Exercice 4 - Galaxy Clusters as Gravitational Lenses

In this practical work, we will make use of deep images of the massive galaxy cluster Abell 370 (A370 in the following). You have been given three frames obtained with the ESO VLT through three filters. They consist of numerous short exposures that have already been reduced as explained in Chapter 1 of this booklet. The reduced exposures have also been combined into one single frame that allow to see very faint objects.

Among the numerous objects in the images are distant galaxies, nearby galactic stars and a massive galaxy cluster. We will use the images in order to estimate the mass of the galaxy cluster and study the color distribution of galaxies in the cluster.

4.1 - The Cluster as a Gravitational Lens

Following Einstein’s equivalence principle between gravity and inertial forces, all bodies even with no mass, are under the influence of gravitation. As a direct consequence, a photon can be attracted by a massive body such as a galaxy or a galaxy cluster. As an optical lens does, a mass modifies the path followed by photons and therefore acts as a gravitational lens. The angle by which the light rays are bended by an object with a mass M can be calculated in the context of Einstein’s theory of general relativity, as:

\hat{\alpha} = \frac{4 G M}{c^2 \xi},

where \xi is the impact parameter of the light ray (see Fig. 1. Compute the value of \hat{\alpha} for the mass of the sun. The impact parameter can be approximated by the radius of the sun.

Fig. 1 gives a schematic description of the phenomenon of gravitational lensing, where a source S is lensed by a massive object in the lens plane (labeled L). The source is not seen at its real angular position \beta by the observer O, but is displaced by an angle \alpha that depends on the true deflection angle \hat{\alpha}.

../../_images/lens.jpg

Fig.1 : Schematic view of the gravitational lensing effect. The observer “O” sees the image “L” of the source “S” at a position \theta on the plane of the sky. The source S is never observable. The lensing galaxy is also called the deflector, hence the indices used to label the different distances.

If we assume that the distances between the different objects are known show that, when the source, the lens and the observer are almost perfectly aligned (i.e., \beta\sim0):

\alpha = \hat\alpha \cdot \frac{D_{ds}}{D_s}

When the source is perfectly aligned with the observer and then lens (i.e., \beta=0), it is imaged into a ring called an Einstein ring. Using Fig. 1 and simple geometrical considerations, show that the Einstein radius is:

\theta_E = \sqrt{\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}}

All the above is true only when the distances involved in the calculations are angular diameter distances. As the reference used to measure distances changes with redshift (i.e., the ruler you are using to measure distances is changing with the redshift), angular diameter distances are given under the form of an integral. The distance between two objects at redshifts z_1 and z_2 (z_1<z_2) is given by:

D_{12} = \frac{1}{1+z_2}\, f_K(x_{12})

with

f_K(x) = \left\{\begin{array}{ll}1/\sqrt{K}\, \sin\left(\sqrt{K}\,x\right) & K>0 \\x & K=0~~~~ ,\\1/\sqrt{-K}\, \sinh\left(\sqrt{-K}\,x\right) & K<0\end{array}\right.

K\approx \left(\frac{H_0}{c}\right)^2\, \left(\Omega_M+\Omega_\Lambda-1\right)

and

x_{12} = \frac{c}{H_0}\int^{z_2}_{z_1}\frac{dz}{\sqrt{(1-\Omega_M-\Omega_{\Lambda})(1+z)^2 + \Omega_M(1+z)^3 + \Omega_{\Lambda}}},

where H_0 is the Hubble parameter, \Omega_M is the density of matter of the universe, normalised to its total density and \Omega_{\Lambda} is the density associated to the vacuum. From the most recent studies using distant supernovae, the current values of these parameters are \Omega_M=0.3 and \Omega_{\Lambda}=0.7. The value of H_0 is still poorly known. The currently more accepted (but probably biased) value is H_0=72\,\rm{km\,s^{-1}\,Mpc^{-1}}.

  • As the integral above can not be calculated explicitly, integrate it numerically to compute the distance between us (z_1=0) and an object at redshift z_2. You could code your own elementary integration, or use one of scipy.integrate’s numerous possibilities.

    Make a graph showing the distance as a function of redshift. Also plot the Hubble relation for comparison. Of course this relation is valid only at very low redshifts. At z>1 it would even imply that galaxies have velocities larger than the speed of light. The Hubble law, in its original form does not take relativistic corrections into account.

  • Using the three VLT images of A370, make a color image, as was done for NGC 613. You can use for instance stiff or the log scale option of ds9. Later you can fine-tune the color scale using the color editor in gimp in order to emphasize the color contrast between the different objects in the field of view.

  • You notice immediately an overdensity of galaxies that seem to have the same color. These galaxies belong to the A370. Give a plausible reason for them being of the same color.

  • An arc-like structure is seen close to the center of the cluster. This arc is not physically in the cluster but corresponds to a gravitationally lensed galaxy in the background of the cluster. The redshift of the arc is z=0.72 while the redshift of the cluster is z=0.37. Compute the angular diameter distances to the cluster and to the arc as well as the distance between the cluster and the arc.

  • The pixel scale in the image is 0.2 arcseconds. Considering the arc as a part of an Einstein ring, measure the angular Einstein radius of the cluster. Give an estimate of the mass of the cluster.

  • If the lensing mass is considered as an isothermal sphere, its Einstein radius can be expressed as a function of the velocity dispersion following:

    \theta_E = \frac{4\pi \sigma_v^2}{c^2}\frac{D_{ds}}{D_s}

    Give an estimate of the velocity dispersion of the lens. How does this compare with the velocity dispersion measured in the second chapter for the same object ? If you find a difference, try to explain it.