Question

Problem 4. Consider a simple solar system that contains one star (mass M) and one planet (mass m) in a circular orbit with radius r. The planet orbits at speed v. (v can be computed in terms of G, M, m, and r using Kepler’s 3rd Law and/or Newton’s Law of Gravitation.) Due to conservation of momentum, as the planet changes its velocity vector, the star’s velocity vector will also change. Astronomers can detect a star’s changing velocity from its spectral lines and infer the presence of a planet. (This is called the ‘radial velocity’ method for exoplanet detection.) (a) What is the difference in the planet’s momentum, ∆~p, from one point on its orbit to the opposite point? Assume that the planet moves in the ±x-direction at these two points. You can leave your answer in terms of v, the orbital speed of the planet. (b) Briefly explain why the momentum of the planet is not constant, but the momentum of the star plus the momentum of the planet IS constant. (c) Relative to v, what is the maximum speed of the star during the orbit of the planet? Assume we are in the ‘center of momentum’ frame of reference, where the total momentum of the system is zero. (d) Plug in numbers to find the speed of our Sun induced by the orbit of Earth. (This is a bit artificial, since Jupiter has a much bigger effect, but just to get a sense of the size of the numbers. . . ) The Sun has MSun = 2 × 1030 kg, and Earth has mEarth = 6 × 1024 kg and orbits at r = 1 AU = 1.5 × 1011 m. (e) Plug in numbers to find the speed of the star Gliese 581 induced by the orbit of its planet, Gliese 581 e, which is the least massive exoplanet detected so far using the radial velocity method. Gliese 581 has M = 0.31MSun, and Gliese 581 e has m = 2.5mEarth and orbits at r = 0.028 AU. (f) We detected Gliese 581 e by measuring its star’s change in velocity; could (hypothetical) alien astronomers on Gliese 581 e plausibly detect Earth using the same method and same equipment?

Answer 1