p09fl3 - Lecture 3: Kepler problem (9 Sep 09) A. Review:...

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A. Review: 2-body relative motion 1. Two-body problem: relative motion with reduced mass μ = m 1 m 2 m 1 + m 2 , momentum p = μ ˙ r , angular momentum l = r × p . Conserved quantities with conservative central forces: 1 and energy E = p 2 2 μ + Φ eff = μ 2 ˙ r 2 + Φ eff with an effective potential Φ eff = Φ( r ) + 2 2 μr 2 2. Motion remains in a plane perpendicular to l and a sketch of Φ eff ( r ) shows the range of r for given E . The turning points r 1 ,r 2 of bounded motion can be set “by inspection” and the period τ evaluated by τ = 2 Z r 2 r 1 dr/ q (2 )[ E - Φ eff ] 3. The gravitational potential Φ = - Γ /r has the special feature that, even for non-circular orbits, the periods of angular and radial motion are equal. 4. The solution for r ( φ ) was obtained by direct integration of the equation for dr/dφ obtained from (2 )[ E - Φ eff ] = ˙ r 2 = [( ‘/r 2 μ ) dr/dφ ] 2 = [( ‘/μ ) d (1 /r ) /dφ ] 2 and is
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p09fl3 - Lecture 3: Kepler problem (9 Sep 09) A. Review:...

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