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RafaPhysics

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- Homework Statement
- I've been trying to solve this problem but I'm pretty sure there's something more that I could do, but I can't see it.

- Relevant Equations
- $$v_r=u sin\frac{2\pi}{T}\left(t-\frac{R}{c}+\frac{Rv_r}{c^2}\right)$$

Hey, I have this problem from the Special Relativity by AP. French . Exercise 3.3, Chapter 3.

The figure shows a double-star system with two stars, A and B, in circular orbits of the same period T about their center of mass. The earth is in the plane definied by these orbits at a distance R of many light-years. Let the speed of A in its orbit be u; then at any instant it has a velocity ##v_r(=ucos\theta)## along the line from the double-star system to the earth. When light emitted from A reaches the earth, its observed Doppler shift (change of wavelength of characteristic spectral lines) reveals the value of vr at the instant of emission.

If the speed of light from A to the earth were modified by the motion of A, so as to be equal to ##c+v_r##, show that the value of ##v_r##, as inferred from spectroscopic observations on earth, would appear to be varying with time in accordance with the following equation if ##u\ll c##: It's the equation above.

I've tried with the doppler efect, but I can't still see when It turns to ##sin##, there's something more but simply I can't see it. Please I need help.

The figure shows a double-star system with two stars, A and B, in circular orbits of the same period T about their center of mass. The earth is in the plane definied by these orbits at a distance R of many light-years. Let the speed of A in its orbit be u; then at any instant it has a velocity ##v_r(=ucos\theta)## along the line from the double-star system to the earth. When light emitted from A reaches the earth, its observed Doppler shift (change of wavelength of characteristic spectral lines) reveals the value of vr at the instant of emission.

If the speed of light from A to the earth were modified by the motion of A, so as to be equal to ##c+v_r##, show that the value of ##v_r##, as inferred from spectroscopic observations on earth, would appear to be varying with time in accordance with the following equation if ##u\ll c##: It's the equation above.

I've tried with the doppler efect, but I can't still see when It turns to ##sin##, there's something more but simply I can't see it. Please I need help.