Question B2 [This question will required roughly two single-sided A4 pages to answer.] A binary star system consists of two stars that revolve about their centre of mass in circular orbits. Suppose that the system is observed edge on. Because of Doppler shift the spectral lines from the two stars (known as a spectroscopic binary system) are observed to shift periodically about a mean to shorter and longer wavelengths as each star moves towards or away from the observer. a) Explain how the orbital period Porb and components of velocity along the line of sight 10,1 and v0.2, for each star, can be found from the observed spectral lines. b) Explain how the velocities of the two stars, v₁ and v2, and the angular velocity of the stars w can be determined. c) Denoting a₁ and a2 as the distances of the two stars from the centre of mass, respectively, find a relationship between the ratio of the masses of the two stars (M₁ and M2), ratio of distances from centre of mass, and ratio of velocities. d) Using the equations of motion for the two stars (for circular motion) show that the total mass of the binary system can be determined from M = M₁ + M₂ = Porb 2π G (V1+V2)³. e) Give expressions for the individual masses of each star as functions of the total mass and velocities.

Question B2 [This question will required roughly two single-sided A4 pages to answer.] A binary star system consists of two stars that revolve about their centre of mass in circular orbits. Suppose that the system is observed edge on. Because of Doppler shift the spectral lines from the two stars (known as a spectroscopic binary system) are observed to shift periodically about a mean to shorter and longer wavelengths as each star moves towards or away from the observer. a) Explain how the orbital period Porb and components of velocity along the line of sight 10,1 and v0.2, for each star, can be found from the observed spectral lines. b) Explain how the velocities of the two stars, v₁ and v2, and the angular velocity of the stars w can be determined. c) Denoting a₁ and a2 as the distances of the two stars from the centre of mass, respectively, find a relationship between the ratio of the masses of the two stars (M₁ and M2), ratio of distances from centre of mass, and ratio of velocities. d) Using the equations of motion for the two stars (for circular motion) show that the total mass of the binary system can be determined from M = M₁ + M₂ = Porb 2π G (V1+V2)³. e) Give expressions for the individual masses of each star as functions of the total mass and velocities.