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Aberration of a particle

  1. Dec 26, 2016 #1
    1. The problem statement, all variables and given/known data

    A particle has speed u‘ in the S’ frame, its track making an angle ## \theta’ ## with the x’ axis. The particle is viewed by an observer in frame S, the two frames having a relative speed parameter ## \beta ##.

    (a) Show that the angle ## \theta ## made by the track of the particle with the x axis is given by

    $$ \tan \theta = \frac {u’ \sin\theta’} { \gamma \{u’ \cos\theta’ + \beta c\} } $$

    (b) Show that this reduces to the standard aberration formula $$ \tan \theta = \frac {\sin \theta’ \sqrt{1-\beta^2} } {\cos \theta’ + \beta} $$ if the “particle” is a light pulse, so that ## u’ = c##.


    2. Relevant equations
    Given in problem statement

    3. The attempt at a solution

    Solving (b) is easy. But (a) is a mystery to me. I think I see how the authors get there, but it seems too simple. I can get there if I argue as follows:

    I believe the authors mean that the relative movement of the two frames is along the common x-x’ axis, though they aren’t explicit. In that case the y-component of velocity is the same as the y’-component of velocity; i.e. ## u_y = u’_y = u’ \sin \theta’.## Also the x-component of velocity is some combination of the relative velocity of the frames (## \beta c##) and the x’-component of velocity (## u’ \cos \theta’##); here is seems clear that the authors just added ## u’ \cos \theta’## and ## \beta c## and expanded by multiplying by ##\gamma##. But I am puzzled as to why that is the right thing to do. After all, why shouldn’t the two velocities be added using the relativistic velocity-addition formula: $$ u_x = \frac {u’_x \cos \theta’ + \beta c} {1+ \frac {u’ \cos \theta’ \beta c} {c^2} }? $$ Or am I misunderstanding completely what they have done to get (a)?

    I also considered trying to derive (a) in the same way the authors derived (b). But they did that assuming a light wave and making use of the fact that that ## \lambda f = \lambda’ f’ = c.## But in (a), we have particle motion, not wave propagation (we haven’t gotten to quantum mechanics yet), so it seems to me that I cannot treat this as a propagating wave.
     
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  3. Dec 26, 2016 #2

    Orodruin

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    This is incorrect. The y-component is frame dependent.

    Do you know the 4-vector formalism? Otherwise you will have to use the Lorentz transformations.
     
  4. Dec 26, 2016 #3
    Oh........as soon as I saw your statement that the y-component is frame dependent, I realized that this is much easier than I thought. Anyway, I'm only vaguely familiar with 4-vectors, but I think I think I see how this can be done with Lorentz transformations. So, I think that part of where I was going wrong was in thinking that the ## \gamma ## in the solution was connected with the x-component of velocity (simply because I saw it in the denominator). Instead it's connected with the y-component, right? That is, by using the Lorentz transformation on ## \frac {\delta y} {\delta t}, ## you get $$ u_y = \frac {u'_y \sqrt{1-\beta^2} } {1+\frac{\beta c u'_x} {c^2}},$$ I believe. And I believe that I am correct about the x-component in my Attempt at Solution (above). Then $$ \tan \theta = \frac {u_y} {u_x}, which yields the answer. Am I understanding now?
    P.S. And thank you for your assistance. You folks are great!
     
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