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Proper motion

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Measure of observed changes in the apparent locations of stars

Not to be confused withproper velocity orstellar parallax.

Relation between proper motion and velocity components of an object.
A year ago the object wasd units of distance from the Sun, and its light moved in a year by angleμ radian/s. If there has been no distortion bygravitational lensing or otherwise then μ = ${\frac {v_{t}}{d}}$ where $v_{t}$ 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.
In this diagram the radial velocity happens to be one of the Sun and object parting, so is positive.

{\displaystyle {\frac {v_{t}}{d}}} — Relation between proper motion and velocity components of an object.
A year ago the object wasd units of distance from the Sun, and its light moved in a year by angleμ radian/s. If there has been no distortion bygravitational lensing or otherwise then μ = ${\frac {v_{t}}{d}}$ where $v_{t}$ 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.
In this diagram the radial velocity happens to be one of the Sun and object parting, so is positive.

Proper motion is theangular speed of acelestial object, such as a star, as it moves across the sky. It is anastrometric measure, giving an object's change in angular position over time relative to thecenter of mass of theSolar System. This parameter is measured relative to thedistant stars or a stable reference such as theInternational Celestial Reference Frame (ICRF).^[1] Patterns in proper motion reveal larger structures likestellar streams, the general rotation of theMilky Way disk, and the random motions of stars in theGalactic halo.^[2]

The components for proper motion in theequatorial coordinate system (of a givenepoch, oftenJ2000.0) are given in the direction ofright ascension (μ_α) and ofdeclination (μ_δ). Their combined value is computed as thetotal proper motion (μ).^[3]^[4] It hasdimensions ofangle pertime, typicallyarcseconds peryear or milliarcseconds per year.

Knowledge of the proper motion, distance, andradial velocity allows calculations of an object's motion from the Solar System'sframe of reference and its motion from the galactic frame of reference – that is motion in respect to the Sun, and bycoordinate transformation, that in respect to theMilky Way.^[5]

Introduction

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The celestial north and south poles are above/belowCNP,*CSP*; theorigin of all 24 hours of Right Ascension (the measure of absolute celestial east–west position), theMarch equinox (center of the sun's position then) at the J2000 epoch, is vectorV.
In red the diagram adds the components of proper motion across thecelestial sphere.
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.
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.
The proper motion vector isμ,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 sameconstellations over historical time. As examples, bothUrsa Major in the northern sky andCrux 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 independentmotion.

This motion is caused by the movement of the stars relative to theSun andSolar System. The Sun travels in a nearly circular orbit (thesolar circle) about the center ofthe galaxy at a speed of about 220 km/s at a radius of 8,000 parsecs (26,000 ly) fromSagittarius A*^[6]^[7] which can be taken as the rate of rotation of the Milky Way itself at this radius.^[8]^[9]

Any proper motion is a two-dimensionalvector (as it excludes the component as to the direction of the line of sight) typically defined by itsposition angle and itsmagnitude. The first is the direction of the proper motion on thecelestial 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 inarcseconds 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'sright ascension (μ_α) anddeclination (μ_δ) with respect to a definedepoch.

Thecomponents 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:^[10] $\mu _{\alpha }={\frac {\alpha _{2}-\alpha _{1}}{\Delta t}},$ $\mu _{\delta }={\frac {\delta _{2}-\delta _{1}}{\Delta t}}\ .$ The magnitude of the proper motionμ is given by thePythagorean theorem:^[11] $\mu ^{2}={\mu _{\delta }}^{2}+{\mu _{\alpha }}^{2}\cdot \cos ^{2}\delta \ ,$ technically abbreviated: $\mu ^{2}={\mu _{\delta }}^{2}+{\mu _{\alpha \ast }}^{2}\ .$ whereδ is the declination. The factor in cos²δ accounts for the widening of the lines (hours) of right ascension away from the poles, cosδ, 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 inα) 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 changeμ_α, which must be multiplied by cosδ to become a component of the proper motion, is sometimes called the "proper motion in right ascension", andμ_δ the "proper motion in declination".^[12]

If the proper motion in right ascension has been converted by cosδ, the result is designatedμ_α*. For example, the proper motion results in right ascension in theHipparcos Catalogue (HIP) have already been converted.^[13] Hence, the individual proper motions in right ascension and declination are made equivalent for straightforward calculations of various other stellar motions.

The position angleθ is related to these components by:^[3]^[14] $\mu \sin \theta =\mu _{\alpha }\cos \delta =\mu _{\alpha \ast }\ ,$ $\mu \cos \theta =\mu _{\delta }\ .$

Motions in equatorial coordinates can be converted to motions ingalactic coordinates.^[15]

Examples

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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 calledhigh-proper motion stars. Two or more stars which are moving in similar directions, exhibit so-called shared orcommon proper motion (or cpm.), suggesting they may share similar motion in space (if the distances and radial velocities are also consistent) and thus be gravitationally linked asbinary stars orstar clusters.

