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. Lecture 10 Transformation of the Boundary-Value Problem #6.7 According to Lambert’s Theorem √µ(t2 − t1)= F (a, r1 + r2,c) the orbit of the boundary-value problem can be transformed to a rectilinear orbit (e =1), keeping the sum of the radii r1 +r2 , the length of the chord c and the semimajor axis a all fixed in value, and the time-of-flight will be unchanged. The transformation is illustrated in the following figure: The flight time for the rectilinear orbit is µ (t2 − t1)=(α − sin α) − (β − sin β) a3 =(E2 − sin E2) − (E1 − sin E1) in terms of the Lagrange parameters and the eccentric anomalies. 16.346 Astrodynamics Lecture 10 Figure by MIT OpenCourseWare.QQQF*F*F*PPPFFFr1r2cc(a)(b)(c)(r1+r2+c)2(r1+r2-c)2PF + QF = r1+r2 c2aTransformation of the Four Basic Ellipses We adopt the convention for assigning quadrants to the Lagrange parameters α and β 0 ≤ α ≤ 2π 0 ≤ β ≤ π for θ ≤ π 0 ≤ α ≤ 2π −π ≤ β ≤ 0 for θ ≥ π which will include all elliptic orbits. 16.346 Astrodynamics Lecture 10 Fig. 6.20 from An Introduction to the Mathematics and Methods of Astrodynamics. Courtesy of AIAA. Used with
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