Masses of Galaxy Clusters: Gravitational Lensing
By: Jacky Lee, Susan Wong, Jeremy Beach
According to Einstein’s theory of general relativity, gravity acts on all matter
and energy, including light. The light ray from a distant star will bend by heavy
objects along their journey which generate a slightly different image on the sky. This
was confirmed by Arthur Eddington during the total solar eclipse of 1919.
The image might be distorted, or even magnified by the cluster's gravitational
lens, depending on the position of the light emitting source relative to the cluster and
to the observer. Cluster gravitational lenses allow us to observe objects that are much
too far or too faint to be seen directly, helping us view the very distant, early universe.
What’s Gravitational Lensing?
Light Deflection
According to Einstein’s theory of general relativity, gravity is in terms of assemblies
of mass and energy curving space-time. So, gravity acts on all matter and energy,
including light. This implies that masses deflect light in a way similar to a convex lens:
they curve light rays towards themselves (See Figure 1).
Figure 1.
The deflection of light rays from their curved straight paths can give the multiple
images because light rays can pass the mass in more than one way. Therefore the
deflection is differential that magnification and de-magnification are possible.
Gravitational Lens
Gravitational lens is a heavy object which bends the light coming from the source
along its journey to the observer.
If the light rays are deflected and have multiple images on the Sky, how can we
choose a real object to be the miconlensed object?
The gravitational lensing will not change the surface brightness, so the apparent
brightness of each image is just proportional to its area. The lensing will only increase
a star’s brightness without changing the colour because gravity will bend radiation of
all wavelengths equally. That’s why the MACHO (Massive Compact Halo Objects)
hunters measured the brightness of their stars through two filters at different
wavelengths. Only stars that varied by the same fraction in both colors were accepted
as microlensed objects.
Geometric Calculation: Masses of Lens by Graviational Lensing
Figure 2.
As we see in figure 2, S’ is the true position of the star and I is the image cause by the
deflection of the light rays from S’. Light ray passing at distance b from a mass M
located at L is bent by an angle α, which given in radians by
(1)
This approximation is only true when the bending is very small, α<<1. By looking at
Figure 2, β = sin-1 (y / ds) ≈ y / ds if ds >> y and Ө ≈ x / ds if ds >>x. Using the
equations above and the equation x – y = α ds, we will get a new equations by
dividing ds to equation (1).
(2)
If a bright source lies exactly along the line of sight to the gravitational les, then it will
changed to an Einstein ring which encircling the lens (β=0). θE is called the Einstein
radius.
(3)
So, we have a quadratic equation for the angular distance Ө between L and the I.
  
d LS
dS

4GM
bc 2
 d LS

 dS
 d LS 


d
S


4GM  d  d L
 2    2  S
c  dS dL
  



4GM
c 2 d L

  0

(4)
From this result, there will be two solutions for Ө and so two images will be formed
by gravitational lens. When β > 0, the image at is further away from the lens and it
lies outside the Einstein radius: Ө+ > β, ӨE. The image at Ө- will be inverted and lies
within the Einstein radius, on the opposite side of the lens.
(5)
where Ho is the Hubble constant and z is the redshift.
Then, the redshift of the source and the lens should be measured and use Hubble’s law
(5) to calculate the ds and dL. So, we can rewrite equation (2) to be,
 1 2 c 2
M
4G
 dS dL

 dS  dL



where Ө1 and Ө2 are Ө+ and Ө+
(6)
Mass of the lens can calculate directly by this equation, but remember that angle
assumes the minus sign is arbitrary.
Now, if we are letting the lens is an entire galaxy cluster, there are few
assumptions have to be considered. We can write equation (1) in terms of a lensing
potential, ΨL:
 b  
d L
db
4GM
where L  2 ln b 
c
(7)
Here, ΨL is defining as the sum of the bending caused by all the mass within the
galaxy cluster. If the cluster radius is << dL, dLS, then the deflection will depend only
on the surface density ∑ (x) of matter within it. Now, assume the cluster is
axisymmetric, so the surface density depends only on the projected distance R from
the cluster’s center. So, the bending of a light ray passing at radius b then depends
only on the mass M(<b) projected within that circle:
(8)
To prove this equation, there are two identities we have to use:
1. A light ray passing through a uniform circular ring is not bent at all.
2. A ray passing outside the ring is bent in the same way as by a point mass at its
center.
The details of this prove is at the P.274-275 or we will talk about it during the
presentation.
We can rewrite equation (2) to relate with the critical density ∑crit for the
lensing and the mass M(<b) projected within the circle:
(9)
where M(<b)/πb2 is the average surface density within radius b. The surface density
∑(R) usually has its peak at the center and decline as it moves away from the center,
so the average surface density will fall too.
1. If the ∑crit >the central density, the image, I, will exactly in line with the
cluster center where the source locate at β = 0. The Einstein ring will form
with the angular size ӨE = bE / dL, where bE is the radius where the average
density falls to the critical value.
2. If the ∑crit >the central density, the cluster cannot produce multiple images of
the source behind it and the ring cannot form as well.
As the source is moving away from the lens, the two images closest to the
center will move together. So, as the source further out, there will be only a single
image produced. Furthermore, as dβ/dӨ-> 0, at β = βr, the images fuse and then
disappear.
Time Delays
The gravitational lensing model above is just a very simple one. As we study
more deeply to the modern modified model, we will find that much more complex to
understand. One of the things which we have to modify the model is time delay.
There is a unique path for each of the light ray coming from different images. The
distances of these paths are proportional to the gravity pull. Time delay will depends
on how deep the paths are in the gravitational potential. As the gravititational
potential increases, the deflection of the path of the light ray will also increase.
Therefore, further the images from the center of the lens, the light arrives faster. By
measuring the angular positions of the images on the sky, and the redshifts of both the
lens and source, the expected time delays can be ccalculated. The value for the time
delay allows us to measure the times when the different images brighten in sequence
if the source is quasars.
Work Cited
1. Bradley W. Carroll, Dale A. Ostlie, An Introduction to Modern
Astrophysics, United States of America: Addison-Wesley Publishing
Company, Inc., ©1996, P.1116-1117 and P.1202-1206
2. Linda S. Sparke, John S. Gallagher, Galaxies in the Universe: An
Introduction, United States of America, Cambridge University Press,
©2000, P.90-93 and P.273-280
3.
http://www.mpa-garching.mpg.de/Lenses/WLRevEls.pdf
4.
Joachim Wambsganss, Living Reviews in Relativity,
http://relativity.livingreviews.org/open?pubNo=lrr-1998-12&page=node5.html
5.
Jcohn, Gravitational Lensing, May 13, 2000.
http://astron.berkeley.edu/~jcohn/lens.html
6.
Christian KaiserI, Gravitational Lensing, 2004-10-04
http://www.astro.soton.ac.uk/~crk/PH227/node43.html