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″ yr⁻¹. Large proper motion usually strongly indicates an object is close to the Sun. This is so for Barnard's Star, about 6light-years away. After the Sun and theAlpha Centauri system, it is thenearest known star. Being ared dwarf with anapparent magnitude of 9.54, it is too faint to see without atelescope or powerful binoculars. Of the stars visible to the naked eye (conservatively limiting unaided visual magnitude to 6.0),61 Cygni A (magnitudeV=5.20) has the highest proper motion at 5.281″ yr⁻¹, discountingGroombridge 1830 (magnitudeV=6.42), proper motion: 7.058″ yr⁻¹.^[16]

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 1992Rho Aquilae became the first star to have itsBayer designation invalidated by moving to a neighbouring constellation – it is now inDelphinus.^[17]

Usefulness in astronomy

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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. ThePalomar Sky Survey is one source of such images. In the past, searches for high proper motion objects were undertaken usingblink comparators to examine the images by eye. More modern techniques such asimage differencing can scan digitized images, or comparisons to star catalogs obtained by satellites.^[18] As anyselection biases 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 asred 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 theLeonard-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.^[19] This now confirmed to exist black hole is calledSgr A*, and has a mass of 4.3 × 10⁶ M_☉ (solar masses).

Proper motions of objects in galaxies in theLocal Group can be used to estimate their distance. In 1999, the proper motion ofwater masers moving very rapidly around the center ofNGC 4258 (M106) galaxy was measured viaVery Long Baseline Interferometry. In combination with their radial motion this yielded an accurate distance to the galaxy of7.2±0.5 Mpc.^[20]^[21] In 2005, the first measurement was made of the proper motion of theTriangulum Galaxy M33, the third largest and only ordinary spiral galaxy in the Local Group, located 0.860 ± 0.028 Mpc beyond the Milky Way.^[22]^[23] The motion of theAndromeda Galaxy was measured in 2012, and anAndromeda–Milky Way collision is predicted in about 4.5 billion years.^[24]

History

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Proper motion was suspected by early astronomers (according toMacrobius,c. AD 400) but a proof was not provided until 1718 byEdmond Halley, who noticed thatSirius,Arcturus andAldebaran were over half a degree away from the positions charted by the ancient Greek astronomerHipparchus roughly 1850 years earlier.^[25]^[26]

The lesser meaning of "proper" used is arguably dated English (but neither historic, nor obsolete when used as apostpositive, 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'saxial precession, and minor deviations, nutations well within the 26,000-year cycle.

Stars with high proper motion

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Main article:List of high-proper-motion stars

