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FOUNDATIONSOFSPACE DYNAMICS

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DesignandDevelopmentofAircraftSystems, 3rdEdition

SabatiniDec2020

TorenbeekJuly2020

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DesignofUnmannedAerialSystemsSadraeyApril2020

FlightDynamicsandControlofAeroandSpace Vehicles

FuturePropulsionSystemsandEnergySourcesin SustainableAviation

YedavalliFebruary2020

FarokhiJanuary2020

ConceptualAircraftDesign:AnIndustrialApproachKunduApril2019

HelicopterFlightDynamics:IncludingaTreatmentof TiltrotorAircraft,3rdEdition

PadfieldNovember2018

SpaceFlightDynamics,2ndEditionKlueverMay2018

PerformanceoftheJetTransportAirplane:Analysis Methods,FlightOperations,andRegulations

DifferentialGameTheorywithApplicationstoMissiles andAutonomousSystemsGuidance

AdvancedUAVAerodynamics,FlightStabilityand Control:NovelConcepts,TheoryandApplications

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FaruqiMay2017

MarquesandDaRonchApril2017

IntroductiontoNonlinearAeroelasticityDimitriadisApril2017

IntroductiontoAerospaceEngineeringwithaFlight TestPerspective

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AircraftControlAllocationDurham,BordignonandBeckJanuary2017 RemotelyPilotedAircraftSystems:AHumanSystems IntegrationPerspective

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TheGlobalAirlineIndustry,SecondEditionBelobaba,Odoniand Barnhart July2015

ModelingtheEffectofDamageinComposite Structures:SimplifiedApproaches

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AircraftAerodynamicDesign:Geometryand Optimization

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TheoreticalandComputationalAerodynamicsSenguptaSeptember2014 AerospacePropulsionLeeOctober2013

AircraftFlightDynamicsandControlDurhamAugust2013 CivilAvionicsSystems,2ndEditionMoir,SeabridgeandJukesAugust2013 ModellingandManagingAirportPerformanceZografos,Andreattaand Odoni July2013

AdvancedAircraftDesign:ConceptualDesign, AnalysisandOptimizationofSubsonicCivil Airplanes

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FOUNDATIONSOFSPACE DYNAMICS

FirstEdition

AshishTewari

IndianInstituteofTechnologyKanpur

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LibraryofCongressCataloging-in-PublicationData

Names:Tewari,Ashish,author.

Title:Foundationsofspacedynamics/AshishTewari.

Description:Firstedition.|Hoboken,NJ:Wiley,[2020]|Series: Aerospaceseries|Includesbibliographicalreferencesandindex.

Subjects:LCSH:Aerospaceengineering.|Astrodynamics.|Orbital mechanics.

Classification:LCCTL545.T3852020(print)|LCCTL545(ebook)|DDC 629.4/11–dc23

LCrecordavailableathttps://lccn.loc.gov/2020033397

LCebookrecordavailableathttps://lccn.loc.gov/2020033398

CoverDesign:Wiley

CoverImage:©PhilipWallick/GettyImages

Typesetin10/12ptTimesLTStdbySPiGlobal,Chennai,India

10987654321

Tothelovingmemoryofmydaughter,Manya(24.1.2000-9.7.2019)

3KeplerianMotion41

3.1TheTwo-BodyProblem41

3.2OrbitalAngularMomentum43

3.3OrbitalEnergyIntegral45

3.4OrbitalEccentricity46

3.5OrbitEquation49

3.5.1EllipticOrbit 53

3.5.2ParabolicOrbit 56

3.5.3HyperbolicOrbit 56

3.5.4RectilinearMotion 58

3.6OrbitalVelocityandFlightPathAngle60

3.7PerifocalFrameandLagrange’sCoefficients63 Exercises65

4TimeinOrbit69

4.1PositionandVelocityinanEllipticOrbit70 4.2SolutiontoKepler’sEquation75

4.2.1Newton’sMethod 76

4.2.2SolutionbyBesselFunctions 78

4.3PositionandVelocityinaHyperbolicOrbit80

4.4PositionandVelocityinaParabolicOrbit84

4.5UniversalVariableforKeplerianMotion86 Exercises88 References89 5OrbitalPlane91 5.1RotationMatrix91

