The Einstein Cross:

How is the mass distributed in the gravitational lens?


  (This page was initially posted in 1999.  After reading the book "Seeing Red" by Halton Arp, I decided to post this page again.  On pages 173-176 of his book, Arp points out that gravitational lensing does not explain the observations satisfactorily.  My question as stated below still remains unanswered.)

   Einstein's prediction that light would be bent in a gravitational field has been shown in many examples taken from our universe.  The effect is small, thus observable only with powerful astronomical instruments.  Despite this difficulty, many observations of light deflected by a massive object are reported.  The interesting consequence of this phenomenon is that a massive object can act as a gravitational lens, a name given by analogy with the refracting lenses encountered in optics.

   One example of the effect of a gravitational lens is an "Einstein Cross".  The picture below shows the objects 2237+0305: a galaxy placed in front of a quasar.
 

Einstein cross 2237+0305
Figure 1.
   The image of an Einstein cross 2237+0305 as an example of a gravitational lens.  The explanation for this pattern claims that it is produced by a galaxy which deflects the light from a quasar into four distinct images (from http://www.nap.edu/readingroom/books/cosmology/4.html).

   The image of the quasar is deflected into four distinct images surrounding the central galaxy core.  The question is: how can this happen?

   The Einstein cross is intriguing because a mass having a spherical symmetry can only give rise to images located on a straight line.  If the mass is concentrated in a small volume, only two images are produced by the gravitational lens.  The following Figure shows what happens:
 
 

Deflection of a light ray by a massive object
Figure 2.

   A ray of light emitted by a source S is indicated by the blue line.  The ray is bent at point I  by the mass M and reaches point E.  The points S, M and O are in the plane of the Figure,  but points I and E may be located outside the plane SMO.  The vectors vector b and vector d are parallel to each other but point in opposite directions.  The angle of deviation is given by the equation delta=4GM/(bc^2).

   For the observer O to see the source, the point E must coincide with O.  The observer will see the source shifted by the angle d (the direction ).  This can only happen if vector b and vector d are in the plane SMO.  Then, vector r is also in the same plane.  Therefore, the observer can only see an image of the source S located somewhere on the y axis.  Careful examination of the equation shows that there are only two solutions, producing an image of S on either side of the mass M.

   This may explain two of the four images of the quasar in the Einstein cross.  However, for most cases of gravitational lensing, the object which serves as the lens is not a point mass.  This could help to explain the other images of the quasar on object 2237+0305, but the lens would need to have a very complex shape that is not apparent on the picture.

- What is the mass distribution that can give such a cross?
- Why is the mass distribution of the visible part of the galaxy so symmetric?
- Is there more than one galaxy in front of the quasar?

If you can help me find the answers, contact me.

Louis Marmet, October 1999, second edition November 2007.

Here are some other sites on gravitational lensing:

http://leo.astronomy.cz/grlens/grl0.html: Gravitational Lensing With Adobe Photoshop