Detecting planets through stellar wobble
A planet orbiting a star exerts gravitational pull on that star, causing it to orbit the planet's center of mass rather than remaining fixed. The star's motion is tiny (Jupiter causes the Sun to wobble by 12 meters per second), but modern spectrographs detect it through the Doppler shift of starlight. When the star moves toward Earth, its light shifts slightly blue (shorter wavelengths); when it moves away, the light redshifts. The magnitude and period of this shift reveal the planet's orbital period and minimum mass.
The radial velocity method is most sensitive to massive planets in tight orbits, because their gravity causes larger stellar wobbles with shorter, easier-to-measure periods. Hot Jupiters (giant planets close to their star) were the first exoplanets discovered in 1995 using this method. Small, distant planets like Earth around a Sun-like star produce wobbles below current detection limits.
Complementary to transit photometry
Radial velocity measures the planet's minimum mass (because the angle of the orbit is unknown). The transit method (observing brightness dips as planets cross their star's disk) provides the planet's radius. Combining both methods gives density and composition constraints: a low-density giant indicates a gas or ice giant; a high-density compact object suggests a rocky terrestrial planet.
Modern planet detection uses both methods routinely. Space telescopes like TESS discover transiting planets, while ground-based spectrographs perform radial velocity follow-up, building a comprehensive picture of planetary systems.