Orbital Mechanics For Engineering Students

951 Pages·2020·43.868 MB·English

by  Howard D. Curtis

Orbital Mechanics for
Engineering Students
Orbital Mechanics for
Engineering Students
Fourth Edition
Howard D. Curtis
Professor Emeritus, AerospaceEngineering
Embry-Riddle AeronauticalUniversity
Daytona Beach,Florida
Butterworth-HeinemannisanimprintofElsevier
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom
50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
#2020ElsevierLtd.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicor
mechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem,without
permissioninwritingfromthepublisher.Detailsonhowtoseekpermission,furtherinformationaboutthe
Publisher’spermissionspoliciesandourarrangementswithorganizationssuchastheCopyrightClearance
CenterandtheCopyrightLicensingAgency,canbefoundatourwebsite:www.elsevier.com/permissions.
ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher
(otherthanasmaybenotedherein).
MATLAB®isatrademarkofTheMathWorks,Inc.andisusedwithpermission.TheMathWorksdoesnot
warranttheaccuracyofthetextorexercisesinthisbook.Thisbook’suseordiscussionofMATLAB®
softwareorrelatedproductsdoesnotconstituteendorsementorsponsorshipbyTheMathWorksofa
particularpedagogicalapproachorparticularuseoftheMATLAB®software.
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperiencebroaden
ourunderstanding,changesinresearchmethods,professionalpractices,ormedicaltreatmentmaybecome
necessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusing
anyinformation,methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethods
theyshouldbemindfuloftheirownsafetyandthesafetyofothers,includingpartiesforwhomtheyhavea
professionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeanyliability
foranyinjuryand/ordamagetopersonsorpropertyasamatterofproductsliability,negligenceorotherwise,
orfromanyuseoroperationofanymethods,products,instructions,orideascontainedinthematerialherein.
LibraryofCongressCataloging-in-PublicationData
AcatalogrecordforthisbookisavailablefromtheLibraryofCongress
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN:978-0-08-102133-0
ForinformationonallButterworth-Heinemannpublications
visitourwebsiteathttps://www.elsevier.com/books-and-journals
Publisher:KateyBirtcher
AcquisitionEditor:SteveMerken
EditorialProjectManager:NathanielMcFadden
ProductionProjectManager:MariaBernard
CoverDesigner:VictoriaPearson
TypesetbySPiGlobal,India
To my beloved wife, Mary
For her patience, encouragement, and love
Preface
Thepurposeofthisbookistoprovideanintroductiontospacemechanicsforundergraduateengineer-
ing students. It is not directed toward graduate students, researchers, and experienced practitioners,
who may nevertheless find useful review material within the book’s contents. The intended readers
are those who are studying the subject for the first time and have completed courses in physics, dy-
namics,andmathematicsthroughdifferentialequationsandappliedlinearalgebra.Ihavetriedmybest
tomakethetextreadableandunderstandabletothataudience.Inpursuitofthatobjective Ihave in-
cludedalargenumberofexampleproblemsthatareexplainedandsolvedindetail.Theirpurposeisnot
tooverwhelmbuttoelucidate.Ifindthatstudentslikethe“teachbyexample”method.Ialwaysassume
thatthematerialisbeingseenforthefirsttimeand,whereverpossible,Iprovidesolutiondetailssoasto
leavelittletothereader’simagination.Thenumerousfiguresthroughoutthebookarealsointendedto
aid comprehension. All of the more labor-intensive computational procedures are accompanied by
®
MATLAB code.
For this, the fourth edition, I have retained the content and style of the previous editions and
corrected all the errors discovered by me or reported to me by readers. Except for the new
Chapter9onbasiclunartrajectoriesandanexpandeddiscussionofquaternionsinChapter11thebook
remainsessentially the same. Adding the new chapter required the following reshuffling:
Topic Thisedition Previousedition
Lunartrajectories Chapter9 Absent
Introductiontoorbitalperturbations Chapter10 Chapter12
Rigidbodydynamics Chapter11 Chapter9
Satelliteattitudedynamics Chapter12 Chapter10
Rocketvehicledynamics Chapter13 Chapter11
The organization of the book remains the same as that of the third edition. Chapter 1 is a review of
vectorkinematicsinthreedimensionsandofNewton’slawsofmotionandgravitation.Italsofocuses
ontheissueofrelativemotion,crucialtothetopicsofrendezvousandsatelliteattitudedynamics.The
materialonordinarydifferentialequationsolverswillbeusefulforstudentswhoareexpectedtocode
numericalsimulationsinMATLABorotherprogramminglanguages.Chapter2presentsthevector-
basedsolutionoftheclassicaltwo-bodyproblem,resultinginahostofpracticalformulasfortheanal-
ysisoforbitsandtrajectoriesofelliptical,parabolic,andhyperbolicshape.Therestrictedthree-body
problemiscoveredtointroducethenotionofLagrangepointsandtopresentthenumericalsolutionofa
lunartrajectory problem. Chapter 3 derives Kepler’s equations,which relate position totime for the
different kinds of orbits. The universal variable formulation is also presented. Chapter 4 is devoted
todescribingorbitsinthreedimensions.CoordinatetransformationsandtheEulerelementaryrotation
sequences are defined. Procedures for transforming back and forth between the state vector and the
classicalorbitalelementsareaddressed.Theeffectoftheearth’soblatenessonthemotionofanorbit’s
ascendingnodeandeccentricityvectorisdescribed,pendingamoredetailedexplanationinChapter10.
