- A gravitational lens is produced in space by a large foreground mass directly in the line of sight of a target object. The light form the target object is bend as it passes through warp space-time that surrounds the large mass. The bending light is not unlike a glass magnifying lens that focuses light and makes objects appear larger to our eyes.
- In this case the massive object is a Galaxy Cluster , RCS2 032727-1342623. The Cluster is 5,400,000 lightyears away in the Constellation Cetus the Whale. The target galaxy behind the Cluster is 9,700,000 lightyears away.
- The target galaxy’s light takes 9,700,000 years to reach us that is 70% back to the beginning of the Universe. Galaxies then were in the early star forming years and astronomers are anxious to study them. Galaxies nearby are fully matured and at the tail end of their star forming ability. Normally the distant galaxies are too far away and too dim to see. The Hubble Telescope could not capture the image if it were not for the intervening Gravitational Lens.
- The image is distorted because the Galaxy Cluster does not make a perfect lens. If the lens were perfect the image would be a perfect circle , a ring around the Cluster. However, once astronomers characterize the Cluster than can recreate the image by removing the distortions. The image has a partial arc and the astronomers use the arc to calculate the diameter of the ring if it existed. This is not unlike looking through the bottom of a wine glass pointed at a candle. You will see multiple images and distorted arcs of the flame created by the glass lens.
- The angular diameter of the ring was measured to be 38 arc seconds.
- The mass of the Galaxy Cluster can be calculated from this information. The mass is directly proportional to the diameter of the ring in radians and to the distance to the Cluster , “D”, and the distance to the target Galaxy ,”d“. It is inversely proportional to the separation of the two, “d - D“.
------------------------- Mass = Constant * ring^2 * d * D / (d - D)
- Any proportion can be transformed into a equality by using the proper constant of proportionality. In this case the Constant is proportional to the constant speed of light squared, c^2”, and inversely proportional to four time the force of gravity, “G”
-------------------- Constant = c^2 / 4 * G
------------------- Mass = (c^2 / 4 * G) * ring^2 * d * D / (d - D)
-------------------- Mass of the Galaxy Cluster
--------------------- ring is the angular diameter of the ring in radians. The ring is 38 arc seconds in diameter. 1 radian = 206,265 arc seconds.
---------------- ring = 38 / 206,265 = 1.84*10^-4
------------------ D is the distance to the Cluster = 5.4*10^9 lightyears. 1 lightyear = 9.5*10^15 meters.
------------------ D = 51.3 * 10^24 meters.
---------------- d = distance to the target Galaxy = 9.7* 10^9 lightyears = 92.15*10^24 meters.
------------------ c = constant speed of light = 38 * 10^8 meters / second
----------------- G = the gravitational constant = 6.67*10^-11 meters^3 / ( kg*sec^2)
------------------- Substituting all this into to the equation above for the mass of the Galaxy Cluster = 132.*10^43 kilograms.
----------------- The mass of the Sun is 66 * 10^13 kilograms
--------------- Therefore the mass of the Galaxy Cluster is 66,000 Solar Mass.
- How to derive these equations is over my head. But, there is a second way to calculate the mass of the Gravitational Lens, the Galaxy Cluster. The galaxies in the cluster are orbiting a common center of gravity. The orbital speed of these galaxies on average is 988 kilometers per second, or 2,210,000 miles per hour.
- The velocity is proportional to the mass and inversely proportional to the radius of the orbit. The constant of proportionality is the force of gravity, G.
------------------ v^2 = G * Mass / R
- The radius is 9.5 million lightyears. Substituting into this formula and solving for mass
---------------------- Mass = 1321*10^42 kilograms
--------------------- Mass = 66,000 Solar Mass.
- Of course, using these equations, if you know the mass of the Galaxy Cluster through other means such as this formula for the velocity and you know the distance to the Cluster you can calculate the distance to the target Galaxy.
- The distances are also calculated using the redshift in the light spectrum due to the expansion of the Universe. There are other reviews available if you want to learn how redshifts are calculated. An announcement will be made shortly , stay tuned.
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707-536-3272, Saturday, March 3, 2012
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