Editing Proper motion (section)

== 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) typically defined by 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 defined [[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 display="block">\mu_{\alpha} = \frac{\alpha_2 - \alpha_1}{\Delta t}, </math>
<math display="block">\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 display="block">\mu^2 = {\mu_\delta}^2 + {\mu_\alpha}^2 \cdot \cos^2 \delta \ , </math>
''technically abbreviated:''
<math display="block">\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 display="block">\mu \sin \theta = \mu_\alpha \cos \delta = \mu_{{\alpha \ast}} \ ,</math>
<math display="block">\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>