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| {{Gravitational Lensing}}
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| '''Gravitational microlensing''' is an [[astronomy|astronomical]] phenomenon due to the [[gravitational lens]] effect. It can be used to detect objects ranging from the mass of a planet to the mass of a star, regardless of the light they emit. Typically, astronomers can only detect bright objects that emit lots of light ([[star]]s) or large objects that block background light (clouds of gas and dust). These objects make up only a tiny fraction of the mass of a galaxy. Microlensing allows the study of objects that emit little or no light. | |
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| When a distant star or [[quasar]] gets sufficiently aligned with a massive compact foreground object, the bending of light due to its gravitational field, as discussed by [[Einstein]] in 1915, leads to two distorted unresolved images resulting in an observable magnification. The time-scale of the transient brightening depends on the mass of the foreground object as well as on the relative proper motion between the background 'source' and the foreground 'lens' object.
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
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| Since microlensing observations do not rely on radiation received from the lens object, this effect therefore allows astronomers to study massive objects no matter how faint. It is thus an ideal technique to study the galactic population of such faint or dark objects as [[brown dwarfs]], [[red dwarfs]], [[planets]], [[white dwarfs]], [[neutron stars]], [[black holes]], and
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| [[Massive compact halo object|Massive Compact Halo Objects]]. Moreover, the microlensing effect is wavelength-independent, allowing study of source objects that emit any kind of electromagnetic radiation.
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| Microlensing by an isolated object was first detected in 1989. Since then, microlensing has been used to constrain the nature of the [[dark matter]], detect [[extrasolar planets]], study [[limb darkening]] in distant stars, constrain the [[binary star]] population, and constrain the structure of the Milky Way's disk. Microlensing has also been proposed as a means to find dark objects like brown dwarfs and black holes, study [[sunspot|starspots]], measure stellar rotation, and probe [[quasars]]<ref name="W2006">{{Cite arxiv| year=2006|title=Gravitational Lensing: Strong, Weak and Micro|periodical=Saas-Fee Lectures, Springer-Verlag|eprint=astro-ph/0604278| author1=Joachim Wambsganss| class=astro-ph| doi=10.1007/978-3-540-30310-7_4| volume=33| pages=453–540| chapter=Gravitational Microlensing| series=Saas-Fee Advanced Courses| isbn=978-3-540-30309-1}}</ref><ref>{{Cite journal| last=Kochanek| year=2004|title=Quantitative Interpretation of Quasar Microlensing Light Curves|journal=The Astrophysical Journal|volume=605 |page=58|arxiv=astro-ph/03074223|doi=10.1086/382180|first1=C. S.|bibcode=2004ApJ...605...58K}}</ref> including their [[accretion disks]].<ref>{{Cite journal|last=Poindexter et al.| year=2008|title=The Spatial Structure of An Accretion Disk|first3=Christopher S.|last3=Kochanek|journal=The Astrophysical Journal,|first2=Nicholas|volume=673|issue= |last2=Morgan| page=34|arxiv=0707.0003|doi=10.1086/524190|first1=Shawn|bibcode=2008ApJ...673...34P}}</ref><ref>{{Cite journal|last=Eigenbrod et al.| year=2008|title=Microlensing variability in the gravitationally lensed quasar QSO 2237+0305 = the Einstein Cross. II. Energy profile of the accretion disk|journal= Astronomy & Astrophysics|volume=490|issue= 3| page=933|arxiv=0810.0011 |bibcode = 2008A&A...490..933E |doi = 10.1051/0004-6361:200810729|first1=A.|last2=Courbin|first2=F.|last3=Meylan|first3=G.|last4=Agol|first4=E.|last5=Anguita|first5=T.|last6=Schmidt|first6=R. W.|last7=Wambsganss|first7=J. }}</ref><ref>{{Cite journal| last=Mosquera et al.| year=2009|title=Detection of chromatic microlensing in Q 2237+0305 A|first3=E.|last3=Mediavilla|journal=The Astrophysical Journal,|first2=J. A.|volume=691|issue= 2| last2=Muñoz| page=1292|arxiv=0810.1626|doi=10.1088/0004-637X/691/2/1292|first1=A. M.|bibcode=2009ApJ...691.1292M}}</ref><ref>{{cite journal| last1=Floyd ''et al.''|year=2009|title=The accretion disc in the quasar SDSS J0924+0219| pages=233–239|volume=398|doi=10.1111/j.1365-2966.2009.15045.x| journal=Monthly Notices of the Royal Astronomical Society|arxiv=0905.2651|bibcode=2009MNRAS.398..233F| first1=David J. E.| last2=Bate| first2=N. F.| last3=Webster| first3=R. L.}}</ref>
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| ==How it works== | | <!--'''PNG''' (currently default in production) |
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| Microlensing is based on the [[gravitational lens]] effect. A massive object (the lens) will bend the light of a bright background object (the source). This can generate multiple distorted, magnified, and brightened images of the background source.<ref>{{cite journal|author1=Refsdal, S.|title=The gravitational lens effect|journal=Monthly Notices of the Royal Astronomical Society|volume=128|page=295|year=1964|bibcode=1964MNRAS.128..295R}}</ref>
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| Microlensing is caused by the same physical effect as strong lensing and weak lensing, but it is studied using very different observational techniques. In strong and weak lensing, the mass of the lens is large enough (mass of a galaxy or a galaxy cluster) that the displacement of light by the lens can be resolved with a high resolution telescope such as the [[Hubble Space Telescope]]. With microlensing, the lens mass is too low (mass of a planet or a star) for the displacement of light to be observed easily, but the apparent brightening of the source may still be detected. In such a situation, the lens will pass by the source in a reasonable amount of time, seconds to years instead of millions of years. As the alignment changes, the source's apparent brightness changes, and this can be monitored to detect and study the event. Thus, unlike with strong and weak gravitational lenses, a microlensing event is a transient phenomenon from a human timescale perspective.<ref>{{cite journal|author1=Paczynski, B.|title=Gravitational microlensing by the galactic halo|journal=Astrophysical Journal|volume=304|page=1|year=1986|doi=10.1086/164140|bibcode=1986ApJ...304....1P}}</ref>
