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The proper motion of a star is the measurement of its change in position in the sky over time after improper motions are accounted for. This contrasts with radial velocity which is the time rate of change in distance toward or away from the viewer.

Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each other, so that they form the same constellations over historical time. Ursa Major, for example, looks nearly the same now as it did hundreds of years ago. However, precise long-term observations show that the constellations change shape, albeit very slowly, and that each star has an independent motion.

This motion is caused by the true movement of the stars relative to the Sun and solar system through space. It is measured by two quantities: the proper motion angle and the proper motion itself. The first quantity indicates the direction of the proper motion on the celestial sphere (with 0 degrees meaning the motion due north, 90 degrees due east, and so on), and the second quantity gives the motion's magnitude, in seconds of arc per year.

Proper motion may also be given by the angular components in the right ascension (μ_α) and declination (μ_δ). The net proper motion (μ) is given by:^[1]

where δ is the declination. The proper motion angle (θ) is related to these components by:[1]

Barnard's Star, showing position every 5 years 1985–2005.

Barnard's star has the largest proper motion of all stars, moving at 10.3 seconds of arc per year. Large proper motion is usually a strong indication that a star is relatively close to the Sun. This is indeed the case for Barnard's Star which, at a distance of about 6 light years, is, after the Sun and the Alpha Centauri system, the nearest known star to Earth (yet, being a red dwarf, too faint to see without a telescope or powerful binoculars, with an apparent magnitude of 9.54).

A proper motion of 1 arcsec per year at a distance of 1 light year corresponds to a relative transverse speed of 1.45 km/s. For Barnard's star this works out to 90 km/s; including the radial velocity of 111 km/s (which is at right angles to the transverse velocity) gives a true motion of 142 km/s. True or absolute motion is more difficult to measure than the proper motion, as the true transverse velocity involves the product of the proper motion times the distance; that is, true velocity measurements depend on distance measurements, which are difficult in general. Currently, the nearby star with the largest true velocity (relative to the Sun) is Wolf 424 which moves at 555 km/s.

Stars with large proper motions tend to be nearby; most stars are far enough away that their proper motions are very small, of order a few thousandths of an arcsecond per year. It is possible to construct nearly complete samples of high proper motion stars by comparing photographic sky survey images taken many years apart. The Palomar Sky Survey is one source of such images. In the past, searches for high proper motion objects were undertaken using blink comparators to examine the images by eye, but modern efforts use techniques such as image differencing to automatically search through digitized image data. Because the selection biases of the resulting high proper motion samples are well-understood and well-quantified, it is possible to use them to construct an unbiased census of the nearby stellar population -- how many stars exist of each true brightness, for example. Studies of this kind show that the local population of stars consists largely of intrinsically faint, inconspicuous stars such as red dwarfs.

Proper motion was discovered in 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.

In research published in 2005, the first measurement of the proper motion of a galaxy (the Triangulum Galaxy) was made.^[2]

The following are the stars with highest proper motion from the Hipparcos catalog.^[3] It does not include stars such as Teegarden's star which are too faint for that catalog.

1. ^ ^{a b} Majewski, Steven R. (2006). Stellar Motions. University of Virginia. Retrieved on 2007-05-14.
2. ^ McKee, Maggie. "Distant galaxy's subtle sidling measured", New Scientist, March 3, 2005. Retrieved on 2007-07-21. 
3. ^ Staff (September 15, 2003). The 150 Stars in the Hipparcos Catalogue with Largest Proper Motion. ESA. Retrieved on 2007-07-21.
4. ^ SIMBAD. Centre de Données astronomiques de Strasbourg. Retrieved on 2007-07-21.

• Hipparcos: High Proper Motion Stars