View source for Proper motion

{{Short description|Measure of observed changes in the apparent locations of stars}}
{{distinguish|proper velocity|stellar parallax}}
[[File:Proper motion.svg|thumb|250px|Relation between proper motion and velocity components of an object.<br>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 = <math>\frac{v_t}{d}</math> where <math>v_t</math> 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.<br>In this diagram the radial velocity happens to be one of the Sun and object parting, so is positive.]]

'''Proper motion''' is the [[astrometry|astrometric]] measure of the observed changes in the [[apparent place]]s of [[star]]s 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 [[fixed stars|distant stars]].<ref name=Koupelis>{{cite book |title=In Quest of the Universe |author=Theo Koupelis |author2=Karl F. Kuhn |page= [https://archive.org/details/inquestofunivers00koup/page/369 369] |url=https://archive.org/details/inquestofunivers00koup |url-access=registration |isbn=978-0-7637-4387-1 |publisher=Jones & Bartlett Publishers |date=2007}}</ref>

The components for proper motion in the [[equatorial coordinate system]] (of a given [[epoch (astronomy)|epoch]], often [[J2000.0]]) are given in the direction of [[right ascension]] (''μ''<sub>α</sub>) and of [[declination]] (''μ''<sub>δ</sub>). Their combined value is computed as the ''total proper motion'' (''μ'').<ref name=Birney75>{{cite book |title=Observational Astronomy |author= D. Scott Birney |author2= Guillermo Gonzalez |author3= David Oesper |page= 75 |url=https://books.google.com/books?id=cc9L8QWcZWsC&q=celestial+sphere+%22right+ascension%22&pg=RA1-PA9 |date=2007 |publisher= Cambridge University Press |isbn=978-0-521-85370-5}}</ref><ref name= Green>{{cite book |title=An Introduction to the Sun and Stars |author=Simon F. Green |author2=Mark H. Jones |page= 87 |url=https://books.google.com/books?id=lb5owLGIQGsC&pg=PA87 |isbn=978-0-521-54622-5 |publisher=Cambridge University Press |date=2004}}</ref> It has [[dimension]]s of [[angle]] per [[time]], typically [[arcsecond]]s per [[year]] or milliarcseconds per year.

Knowledge of the proper motion, distance, and [[radial velocity]] allows calculations of an object's motion from the Solar System's [[frame of reference]] and its motion from the galactic frame of reference &ndash; that is motion in respect to the Sun, and by [[coordinate transformation]], that in respect to the [[Milky Way]].<ref name=Birney73>{{cite book |title=Observational Astronomy |author= D. Scott Birney |author2= Guillermo Gonzalez |author3= David Oesper |page= 73 |url=https://books.google.com/books?id=cc9L8QWcZWsC&pg=RA1-PA73 |isbn=978-0-521-85370-5 |date=2007 |publisher=Cambridge University Press}}</ref>

== Introduction ==
[[File:Components of proper motion.svg|thumb| 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 [[Equinox (celestial coordinates)|March equinox]] (center of the sun's position then) at the J2000 epoch, is vector ''V''.<br>In red the diagram adds the components of proper motion across the [[celestial sphere]].<br>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.<br>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.<br>The proper motion vector is '''''μ''''',  ''α'' = [[right ascension]], ''δ'' = [[declination]], ''θ'' = [[position angle]].]]
Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each other, so that they form the same [[constellation]]s over historical time. As examples, both [[Ursa Major]] in the northern sky and [[Crux]] in the southern sky, look nearly the same now as they did hundreds of years ago. However, precise long-term observations show that such constellations change shape, albeit very slowly, and that each star has an independent [[motion (physics)|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 [[Milky Way|the galaxy]] at a speed of about 220&nbsp;km/s at a radius of {{convert|8,000|pc}} from [[Sagittarius A*]]<ref name=Smith>{{cite book |author=Horace A. Smith |isbn=978-0-521-54817-5 |date=2004 |publisher=Cambridge University Press |title=RR Lyrae Stars |url=https://books.google.com/books?id=dMv_r82moCQC&q=Galactocentric+%22solar+circle%22&pg=PA80  |page= 79 }}</ref><ref name=Combes>{{cite book |title=Mapping the Galaxy and Nearby Galaxies |author=M Reid |author2=A Brunthaler |author3=Xu Ye |display-authors=etal |chapter=Mapping the Milky Way and the Local Group |editor=F Combes |editor2=Keiichi Wada |isbn= 978-0-387-72767-7 |date=2008 |publisher=Springer |chapter-url=https://books.google.com/books?id=bP9hZqoIfhMC&q=rotation+%22proper+motion%22+galaxy+OR+galactic&pg=PA24}}</ref> which can be taken as the rate of rotation of the Milky Way itself at this radius.<ref name=Sofue>{{cite journal |arxiv=astro-ph/0010594 |author=Y Sofu |author2=V Rubin |name-list-style=amp |title=Rotation Curves of Spiral Galaxies |date=2001 |journal=Annual Review of Astronomy and Astrophysics |volume=39 |pages=137–174 |doi=10.1146/annurev.astro.39.1.137 |bibcode=2001ARA&A..39..137S|s2cid=11338838 }}</ref><ref name=Loeb>{{cite journal |title=Constraints on the proper motion of the Andromeda galaxy based on the survival of its satellite M33 |pages=894–898 |author=Abraham Loeb |author2=Mark J. Reid |author3=Andreas Brunthaler |author4=Heino Falcke |journal=The Astrophysical Journal |volume=633 |date=2005 |url=http://www.mpifr-bonn.mpg.de/staff/abrunthaler/pub/loeb.pdf |doi=10.1086/491644 |bibcode=2005ApJ...633..894L|arxiv = astro-ph/0506609 |issue=2 |s2cid=17099715 }}</ref>