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| Unlike with strong and weak lensing, no single observation can establish that microlensing is occurring. Instead the rise and fall of the source brightness must be monitored over time using [[photometry (astronomy)|photometry]]. This function of brightness versus time is known as a [[light curve]]. A typical microlensing light curve is shown below: [[Image:Gravitational.Microlensing.Light.Curve.OGLE-2005-BLG-006.png|700px|center|Typical light curve of gravitational microlensing event (OGLE-2005-BLG-006) with its model fitted (red)]]
| | ==Demos== |
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| A typical microlensing event like this one has a very simple shape, and only one physical parameter can be extracted: the time scale, which is related to the lens mass, distance, and velocity. There are several effects, however, that contribute to the shape of more atypical lensing events:
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| * Lens mass distribution. If the lens mass is not concentrated in a single point, the light curve can be dramatically different, particularly with [[Caustic (optics)|caustic]]-crossing events, which may exhibit strong spikes in the light curve. In microlensing, this can be seen when the lens is a [[binary star]] or a [[Extrasolar planet|planetary system]].
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| * Finite source size. In extremely bright or quickly-changing microlensing events, like caustic-crossing events, the source star cannot be treated as an infinitesimally small point of light: the size of the star's disk and even [[limb darkening]] can modify extreme features.
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| * [[Parallax]]. For events lasting for months, the motion of the Earth around the Sun can cause the alignment to change slightly, affecting the light curve.
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| Most focus is currently on the more unusual microlensing events, especially those that might lead to the discovery of extrasolar planets. Although it has not yet been observed, another way to get more information from microlensing events that may soon be feasible involves measuring the [[Astrometry|astrometric]] shifts in the source position during the course of the event<ref>{{cite journal|author1=Boden, A. F.|author2=Shao, M.|author3=van Buren, D.|title=Astrometric Observation of MACHO Gravitational Microlensing|journal=Astrophysical Journal|volume=502|issue=2|page=538|year=1998|doi=10.1086/305913|bibcode=1998ApJ...502..538B|arxiv = astro-ph/9802179 }}</ref> and even resolving the separate images with [[Interferometry#astro|interferometry]].<ref>{{cite journal|bibcode=2001A&A...375..701D|arxiv = astro-ph/0108178 |doi = 10.1051/0004-6361:20010783|title=Resolving gravitational microlensing events with long-baseline optical interferometry|year=2001|last1=Delplancke|first1=F.|last2=Górski|first2=K. M.|last3=Richichi|first3=A.|journal=Astronomy and Astrophysics|volume=375|issue=2|pages=701–710 }}</ref>
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
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| ==Observing microlensing== | | ==Test pages == |
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| In practice, because the alignment needed is so precise and difficult to predict, microlensing is very rare. Events, therefore, are generally found with surveys, which photometrically monitor tens of millions of potential source stars, every few days for several years. Dense background fields suitable for such surveys are nearby galaxies, such as the Magellanic Clouds and the Andromeda galaxy, and the Milky Way bulge. In each case, the lens population studied comprises the objects between Earth and the source field: for the bulge, the lens population is the Milky Way disk stars, and for external galaxies, the lens population is the Milky Way halo, as well as objects in the other galaxy itself. The density, mass, and location of the objects in these lens populations determines the frequency of microlensing along that line of sight, which is characterized by a value known as the optical depth due to microlensing. (This is not to be confused with the more common meaning of [[optical depth]], although it shares some properties.) The optical depth is, roughly speaking, the average fraction of source stars undergoing microlensing at a given time, or equivalently the probability that a given source star is undergoing lensing at a given time. The MACHO project found the optical depth toward the LMC to be 1.2×10<sup>−7</sup> or about 1 in 8,000,000,<ref>{{cite journal |author1=The MACHO collaboration |author2=Alcock |author3=Allsman |author4=Alves |author5=Axelrod |author6=Becker |author7=Bennett |author8=Cook |author9=Dalal |displayauthors=9 |title=The MACHO Project: Microlensing Results from 5.7 Years of LMC Observations |year=2000 |pages=281–307 |volume=542 |doi=10.1086/309512 |journal=Astrophys.J. |arxiv=astro-ph/0001272 |bibcode=2000ApJ...542..281A}}</ref> and the optical depth toward the bulge to be 2.43×10<sup>−6</sup> or about 1 in 400,000.<ref>{{cite journal |author1=Alcock |author2=Allsman |author3=Alves |author4=Axelrod |author5=Becker |author6=Bennett |author7=Cook |author8=Drake |author9=Freeman |displayauthors=9 |title=The MACHO project: Microlensing Optical Depth towards the Galactic Bulge from Difference Image Analysis |year=2000 |doi=10.1086/309484 |journal=Astrophysical Journal |bibcode=2000ApJ...541..734A |arxiv=astro-ph/0002510 |volume=541 |issue=2 |pages=734–766}}</ref>
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
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| Complicating the search is the fact that for every star undergoing microlensing, there are thousands of stars changing in brightness for other reasons (about 2% of the stars in a typical source field are naturally [[variable stars]]) and other transient events (such as [[nova]]e and [[supernovae]]), and these must be weeded out to find true microlensing events. After a microlensing event in progress has been identified, the monitoring program that detects it often alerts the community to its discovery, so that other specialized programs may follow the event more intensively, hoping to find interesting deviations from the typical light curve. This is because these deviations – particularly ones due to exoplanets – require hourly monitoring to be identified, which the survey programs are unable to provide while still searching for new events. The question of how to prioritize events in progress for detailed followup with limited observing resources is very important for microlensing researchers today.
