lecture15 - Bi-elliptic detailsUse the Hohmann transfer...

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Unformatted text preview: Bi-elliptic 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 Hohmanndelta-vs each:transfer ellipse 1vA=qrAr2rBrA+rB-1vB1=qrB1-r2rArA+rBtransfer ellipse 2vB2=qrBr2rCrB+rC-1vC=qrC1-r2rBrB+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+rB-1vB=srBs2rCrB+rC-s2rArA+rBvC=srC1-s2rBrB+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 delta-v. How-ever, the total transfer time becomes larger.L. Healy ENAE404 Spring 2007 Lecture 15 (Mar. 15)2Bi-elliptic 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)3Bi-elliptic speed changes, transfer timeThe delta-vs arevA= 2.614km/svB= 1.276km/svC= 0.187km/s.Summing the all delta-vs,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 Bi-elliptic TransfersThese two examples show Hohmann has lowerv, but thats not always the case.When is it better to use Hohmann transfer,and when a bi-elliptic transfer? Just consid-ering the minimization of v(fuel consump-tion), 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; bi-elliptic typically has longer trans-fer times.L. Healy ENAE404 Spring 2007 Lecture 15 (Mar. 15)5Introduce radius ratiosWrite the delta-vs in terms of the radius ratio=rCrA,and, for the bi-elliptic transfers, the ratio=rBrA,The delta-vs for Hohmann and bi-elliptic maybe expressed as a fraction of the initial circularspeed;q/rA; assuming, >1Hohmann1-2(1-)q(1 +)-1Bi-elliptics2(...
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lecture15 - Bi-elliptic detailsUse the Hohmann transfer...

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