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

This preview shows pages 1–7. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 non-elliptical 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....
View Full Document

## This document was uploaded on 04/27/2011.

### Page1 / 20

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

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online