Chapter5isanintroductiontopreliminaryorbitdetermination,includingGibbs’andGauss’methods
andthesolutionofLambert’sproblem.Auxiliarytopicsincludetopocentriccoordinatesystems,Julian
xiii
xiv Preface
daynumbering,andsiderealtime.Chapter6presentsthecommonmeansoftransferringfromoneorbit
to another by impulsive delta-v maneuvers, including Hohmann transfers, phasing orbits, and plane
changes.Chapter7isabriefintroductiontorelativemotioningeneralandtothetwo-impulserendez-
vousprobleminparticular.Thelatter isanalyzed usingtheClohessy-Wiltshireequations,whichare
derived in this chapter. Chapter 8 is an introduction to interplanetary mission design using patched
conics. Chapter 9 extends the patched conic method and the restricted three-body approach to lunar
trajectoryanalysis.Chapter10isanintroductiontocommonorbitalperturbations:drag,nonspherical
gravitationalfield,solarradiationpressure,andlunarandsolargravity.Chapter11presentsthoseel-
ementsofrigidbodydynamicsrequiredtocharacterizetheattitudeofaspacevehicle.Euler’sequa-
tionsofrotationalmotionarederivedandappliedinanumberofexampleproblems.Eulerangles,yaw-
pitch-roll angles, and quaternions are presented as ways to describe the attitude of rigid body.
Chapter 12 describes the methods of controlling, changing, and stabilizing the attitude of spacecraft
bymeansofthrusters,gyros,andotherdevices.Chapter13isabriefintroductiontothecharacteristics
and design ofmultistage launchvehicles.
Chapters 1 through 4 form the core of a first orbital mechanics course. The time devoted to
Chapter1dependsonthebackgroundofthestudent.Itmightbesurveyedbrieflyandusedthereafter
simply as areference. What followsChapter4 depends onthe objectives of the course.
Chapters5through10carryonwiththesubjectoforbitalmechanics.Chapter6onorbitalmaneu-
vers should be included in any case. Coverage of Chapters 5, 7, 8, and 9 is optional. However, if
Chapters8and9oninterplanetaryandlunarmissionsistoformapartofthecourse,thenthesolution
ofLambert’s problem (Section 5.3) must bestudied beforehand.
Chapter10isappropriateforacoursedevotedexclusivelytoorbitalmechanicswithanintroduction
toperturbations, which is awhole topic unto itself.
Chapters11and12mustbecoveredifthecourseobjectivesincludeanintroductiontospacecraft
dynamics. Inthat case Chapters 5, 7,8, and 9wouldprobablynot bestudied indepth.
Chapter13isoptionaliftheengineeringcurriculumrequiresaseparatecourseinpropulsioninclud-
ingrocketdynamics.
Theimportanttopicofspacecraftcontrolsystemsisomitted.However,thematerialinthisbookand
a course incontrol theoryprovidethe basisfor the study ofspacecraft attitude control.
Tounderstandthematerialandtosolveproblemsrequiresusingalotofundergraduatemathemat-
ics.Mathematics,ofcourse,isthelanguageofengineering.StudentsmustnotforgetthattheEnglish
mathematicianandphysicistSirIsaacNewton(1642–1727)hadtoinventcalculussohecouldsolve
orbitalmechanicsproblemsinmorethanjustaheuristicway.Newton’s1687publicationMathemat-
icalPrinciplesofNaturalPhilosophy(“thePrincipia”)isoneofthemostinfluentialscientificworksof
all time. It must be noted that his contemporary, the German mathematician Gottfried Wilhelm von
Leibnitz (1646–1716) is credited with inventing infinitesimal calculus independently of Newton in
the 1670s.
