Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S30 / 3

During a exoplanet transit, starlight traverses the planetary limb. The effective opaque radius varies with wavelength because the altitude of slant optical depth near unity depends on molecular absorption and scattering. Differencing the in-transit and out-of-transit stellar spectra measures this wavelength-dependent blocked area, the exoplanet transmission spectrum. Stellar spots and faculae can contaminate that difference and require control.

During a exoplanet secondary eclipse, the disappearance of planetary light permits subtraction of the stellar contribution. Visible planetary light is predominantly reflected starlight, sensitive to clouds, hazes, Rayleigh scattering, and atomic sodium and potassium absorption. Infrared planetary emission and limb absorption show rotational-vibrational features of species such as water, carbon monoxide, carbon dioxide and methane, together with collision-induced absorption involving molecular hydrogen. Thermal emission also depends on the atmospheric temperature-pressure profile; a temperature inversion can turn a molecular absorption feature into emission. Which species are prominent depends on composition, temperature and the observed band, rather than being universal to every planet.

The measured mass and radius implyCompared with the supplied composition curves, this is inconsistent with a bare iron-rich or ordinary rock-only planet. A volatile-rich interior and/or an extended low-density gas envelope is needed. A water-rich body with an atmosphere and a rocky core with a modest H/He envelope are different plausible families; mass and radius do not uniquely determine the composition.^[ref]

For the requested H/He estimate, assume a hydrostatic, approximately isothermal terminator at the zero-Bond-albedo, fully redistributed equilibrium temperature. The orbit and temperature areSurface gravity is $g=Gm/R_p^2\simeq 9.58\mathrm{m}{\mathrm{s}}^{-2}$. With molar mass $\mathcal{M}=2.2\times 10^{-3}\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$, the atmospheric scale height isA spectral feature spanning $N$ scale heights changes the blocked stellar fraction by the atmospheric annulus area:Thus one scale height is about $338$ parts per million in transit depth; a clear strong feature spanning roughly five scale heights is of order $1.7\times 10^{-3}$. The number of scale heights requires the actual opacity ratio; $N\simeq \ln ({\kappa }_{\mathrm{line}}/{\kappa }_{\mathrm{cont}})$ is a useful isothermal estimate, not a supplied measured value. The true terminator temperature need not equal this redistribution estimate.

A nearly flat spectrum can result from a high-mean-molecular-mass atmosphere with a much smaller scale height, or from optically thick clouds/hazes that hide the deeper absorption features. These are the two qualitative possibilities; detailed precision data can discriminate them. In particular, the pre-2015 HST study favored clouds over the clear high-molecular-mass models it tested, so listing both physical mechanisms is not a claim that all such models fit those data equally well.^[ref]

The circumstellar habitable zone is the range of stellar irradiation at which a suitable rocky planet could maintain liquid water at its surface under specified atmospheric conditions. It is not a guarantee of life or even of liquid water on every planet at that distance. Its boundaries depend on greenhouse gases, clouds, humidity, pressure, albedo, rotation and the stellar spectrum; stellar evolution, ultraviolet activity, atmospheric loss and tidal locking can complicate applications to cool stars.

Approximating both stars as blackbodies, the host luminosity relative to the Sun isThe Earth-irradiance orbit therefore hasUse the largest projected separation and a visible reference wavelength of $550\mathrm{nm}$. The Rayleigh criterion for a circular aperture givesThis is only a diffraction-based minimum: detecting a faint planet beside its star additionally requires contrast suppression and adequate photon counts. An interferometer baseline, rather than its collecting-aperture diameter, sets its angular resolution.

For reflected light the reflected-light planetary phase curve gives the flux ratio $ϵ=A_g\Phi (\alpha )(R_{\oplus }/a_{\mathrm{E}}{)}^2$, where $A_g$ is geometric albedo and $\Phi (0)=1$. Numerically,A unit-geometric-albedo reference gives about $6\times 10^{-7}$; an Earth-like chosen $A_g=0.3$ gives $1.7\times 10^{-7}$. Albedo and phase function were not specified, so an exact contrast cannot be inferred just from the radius. Full-phase brightness occurs close to zero projected separation for an edge-on orbit; at the best-resolved quadrature, a Lambert sphere has $\Phi (\pi /2)=1/\pi$, reducing the Earth-albedo example to $5.4\times 10^{-8}$. Thus the best angular separation and maximum reflected contrast are not generally achieved at the same phase.