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| ==History==
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| In 1704 [[Isaac Newton]] suggested that a light ray could be deflected by gravity. In 1801 [[Johann Georg von Soldner]] calculated the amount of deflection of a light ray from a star under Newtonian gravity. In 1915 [[Einstein]] correctly predicted the amount of deflection under [[General Relativity]], which was twice the amount predicted by von Soldner. Einstein's prediction was validated by a 1919 expedition led by [[Arthur Stanley Eddington|Arthur Eddington]], which was a great early success for General Relativity.<ref>Schneider, Ehlers, and Falco. ''Gravitational Lenses''. 1992.</ref> In 1924 [[Orest Chwolson]] found that lensing could produce multiple images of the star. A correct prediction of the concomitant brightening of the source, the basis for microlensing, was published in 1936 by Einstein.<ref>{{cite journal|last1=Einstein|first1=A.|title=Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field|journal=Science|volume=84|issue=2188|year=1936|pmid=17769014|doi=10.1126/science.84.2188.506|bibcode = 1936Sci....84..506E|pages=506–7 }}</ref> Because of the unlikely alignment required, he concluded that "there is no great chance of observing this phenomenon". Gravitational lensing's modern theoretical framework was established with works by Yu Klimov (1963), Sidney Liebes (1964), and [[Sjur Refsdal]] (1964).<ref name="W2006"/>
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| Gravitational lensing was first observed in 1979, in the form of a quasar lensed by a foreground galaxy. That same year Kyongae Chang and Sjur Refsdal showed that individual stars in the lens galaxy could act as smaller lenses within the main lens, causing the source quasar's images to fluctuate on a timescale of months.<ref>{{cite journal|last1=Chang|first1=K.|last2=Refsdal|first2=S.|title=Flux variations of QSO 0957 + 561 A, B and image splitting by stars near the light path|journal=Nature|volume=282|issue=5739|page=561|year=1979|doi=10.1038/282561a0|bibcode = 1979Natur.282..561C }}</ref> [[Bohdan Paczyński]] first used the term "microlensing" to describe this phenomenon. This type of microlensing is difficult to identify because of the intrinsic variability of quasars, but in 1989 Mike Irwin et al. published detection of microlensing in [[Huchra's Lens]].
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| In 1986, Paczyński proposed using microlensing to look for [[dark matter]] in the form of massive compact halo objects (MACHOs) in the [[dark matter halo|Galactic halo]], by observing background stars in a nearby galaxy. Two groups of particle physicists working on dark matter heard his talks and joined with astronomers to form the Anglo-Australian MACHO collaboration<ref>[http://wwwmacho.mcmaster.ca mcmaster.ca]</ref> and the French EROS<ref>[http://eros.in2p3.fr/ eros.in2p3.fr]</ref> collaboration.
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| In 1986, [[Robert J. Nemiroff]] predicted the likelihood of microlensing<ref>{{cite journal |last=Nemiroff |first=Robert J. |title=Random gravitational lensing |journal=Astrophysics and Space Science |date=June 1986 |volume=123 |issue=2 |pages=381-387 |doi=10.1007/BF00653957 |url=http://articles.adsabs.harvard.edu/full/1986Ap%26SS.123..381N |accessdate=27 January 2014}}</ref> and calculated basic microlensing induced light curves for several possible lens-source configurations in his 1987 thesis.<ref>{{cite journal|last=Nemiroff|first=Robert J.|title=Prediction and analysis of basic gravitational microlensing phenomena|date=December 1987 |url=http://adsabs.harvard.edu/abs/1987PhDT........12N|accessdate=27 January 2014}}</ref>
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| In 1991 Mao and Paczyński suggested that microlensing might be used to find binary companions to stars, and in 1992 Gould and Loeb demonstrated that microlensing can be used to detect exoplanets. In 1992, Paczyński founded the OGLE microlensing experiment,<ref>[http://ogle.astrouw.edu.pl/ OGLE homepage at ogle.astrouw.edu.pl]</ref> which began searching for events in the direction of the [[Galactic center|Galactic bulge]].