Inadditiontohoningtheirmathskills,studentsareurgedtotakeadvantageofcomputers(which,
incidentally, use the binary numeral system developed by Leibnitz). There are many commercially
available mathematics software packages for personal computers. Wherever possible they should be
used to relieve the burden of repetitive and tedious calculations. Computer-programming skills can
andshouldbeputtogooduseinthestudyoforbitalmechanics.TheelementaryMATLABprograms
referredtoinAppendixDofthisbookillustratehowmanyoftheproceduresdevelopedinthetextcan
Preface xv
be implemented in software. All the scripts were developed and tested using MATLAB version 9.2
(release 2017a). Information about MATLAB, which is a registered trademark of The MathWorks,
Inc., may beobtained from
TheMathWorks, Inc.
3 Apple Hill Drive
Natick,MA 01760-2089,USA
www.mathworks.com
AppendixApresentssometablesofphysicaldataandconversionfactors.AppendixBisaroadmap
throughthefirstthreechapters,showinghowthemostfundamentalequationsoforbitalmechanicsare
related. Appendix C shows how to set up the n-body equations of motion and program them in
MATLAB. Appendix D contains listings of all the MATLAB algorithms and example problems
presentedinthe text.Appendix Eshowsthatthe gravitational field ofasphericallysymmetric body
is the same as if the mass were concentrated at its center. Appendix F explains how to deal with a
computational issuethat arises in some perturbation analyses.
SUPPLEMENTS TO THE TEXT
Forpurchasersofthebook,copiesoftheMATLABM-fileslistedinAppendixDcanbefreelydown-
loadedfromthisbook’scompanionwebsite.Alsoavailableonthecompanionwebsiteareasetofan-
imationsthataccompanythetext.Toaccessthesefiles,pleasevisithttps://www.elsevier.com/books-
and-journals/book-companion/9780081021330.
Forinstructorsusingthisbookforacourse,pleasevisitwww.textbooks.elsevier.comtoregisterfor
accesstothe solutions manual,PowerPoint lecture slides, and other resources.
ACKNOWLEDGEMENTS
Sincethepublicationofthefirstthreeeditionsandduringthepreparationofthisone,Ihavereceived
helpfulcriticism,suggestions,andadvicefrommanysourceslocallyandworldwide.Ithankthemall
and regret that time and space limitations prohibited the inclusion of some recommended additional
topics thatwouldhave enhancedthe book.
IthasbeenapleasuretoworkwiththepeopleatElsevier,inparticularJosephP.Hayton,Publisher;
Steve Merken, Senior Acquisitions Editor; and Nate McFadden, Senior Developmental Editor.
Iappreciatetheirenthusiasmforthebook,theirconfidenceinme,andalltheworktheydidtomove
this project tocompletion.
Finallyandmostimportantly,Imustacknowledgethepatienceandsupportofmywife,Mary,who
was a continuous source ofoptimismand encouragement throughout the revision effort.
Howard D.Curtis
DaytonaBeach,FL,UnitedStates
CHAPTER
1
DYNAMICS OF POINT MASSES
1.1 INTRODUCTION
Thischapterservesasaself-containedreferenceonthekinematicsanddynamicsofpointmassesas
well as some basic vector operations and numerical integration methods. The notation and concepts
summarizedherewillbeusedinthefollowingchapters.Thosefamiliarwiththevector-baseddynamics
ofparticlescansimplypagethroughthechapterandthenreferbacktoitlaterasnecessary.Thosewho
need abitmoreinthewayofreview willfindthatthechaptercontainsallthematerialtheyneed to
followthe development oforbitalmechanicstopics inthe upcomingchapters.
Webeginwithareviewofvectorsandsomevectoroperations,afterwhichweproceedtotheprob-
lem of describingthe curvilinear motion ofparticles in three dimensions. The concepts of force and
mass are considered next, along with Newton’s inverse-square law of gravitation. This is followed
by a presentation of Newton’s second law of motion (“force equals mass times acceleration”) and
the important conceptofangular momentum.
Asapreludetodescribingmotionrelativetomovingframesofreference,wedevelopformulasfor
calculating the time derivatives of moving vectors. These are applied to the computation of relative
velocityandacceleration.Exampleproblemsillustratetheuseoftheseresults,asdoesadetailedcon-
siderationofhowtheearth’srotationandcurvatureinfluenceourmeasurementsofvelocityandaccel-
eration.ThisbringsinthecuriousconceptofCoriolisforce.Embeddedinexercisesattheendofthe
chapterispracticeinverifyingseveralfundamentalvectoridentitiesthatwillbeemployedfrequently
throughout the book.
