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Doppler spectroscopy detects an exoplanet through the host star's changing Doppler shift. Both bodies orbit their shared astronomical barycenter: the star undergoes stellar reflex motion even when the exoplanet itself is unseen. Its velocity along the line of sight shifts its spectral lines; at speeds small compared with the speed of light, $\Delta \lambda /\lambda \simeq v_r/c$, with positive $v_r$ for recession. Repeated measurements can fit a Kepler orbit, rather than merely detect one shifted spectrum. Starspots and other changes in stellar spectral lines must be distinguished from actual orbital motion.

For one companion in an isolated two-body problem, the stellar radial-velocity semi-amplitude isHere $G$ is the gravitational constant, $P$ the orbital period, $e$ the orbital eccentricity, and $M_{\star },M_p$ the star and exoplanet masses. The orbital inclination $i$ is measured relative to the sky plane: $i=0$ is face-on and $i=90^{\circ }$ is an edge-on orbit. The exact binary mass function measured from the fitted orbit isWith known $M_{\star }$ and $M_p\ll M_{\star }$, this is usually reported as $M_p\sin i$. Taking $\sin i=1$ gives the minimum companion mass; a smaller orbital inclination allows a larger mass. A detected exoplanet transit constrains $i$ close to $90^{\circ }$ for a compact planet orbiting well outside the star's surface, substantially resolving this ambiguity. Combining Doppler spectroscopy and exoplanet transit photometry then supplies a mass and radius for exoplanet interior studies.

For a computed circular example, let $q=M_p/M_{\star }=0.001$ and choose a relative orbital speed $V_{\mathrm{rel}}=100\mathrm{km}{\mathrm{s}}^{-1}$: this is the speed of the planet relative to the star, not its speed about the astronomical barycenter. Conservation of momentum givesThe barycentric velocities point in opposite directions, so their speeds add to $V_{\mathrm{rel}}$. Here $e=0$ and $K=V_{\star }\sin i$: $K=99.9001\mathrm{m}{\mathrm{s}}^{-1}$ at $i=90^{\circ }$, $49.9500\mathrm{m}{\mathrm{s}}^{-1}$ at $i=30^{\circ }$, and zero at $i=0$. The star still orbits in the face-on case, but its orbital velocity has no line-of-sight component. The plotted circular signal is $v_r(t)-\gamma =K\sin (2\pi t/P)$ for a chosen phase origin and constant system velocity $\gamma$.

Methods of detecting exoplanets infer or resolve an exoplanet through its effects on the host star's light or through its own light. The observable determines what can be learned:

An exoplanet secondary eclipse measures the loss of the exoplanet's light when it passes behind its star, complementing exoplanet transit photometry. A detection method also selects which systems are easiest to find: deeper exoplanet transits favour larger radius ratios, and larger stellar radial-velocity semi-amplitudes favour more massive companions, shorter orbital periods, and larger $\sin i$. These preferences must be accounted for when inferring the underlying exoplanet population.

Stellar reflex motion. A planet and its host orbit their common barycentre. The star's line-of-sight velocity shifts its absorption lines by $\Delta \lambda /\lambda \simeq v_r/c$ at low speeds. A time series can be fitted withHere $\nu (t)$ is true anomaly, $\varpi$ the argument of periastron and $\gamma$ an arbitrary systemic velocity offset. At small companion mass and known stellar mass, the orbit supplies $M_p\sin i$, not the true planet mass without an inclination constraint. Long monitoring helps distinguish a reproducible Keplerian signal from stellar oscillations, magnetic activity, spots and instrumental trends.

The Lick reference-cell implementation. Starlight passes through a temperature-controlled cell of molecular iodine before entering the Hamilton high-resolution echelle spectrograph. The cell imprints a dense pattern of reference absorption lines on the stellar spectrum, predominantly in the approximately $500$–$630\mathrm{nm}$ interval. These lines share the same exposure and downstream optical path as the stellar light. A calibrated high-resolution Fourier-transform iodine spectrum and an iodine-free stellar template allow the wavelength solution and instrumental line-spread function to be fitted along with the stellar Doppler shift. The method corrects shifts and line-shape changes that a separate calibration exposure would not necessarily follow.^[ref]

A useful forward model for one short spectral segment isfollowed by pixel integration, continuum scaling and a fitted wavelength calibration. Here $S$ is the intrinsic stellar spectrum, $T_{{\mathrm{I}}_2}$ the cell transmission and $\mathcal{P}$ the instrumental line-spread function. The spectra multiply before convolution; simply dividing a recorded blended spectrum by the laboratory iodine spectrum is not equivalent to this model. The iodine pattern is stationary in the observatory frame while stellar lines move relative to it. Separate spectral segments accommodate wavelength dependence of the instrumental profile.

Earth's rotational and orbital velocities must then be removed with a barycentric correction using accurate timing, ideally weighted by photon arrival rate. Many stellar lines constrain a shift smaller than one instrumental resolution element; spectral resolution alone is not the velocity error. Photon statistics, template quality, instrumental-profile modelling and intrinsic stellar variability limit the precision. The method measures changes relative to a template and does not by itself provide a perfectly absolute centre-of-mass velocity. No particular discovery, modern precision floor or historical observing performance is assumed here.