Online Astronomy eText: Stars and Stellar Systems
Proper Motion
(also see Stellar Motions and The Changeable Constellations)
The Motion of Stars
Most stars have nearly fixed positions in the sky relative to each other. In fact they are often referred to as the "fixed stars" in contrast with the planets, which are always in motion relative to each other and the stellar background.
In reality, all stars are in motion relative to each other and our Galaxy, with typical velocities of a hundred fifty miles per second or so relative to the center of the Galaxy, and a few tens of miles per second relative to each other. The difference between their rapid motion relative to the Galaxy, and their much slower motion relative to each other is due to the fact that they are mostly going around the Galaxy in nearly the same direction, at nearly the same speed, much as cars on a freeway are heading in nearly the same direction, at nearly the same speed. Under such circumstances their speeds relative to each other are much smaller than their speeds relative to objects not moving with them.
As discussed in Stellar Motions, the motions of the stars are divided into specific parts, each with its own name and measurement technique, according to its direction. Motions toward or away from the Sun are measured by Doppler effects -- changes in the apparent wavelength of the light they emit caused by their radial motion. The motion measured in this way is called radial velocity, and is considered positive if the radius vector from the Sun to the star is increasing (the star is moving away from us), and negative if the radius vector is decreasing (the star is moving toward us). Radial velocity can be measured for any star, no matter how distant it is, as long as it is bright enough to spread its light out into a spectrum and measure the Doppler effect of its motion.
 The motion of a star relative to the Sun.Motion toward or away from the Sun is called radial velocity.Motion perpendicular to the direction to the Sun is called tangential velocity.The combination of the two motions is the star's space velocity.
That part of a star's motion which is not toward or away from us is called its tangential motion, and is perpendicular to its radial motion. The tangential motion is measured by the change in the star's direction in space over a period of time. If the star is far away or its tangential motion is small it may take millennia to observe any change in its position; but if the star is very close to us, so that any motion it has looks relatively large, or if it is at a more moderate distance but is moving sideways at a rapid rate we may be able to observe a change in its position and calculate its tangential motion in just a few decades.
The change in a star's position, measured in seconds of arc per year or per century, is referred to as proper motion. As explained above proper motion cannot be measured for all stars -- only for stars that are unusually close or moving unusually fast relative to the Sun, or both -- and even then it takes decades or centuries to measure it, whereas radial velocity may be measured as quickly as a spectrum of the star can be obtained and analyzed. As a result most stars (and galaxies) have known radial velocities, whereas for most stars (and all galaxies) the proper motion is unknown.

Faster Than A Speeding Bullet -- Barnard's Star
Since stars that are close to the Sun can have relatively small velocities and still have noticeable proper motions, one of the easiest ways to recognize nearby stars is to look for stars with large proper motions. In fact the star with the fastest proper motion, Barnard's Star, with a proper motion of 10.3 seconds of arc per year, is the second closest stellar system to the Sun, and because it is rapidly approaching the Sun will be the closest star to the Sun, with a distance of less than four light years, in less than ten thousand years.

(2007 note: The following material was written before the preceding material, and will eventually be changed to fit the new layout)

In the case of the stars shown below the reason for the larger than normal proper motion is partly their distance (Barnard's Star is one of the closest stars to the Sun, and Mira, though more distant, is closer than most stars), and partly a larger than usual motion of the stars relative to the other stars in their vicinity. Barnard's Star is moving at about the same speed as the stars near it, but they are moving around the Galaxy in more nearly circular paths than not, while Barnard's Star is moving almost perpendicular to that direction, more nearly toward the center of the Galaxy. The stars moving around the Galaxy with the Sun have small motions relative to us, and small proper motions. Barnard's Star, moving in a very different direction, has a motion relative to us which is a combination of its and our overall motion, so it appears to move much faster relative to the background of stars than any other star.