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| The first two microlensing events in the direction of the [[Large Magellanic Cloud]] that might be caused by dark matter were reported in back to back [[Nature (journal)|Nature]] papers by MACHO<ref>{{cite journal|doi=10.1038/365621a0|title=Possible gravitational microlensing of a star in the Large Magellanic Cloud|year=1993|last1=Alcock|first1=C.|last2=Akerlof|first2=C. W.|last3=Allsman|first3=R. A.|last4=Axelrod|first4=T. S.|last5=Bennett|first5=D. P.|last6=Chan|first6=S.|last7=Cook|first7=K. H.|last8=Freeman|first8=K. C.|last9=Griest|first9=K. |displayauthors=9 |journal=Nature|volume=365|issue=6447|pages=621–623|arxiv = astro-ph/9309052 |bibcode = 1993Natur.365..621A }}</ref> and EROS<ref>{{cite journal|doi=10.1038/365623a0|title=Evidence for gravitational microlensing by dark objects in the Galactic halo|year=1993|last1=Aubourg|first1=E.|last2=Bareyre|first2=P.|last3=Bréhin|first3=S.|last4=Gros|first4=M.|last5=Lachièze-Rey |first5=M. |last6=Laurent |first6=B. |last7=Lesquoy |first7=E. |last8=Magneville |first8=C. |last9=Milsztajn |first9=A. |displayauthors=9 |journal=Nature|volume=365|issue=6447|pages=623–625|bibcode = 1993Natur.365..623A }}</ref> in 1993, and in the following years, events continued to be detected. The MACHO collaboration ended in 1999. Their data refuted the hypothesis that 100% of the dark halo comprises MACHOs, but they found a significant unexplained excess of roughly 20% of the halo mass, which might be due to MACHOs or to lenses within the Large Magellanic Cloud itself.<ref>{{cite journal |author1=Alcock, C. |author2=Allsman, R. A. |author3=Alves, D. R. |author4=Axelrod, T. S. |author5=Becker, A. C. |author6=Bennett, D. P. |author7=Cook, K. H. |author8=Dalal, N. |author9=Drake, A. J. |displayauthors=9 |title=The MACHO Project: Microlensing Results from 5.7 Years of Large Magellanic Cloud Observations |journal=The Astrophysical Journal |volume=542 |page=281 |year=2000 |doi=10.1086/309512 |bibcode=2000ApJ...542..281A |arxiv=astro-ph/0001272}}</ref>
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| EROS subsequently published even stronger upper limits on MACHOs,<ref>{{cite journal |bibcode=2007A&A...469..387T |arxiv=astro-ph/0607207 |doi=10.1051/0004-6361:20066017 |title=Limits on the Macho content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds |year=2007 |last1=Tisserand |first1=P. |last2=Le Guillou |first2=L. |last3=Afonso |first3=C. |last4=Albert |first4=J. N. |last5=Andersen |first5=J. |last6=Ansari |first6=R. |last7=Aubourg |first7=É. |last8=Bareyre |first8=P. |last9=Beaulieu |first9=J. P. |displayauthors=9 |journal=Astronomy and Astrophysics|volume=469|issue=2|pages=387–404}}</ref> and it is currently uncertain as to whether there is any halo microlensing excess that could be due to dark matter at all. The SuperMACHO project<ref>[http://www.ctio.noao.edu/supermacho/ An NOAO Long Term Survey with the MOSAIC Imager on the Blanco 4 meter telescope]. Ctio.noao.edu (2005-01-03). Retrieved 2011-05-22.</ref> currently underway seeks to locate the lenses responsible for MACHO's results.