The chapter concludes with an introduction to numerical methods, which can be called upon to
solvethe equations ofmotion when an analyticalsolution is not possible.
1.2 VECTORS
Avectorisanobjectthatisspecifiedbybothamagnitudeandadirection.Werepresentavectorgraph-
icallybyadirectedlinesegment(i.e.,anarrowpointinginthedirectionofthevector).Theendopposite
thearrowiscalledthetail.Thelengthofthearrowisproportionaltothemagnitudeofthevector.Ve-
locityisagoodexampleofavector.Wesaythatacaristravelingeastwardat80km/h.Thedirectionis
eastandthemagnitude,orspeed,is80km/h.Wewilluseboldfacetypetorepresentvectorquantities
and plain type todenotescalars.Thus,whereas B is ascalar, Bis avector.
1
OrbitalMechanicsforEngineeringStudents.https://doi.org/10.1016/B978-0-08-102133-0.00001-5
#2020ElsevierLtd.Allrightsreserved.
2 CHAPTER 1 DYNAMICS OF POINT MASSES
FIG.1.1
AllofthesevectorsmaybedenotedA,sincetheirmagnitudesanddirectionsarethesame.
FIG.1.2
Parallelogramruleofvectoraddition.A+B¼C.
Observe thatavector is specified solely by itsmagnitude anddirection. If Ais avector,then all
vectorshavingthesamephysicaldimensions,thesamelength,andpointinginthesamedirectionasA
aredenotedA,regardlessoftheirlineofaction,asillustratedinFig.1.1.Shiftingavectorparallelto
itselfdoesnotmathematicallychangethevector.However,theparallelshiftofavectormightproduce
a different physical effect.Forexample, an upward 5-kN load (force vector) applied to the tip ofan
airplanewinggivesrisetoquiteadifferentstressanddeflectionpatterninthewingthanthesameload
acting at the wing’s midspan.
The magnitude ofa vector Ais denotedkAk, or,simplyA.
MultiplyingavectorBbythereciprocalofitsmagnitudeproducesavectorthatpointsinthedi-
rectionofB,butitisdimensionlessandhasamagnitudeofone.Vectorshavingdimensionlessmag-
nitudearecalledunitvectors.Weputahat(^)overtheletterrepresentingaunitvector.Thenwecantell
simply by inspectionthat, for example,u^ is a unit vector, as are B^ and^e.
ItisconvenienttodenotetheunitvectorinthedirectionofthevectorAasu^ .Aspointedoutabove,
A
we obtainthis vector from Aas follows:
A
u^ ¼ (1.1)
A A
Likewise,u^ ¼C=C,u^ ¼F=F, etc.
C F
Thesumorresultantoftwovectorsisdefinedbytheparallelogramrule(Fig.1.2).LetCbethesum
ofthe two vectors AandB.To formthat sum using the parallelogram rule, the vectors Aand Bare
1.2 VECTORS 3
FIG.1.3
Three-dimensional,right-handedCartesiancoordinatesystem.
shiftedparalleltothemselves(leavingthemunaltered)untilthetailofAtouchesthetailofB.Drawing
dotted lines through the head of each vector parallel to the other completes a parallelogram. The
diagonal from the tails of A and B to the opposite corner is the resultant C. By construction, vector
addition is commutative;that is,
A+B¼B+A (1.2)
A Cartesian coordinate system in three dimensions consists of three axes, labeled x, y, and z,
which intersect at the origin O. We will always use a right-handed Cartesian coordinate system,
which means if you wrap the fingers of your right hand around the z axis, with the thumb
pointing in the positive z direction, your fingers will be directed from the x axis toward the y axis.
Fig.1.3illustratessuchasystem.Notethattheunitvectorsalongthex,y,andzaxesare,respectively,
^ ^ ^
i, j, and k.
IntermsofitsCartesiancomponents,andinaccordancewiththeabovesummationrule,avectorA
is written interms of itscomponents A , A , and A as
x y z
^ ^ ^
A¼A i+A j+Ak (1.3)
x y z
TheprojectionofA onthe xyplane is a vector denoted A . It follows that
xy
^ ^
A ¼A i+A j
xy x y
AccordingtothePythagoreantheorem,themagnitudeofAintermsofitsCartesiancomponentsis
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A¼ A2+A2+A2 (1.4)
x y z
From Eqs.(1.1) and (1.3),the unitvectorinthe direction of Ais
u^ ¼cosθ^i+cosθ^j+cosθ k^ (1.5)
A x y z