5.2EulerAxisandPrincipalAngle94

5.3ElementaryRotationsandEulerAngles97

5.4Euler-AngleRepresentationoftheOrbitalPlane101

5.4.1CelestialReferenceFrame 103

5.4.2Local-HorizonFrame 104

5.4.3ClassicalEulerAngles 106

5.5Planet-FixedCoordinateSystem111 Exercises114

6OrbitalManoeuvres117

6.1Single-ImpulseOrbitalManoeuvres119

6.2Multi-impulseOrbitalTransfer123

6.2.1HohmannTransfer 124

6.2.2RendezvousinCircularOrbit 127

6.2.3OuterBi-ellipticTransfer 130

6.3ContinuousThrustManoeuvres133

6.3.1PlanarManoeuvres 134

6.3.2ConstantRadialAccelerationfromCircularOrbit 135

6.3.3ConstantCircumferentialAccelerationfromCircularOrbit

6.3.4ConstantTangentialAccelerationfromCircularOrbit 139 Exercises141 References143

7RelativeMotioninOrbit145

7.1Hill-Clohessy-WiltshireEquations148

7.2LinearState-SpaceModel151

7.3ImpulsiveManoeuvresAboutaCircularOrbit153

7.3.1OrbitalRendezvous

7.4KeplerianRelativeMotion155 Exercises158

8Lambert’sProblem161 8.1Two-PointOrbitalTransfer161

8.1.1TransferTriangleandTerminalVelocityVectors

8.2.1LocusoftheVacantFocii

8.2.2Minimum-EnergyandMinimum-EccentricityTransfers

8.3Lambert’sTheorem168

8.3.1TimeinEllipticTransfer

8.3.2TimeinHyperbolicTransfer

8.3.3TimeinParabolicTransfer

8.4SolutiontoLambert’sProblem177

8.4.1ParameterofTransferOrbit

8.4.2StumpffFunctionMethod

8.4.3HypergeometricFunctionMethod

9OrbitalPerturbations191

9.1PerturbingAcceleration191

9.2OsculatingOrbit192

9.3VariationofParameters194

9.3.1LagrangeBrackets

9.4LagrangePlanetaryEquations199

9.5GaussVariationalModel209

9.6VariationofVectors214

9.7MeanOrbitalPerturbation219

9.8OrbitalPerturbationDuetoOblateness220

9.8.1Sun-SynchronousOrbits

9.8.2MolniyaOrbits

9.9EffectsofAtmosphericDrag227

9.9.1LifeofaSatelliteinaLowCircularOrbit

9.9.2EffectonOrbitalAngularMomentum

9.9.3EffectonOrbitalEccentricityandPeriapsis

Preface

FoundationsofSpaceDynamics iswrittenasatextbookforstudents,aswellasareadyreference coveringtheessentialconceptsforpracticingengineersandresearchers.Itintroducesareaderto thebasicaspectsofbothorbitalmechanicsandattitudedynamics.Whilemanygoodtextbooks areavailableonorbitalmechanicsandattitudedynamics,thereisaneedforadirect,concise, yetrigoroustreatmentofboththetopicsinasingletextbook.Importantderivationsfrombasic principlesarehighlighted,whileofferinginsightsintothephysicalprincipleswhichcanoften behiddenbymathematicaldetails.Whiletheemphasisisonanalyticalderivations,theessential computationaltoolsarepresentedwhereverrequired,suchastheiterativeroot-findingmethods andthenumericalintegrationofordinarydifferentialequations.

Theobjectiveofthisbookistoprovideaphysicallyinsightfulpresentationofspacedynamics. Theusageofsimpleideasandnumericaltoolstoillustrateadvancedconceptsisinspiredbythe workoftheoriginalmasters(Newton,Liebnitz,Laplace,Gauss,etc.),andiscombinedwiththe applicationandterminologyofmodernspacedynamics.

Astudentofspacedynamicsinthepastgenerallypossessedastrongbackgroundinanalyticalmechanics,oftenreinforcedbysuchclassicaltreatisesasthosebyWhittaker,Lanczos, Truesdell,andMach.Today,theexposuretoanalyticaldynamicsisoftenbaseduponasingle undergraduatecourse.Thisbookthereforeincludesabasicintroductiontoanalyticalmechanics bybothNewtonianandLagrangianapproaches.