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| Despite not solving the dark matter problem, microlensing has been shown to be a useful tool for many applications. Hundreds of microlensing events are detected per year toward the [[Galactic bulge]], where the microlensing optical depth (due to stars in the Galactic disk) is about 20 times greater than through the Galactic halo. In 2007, the OGLE project identified 611 event candidates, and the MOA project (a Japan-New Zealand collaboration)<ref>[http://www.phys.canterbury.ac.nz/moa/index.html Microlensing Observations in Astrophysics]</ref> identified 488 (although not all candidates turn out to be microlensing events, and there is a significant overlap between the two projects). In addition to these surveys, follow-up projects are underway to study in detail potentially interesting events in progress, primarily with the aim of detecting extrasolar planets. These include MiNDSTEp,<ref>[http://www.mindstep-science.org/]</ref> RoboNet,<ref>[http://robonet.lcogt.net/ RoboNet-II]</ref> MicroFUN <ref>[http://www.astronomy.ohio-state.edu/~microfun/ Microlensing Follow-up Network]</ref> and PLANET.<ref>[http://planet.iap.fr/ μFUN-PLANET collaboration]</ref>
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| ==Mathematics==
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| The mathematics of microlensing, along with modern notation, are described by Gould<ref>{{cite journal|last1=Gould|first1=Andrew|title=A Natural Formalism for Microlensing|journal=The Astrophysical Journal|volume=542|issue=2|page=785|year=2000|doi=10.1086/317037|bibcode=2000ApJ...542..785G|arxiv = astro-ph/0001421 }}</ref> and we use his notation in this section, though other authors have used other notation. The [[Einstein radius]], also called the Einstein angle, is the [[Angular diameter|angular radius]] of the [[Einstein ring]] in the event of perfect alignment. It depends on the lens mass M, the distance of the lens d<sub>L</sub>, and the distance of the source d<sub>S</sub>:
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| <math>\theta_E = \sqrt{\frac{4GM}{c^2} \frac{d_S - d_L}{d_S d_L}}</math> (in radians)
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| For M equal to the mass of the Sun, d<sub>L</sub> = 4000 parsecs, and d<sub>S</sub> = 8000 parsecs (typical for a Bulge microlensing event), the Einstein radius is 0.001 [[arcsecond]]s (1 milliarcsecond). By comparison, ideal Earth-based observations have [[Astronomical seeing|angular resolution]] around 0.4 arcseconds, 400 times greater. Since <math>\theta_E</math> is so small, it is not generally observed for a typical microlensing event, but it can be observed in some extreme events as described below.
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| Although there is no clear beginning or end of a microlensing event, by convention the event is said to last while the angular separation between the source and lens is less than <math>\theta_E</math>. Thus the event duration is determined by the time it takes the apparent motion of the lens in the sky to cover an angular distance <math>\theta_E</math>. The Einstein radius is also the same order of magnitude as the angular separation between the two lensed images, and the astrometric shift of the image positions throughout the course of the microlensing event.
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| During a microlensing event, the brightness of the source is amplified by an amplification factor A. This factor depends only on the closeness of the alignment between observer, lens, and source. The unitless number u is defined as the angular separation of the lens and the source, divided by <math>\theta_E</math>. The amplification factor is given in terms of this value:
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| <math>A(u) = \frac{u^2 + 2}{u \sqrt{u^2 + 4}}</math>
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| This function has several important properties. A(u) is always greater than 1, so microlensing can only increase the brightness of the source star, not decrease it. A(u) always decreases as u increases, so the closer the alignment, the brighter the source becomes. As u approaches infinity, A(u) approaches 1, so that at wide separations, microlensing has no effect. Finally, as u approaches 0, A(u) approaches infinity as the images approach an Einstein ring. For perfect alignment (u = 0), A(u) is theoretically infinite. In practice, finite source size effects will set a limit to how large an amplification can occur for very close alignment, but some microlensing events can cause a brightening by a factor of hundreds.
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| Unlike gravitational macrolensing where the lens is a galaxy or cluster of galaxies, in microlensing u changes significantly in a short period of time. The relevant time scale is called the Einstein time <math>t_E</math>, and it's given by the time it takes the lens to traverse an angular distance <math>\theta_E</math> relative to the source in the sky. For typical microlensing events, <math>t_E</math> is on the order of a few days to a few months. The function u(t) is simply determined by the Pythagorean theorem:
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| <math>u(t) = \sqrt{u_{min}^2 + \left ( \frac{t-t_0}{t_E} \right )^2}</math>
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| The minimum value of u, called u<sub>min</sub>, determines the peak brightness of the event.
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| In a typical microlensing event, the light curve is well fit by assuming that the source is a point, the lens is a single point mass, and the lens is moving in a straight line: the ''point source-point lens'' approximation. In these events, the only physically significant parameter that can be measured is the Einstein timescale <math>t_E</math>. Since this observable is a [[Degeneracy (mathematics)|degenerate]] function of the lens mass, distance, and velocity, we cannot determine these physical parameters from a single event.
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| However, in some extreme events, <math>\theta_E</math> may be measurable while other extreme events can probe an additional parameter: the size of the Einstein ring in the plane of the observer, known as the ''Projected Einstein radius'': <math>\tilde{r}_E</math>. This parameter describes how the event will appear to be different from two observers at different locations, such as a satellite observer. The projected Einstein radius is related to the physical parameters of the lens and source by
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| <math>\tilde{r}_E = \sqrt{\frac{4GM}{c^2} \frac{d_S d_L}{d_S - d_L}}</math>.
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| It is mathematically convenient to use the inverses of some of these quantities. These are the Einstein [[proper motion]]
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| <math>\vec{\mu}_E = {t_E}^{-1}</math>
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| and the Einstein [[parallax]]
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| <math>\vec{\pi}_E = {\tilde{r}_E}^{-1}</math>.
