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Unformatted text preview: Lecture 5: Kepler hyperbolas (14 Sep 09) A. Review: Orbit equation 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 = μ Γ ‘ 2 [1- cos( φ )] after choosing an origin for φ and with eccentricity = q 1 + (2 E‘ 2 /μ Γ 2 ) 3. Bounded motion ( E < 0, < 1), this is the equation of an ellipse with distance r measured from one of the foci. The semi-major axis is the “long” one, b 2 = a 2 (1- 2 ). 4. For E > 0, > 1, there is unbounded motion (or: r > r min – semi- infinite range) [Sketch V eff ]. This limits cos φ < 1 / for r > 0. B. Scattering problem 1. Generic “probe” experiment: particle “in” and observe where it comes out. In spherical polar coordinates specify an element of solid angle where it goes. 2. Incident flux F = number of particles crossing unit ( ⊥ ) area in time and count number per unit time ˙ N in the final solid angle d Ω. The differential cross section dσ is the proportionality constant relating the two quantities, dimension = area.two quantities, dimension = area....
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