MicrolensingENCYCLOPEDIA OF ASTRONOMY AND ASTROPHYSICSMicrolensingMicrolensing refers to the special case of GRAVITATIONALLENSINGwhere the multiple images produced are too closetogether on the sky to be observed as separate images.However, the lensing can still be detected because thesemultiple images appear as a single object of increasedapparent brightness. Although this is not detectable in aone-off observation (since we do not know the ‘normal’brightness of the source), with the passage of time thelens moves across the Earth–source line and the amount ofbrightening changes. Typically the source will appear tobrighten, reach a maximum and then fade symmetricallyback to normal over the course of a few weeks or months;this is called a ‘microlensing event’.The major application of microlensing, suggested byPaczynski in 1986, is in the search for theDARK MATTER whichis strongly believed to exist from rotation curves of spiralgalaxies etc. Since the lensing effect depends only on lensmass, it can be used to search for very faint or invisibleobjects such as brown dwarfs, neutron stars, old whitedwarfs or black holes, which might make up the darkmatter. These are collectively known as massive compacthalo objects or MACHOs, in contrast to the hypotheticalweakly interacting massive particles or WIMPs.To understand the basics of microlensing, considera small massive object (the lens) situated exactly on theline of sight from Earth to a background star and considera number of light rays radiating from the star passingthe lens at different distances and being bent towards thelens. Since the bending angle for a light ray increases withdecreasing distance from the lens, it is clear that there is aunique ‘miss distance’ such that the ray will be deflectedjust enough to hit the Earth; this distance is called theEinstein radius. By rotational symmetry about the Earth–star axis, an observer on Earth with perfect resolutionwould see the star lensed into an annulus centered onits ‘true’ position, called an Einstein ring. As the lens ismoved slightly off the line of sight (e.g. by 0.1 Einsteinring radii), the Einstein ring splits into two banana-shapedarcs, one on the same side of the lens as the source, one onthe opposite side. As the lens moves further off (morethan 1 Einstein radius), the arcs become more circular, the‘opposite-side’ arc fades very rapidly and the ‘same-side’arc turns into a slightly deflected and nearly circular imageof the star. Figure 1 illustrates a sequence of such imagesfor a typical microlensing event.Although the perfect alignment giving the Einsteinring will rarely occur in practice, it is still a very importantconcept because the size of the hypothetical Einstein ringsets the length scale over which substantial brighteningwill occur. As we will see, for a typical lens in ourGalaxy the radius of the Einstein ring rEis roughly8(M/M)1/2AU (astronomical units), where M is the lensmass. Knowing this scale allows us to understand most ofthe general characteristics of microlensing: it is extremelysmall compared with the typical distance to a lens, sothe angular separation of the two images will be tooFigure 1. A microlensing event seen at ‘perfect’ resolution. Theaxes show angular offsets on the sky from the lens (central dot)in units of the Einstein angle; the dashed circle is the Einsteinring. The series of small open circles shows the ‘true’ sourceposition at successive timesteps. For each source position, thereare two images (solid blobs) collinear with the lens and source,as indicated by the dotted line; the arrows illustrate their motion.small to resolve, hence the ‘micro’lensing. However, itis considerably larger than either the size of a star or thesize of a MACHO, so we can usually approximate the lensand source as pointlike, which leads to a simple predictionfor the lightcurve shape. Also, rEis very small comparedwith the typical separation of objects in the Galaxy, whichimplies that microlensing will be a very rare phenomenon.Another notable feature is that rEis proportional to thesquare root of the lens mass. This means that the area ofsky ‘covered’ by a lens (at fixed distance) is proportionalto its mass, so the total fraction of sky covered dependsonly on the total mass density in lenses, not the individuallens masses. This fraction is called the ‘optical depth’ τ ,and is ∼10−6for Galactic microlensing. The duration for amicrolensing event is given by the time for the lens to moveby 2rErelative to the Earth–star line; for typical Galacticspeeds of 200 km s−1, this is ∼130 days × (M/M)1/2.For perfect alignment, simple geometry gives the(small) deflection angle of the light ray meeting Earthas α = rE/Dol+ rE/Dls, where Dolis the observer–lensdistance, Dlsis the lens–source distance etc. Requiring thisto equal the general relativity deflection, α = 4GM/c2rE,we obtainrE=4GMc2DolDlsDos0.5.The angular Einstein radius is just θE≡ rE/Dol.Ifwenowintroduce a small offset of the lens by a distance b fromCopyright © Nature Publishing Group 2001Brunel Road, Houndmills, Basingstoke, Hampshire, RG21 6XS, UK Registered No. 785998and Institute of Physics Publishing 2001Dirac House, Temple Back, Bristol, BS1 6BE, UK1MicrolensingENCYCLOPEDIA OF ASTRONOMY AND A STROPHYSICSFigure 2. Microlensing event lightcurves (magnification versustime) for six values of the impact parameterumin= 0.0, 0.2,...,1.0 as labelled. Time is in units of theEinstein radius crossing time rE/v⊥. The inset illustrates theEinstein ring (dotted circle) and the source paths relative to thelens (dot) for the six curves.the Earth–source line, i.e. an angle β ≡ b/Dol, a simplegeneralization gives the two image angular positions(relative to the lens) asθ±= 0.5[β ± (β2+4θ2E)1/2].Since lensing preserves surface brightness, the magnifi-cation Aiof each image is given by the ratio of image tosource areas, which for a ‘small’ source and any axisym-metric lens is justAi= θiβdθidβ .For a point lens, this leads to a total observed magnificationas the sum of the two image magnifications,A = A++ A−=u2+2u(u2+4)1/2(1)where u ≡ β/θE= b/ rEis the misalignment in units of theEinstein radius. This behaves as A ≈ u−1for u  0.5, sothe magnification may be large, but as A ≈ 1+2u−4foru  2, so the magnification rapidly becomes negligibleat large u. For uniform motions, we will have u(t) ={u2min+[v⊥(t