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Unformatted text preview: Astronomy 210 Spring 2011 Homework Set #7: Solutions 1. A Transiting Exoplanet . (a) [5 points] Since a 3 = GMP 2 / (4 2 ), we have a = parenleftbigg GM 4 2 parenrightbigg 1 / 3 P 2 / 3 (1) It is fine to just use this formula, being careful with units. A shortcut is to divide this result by the value for the earth a = parenleftbigg GM 4 2 parenrightbigg 1 / 3 P 2 / 3 (2) where of course P = 1 year and a = 1 AU. Dividing, we get a 1AU = parenleftbigg M M parenrightbigg 1 / 3 parenleftbigg P 1yr parenrightbigg 2 / 3 (3) and thus for M = 1 . 35 M and P = 4 . 89 day = 0 . 0134 yr, we get a = 0 . 062 AU, in good agreement with the professional value. (b) [5 points] We want the stars speed v about the center of mass. Since both the star and the planet move about the center of mass with period P and angular speed = 2 /P , the stars speed is v = r (4) where r s tar is the stars distance from the center of mass, which we need to find. We have r + r p = a (5) and m r = m p r p (6) Since we want r , we eliminate r p : m r = m p ( a r ) (7) ( m + m p ) r = m p a (8) r = m p m + m p a (9) and thus we have v = m p m + m p 2 a P = m p m + m p v (10) where v is the speed of the planet relative to the star ....
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