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| These vector quantities point in the direction of the relative motion of the lens with respect to the source. Some extreme microlensing events can only constrain one component of these vector quantities. Should these additional parameters be fully measured, the physical parameters of the lens can be solved yielding the lens mass, parallax, and proper motion as
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| <math>M=\frac{c^2}{4G}\theta_E \tilde{r}_E</math>
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| <math>\pi_L=\pi_E\theta_E + \pi_S</math>
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| <math>\mu_L=\mu_E\theta_E + \mu_S</math>
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| ==Extreme microlensing events==
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| In a typical microlensing event, the light curve is well fit by assuming that the source is a point, the lens is a single point mass, and the lens is moving in a straight line: the ''point source-point lens'' approximation. In these events, the only physically significant parameter that can be measured is the Einstein timescale <math>t_E</math>. However, in some cases, events can be analyzed to yield the additional parameters of the Einstein angle and parallax: <math>\theta_E</math> and <math>\pi_E</math>. These include very high magnification events, binary lenses, parallax, and xallarap events, and events where the lens is visible.
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| ===Events yielding the Einstein angle===
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| Although the Einstein angle is too small to be directly visible from a ground-based telescope, several techniques have been proposed to observe it.
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| If the lens passes directly in front of the source star, then the finite size of the source star becomes an important parameter. The source star must be treated as a disk on the sky, not a point, breaking the point-source approximation, and causing a deviation from the traditional microlensing curve that lasts as long as the time for the lens to cross the source, known as a ''finite source light curve''. The length of this deviation can be used to determine the time needed for the lens to cross the disk of the source star <math>t_S</math>. If the angular size of the source <math>\theta_S</math> is known, the Einstein angle can be determined as
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| <math>\theta_E = \theta_S \frac{t_E}{t_S}</math> .
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| These measurements are rare, since they require an extreme alignment between source and lens. They are more likely when <math>\theta_S/\theta_E</math> is (relatively) large, i.e., for nearby giant sources with slow-moving low-mass lenses close to the source.
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| In finite source events, different parts of the source star are magnified at different rates at different times during the event. These events can thus be used to study the [[limb darkening|limb-darkening]] of the source star.
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| ===Binary lenses===
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| If the lens is a binary star with separation of roughly the Einstein radius, the magnification pattern is more complex than in the single star lenses. In this case, there are typically three images when the lens is distant from the source, but there is a range of alignments where two additional images are created. These alignments are known as ''caustics''. At these alignments, the magnification of the source is formally infinite under the point-source approximation.
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| Caustic crossings in binary lenses can happen with a wider range of lens geometries than in a single lens. Like a single lens source caustic, it takes a finite time for the source to cross the caustic. If this caustic-crossing time <math>t_S</math> can be measured, and if the angular radius of the source is known, then again the Einstein angle can be determined.
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| As in the single lens case when the source magnification is formally infinite, caustic crossing binary lenses will magnify different portions of the source star at different times. They can thus probe the structure of the source and its limb darkening.
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| An animation of a binary lens event can be found at [http://www.youtube.com/watch?v=_0u7DVbw4o4 this YouTube video].
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| ===Events yielding the Einstein parallax===
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| In principle, the Einstein parallax can be measured by having two observers simultaneously observe the event from different locations, e.g., from the earth and from a distant spacecraft.<ref>{{cite journal |last1=Gould |first1=Andrew |title=MACHO velocities from satellite-based parallaxes |journal=The Astrophysical Journal |volume=421 |pages=L75 |year=1994 |doi=10.1086/187191 |bibcode=1994ApJ...421L..75G}}</ref> The difference in amplification observed by the two observers yields the component of <math>\vec{\pi}_E</math> perpendicular to the motion of the lens while the difference in the time of peak amplification yields the component parallel to the motion of the lens. This direct measurement was recently reported<ref>{{cite journal |last1=Dong |first1=Subo |last2=Udalski |first2=A. |last3=Gould |first3=A. |last4=Reach |first4=W. T. |last5=Christie |first5=G. W. |last6=Boden |first6=A. F. |last7=Bennett |first7=D. P. |last8=Fazio |first8=G. |last9=Griest |first9=K. |displayauthors=9 |title=First Space‐Based Microlens Parallax Measurement:SpitzerObservations of OGLE‐2005‐SMC‐001 |journal=The Astrophysical Journal |volume=664 |issue=2 |page=862 |year=2007 |doi=10.1086/518536 |bibcode=2007ApJ...664..862D |arxiv = astro-ph/0702240}}</ref> using the [[Spitzer Space Telescope]]. In extreme cases, the differences may even be measurable from small differences seen from telescopes at different locations on the earth.<ref>{{cite journal |author1=Hardy, S. J. |author2=Walker, M. A. |title=Parallax effects in binary microlensing events|journal=Monthly Notices of the Royal Astronomical Society |volume=276 |pages=L79 |year=1995 |bibcode=1995MNRAS.276L..79H}}</ref>
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| More typically, the Einstein parallax is measured from the non-linear motion of the observer caused by the rotation of the earth about the sun. It was first reported in 1995 <ref>{{cite journal |last1=Alcock |first1=C. |last2=Allsman |first2=R. A. |last3=Alves |first3=D. |last4=Axelrod |first4=T. S. |last5=Bennett |first5=D. P. |last6=Cook |first6=K. H. |last7=Freeman |first7=K. C. |last8=Griest |first8=K. |last9=Guern |first9=J. |displayauthors=9 |title=First Observation of Parallax in a Gravitational Microlensing Event|journal=The Astrophysical Journal |volume=454 |issue=2 |year=1995 |doi=10.1086/309783 |bibcode=1995ApJ...454L.125A |arxiv=astro-ph/9506114}}</ref> and has been reported in a handful of events since. Parallax in point-lens events can best be measured in long-timescale events with a large <math>\pi_E</math>—from slow-moving, low mass lenses which are close to the observer.
