Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S30 / 1

A exoplanet transit occurs when a planet crosses the projected stellar disk and blocks some stellar flux. For a circular orbit with isotropically distributed angular-momentum directions, $\cos i$ is uniform on $[0,1]$. A transit requires $a|\cos i|<R_{\star }+R_p$, soFor a dark planet, a uniformly bright stellar disk and negligible dilution, the transit exoplanet transit depth is $\boxed{\delta \simeq (R_p/R_{\star }{)}^2}$. For $R_{\star }/a\ll 1$, a small planet and impact parameter $b=a\cos i/R_{\star }$,The central-transit result sets $b=0$. A first-to-fourth-contact duration retaining planet size replaces $\sqrt{1-b^2}$ by $\sqrt{(1+R_p/R_{\star }{)}^2-b^2}$ and includes the exact projected-speed geometry when needed. Eccentricity, limb darkening and grazing geometry modify these expressions.

The common one-per-cent exoplanet transit depth gives $R_p=0.1R_{\star }$. Apply the supplied main-sequence mass-radius relation, $R_{\star }/R_{\odot }=M_{\star }/M_{\odot }$, and Kepler's third law, neglecting the planet mass. For $P=6$ days,The companions are approximately Jupiter-sized and Earth-sized respectively; these are sizes, not independently determined masses. The duration uses the small-planet chord approximation. Counting the finite planet radius would increase first-to-fourth-contact durations by about ten per cent for central transits.

For equal distance, passband, cadence and noise dominated by stellar photon counting, the supplied mass-luminosity relation makes the solar-mass host about $10^3$ times more luminous, with a transit about $4.64$ times longer. At equal fractional exoplanet transit depth, signal-to-noise per event scales as $\delta \sqrt{{\dot{N}}_{\gamma }{t}_{\mathrm{tr}}}$, favouring the solar-mass host by about $\sqrt{1000\times 4.64}\simeq 68$. Equal apparent brightness would remove the luminosity advantage but retain the duration advantage. Without distances and noise properties the relative detectability is conditional. Transit probability is also larger for the solar-mass case, about $7.2\%$ versus $1.55\%$.

To distinguish the companion, measure the host's radial velocity reflex motion to infer a mass; a low-mass star generally produces a much larger signal than a planet. Also test for a detectable secondary eclipse, colour-dependent dilution, doubled spectral lines and out-of-eclipse ellipsoidal variation. High angular-resolution imaging and shifts of the image centroid can identify a blended background eclipsing binary. A Jupiter-sized radius alone is not decisive because low-mass stars and brown dwarfs can have comparable radii. The three labelled effects on the transit are treated individually below.

Reducing orbital inclination from exactly edge-on increases the impact parameter and shortens the chord, giving $t_{\mathrm{tr}}\propto \sqrt{1-b^2}$ in the small-planet, circular approximation. For a uniformly bright disk, a fully internal transit retains approximately the same exoplanet transit depth $(R_p/R_{\star }{)}^2$. Once grazing, the occulted area and exoplanet transit depth decrease; for $b>1+R_p/R_{\star }$ the exoplanet transit disappears. With limb darkening, a higher-$b$ path is also typically shallower before grazing.

Limb darkening makes the central stellar surface brighter than the disk mean. A small planet at projected position $\mathbf{x}$ blocks fractional flux approximately $\delta (\mathbf{x})=(R_p/R_{\star }{)}^{2}I(\mathbf{x})/⟨I⟩$. A central transit is therefore deepest near mid-transit and rounded rather than flat-bottomed; a near-limb chord can be shallower. The geometrical contact times do not change for fixed radii and orbit, although a threshold-based measured duration can change because ingress and egress are fainter.

As the star expands onto the red giant branch, a planet of fixed radius produces a shallower exoplanet transit, $\delta \propto R_{\star }^{-2}$. For a fixed circular orbit and an approximately central chord, duration grows approximately as $R_{\star }$, and transit probability also rises. At fixed orbital inclination the impact parameter in stellar-radius units decreases, further changing the chord. These statements neglect orbital evolution, stellar mass loss, tidal effects and eventual engulfment, all of which can matter for a very close planet.

The habitable zone is the range of orbital distances where an appropriate planetary atmosphere could sustain liquid water at the surface; it is not a guarantee of inhabited or even habitable conditions for every planet. At a fixed habitable incident flux, $a_{\mathrm{HZ}}\propto L_{\star }^{1/2}$. With the supplied $L_{\star }\propto M_{\star }^3$, Kepler's third law givesTaking Earth as a rough central reference, a $0.1M_{\odot }$ star has $a_{\mathrm{HZ}}\sim 0.0316\mathrm{AU}$ and $P_{\mathrm{HZ}}\sim 6.5$ days. Its six-day, Earth-sized companion receives about $1.11$ times terrestrial irradiance and could therefore lie in the habitable zone in this simplified model. Atmosphere, stellar activity, tidal locking and volatile retention remain unknown. The six-day companion around the Sun-like star receives about $239$ times terrestrial irradiance and is approximately Jupiter-sized; it is not a plausible surface-water Earth analogue. The stellar spectral dependence of actual habitable-zone edges is omitted by the simple scaling.