Graphing the Potential Impact Parameter

[Airsteve0]

Homework Statement

In class we derived the Schwarzschild interior solution (constant
density). Examine the behaviour of non-radial null geodesics in this
spacetime.

Homework Equations

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

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

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) $R=\alpha M$

(2) $r=Rx$ where 'x' is what I will plot as the horizontal axis

(3)$B=\frac{2xR}{3\sqrt{1-\frac{2}{\alpha}}-\sqrt{1-\frac{2x^2}{\alpha}}}$

 

[202250]

Just plot B/R vs r/R. I'm assuming that R is a constant?