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proper motion
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{{distinguish|proper velocity}}(File:Proper motion.JPG|thumb|250px|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 Î¼ radian/s, that is, Î¼ = vt / d with vt = the component of velocity transverse to line of sight from the Sun. (The diagram illustrates an angle Î¼ 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.BOOK, In Quest of the Universe, Theo Koupelis, Karl F. Kuhn, 369,weblink 978-0-7637-4387-1, Jones & Bartlett Publishers, 2007, 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 (Î¼Î±) and of declination (Î¼Î´). Their combined value is computed as the total proper motion (Î¼).BOOK, Observational Astronomy, D. Scott Birney, Guillermo Gonzalez, David Oesper, 75,weblink 2007, 978-0-521-85370-5, BOOK, An Introduction to the Sun and Stars, Simon F. Green, Mark H. Jones, 87,weblink 978-0-521-54622-5, Cambridge University Press, 2004, 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 "proper" (that is, intrinsic to the celestial body or star), because it includes a component due to the motion of the Solar System itself.BOOK, Observational Astronomy, D. Scott Birney, Guillermo Gonzalez, David Oesper, 73,weblink 978-0-521-85370-5, 2007, Cambridge University Press,

## Introduction

File:Twilight Rays at La Silla.jpg|thumb|Proper motion is part of the intrinsic "property" of a star and involves its actual movement through space. By contrast, these photographic star trails are due to the Earth's rotation while using a long exposure. This image of star trails at twilighttwilightOver 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 {{val|8|u=kPc}} from the center,BOOK, Horace A. Smith, 978-0-521-54817-5, 2004, Cambridge University Press, RR Lyrae Stars,weblink 79, BOOK, Mapping the Galaxy and Nearby Galaxies, M Reid, A Brunthaler, Xu Ye, etal, Mapping the Milky Way and the Local Group, F Combes, Keiichi Wada, 978-0-387-72767-7, 2008, Springer,weblink which can be taken as the rate of rotation of the Milky Way itself at this radius.JOURNAL, astro-ph/0010594, Y Sofu, V Rubin, yes, Rotation Curves of Spiral Galaxies, 2001, Annual Review of Astronomy and Astrophysics, 39, 137â€“174, 10.1146/annurev.astro.39.1.137, 2001ARA&A..39..137S, JOURNAL, Constraints on the proper motion of the Andromeda galaxy based on the survival of its satellite M33, 894â€“898, Abraham Loeb, Mark J. Reid, Andreas Brunthaler, Heino Falcke, The Astrophysical Journal, 633, 2005,weblink 10.1086/491644, 2005ApJ...633..894L, astro-ph/0506609, 2, 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).File:Components of proper motion.svg|thumb |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 Î¼, Î± = right ascension, Î´ = declination, Î¸ = position angleposition angleProper motion may alternatively be defined by the angular changes per year in the star's right ascension (Î¼Î±) and declination (Î¼Î´), 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 (Î±1, Î´1) to coordinates (Î±2, Î´2) in a time Δt. The proper motions are given by:BOOK, Textbook on Spherical Astronomy, 252,weblink William Marshall Smart, William Marshall Smart, Robin Michael Green, 978-0-521-29180-4, Cambridge University Press, 1977,
mu_{alpha} = frac{alpha_2 - alpha_1}{Delta t} , mu_{delta}= frac{delta_2-delta_1}{Delta t} .
The magnitude of the proper motion Î¼ is given by the Pythagorean theorem:BOOK, A Treatise on Practical Astronomy, as Applied to Geodesy and Navigation, 583, Charles Leander Doolittle,weblink Wiley, 1890,
mu^2 = {mu_delta}^2 + {mu_alpha}^2 cdot cos^2 delta , mu^2 = {mu_delta}^2 + {mu_{{alpha ast}}}^2 ,
where Î´ is the declination. The factor in cos2Î´ accounts for the fact that the radius from the axis of the sphere to its surface varies as cosÎ´, becoming, for example, zero at the pole. Thus, the component of velocity parallel to the equator corresponding to a given angular change in Î± is smaller the further north the object's location. The change Î¼Î±, which must be multiplied by cosÎ´ to become a component of the proper motion, is sometimes called the "proper motion in right ascension", and Î¼Î´ the "proper motion in declination".BOOK, The Stars: A study of the Universe, Simon Newcomb, 287â€“288,weblink 1904, Putnam, If the proper motion in right ascension has been converted by cosÎ´, the result is designated Î¼Î±*. For example, the proper motion results in right ascension in the Hipparcos Catalogue (HIP) have already been converted.WEB
, Matra Marconi Space, Alenia Spazio
, September 15, 2003
, Mar 3, 2016
, The Hipparcos and Tycho Catalogues : Astrometric and Photometric Star Catalogues derived from the ESA Hipparcos Space Astrometry Mission
, 25
, ESA
, 2015-04-08, Hence, the individual proper motions in right ascension and declination are made equivalent for straightforward calculations of various other stellar motions.
The position angle Î¸ is related to these components by:See WEB
, Majewski, Steven R., 2006
, Stellar motions: parallax, proper motion, radial velocity and space velocity, University of Virginia
, 2008-12-31
,
mu sin theta = mu_alpha cos delta = mu_{{alpha ast}} , mu cos theta = mu_delta .
Motions in equatorial coordinates can be converted to motions in galactic coordinates.See weblink" title="web.archive.org/web/20160827130313weblink">lecture notes by Steven Majewski.