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| If the source star is a [[binary star]], then it too will have a non-linear motion which can also cause slight, but detectable changes in the light curve. This effect is known as [[Xallarap]] (parallax spelled backwards).
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| ==Detection of extrasolar planets==
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| {{see also|Methods of detecting extrasolar planets#Gravitational microlensing}}
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| [[Image:Gravitational micro rev.svg|thumb|right|250px|Gravitational microlensing of an extrasolar planet]]
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| If the lensing object is a star with a planet orbiting it, this is an extreme example of a binary lens event. If the source crosses a caustic, the deviations from a standard event can be large even for low mass planets. These deviations allow us to infer the existence and determine the mass and separation of the planet around the lens. Deviations typically last a few hours or a few days. Because the signal is strongest when the event itself is strongest, high-magnification events are the most promising candidates for detailed study. Typically, a survey team notifies the community when they discover a high-magnification event in progress. Follow-up groups then intensively monitor the ongoing event, hoping to get good coverage of the deviation if it occurs. When the event is over, the light curve is compared to theoretical models to find the physical parameters of the system. The parameters that can be determined directly from this comparison are the mass ratio of the planet to the star, and the ratio of the star-planet angular separation to the Einstein angle. From these ratios, along with assumptions about the lens star, the mass of the planet and its orbital distance can be estimated.
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| [[File:Exoplanet Discovery Method Bar ML.png|thumb|250px|Exoplanets discovered using microlensing, by year, through 2010-01-13.]]
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| The first success of this technique was made in 2003 by both OGLE and MOA of the microlensing event [[OGLE-2003-BLG-235/MOA-2003-BLG-53|OGLE 2003–BLG–235 (or MOA 2003–BLG–53)]]. Combining their data, they found the most likely planet mass to be 1.5 times the mass of Jupiter.<ref>{{cite journal |author1=Bond |author2=Udalski |author3=Jaroszynski |author4=Rattenbury |author5=Paczynski |author6=Soszynski |author7=Wyrzykowski |author8=Szymanski |author9=Kubiak |displayauthors=9 |title=OGLE 2003-BLG-235/MOA 2003-BLG-53: A planetary microlensing event |year=2004 |pages=L155–L158 |issue=2 |volume=606 |doi=10.1086/420928 |journal=Astrophys.J. |arxiv=astro-ph/0404309 |bibcode=2004ApJ...606L.155B}}</ref> As of January 2011, eleven exoplanets have been detected by this method, including [[OGLE-2005-BLG-071Lb]],<ref>{{cite journal |author1=Udalski |author2=Jaroszynski |author3=Paczynski |author4=Kubiak |author5=Szymanski |author6=Soszynski |author7=Pietrzynski |author8=Ulaczyk |author9=Szewczyk |displayauthors=9 |title=A Jovian-mass Planet in Microlensing Event OGLE-2005-BLG-071 |year=2005 |doi=10.1086/432795 |journal=The Astrophysical Journal |volume=628 |issue=2 |pages=L109–L112 |arxiv=astro-ph/0505451 |bibcode=2005ApJ...628L.109U}}</ref> [[OGLE-2005-BLG-390Lb]],<ref>[http://ogle.astrouw.edu.pl/cont/4_main/epl/blg390/blg390.html OGLE website]</ref> [[OGLE-2005-BLG-169Lb]],<ref>{{cite journal |author1=Gould |author2=Udalski |author3=An |author4=Bennett |author5=Zhou |author6=Dong |author7=Rattenbury |author8=Gaudi |author9=Yock |displayauthors=9 |title=Microlens OGLE-2005-BLG-169 Implies Cool Neptune-Like Planets are Common |year=2006 |pages=L37–L40 |volume=644 |doi=10.1086/505421 |journal=Astrophys.J. |arxiv=astro-ph/0603276 |bibcode=2006ApJ...644L..37G}}</ref> two exoplanets around [[OGLE-2006-BLG-109L]],<ref>{{cite journal |author1=Gaudi |author2=Bennett |author3=Udalski |author4=Gould |author5=Christie |author6=Maoz |author7=Dong |author8=McCormick |author9=Szymanski |displayauthors=9 |title=Discovery of a Jupiter/Saturn Analog with Gravitational Microlensing |year=2008 |pages=927–930 |issue=5865 |volume=319 |doi=10.1126/science.1151947 |journal=Science |arxiv=0802.1920 |pmid=18276883 |bibcode=2008Sci...319..927G}}</ref> and [[MOA-2007-BLG-192Lb]].<ref>Paul Rincon, [http://news.bbc.co.uk/2/hi/science/nature/7432114.stm Tiniest extrasolar planet found], BBC, 2 June 2008</ref> Notably, at the time of its announcement in January 2006, the planet OGLE-2005-BLG-390Lb probably had the lowest mass of any known exoplanet orbiting a regular star, with a median at 5.5 times the mass of the Earth and roughly a factor two uncertainty. This record was contested in 2007 by [[Gliese 581 c]] with a minimal mass of 5 Earth masses, and since 2009 [[Gliese 581 e]] is the lightest known "regular" exoplanet, with minimum 1.9 Earth masses.
