A binary star is a pair of stars which move in circular orbits around their common centre of mass. In this question, consider the stars to be point masses situated at their centres.
Explain why the stars of equal mass must always be diametrically opposite as they travel in the circular orbit.
The centres of the two stars are separated by a distance of 2R 2R\,2R equal to 6.4 × 1010 m, where R R\,R is the radius of the orbit. The stars have an orbital period T T\,T of 30.0 days30.0\text{ days}30.0 days. The mass of each star is given by the equation
m=16π2R3GT2 m = \frac{16\pi^2 R^3}{GT^2} m=GT216π2R3where G G\,G is the gravitational constant (G=6.67×10−11 m3 kg−1 s−2G = 6.67 \times 10^{-11}\text{ m}^3\text{ kg}^{-1}\text{ s}^{-2}G=6.67×10−11 m3 kg−1 s−2).
Calculate the mass m m\,m of each star in terms of the mass M⊙ M_\odot\,M⊙ of the Sun.
The stars are viewed from Earth in the plane of rotation. The stars are observed using light that has a laboratory wavelength of 656 nm656\text{ nm}656 nm. The observed light from the stars is Doppler shifted.
Calculate the maximum change in the observed wavelength Δλ \Delta \lambda\,Δλ of this light from the orbiting stars, giving your answer in nm\text{nm}nm. (Speed of light c=3.00×108 m s−1c = 3.00 \times 10^8\text{ m s}^{-1}c=3.00×108 m s−1).
For another binary star system where the stars have masses of 4m and mmm, describe the characteristics of their circular orbits around their common centre of mass.
Explain why the smaller mass star travels faster in its orbit than the larger mass star.
