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 / c

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.