Any proper motion is a two-dimensional [[Euclidean vector|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 [[Euclidean vector|magnitude]]. The first is the direction of the proper motion on the [[celestial sphere]] (with 0&nbsp;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 [[Minute of arc#Symbols and abbreviations|arcseconds per year]] (symbols: arcsec/yr, as/yr, ″/yr, ″&nbsp;yr<sup>&minus;1</sup>) or milliarcseconds per year (symbols: mas/yr, mas&nbsp;yr<sup>&minus;1</sup>).

Proper motion may alternatively be defined by the angular changes per year in the star's [[right ascension]] (''μ<sub>α</sub>'') and [[declination]] (''μ<sub>δ</sub>'') with respect to a constant [[epoch (astronomy)|epoch]].

The [[Vector component|component]]s of proper motion by convention are arrived at as follows. Suppose an object moves from coordinates (α<sub>1</sub>, δ<sub>1</sub>) to coordinates (α<sub>2</sub>, δ<sub>2</sub>) in a time &Delta;''t''. The proper motions are given by:<ref name=Smart>{{cite book |title=Textbook on Spherical Astronomy |page= 252 |url=https://books.google.com/books?id=W0f2vc2EePUC&pg=PA252 |author=William Marshall Smart |author-link=William Marshall Smart |author2=Robin Michael Green |isbn=978-0-521-29180-4 |publisher=Cambridge University Press |date=1977}}</ref>
:<math>\mu_{\alpha} = \frac{\alpha_2 - \alpha_1}{\Delta t}, </math>
:<math>\mu_{\delta}= \frac{\delta_2-\delta_1}{\Delta t} \ .</math>
The magnitude of the proper motion ''μ'' is given by the [[Pythagorean theorem]]:<ref name=Doolittle>{{cite book |title=A Treatise on Practical Astronomy, as Applied to Geodesy and Navigation |page= [https://archive.org/details/atreatiseonprac02doolgoog/page/n611 583] |author=Charles Leander Doolittle |url=https://archive.org/details/atreatiseonprac02doolgoog |publisher=Wiley |date=1890}}</ref>
:<math>\mu^2 = {\mu_\delta}^2 + {\mu_\alpha}^2 \cdot \cos^2 \delta \ , </math>
:''technically abbreviated:''
:<math>\mu^2 = {\mu_\delta}^2 + {\mu_{{\alpha \ast}}}^2 \ . </math>
where ''δ'' is the declination. The factor in cos<sup>2</sup>''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''<sub>α</sub>, 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''<sub>δ</sub> the "proper motion in declination".<ref name=Newcomb>{{cite book |title=The Stars: A study of the Universe |author=Simon Newcomb |pages= [https://archive.org/details/starsastudyuniv02newcgoog/page/n313 287]–288 |url=https://archive.org/details/starsastudyuniv02newcgoog | date=1904 |publisher=Putnam }}</ref>

If the proper motion in right ascension has been converted by cos''δ'', the result is designated ''μ''<sub>α*</sub>. For example, the proper motion results in right ascension in the [[Hipparcos Catalogue]] (HIP) have already been converted.<ref>{{cite web
 | author=Matra Marconi Space, Alenia Spazio
 | date=September 15, 2003
 |archive-url=https://web.archive.org/web/20160303180237/http://www.rssd.esa.int/SA/HIPPARCOS/docs/vol1_all.pdf
 |archive-date=March 3, 2016
 | url=http://www.rssd.esa.int/SA/HIPPARCOS/docs/vol1_all.pdf
 | title=The Hipparcos and Tycho Catalogues : Astrometric and Photometric Star Catalogues derived from the ESA Hipparcos Space Astrometry Mission
 | page=25
 | publisher=ESA
 | access-date=2015-04-08 }}</ref>  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:<ref name=Birney75/><ref name=MajewskiNotes>See {{cite web | last = Majewski | first = Steven R. | date = 2006 | url = http://www.astro.virginia.edu/class/majewski/astr551/lectures/VELOCITIES/velocities.html | title = Stellar motions: parallax, proper motion, radial velocity and space velocity | publisher = University of Virginia | access-date = 2008-12-31 | archive-url = https://archive.today/20120125061201/http://www.astro.virginia.edu/class/majewski/astr551/lectures/VELOCITIES/velocities.html | archive-date = 2012-01-25 | url-status = dead }}</ref>

