propermotion

Proper motion is the observed angular change in a star’s position on the celestial sphere over time, caused by the star’s actual motion through space relative to the Sun. It is measured as an angular velocity, typically in milliarcseconds per year (mas/yr). Proper motion is a two‑dimensional vector on the sky, usually described by components in right ascension and declination, often expressed as mu_alpha cos delta and mu_delta. The total proper motion mu is the length of this vector.

The relation to a star’s transverse velocity is given by v_t = 4.74 * mu * d, where mu is the total proper motion in arcseconds per year and d is the distance in parsecs, yielding v_t in kilometers per second. Nearby stars can exhibit large proper motions; for example, Barnard’s Star moves about 10.3 arcseconds per year, while many distant stars show only a few milliarcseconds per year. Proper motion is an intrinsic motion of the star and is distinct from parallax, the apparent shift due to Earth’s orbit around the Sun.

Proper motion complements parallax: parallax measures distance, whereas proper motion measures lateral motion. When combined with distance, proper motion provides the transverse component of a star’s space velocity; together with radial velocity, it yields the full three-dimensional motion through the Galaxy.

Measurements rely on long time baselines and precise astrometry. Space missions such as Hipparcos and Gaia have provided high-precision proper motions for millions of stars, enabling studies of Galactic kinematics, stellar populations, and cluster membership. Limitations include small angular shifts for distant objects, measurement errors, and projection effects that must be corrected for accurate results.