Exoplanet Eclipse Mapping

A line scanner you didn't build

You will never see a hot Jupiter as anything but a single unresolved point of light. The planet WASP-43b is about 87 parsecs away; its disk subtends roughly 10 micro-arcseconds, thousands of times smaller than what any telescope ever built can resolve. And yet we have a map of it — a crude one, but a real two-dimensional picture of which parts of its surface are hot and which are cold. The trick is that nature hands us a scanning slit for free, twice per orbit.

That slit is the edge of the star. Every orbit the planet passes behind its host — the secondary eclipse — and for a few minutes at the start (ingress) and a few minutes at the end (egress) the star's limb lies partway across the planetary disk. As the planet moves, that sharp edge advances across the disk at a known speed, covering one thin strip of the planet at a time. Each instant the light you lose is exactly the brightness of the strip that just went dark. The derivative of the light curve during ingress and egress is a brightness scan across the planet. Eclipse mapping is the art of reading those scans and inverting them into a map. That's the whole idea: watch how the light dims as the planet slips behind the star, and reconstruct the face you can't resolve.

The occultation geometry

Treat the planet as a uniform disk of radius R_p and the star as a much larger disk of radius R_★ whose limb, over the few minutes of ingress, is locally a straight edge moving at the relative sky-plane velocity v. For a hot Jupiter like WASP-43b, R_p ≈ 1.04 R_Jup and the orbital speed projects to roughly

v_sky ≈ 2π a / P ≈ 2π (0.0153 AU) / (0.813 d) ≈ 205 km/s

The full disk takes 2R_p/v to be covered. With R_p ≈ 7.4 × 10⁴ km that is about 12 minutes — the ingress (and egress) duration. During those 12 minutes the fraction of the planet hidden, f(t), is the area of the disk lying behind the stellar limb. If the planet's surface brightness were uniform, the lost flux would simply track f(t): a smooth, symmetric S-curve. The signature of structure is the departure of the real ingress/egress shape from that uniform-disk curve.

Concretely, let I(x, y) be the planet's specific intensity across the sky-projected disk, and let the limb at time t be the line that has swept to position s(t). The lost flux is

F_lost(t) = ∬_(covered region) I(x, y) dx dy

dF_lost/dt = v ∫ I(s(t), y) dy        (integral along the strip just covered)

So the time derivative of the eclipse light curve equals v times the brightness integrated along the strip currently at the limb. That is a one-dimensional projection of the 2D map — a single "slice." This is exactly the projection-slice relationship that underlies computed tomography; eclipse mapping is tomography of an exoplanet, with the stellar limb playing the role of the X-ray beam.

Two slices, and why the map is crude

One ingress gives one slice: the brightness summed along strips parallel to the limb. From a single slice you cannot tell where along each strip the light originated — north or south within the strip is invisible. You need a second projection at a different angle to break that degeneracy.

The geometry obliges, just barely. The stellar limb crosses the planet at one position angle during ingress and a different one during egress, because the planet's chord across the stellar disk is not symmetric about the disk center unless the impact parameter is zero. In addition, the planet rotates a little during the eclipse. The result is two projections at two angles — like a CT scan with exactly two views. Two views are nowhere near enough to reconstruct a sharp image. What you can recover is the low-order structure: the position of the brightest region (east-west offset and north-south offset), the overall day-night contrast, and a rough sense of how concentrated the hot spot is. Practitioners parameterise the map in low-order spherical harmonics (typically up to ℓ = 2 or 3) precisely because higher orders are unconstrained and would just fit noise.

This is the honest limitation behind the word "crude" in every description of the technique: a hot Jupiter eclipse map is real, but it is a handful of numbers about where the light is concentrated, not a photograph.

The eastward hot spot — what the map is for

The headline science result is the location of the thermal hot spot. A tidally locked hot Jupiter is heated only on the dayside, so naively the hottest point should be the substellar point, dead-center on the dayside. It isn't. Atmospheric circulation on these worlds is dominated by a fast, broad eastward equatorial jet — superrotation — with wind speeds of order 1–5 km/s. That jet advects the hottest gas downwind before it can radiate away its heat, displacing the thermal peak east of the substellar point, typically by 10°–40° of longitude.