:<math>\mu \sin \theta = \mu_\alpha \cos \delta = \mu_{{\alpha \ast}} \ ,</math>
:<math>\mu \cos \theta = \mu_\delta \ . </math>

Motions in equatorial coordinates can be converted to motions in [[galactic coordinates]].<ref>See [https://web.archive.org/web/20160827130313/http://www.faculty.virginia.edu/ASTR5610/lectures/VELOCITIES/velocities.html lecture notes] by Steven Majewski.</ref>

== 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]]s or [[Open cluster|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.

[[Image:Barnard2005.gif|thumb|[[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″&nbsp;yr<sup>&minus;1</sup>. Large proper motion usually strongly indicates an object is close to the Sun. This is so for Barnard's Star, about 6 [[light-year]]s away. After the Sun and the [[Alpha Centauri]] system, it is the [[List of nearest stars|nearest]] known star. Being 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 (conservatively limiting unaided visual magnitude to 6.0), [[61 Cygni A]] (magnitude [[UBV photometric system|V=]]5.20) has the highest proper motion at 5.281″&nbsp;yr<sup>&minus;1</sup>, discounting [[Groombridge 1830]] (magnitude [[UBV photometric system|V=]]6.42), proper motion: 7.058″&nbsp;yr<sup>&minus;1</sup>.<ref>Hipparcos: Catalogues: The Millennium Star Atlas: [https://www.cosmos.esa.int/web/hipparcos/high-proper-motion The Top 20 High Proper Motion], [[European Space Agency]], retrieved 2019-06-27</ref>

A proper motion of 1&nbsp;arcsec per year 1&nbsp;light-year away corresponds to a relative transverse speed of 1.45&nbsp;km/s. Barnard's Star's transverse speed is 90&nbsp;km/s and its radial velocity is 111&nbsp;km/s (perpendicular (at a right, 90° angle), which gives a true or "space" motion of 142&nbsp;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 &ndash; it is now in [[Delphinus]].<ref name="jrasc92">{{cite journal
 | title = Book-Review – Sky Catalogue 2000.0 – V.1 – Stars to Magnitude 8.0 ED.2
 | date = 1992
 | bibcode = 1992JRASC..86..221L
 | last = Lemay
 | first = Damien<!-- book authors:
 | last1 = Hirshfeld
 | first1 = Alan
 | last2 = Sinnott
 | first2 = Roger W.
 | last3 = Ochsenbein
 | first3 = François -->
 | volume = 86
 | page = 221
 | journal = Journal of the Royal Astronomical Society of Canada
}}</ref>

== 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 [[National Geographic Society – Palomar Observatory Sky Survey|Palomar Sky Survey]] is one source of such images. In the past, searches for high proper motion objects were undertaken using [[blink comparator]]s to examine the images by eye. More modern techniques such as [[image differencing]] can scan digitized images, or comparisons to star catalogs obtained by satellites.<ref>{{Cite journal |last1=Akhmetov |first1=V. S. |last2=Fedorov |first2=P. N. |last3=Velichko |first3=A. B. |last4=Shulga |first4=V. M. |date=2017-07-21 |title=The PMA Catalogue: 420 million positions and absolute proper motions |journal=Monthly Notices of the Royal Astronomical Society |volume=469 |issue=1 |pages=763–773 |doi=10.1093/mnras/stx812 |doi-access=free |issn=0035-8711}}</ref> As any [[selection bias]]es of these surveys are well understood and quantifiable, studies have confirmed more and inferred approximate quantities of unseen stars &ndash; 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 [[red dwarf]]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]], 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.<ref name=Ghez>{{cite journal |journal=Astrophysical Journal |title=The First Measurement of Spectral Lines in a Short-Period Star Bound to the Galaxy's Central Black Hole: A Paradox of Youth |first1=Andrea M. |last1=Ghez |display-authors=etal <!-- Gaspard Duchêne, Keith Matthews, Seth D. Hornstein, Angelle Tanner, James E. Larkin, Margaret Morris, Eric E. Becklin, Samir Salim, Ted Kremenek, David R. Thompson, Baruch T. Soifer, Gerry Neugebauer, Ian S. McLean -->|volume=586 |pages=L127–L131 |date=2003 |arxiv=astro-ph/0302299 |doi=10.1086/374804 |bibcode=2003ApJ...586L.127G |issue=2 |s2cid=11388341 }}</ref> This now confirmed to exist black hole is called [[Sgr A*]], and has a mass of 4.3 × {{Solar mass|10<sup>6</sup>|link=y}} (solar masses).

