Many of the stars in the sky are actually $binary\space stars$, in which two stars orbit about their common center of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they emit. Stars for which this is the case are called $spectroscopic\space binary\space stars. \textbf{Figure P37.68}$ shows the simplest case of a spectroscopic binary star: two identical stars, each with mass $m$, orbiting their center of mass in a circle of radius $R$. The plane of the stars' orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of 4.568110 $\times$ 10$^{14}$ Hz. In the light received from the stars by a telescope on the earth, hydrogen light is observed to vary in frequency between 4.567710 $\times$ 10$^{14}$ Hz and 4.568910 $\times$ 10$^{14}$ Hz. Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. ($Hint$: The speeds involved are much less than $c$, so you may use the approximate result $\Delta f/f = u/c$ given in Section 37.6.) (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in 11.0 days. Determine the orbital radius $R$ and the mass $m$ of each star. Give your answer for $m$ in kilograms and as a multiple of the mass of the sun, 1.99 $\times$ 10$^{30}$ kg. Compare the value of $R$ to the distance from the earth to the sun, 1.50 $\times$ 10$^{11}$ m. (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.)