# Proper motion

Relation between proper motion and velocity components of an object. At emission, the object was at distance d from the Sun, and moved at angular rate μ radian/s, that is, μ = vt / d with vt = the component of velocity transverse to line of sight from the Sun. (The diagram illustrates an angle μ swept out in unit time at tangential velocity vt.)

Proper motion is the astronomical measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more distant stars.[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). Their combined value is computed as the total proper motion (μ).[2][3] It has dimensions of angle per time, typically arcseconds per year or milliarcseconds per year. Knowledge of the proper motion, distance, and radial velocity allows calculations of true stellar motion or velocity in space in respect to the Sun, and by coordinate transformation, the motion in respect to the Milky Way. Proper motion is not entirely intrinsic to the celestial body or star, because it includes a component due to the motion of the Solar System itself.[4]

## Introduction

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 or Crux, for example, look nearly the same now as they 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 movement of the stars relative to the Sun and Solar System. The Sun travels in a nearly circular orbit (the solar circle) about the center of the Milky Way at a speed of about 220 km/s at a radius of kPc from the center,[5][6] which can be taken as the rate of rotation of the Milky Way itself at this radius.[7][8]

The proper motion is a two-dimensional vector (because it excludes the component in the direction of the line of sight) and is thus defined by two quantities: its position angle and its magnitude. The first quantity indicates the direction of the proper motion on the celestial sphere (with 0 degrees meaning the motion is due north, 90 degrees meaning the motion is due east, and so on), and the second quantity is the motion's magnitude, typically expressed in arcseconds per year (symbol arcsec/yr, as/yr) or milliarcsecond per year (mas/yr).

Components of proper motion on the Celestial sphere. The celestial north pole is CNP, the vernal equinox is V, the star path on the celestial sphere is indicated by arrows. The proper motion vector is μ, α = right ascension, δ = declination, θ = position angle.

Proper motion may alternatively be defined by the angular changes per year in the star's right ascension (μα) and declination (μδ), using a constant epoch in defining these.

The components of proper motion by convention are arrived at as follows. Suppose an object moves from coordinates (α1, δ1) to coordinates (α2, δ2) in a time Δt. The proper motions are given by:[9]

${\displaystyle \mu _{\alpha }={\frac {\alpha _{2}-\alpha _{1}}{\Delta t}}\ ,}$
${\displaystyle \mu _{\delta }={\frac {\delta _{2}-\delta _{1}}{\Delta t}}\ .}$

The magnitude of the proper motion μ is given by the Pythagorean theorem:[10]

${\displaystyle \mu ^{2}={\mu _{\delta }}^{2}+{\mu _{\alpha }}^{2}\cdot \cos ^{2}\delta \ ,}$
${\displaystyle \mu ^{2}={\mu _{\delta }}^{2}+{\mu _{\alpha \ast }}^{2}\ ,}$

where δ is the declination. The factor in cos2δ accounts for the fact that the radius from the axis of the sphere to its surface varies as cosδ, becoming, for example, zero at the pole. Thus, the component of velocity parallel to the equator corresponding to a given angular change in α is smaller the further north the object's location. The change μα, which must be multiplied by cosδ to become a component of the proper motion, is sometimes called the "proper motion in right ascension", and μδ the "proper motion in declination".[11]

If the proper motion in right ascension has been converted by cosδ, the result is designated μα*. For example, the proper motion results in right ascension in the Hipparcos Catalogue (HIP) have already been converted.[12] Hence, the individual proper motions in right ascension and declination are made equivalent for straightforward calculations of various other stellar motions.

The position angle θ is related to these components by:[2][13]

${\displaystyle \mu \sin \theta =\mu _{\alpha }\cos \delta =\mu _{\alpha \ast }\ ,}$
${\displaystyle \mu \cos \theta =\mu _{\delta }\ .}$

Motions in equatorial coordinates can be converted to motions in galactic coordinates.[14]

## Examples

For the majority of stars seen in the sky, the observed proper motions are usually small and unremarkable. Such stars are often either faint or are significantly distant, have changes of below 10 milliarcseconds per year, and do not appear to move appreciably over many millennia. A few do have significant motions, and are usually called high-proper motion stars. Motions can also be in almost seemingly random directions. Two or more stars, double stars or open star clusters, which are moving in similar directions, exhibit so-called shared or common proper motion (or cpm.), suggesting they may be gravitationally attached or share similar motion in space.

