The most reliable way to measure stellar masses is to analyze binary star orbits. The most massive pair in our galaxy yet revealed by this method was reported in a 2008 study of an eclipsing double-lined spectroscopic binary in the young (1-2 million year-old) star cluster NGC 3603. They report the a period of 3.77 days, an inclination of 71 degrees, and present the following velocity curve obtained from Doppler measurements of spectral lines from the two stars: XXXXX 05 +(374) (i) Judging from this graph, what are the shapes of the orbits of the two stars around the system's center of mass? Justify your answer. (ii) Is the binary system as a whole (i.e., its center of mass) moving toward or away from the Sun? At what speed? Justify your answer. Use a ruler to draw a straight line on the graph to help you judge this center-of-mass velocity. (iii) Relative to the system's center of mass motion, what is the observed maximum velocity of the more massive of the two stars? Use a ruler to draw lines in the graph to help you make this measurement. Repeat for the less massive of the two stars. (iv) What is the ratio of masses of the two stars? (v) Given the inclination of the system, and your results in part (iii), compute the actual orbital speeds of each of the two stars. Is the true orbital speed faster or slower than the speed deduced from the Doppler shifts? Explain. (vi) Given what you know about the orbital period of the system and the orbital velocities of the stars, compute the distance of each star from the center of mass of the system. (vii) Compute the sum of the masses of the two stars. Then compute the individual masses. (ix) Compute estimated radii for the two stars, assuming R is proportional to M.6 (x) Make a detailed sketch of the system, to scale, on a separate sheet. Indicate the direction to the Earth on your sketch (remember the inclination!). Does it make sense that system has eclipses?