# Doppler shift for an observer in circular motion

• I
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?

PeterDonis
Mentor
Doppler shift of what? Light coming from the star?

Gold Member
Yes.

PeterDonis
Mentor

• BiGyElLoWhAt
Gold Member
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.

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