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The change of an optical propagation angle with wavelength. A diffraction grating at normal incidence has $d\sin \theta =m\lambda$, so differentiating gives $d\theta /d\lambda =m/(d\cos \theta )$. This converts a small spectral-line separation into angular separation, but resolving the two images also requires sufficient aperture and a narrow source image.

A spectrometer that resolves spectral lines using the angular dispersion of a diffraction grating. A narrow slit at a collimator focal plane supplies nearly parallel illumination; a telescope or focusing lens forms separate images for different outgoing angles. At normal incidence, the grating equation is $d\sin \theta =m\lambda$. The slit width, illuminated groove count, aberrations and detector sampling jointly determine whether a calculated angular separation is actually resolved.

Treat the given orbital separation as the projected separation on the sky. The small angular separation isFor an ideal circular telescope aperture of diameter $D$, the Rayleigh criterion requires $1.22\lambda /D\le \theta$. ThereforeThis is the diffraction limit, assuming comparable image brightness and adequate sampling. Atmospheric image blurring can require correction or space-based observing. A smaller actual projected separation of the binary star would require a larger aperture.

Transit photometry. A planet crossing the stellar disc produces a periodic reduction in flux, approximately $\delta =(R_p/R_{\ast }{)}^2$ for a small, dark planet away from the limb. The orbital period and stellar radius allow an inferred planetary radius, but not its mass by this method alone. The geometric geometric transit probability is roughly $(R_{\ast }+R_p)/a$ for a circular orbit, so the method strongly favours close orbits and large planets. Small signals compete with instrumental noise and stellar activity; eclipsing stellar binaries and blended sources can mimic a planetary exoplanet transit, and long orbital periods require long observing baselines.

Doppler spectroscopy. Periodic shifts of stellar spectral lines reveal the stellar radial velocity caused by motion around the centre of mass. For a light companion, $K\propto M_p\sin i{P}^{-1/3}{M}_{\ast }^{-2/3}/\sqrt{1-e^2}$, favouring massive, short-period companions and low-mass host stars. Unknown orbital inclination gives a minimum mass; stellar activity, line broadening, and instrumental calibration limit precision. Long orbital periods require years of measurements.

Direct imaging. Spatially resolved reflected light or thermal emission separates a planet from its host. The large star–planet brightness contrast and small angular separation demand starlight suppression and precise correction of optical aberrations. The method favours wide orbits around nearby stars, and young, hot, massive planets with stronger thermal emission. Translating brightness to mass depends on planetary cooling models and age; it is difficult to image an Earth analogue close to a bright star.