Angular Velocity of a source of light

[Felipe Doria]

This is not schoolwork.
Imagine that light did not have a constant speed, but behaved in the manner expected from experience. Namely, if the source of the light is rushing toward you, the light will approach you faster; if the source is rushing away from you, the light will approach you slower. This is incorrect, of course, but it's worth investigating the consequences of a non-constant speed of light because the failure to observe those consequences is evidence that the speed of light is constant. With that backdrop, consider a binary star system situated a very large distance L from Earth. Let the angular velocity of the smaller star be ω, as it orbits the larger star in a circle of radius r.
What value of ω=v/r will result in the light emitted when the smaller star is traveling directly away from Earth reaching us at the same moment as the light emitted later, when the smaller star's orbit has it moving directly toward earth? (choose one)
a) ω=(c/r)*sqrt((πr)/(2L+2πr))
b) ω=(c/r)*((πr)/(2L+πr))
c) ω=(c/r)*sqrt((πr)/(2L+πr))

I think that the time it takes for the light emitted from the smaller star when it is traveling directly away from Earth has to be equal to the time it takes for the light emitted later plus the time it takes to complete half an orbit. So:
t1 = t2 + t3
L/(c-v) = L/(c+v) + pi*r/v

How can I get the answer from this? Thank you for your help.

 

[Simon Bridge]

This is not schoolwork.
Imagine that light did not have a constant speed, but behaved in the manner expected from experience.

"from experience" the speed of light in invariant - but I get what you mean:

Namely, if the source of the light is rushing toward you, the light will approach you faster; if the source is rushing away from you, the light will approach you slower.

... you mean, what if light obeyed Galilean relativity.

This is incorrect, of course, but it's worth investigating the consequences of a non-constant speed of light because the failure to observe those consequences is evidence that the speed of light is constant.

See "Michealson-Morely experiment" for an example of this sort of calculation.

With that backdrop, consider a binary star system situated a very large distance L from Earth. Let the angular velocity of the smaller star be ω, as it orbits the larger star in a circle of radius r.
What value of ω=v/r will result in the light emitted when the smaller star is traveling directly away from Earth reaching us at the same moment as the light emitted later, when the smaller star's orbit has it moving directly toward earth? (choose one)
a) ω=(c/r)*sqrt((πr)/(2L+2πr))
b) ω=(c/r)*((πr)/(2L+πr))
c) ω=(c/r)*sqrt((πr)/(2L+πr))

I think that the time it takes for the light emitted from the smaller star when it is traveling directly away from Earth has to be equal to the time it takes for the light emitted later plus the time it takes to complete half an orbit.

The time is also the distance to the Earth divided by the velocity of the light.