Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 / 1

For the two-zone cloudy annulus transmission model, assume a uniform stellar disc, an opaque planet below $R_p$, $0\le H_c\le H$, and constant total slant optical depth within each radial zone. Neglect scattered light entering the beam and the planet's own emission. A ray transmits the fraction $e^{-\tau }$, so each zone blocks its projected area multiplied by $1-e^{-\tau }$.

The opaque disc, cloudy annulus and clear annulus have areas $\pi R_p^2$, $\pi [(R_p+H_c{)}^2-R_p^2]$ and $\pi [(R_p+H{)}^2-(R_p+H_c{)}^2]$, respectively. Thus the exoplanet transmission spectrum, expressed as total transit depth, isHere ${\tau }_{\lambda ,c}$ means the total optical depth on rays assigned to the cloudy zone. If it denotes cloud extinction alone, its exponent must instead contain the sum of cloud and gas optical depths. For a geometrically thin exoplanet atmosphere, the two atmospheric prefactors become $2R_p{H}_c/R_s^2$ and $2R_p(H-H_c)/R_s^2$.

An opaque exoplanet cloud deck gives the simpler expressionThe exoplanet cloud deck raises the wavelength-independent occulting radius and suppresses the clear atmospheric contribution. If $H_c=H$, the spectrum is flat at $(R_p+H{)}^2/R_s^2$; if $H_c=0$, the ordinary annulus model for transmission spectroscopy is recovered. Stellar limb darkening and varying tangent-ray optical depth would require an intensity-weighted radial integral rather than this constant-depth model.

Assume an isothermal atmosphere in hydrostatic equilibrium, with constant gravity $g$, mean molecular weight $\mu$, and an ideal gas equation of state. Use $h$ for the atmospheric scale height, to distinguish it from the total observable height $H$. Thenso the cloud-top pressure in an isothermal atmosphere isAn opaque, approximately grey exoplanet cloud deck can flatten the exoplanet transmission spectrum if the remaining gas above it has little wavelength-dependent extinction. If its top nearly reaches the observable atmospheric extent, $H_c\simeq H$, this gives $P_c\simeq P_s{e}^{-H/h}$. More generally, a top at $N$ atmospheric scale heights has $P_c/P_s=e^{-N}$; illustrative values $N=5$ and $10$ give $6.7\times 10^{-3}$ and $4.5\times 10^{-5}$.

A flat exoplanet transmission spectrum and $P_s$ alone do not determine a unique cloud-top pressure. The temperature, gravity, mean molecular weight and cloud altitude, or equivalent opacity information, are also needed. A large mean molecular weight can itself suppress spectral features, so the exoplanet cloud deck interpretation is conditional.

Assume that the optical continuum is dominated by Rayleigh scattering by particles much smaller than the wavelength, and that their column abundance is fixed. Their cross-section, hence the slant optical depth, scales as ${\tau }_{\lambda }\propto {\lambda }^{-4}$. Differentiating the annulus model for transmission spectroscopy exactly gives the logarithmic slope of total transit depthFor $A\ge 0$ and $B>0$, the slope lies between $-4$ and zero, but generally depends on wavelength. In the optically thin limit, ${\delta }_{\lambda }\simeq A+B{\tau }_{\lambda }$, so its slope is $-4B{\tau }_{\lambda }/(A+B{\tau }_{\lambda })$. It vanishes as the atmospheric signal becomes negligible compared with the opaque disc; it also approaches zero in the saturated optically thick limit.

The familiar value $m\simeq -4$ applies to the optically thin atmospheric excess,or formally to a model in which $A$ is negligible. For an ordinary thin exoplanet atmosphere, however, $B/A\simeq 2H/R_p\ll 1$, so neglecting the opaque planetary baseline in the total transit depth is not usually justified. A universal $m=-4$ for the logarithm of the total transit depth does not follow from the stated model.

In a stratified isothermal atmosphere, the related scattering slope of a transmission spectrum is instead $d{R}_{\mathrm{tr}}/d\ln \lambda =-4h$, obtained by moving the effective tangent level to keep its slant optical depth of order unity. Consequently $d\ln \delta /d\ln \lambda =-8h/R_{\mathrm{tr}}$ there. The radius slope, atmospheric-excess slope and total-depth slope are different observables.