A planet doesn't just orbit its star — the star orbits the shared centre of mass too, tracing a small ellipse of its own. As the star swings toward and away from Earth, its spectral lines shift blue then red by the Doppler effect. This is how the first exoplanet around a sun-like star (51 Pegasi b) was found in 1995, and it is still how most confirmed masses are measured today.
Radial velocity: v_r(t) = K [cos(ν(t)) + e] (argument of periastron ω = 0)
Semi-amplitude: K = (2πG/P)^(1/3) · (M_p sin i) / (M_★+M_p)^(2/3) · 1/√(1-e²)
Kepler's eqn: E − e·sin E = M(t) = 2πt/P (mean anomaly → eccentric anomaly E)
True anomaly: ν = 2·atan2(√(1+e)·sin(E/2), √(1-e)·cos(E/2))
- Mp — heavier planets pull the star around a wider, faster ellipse, raising K.
- P — a tighter, faster orbit (short P) gives a larger K for the same mass.
- e — eccentric orbits produce a sharp, asymmetric "sawtooth" RV curve instead of a smooth sine.
- i — only the line-of-sight component of the star's velocity is measured. At i = 90° (edge-on) the full wobble shows; at i → 0° (face-on) the star moves entirely across the sky and K → 0. Because i is usually unknown, spectroscopy alone only ever measures Mp sin i — a true lower limit on the planet's mass, the "sin i degeneracy".
The star's orbit here is exaggerated for visibility — a real Jupiter-mass planet moves its Sun-like host by only a few metres per second, tens of times smaller than a pixel at this scale, which is exactly why detecting it takes a precision spectrograph, not a telescope image.