Thecontentsofthetextbookarearrangedsuchthattheymaybecoveredintwosuccessive courses: SpaceDynamicsI couldfocusonChaps.1–7and11,whilethefollowingcourse, Space DynamicsII,couldcoverChaps.8–10and12,supplementedbyasemesterprojectexploringa specificresearchtopic.However,thearrangementofthechaptersinthebookofferssufficient flexibilityforthemtobecoveredinasinglecomprehensivecourse,ifsorequired.Therearea multitudeofexercisesattheendofthechapterswhichcanserveashomeworkassignmentsand quizproblems.Solutionstoselectedexercisesisalsoprovided.

IwouldliketothanktheeditorialandproductionstaffofWiley,Chichester,fortheir constructivesuggestionsandvaluableinsightsduringthepreparationofthemanuscript.

May2020

AshishTewari

1 Introduction

Thischaptergivesanintroductiontothebasicfeaturesofspaceflight,whichispredominatedby thequietspaceenvironmentandgravity.Theessentialdifferenceswithatmosphericflightare discussed,andtheimportanttimescalesandframesofreferenceforspaceflightaredescribed. Topicsinspacedynamicsareclassifiedasthetranslationalmotion(orbitalmechanics)androtationalmotion(attitudedynamics)ofarigidspacecraft.Classificationofthevariouspractical spacecraftisgivenaccordingtotheirmissions.

1.1SpaceFlight

Spaceflight referstomotionoutsidetheconfinesofaplanetaryatmosphere.Itisdifferentfrom atmosphericflightinthatnoassistancecanbederivedfromtheatmosphericforcestosupport avehicle,andnobenefitofplanetaryoxygencanbeutilizedforpropulsion.Apartfromthese majordisadvantages,spaceflighthastheadvantageofexperiencingno(orlittle)dragdueto theresistanceoftheatmosphere;henceaspacecraftcanachieveamuchhigherflightvelocity thananaircraft.Sinceatmosphericliftisabsenttosustainspaceflight,aspacecraftrequires suchhighvelocitiestobalancetheforceofgravitybyacentrifugalforceinordertoremain inflight.Thetrajectoriesofspacecraft(called orbits)–beinggovernedsolelybygravity–are thusmuchbetterdefinedthanthoseofaircraft.Sincegravityisaconservativeforce,spaceflight involvesaconservationofthesumofkineticandpotentialenergies,aswellasthatoftheangular momentumaboutafixedpoint.Therefore,spaceflightismucheasiertoanalyzemathematically whencomparedtoatmosphericflight.

1.1.1AtmosphereasPerturbingEnvironment

Whencantheeffectsoftheatmospherebeconsiderednegligiblesothatspaceflightcancome intoexistence?Theatmosphereofaplanetarybody–beingboundbygravity–becomes lessdenseasthedistancefromtheplanetarysurface(called altitude)increases,owingto theinverse-squarediminishingoftheaccelerationduetogravityfromtheplanetarycentre. Foranatmospherecompletelyatrest,thisrelationshipbetweentheatmosphericdensity,

��,andthealtitude, z,canbederivedfromthefollowingdifferentialequationof aerostatic equilibrium (Tewari,2006):

where p referstotheatmosphericpressure,and g the accelerationduetogravity prevailing atagivenaltitude.Forasphericalbodyofradius r0 ,thegravityobeystheinverse-squarelaw discoveredbyNewton,givenby

where g0 istheaccelerationduetogravityatthesurfaceofthebody(i.e.,at z = 0).When Eq.(1.2)issubstitutedintoEq.(1.1),andthethermodynamicpropertiesoftheatmosphericgases aretakenintoaccount,thedifferentialequation,Eq.(1.1),canbeintegratedtoyieldanalgebraic relationshipbetweenthe atmosphericdensity, ��,andthealtitude, z,calledanatmosphericmodel. ForEarth’satmosphere,onesuchmodelisthe U.S.StandardAtmosphere1976 (Tewari,2006), whosepredicteddensityvariationwiththealtitudeintherange0 ≤ z ≤ 250kmislistedin Table1.1.ItisevidentfromTable1.1thattheatmosphericdensity, ��,canbeconsideredto benegligibleforaflightfor z ≥ 120kmaroundEarth.Asimilar(albeitsmaller)valueof �� is obtainedonMarsat z = 120km.Hence,forbothEarthandMars, z = 120kmcanbetakento betheboundaryabovewhichthe space begins.

