# Calculating impact parameter

1. Oct 28, 2009

### nocks

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

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

3. The attempt at a solution

2. Oct 28, 2009

### 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?

3. Oct 28, 2009

### nocks

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

4. Oct 29, 2009

### tiny-tim

(just got up :zzz: …)

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

which is … ?

5. Oct 29, 2009

### 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 im not a physicist)

6. Oct 29, 2009

### 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" [Broken], including some plots of trajectories for different impact parameters.

Last edited by a moderator: May 4, 2017
7. Oct 30, 2009

### nocks

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.