Transit depth. A planet of radius Rp in front of a star of radius Rs blocks δ = (Rp/Rs)² of the light for a uniform disc. The star is limb-darkened (quadratic law I(μ)=1−u1(1−μ)−u2(1−μ)²), so the light curve here integrates the blocked flux numerically ring by ring; for a small planet the mid-transit depth is δ·I(μb)/⟨I⟩.
Scale height. An isothermal atmosphere thins exponentially, n(z)=n0e−z/H, with H = kBT/(μ mH g), g = GMp/Rp². Hot, puffy, light-gas worlds have large H; cool, massive, metal-rich worlds have small H.
Effective radius. Starlight grazing the limb is dimmed when the slant optical depth reaches about 0.56. For an exponential atmosphere this gives (Lecavelier des Etangs et al. 2008; Fortney 2005) z(λ) = H ln( n0σ(λ) √(2πRpH) / 0.56 ). A molecule with a stronger cross-section σ at some wavelength raises the opaque radius there, so the spectrum shows its bands. Between two wavelengths ΔR = H ln(σ1/σ2) and Δδ ≈ 2RpΔR/Rs² — this is why a feature is a few scale heights tall and why it scales with H. The experiment below checks the closed form against a direct integral of the optical depth along a chord.
Molecules. The H2O (1.4, 1.9, 2.7 µm), CH4 (1.7, 2.3, 3.3 µm), CO2 (4.3 µm) and CO (4.6 µm) bands are schematic: Gaussians at the right wavelengths with band-averaged strengths of a plausible order of magnitude, not line lists. Mixing ratios come from a simple metallicity scaling with a temperature switch between CH4 and CO; real abundances depend on pressure, C/O and disequilibrium chemistry.
Clouds and haze. A grey opaque deck at pressure Pc sits at altitude H ln(1 bar/Pc) and hides everything below, flattening the bands (Sing et al. 2016; Kreidberg et al. 2014). Haze adds a steep Rayleigh-like slope at short wavelengths.
Lopsided limbs. A tidally locked planet has hot day and cooler night sides; winds shift the hot spot east. The two terminators that the starlight grazes then differ: T(λlon) = T0[1 + (c/2) cos(λlon − φoff)] gives a hotter evening limb and a cooler morning limb (3D circulation models: Fortney et al. 2010; Kempton & Rauscher 2012). The cool limb can form clouds and has a smaller H, the hot one is puffier and clearer. The measured spectrum is the area average of the two; JWST has begun to resolve such morning/evening differences (e.g. WASP-39 b, Espinoza et al. 2024, reported). In this model the leading (ingress) limb is the morning limb.
Noise. Photon-limited error per wavelength bin and transit is σ1 = √(1/Nin + 1/Nout); averaging N transits gives σ1/√N. The photon scale is calibrated so that a K=8 star at R=100 gives about 100 ppm per bin per transit near 2 µm, similar to a NIRSpec-class instrument — it is a teaching calibration, not an exposure-time calculator. The optional systematic floor does not average down.
Retrieval. The button fits log10 of the H2O mixing ratio on a grid, minimising χ² with the unknown baseline radius (a constant depth offset) removed analytically. All other parameters are held at their true values, so the quoted error is optimistic; real retrievals marginalise temperature, clouds and C/O and show strong degeneracies.
Assumptions: isothermal, hydrostatic limbs with constant g and a 1-bar reference radius equal for both limbs; ring heights in the 3D view are exaggerated (slider) because H/Rp is only about 0.5%; JWST and orbit distances are not to scale; no stellar contamination, spots or instrument systematics beyond the floor.