eaa iop org DOI 10 1888 0333750888 2632 Microlensing Will Sutherland From Encyclopedia of Astronomy Astrophysics P Murdin IOP Publishing Ltd 2006 ISBN 0333750888 Institute of Physics Publishing Bristol and Philadelphia Downloaded on Tue Jan 31 17 08 16 GMT 2006 127 0 0 1 Terms and Conditions Microlensing E N C Y C LO P E D IA O F A S T R O N O M Y AN D A S T R O P H Y S I C S Microlensing Microlensing refers to the special case of GRAVITATIONAL LENSING where the multiple images produced are too close together on the sky to be observed as separate images However the lensing can still be detected because these multiple images appear as a single object of increased apparent brightness Although this is not detectable in a one off observation since we do not know the normal brightness of the source with the passage of time the lens moves across the Earth source line and the amount of brightening changes Typically the source will appear to brighten reach a maximum and then fade symmetrically back to normal over the course of a few weeks or months this is called a microlensing event The major application of microlensing suggested by Paczynski in 1986 is in the search for the DARK MATTER which is strongly believed to exist from rotation curves of spiral galaxies etc Since the lensing effect depends only on lens mass it can be used to search for very faint or invisible objects such as brown dwarfs neutron stars old white dwarfs or black holes which might make up the dark matter These are collectively known as massive compact halo objects or MACHOs in contrast to the hypothetical weakly interacting massive particles or WIMPs To understand the basics of microlensing consider a small massive object the lens situated exactly on the line of sight from Earth to a background star and consider a number of light rays radiating from the star passing the lens at different distances and being bent towards the lens Since the bending angle for a light ray increases with decreasing distance from the lens it is clear that there is a unique miss distance such that the ray will be deflected just enough to hit the Earth this distance is called the Einstein radius By rotational symmetry about the Earth star axis an observer on Earth with perfect resolution would see the star lensed into an annulus centered on its true position called an Einstein ring As the lens is moved slightly off the line of sight e g by 0 1 Einstein ring radii the Einstein ring splits into two banana shaped arcs one on the same side of the lens as the source one on the opposite side As the lens moves further off more than 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 image of the star Figure 1 illustrates a sequence of such images for a typical microlensing event Although the perfect alignment giving the Einstein ring will rarely occur in practice it is still a very important concept because the size of the hypothetical Einstein ring sets the length scale over which substantial brightening will occur As we will see for a typical lens in our Galaxy the radius of the Einstein ring rE is roughly 8 M M 1 2 AU astronomical units where M is the lens mass Knowing this scale allows us to understand most of the general characteristics of microlensing it is extremely small compared with the typical distance to a lens so the angular separation of the two images will be too Figure 1 A microlensing event seen at perfect resolution The axes show angular offsets on the sky from the lens central dot in units of the Einstein angle the dashed circle is the Einstein ring The series of small open circles shows the true source position at successive timesteps For each source position there are 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 it is considerably larger than either the size of a star or the size of a MACHO so we can usually approximate the lens and source as pointlike which leads to a simple prediction for the lightcurve shape Also rE is very small compared with the typical separation of objects in the Galaxy which implies that microlensing will be a very rare phenomenon Another notable feature is that rE is proportional to the square root of the lens mass This means that the area of sky covered by a lens at fixed distance is proportional to its mass so the total fraction of sky covered depends only on the total mass density in lenses not the individual lens masses This fraction is called the optical depth and is 10 6 for Galactic microlensing The duration for a microlensing event is given by the time for the lens to move by 2rE relative to the Earth star line for typical Galactic speeds 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 Earth as rE Dol rE Dls where Dol is the observer lens distance Dls is the lens source distance etc Requiring this to equal the general relativity deflection 4GM c2 rE we obtain 4GM Dol Dls 0 5 rE c2 Dos The angular Einstein radius is just E rE Dol If we now introduce a small offset of the lens by a distance b from Copyright Nature Publishing Group 2001 Brunel Road Houndmills Basingstoke Hampshire RG21 6XS UK Registered No 785998 and Institute of Physics Publishing 2001 Dirac House Temple Back Bristol BS1 6BE UK 1 Microlensing E N C Y C LO P E D IA O F A S T R O N O M Y AN D A S T R O P H Y S I C S Figure 2 Microlensing event lightcurves magnification versus time for six values of the impact parameter umin 0 0 0 2 1 0 as labelled Time is in units of the Einstein radius crossing time rE v The inset illustrates the Einstein ring dotted circle and the source paths relative to the lens dot for the six curves the Earth source line i e an angle b Dol a simple generalization gives the two image angular positions relative to the lens as 0 5 2 4 E2 1 2 Since lensing preserves surface brightness the magnification Ai of each image is given by the ratio of image to source areas which for a small source and any axisymmetric lens is just i i d i Ai d For a point lens this leads to a total observed magnification as the sum of the two image magnifications A A A u2 2 u u2 4 1 2 be far enough away to give a good path length for …