Proper motions of the galaxies in the [[Local Group]] are discussed in detail in Röser.<ref name="Röser">{{cite book  |title=Reviews in Modern Astronomy: From Cosmological Structures to the Milky Way |author=Andreas Brunthaler |editor=Siegfried Röser |pages=179–194 |chapter= M33 – Distance and Motion|chapter-url=https://books.google.com/books?id=P3FAxGsJ_B8C&q=M33+%22proper+motion%22&pg=PA189 |isbn=978-3-527-40608-1 |publisher=Wiley |date=2005}}</ref> 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.<ref name=Brunthaler>{{cite journal |journal=Science |volume=307 |pages= 1440–1443 |date=2005 |title=The Geometric Distance and Proper Motion of the Triangulum Galaxy (M33) |author=A. Brunthaler |author2=M.J. Reid |author3=H. Falcke |author4=L.J. Greenhill |author5=C. Henkel  |arxiv=astro-ph/0503058 |doi=10.1126/science.1108342 |pmid=15746420 |issue=5714 |bibcode=2005Sci...307.1440B|s2cid=28172780 }}</ref> The motion of the [[Andromeda Galaxy]] was measured in 2012, and an [[Andromeda–Milky Way collision]] is predicted in about 4.5 billion years.<ref name="Universe Today">{{cite web |last1=Gough |first1=Evan |title=Universe Today |date=12 February 2019 |url=https://www.universetoday.com/141471/thanks-to-gaia-we-now-know-exactly-when-well-be-colliding-with-andromeda/ |publisher=The Astrophysical Journal |access-date=12 February 2019 |ref=utgaia}}</ref> Proper motion of the [[Messier 106|NGC 4258 (M106) galaxy]] in the M106 group of galaxies was used in 1999 to find an accurate distance to this object.<ref name=Weinberg>{{cite book |title=Cosmology |author= Steven Weinberg |page= 17 |url=https://books.google.com/books?id=48C-ym2EmZkC&q=M33+%22proper+motion%22+date:2007-2010&pg=PA17 |isbn=978-0-19-852682-7 |date=2008 |publisher=Oxford University Press}}</ref> Measurements were made of the radial motion of objects in that galaxy moving directly toward and away from Earth, 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 {{val|7.2|0.5|u=Mpc}}.<ref name=Hernnstein>{{cite journal |journal=Nature |date= 1999 |volume=400 |pages=539–541 |title=A geometric distance to the galaxy NGC4258 from orbital motions in a nuclear gas disk |author=J. R. Herrnstein |display-authors=etal | doi= 10.1038/22972|arxiv = astro-ph/9907013 |bibcode = 1999Natur.400..539H |issue=6744|s2cid= 204995005 }}</ref>

== History ==

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.<ref name=Neugebauer>{{cite book |title=A History of Ancient Mathematical Astronomy |author=Otto Neugebauer |url=https://books.google.com/books?id=vO5FCVIxz2YC&q=proper+motion+angle&pg=PA1085 |page= 1084 |isbn=978-3-540-06995-9 |publisher=Birkhäuser |date=1975}}</ref><ref>{{Cite journal |date=1719-12-31 |title=I. Considerations on the change of the latitudes of some of the principal fixt stars |url=https://royalsocietypublishing.org/doi/10.1098/rstl.1717.0025 |journal=Philosophical Transactions of the Royal Society of London |language=la |volume=30 |issue=355 |pages=736–738 |doi=10.1098/rstl.1717.0025 |issn=0261-0523}}</ref>

The lesser meaning of "proper" used is arguably dated English (but neither historic, nor obsolete when used as a [[postpositive adjective|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]], and minor deviations, nutations well within the 26,000-year cycle.

== Stars with high proper motion ==
{{Main article|List of high-proper-motion stars}}

==See also==
* [[Astronomical coordinate systems]]
* [[Galaxy rotation curve]]
* [[Leonard–Merritt mass estimator]]
* [[Milky Way]]
* [[Peculiar velocity]]
* [[Radial velocity]]
* [[Relative velocity]]
* [[Solar apex]]
* [[Space velocity (astronomy)]]
* [[Stellar kinematics]]
* [[Very-long-baseline interferometry]]

==References==
{{reflist}}

== External links ==
* [http://www.rssd.esa.int/index.php?project=HIPPARCOS&page=high_p Hipparcos: High Proper Motion Stars]
* [http://www.leosondra.cz/en/halley-proper-motions/ Edmond Halley: Discovery of proper motions]

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[[Category:Motion (physics)]]
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