Impact parameter of a photon in Schwarzchild metric

[Big Guy]

Hi, I'm having trouble answering Question 9.20 in Hobson's book (Link: http://tinyurl.com/pjsymtd). This asks to prove that a photon will just graze the surface of a massive sphere if the impact parameter is $$b = r(\frac{r}{r-2\mu})^\frac{1}{2}$$

So far I have used the geodeisic equations $$(1-\frac{2\mu}{r})\dot{t} = k$$ and $$r^2\dot{\phi} = h$$ to give $$\frac{d\phi}{dt} = \frac{b(1-\frac{2\mu}{r})}{r^2}$$ and b = h/k due to the argument given here http://www.physicspages.com/2013/06/13/photon-equations-of-motion/

This is extremely close to the actual result but I can't figure out why $$\frac{d\phi}{dt}=\frac{1}{b}$$.

Any help? Thank you!

 

[Big Guy]

I solved it myself. The metric for lightlike separation implies $$g_{00}\dot{t}^2 +g_{11}\dot{r}^2+g_{22}\dot{\phi}^2 =0$$ and we have expressions for phi dot and t dot from the OP. Just plug them in and since the expression is true everywhere we evaluate it on the surface of the star i.e where motion is purely tangential -> r dot is zero. So we just arrange the above equation for b = h/k to get the required answer.