lecture10 - Summary of true, eccentric, and mean anomalies...

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Unformatted text preview: Summary of true, eccentric, and mean anomalies tan 2 = s 1 + e 1- e tan E 2 M = E- e sin E M = n r a 3 Note quadrant of , E , and M ; at 0 and , they are all the same, = E = M = 0 and = E = M = . Quadrant is never ambiguous: If any of , E , M is between 0, , they all are. If any of , E , M is between , 2 , they all are. See Curtis Fig. 3.2 for a plot of M vs. . For circular orbits, all three angles are always the same, = E = M anywhere on the orbit. L. Healy ENAE404 Spring 2007 Lecture 10 (Feb. 27) 1 Differential formulation of eccentric anomaly Ref.: Vallado Section 2.2.1 For doing non-elliptical orbits, and to under- stand angular relations, an alternative deriva- tion of Keplers equation is useful. We can differentiate the cos = cos E- e 1- e cos E expression with respect to E ,- sin d dE =- sin E (1- e cos E )- (cos E- e ) e sin E (1- e cos E ) 2 . L. Healy ENAE404 Spring 2007 Lecture 10 (Feb. 27) 2 Eccentric anomaly contd. Rearrange to get d = (1- e 2 )sin E (1- e cos E ) 2 sin dE. From before, r sin = b sin E , or sin = a q 1- e 2 r sin E, and using the square of the distance eccen- tric anomaly relation gives d = (1- e 2 )sin E r 2 a 2 a 1- e 2 r sin E dE = a q 1- e 2 r dE. L. Healy ENAE404 Spring 2007 Lecture 10 (Feb. 27) 3 Time evolution of eccentric anomaly Angular momentum is a differential relation of true anomaly and time, (recall from Lecture 1) h = r 2 d dt . We can split into two differentials and inte- grate, Z t t hdt = Z r 2 d , where t is the time of perigee = 0, and t is the time of interest for = . L. Healy ENAE404 Spring 2007 Lecture 10 (Feb. 27) 4 Relation of time to eccentric anomaly Substitute our d relation and integrate, noting h constant, h ( t- t ) = Z E r 2 a q 1- e 2 r dE and use the distance eccentric anomaly re- lation h ( t- t ) = a 2 q 1- e 2 Z E (1- e cos E ) dE or using h = p = q a (1- e 2 ), t- t = v u u t a 3 | {z } 1 /n ( E- e sin E ) . This is Keplers equation, but now without an explicit mean anomaly. L. Healy ENAE404 Spring 2007 Lecture 10 (Feb. 27) 5 Time of flight worked example Using the linearity of M ( t ), we can find the time of flight for a satellite. Curtis Example 3.1, p. 115....
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lecture10 - Summary of true, eccentric, and mean anomalies...

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