Angular Resolution Calculator

The small-angle formula relates angular size (theta), linear size (d), and distance (D): theta_rad = d / D. For small angles (theta less than ~10 degrees), this approximation is very accurate. In arcseconds: theta_arcsec = 206,265 * d / D, where d and D must use the same unit. In parsecs and AU: theta_arcsec = d_AU / D_pc (the definition of the parsec).

Angular diameter (also apparent size or angular size) is the angle subtended by an object at the observer's location. It depends on both the actual size and the distance. The Moon and Sun happen to have nearly equal angular diameters (~0.5 degrees or 1,800 arcseconds) from Earth even though the Sun is 400 times larger, because it is also 400 times farther away. This coincidence makes total solar eclipses possible.

Angular sizes are measured using: (1) direct imaging with telescopes, limited by angular resolution; (2) interferometry for very small angles (milli- or microarcseconds); (3) lunar occultations (timing when the Moon covers a star); (4) speckle interferometry for binary stars. The Hipparcos satellite measured stellar angular diameters down to microarcseconds via indirect means.

The human eye has an angular resolution of approximately 1 arcminute (60 arcseconds = 0.017 degrees) under good conditions. This corresponds to about 0.3 mm at 1 meter distance. The diffraction limit of the eye's pupil (about 5 mm diameter at night) gives theta = 1.22 * 550e-9 / 0.005 = 0.000134 rad = 27.7 arcseconds, so the eye operates close to its diffraction limit.

Hubble Space Telescope: aperture 2.4 m, diffraction limit at 500 nm = 0.05 arcseconds. James Webb Space Telescope: aperture 6.5 m, diffraction limit at 2 microns = 0.065 arcseconds. JWST has a larger aperture but works at longer infrared wavelengths, so its diffraction-limited resolution is similar to Hubble's at its designed wavelength range. JWST's large mirror mainly gives it much greater light-gathering power.