Proper motion is the

astrometric Astrometry is a branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. It provides the kinematics and physical origin of the Solar System and this galaxy, the Milky Way. Histor ...

measure of the observed changes in the

apparent place The apparent place of an object is its position in space as seen by an observer. Because of physical and geometrical effects it may differ from the "true" or "geometric" position. Astronomy In astronomy, a distinction is made between the ''mean ...

s of

star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make ...

s or other

celestial objects An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often use ...

in the sky, as seen from the center of mass of the

Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...

, compared to the abstract background of the more distant stars. The components for proper motion in the

equatorial coordinate system The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fun ...

(of a given

epoch In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured. The moment of epoch is usually decided by ...

, often

J2000.0 In astronomy, an epoch or reference epoch is a moment in time used as a reference point for some time-varying astronomical quantity. It is useful for the celestial coordinates or orbital elements of a celestial body, as they are subject to pertu ...

) are given in the direction of

right ascension Right ascension (abbreviated RA; symbol ) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the ( hour circle of the) point in question above the earth. When pai ...

(''μ''_α) and of

declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of t ...

(''μ''_δ). Their combined value is computed as the ''total proper motion'' (''μ''). It has

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

s of

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...

per

time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...

, typically

arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The na ...

s per

year A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the ...

or milliarcseconds per year. Knowledge of the proper motion, distance, and

radial velocity The radial velocity or line-of-sight velocity, also known as radial speed or range rate, of a target with respect to an observer is the rate of change of the distance or range between the two points. It is equivalent to the vector projection ...

allows calculations of an object's motion from our star system's

frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathem ...

and its motion from the galactic frame of reference – that is motion in respect to the Sun, and by

coordinate transformation In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is signi ...

, that in respect to the

Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked ey ...

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 Ursa Major (; also known as the Great Bear) is a constellation in the northern sky, whose associated mythology likely dates back into prehistory. Its Latin name means "greater (or larger) bear," referring to and contrasting it with nearby Ursa ...

Crux Crux () is a constellation of the southern sky that is centred on four bright stars in a cross-shaped asterism commonly known as the Southern Cross. It lies on the southern end of the Milky Way's visible band. The name ''Crux'' is Latin for ...

, 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 In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...

. This motion is caused by the movement of the stars relative to the

Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared rad ...

and

. The Sun travels in a nearly circular orbit (the ''

solar circle Solar may refer to: Astronomy * Of or relating to the Sun ** Solar telescope, a special purpose telescope used to observe the Sun ** A device that utilizes solar energy (e.g. "solar panels") ** Solar calendar, a calendar whose dates indicate ...

'') about the center of the galaxy at a speed of about 220 km/s at a radius of from

Sagittarius A* Sagittarius A* ( ), abbreviated Sgr A* ( ), is the supermassive black hole at the Galactic Center of the Milky Way. It is located near the border of the constellations Sagittarius and Scorpius, about 5.6° south of the ecliptic, ...

which can be taken as the rate of rotation of the Milky Way itself at this radius. Any proper motion is a two-dimensional

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

(as it excludes the component as to the direction of the line of sight) and it bears two quantities or characteristics: its

position angle In astronomy, position angle (usually abbreviated PA) is the convention for measuring angles on the sky. The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the ...

and its

magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...

. The first is the direction of the proper motion on the

celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...

(with 0 degrees meaning the motion is north, 90 degrees meaning the motion is east, (left on most sky maps and space telescope images) and so on), and the second is its magnitude, typically expressed in arcseconds per year (symbols: arcsec/yr, as/yr, ″/yr, ″ yr⁻¹) or milliarcseconds per year (symbols: mas/yr, mas yr⁻¹). Proper motion may alternatively be defined by the angular changes per year in the star's

(''μ_α'') and

(''μ_δ'') with respect to a constant

. The

component Circuit Component may refer to: •Are devices that perform functions when they are connected in a circuit. In engineering, science, and technology Generic systems * System components, an entity with discrete structure, such as an assem ...

s 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 Δ''t''. The proper motions are given by: :

\mu_ = \frac,

\mu_= \frac \ .

