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I am trying to understand the details of the full treatment of synchrotron radiation. I am using Rybicki & Lightman (1979), along with the more detailed treatment given by Longair (1992).

For instance, in Longair, chapter 18 (p.240 in the Second Edition), I see that the radiated energy per unit solid angle per unit angular frequency is evaluated at the retarded time, and a change of variable is operated: going from time t to retarded time t',

with t' = t - R(t')/c.

and

R(t') = r - n.r_o(t')

(n is the unit vector along the direction joining the particle to the point where the radiation is measured, and r_o(t') is the position vector of the particle at t')

In the exponential factor exp(i w t) coming from the Fourier transform, the change of variable leads to

exp(i w ( t'+R(t')/c )).

It is then said that r_o(t') << r (I agree with that, as the source is at a quite large distance), and finally the exponential factor becomes

exp(i w ( t' - n.r_o(t')/c ))

I can't understand how to obtain this last result. Probably a first order expansion could be applied somehow but I don't see where and how.

Please, could someone give me an explanation for that?

Thank you so much in advance for your help.

Best regards.

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# Full treatment of synchrotorn radiation

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