arXiv:0811.0441v1 [astro-ph] 4 Nov 2008Introduction to Gravitational MicrolensingShude Mao∗Jodrell Bank Centre for Astrophysics, University of Manchester, Manchester M13 9PL, UKE-mail: [email protected] basic concepts of gravitational microlensing are introduced. We start with the lens equation,and then derive the image positions and magnifications. The statistical quantities of optical depthand event rate are then described. We finish with a summary and a list of challenges and openquestions. A problem set is given for students to practice.The Manchester Microlensing Conference: The 12th International Conference and ANGLES MicrolensingWorkshopJanuary 21-25 2008Manchester, UK∗Speaker.c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/Introduction to Gravitational Microlensing Shude Mao1. IntroductionGravitational microlensing (in the local group) refers to the temporal brightening of a back-ground star due to intervening objects. Einstein (1936) first examined (micro)lensing by a singlestar, and concluded that “there is no great chance of observing this phenomenon.” Some importantworks were performed in intervening years by [46] and [34], but the research topic was revitalisedby Paczy´nski (1986) who proposed it as a method to detect compact dark matter objects in theGalactic halo.The original goal is now out of favour, since we know with high precision that most of thedark matter must be non-baryonic, e.g. from observations of microwave background radiation andnucleosynthesis (at the time of his paper, this was, however, unclear). Nevertheless, gravitationalmicrolensing has turned into a powerful technique with diverse applications in astrophysics, includ-ing the study of the structure of the Milky Way, stellar atmospheres and the detection of extrasolarplanets and stellar-mass black hole candidates. The field has made enormous progress in the lasttwo decades. There have been a number of reviews on this topic (e.g. [39, 36, 20, 54]), the mostrecent highlight was given in [26]. This article gives an introduction to microlensing, aimed at alevel for a starting PhD student. Together with other talks in the workshop and proceedings1, onecan gain a thorough feeling about the state-of-the-art research in this field (as of 2008).The reference list given here is seriously incomplete (and biased). For more complete refer-ences and information about ongoing microlensing surveys, see the review papers mentioned aboveand the web site: http://mlens.net/ (built by Szymon Kozłowski, Subo Dong and LukaszWyrzykowski).2. What is gravitational microlensing?The light from a background source is deflected, distorted and (de)magnified by interveningobjects along the line of sight. If the lens, source and observer are sufficiently well aligned, thenstrong gravitational lensing can occur. Depending on the lensing object, strong gravitational lensingcan be divided into three areas: microlensing by stars, multiple-images by galaxies, and giant arcsand large-separation lenses by clusters of galaxies. For microlensing, the lensing object is a stellar-mass compact object (e.g. normal stars, brown dwarfs or stellar remnants [white dwarfs, neutronstars and black holes]); the image splitting in this case is usually too small (of the order of milli-arcsecond in the local group) to be resolved by ground-based telescopes, thus we can only observethe magnification change as a function of time.The left panel in Fig. 1 illustrates the microlensing geometry. A stellar-mass lens movesacross the line of sight towards a background star. As the lens moves closer to the line of sight, itsgravitational focusing increases, and the background star becomes brighter. As the source movesaway, the star falls back to its baseline brightness. If the motions of the lens, the observer andthe source can be approximately taken as linear, then the light curve is symmetric. Since thelensing probability for microlensing in the local group is of the order of 10−6(see section 5), themicrolensing variability usually should not repeat. Since photons of different wavelengths followthe same propagation path (geodesics), the light curve (for a point source) should not depend on1available at http://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=542Introduction to Gravitational Microlensing Shude MaoFigure 1: The left panel shows a side-on view of the geometry of microlensing where a lens moves acrossthe line of sight towards a background source. The right panel shows two light curves corresponding to twodimensionless impact parameters, u0= 0.1 and 0.3. The time on the horizontal axis is centred on the peaktime t0and is normalised to the Einstein radius crossing time tE. The lower the value of u0, the higher thepeak magnification. For the definitions of u0and tEsee section 4.1.the colour. The characteristic symmetric shape, non-repeatability, and achromaticity can be usedas criteria to separate microlensing from other types of variable stars (exceptions to these rules willbe discussed in section 4.2).3. Lens equation, image positions and magnificationsTo derive the characteristic light curve shape shown in the right panel of Fig. 1, we must lookclosely at the lens equation, and the resulting image positions and magnifications for a point source.3.1 Lens equationThe lens equation is straightforward to derive. Figure 2 illustrates a side-on view of the lensingconfiguration. Simple geometry yields~η+ Ddsˆ~α=~ξ·DsDd, (3.1)where Dd, Dsand Ddsare the distance to the lens (deflector), distance to the source and distancebetween the lens (deflector) and the source,~ηis the source position (distance perpendicular to theline connecting the observer and the lens),~ξis the image position, andˆ~αis the deflection angle. Forgravitational microlensing in the local group, Dds= Ds−Dd.2Mathematically, the lens equationprovides a mapping between the source plane to the lens plane. The mapping is not necessarilyone-to-one.2For cosmological microlensing in an expanding universe, the distances should be taken as angular diameter dis-tances, and in general Dds6= Ds−Dd. See the review by Wambsganss in these proceedings on cosmological microlens-ing.3Introduction to Gravitational Microlensing Shude MaoFigure 2: Illustration of various distances and angles in the lens equation (eqs. 3.1 and 3.2).Dividing both sides of eq. (3.1) by Ds, we obtain
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