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Unformatted text preview: Lecture 4: Kepler orbits (11 Sep 09) A. Review 1. Relative motion with reduced mass = m 1 m 2 / ( m 1 + m 2 ), potential = /r , relative angular momentum . 2. Integration of equation of motion gives 1 r = C [1- cos( )] after choosing an origin for , C / 2 , eccentricity = q 1 + (2 E 2 / 2 ) 3. Virial theorem relates average potential and kinetic energies: 2 h K i = h r i 2 h K i =-h i E = h i / 2 h ... i is a time average, so we will need to evaluate some trigonometric integrals. For the period and time average over a period, use r 2 = : = Z 2 d/ = Z 2 r 2 d h 1 r i = 1 Z dt r = 1 Z 2 d r = Z 2 rd B. Analytical geometry of an ellipse 1. Ellipse has foci at f = f x and is traced out as the points for which sum of the distances from the foci is a constant: d 1 + d 2 = 2 a a is the semimajor axis of the ellipse. Define the focal position f = a ; is found to be the eccentricity already defined. 1 2. Vector relation d 2 = d 1 + 2 f x . Then calculate d 2 1 from this equation and from the B.1 equation using scalar product d 2 x = r cos , with plane polar coordinates r, for d 2 . The result of the algebra is: r (1- cos ) = a (1- 2 ) 3. The correspondence to the Kepler orbit is 1 /C = a (1- 2 ). The center of force is at one of the foci; r is measured from that focus.is measured from that focus....
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