PS05_binary_stars_2bodytheory_solutions

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Rutgers University *

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341

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Physics

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Dec 6, 2023

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Physics 341: Problem Set #5 due October 18 by 2:00 PM in PDF format on Canvas [40 pts] You are encouraged to work in groups on these problems, but each student must write up the solutions individually. You must also list your collaborators on your solutions, and cite any external sources you used (other than the course notes or textbook). Using online solutions to these or similar problems is not allowed. Show your work and explain your reasoning. The number of points for each problem is listed; partial credit will be given. Be careful with units. Questions asking about concepts require at most a few sentences to answer. Please make your answers clear and concise . 1. Which of these is a true statement about typical visual binaries and typical spectro- scopic binaries that we observe? [5 pts] A. The typical spectroscopic binaries have higher eccentricities B. The typical spectroscopic binaries have lower inclinations (closer to face-on) C. The typical visual binaries have longer periods D. The typical visual binaries have faster orbital speeds Answer: C 2. In a single-lined spectroscopic binary, where we only measure the period P and one radial velocity amplitude K 1 , what can we determine about the unseen companion? [5 pts] A. Its minimum mass B. Its maximum mass C. Its minimum radius D. Its maximum radius E. none of the above Answer: E. 3. Below is a sketch of the orbits of two stars orbiting each other about a common focal point marked by the . The locations of star 1 are marked at times A , B , C , and D (in 1
order of earliest to latest). The distance x is 0 . 1 AU, and the eccentricity is e = 0 . 4. [15 pts] (a) In the diagram above, mark the location of star 2 at times A , B , C , and D . Star 2 must be on the opposite side of the focal point from star 1 at each time. Therefore, the locations are: (b) Sketch a plot of the kinetic and potential energies of star 1 as a function of time. On a separate plot, sketch the kinetic and potential energies of star 2 as a function of time. Indicate on the time axes where times A , B , C , and D are. You do not need to place numeric values on the axes, but you do need to indicate the location of zero on the energy axis. Star 1 has the lowest kinetic energy at time B, and the highest (least negative) 2
potential energy at B. It has the lowest potential and kinetic energies at D. A must be prior to B and C between B and D, with A closer to B than C is to B. Star 2’s kinetic and potential energy curves should match the general shape of star 1, as the times when star 1 are nearest to the focal point correspond to the times when star 2 is as well, and vice versa. (c) Star 1 completes an orbit in 25 days. What are the masses of star 1 and star 2? Answer in units of M . In the equivalent 1-body problem, the period P is related to the mass sum M = m 1 + m 2 via P 2 = 4 π 2 a 3 GM . We now need both a and a relation between m 1 and m 2 . For a , we know that a = a 1 + a 2 , The distance x is the distance to aphelion for star 2, and 3 x is the distance to aphelion for star 2. These are related to the a 1 and a 2 by: x = a 2 (1 + e ) a 2 = x (1 + e ) = 0 . 0714 AU = 1 . 07 × 10 10 m 3 x = a 1 (1 + e ) a 1 = 3 x (1 + e ) = 0 . 214 AU = 3 . 21 × 10 10 m a = a 1 + a 2 = 0 . 286 AU = 4 . 27 × 10 10 m Thus, m 1 + m 2 = 4 π 2 ( a 1 + a 2 ) 3 GP 2 3
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