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I am interested in gravitational lensing caused by a cluster of galaxies (say it has a diameter of 1 Mpc and mass of $10^{12}$ solar masses). How close must a light travel as it passes by to be notably bent - if compared to a case where there would be much smaller object in a spot where is now a center of cluster, like one isolated galaxy?

I am trying to determine angle of deflection for a special case of gravitational potential and I have no feeling for the distances at which lens still has an effect on light.

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I'd be curious, what is the application of this? Astronomy? Or some theoretical physics? –  Tomas Sep 22 '11 at 13:04
    
Observing clusters of galaxies could potentially give many relevant cosmological results and it turns out in some cases gravitational potential is more useful information than mass distribution (path of bent light directly depends on potential and indirectly on mass). General results that describe trajectory and deflection angle for light bent by massive object, are derived for point-like mass and it's potential 1/r. So it is an approximation. Cluster of galaxies can be better described with different potential and therefore gives different results for path of light. –  Ivana Sep 22 '11 at 18:19
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up vote 2 down vote accepted

The bending of light in GR is by an angle which is twice the ratio of the Schwartzschild radius of the mass divided by the distance to the light. It falls off as 1/distance, so it is a very slow falloff, and essentially if you get noticible bending at one distance, you get noticible bending at any distance.

A solar mass has a Schwarzschild radius of 3km, so 1012 solar masses is 3 x 1012 km, or in term of light-seconds 104 light-seconds, or three light-hours, 1/3000 light-year, so the deflection of light passing 105 light years out will be deflected by 2 x 10-8 turns, or about two hundredth of an arcsecond.

Many cases are worked out in this answer: How does gravitational lensing account for Einstein's Cross?

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