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Calculating impact parameter

  1. Oct 28, 2009 #1
    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

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

    3. The attempt at a solution
     
  2. jcsd
  3. Oct 28, 2009 #2

    tiny-tim

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    Welcome to PF!

    Hi nocks! Welcome to PF! :wink:

    From the PF Library on photon sphere …​

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

    Does that help? :smile:
     
  4. Oct 28, 2009 #3
    Thanks for the link but there's no mention of the impact parameter
     
  5. Oct 29, 2009 #4

    tiny-tim

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    (just got up :zzz: …)

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

    which is … ? :smile:
     
  6. Oct 29, 2009 #5
    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)
     
  7. Oct 29, 2009 #6

    tiny-tim

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