Proper motion

[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.

Relation between proper motion and velocity components of an object.
A year ago the object was d units of distance from the Sun, and its light moved in a year by angle μ radian/s. If there has been no distortion by gravitational lensing or otherwise then μ = where is the distance (usually expressed as annual velocity) transverse (tangential or perpendicular) to line of sight from the Sun. The angle is shaded light blue from the Sun to the object's start point and its year later position as if it had no radial velocity.
In this diagram the radial velocity happens to be one of the Sun and object parting, so is positive.
The celestial north and south poles are above/below CNP , CSP ; the origin of all 24 hours of Right Ascension (the measure of absolute celestial east–west position), the March equinox (center of the sun's position then) at the J2000 epoch, is vector V .
In red the diagram adds the components of proper motion across the celestial sphere .
An ideal time to measure exactly such a small annual shift is at culmination. The culmination of the star is daily reached when the observer (and Earth) passes as shown by the blue arrows "beneath" the star.
The positive axes of the two components of its usually annually measured or published shift in proper motion are the exaggerated red arrows, note: the right arrows point to the east horizon. One red annotation is subtly shorter as the cosine of a star resting at 0° declination is 1, so such a star's east or west shift would not need to be multiplied by the cosine of its declination.
The proper motion vector is μ , α = right ascension , δ = declination , θ = position angle .
Barnard's Star , showing position every 5 years 1985–2005.