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Unformatted text preview: Bielliptic detailsUse the Hohmann transfer computation, butapply it twice. CallrBthe geocentric distanceof the intermediate orbit. The semimajoraxes of the two transfer ellipses:at1=rA+rB2,at2=rB+rC2.On the two transfer ellipses, we have two Hohmanndeltavs each:transfer ellipse 1vA=qrAr2rBrA+rB1vB1=qrB1r2rArA+rBtransfer ellipse 2vB2=qrBr2rCrB+rC1vC=qrC1r2rBrB+rCIn reality, the middle two burns are one burn.L. Healy ENAE404 Spring 2007 Lecture 15 (Mar. 15)1Three changes of speedSo there are three changes of speed:vA=srAs2rBrA+rB1vB=srBs2rCrB+rCs2rArA+rBvC=srC1s2rBrB+rC,with v= vA+vB+vC, and flight timeis calculated as the sum of the two individualflight times:t=8(rA+rB)3/2+ (rB+rC)3/2.Note that there is a parameter to be chosen:rB. The larger this is, the smaller vBis,which reduce the overall total deltav. However, the total transfer time becomes larger.L. Healy ENAE404 Spring 2007 Lecture 15 (Mar. 15)2Bielliptic worked exampleUse the same initial and final conditions as forthe Hohmann example, with an intermediatedistance:Initial orbit altitude: 191km.Intermediate B orbit altitude: 47836km.Final orbit altitude: 35781km.Find the total vand total time.L. Healy ENAE404 Spring 2007 Lecture 15 (Mar. 15)3Bielliptic speed changes, transfer timeThe deltavs arevA= 2.614km/svB= 1.276km/svC= 0.187km/s.Summing the all deltavs,v= vA+ vB+ vC= 4.077km/s;notice this is slightly higher than the straightHohmann transfer (3.935km/s).The total transfer time isTt=8(rA+rB)3/2+ (rB+rC)3/2=21h56m39.106swhich is considerably longer than the Hohmanntransfer time.L. Healy ENAE404 Spring 2007 Lecture 15 (Mar. 15)4Comparing Hohmann and Bielliptic TransfersThese two examples show Hohmann has lowerv, but thats not always the case.When is it better to use Hohmann transfer,and when a bielliptic transfer? Just considering the minimization of v(fuel consumption), sometimes one is better, and sometimesthe other. We will analyze to see the caseswhere each is superior.Remember also that transfer time may be aconsideration; bielliptic typically has longer transfer times.L. Healy ENAE404 Spring 2007 Lecture 15 (Mar. 15)5Introduce radius ratiosWrite the deltavs in terms of the radius ratio=rCrA,and, for the bielliptic transfers, the ratio=rBrA,The deltavs for Hohmann and bielliptic maybe expressed as a fraction of the initial circularspeed;q/rA; assuming, >1Hohmann12(1)q(1 +)1Bielliptics2(...
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This document was uploaded on 04/27/2011.
 Spring '11

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