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 with some modifications.  On pages 173-176 of his book, Arp points out that gravitational lensing does not explain the observations satisfactorily.)

Simple point mass model for a gravitational lens

   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, but 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 located 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 The angular separation between the upper and lower quasar images is 1.6 arcseconds.

   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 aligned with the lensing galaxy.  If the mass is concentrated in a small volume, only two images are produced by the gravitational lens (and if the quasar is centered on the lensing mass, a ring appears).  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 explains 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.

  More information is available on more complex mass distributions producing gravitational lensing.  The web page Gravitational Lensing With Adobe Photoshop, gives many links to web pages.  Also, Chapter 1 in the thesis discusses some important points in the calculation of gravitational lensing.

  The question about the mass distribution of the lensing galaxy of the Einstein cross is discussed in a paper at  This paper answers the following questions:

- What is the mass distribution that can give such a cross? (An oval (quadrupole) mass distribution (such as the bulge of a galaxy) typically produces 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?

Concerns expressed by Arp

  In his book "Seeing Red", Halton Arp expresses some concerns about 2237+0305 which are:

- The low probability, estimated at 2 in a million, that this near perfect alignment occurs.
- The gaseous connection between quasar component D (left from the galaxy in Fig. 1) seen in the 3400 Angstrom wavelength band that includes the Lyman alpha line of the redshifted quasars (Fig. 3). While image A is well resolved, image D is attached to the galaxy.  Note that the connection is not a result of the point spread function which also causes 'bridges' between sources with little angular separation (Fig. 4).
- High redshift gas is detected in the filament between image D and the galaxy nucleus (see Phys. Lett. A 168, 1992 ).
- The estimated M/L of the lensing galaxy is unacceptably high.
- If resolved, the luminous isophotes should be extended along a direction parallel to a circumference (an arc).The images of the quasars are well resolved and do not show the extension into arcs they should have.

Figure 3.
  Hubble Space Telescope picture, in false color, of the Einstein Cross at the wavelength of redshifted hydrogen Lyman alpha emission.  Connecting material is seen between the quasar D and the central galaxy core.  (The original image was mirrored(?) to match the orientation in the Figure above.) From: 


Figure 4.
  Simulated isophotes (false color) generated from five point sources blurred to simulate a point spread function.  The five sources were places at the locations of the four quasar images and the lensing galaxy of the Einstein cross in Fig. 1.

Louis Marmet, October 1999, second edition November 2007, third edition January 2013.