Phase curves discovered this offset first: the system's combined brightness peaks slightly before secondary eclipse, because the hot region has rotated into view ahead of the substellar point. But a phase curve measures one number per orbital phase — it constrains longitude only, and it cannot separate an east-west offset from a north-south one. Eclipse mapping is the upgrade: because ingress and egress scan the disk along different directions, the map separates the longitudinal (east-west) offset from the latitudinal (north-south) distribution. For WASP-43b, JWST mapping confirmed the eastward offset and placed the hot spot near the equator, consistent with the jet picture and inconsistent with off-equatorial heating.

How small is the signal

Eclipse mapping is hard because every relevant signal is tiny and they nest inside each other like Russian dolls.

Quantity	Typical hot-Jupiter value	What it tells you
Transit depth (Rp/R★)²	~1–2.5 % (10,000–25,000 ppm)	Planet radius
Secondary-eclipse depth (mid-IR)	~0.1–0.5 % (1000–5000 ppm)	Disk-averaged dayside flux / temperature
Hot-spot map signal	~10–50 ppm	Where the dayside is brightest (the map)
Ingress / egress duration	~10–20 min	Window the slit scan happens in
Required precision	tens of ppm per minute	Why Spitzer couldn't, JWST can
Recovered resolution	~a few % of Rp (best case)	Coarse, low-ℓ harmonics only

The map information is roughly two orders of magnitude smaller than the eclipse it lives on, and it is concentrated in two windows totalling perhaps half an hour out of a ~20-hour orbit. That budget — tens of parts per million per minute of integration — is precisely the regime where Spitzer's photometry hit a systematics floor and JWST's does not. It is no accident that the first real exoplanet eclipse maps appeared in 2023–2024, within two years of JWST's first science.

The math: from light curve to map

The forward model expresses the planet's intensity as a sum of orthogonal spatial modes (spherical harmonics or an equivalent map basis), each multiplied by an unknown coefficient:

I(θ, φ) = Σ_ℓ Σ_m  c_ℓm  Y_ℓm(θ, φ)

Each harmonic Y_ℓm, when run through the occultation geometry, produces a specific, computable contribution to the eclipse light curve — a "mapping kernel" or eigencurve. Because the occultation is a linear operator on the brightness, the observed light curve is a linear combination of these eigencurves with the same coefficients c_ℓm. Fitting the light curve therefore means solving a linear inverse problem for the c_ℓm:

d = K c + n         d = data, K = design matrix of eigencurves, n = noise
ĉ = (KᵀN⁻¹K + λR)⁻¹ KᵀN⁻¹ d   (regularised least squares)

The key practical insight, due to the "eigenmapping" approach of Rauscher et al. and the analytic framework of Luger et al.'s starry package, is that the eigencurves are highly correlated: many harmonics produce nearly identical light-curve signatures and so are individually unconstrained. A principal-component analysis of K reveals only a handful of well-measured combinations ("eigenmaps"). You fit those, regularise the rest toward zero (the λR term), and quote the recovered eigenmaps rather than pretending you measured every c_ℓm. Honest eclipse-mapping papers report the number of statistically significant eigenmaps — usually two to five — not a glossy rendered globe.

Worked example: locating WASP-43b's hot spot

Take WASP-43b's numbers: orbital period P = 0.813 d, semi-major axis a = 0.0153 AU, planet radius R_p = 1.04 R_Jup = 7.4 × 10⁴ km, star radius R_★ = 0.667 R_☉. The sky-plane velocity is

v_sky = 2π a / P = 2π (2.29 × 10⁶ km) / (7.02 × 10⁴ s) ≈ 205 km/s

(this is the full orbital sky velocity; the effective speed at which the limb actually scans across the planet is somewhat lower and depends on the impact parameter b ≈ 0.66, since the chord across the stellar disk is not through its center — but 205 km/s sets the right order of magnitude). The time to cross the full planetary disk is

t_cross = 2 R_p / v_sky = 2 (7.4 × 10⁴) / 205 ≈ 720 s ≈ 12 min

Now suppose the hot spot is offset 20° east of the substellar point. Twenty degrees of longitude on a disk of radius R_p projects to a sky displacement of about R_p sin(20°) ≈ 0.34 R_p ≈ 2.5 × 10⁴ km. The stellar limb reaches that displaced bright spot earlier on one side of the eclipse and later on the other, shifting the time at which ingress (and egress) is steepest by

Δt ≈ (0.34 R_p) / v_sky ≈ (2.5 × 10⁴ km) / 205 km/s ≈ 120 s ≈ 2 min

A two-minute asymmetry between ingress and egress timing, riding on a 0.3%-deep eclipse, is the observable. To detect a centroid shift of order 2 minutes you need the light-curve slope sampled to tens of ppm every minute or two — exactly JWST/MIRI's demonstrated performance, and exactly why the measurement was impossible before 2022. The recovered offset for WASP-43b, ~10°–20° east depending on wavelength, matches general-circulation-model predictions for its ~1500 K dayside.