References

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^Theo Koupelis; Karl F. Kuhn (2007).In Quest of the Universe. Jones & Bartlett Publishers. p. 369.ISBN 978-0-7637-4387-1.
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^^a ^bD. Scott Birney; Guillermo Gonzalez; David Oesper (2007).Observational Astronomy. Cambridge University Press. p. 75.ISBN 978-0-521-85370-5.
^Simon F. Green; Mark H. Jones (2004).An Introduction to the Sun and Stars. Cambridge University Press. p. 87.ISBN 978-0-521-54622-5.
^D. Scott Birney; Guillermo Gonzalez; David Oesper (2007).Observational Astronomy. Cambridge University Press. p. 73.ISBN 978-0-521-85370-5.
^Horace A. Smith (2004).RR Lyrae Stars. Cambridge University Press. p. 79.ISBN 978-0-521-54817-5.
^M Reid; A Brunthaler; Xu Ye; et al. (2008)."Mapping the Milky Way and the Local Group". In F Combes; Keiichi Wada (eds.).Mapping the Galaxy and Nearby Galaxies. Springer.ISBN 978-0-387-72767-7.
^Y Sofu & V Rubin (2001). "Rotation Curves of Spiral Galaxies".Annual Review of Astronomy and Astrophysics.39:137–174.arXiv:astro-ph/0010594.Bibcode:2001ARA&A..39..137S.doi:10.1146/annurev.astro.39.1.137.S2CID 11338838.
^Abraham Loeb; Mark J. Reid; Andreas Brunthaler; Heino Falcke (2005)."Constraints on the proper motion of the Andromeda galaxy based on the survival of its satellite M33"(PDF).The Astrophysical Journal.633 (2):894–898.arXiv:astro-ph/0506609.Bibcode:2005ApJ...633..894L.doi:10.1086/491644.S2CID 17099715.
^William Marshall Smart; Robin Michael Green (1977).Textbook on Spherical Astronomy. Cambridge University Press. p. 252.ISBN 978-0-521-29180-4.
^Charles Leander Doolittle (1890).A Treatise on Practical Astronomy, as Applied to Geodesy and Navigation. Wiley. p. 583.
^Simon Newcomb (1904).The Stars: A study of the Universe. Putnam. pp. 287–288.
^Matra Marconi Space, Alenia Spazio (September 15, 2003)."The Hipparcos and Tycho Catalogues : Astrometric and Photometric Star Catalogues derived from the ESA Hipparcos Space Astrometry Mission"(PDF). ESA. p. 25. Archived fromthe original(PDF) on March 3, 2016. Retrieved2015-04-08.
^SeeMajewski, Steven R. (2006)."Stellar motions: parallax, proper motion, radial velocity and space velocity". University of Virginia. Archived fromthe original on 2012-01-25. Retrieved2008-12-31.
^Seelecture notes by Steven Majewski.
^Hipparcos: Catalogues: The Millennium Star Atlas:The Top 20 High Proper Motion,European Space Agency, retrieved 2019-06-27
^Lemay, Damien (1992). "Book-Review – Sky Catalogue 2000.0 – V.1 – Stars to Magnitude 8.0 ED.2".Journal of the Royal Astronomical Society of Canada.86: 221.Bibcode:1992JRASC..86..221L.
^Akhmetov, V. S.; Fedorov, P. N.; Velichko, A. B.; Shulga, V. M. (2017-07-21)."The PMA Catalogue: 420 million positions and absolute proper motions".Monthly Notices of the Royal Astronomical Society.469 (1):763–773.doi:10.1093/mnras/stx812.ISSN 0035-8711.
^Ghez, Andrea M.; et al. (2003). "The First Measurement of Spectral Lines in a Short-Period Star Bound to the Galaxy's Central Black Hole: A Paradox of Youth".Astrophysical Journal.586 (2):L127 –L131.arXiv:astro-ph/0302299.Bibcode:2003ApJ...586L.127G.doi:10.1086/374804.S2CID 11388341.
^Steven Weinberg (2008).Cosmology. Oxford University Press. p. 17.ISBN 978-0-19-852682-7.
^J. R. Herrnstein; et al. (1999). "A geometric distance to the galaxy NGC4258 from orbital motions in a nuclear gas disk".Nature.400 (6744):539–541.arXiv:astro-ph/9907013.Bibcode:1999Natur.400..539H.doi:10.1038/22972.S2CID 204995005.
^A. Brunthaler; M.J. Reid; H. Falcke; L.J. Greenhill; C. Henkel (2005). "The Geometric Distance and Proper Motion of the Triangulum Galaxy (M33)".Science.307 (5714):1440–1443.arXiv:astro-ph/0503058.Bibcode:2005Sci...307.1440B.doi:10.1126/science.1108342.PMID 15746420.S2CID 28172780.
^Andreas Brunthaler (2005)."M33 – Distance and Motion". In Siegfried Röser (ed.).Reviews in Modern Astronomy: From Cosmological Structures to the Milky Way. Wiley. pp. 179–194.ISBN 978-3-527-40608-1.
^Gough, Evan (12 February 2019)."Universe Today". The Astrophysical Journal. Retrieved12 February 2019.
^Otto Neugebauer (1975).A History of Ancient Mathematical Astronomy. Birkhäuser. p. 1084.ISBN 978-3-540-06995-9.
^"I. Considerations on the change of the latitudes of some of the principal fixt stars".Philosophical Transactions of the Royal Society of London (in Latin).30 (355):736–738. 1719-12-31.doi:10.1098/rstl.1717.0025.ISSN 0261-0523.

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Stars

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Formation

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Classification

Early Late Main sequence O B A F G K M Subdwarf O B WR OB Subgiant Giant Blue Red Yellow Bright giant Supergiant Blue Red Yellow Hypergiant Yellow Carbon S CN CH White dwarf Chemically peculiar Am Ap/Bp CEMP HgMn He-weak Barium Lambda Boötis Lead Technetium Be Shell B[e] Helium Extreme Blue straggler
Remnants	Compact star Parker's star White dwarf Helium planet Neutron Radio-quiet Pulsar Binary X-ray Magnetar Stellar black hole X-ray binary Burster SGR
Hypothetical	Blue dwarf Black dwarf Exotic Boson Electroweak Strange Preon Planck Dark Dark-energy Quark Q Black hole star Black Hawking Quasi-star Gravastar Thorne–Żytkow object Iron Blitzar White hole