Friday, 1 May 2015

homework and exercises - Deriving relativistic Doppler shift in terms of wavelength?





Consider a star moving with velocity v at an angle θ with respect to its line of sight to Earth. Show that the relativistic Doppler shift is


λobs=1vccos(θ)1v2c2λem


in which c is the speed of light, λobs is the observed wavelength, and λem is the emitted wavelength.



Can someone show me to derive this equation? So far, I have been using a reference frame S for a certain angle θ in which the y=ctsin(θ) and x=ctcos(θ). I used the Lorentz transformation to find that


x=x+vt1v2c2=ctcos(θ)1v2c2=ct(cos(θ)+vc)1v2c2


I am not sure what do from here. Also, what happens for velocities that are much smaller than c? How can I use this equation to write how at vc the equation reduces to the usual expression for a Doppler shift such that


λobs=(1+vrc)λem



in which vr is the radial velocity?




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