Starlight through an atmosphere

Exoplanet Transmission Spectra, Clouds and Starspots

Does a spectral bump prove that a planet has water? Water leaves several absorption bands at related wavelengths. Their joint shape and strength can identify it more reliably than one bump. Cloud height, atmospheric temperature and cool regions on the star change the same spectrum, so the interpretation must compare those possibilities together.

Drag the cloud top or warm the atmospheric ring. Follow grazing starlight into a water absorption spectrum, then change the unocculted stellar surface.

Scale height at the reference radius $H_0=828\,\mathrm{km}$

Depth range across retained spectral bins $\Delta D=2.27\times10^3\,\mathrm{ppm}$

Cloud radius relative to the reference radius $R_c/R_0=1.02$

Largest unocculted-region correction $\delta D=0\,\mathrm{ppm}$

Change the absorbing ring

Infrared ray shown in the scene

Atmospheric temperature

1.10\times10^3\,\mathrm K

T\;[\mathrm K]

Water number fraction

10^{-4}

x_{\mathrm{H_2O}}

Cloud-top pressure

1.00\,\mathrm{bar}

P_c\;[\mathrm{bar}]

Unocculted cool-region fraction

0\%

f

Drag the gold cloud ring. The illuminated rays and the spectrum share the same atmosphere.

Saved synthetic atmosphere is visible.

Reference radius, stellar spectrum and model range

Radius at ten bar

0.133

R_0/R_*

Cool-region source spectrum

Finite isothermal hydrogen–helium gas with water absorption and an opaque gray cloud. Constant composition; inverse-square gravity. CIA, other gases and non-gray aerosols are omitted.

Cool regions are outside the transit chord. Their coverage is a synthetic assumption. The frozen NIRISS points and the conditional retrieval are separate from this interactive prediction.

Against the declared independent 1280-layer cloud-bounded reference, checked solver and inference-grid differences are below two ppm only within the complete frozen R15000 opacity, specified temperature–pressure interpolation and physical assumptions. Halving source wavelength samples changes five presets by about 40–145 ppm; omitting a known temperature row changes them by about 3.4–31.8 ppm, comparable to measurement errors. These source sensitivities are neither Gaussian noise nor a total accuracy bound. The tutorial retains poor fits and makes no water or starspot detection claim.

Follow the derivation and rerun the measured spectrum

Model assumptions

Synthetic, spherical, isothermal hydrostatic atmosphere with inverse-square gravity and a finite top. A prescribed hydrogen–helium mixture contains a constant water number fraction; the molecular mass changes with that fraction. Actual ExoMolOP POKAZATEL cross sections retain the dominant water isotopologue and their stated sampled resolution, temperature and pressure grid. Below the minimum source pressure the cross section is frozen; sensitivity to that choice is reported separately from numerical accuracy. The opaque reference radius is defined at ten bar and a gray cloud supplies a second opaque boundary. CIA, other gases, wavelength-dependent aerosol scattering, refraction, thermal emission, escape and three-dimensional terminator structure are omitted. Native wavelength-dependent blocked areas are solved before photon and stellar weighting in the reported observation bins. PHOENIX photosphere and cool-region spectra describe a uniform-intensity stellar disk with an immaculate transit chord; active regions are unocculted. The measured NIRISS points are an independently fitted radius-ratio spectrum with an upstream limb-darkening treatment. This experiment does not refit those original light curves. The planet and gas retain their radial proportions; the star, separation and photon-marker speed are schematic. The public tutorial compares restricted proper posteriors and retains their systematic residuals.

From grazing rays to a transit spectrum

Gold: planetary prediction. Coral: the same atmosphere behind the chosen stellar surface. White points: frozen WASP-39 b NIRISS measurements with reported errors.

Photon-weighted depth in the observed bins

Water opacity is evaluated before the nonlinear transmitted fraction and annular area. The supplied bin halfwidths define an approximate top-hat response. This is a fitted-radius spectrum comparison, with the original light-curve limb-darkening treatment upstream.

How each grazing ray becomes transparent

Four actual native wavelengths from the same gas column. Height starts at the ten-bar radius. The gold marker locates the opaque cloud; paths below it are hidden by the cloud.

Real-data conditional inference and source files

The public exercise fits all 74 frozen bins with normalized priors, compares gray-cloud and stellar-contamination models, and saves posterior predictions and residuals. The interactive controls are synthetic physical conditions, not an automatic retrieval.

Rerun the frozen NIRISS analysis

Start with the essentials

Focus question: Can a higher cloud hide water without removing the water?
One-sentence intuition: A cloud hides the deepest grazing paths. Molecular bands that already become opaque higher up can remain visible after weaker features have flattened.

Core mathematical model

Scale height near the reference radius

H_0=\frac{k_B T}{\bar m g_0}

Temperature raises the vertical distance over which pressure changes appreciably; heavier molecules and stronger gravity reduce it. The complete calculation lets gravity decrease with radius.

A grazing ray loses photons along its whole path

\tau_\lambda(b)=\int_{\mathrm{ray}}n(r)x_{\mathrm{H_2O}}\sigma_\lambda[T,P(r)]\,ds

The dimensionless optical depth adds number density times molecular fraction times cross section over path length. The same value sets the light markers and the absorbed annulus.

Partially transparent rings add an effective area

A_\lambda=\pi R_c^2+2\pi\int_{R_c}^{r_{\mathrm{top}}}(1-e^{-\tau_\lambda})b\,db

The opaque cloud contributes a disk. Each surrounding annulus contributes its area multiplied by the fraction of light it removes.

Common difficulties

An absorption band can raise a transit spectrum

Typical misconceptionAbsorption should always point downward.

Better mental modelThis plot measures blocked light, not transmitted intensity. A stronger molecular absorption makes the planet appear larger and raises the depth.

A cold patch changes the reference starlight

Typical misconceptionAnything outside the transit chord cancels from a normalized spectrum.

Better mental modelThe occulted light comes from the normal photosphere, but the out-of-transit denominator includes the cooler spectrum. Their wavelength-dependent ratio changes the measured depth.

Run the experiment

01
Lift the cloud
Start with Clear water bands. Predict which bands will survive, then drag the gold cloud ring outward and compare the band shapes.
What to observe: The weak windows fill in first. The strongest water bands stay above the opaque boundary longer.
02
Change temperature at the same composition
Reset, then raise Atmospheric temperature. Wait for the native gas calculation to finish and compare the complete spectrum and ray profile together.
What to observe: The atmospheric scale and the real temperature-dependent molecular opacity both change.
03
Move the stellar denominator
Load Spotted star, then reduce Unocculted cool-region fraction to zero. Keep the planetary spectrum fixed and compare the coral and gold curves.
What to observe: The stellar correction disappears without changing the gas column. The 3500-kelvin stress preset is a synthetic active surface.

Exoplanet Transmission Spectra, Clouds and Starspots

From grazing rays to a transit spectrum

Photon-weighted depth in the observed bins

How each grazing ray becomes transparent

Why an atmosphere changes a planet’s apparent size

Core mathematical model

Scale height near the reference radius

A grazing ray loses photons along its whole path

Partially transparent rings add an effective area

Common difficulties

An absorption band can raise a transit spectrum

A cold patch changes the reference starlight

Run the experiment

Lift the cloud

Change temperature at the same composition

Move the stellar denominator

Related experiments and articles

Starlight through a planet’s atmosphere