Full treatment of synchrotorn radiation

[MikeDB]

Hello,

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.

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[MikeDB]

Hello,

Thanks a lot for your reaction.

I possibly found an answer to my question quite recently.

The exponential factor can be written (considering the expression of R(t') in my first post)

exp(i w ( t'+R(t')/c )) = exp(i w ( t' + r/c - n.r_o(t')/c ))

The term r/c in the exponential is not a function of retarded time. As a result, this term plays the role of a phase which does not play a significant role in physics of the process (see Jackson 1975). The important terms are those with a dependence with respect to retarded time t'.

So, the exponential factor reduces to

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

I hope this is convincing enough.

Best regards.

 

[e.bar.goum]

Hi Mike!

Thanks for coming back with your answer, I read your OP and was intrigued, but couldn't answer it myself.

 

[MikeDB]

Hi e.bar.goum,

As a complementary information following my previous post, the quantity r/c is constant as r is simply the distance between the point where the radiation is measured and the origin from where the position vectors are defined.

In addition, I also found in the paper by Blumenthal & Gould (1970, Rev. Mod. Phys., 42, 237) the same justification for neglecting the constant term in the exponential (p.258): "... a constant term has been ignored as contributing only an over-all phase factor..."

It seems the solution is there!

Best regards.