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| Comparing this method of detecting extrasolar planets with other techniques such as the [[Astronomical transit|transit]] method, one advantage is that the intensity of the planetary deviation does not depend on the planet mass as strongly as effects in other techniques do. This makes microlensing well suited to finding low-mass planets. It also allows to detect planets further away from the host star than most of the other methods. One disadvantage is that followup of the lens system is very difficult after the event has ended, because it takes a long time for the lens and the source to be sufficiently separated to resolve them separately.
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| ==Microlensing experiments==
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| There are two basic types of microlensing experiments. "Search" groups use large-field images to find new microlensing events. "Follow-up" groups often coordinate telescopes around the world to provide intensive coverage of select events. The initial experiments all had somewhat risqué names until the formation of the PLANET group. There are current proposals to build new specialized microlensing satellites, or to use other satellites to study microlensing.
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| ===Search collaborations===
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| * {{cite arxiv|eprint=astro-ph/9506101|author1=Alard|author2=Mao|author3=Guibert|title=Object DUO 2: A New Binary Lens Candidate|class=astro-ph|year=1995}} Photographic plate search of bulge. Remarkable for largely being the work of a single graduate student, Christophe Alard, for his Ph.D. Thesis.
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| * [http://eros.in2p3.fr/ Experience de Recherche des Objets Sombres (EROS)] (1993–2002) Largely French collaboration. EROS1: Photographic plate search of LMC: EROS2: CCD search of LMC, SMC, Bulge & spiral arms.
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| * [http://wwwmacho.mcmaster.ca/ MACHO] (1993–1999) Australia & US collaboration. CCD search of bulge and LMC.
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| * [[Optical Gravitational Lensing Experiment|Optical Gravitational Lensing Experiment (OGLE)]] ( 1992 – ), Polish collaboration established by Paczynski and Udalski. Dedicated 1.3m telescope in Chile run by the University of Warsaw. Targets on bulge and Magellanic Clouds.
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| * [[Microlensing Observations in Astrophysics|Microlensing Observations in Astrophysics (MOA)]] (1998 – ), Japanese-New Zealand collaboration. Dedicated 1.8m telescope in New Zealand. Targets on bulge and Magellanic Clouds.
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| * [http://www.ctio.noao.edu/supermacho/ SuperMACHO] (2001 – ), successor to the MACHO collaboration used 4 m CTIO telescope to study faint LMC microlenses.
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| ===Follow-up collaborations===
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| * [[Probing Lensing Anomalies Network|Probing Lensing Anomalies Network (PLANET)]] Multinational collaboration.
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| * [[MicroFUN]], Microlensing Follow Up Network
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| * [http://bustard.phys.nd.edu/MPS/index.html Microlensing Planet Search (MPS)]
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| * [http://www.mindstep-science.org/ Microlensing Network for the Detection of Small Terrestrial Exoplanets, MiNDSTEp]
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| * [http://robonet.lcogt.net/ RoboNet-II. Searching for planets using a global network of robotic telescopes]
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| ===Andromeda galaxy pixel lensing collaborations===
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| * [http://www.astro.rug.nl/~jdejong/mega/ MEGA]
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| * [http://cdfinfo.in2p3.fr/Experiences/AGAPE/ AGAPE] (in French)
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| * [http://www.usm.uni-muenchen.de/people/fliri/wecapp.html WeCAPP]
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| * [http://www.astro.livjm.ac.uk/angstrom/ The Angstrom Project]
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| * [http://plan.physics.unisa.it/ PLAN]
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| ===Proposed satellite experiments===
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| * [http://bustard.phys.nd.edu/GEST/ Galactic Exoplanet Survey Telescope (GEST)]
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| * [http://planetquest.jpl.nasa.gov/Navigator/ao_support/gould.pdf SIM Microlensing Key Project] would have used the extremely high precision [[astrometry]] of the [[Space Interferometry Mission]] satellite to break the microlensing degeneracy and measure the mass, distance, and velocity of lenses. This satellite was postponed several times and finally cancelled in 2010.
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| * [http://wfirst.gsfc.nasa.gov/ Wide-Field Infrared Survey Telescope - Astrophysics Focused Telescope Assets (WFIRST - AFTA)] is to combine a microlensing survey with several other missions. The microlensing data will complement data from Kepler, with better sensitivity to planets like Earth that are not close in to their suns.
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| ==References==
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| {{reflist|colwidth=30em}}
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| ==External links==
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| * [http://skyandtelescope.com/news/article_1667_1.asp Discovery of planet five times as massive as earth orbiting a star 20,000 light-years away]
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| {{Exoplanet}}
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| {{DEFAULTSORT:Gravitational Microlensing}}
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| [[Category:Effects of gravitation]]
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| [[Category:Gravitational lensing]]
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