve delivers is not a curiosity but the one a stellar model most wants. When Kepler’s candidate list was being sifted, the density derived from each light curve could be compared with the density implied by the catalogued stellar parameters — and a large disagreement meant either that the star was not the kind of star it was thought to be — a subgiant catalogued as a dwarf, which was common before Gaia and which inflates every planetary radius derived from it — or that the transit was not around that star at all. Many false positives were caught exactly there, by a mismatch between two densities.

What the ingress alone says #

The two shoulders deserve separating out, because they carry a quantity of their own.

An ingress lasts as long as it takes the planet to move its own diameter across the limb, projected along the chord. For a central crossing that is simply $2R_p$ divided by the orbital speed, so

$$\tau \approx T_{14}\,\frac{k}{\sqrt{1-b^2}}$$

to a good approximation. The ingress is therefore roughly the fraction $k$ of the whole event: a hot Jupiter with $k = 0.1$ crossing in three hours spends about twenty minutes on each shoulder, and an Earth-sized planet with $k = 0.01$ crossing in thirteen hours spends about eight minutes.

That short interval is where a surprising amount of the information sits. The depth is degraded by any dilution of the light — a companion star in the aperture, a background star, scattered light — but the ratio of the ingress to the total is not, because both are times. A transit whose depth says $k = 0.05$ and whose shape says $k = 0.10$ is a diluted transit, and comparing the two is one of the standard tests for a blend. The shape is harder to counterfeit than the amplitude, which is the theme of this whole essay stated in one line.

The ingress is also the shortest feature a light curve contains, so it is what sets the cadence a survey must observe at. Kepler’s 29.4-minute long cadence does not resolve the eight-minute ingress of an Earth analogue at all, which is why the mission also recorded a small number of targets at one-minute cadence, and why those targets are the ones with the best-determined geometries.

What limb darkening does to all of this #

The clean version above assumes the shape can be read off unambiguously. It cannot, and the reason is the sag in the figure at the top of this essay.

A limb-darkened disc has no sharp edge in brightness, so second and third contact are not sharp features of the light curve; the curve rounds into them. Worse, limb darkening and impact parameter are partly degenerate: a high-$b$ chord across a uniform disc and a low-$b$ chord across a strongly darkened disc both give a rounded, short-floored transit. Fitting a light curve therefore means fitting $k$, $b$, $a/R_\star$ and two limb-darkening coefficients together, and the last two are usually fixed at values predicted by a stellar atmosphere model rather than measured — which quietly imports a model into a measurement that was advertised as geometric.

The degeneracy is broken by wavelength. Limb darkening is strong in the blue and weak in the infrared, while the geometry is the same in both. So the same transit observed in two colours separates the two effects, and this is one of several reasons why the best-characterised transiting systems have been observed with instruments spanning as wide a wavelength range as possible.

The grazing case, where the information runs out #

When $b > 1 - k$ there is no second or third contact, the flat floor vanishes, and the light curve becomes a shallow rounded notch. Any combination of a large planet grazing the very edge and a smaller planet crossing slightly further in produces nearly the same curve. Grazing transits therefore give a radius with an enormous uncertainty, and they are also the geometry most easily mimicked by a grazing stellar binary, which is why they are treated with suspicion.

That is a satisfying place for a method to fail, because it fails where the assumption fails rather than somewhere unrelated: the inference in this essay is built on the existence of four contacts, and at $b > 1-k$ there are only two.

The eccentricity that hides in the duration #

The derivation of $\rho_\star$ assumed a circular orbit, and the assumption enters through the speed. A planet on an eccentric orbit crosses its star at whatever speed it happens to have at conjunction, which may be much faster or slower than the circular value — the vis-viva relation gives it exactly, and near periapsis on a moderately eccentric orbit it is easily half again the circular speed. The elements that fix where in the orbit conjunction falls are exactly the ones a transit is blind to, so the two enter the duration together and cannot be separated by photometry alone. What can be done instead is to turn the inference around. A transit duration that is inconsistent with the star’s known density is evidence of an eccentric orbit rather than of a broken method. This is the photoeccentric effect: with an independent density — from spectroscopy, from asteroseismology, or from Gaia — the discrepancy between the density implied by the light curve and the density of the star becomes a measurement of $e$. A method that fixes stellar density when the orbit is circular becomes a method that fixes eccentricity when the density is known, and the same equation does both jobs depending on which side of it is trusted.

What was actually measured #

The formalism, 2003. Sara Seager and Gabriela Mallén-Ornelas wrote down the inversion in closed form: given the depth, the total duration, the flat duration and the period, and assuming a circular orbit and a uniform source, the four observables give $k$, $b$, $a/R_\star$ and $\rho_\star$ uniquely. The paper was written before there was a second transiting planet to apply it to, and it defined how the field would read a light curve.

The asteroseismic cross-check. The strongest test of the density inference is against a completely independent measurement of the same quantity. The mean density of a star is also encoded in the frequencies of its acoustic oscillations — the large frequency separation between consecutive overtones scales as $\sqrt{\rho_\star}$. Kepler measured both for a few hundred bright stars: densities from the transit shape and densities from the star’s own ringing, two measurements sharing no assumptions except the harmonic law. They agree, typically at the few-per-cent level. That agreement is the reason the transit-derived densities are trusted for the tens of thousands of stars too faint to oscillate detectably.

HD 209458 b, again. The Hubble photometry of 2001 reached about $10^{-4}$ precision and resolved the shoulders properly for the first time: $k = 0.1207$, $b = 0.507$, $a/R_\star = 8.76$, all from the shape. The implied stellar density matched the spectroscopic classification, and the planetary radius — 1.35 Jupiter radii for a mass of 0.69 — was the first clear statement that this class of object is larger than a cold hydrogen sphere has any right to be.

