Past exam of the frontiers of Observational Astrophysics course of the University of Cambridge / 2012 / Paper 1 / 3 / b

The radial-velocity method measures repeated Doppler shifts of stellar spectral lines as a star responds to an orbiting companion. A stable high-resolution spectrograph and wavelength reference distinguish stellar motion from instrumental drift; many lines or a spectral template improve precision. A Keplerian fit yields period, eccentricity, phase and velocity semi-amplitude. For a companion mass $m_p$, stellar mass $M_{\ast }$ and inclination $i$,With a stellar-mass estimate and $m_p\ll M_{\ast }$, this measures approximately $m_p\sin i$, a minimum mass unless the inclination is independently constrained. A nearly face-on orbit can hide a massive companion. The method preferentially detects larger masses, shorter periods and favorable inclinations.

Starspots, activity, pulsations, rotation, blended spectra and sampling aliases can imitate or distort a periodic signal. Line-shape diagnostics, activity indicators, repeated observations and independent constraints help test a planetary explanation. Photon noise and spectral line width constrain precision even for a perfectly calibrated spectrograph. Multiple planets or incomplete orbital coverage require an adequate model and longer monitoring; a periodic centroid shift alone is not a proof of a particular planetary system.