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p09fl4 - Lecture 4 Kepler orbits(11 Sep 09 A Review 1...

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Lecture 4: Kepler orbits (11 Sep 09) A. Review 1. Relative motion with reduced mass μ = m 1 m 2 / ( m 1 + m 2 ), potential Φ = Γ /r , relative angular momentum . 2. Integration of equation of motion gives 1 r = C [1 - cos( φ )] after choosing an origin for φ , C μ Γ /‘ 2 , eccentricity = q 1 + (2 E‘ 2 Γ 2 ) 3. Virial theorem relates average potential and kinetic energies: 2 h K i = h r · ∇ Φ i ⇒ 2 h K i = -h Φ i E = h Φ 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 μr 2 ˙ φ = : τ = Z 2 π 0 dφ/ ˙ φ = μ Z 2 π 0 r 2 h 1 r i = 1 τ Z τ 0 dt r = 1 τ Z 2 π 0 r ˙ φ = μ ‘τ Z 2 π 0 rdφ B. Analytical geometry of an ellipse 1. Ellipse has foci at f = ± f ˆ 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
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2. Vector relation d 2 = d 1 + 2 f ˆ x . Then calculate d 2 1 from this equation and from the B.1 equation using scalar product d 2 · ˆ x = r cos φ , with plane polar coordinates r, φ for d 2 . The result of the algebra is: r (1 - cos φ ) = 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.
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