Special Relativity and Binary Star Systems

[BeRiemann]

Homework Statement

Suppose that you decide to look at a known binary star system. The system is too far away to resolve the individual stars, so it appears to be just one point of light. By looking at the spectrum of the system, though, you should be able to use the Doppler shift to determine some parameters of the two-star system.

The first thing that one notices is that there appear to be two hydrogen spectra shifted relative to each other owing to the motion of the stars relative to each other. Furthermore, one notices that the position of these lines will shift over time as the stars orbit around each other.

Consider a binary star system that has bright lines at 656.72 and 656.86 nm. Over the course of six months the 656.72-nm line moves to longer wavelength and the 656.86-nm line moves to shorter wavelength, until finally the two have swapped (i.e., the spectrum of the star system again shows bright lines at 656.72 and 656.86 nm). Assume that the stars are of roughly equal mass and moving in a circular orbit with axis perpendicular to the line connecting the star system to the Earth as shown in the figure.

Assuming that these spectral lines correspond to the 656.46-nm hydrogen line in the rest frame, estimate the speed V of the center of mass of the binary system.

Homework Equations

Two tried approaches: Doppler shift for light - f₁=f₀√(c±v)/√(c±v)
Dilation - L₁ = L₀/γ

The Attempt at a Solution

I first said that one star is traveling with some v+a, where v is the center of mass velocity and a is the linear velocity due to rotation. The other star would then be traveling with v-a. Given these two unknowns and two equations (tried both with dopp. shift and both with dilation, neither worked out), I could isolate v algebraically. This was given right after our first day of relativity, so I'm still a little lost.

 

[_coyote_]

The other approach I tried was to use the equation ω=(2π/T), where T is the period of the orbit, and the angular velocity of one star is ω+Δω and the other is ω-Δω. Again, two unknowns and two equations, but I couldn't work out the algebra to get v.I'm sure there's a simple answer I'm missing, so any help would be much appreciated!