Unformatted text preview: integrate both sides: v x h = GM u C . Without loss of generality (WLOG), let h point in the direction of k (standard basis vector for zaxis). Thus r , u , v , c h and thus lie in the xy plane. ( C is perp. because v x h equals a constant times u (which is perp. to h ) plus C , and if C wasn’t perp. to h then u x C couldn’t be perp. to h . From Fig. XD, r , is the polar coordinate of the planet. r v x h = r GM u C = GM r u r C = GM r u r C cos = GM r u r t C cos , where C = C . Solve above eqn. for r t . r v x h = GM r u r t C cos = GM r u rC cos = GMr rC cos . r v x h GMtC cos = r x v h GMtC cos = h h GMtC cos = h 2 GMtC cos = h 2 GMtC cos = h 2 GM 1 C GM cos . Let e = C GM and d = h 2 C = ed 1 e cos , thus it’s of form Thm 14.4.1. Since orbits are closed, we must have the above equation representing an ellipse....
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 Fall '07
 Bumpus
 Kepler's laws of planetary motion, GMtC cos, cos GMtC cos

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