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 (arcsec/a). 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, located at a distance of about 6 light-years. After the Sun and the Alpha Centauri system, it is the nearest known star to Earth. Because it is a red dwarf with an apparent magnitude of 9.54, it is too faint to see without a telescope or powerful binoculars. Of the stars visible to the naked eye (by convention, limiting visual magnitude of 6.0), 61 Cygni A (magnitude V=5.20) has the highest proper motion at 5.281 arcsec/a, although Groombridge 1830 (magnitude V=6.42), proper motion 7.058 arcsec/a, might be visible for an observer with exceptionally keen vision.[15]

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. Barnard's Star's transverse speed is 90 km/s and its radial velocity is 111 km/s (which is at right angles to the transverse velocity), which gives a true motion of 142 km/s. True or absolute motion is more difficult to measure than the proper motion, because the true transverse velocity involves the product of the proper motion times the distance. As shown by this formula, true velocity measurements depend on distance measurements, which are difficult in general.

In 1992, Rho Aquilae became the first star to have its Bayer designation invalidated by moving to a neighbouring constellation – it is now a star of the constellation Delphinus.[16]

## Usefulness in astronomy

Stars with large proper motions tend to be nearby; most stars are far enough away that their proper motions are very small, on the order of 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.

Measurement of the proper motions of a large sample of stars in a distant stellar system, like a globular cluster, can be used to compute the cluster's total mass via the Leonard-Merritt mass estimator. Coupled with measurements of the stars' radial velocities, proper motions can be used to compute the distance to the cluster.

Stellar proper motions have been used to infer the presence of a super-massive black hole at the center of the Milky Way.[17] This black hole is suspected to be Sgr A*, with a mass of 4.2 × 106 M, where M is the solar mass.

Proper motions of the galaxies in the Local Group are discussed in detail in Röser.[18] In 2005, the first measurement was made of the proper motion of the Triangulum Galaxy M33, the third largest and only ordinary spiral galaxy in the Local Group, located 0.860 ± 0.028 Mpc beyond the Milky Way.[19] The motion of the Andromeda Galaxy was measured in 2012, and an Andromeda–Milky Way collision is predicted in about 4 billion years.[20][failed verification] Proper motion of the NGC 4258 (M106) galaxy in the M106 group of galaxies was used in 1999 to find an accurate distance to this object.[21] Measurements were made of the radial motion of objects in that galaxy moving directly toward and away from us, and assuming this same motion to apply to objects with only a proper motion, the observed proper motion predicts a distance to the galaxy of 7.2±0.5 Mpc.[22]

## History

Proper motion was suspected by early astronomers (according to Macrobius, 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.[23]

The term "proper motion" derives from the historical use of "proper" to mean "belonging to" (cf, propre in French and the common English word property). "Improper motion" would refer to "motion" common to all stars, such as due to axial precession.

## Stars with high proper motion

The following are the stars with highest proper motion from the Hipparcos catalog.[24] It does not include stars such as Teegarden's star, which are too faint for that catalog. A more complete list of stellar objects can be made by doing a criteria query at the SIMBAD astronomical database.

Proper motion of 61 Cygni in one year intervals.
Highest proper motion stars[25]
velocity
(km/s)
Parallax
(mas)
μα · cos δ
(mas/yr)
μδ
(mas/yr)
1 Barnard's Star −798.58 10328.12 −110.51 548.31
2 Kapteyn's star 6505.08 −5730.84 +245.19 255.66
3 Groombridge 1830 4003.98 −5813.62 −98.35 109.99
4 Lacaille 9352 6768.20 1327.52 +8.81 305.26
5 Gliese 1 (CD −37 15492) (GJ 1) 5634.68 −2337.71 +25.38 230.42
6 HIP 67593 2118.73[26] 5397.57[26] -4.4 187.76
7 61 Cygni A & B 4133.05 3201.78 −65.74 286
8 Lalande 21185 −580.27 −4765.85 −84.69 392.64
9 Epsilon Indi 3960.93 −2539.23 −40.00 276.06

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