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Unformatted text preview: Solutions for Homework # 7 1. In 1959, Kopal determined that the Roche lobe radius R R with nearly the same volume as the Roche lobe surrounding the star m which is orbiting the star M is R R = 0 . 46 D m M + m 1 / 3 . D is the distance between the stars and they are assumed to be in a circular orbit. The Roche lobe is the volume defined by the potential surface that defines the Roche limit, i.e., the potential surface that passes through the inner Lagrangian point. The Ja- cobi radius r J , which is the distance from the center of m to the inner Lagrangian point when m << M , is r J = D m 3 M + m 1 / 3 . Which, r J or R R , is larger, and why? When m << M , r R /R J = 0 . 46 3 1 / 3 = 0 . 66 so r J is big- ger. This is because R R is the radius that would result if the volume of the Roche lobe was spherical. An ellipsoid with the x, y semi-major axes of r J has a z semi-major axis r z , and its volume satisfies r 2 J r z = R 3 R . Since r z < r J , we have r J > R R ....
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- Spring '11