Barnard's Star
 (Image credit POSS1, POSS2, Digitized Sky Survey)      At top, two images of the same area of the sky (centered near RA 17 58, Dec +04 36) taken about 50 years apart show the proper motion of Barnard's Star (the bright star near the bottom of the left image, and the top of the right image. The left image was taken in the 1950's as part of the Palomar Observatory Sky Survey; the right image recently, as part of a second POSS. Most of the stars appear to be in the same position relative to their neighbors in both fields of view, but if the positions of individual stars are carefully examined some can be seen to move just a little during the half century between the two images. To make that comparison easier, the animated image below the pair alternates them, so that changes in the positions of stars are easier to notice. (Hint: check the area just to the left of the second image of Barnard's Star.)

An animated view of Barnard's Star's proper motion
(Modified version of Wikipedia Commons image created by Steve Quirk)
An animation showing the motion of Barnard's Star from 1985 to 2005, at about the same scale as the Palomar Survey images, but taken with a considerably smaller telescope, so the fainter stars visible in the Palomar images are not visible. Most stars hardly seem to move over periods as long as a millennium; but Barnard's Star moves a distance equal to the diameter of the full moon in less than two centuries.

The rapid apparent movement of Barnard's Star is not due to its having an exceptionally large velocity. It just happens to be moving in a different direction from the Sun and most of its neighbors. Most stars near the Sun are moving in the same direction with nearly the same speed, and have relative motions that are only about 10 to 15 percent of their actual motion, while Barnard's Star is moving in a different direction, so although its actual speed is about the same, its speed relative to the Sun is larger (about 85 mi/sec, compared to 15 to 25 mi/sec for other stars). Even this motion wouldn't be particularly noticeable, however, if not for the fact that Barnard's Star is one of the closest stars to the Sun. At just over 5 light years distance it is only a light year further than the nearest star, Alpha Centauri, and is the closest star north of the Celestial Equator. Because it is approaching the Sun, Barnard's Star will pass less than 4 light years from us about ten thousand years from now, and for some time before and after that will be the closest star to the solar system.

Mira -- A Whale of a Tail
Mira's motion is not as different from that of its neighbors as Barnard's Star, but has recently attracted attention because of its spectacular, thirteen-light-year long "tail". (Mira is in the constellation of Cetus, the Whale, hence the pun commonly used to refer to its tail.)
Mira is a red giant near the end of its life as an easily visible star. Because of its large size it has a low surface gravity, and gas is easily ejected from its outer layers into space. If it were stationary relative to the material surrounding it, the gas leaving Mira would gradually expand in all directions, forming a "pre-planetary" nebula. But it is moving through space relative to nearby stars and interstellar gas at about 80 miles per second (most such motions are only a quarter that speed), and as the star and the gas leaving it plow through the interstellar medium a "bow shock" is formed, like the bow wave in front of a speedboat, and a long "wake" or "tail" has formed behind the star. The collision of the stellar and interstellar gas heats them to very high temperatures, and as the gas streams behind the star the hydrogen which makes up most of the gas emits radiation (almost entirely in the ultraviolet), as shown below.
Mira's thirteen-light-year long ultraviolet tail, with dates showing its position at different times in the past. (NASA, JPL-Caltech, C. Martin (Caltech), M. Seibert(OCIW), GALEX)

A closeup of the area near Mira, showing the bow shock to the right of the star, in ultraviolet radiation (top image), and the same region in visible light (bottom image). Because Mira is very cool, the star is much brighter in visible light than in ultraviolet; but because the gas in its tail is very hot, it is much brighter in ultraviolet and cannot be seen in visible light at all. (NASA/JPL-Caltech/POSS-II/DSS/C. Martin (Caltech)/M. Seibert(OCIW), GALEX)

Four frames from an animation of Mira's motion. (Most browsers will not play the animation, and if so right-click on the image, "Save Target As" miraani.mov, then use your media player to view it.) (NASA/JPL-Caltech/R. Hurt (SSC), GALEX)