 We've seen how massive galaxies can act as gravitational lenses to create Einstein rings. You can imagine how massive clusters with thousands of galaxies would produce much larger and more powerful gravitational lenses. But before a gravitational lens can provide accurate detailed information about the object being lensed, we need to know its light bending characteristics. Clusters of galaxies don't provide smooth lenses. They are quite complex and no two are alike. The source object's light is bent by the cluster's clumpy matter distribution. And to make it harder, most of the cluster is dark matter that we can't see directly. All this makes cluster lens analysis very difficult and time consuming. But with these magnifying glasses in space being our only window into the early universe, astronomers and astrophysicists have been analyzing galaxy cluster lensing for a half a century now. We'll start with the Coma Cluster to illustrate how they do it. The first step is to measure the radial velocities of every galaxy in the cluster from their Doppler shifts. This is then generalized into their three-dimensional velocity dispersion statistical equivalence. This galaxy motion gives us the kinetic energy of the cluster. This allows us to solve for the mass of the cluster. In addition, each lens object found in the cluster is analyzed to calculate the magnitude of the mass needed to create the observed distortion. The end result is a model for what the cluster will do with the light passing through it. To illustrate the basic geometry of gravitational lensing, we'll use the giant galaxy cluster WHL0137-08's lensing of the sunrise arc. Here's the basic geometry of gravitational lensing. This is the sunrise arc light on a direct path to us as if the galaxy cluster was not there. We could not detect this light. It is too dim for even our most powerful space telescopes. Here's its light heading in another direction. As it encounters the cluster, it is bent towards us by the cluster's mass. We can estimate the deflection angle once we know the mass and center of mass of the cluster. On Earth, we observe the image to be on a straight line at an angle from its actual direction. The lens equation gives us these angles. This geometry enables us to map points seen on the lens plane back to its position on a source plane. Note that the magnitude of the light bending changes for objects closer to the lens. The lens magnification will decrease as the ratio of the distance from the lens to the object over the distance of the lens to the observer decreases. Here's an example. In this image, a remote galaxy has been distorted and magnified by a factor of 20 by the gravitational lensing. Lensing effects also created multiple operations around the curved arc of the single background magnified galaxy. The object was nicknamed the molten ring because of its appearance. The typical magnification created by this lensing ranges from 20 to 40 times the source size. But there is one more additional optical process involved that can magnify an object thousands of times over. Here's how it works. Picture a set of uniformly distributed particles on a line, each with slightly different velocities. They start out with a uniform particle density. But because of the small velocity differences, the particle density will vary as time goes by. Areas of high and low density will develop. The density at a later time, T, is described by an equation. The equation has hotspots when the denominator approaches zero. Expanding this to two dimensions, we get density peaks along curved lines that themselves intersect at points with maximum intensity. I see this phenomena in my own backyard swimming pool. Sunlight is evenly distributed as it reaches the water's surface. All waves on the surface are creating small changes in the sunlight's direction. It's the caustic process that generates the lines at the bottom of the pool. These lines are not ripples in space-time. They are simply lines of intense light magnification. Light passing through a galaxy cluster is impacted in exactly the same way. Lines of extreme magnification, referred to as lensing critical curves, are created by the caustic process. As a distant object approaches a critical curve, its image will be duplicated side by side and under some circumstances will develop yet a third image. Astrophysicists and astronomers model these lines and use the lens equations to map them back to their source. They can then reconstruct the shape and dimensions of an object, determine its deformation, calculate its observed magnification, and deduce its intrinsic luminosity. Here's an example. This galaxy is visible twice because its light has approached a critical curve, or sometimes referred to as a ripple of dark matter, enabled 68. The galaxy cluster Max 1206, 4.6 billion light years away, has produced 47 multiple images of 12 newly identified, more distant galaxies as seen in this Hubble picture. This dashed line identifies a calculated critical curve for a galaxy 3.2 billion light years behind the cluster. This solid line identifies a calculated critical curve for a galaxy 8.3 billion light years behind the cluster. You can see how the gravitational lens characteristics change with the distance to the far away sources.