The magnitude of the proper motion ''μ'' is given by the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...

: :

\mu^2 = ^2 + ^2 \cdot \cos^2 \delta \ ,

:''technically abbreviated:'' :

\mu^2 = ^2 + ^2 \ .

where ''δ'' is the declination. The factor in cos²''δ'' accounts for the widening of the lines (hours) of right ascension away from the poles, cos''δ'', being zero for a hypothetical object fixed at a celestial pole in declination. 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. The change ''μ''_α, which must be multiplied by cos''δ'' to become a component of the proper motion, is sometimes called the "proper motion in right ascension", and ''μ''_δ the "proper motion in declination". If the proper motion in right ascension has been converted by cos''δ'', the result is designated ''μ''_α*. For example, the proper motion results in right ascension in the

Hipparcos Catalogue ''Hipparcos'' was a scientific satellite of the European Space Agency (ESA), launched in 1989 and operated until 1993. It was the first space experiment devoted to precision astrometry, the accurate measurement of the positions of celestial ob ...

(HIP) have already been converted. 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:See :

\mu \sin \theta = \mu_\alpha \cos \delta = \mu_ \ ,

\mu \cos \theta = \mu_\delta \ .

Motions in equatorial coordinates can be converted to motions in

galactic coordinates The galactic coordinate system is a celestial coordinate system in spherical coordinates, with the Sun as its center, the primary direction aligned with the approximate center of the Milky Way Galaxy, and the fundamental plane parallel to an ap ...

Examples

For most stars seen in the sky, the observed proper motions are small and unremarkable. Such stars are often either faint or are significantly distant, have changes of below 0.01″ 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 star In observational astronomy, a double star or visual double is a pair of stars that appear close to each other as viewed from Earth, especially with the aid of optical telescopes. This occurs because the pair either forms a binary star (i.e. a ...

s or open star clusters, which are moving in similar directions, exhibit so-called shared or ''

common proper motion This glossary of astronomy is a list of definitions of terms and concepts relevant to astronomy and cosmology, their sub-disciplines, and related fields. Astronomy is concerned with the study of celestial objects and phenomena that originate outsi ...

'' (or cpm.), suggesting they may be gravitationally attached or share similar motion in space.

Barnard's Star Barnard's Star is a red dwarf about six light-years from Earth in the constellation of Ophiuchus. It is the fourth-nearest-known individual star to the Sun after the three components of the Alpha Centauri system, and the closest star in the ...

has the largest proper motion of all stars, moving at 10.3″ yr⁻¹. Large proper motion usually strongly indicates an object is close to the Sun. This is so for Barnard's Star, about 6

light-year A light-year, alternatively spelled light year, is a large unit of length used to express astronomical distances and is equivalent to about 9.46 trillion kilometers (), or 5.88 trillion miles ().One trillion here is taken to be 101 ...

s away. After the Sun and the

Alpha Centauri Alpha Centauri ( Latinized from α Centauri and often abbreviated Alpha Cen or α Cen) is a triple star system in the constellation of Centaurus. It consists of 3 stars: Alpha Centauri A (officially Rigil Kentaurus), Alpha Centau ...

system, it is the nearest known star. Being a

red dwarf ''Red Dwarf'' is a British science fiction comedy franchise created by Rob Grant and Doug Naylor, which primarily consists of a television sitcom that aired on BBC Two between 1988 and 1999, and on Dave (TV channel), Dave since 2009, gaining a ...

with an

apparent magnitude Apparent magnitude () is a measure of the brightness of a star or other astronomical object observed from Earth. An object's apparent magnitude depends on its intrinsic luminosity, its distance from Earth, and any extinction of the object's li ...

of 9.54, it is too faint to see without a

telescope A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to obse ...

or powerful binoculars. Of the stars visible to the naked eye (conservatively limiting unaided visual magnitude to 6.0), 61 Cygni A (magnitude V=5.20) has the highest proper motion at 5.281″ yr⁻¹, discounting

Groombridge 1830 Groombridge 1830 (also known as 1830 Groombridge or Argelander's Star)Peters, C. A. F.; "On the Parallax of Argelander's Star", Monthly Notices of the Royal Astronomical Society, December 1853, v.50, p.302, is a star in the constellati ...