Where it has actually been done

• WASP-43b (JWST/MIRI, 2024). The flagship demonstration. A full-orbit MIRI/LRS phase curve plus eclipse analysis produced a longitude-and-latitude brightness-temperature map: dayside ≈ 1525 K, nightside ≈ 860 K, eastward hot-spot offset of order 15°, and evidence for a thick nightside cloud deck suppressing nightside emission. The largest contiguous JWST exoplanet-climate dataset to date.
• WASP-18b (JWST/NIRISS, 2023). An ultrahot Jupiter (dayside ~2900 K). Eclipse mapping found the hot spot essentially at the substellar point, with little eastward offset — because radiative cooling is so fast at those temperatures that the jet cannot carry heat downwind before it re-radiates. The contrast with WASP-43b is itself the result: offset tracks the competition between advection and radiation.
• HD 189733b and HD 209458b. The historical proving ground. Spitzer phase curves (2007 onward) first measured hot-spot offsets statistically; HST and Spitzer eclipse data drove the development of the inversion machinery even when the per-point precision was marginal.
• Kepler/CHEOPS optical mapping. For the hottest planets, reflected-plus-thermal optical light curves from Kepler and CHEOPS (e.g. Kepler-7b) yielded longitudinal brightness maps showing inhomogeneous cloud cover — bright cloud banks on the cooler western dayside — a different physical map (clouds, not just temperature) made with the same eclipse-shape logic.

Eclipse mapping vs the techniques it builds on

Technique	What it measures	Spatial info	Lives in the light curve at	Instrument era
Secondary-eclipse depth	Disk-averaged dayside flux	None (1 number)	Eclipse bottom	Spitzer 2005+
Phase curve	Brightness vs orbital phase	Longitude only (1D)	Whole orbit, slow sinusoid	Spitzer 2007+, JWST
Eclipse mapping	2D dayside brightness map	Longitude + latitude (low-ℓ 2D)	Ingress/egress slope curvature	JWST 2023+
Transmission spectroscopy	Atmospheric composition	Day-night terminator ring	Transit depth vs wavelength	HST/JWST
Direct imaging	Resolved point/photometry	None for close-in planets	Spatially separated source	VLT/JWST (wide orbits only)
Doppler imaging (future)	Surface map via line shapes	2D in principle	Spectral line profiles	ELT era

The chain is incremental: depth gives you a temperature, the phase curve smears that into longitude, and eclipse mapping adds the second spatial dimension by exploiting the one moment when a hard edge crosses the planet. Each step costs roughly an order of magnitude in required photometric precision.

Common misconceptions and edge cases

• "The map comes from the eclipse depth." No — the depth is just the disk average. All the spatial information is in the curvature of the ingress and egress slopes. A perfectly trapezoidal eclipse (straight slopes, flat bottom) corresponds to a uniform disk and carries zero map information.
• "It's a photograph of the planet." It is a low-order harmonic reconstruction — typically two to five statistically significant eigenmaps. The pretty rendered globes in press releases are model interpolations constrained by those few numbers, not pixels.
• "Any transiting planet can be mapped." Only if the eclipse signal is large enough. Hot and ultrahot Jupiters work; an Earth-size temperate planet's eclipse is tens of ppm total, below the per-minute precision the technique needs.
• "The hot spot is always east." Usually, because of superrotation — but the offset shrinks toward zero for the hottest planets (WASP-18b), where radiation beats advection, and circulation models even predict westward offsets in some magnetised or low-pressure regimes. The offset is a measurement, not an assumption.
• "Impact parameter doesn't matter." It is central. A central eclipse (b = 0) makes ingress and egress nearly degenerate — the limb crosses at almost the same angle both times — and the latitudinal information collapses. Grazing-to-moderate impact parameters give the two well-separated projection angles that make 2D mapping possible. WASP-43b's b ≈ 0.66 is, in this sense, lucky geometry.
• "Clouds and temperature are the same map." In the thermal infrared you map emission (temperature); in the optical you map reflection (clouds/albedo). Kepler-7b's optical map of bright western clouds and WASP-43b's IR thermal map are physically different pictures of the same kind of world.