## 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.
missing image!
- Barnard2005.gif -
Barnard's Star, showing position every 5 years 1985â€“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.Hipparcos: Catalogues: The Millennium Star Atlas: The Top 20 High Proper Motion, European Space Agency, retrieved 2019-06-27A 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 â€“ it is now a star of the constellation Delphinus.JOURNAL
, Book-Review â€“ Sky Catalogue 2000.0 â€“ V.1 â€“ Stars to Magnitude 8.0 ED.2
, 1992
, 1992JRASC..86..221L
, Lemay
, Damien
, 86
, 221
, Journal of the Royal Astronomical Society of Canada
,

## 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 â€” 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.JOURNAL, Astrophysical Journal, The First Measurement of Spectral Lines in a Short-Period Star Bound to the Galaxy's Central Black Hole: A Paradox of Youth, Andrea M., Ghez, etal, 586, L127â€“L131, 2003, astro-ph/0302299, 10.1086/374804, 2003ApJ...586L.127G, 2, This black hole is suspected to be Sgr A*, with a mass of 4.2 Ã— {{Solar mass|106|link=y}}, where {{Solar mass}} is the solar mass.Proper motions of the galaxies in the Local Group are discussed in detail in RÃ¶ser.BOOK, Reviews in Modern Astronomy: From Cosmological Structures to the Milky Way, Andreas Brunthaler, Siegfried RÃ¶ser, 179â€“194, M33 â€“ Distance and Motion,weblink 978-3-527-40608-1, Wiley, 2005, 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 Â± 0.028 Mpc beyond the Milky Way.JOURNAL, Science, 307, 1440â€“1443, 2005, The Geometric Distance and Proper Motion of the Triangulum Galaxy (M33), A. Brunthaler, M.J. Reid, H. Falcke, L.J. Greenhill, C. Henkel, astro-ph/0503058, 10.1126/science.1108342, 15746420, 5714, 2005Sci...307.1440B, The motion of the Andromeda Galaxy was measured in 2012, and an Andromedaâ€“Milky Way collision is predicted in about 4 billion years.JOURNAL, Sangmo Tony Sohn, Jay Anderson, Roeland van der Marel, The M31 velocity vector. I. Hubble Space Telescope proper-motion measurements, The Astrophysical Journal, 753, 1, 7, Jul 1, 2012, 10.1088/0004-637X/753/1/7, 1205.6863, 2012ApJ...753....7S, {{not in ref|date=February 2018}} 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.BOOK, Cosmology, Steven Weinberg, 17,weblink 978-0-19-852682-7, 2008, Oxford University Press, 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 {{val|7.2|0.5|u=Mpc}}.JOURNAL, Nature, 1999, 400, 539â€“541, A geometric distance to the galaxy NGC4258 from orbital motions in a nuclear gas disk, J. R. Herrnstein, etal, 10.1038/22972, astro-ph/9907013, 1999Natur.400..539H, 6744,

## 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.BOOK, A History of Ancient Mathematical Astronomy, Otto Neugebauer,weblink 1084, 978-3-540-06995-9, BirkhÃ¤user, 1975, 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.WEB
, Staff
, September 15, 2003
, The 150 Stars in the Hipparcos Catalogue with Largest Proper Motion
, ESA
, 2007-07-21, 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.
File:61 Cygni Proper Motion.gif|thumb|Proper motion of 61 Cygni61 Cygni{| class="wikitable"|+ Highest proper motion starsWEB
, Centre de DonnÃ©es astronomiques de Strasbourg
, 2016-04-13, !rowspan="2"| #!rowspan="2"| Star!colspan="2"| Proper motion!rowspan="2"| Radialvelocity(km/s)!rowspan="2"| Parallax(mas)
! Î¼Î± Â· cos Î´(mas/yr)! Î¼Î´(mas/yr)
| 1| Barnard's Star
âˆ’798.58 10328.12 âˆ’110.51 548.31
| 2| Kapteyn's star
6505.08 âˆ’5730.84 +245.19 255.66
| 3| Groombridge 1830
4003.98 âˆ’5813.62 âˆ’98.35 109.99
| 4| Lacaille 9352
6768.20 1327.52 +8.81 305.26
| 5| Gliese 1 (CD âˆ’37 15492) (GJ 1)
5634.68 âˆ’2337.71 +25.38 230.42
| 6| HIP 67593
2118.73JOURNAL, Fabricius, C., Makarov, V.V., May 2000, Hipparcos astrometry for 257 stars using Tycho-2 data, Astronomy and Astrophysics Supplement, 144, 45â€“51, 10.1051/aas:2000198, 2000A&AS..144...45F, 5397.57 -4.4 187.76
| 7| 61 Cygni A & B
4133.05 3201.78 âˆ’65.74 286
| 8| Lalande 21185
âˆ’580.27 âˆ’4765.85 âˆ’84.69 392.64
| 9| Epsilon Indi
3960.93 âˆ’2539.23 âˆ’40.00 276.06

## Software

There are a number of software products that allow a person to view the proper motion of stars over differing time scales. Free ones include:
• HippLiner Windows â€“ moderately sophisticated with some pretty displays. Still under development, needs some more navigation and configuration features.
• XEphem Linux and MacOS â€“ complete astrometry package, can view a region of the sky, set a time step, and watch stars move over time.
• Proper Motion Simulator Website â€“ runs in-browser. Watch the positions of stars change with time and fly through constellations to get a sense of their volume.

## References

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