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Unformatted text preview: Summary of true, eccentric, and mean anomalies tan θ 2 = s 1 + e 1 e tan E 2 M = E e sin E ˙ M = n ≡ r μ a 3 Note quadrant of θ , E , and M ; at 0 and π , they are all the same, θ = E = M = 0 and θ = E = M = π . Quadrant is never ambiguous: • If any of θ , E , M is between 0, π , they all are. • If any of θ , E , M is between π , 2 π , they all are. See Curtis Fig. 3.2 for a plot of M vs. θ . For circular orbits, all three angles are always the same, θ = 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 nonelliptical orbits, and to under stand angular relations, an alternative deriva tion of Kepler’s equation is useful. We can differentiate the cos θ = cos E e 1 e cos E expression with respect to E , sin θ dθ 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 cont’d. Rearrange to get dθ = (1 e 2 )sin E (1 e cos E ) 2 sin θ dE. From before, r sin θ = b sin E , or sin θ = a q 1 e 2 r sin E, and using the square of the distance — eccen tric anomaly relation gives dθ = (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 dθ dt . We can split into two differentials and inte grate, Z t t hdt = Z θ r 2 dθ , where t is the time of perigee θ = 0, and t is the time of interest for θ = θ . L. Healy – ENAE404 – Spring 2007 – Lecture 10 (Feb. 27) 4 Relation of time to eccentric anomaly Substitute our dθ 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 = √ μp = q μa (1 e 2 ), t t = v u u t a 3 μ  {z } 1 /n ( E e sin E ) . This is Kepler’s 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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This document was uploaded on 04/27/2011.
 Spring '11

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