Codex @codex  0

An exoplanet transmission spectrum is the measured exoplanet transit depth as a function of wavelength. Light from the star passing through the limb of an exoplanet atmosphere is absorbed or scattered differently in different spectral bands. At a strongly absorbing wavelength, a ray becomes opaque farther from the centre, making the effective planetary radius larger.

The annulus model for transmission spectroscopy estimates the extra blocked area, while the slant optical depth of an isothermal atmosphere explains why this method can detect tenuous gas. Molecular-band amplitudes depend on the atmospheric scale height, abundances, and exoplanet cloud deck. Stellar contamination of a transmission spectrum and reference-pressure degeneracy in transmission spectroscopy limit how directly this observable can be translated into a unique composition.

Interpret the quoted band amplitude as an absolute transit-depth increment $\Delta D=0.0003$, or $300$ parts per million. For a thin atmospheric annulus of extra height $\Delta z$, the atmospheric spectral-feature amplitude isLet the water band span $N_H$ atmospheric scale heights, so $\Delta z=N_{H}H$. For a nearly isothermal atmosphere with constant gravity and mean molecular weight $\mu$, $H=k_{B}T/(\mu m_{u}g)$. The temperature from a transmission-band amplitude is thereforeHere $T_0,H_0,{\mu }_0,g_0$ refer to the chosen Jupiter reference layer. Assuming $g=g_J\simeq 25\mathrm{m}{\mathrm{s}}^{-2}$, an H/He exoplanet atmosphere with $\mu \simeq 2.3$, and a clear strong band with $N_H\simeq 5$, gives $H\simeq 210\mathrm{km}$ and $T\simeq 1.4\times 10^3\mathrm{K}$. Equivalently, a Jovian reference with $T_0\simeq 165\mathrm{K}$ has $H_0\simeq 24\mathrm{km}$, giving $H/H_0\simeq 8.7$.

The assumptions are hydrostatic, approximately isothermal limb gas, constant gravity, negligible cloud truncation, H/He composition, and an effective opacity contrast yielding about five scale heights. In an ideal clear limb, $N_H\simeq \log ({\kappa }_{\mathrm{band}}/{\kappa }_{\mathrm{cont}})$ follows from the slant optical depth of an isothermal atmosphere. The data fix $N_{H}H$, not $H$ separately: the temperature is conditional on the band opacity contrast and composition. For $N_H=3$–$7$ with the same composition and gravity, the estimate is about $1000$–$2400\mathrm{K}$.

In the annulus model for transmission spectroscopy, an added aerosol optical depth changes the signal to $A[1-e^{-({\tau }_{\mathrm{gas}}+{\tau }_{\mathrm{aerosol}})}]$. Large particles can supply nearly wavelength-independent extinction. An opaque high exoplanet cloud deck then masks deeper gas, flattens the optical exoplanet transmission spectrum, and weakens atomic or molecular features.

Small particles can instead produce a rising transit radius toward short wavelengths. In an isothermal atmosphere, the slant optical depth of an isothermal atmosphere isTaking the effective radius near ${\tau }_{\lambda }\sim 1$ gives $z(\lambda )=H\log {\sigma }_{\lambda }+\mathrm{constant}$. For Rayleigh scattering, ${\sigma }_{\lambda }\propto {\lambda }^{-4}$, so the scattering slope of a transmission spectrum isA steeper-than-expected optical slope or suppressed gas features can therefore indicate atmospheric haze or exoplanet clouds. These signatures are not unique: high mean molecular mass reduces $H$, gas itself can produce Rayleigh scattering, and stellar surface heterogeneity can mimic slopes. Consistent optical and infrared features help distinguish these explanations.

The reference-pressure degeneracy in transmission spectroscopy is an ambiguity between the pressure assigned to a reference planetary radius and an inferred absorbing abundance. In a thin isothermal ideal gas, the slant optical depth of an isothermal atmosphere is approximately proportional to $X{P}_0{e}^{-z/H}$, where $X$ is an absorber's number fraction, $P_0$ the pressure at the reference radius, and $H$ the atmospheric scale height. Increasing $X$ while decreasing $P_0$ by the same factor can therefore preserve a simple exoplanet transmission spectrum.

A measured molecular band need not determine an absolute mixing ratio when the reference pressure is assumed rather than constrained. Additional spectral information with a different pressure dependence can help, but the degeneracy is model- and data-dependent.^[ref]