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Two planets each of mass m are in counterclockwise circular orbits around a star of mass M (see the figure below). The inner planet P₁ is in an orbit of radius R, while the outer planet P₂ is in an orbit of radius 2R. Express your answer to all parts of this problem in terms of R, M, m and G, Newton's gravitational constant. Assume m << M. P₁ P₂ a) (2 points) What is the ratio T₂/T₁ of the orbital period of planet P₂ to that of P₁? What is the value of T₁? b) (2 points) What is the ratio of E₂/E₁ of the total energies of the orbits? What is the value of E₁? c) (2 points) What is the ratio L₂/L₁ of the orbital angular momenta of the orbits? What is the value of L₁? Now suppose that we send a spacecraft of mass m, from P₁ to P₂ along the elliptical transfer orbit shown below. The orbit has a "periastron" (the distance of closest approach to the star about which the planets are orbiting) of R and an "apastron" (the greatest distance to the star) of 2R. d) (4 points) What are the semimajor axis a, the energy E, the angular momentum L and the eccentricity e of the transfer orbit? Transfer Orbit Two planets each of mass m are in counterclockwise circular orbits around a star of mass M (see the figure below). The inner planet P₁ is in an orbit of radius R, while the outer planet P₂ is in an orbit of radius 2R. Express your answer to all parts of this problem in terms of R, M, m and G, Newton's gravitational constant. Assume m << M. P₁ P₂ a) (2 points) What is the ratio T₂/T₁ of the orbital period of planet P₂ to that of P₁? What is the value of T₁? b) (2 points) What is the ratio of E₂/E₁ of the total energies of the orbits? What is the value of E₁? c) (2 points) What is the ratio L₂/L₁ of the orbital angular momenta of the orbits? What is the value of L₁? Now suppose that we send a spacecraft of mass m, from P₁ to P₂ along the elliptical transfer orbit shown below. The orbit has a "periastron" (the distance of closest approach to the star about which the planets are orbiting) of R and an "apastron" (the greatest distance to the star) of 2R. d) (4 points) What are the semimajor axis a, the energy E, the angular momentum L and the eccentricity e of the transfer orbit? Transfer Orbit
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University Physics with Modern Physics
ISBN: 9780321696861
13th edition
Authors: Hugh D. Young, Roger A. Freedman, A. Lewis Ford
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