The planet is seen when it disappears

A transiting planet passes in front of its star once per orbit. Half an orbit later it passes behind it — and for the hour or two it is hidden, the system’s total brightness is the star’s alone.

The difference between hidden and not hidden is the planet’s own light. Nothing has been resolved, nothing has been separated spatially, and yet what has been measured belongs to the planet and not to the star.

The occultation is a subtraction, and that is its whole virtue #

During the secondary eclipse the planet contributes nothing. Immediately before and after, it contributes everything it has. So

$$\delta_{\text{sec}} = \frac{F_p}{F_\star},$$

directly, with no model in between. This is the same geometry as a transit with the roles exchanged: there the planet blocks the star, here the star blocks the planet.

The size of the effect is set by the same two ingredients as everything else: the area ratio and the brightness ratio.

$$\frac{F_p}{F_\star} = \left(\frac{R_p}{R_\star}\right)^2 \frac{B_\lambda(T_p)}{B_\lambda(T_\star)}.$$

In the optical, where the star is enormously brighter per unit area, that is $10^{-5}$ or less unless the planet is very hot. In the mid-infrared, on the Rayleigh–Jeans tail of the star and near the peak of the planet, the brightness ratio climbs by orders of magnitude — and the measurement becomes possible.

What the depth is a temperature of #

Converting a depth into a temperature requires assuming the planet radiates as a blackbody at the wavelength observed, which gives a brightness temperature — the temperature a blackbody would need to produce that flux at that wavelength.

Brightness temperatures at different wavelengths differ, and the differences are the emission spectrum. A molecular band in absorption makes the planet fainter at that wavelength, so the brightness temperature is lower there and the observation is probing higher, cooler layers; a band in emission means the temperature rises with height, which happens where a strong absorber high in the atmosphere is heated directly by the star. The presence or absence of such an inversion is one of the main things secondary eclipse spectroscopy measures.

That is a different measurement from transmission spectroscopy, and the difference is worth being precise about. Transmission probes the terminator, in slant geometry, at millibar pressures. Emission probes the day side, looking down, at pressures of order a bar. The two see different parts of the same atmosphere and are sensitive to different things — transmission to the composition of a thin high shell, emission to the temperature structure of the bulk.

The phase curve, which measures winds #

Between the two eclipses, the planet’s illuminated fraction changes exactly as the Moon’s phases do, and the system’s total brightness rises and falls with it.

The amplitude of that variation is the difference between the planet’s day-side and night-side brightness. If the atmosphere transported no heat at all, the night side would be at absolute zero and the phase curve amplitude would equal the secondary eclipse depth. If it transported heat perfectly, the two hemispheres would be equally bright and the phase curve would be flat.

Real planets are in between, and where they sit is a measurement of the wind.

And there is a further, sharper measurement in the same data. If the hottest point on the planet were directly beneath the star, the phase curve’s maximum would fall exactly at the secondary eclipse. It does not: on most hot Jupiters the maximum arrives early, meaning the hottest region is displaced eastward of the substellar point.

The explanation is a superrotating equatorial jet — a band of atmosphere moving faster than the planet rotates, carrying heat downwind before it can be radiated away. The offset is typically 10–30 degrees, and it converts directly into a ratio of radiative to advective timescales, and thence to a wind speed of a few kilometres per second.

A jet stream on a planet 150 parsecs away, measured from the phase of a sinusoid.

The physical reasoning behind the offset is worth one more sentence, because it makes the measurement interpretable rather than merely striking. A parcel of gas at the substellar point is heated on the radiative timescale and blown downwind on the advective one; if radiation wins, the hot spot sits under the star, and if advection wins, it is carried east. The measured offset is therefore a ratio of two timescales, and since the radiative timescale can be computed from the temperature and the opacity, the advective one — and with it the wind speed — falls out.

The timing of the eclipse measures the orbit #

There is a second measurement buried in the same event, and it is precise enough to have been used as a test of general relativity’s less exotic predictions.

For a circular orbit the secondary eclipse falls exactly halfway between transits. For an eccentric one it does not: the planet covers the two halves of its orbit at different speeds, so the eclipse is displaced by an amount proportional to $e\cos\omega$ — the eccentricity times the cosine of the argument of periapsis. Measuring that offset, in a system where the eclipse can be timed to a minute, constrains a combination of orbital elements that photometry has no other access to.