(magnitude V=6.42), proper motion: 7.058″ yr⁻¹. A proper motion of 1 arcsec per year 1 light-year away 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 (perpendicular (at a right, 90° angle), which gives a true or "space" 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 Rho Aquilae, ρ Aquilae, is the Bayer designation for a star in the northern constellation of Delphinus. It has an apparent visual magnitude of 4.94 and is bright enough to be seen with the naked eye in good conditions. Distance, proper ...

became the first star to have its

Bayer designation A Bayer designation is a stellar designation in which a specific star is identified by a Greek or Latin letter followed by the genitive form of its parent constellation's Latin name. The original list of Bayer designations contained 1,564 stars. T ...

invalidated by moving to a neighbouring constellation – it is now in

Delphinus Delphinus (Pronounced or ) is a small constellation in the Northern Celestial Hemisphere, close to the celestial equator. Its name is the Latin version for the Greek word for dolphin (). It is one of the 48 constellations listed by the 2nd ce ...

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 Palomar may refer to: Places * Any of several locations in San Diego County, California: ** Palomar Mountain ** Palomar Observatory, located on Palomar Mountain ** Palomar College in San Marcos, California ** Palomar Medical Center in Escondido, C ...

is one source of such images. In the past, searches for high proper motion objects were undertaken using

blink comparator A blink comparator is a viewing apparatus formerly used by astronomers to find differences between two photographs of the night sky. It permits rapid switching from viewing one photograph to viewing the other, "blinking" back and forth between th ...

s to examine the images by eye. More modern techniques such as

image differencing Image differencing is an image processing technique used to determine changes between images. The difference between two images is calculated by finding the difference between each pixel in each image, and generating an image based on the result. Fo ...

can scan digitized images, or comparisons to star catalogs obtained by satellites. As any

selection bias Selection bias is the bias introduced by the selection of individuals, groups, or data for analysis in such a way that proper randomization is not achieved, thereby failing to ensure that the sample obtained is representative of the population int ...

es of these surveys are well understood and quantifiable, studies have confirmed more and inferred approximate quantities of unseen stars – revealing and confirming more by studying them further, regardless of brightness, for instance. Studies of this kind show most of the nearest stars are intrinsically faint and angularly small, such as

s. 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 The radial velocity or line-of-sight velocity, also known as radial speed or range rate, of a target with respect to an observer is the rate of change of the distance or range between the two points. It is equivalent to the vector projection o ...

, 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. This black hole is suspected to be

Sgr A* Sagittarius A* ( ), abbreviated Sgr A* ( ), is the supermassive black hole at the Galactic Center of the Milky Way. It is located near the border of the constellations Sagittarius and Scorpius, about 5.6° south of the ecliptic, vis ...

, with a mass of 4.2 × (solar masses). Proper motions of the galaxies in the

Local Group The Local Group is the galaxy group that includes the Milky Way. It has a total diameter of roughly , and a total mass of the order of . It consists of two collections of galaxies in a "dumbbell" shape: the Milky Way and its satellites form ...

are discussed in detail in Röser. In 2005, the first measurement was made of the proper motion of the

Triangulum Galaxy The Triangulum Galaxy is a spiral galaxy 2.73 million light-years (ly) from Earth in the constellation Triangulum. It is catalogued as Messier 33 or NGC (''New General Catalogue)'' 598. With the D25 isophotal diameter of , the Triangulum Galax ...

