Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 3

For a plane-parallel atmosphere with optical depth increasing downward, the radiative transfer equation isIntegrating over solid angle gives the first radiation-field momentIn radiative equilibrium, matter has no net local radiative heating, so the opacity-weighted frequency integral of $J_{\nu }-S_{\nu }$ vanishes. After converting each optical-depth derivative to physical depth and integrating over frequency,Thus the bolometric internal flux is constant with depth, as stated by constant flux in a plane-parallel radiative-equilibrium atmosphere.

In local thermal equilibrium $S_{\nu }=B_{\nu }(T)$. Define the planet's internal effective temperature of a planet by the constant outward flux. The standard Planck law integral givesThe corresponding intrinsic luminosity is $L_{\mathrm{int}}=4\pi R_p^2\sigma T_{\mathrm{eff}}^4$.

Let the atmosphere occupy a thin annulus from $R_p$ to $R_a=R_p+\Delta R$. With mass extinction coefficient $k_{\nu }$, constant density $\rho$, and the assumed common chord length $l$, its optical depth isThe opaque solid planet removes area $\pi R_p^2$, while the annulus removes the fraction $1-e^{-{\tau }_{\nu }}$ of the incident specific intensity. Neglecting limb darkening, the exoplanet transmission spectrum is thereforeFor a thin annulus,If the geometrical thickness is estimated as $\Delta R=N_{H}H$, then the atmospheric scale height is $H=k_B{T}_p/(\mu m_{H}G{M}_p/R_p^2)$. A wavelength-independent $k_{\nu }$ makes this idealized spectrum flat; real molecular opacities create its features.