Calculating impact parameter

[nocks]

Homework Statement

I'm trying to plot the trajectory of a photon near a schwarzschild black hole. I have the equation for the trajectory but I'm stumped by the impact parameter and cannot find out how to calculate it.

Homework Equations

\frac{d\phi}{dr} = \pm \frac{b}{r^{2}\sqrt{1 - \frac{b^{2}}{r^{2}} (1 - \frac{r_{s}}{r}})}

The Attempt at a Solution

 

[tiny-tim]

Welcome to PF!

Hi nocks! Welcome to PF!

From the PF Library on photon sphere …​

The usual Schwarzschild coordinates, are related to the "age", \tau, of a photon (measured as number of wavelengths, since of course the "proper time" of a photon does not change) by the equations:
\frac{dt}{d\tau}\ =\ E/(1\ -\ 2M/r)
\frac{d\phi}{d\tau}\ =\ L/r^2
\frac{dr}{d\tau}\ =\ \pm E\sqrt{1\ -\ (1\ -\ 2M/r)L^2/E^2r^2}

Does that help? ​

 

[nocks]

Thanks for the link but there's no mention of the impact parameter

 

[tiny-tim]

Thanks for the link but there's no mention of the impact parameter

(just got up :zzz: …)

Well, if you mean the "sideways distance at infinity", that'll be lim_r->∞ r sin(φ - φ_∞),

which is … ?

 

[nocks]

Oh so the impact parameter is the distance parallel to the centre of the black hole at approach from infinty?
Now to attempt plotting the trajectory. Is the equation I mentioned above enough for this? (i should mention I am not a physicist)

 

[tiny-tim]

You're not a physicist? Then whyever are you doing this?

You may find some assistance at http://Albert51.tripod.com/non.html" , including some plots of trajectories for different impact parameters.

 

[nocks]

You're not a physicist? Then whyever are you doing this?

Just an interest of mine. It's taking a while to get my head around the maths but I guess I'm slowly getting there.