The offset is not small. A hot Jupiter with $e = 0.01$ on a three-day orbit has its eclipse displaced by about twenty minutes, so an eccentricity of a per cent is a comfortably measurable timing shift — and several hot Jupiters that were assumed circular turn out to have small but definite eccentricities this way.

There is also a light-travel correction, and it is the kind of detail that is satisfying to have to include: the eclipse is further away than the transit by twice the orbital radius, so its light arrives late by $2a/c$. For a hot Jupiter that is about a minute, and for a planet at 1 AU it is sixteen minutes — the same light-time that Rømer used to measure the speed of light from the moons of Jupiter, appearing here as a systematic that has to be subtracted before the eccentricity can be read.

What was actually measured #

HD 209458 b and TrES-1, 2005. The first secondary eclipses, both with Spitzer at 24 and 4.5 µm. Depths of a few hundred to a few thousand parts per million, giving day-side brightness temperatures around 1,100 K. The measurement established that these objects could be characterised at all, six years after the first transit.

HD 189733 b, 2007. The earliest such curve: Spitzer watched the system continuously for 33 hours at 8 µm, and the brightness varied by 1.2 per cent of the star’s total, with a maximum 16 degrees before the secondary eclipse. Day side about 1,210 K, night side about 970 K — a difference of only 240 K on a tidally locked planet, which means the winds are carrying a great deal of heat. That single observation established both the redistribution efficiency and the hot-spot offset.

WASP-43 b, 2014 and 2023. Observed by Hubble and then by JWST with full spectral coverage across a whole orbit. The JWST data gave a night side at about 700 K against a day side at 1,500 K, and — the striking result — the night side is covered in thick cloud, which was inferred from the absence of the expected methane emission rather than from any direct detection.

LHS 3844 b, 2019. A rocky planet observed in secondary eclipse: the phase curve amplitude was as large as it could be, meaning essentially no heat transport, meaning no appreciable atmosphere. A null result, and one of the cleanest measurements in the field — the absence of an atmosphere established from the shape of a phase curve rather than from a spectrum.

And the ones where the light was reflected. Kepler measured optical phase curves for dozens of hot Jupiters, where the signal is reflected starlight rather than thermal emission. The geometric albedos come out low — typically below 0.1, darker than coal — which is another way of saying these atmospheres are dominated by absorbers rather than by cloud, on the day side at least.

Where the picture stops #

A brightness temperature is not a temperature. It is what a blackbody would need to be. A real atmosphere has a temperature that varies with depth, and the emission at each wavelength comes from a range of depths weighted by the opacity — so a single brightness temperature is a weighted average over a region whose extent is itself model-dependent.

Thermal and reflected light are entangled. In the near infrared both contribute, and separating them requires either a wide wavelength coverage or an assumption about albedo. Several early “high albedo” claims turned out to be thermal emission from hotter-than-expected day sides.

A phase curve requires continuous observation of a whole orbit. That is a day or more of uninterrupted telescope time on one target, which is expensive enough that fewer than a hundred exist. It also means the instrument’s systematics must be stable over a day, which is precisely the timescale on which they usually are not.

Instrumental systematics are the same size as the signal. Spitzer’s detectors showed a ramp in response over the first hours of every observation, and pointing jitter moved the star across pixels of different sensitivity — both at the level of the eclipses being measured. A decade of the field’s literature is the story of learning to remove them, and several early results moved by more than their quoted errors when better decorrelation methods arrived. That is why a null result like an absent atmosphere is trusted only when the systematic is much smaller than the effect.

And the geometry restricts everything to transiting systems. A secondary eclipse needs an occultation, and an occultation needs an alignment — the same $R_\star/a$ lottery that limits the transit method. A non-transiting planet still shows a phase curve, in principle, and a handful have been claimed; without the eclipse to set the zero point, the amplitude and the albedo cannot be separated.

What an eclipse can resolve #

There is one more thing hidden in the shape of the event, and it is the closest the field comes to an image.

The secondary eclipse is not instantaneous. The star’s limb sweeps across the planet’s disc over the ingress, covering first one edge and then the rest, and the rate at which light disappears therefore depends on how the planet’s brightness is distributed across its face. A uniformly bright disc gives one ingress shape; a disc with a bright spot displaced eastward gives another, slightly asymmetric between ingress and egress.

