# lecture17 - Solve for intersection of rotated ellipses An...

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Solve for intersection of rotated ellipses An equation of the form a cos θ + b sin θ = c can be solved by θ = φ ± arccos ± c a cos φ ² where tan φ = b/a . (Variables a , b , c are just coeﬃcients, not semimajor axis etc.) The two signs correspond to the two intersection points of the ellipses. In our case, a = e 1 p 2 - e 2 p 1 cos η b = - e 2 p 1 sin η c = p 1 - p 2 with all quantities on the right side known. L. Healy – ENAE404 – Spring 2007 – Lecture 17 (Mar. 29) 1

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Apse line rotation example Curtis Example 6.7 p. 280 An earth satellite has perigee distance 8000km and apogee distance 16000km. Calculate the delta-v and true anomaly needed to make the orbit 7000km and 21000km whose apse line is rotated by 25 counterclockwise. Compute e 1 = 0 . 3333 e 2 = 0 . 5 p 1 = 10666km p 2 = 10500km a = - 1334km b = - 2254km c = 166 . 7km . L. Healy – ENAE404 – Spring 2007 – Lecture 17 (Mar. 29) 2
Apse line rotation angle Thus φ = arctan b a = 59 . 39 . Note that we do not use the two-argument arctangent here, we want the angle to be in the ﬁrst or fourth quadrant. Intersection true anomaly θ 1 = φ ± arccos ± c a cos φ ² = 59 . 39 ± 93 . 65 so the two intersection points are at true anoma- lies on the initial orbit θ 1 = 153 . 04 and θ 1 = 325 . 74 . L. Healy – ENAE404 – Spring 2007 – Lecture 17 (Mar. 29) 3

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Apse line rotation example cont’d. In order to ﬁgure out the delta-v, we need to do a vector analysis and apply the law of cosines. Assume the delta-v occurs at the ﬁrst point, and compute its magnitude. In the perifocal frame of the initial orbit, ˙ r = s μ p 1 h - sin θ 1 ˆ P 1 + (cos θ 1 + e 1 ) ˆ Q 1 i = - 2 . 771 ˆ P 1 - 3 . 411 ˆ Q 1 km / s . In the perifocal frame of the second orbit, ˙ r = s μ p 2 h - sin θ 2 ˆ P 2 + (cos θ 2 + e 2 ) ˆ Q 2 i = - 4 . 853 ˆ P 2 - 0 . 7160 ˆ Q 2 km / s with θ 2 = θ 1 - η = 128 . 04 . L. Healy – ENAE404 – Spring 2007 – Lecture 17 (Mar. 29) 4
We can compute the delta-v if we’re in the same reference frame; do a 2D rotation of the second orbit perifocal velocity by the negative of the apse rotation back to the ﬁrst perifocal frame, R ( - 25 ) " - 4 . 853 - 0 . 7160 # = " - 4 . 096 - 2 . 700 # The delta v is the diﬀerence from the ﬁrst orbit perifocal velocity Δ v = - 4 . 096 ˆ P 1 + - 2 . 700 ˆ Q 1 - ( - 2 . 771 ˆ P 1 + - 3 . 411 ˆ Q 1 ) = - 1 . 324 ˆ P 1 + 0 . 7111 ˆ Q 1 km / s . and the magnitude is

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lecture17 - Solve for intersection of rotated ellipses An...

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