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# lecture15 - Bi-elliptic details Use the Hohmann transfer...

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Bi-elliptic details Use the Hohmann transfer computation, but apply it twice. Call r B the geocentric distance of the intermediate “orbit”. The semimajor axes of the two transfer ellipses: a t 1 = r A + r B 2 , a t 2 = r B + r C 2 . On the two transfer ellipses, we have two Hohmann delta-vs each: transfer ellipse 1 Δ v A = μ r A 2 r B r A + r B - 1 Δ v B 1 = μ r B 1 - 2 r A r A + r B transfer ellipse 2 Δ v B 2 = μ r B 2 r C r B + r C - 1 Δ v C = μ r C 1 - 2 r B r B + r C In reality, the middle two burns are one burn. L. Healy – ENAE404 – Spring 2007 – Lecture 15 (Mar. 15) 1

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Three changes of speed So there are three changes of speed: Δ v A = μ r A 2 r B r A + r B - 1 Δ v B = μ r B 2 r C r B + r C - 2 r A r A + r B Δ v C = μ r C 1 - 2 r B r B + r C , with Δ v = Δ v A + Δ v B + Δ v C , and flight time is calculated as the sum of the two individual flight times: Δ t = π 8 μ ( r A + r B ) 3 / 2 + ( r B + r C ) 3 / 2 . Note that there is a parameter to be chosen: r B . The larger this is, the smaller Δ v B is, which reduce the overall total delta-v. How- ever, the total transfer time becomes larger. L. Healy – ENAE404 – Spring 2007 – Lecture 15 (Mar. 15) 2
Bi-elliptic worked example Use the same initial and final conditions as for the Hohmann example, with an intermediate distance: Initial orbit altitude: 191 km. Intermediate “B” orbit altitude: 47836 km. Final orbit altitude: 35781 km. Find the total Δ v and total time. L. Healy – ENAE404 – Spring 2007 – Lecture 15 (Mar. 15) 3

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Bi-elliptic speed changes, transfer time The delta-vs are Δ v A = 2 . 614 km / s Δ v B = 1 . 276 km / s Δ v C = 0 . 187 km / s . Summing the all delta-vs, Δ v = Δ v A + Δ v B + Δ v C = 4 . 077 km / s ; notice this is slightly higher than the straight Hohmann transfer (3 . 935 km / s ). The total transfer time is T t = π 8 μ ( r A + r B ) 3 / 2 + ( r B + r C ) 3 / 2 =21h56m39.106s which is considerably longer than the Hohmann transfer time. L. Healy – ENAE404 – Spring 2007 – Lecture 15 (Mar. 15) 4
Comparing Hohmann and Bi-elliptic Transfers These two examples show Hohmann has lower Δ v , but that’s not always the case. When is it better to use Hohmann transfer, and when a bi-elliptic transfer? Just consid- ering the minimization of Δ v (fuel consump- tion), sometimes one is better, and sometimes the other. We will analyze to see the cases where each is superior. Remember also that transfer time may be a consideration; bi-elliptic typically has longer trans- fer times.

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