= Solution
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> $M_*$ and inclination $i$,
$$
\boxed{K=\left(\frac{2\pi G}{P}\right)^{1/3}
\frac{m_p\sin i}{(M_*+m_p)^{2/3}\sqrt{1-e^2}}.}
$$
With a stellar-mass estimate and $m_p\ll M_*$, 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.
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