That is eclipse mapping, and it recovers a coarse two-dimensional brightness map of the day side from a one-dimensional light curve. The information content is small — a handful of spatial modes at best — but it is genuinely spatial, and it has been done: JWST observations of WASP-43 b resolve the day side into a bright region offset from the substellar point, in agreement with the offset measured independently from the phase curve.

The technique is old in a different form. Occultations of stars by the Moon give angular diameters from the rate at which the light vanishes, and the same reasoning gives asteroid shapes from timings across a shadow track. The planet case is harder by every measure and identical in principle: a moving edge across an unresolved source turns time into position. The occultation depth is a ratio of two brightnesses, so it is worth drawing at a much larger ratio and at a much smaller planet, which are the two ways the measurement becomes possible or impossible.

The generalisation #

Isolating a source by removing it and measuring the difference is one of the oldest tricks in observational astronomy, and it works whenever the removal can be timed.

The solar corona is measured by subtracting an eclipsed Sun from an uneclipsed one — the occulting body being the Moon, and the same geometry as here. Lunar occultations of stars give angular diameters from the diffraction pattern as the star vanishes. In radio astronomy, position switching and beam switching remove the sky background by pointing on and off the source. Lock-in amplification, which is the same idea in electronics, chops the signal at a known frequency and detects only what varies at that frequency.

The common structure is that a difference is measurable far below the level at which an absolute value is. Nothing about a hot Jupiter’s 1,800 ppm could be established by measuring the system’s brightness and subtracting a model of the star. It is established because the planet is removed on a schedule known in advance, and everything that does not follow that schedule cancels.

That is the same principle that runs through this whole field. A transit is a difference; a radial velocity is a difference; a transmission spectrum is a difference of differences. In a subject where nothing can be visited, resolved or repeated at will, the thing that makes measurement possible at all is that the geometry keeps a timetable.

The clock the eclipse keeps #

The measurements above use the eclipse’s depth and the phase curve’s shape. Its timing carries something else entirely, and it is the cleanest orbital measurement a transiting system offers.

For a circular orbit the secondary eclipse falls exactly halfway between two transits. For an eccentric one it does not, because the planet covers the two halves of its orbit at different rates and the line of sight cuts the orbit asymmetrically. The offset from phase 0.5 is, to good approximation,

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

so a timing measured to a minute on a three-day orbit constrains $e\cos\omega$ to a few parts in ten thousand. The durations of the transit and the eclipse give the orthogonal combination, $e\sin\omega$, from the different orbital speeds at the two crossings. Together they give the eccentricity and the orientation, from timings alone, with no spectroscopy at all.

That matters because the eccentricity of a hot Jupiter is a statement about its history. Tides circularise a close-in orbit on a timescale that depends steeply on the planet’s own dissipation, so a measured eccentricity of a few thousandths where zero was expected is evidence either that the circularisation is slower than assumed or that something is still pumping it — a companion, most often. A secondary eclipse arriving four minutes early is a constraint on the interior of a planet nobody can see, through the chain from timing to eccentricity to tidal quality factor.

One correction has to be applied before any of that is believed, and it is a pleasing one. The light from the secondary eclipse crosses the orbit’s diameter more than the light from the transit does, so the eclipse is observed late by the light-travel time across the system — about 45 seconds for a typical hot Jupiter. That is the same Rømer delay by which the eclipses of Jupiter’s moons run late when Jupiter is far, measured three and a half centuries later on a system whose planet has never been seen.

Both of those corrections are smaller than the eclipse itself by orders of magnitude and neither can be neglected, which is the ordinary condition of a measurement made by differencing two large numbers that are nearly equal.

And the phase curve with no hot-spot offset at all, which is the null case the wind measurement is made against.

Where this goes next #

The measurements in this essay are made on hot giant planets because those are the ones bright enough to subtract. Applying them to a temperate rocky planet requires either a much larger telescope or a much smaller star, and the field is currently attempting both.

Later rungs on this anchor: emission spectra and thermal inversions. Brightness temperature against physical temperature. Phase curves and general circulation models. The hot-spot offset and equatorial jets. Optical phase curves and albedos. Eclipse mapping, which resolves the day side from the shape of ingress. Secondary eclipses of rocky planets. Nightside clouds. Thermal emission from non-transiting planets. And the day–night contrast of a tidally locked temperate planet, which is the measurement that would say whether such a world can hold an atmosphere at all.