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Unformatted text preview: Astro 405/505, fall semester 2005 Homework, 3rd set, solutions Problem 5: Gravitational collapse The important principle is energy conservation between the potential energy and the kinetic energy associated with the radial infall. It suffices to follow the fall of one atom at the surface of the collapsing gas cloud. The gravitationally effective mass is then always the total mass of the cloud, and the acceleration is purely radial, for the angular momentum is zero. Then we have for the gravitational acceleration R =- G M R 2 R R =- G M R 2 R d dt R 2 2- G M R ! = 0 R 2 2 = G M R + C where C is a constant that we can fix using an initial condition, for example that the system was at rest at some earlier time when it had the radius R . Then the remaining differential equation can be solved using standard methods. R 2 = 2 G M 1 R- 1 R dR dt = s 2 G M R s R R- 1 Given that the acceleration is always negative, the minus-sign is the only relevant case.Given that the acceleration is always negative, the minus-sign is the only relevant case....
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