Exoplanet transmission spectrum

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.

Stellar contamination of a transmission spectrum arises when the spectrum behind a transiting planet differs from the spectrum averaged over the whole stellar disc. For a small opaque planet, neglecting limb darkening, the approximate depth isHere $I_{\lambda }$ is specific intensity. Unocculted cool starspots change the denominator in a wavelength-dependent way, creating apparent planetary features even if $R_p$ is constant. Crossing a starspot changes the numerator and can produce a visible anomaly in the light curve.^[ref]

Repeated exoplanet transits, monitoring of the host star, and joint stellar and planetary modelling help test this effect. Small rotational variability alone does not establish that the stellar surface is uniform.

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]

Take a planet with opaque radius $R_p$ and an atmosphere of thickness $z\ll R_p$, projected against a uniform stellar disc of radius $R_{\star }$. If the atmospheric annulus has one representative slant optical depth ${\tau }_{\lambda }$, its additional transit depth isThe general exoplanet transmission spectrum instead uses an integral over impact parameter $b$,The annulus model replaces this varying slant opacity by an effective constant. The atmospheric signal scales with optical depth when optically thin and saturates at the annular area when opaque.

An exoplanet cloud deck divides the projected atmospheric annulus into cloudy and clear radial zones. Their contributions to the exoplanet transmission spectrum add as projected areas multiplied by $1-e^{-{\tau }_{\lambda }}$ for each zone's slant optical depth.

The atmospheric scale height measures how quickly pressure decreases upward. In an isothermal ideal gas with nearly constant gravitational acceleration, hydrostatic equilibrium givesHere $k_B$ is the Boltzmann constant, $T$ the temperature, $\mu m_H$ the mean particle mass, and $g$ the gravitational acceleration. Rising by one atmospheric scale height reduces pressure and number density by a factor $e$. Hotter, lighter gas has a larger atmospheric scale height; stronger gravity compresses it.

For a thin exoplanet atmosphere, a feature spanning $N_H$ such heights contributes roughly $2R_p{N}_{H}H/R_{\star }^2$ to the exoplanet transit depth. This explains why light molecular hydrogen-rich envelopes are favourable for transit spectroscopy. It is the atmospheric application of the general scale height.

For an isothermal thin atmosphere with number density $n(z)=n_0{e}^{-z/H}$ and extinction cross-section ${\sigma }_{\lambda }$, a tangent ray near radius $R_p+z$ has height $z+{\ell }^2/(2R_p)$ at distance $\ell$ along the ray. The resulting Gaussian integral gives slant optical depthThe effective transit height is near ${\tau }_{\lambda }\sim 1$, so $z(\lambda )=H\log {\sigma }_{\lambda }+\mathrm{constant}$ if composition is constant. This explains the scattering slope of a transmission spectrum.

Let $R_p$ be a reference radius at pressure $P_0$ in an isothermal atmosphere. It can be a reference isobar on a gas giant, rather than a solid surface. Assume constant temperature $T$, constant mean molecular weight $\mu$ in units of hydrogen mass $m_H$, and approximately constant gravitational acceleration $g$. The ideal gas law and hydrostatic equilibrium giveIntegrating from the reference isobar yieldsReducing the pressure by a factor $e$ therefore increases the pressure-level radius by one atmospheric scale height. Constant gravity is a thin-atmosphere approximation over the height range being modeled.

A pressure-level radius is not automatically the observed transit radius. Transmission depends on the line-of-sight optical depth, which also requires composition and opacity. Different reference pressures or cloud assumptions can therefore assign different pressure levels to the same observed exoplanet transmission spectrum.^[ref]

The column mass between pressures $P_0$ and $P_{\mathrm{top}}$ is $(P_0-P_{\mathrm{top}})/g$. A thin atmosphere over a spherical planet therefore haswhen its top pressure is negligible.

If extinction scales as $\sigma \propto {\lambda }^{\gamma }$, an isothermal hydrostatic atmosphere has $d{R}_p/d\log \lambda =\gamma H$. The measured continuum slope can therefore estimate atmospheric temperature when gravity and mean molecular mass are known.

For $\delta =A+B(1-e^{-\tau })$ and $\tau \propto {\lambda }^{-p}$, the logarithmic slope is $-pB\tau e^{-\tau }/\delta$. The opaque planetary baseline generally dilutes the slope. In the optically thin limit, the atmospheric excess $\delta -A$ has logarithmic slope $-p$.

The day-night terminator is the boundary between the illuminated and dark hemispheres of a planet. During transit, stellar rays probe morning and evening limbs that can have different temperatures, clouds, and compositions.

If the morning and evening limbs have effective radii $R_m(\lambda )$ and $R_e(\lambda )$ and each occupies half the annulus, the transit area is $\pi [R_m^2(\lambda )+R_e^2(\lambda )]/2$.