Theflightofaspacecraftaroundalargesphericalbodyofradius r0 isassumedtotakeplace outsidetheatmosphere,(suchas z > 120kmforEarthandMars),andisgovernedbythegravity ofthebody,withaccelerationgivenbyEq.(1.2).Space-flighttrajectoriesarewelldefined orbits

Altitude, z (km)Density, �� kg∕m3

Table1.1 VariationofdensitywithaltitudeinEarth’s atmosphere

duetothesimplenatureofEq.(1.2).However,sincetheatmosphericdensityinaverylow orbit(e.g.,120 < z < 250kmonEarth),albeitquitesmall,isnotexactlyzero,theflightof aspacecraftcanbegraduallyaffected,tocausesignificantdeviationsoveralongperiodof timefromtheorbitspredictedbyEq.(1.2).Thisisduetothefactthattheatmosphericforces andmomentsaredirectlyproportionaltotheflightdynamicpressure,1∕2��v2 ,where v isthe flightspeed.Thehighorbitalspeed, v,requiredforspaceflightmakesthedynamicpressure appreciable,eventhoughthedensity, ��,isbyitselfnegligible.Theatmospheric drag (theforce resistingthemotion)causesaslowbutsteadydeclineintheflightspeed,untilthelatterfalls belowthemagnitudewhereanorbitalmotioncanbesustained.Thusatmosphericdragcan causealow-orbitingsatellitetoslightlydecayinaltitudeaftereveryorbit,andtoultimately enterthelower(dense)portionsoftheatmosphere,wherethemechanicalstresscreatedbythe everincreasingdynamicpressure,aswellastheheatgeneratedbyatmosphericfriction,lead toitsdestruction.Therefore,forpredictingthelifeofasatelliteinaloworbit,theatmospheric effectsmustbeproperlytakenintoaccount.Figure1.1showsanexampleofthedecayinthe orbitofaspacecraftinitiallyplacedintoacircularorbitof z = 200kmaroundEarth.Inthis simulationobtainedbya Runge-Kuttamethod (AppendixA),thespacecraftisassumedtobea sphereof1mdiameter,withaconstantfree-moleculardragcoefficientof2.0(Tewari,2006). Asseeninthefigure,thealtitudedecaysquiterapidlyasthenumberoforbits, N ,increases. TheinitialaveragerateofaltitudelossseeninFig.1.1–1kmper4orbits–islikelytoincrease asthespacecraftdescendslower,therebyencounteringahigherdensity.Whenthespacecraft isplacedinacircularorbitof z = 180km,itsaltitudedecaysveryrapidly,anditre-entersthe

Figure1.1 Decayintheorbitduetoatmosphericdragforaspacecraftinitiallyplacedinacircularorbit of z = 200kmaroundEarth.

Figure1.2 Decayintheorbitduetoatmosphericdragforaspacecraftinitiallyplacedinacircularorbit of z = 180kmaroundEarth. atmosphereafteronly3.5orbits(Fig.1.2).Hence,thelifeofthespacecraftisonlyabout3.5 revolutionsinacircularorbitofaltitude180kmaboveEarth.AsFigs.1.1and1.2indicate,a stableorbitaroundEarthforthisspacecraftshouldhave z > 200kmatalltimes.

Apartfromtheatmosphericeffects,thereareotherenvironmentalperturbationstoaspacecraft’sflightaroundacentralbody,whichisassumedtobesphericalasrequiredbyEq.(1.2). Thesearethegravityoftheactual(non-spherical)shapeofthecentralbody,aswellasthe gravityofotherremotelargebodies,andthesolarradiationpressure.However,sucheffectsare typicallysmallenoughtobeconsideredsmallperturbationswhencomparedtothespherical gravityfieldofthecentralbodygivenbyEq.(1.2).SucheffectscanberegardedassmallperturbationsappliedtotheorbitgovernedbyEq.(1.2),andshouldbecarefullymodelledinorder topredicttheactualmotionofthespacecraft.

