14-superfluous

# 14-superfluous - integrate both sides v x h = GM u C...

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Kepler’s Laws : 1. A planet revolves around the sun in an elliptical orbit with the sun at one focus. 2. A line joining the sun to a planet sweeps out equal areas in equal times. 3. The square of the period of revolution of a planet is proportional to the cube of the length of the major axis of its orbit. Proof of 1: goal is to show the orbit is of the form 14.4.1 Aside: Prove orbit is planar. Using Newton’s Laws and equating them, we have F = m a = GMm r 3 r . Thus a and r are constant they are parallel. r x a = 0. d d t r x v = r x v r x v ’= v x v r x a = 0 . Thus r x v = h , a constant. Assume h 0. Then r, v are both to h . Since r t h for all t , r t lies in one plane perpendicular to h . End aside. Since r h we also have u h . h = r x v = r x r ’= r u x r u ’ = r u x r u r u = rr u x u r 2 u x u . r is not a scalar but a scalar function . Consider a x h. a = GM r 3 r = GM r 2 u . a x h = GM r 2 u x r 2 u x u = GM u u u u u u . Note u = 1 i.e. has constant length. Then u u = 0 (Thm 14.2.3, since u and u’ are orthogonal). So GM u u u u u u = GM u ’. Consider v x h ’ = v x h v x h = a x h = GM u . Thus
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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 z-axis). 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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