MIT OpenCourseWare http://ocw.mit.edu 16.346 Astrodynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Exercises 02 1. The angular momentum and eccentricity vectors of an orbit are h =2 µ 3 i z e = − 1 3 (2 i x + i y) Find the position and velocity vectors r and v when the direction of the position vector is i r = i x . (Use µ =4π2 ) πAnswer: r =4 i x and v = √3 (i x + i y) 2. Prob. 4–8 To derive the polar equation of an ellipse with the origin of coordinates at the center of the ellipse (See Lecture 2, Page 3), we may consider the triangle CPF where r is the radius from the center C to a point P on the ellipse and F is the focus of the ellipse. The sides of the triangle are PF = a − ex = a − er cos θ CF = ae CP = r We can use the Law of Cosines for the triangle (a − ex)2 =(a − er cos θ)2 = r 2 + a 2 e 2 − 2aer cos θ which gives r2(1 − e2 cos2 θ)= a2(1 − e2)= b2 or b r = √1 − e2 cos2 θ 1 mile = 1.609347221 km 1 au = 149, 597, 870.00 km 1au = 92, 955, 620.79 miles 1 au/day = 1078.822025 miles/sec 1 au/day = 5, 696, 180.29
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