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-------------------- 2533 - DISTANCES - using a gravity lens?
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- Astronomers are observing a Cluster of Galaxies and want to use their enormous mass for a gravitational lens in order to see even more distant galaxies directly behind the Cluster. See 2532 - “DISTANCES - how to measure them” for many of the methods to measure the distances to galaxies. But, this is a Cluster of Galaxies. To measure the distance we use the redshifts for five of the galaxies in the Cluster to measure their recession velocities:
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----------------- Recession velocity ----------------- distance
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----------------- 9700 km/sec ----------------- 440 million lightyears
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----------------- 8600 km/sec ----------------- 390 million lightyears
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----------------- 8500 km/sec ----------------- 389 million lightyears
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----------------- 10,000 km/sec --------------- 450 million lightyears
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----------------- 8200 km/sec ----------------- 370 million lightyears
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- Using the recession velocity we can calculate the distance using the Hubble Constant for the expansion of the Universe of 22 km/sec per million lightyears. You can see a problem here. Our distances range over 80 million lightyears. The average recession velocity is 9000 km/second and this represents the expansion of the Universe where the Hubble Constant would apply. The distance to the Cluster is therefore 409 million lightyears.
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- The other velocities that are plus and minus this average velocity represent the rotational velocities of the individual galaxies orbiting the center of the Cluster. The rotational velocities are therefore + 900 km/sec when the galaxies is rotating away from us (redshift) and -900 km/ sec when the galaxy is rotating towards us (blue shift).
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- To calculate the mass of the Cluster of Galaxies we start with Kepler’s formula that states the period^2 is directly proportional to the radius^3 of the orbiting body. The formula is:
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--------------- period^2 = 4*pi^2 * radius^3 / G * Mass
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-------------- where: G is the force of gravity = 6.67 *10^-11 m^3/kg*sec^2
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-------------- where: period = 2*pi*radius / velocity, assuming the orbit is a circle.
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--------------- Mass = radius * velocity^2 / G , substituting into Kepler’s formula
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- To calculate the Mass we need to know the radius. Astronomers projected every galaxy and determined each to be roughly 0.5 degrees from the Cluster center. The sine of 0.5 degrees is 8.7*10^-3. The radius is 8.7*10^-3 * 409 million lightyears. Radius is 33.5*10^21 meters, or 3.5 million lightyears.
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-------- Mass = radius * velocity^2 / G
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-------- Mass = 33.5^10^21 m * ( 9*10^5 m/sec)^2 / 6.67 *10^-11 m^3/kg*sec^2
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-------- Mass = 4.6*10^45 kg
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-------- Mass = 2.3*10^15 Solar Mass.
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- Knowing the mass of the Cluster of Galaxies will allow us to calculate the angle that light will be deflected by the gravitational lens created by the enormous mass. The angle that light is bent = 2* Schwarzchild radius / grazing distance to the mass.
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- This formula only works if the grazing radius is much, much larger than the Schwarzchild radius, and, the grazing distance is much, much smaller than the distance to the object. The Schwarzchild radius is the smallest radius a mass can have before it turns into a Black Hole. These conditions apply in our case:
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--------------- Angle = 2*Schwarzchild radius / b
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---------------- Schwarzchild radius = 2* G* Mass / c^2
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---------------- Angle = 4*G*Mass / c^2 * b
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----------- where: b = the grazing distance to the Mass = 1 lightyear = 9.5*10^15 meters
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---------------- where: c = the speed of light = 3*10^8 m/sec
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-------------- Schwarzchild radius = 2* G* Mass / c^2
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------------- Schwarzchild radius = 2* 6.67*(4.6*10^45) / (9*10^16)
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------------- Schwarzchild radius = 72 * 10^12 meters
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--------------Angle = 2*Schwarzchild radius / b
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------------- Angle = 2 * 72*10^12m / 9.5*10^15 m
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------------- Angle = 0.015 radians
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------------- Angle = 0.86 degrees.
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- Therefore, light passing within one lightyear of the Cluster of Galaxies will bend almost one degree (0.86). This angle can now be used to calculate the distance of the object behind the Cluster and the degree of magnification created by the gravitational lens.
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- December 9, 2019 2533 864
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--------------------- Tuesday, December 17, 2019 --------------------
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