TRAPPIST-1, 2016–2021. Seven planets transiting one star of 0.089 solar masses, with radii between 0.75 and 1.13 Earth radii. The stellar radius is small enough that each transit is around half a per cent deep — a signal an amateur telescope can see, on a star far too faint for a spectrograph to weigh its planets. Every radius in the system comes from a depth, every orbital distance from a duration, and the masses come from the planets pulling on each other rather than on the star. It is the clearest demonstration that the geometry in this essay is not a preliminary to the real measurement; on the systems that matter most it is the measurement.

The other eclipse, and the quantity it gives away #

A transiting planet passes behind its star half an orbit later, and that second event has the same four contacts and a different pair of uses.

The depth of the secondary eclipse is the planet’s own brightness relative to the star’s, since what disappears is the planet rather than a piece of the star. In the infrared that is thermal emission and gives a temperature; in the optical it is reflected light and gives an albedo. Neither is geometry, and neither is this essay’s subject.

The timing of it is. For a circular orbit the secondary eclipse falls exactly halfway between transits, because the planet takes as long to go from conjunction to conjunction one way as the other. For an eccentric orbit it does not, and the offset is a direct measurement.

The reason is the same speed argument that made the duration ambiguous. On an eccentric orbit the planet covers the two halves of its path at different rates, so the two conjunctions are not half a period apart. Working out the offset gives, to first order,

$$\frac{\Delta t}{P} \approx \frac{2}{\pi}\,e\cos\omega,$$

which is one combination of the eccentricity and the argument of periapsis, measured from two timings and nothing else.

That is a strikingly clean result. The transit duration carries the eccentricity entangled with the stellar density; the eclipse offset carries it entangled with nothing at all. And the two are complementary rather than redundant, because the duration is most sensitive to $e\sin\omega$ while the offset gives $e\cos\omega$ — so a system with both measured has the eccentricity vector in the plane, and no assumption about the star is required for half of it.

The offset is small: for $e = 0.1$ on a three-day orbit it is about two hours, and for $e = 0.01$ about twelve minutes. Both are comfortably measurable, which is why the eccentricities of hot Jupiters are known far better than those of planets found by any other means.

The four contacts occur twice per orbit, and reading the second set is how a light curve says something about the shape of the orbit rather than only about the geometry of one crossing.

Where the picture stops #

Every shape statement here assumes the star is a disc. A rapidly rotating star is oblate and gravity-darkened, with poles hotter than the equator, so a transit chord across it crosses a surface that is neither circular nor uniform — the equator can be a thousand kelvin cooler than the poles, which is a large excursion on the diagram that sorts stars by temperature — and the light curve becomes asymmetric. Kepler saw this in a handful of hot, fast rotators, and the asymmetry gives the spin–orbit angle — which no other photometric measurement does.

Spots and plages move the shape as well as the depth. A planet crossing a spot group produces a bump; a transit chord that misses the spots but crosses a star whose visible face is spotted has a slightly wrong baseline. Both bias the fitted $b$.

A single transit constrains almost nothing. Every quantity here improves as the square root of the number of events, because a transit is a few hours out of a period of days and almost all of the observing time is spent on the baseline. A planet observed twice has a period and no shape; the well-measured systems have hundreds of transits stacked, and stacking assumes the geometry did not change between them — which for a system with more than one planet in it is not quite true.

Finite integration blurs the shoulders. Kepler’s long-cadence data is a 29.4-minute average, which is a substantial fraction of an ingress — so fitting long-cadence photometry without integrating the model over the exposure systematically flattens the shoulders and biases $b$ upward. The correction is arithmetic, and forgetting it was a recognised early error.

Each of the transit’s three shape parameters can be moved on its own, and each produces a change no combination of the others can imitate — which is the whole reason four contact points fix four numbers.

The generalisation #

Reading a geometry out of the shape of a time series rather than its amplitude is a move that recurs whenever the source cannot be resolved.

An eclipsing binary’s light curve gives both stars’ radii in units of the orbital separation, from exactly this analysis with two occultations rather than one; combined with a double-lined spectroscopic orbit it gives absolute masses and radii to a per cent or better, which is how the relation between a star’s mass and everything else about it is calibrated at all. A lunar occultation’s diffraction fringes give an angular diameter. The rise and fall of a Cepheid’s light curve carries its class in the asymmetry rather than in the amplitude.

In each case the same principle applies: an amplitude is one number and easily faked, while a shape is a function, and a function is much harder to counterfeit by accident.

And a longer-period transit at an intermediate impact parameter, which is the shape the surveys actually spend most of their time fitting.

One more reading shows what the four contact points cost when the limb darkening has to be fitted alongside them.

Where this goes next #

The shape has been used here to fix the geometry of a single, unchanging orbit. The next question is what happens when the shape and the timing do not stay the same from one transit to the next — which is a planet nobody has seen pulling on the one that can be.

Later rungs on this anchor: limb darkening as stellar physics rather than nuisance. The Rossiter–McLaughlin effect and spin–orbit alignment. Gravity darkening on rapid rotators. Transit timing and duration variations. The photoeccentric effect worked through. Eclipsing binaries as absolute calibrators. Spot crossings and stellar surface mapping. Multiplanet systems and mutual inclinations. Circumbinary transits, which are neither periodic nor of fixed duration. And the transits of the solar system’s own planets, timed from Earth, which is where the four contacts got their names.