# Doppler shift for an observer in circular motion

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• BiGyElLoWhAt

#### BiGyElLoWhAt

Gold Member
Say we have an observer in perfectly circular motion around a source, like a star.

Is it reasonable to apply the angle change formula ##cos \theta_o = \frac{cos \theta_s - \frac{v}{c}}{1-\frac{v}{c}cos \theta_s}## and then take the component of the motion parallel to the light wave in the observers frame and apply the doppler shift formula to it in order to obtain a doppler shift?

Doppler shift of what? Light coming from the star?

Yes.

What answer do you get when you apply your method?

BiGyElLoWhAt
Well, the velocity component parallel to light comes out to be ##-v^2/c## using a coordinate system such that the light from the source always points along the y-axis (and thus the observer is moving along the x axis).
Plugging that into the doppler shift formula ##f_s/f_0 = \sqrt{\frac{1+\beta}{1-\beta}}## you get ##\frac{f_s}{f_0} = \frac{1+\frac{-v^2}{c^2}}{1-\frac{-v^2}{c^2}}## whiiichhh appears to be the transverse doppler shift. That's cool. Thanks.

*-x axis, because convention, +theta direction.

I suppose I should have just worked it out. I didn't realize that they led to the same thing.