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Graphing the Potential Impact Parameter

  1. Oct 29, 2012 #1
    1. The problem statement, all variables and given/known data
    In class we derived the Schwarzschild interior solution (constant
    density). Examine the behaviour of non-radial null geodesics in this
    spacetime.


    2. Relevant equations

    [itex]\Phi(r)=ln\left(\frac{3\sqrt{1-\frac{2M}{R}}}{2}-\frac{\sqrt{1-\frac{2Mr^2}{R^3}}}{2}\right)[/itex]

    [itex]B(r)=re^{-\Phi}=\frac{2xR}{3\sqrt{1-\frac{2M}{R}}-\sqrt{1-\frac{2Mr^2}{R^3}}}[/itex]

    3. The attempt at a solution
    I have talked to my professor and what he is looking for is a plot of the potential impact parameter "B" as a function of "r/R". In order to do this I made the substitutions shown below and simplified the equation; however, I am left with an annoying 'R' in the top of the fraction. I'm not sure if my substitution is incorrect or I'm not noticing a way to get rid of it. Thanks for any help!

    (1) [itex] R=\alpha M[/itex]

    (2) [itex] r=Rx [/itex] where 'x' is what I will plot as the horizontal axis

    (3)[itex] B=\frac{2xR}{3\sqrt{1-\frac{2}{\alpha}}-\sqrt{1-\frac{2x^2}{\alpha}}}[/itex]
     
    Last edited: Oct 29, 2012
  2. jcsd
  3. Nov 4, 2012 #2
    Just plot B/R vs r/R. I'm assuming that R is a constant?
     
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