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Strong Lensing

Page history last edited by Tom 14 years, 8 months ago

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When the light from a distant object is bent to such a great extent that we, the observer, can see more than one image of single source then we are in the strong lensing regime. Technically, this can be defined as the region where the 2D mass distribution is greater than the critical density (for more details see General Lensing). This page is designed to give a brief overview of some simple strong lensing systems and to show some well known results. These will all use the thin lens approximation, where the observer, lens and source are on fixed planes.


In strong lensing, we can use the information coming from the multiple images to study the lens, the source and some general cosmology to do with the geometry of the Universe.


Examples of Strong Lenses


Astronomy observations are full of examples of multiple images systems. In fact, due to the extent of the lensing signal, strong lensing systems produce some of the most striking images in astronomy. Most of us working in this field usually like to illustrate the lensing effect but showing images of galaxy clusters. Clusters are the most massive (collapsed) objects in the Universe. They typically have a mass of 1015 times that of the mass of our sun. This makes them over 1000 times more massive than our own galaxy, the Milky Way. Figure SL1 shows the image of one such galaxy cluster (Abell 2218), which was taken using the Hubble Space Telescope (HST). In this image, we see a large number of giant arcs that seem to form a circular feature surrounding the brightest galaxy in the image. These arcs are the images of galaxies that are behind the cluster. In the absence of lensing, the images from these galaxies would have been fairly round. However, the mass of the cluster has bent and distorted the light from these galaxies to such an extent that we see them as long stretched arcs.



Figure  SL1: Example of a strong lensing by clusters. This is an image taken with the Hubble Space Telecope (HST) of a cluster called ‘Abell 2218’.


Since clusters are the most massive compact objects in the Universe, we can expect their lensing effect to be extreme, but strong lensing is not limited to these extremely massive objects. In fact, we have also seen strong lensing from isolated individual galaxies. Figure SL2 shows images of isolated galaxies taken with the HST. These are examples of Einstein ring systems. At the centre of each image, we see a yellow/orange blob of light. This is the lens galaxy (i.e. the one that is causing the light distortion). In blue, we can see the background source whose image has been distorted into a ring.



Figure SL2: Examples of strong lensing by galaxies taken with the HST.


Singular Isothermal Sphere (SIS)


To mathematically explain this lensing effect, we begin by considering circularly symmetric lenses. We shall see that due to this symmetry that such lenses can typically produce either 1 or 3 images depending on the position of the source. We begin by studying the Singular Isothermal Sphere (SIS), which is a convenient model of galaxies. Galaxies are often modeled as SIS's since this is the simplest model that best reproduces their observed flat rotation curves. The 3D density distribution of a SIS is


 Formula SL1


where σ2 is a measure of the temperature of the SIS. If the mass, M, enclosed within a radius, R, is known, then σ2 can be set by


Formula SL2


In order to use the thin lens approximation, we need to integrate along one of the spatial directions to convert the 3D SIS density into a 2D surface density.


Formula SL3


Placing the lens surface at a distance DL and converting the surface density into a angular dependent convergence (see General Lensing),


Formula SL4


where k0 is given by


Formula SL5


and r=DLθ. Solving the 2D Poisson Equation, ∇2ψ(θ)=2k(θ), we find that the deflection potential for a SIS is (see General Lensing) 

Formula SL6


With the deflection potential, we can now easily calculate the deflection angles using equation GL3, giving us


Formula SL7


Note that this is a radial deflection that acts towards the density peak of the SIS. Furthermore, we can calculate the second order differential of the potential to obtain the shear,


Formula SL8


Finally we can calculate the magnification,


Formula SL9


We see that the magnification goes singular at a distance of θ=k0. This curve (specifically here it is a circle) where the magnification goes to infinity is known as the critical curve. If we map this curve back to the source plane, we would produce a new curve which is known as the caustic curve.


Softened Isothermal Sphere


The fact that the SIS is singular (density goes to infinity) in the inner region also causes some concern. A simple extension of the SIS that overcomes the problem of the central singularity is the softened isothermal sphere, which is described by the following density profile:


Formula SL10


Following the same procedure as above we find the deflection potential is


Formula SL11


where k0 is the same as the one used to describe the SIS. Using this deflection potential we can calculate the deflection angle


Formula SL12


The critical and caustic curves for the softened isothermal sphere are shown in Figure SL3.


Figure SL3. The images formed by three sources on the source plane due to gravitational lensing by a non-singular Isothermal sphere. We see that the number of images formed by a source depends on its position relative to the caustic curves (blue). The orange source lies outside the caustic and so produces only one image. The blue source lies inside the radial caustic (dashed blue line), producing three images, two of which are distorted in the radial direction. Due to the radial symmetry of the lens, we see that the tangential critical curve (solid red) maps back to a single degenerate point. The lime green source close to the tangential caustic point forms three images, two of which are stretched tangentially and one is a central demagnified image.


Singular Isothermal Ellipsoid (SIE)


We now explore the consequences of breaking the circular symmetry of the lens. We do this by studying the Singular Isothermal Ellipsoid (SIE), whose convergence is described by


Formula SL13


The deflection angles and magnification for this system can also be calculated by


Formula SL14

Formula SL15


The properties of this lens are illustrated in Figure SL4, where we see that breaking the circular symmetry stops the tangential caustic from being mapped back to a degenerate point, thus leading instead to a tangential caustic curve.



Figure SL4: Diagram illustrating some examples of the images formed due to lensing from a non-singular isothermal ellipsoid. Breaking the radial symmetry causes both the critical curves (red) to be mapped back to caustic curves (blue). The three sources shown here all lie within both caustic curves, leading to formation of five images. Three basic configurations are shown: Einstein cross, cusp caustic (three images close together) and fold caustic (two images close together).


Five Easy Steps


  1. Exercise 1
  2. Exercise 2
  3. Exercise 3
  4. Exercise 4
  5. Exercise 5


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