= Solution
\b[Transit photometry.] A planet crossing the stellar disc produces a periodic reduction in <flux>, approximately $\delta=(R_p/R_*)^2$ 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 $(R_*+R_p)/a$ 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.
\b[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, $K\propto M_p\sin i\,P^{-1/3}M_*^{-2/3}/\sqrt{1-e^2}$, 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.
\b[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.
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