# Photon "escaping" from photon sphere in Schwarzchild space

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1. Jan 1, 2015

### N00813

1. The problem statement, all variables and given/known data
Close to a Schwarzchild black hole, a photon is emitted between r = 2(mu) and 3(mu), where $\mu = \frac{GM}{c^2}$. The photon is emitted at an angle (alpha) to the radial direction. At r = 2(mu), the highest angle that the photon can escape at is (alpha) = 0; at r = 3(mu), the highest angle is (pi/2). For a general r between 2(mu) and 3(mu), what is the highest angle (alpha) in which the photon can escape to infinity?

2. Relevant equations
Eqns of motion (EQM):
$(1-\frac{2\mu}{r}) \frac{dt}{d\sigma} = k$

$c^2 (1-\frac{2\mu}{r}) (\frac{dt}{d\sigma})^2 - (1-\frac{2\mu}{r})^(-1) (\frac{dr}{d\sigma})^2 -r^2 ((\frac{d\phi}{d\sigma})^2 =0$
$r^2 (\frac{d\phi}{d\sigma}) =h$

Impact parameter:
$b = \frac{h}{ck} = r sin (\phi)$

3. The attempt at a solution
I guessed that the condition to escape was that $\frac{dr}{d\phi} =0$ at r = 3 mu to give b, and then using that to give (phi), but that didn't get the right result for r = 2(mu).
I think that I don't really know what the condition for escape to infinity is.

2. Jan 6, 2015