M33, the third largest and only ordinary spiral galaxy in the Local Group, located 0.860 ± 0.028 Mpc beyond the Milky Way. The motion of the

Andromeda Galaxy The Andromeda Galaxy (IPA: ), also known as Messier 31, M31, or NGC 224 and originally the Andromeda Nebula, is a barred spiral galaxy with the diameter of about approximately from Earth and the nearest large galaxy to the Milky Way. The gal ...

was measured in 2012, and an

Andromeda–Milky Way collision The Andromeda–Milky Way collision is a galactic collision predicted to occur in about 4.5 billion years between the two largest galaxies in the Local Group—the Milky Way (which contains the Solar System and Earth) and the Andromeda Galaxy. ...

is predicted in about 4.5 billion years. 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. 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 .

History

Proper motion was suspected by early astronomers (according to

Macrobius Macrobius Ambrosius Theodosius, usually referred to as Macrobius (fl. AD 400), was a Roman provincial who lived during the early fifth century, during late antiquity, the period of time corresponding to the Later Roman Empire, and when Latin was ...

, AD 400) but a proof was not provided until 1718 by

Edmund Halley Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720. From an observatory he constructed on Saint Helena in 1676–77, ...

, who noticed that

Sirius Sirius is the brightest star in the night sky. Its name is derived from the Greek word , or , meaning 'glowing' or 'scorching'. The star is designated α Canis Majoris, Latinized to Alpha Canis Majoris, and abbreviated Alpha CMa ...

Arcturus , - bgcolor="#FFFAFA" , Note (category: variability): , , H and K emission vary. Arcturus is the brightest star in the northern constellation of Boötes. With an apparent visual magnitude of −0.05, it is the third-brightest of ...

and

Aldebaran Aldebaran (Arabic: “The Follower”, "الدبران") is the brightest star in the zodiac constellation of Taurus. It has the Bayer designation α Tauri, which is Latinized to Alpha Tauri and abbreviated Alpha Tau or α Tau. Aldeb ...

were over half a degree away from the positions charted by the ancient Greek astronomer

Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos''; BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equ ...

roughly 1850 years earlier. The lesser meaning of "proper" used is arguably dated English (but neither historic, nor obsolete when used as a postpositive, as in "the city proper") meaning "belonging to" or "own". "Improper motion" would refer to perceived motion that is nothing to do with an object's inherent course, such as due to earth's

axial precession In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In the absence of precession, the astronomical body's orbit would show axial parallelism. In particu ...

, and minor deviations, nutations well within the 26,000-year cycle.

Stars with high proper motion

The following are the stars with highest proper motion from the ''

Hipparcos ''Hipparcos'' was a scientific satellite of the European Space Agency (ESA), launched in 1989 and operated until 1993. It was the first space experiment devoted to precision astrometry, the accurate measurement of the positions of celestial obj ...

'' catalog. 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 criterion query at the

SIMBAD SIMBAD (the Set of Identifications, Measurements and Bibliography for Astronomical Data) is an astronomical database of objects beyond the Solar System. It is maintained by the Centre de données astronomiques de Strasbourg (CDS), France. SIMB ...

astronomical database. 61 Cygni Proper Motion

The figure for HIP 67593 is almost certainly an error, probably because the star has a relatively nearby brighter visual binary companion; the movement between the DSS2 and SDSS9 images is less than it.

Gaia In Greek mythology, Gaia (; from Ancient Greek , a poetical form of , 'land' or 'earth'),, , . also spelled Gaea , is the personification of the Earth and one of the Greek primordial deities. Gaia is the ancestral mother—sometimes parthe ...

measured a much smaller proper motion for its Data Release 2, yet a 15-fold parallax between it and its likely common-proper-motion companion HIP 67594. Reconciling its distance and motion will have to wait for Data Release 3 expected to analyse well very high proper motion objects.

References

External links

Hipparcos: High Proper Motion Stars

Edmond Halley: Discovery of proper motions
{{DEFAULTSORT:Proper Motion Astrometry Stellar astronomy Motion (physics) Concepts in astronomy