Past exam of the short options paper of the physics Prelims of the University of Oxford / 2021 / S02 / 2

The intact source figure gives orbital period $P=6.0\mathrm{yr}$, stellar radial velocity semiamplitude $K=30.0\mathrm{m}{\mathrm{s}}^{-1}$, and orbital eccentricity $e=0$. A planetary exoplanet transit implies a nearly edge-on orbit, so $\sin i\simeq 1$. Adopt a solar mass for the Solar-type star and initially neglect the companion's mass relative to it.

For a circular solution of the two-body problem, the star's distance from the centre of mass is $a_{\ast }=a{M}_p/(M_{\ast }+M_p)$, and the measured semiamplitude is $K=2\pi a_{\ast }\sin i/P$. Kepler's third law givesHenceThe inferred binary mass ratio is only $0.00183$, validating the approximation. Since $e=0$, the star–planet separation is constant and equals the orbital semimajor axis. Without the exoplanet transit, Doppler spectroscopy would generally give only a minimum companion mass through $M_p\sin i$.

Assume uniform reradiation over the planetary surface, unit thermal emissivity, and zero Bond albedo. The planet absorbs over its projected disc and emits over its full sphere. Radiative equilibrium and the Stefan–Boltzmann law giveUsing the liquid-water range $273.15\le T_p\le 373.15\mathrm{K}$ as the crude thermal criterion givesFor nonzero Bond albedo $A_B$, both radii acquire the factor $\sqrt{1-A_B}$. For example $A_B=0.3$ gives approximately $0.470$ to $0.877\mathrm{AU}$. An airless planet need not have uniform surface temperature; inefficient heat redistribution makes the dayside hotter and the nightside colder, so a single equilibrium temperature is an averaging approximation.

This simple habitable zone contains the Earth's orbit under the zero-albedo assumption, but also contains Venus's orbit, although its greenhouse effect produces an uninhabitably hot surface. Mars lies beyond the calculated outer boundary. These comparisons show the usefulness of the orbital scale and the limitations of a temperature-only criterion. The stipulated airless model also omits the atmospheric pressure needed for stable surface liquid water: this is an effective thermal window, not a sufficient habitability condition.

Transit photometry. A planet crossing the stellar disc produces a periodic reduction in flux, approximately $\delta =(R_p/R_{\ast }{)}^2$ for a small, dark planet away from the limb. The orbital period and stellar radius allow an inferred planetary radius, but not its mass by this method alone. The geometric geometric transit probability is roughly $(R_{\ast }+R_p)/a$ for a circular orbit, so the method strongly favours close orbits and large planets. Small signals compete with instrumental noise and stellar activity; eclipsing stellar binaries and blended sources can mimic a planetary exoplanet transit, and long orbital periods require long observing baselines.

Doppler spectroscopy. Periodic shifts of stellar spectral lines reveal the stellar radial velocity caused by motion around the centre of mass. For a light companion, $K\propto M_p\sin i{P}^{-1/3}{M}_{\ast }^{-2/3}/\sqrt{1-e^2}$, favouring massive, short-period companions and low-mass host stars. Unknown orbital inclination gives a minimum mass; stellar activity, line broadening, and instrumental calibration limit precision. Long orbital periods require years of measurements.

Direct imaging. Spatially resolved reflected light or thermal emission separates a planet from its host. The large star–planet brightness contrast and small angular separation demand starlight suppression and precise correction of optical aberrations. The method favours wide orbits around nearby stars, and young, hot, massive planets with stronger thermal emission. Translating brightness to mass depends on planetary cooling models and age; it is difficult to image an Earth analogue close to a bright star.