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2. Binary Stars: Suppose you discover a new pair of binary stars, which you decide to name Kim and Khloe.(a) You measure an emission line from star Kim to have a wavelength that varies as it moves around its orbit, but it has a maximum value of 600.1 nm. From laboratory measurements, you know that the real wavelength of the line is 600 nm. How fast is the star moving along its orbit? (You may assume the orbit is not inclined if you are worried about that.) Explain why the observed wavelength changes with time. (b) The orbit takes 40 days to complete. If it is a circular orbit, what is its circumference? Hint: For a circular orbit, Kepler’s Second Law then tells you that the orbital speed is constant. (c) What is the orbit’s radius? (d) What is the total combined mass of star Kim and star Khloe? Hint: What units should you be using for the proper equation? (e) From similar measurements of star Khloe, you determine that its orbit has a semimajor axis just one tenth the size of star Kim’s. Which star is more massive? Explain your reasoning. 3. Fusion in a Massive Star: (a) The star Sirius A has a luminosity of L = 25L. How many helium atoms must it form per second to maintain this luminosity? You may begin with the mass difference between four protons and one helium atom, ∆m = 4.8 × 10−29 kg. (The “A” notation is there because Sirius is actually a binary star, with the two stars named “Sirius A” and “Sirius B.” Sirius A is the brightest star in the night sky; Sirius B is a white dwarf not visible to the naked eye.) – 2 – (b) Given the number of helium atoms formed per second to maintain the star’s luminosity, what is the total mass of hydrogen that it fuses each second? (c) If Sirius A is 300 Myr old, how much hydrogen mass has it fused over its entire history? Express your answer in units of the earth’s mass, MEarth = 6.0 × 1024 kg. (d) If the star has M = 0.2M of hydrogen to fuse, how much longer will it live, in Myr? (Note this is not the total mass of the star, because it also has helium and because only about 10% of the star’s hydrogen is near enough to the star’s center to ever undergo fusion.)
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