lec07 - Lecture 7 Planetary Motion Lecture 7 Purdue...

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Lecture 7 Planetary Motion Lecture 7 Purdue University, Astronomy 364 1
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General Solutions Lecture 7 Purdue University, Astronomy 364 2 F = G Mm r 2 ˆ r = m d v dt r = L GMm 2 (1 + e cos θ ) 0 < e < 1 e > 1 e = 1 e = 0 conic section
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Kepler’s 1 st Law Lecture 7 Purdue University, Astronomy 364 3 Close orbits: ellipses or circles a 2 = b 2 + c 2 = b 2 + ( ae ) 2 e = 1 b 2 / a 2 ( ) 1/2 r = a (1 e 2 ) 1 + e cos θ L 2 m 2 = GMa (1 e 2 ) r 2 v t 2 = GMa (1 e 2 )
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Velocities Lecture 7 Purdue University, Astronomy 364 4 At perihelion and aphelion, the velocity of a planet is entirely tangential. At perihelion, r = a-c = a (1-e) a 2 (1 e ) 2 v pe 2 = GMa (1 e 2 ) v pe = GM a 1 + e 1 e 1/2 At aphelion, r = a+c = a (1+e) a 2 (1 + e ) 2 v ap 2 = GMa (1 e 2 ) v ap = GM a 1 e 1 + e 1/2
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Kepler’s 3 rd Law Lecture 7 Purdue University, Astronomy 364 5 The rate at which the area swept out by the planet-Sun line is a constant: dA dt = L 2 m = π ab P π 2 a 4 (1 e 2 ) P 2 = L 2 4 m 2 L 2 m 2 = GMa (1 e 2 )
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