[1] The components for proper motion in the equatorial coordinate system (of a given epoch, often J2000.0) are given in the direction of right ascension (μα) and of declination (μδ).
However, precise long-term observations show that such constellations change shape, albeit very slowly, and that each star has an independent motion.
[7][8] Any proper motion is a two-dimensional vector (as it excludes the component as to the direction of the line of sight) and it bears two quantities or characteristics: its position angle and its magnitude.
Proper motion may alternatively be defined by the angular changes per year in the star's right ascension (μα) and declination (μδ) with respect to a constant epoch.
Thus, a co-efficient is given to negate the misleadingly greater east or west velocity (angular change in α) in hours of Right Ascension the further it is towards the imaginary infinite poles, above and below the earth's axis of rotation, in the sky.
Barnard's Star's transverse speed is 90 km/s and its radial velocity is 111 km/s (perpendicular (at a right, 90° angle), which gives a true or "space" motion of 142 km/s.
In 1992 Rho Aquilae became the first star to have its Bayer designation invalidated by moving to a neighbouring constellation – it is now in Delphinus.
It is possible to construct nearly complete samples of high proper motion stars by comparing photographic sky survey images taken many years apart.
In the past, searches for high proper motion objects were undertaken using blink comparators to examine the images by eye.
Studies of this kind show most of the nearest stars are intrinsically faint and angularly small, such as red dwarfs.
[23] Proper motion was suspected by early astronomers (according to Macrobius, c. AD 400) but a proof was not provided until 1718 by Edmund Halley, who noticed that Sirius, Arcturus and Aldebaran were over half a degree away from the positions charted by the ancient Greek astronomer Hipparchus roughly 1850 years earlier.