Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius rs) (see this for illustration). Its impact parameter is b and its distance of closest approach is r0 with b2=r30r0−rs
Then its trajectory in polar coordinates is defined by :
dφdr=1r2√1b2−(1−rsr)1r2
Consequently : φ(r)=∫rr0dpp2√1b2−(1−rsp)1p2
and one can compute the total deviation using : Δφ=2×(limr→+∞φ(r)−π2)
But my question is : how can I plot/draw the trajectory using the integral expression of φ(r) ?
Because if I compute : f(r)=2×(∫rr0dpp2√1b2−(1−rsp)1p2−π2)
If I compute : g(r)=∫rr0dpp2√1b2−(1−rsp)1p2
I would like to compute the trajectory in the (x,y) plane, so how to use the values of f(r) or g(r) to compute the function y(x) ?
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