1.1.2GravityastheGoverningForce

Spaceflightisprimarilygovernedbygravity.“Governing”impliesdictatingthepathagiven bodydescribesinathree-dimensionalspace.Aircraftandrocketflightsare not primarilygovernedbygravity,becausethereareotherforcesactingonthebody,suchastheliftandthethrust, whichareofcomparablemagnitudestothatofgravityandthereforedeterminetheflightpath. DiscoveredandproperlyanalyzedforthefirsttimebyNewtoninthelate17th century,gravity canbeexpressedsimply,buthasprofoundconsequences.Forexample,byapplyingNewton’s

lawofgravitation,itcouldhavebeeninferredthattheuniversecannotbestatic,becausegravity wouldcausealltheobjectstocollapsetowardsasinglepoint.However,thissimplefactescaped thenoticeofallphysicistsrangingfromNewtonhimselftoEinstein,untilitwasobservedby Hubblein1924thattheuniverseisexpandingataratewhichincreaseswiththedistancebetween anytwoobjects.Areadermaybecautionedagainstthecomplacencywhichoftenarisesby treatingthemotiongovernedbygravityassimple(eventrivial)tounderstand.Therearemany surprisingandinterestingconsequencesofgravitybeingthegoverningforceinflight,suchas Kepler’sthirdlawofplanetarymotion,whichimpliesthatthetimeperiodofanorbitingbody dependsonlyuponthemeanradius,andisindependentoftheshapeoftheorbit.Alargerpart ofacourseonspacedynamicsinvolvesunderstandinggravityanditseffectsonthemotionofa bodyinspace.

1.1.3TopicsinSpaceDynamics

Spacedynamicsconsistsoftwoparts:(a) orbitalmechanics,whichdescribesthetranslation inspaceofthecentreofmassofarigidbodyprimarilyundertheinfluenceofgravity,and(b) attitudedynamics,whichisthedescriptionoftherotationoftherigidbodyaboutitsowncentre ofmass.Whilethesetwotopicsarelargelystudiedseparately,insomecasesorbitalmechanics andattitudedynamicsareintrinsicallycoupled,suchaswhentherigidbodyexperiencesan appreciablegravity-gradienttorqueduringitsorbit.Furthermore,whendesigninganattitude controlsystemforaspacecraft,itisnecessarytoaccountforitsorbitalmotion.Therefore,while elementsoforbitalmechanicsandattitudedynamicscanbegraspedseparately,theirpractical applicationinvolvesacombinedapproach.

1.2ReferenceFramesandTimeScales

Spaceflightrequiresadefinitebackgroundofobjectstomeasuredistances,aswellastoorient thespacecraftinspecificdirections.Sincefixedobjectsarehardtocomebyinpractice,navigationandattitudedeterminationarenon-trivialproblemsinspaceflight.Suchaproblemdoes notexistforthemotiontakingplaceon,orverycloseto,asolidsurface,whereground-fixed objectscanserveasusefulreferencesforbothnavigationandorientationofthevehicles.

1.2.1SiderealFrame

Threemutuallyperpendicularstraightlinesjoiningdistantobjectsconstitutea referenceframe. Generally,distantobjectsintheuniversearemovingwithrespecttooneanother;hencethe straightlinesjoiningthemwouldrotate,aswellaseitherstretchoutorcontractwithtime.Supposeonecanfindtwoobjectswhicharefixedrelativetoeachother.Thenastraightlinejoining themwouldbefixedinlength,andavectorpointingfromoneobjecttotheotherwouldalways haveaconstantdirection.Areferenceframeconsistingofaxeswhichhavefixeddirectionsis saidtobea siderealframe.Therearecertaindirectionswhichcanbeusedtoorientasidereal frame.Forexample,theorbitalplaneofEartharoundthesun,calledthe ecliptic,intersects Earth’sequatorialplanealongastraightlinecalledthe lineofnodes.The nodes arethetwo specificpointswherethislineintersectsEarth’sorbit,asshowninFig.1.3.Oneofthetwo

Distant star

Equatorial plane

meridian

Obliquityof ecliptic

Eclipticplane Decending node

Earth Longitude of equinox

Ascending node Vernal equinox Apparent sun

Figure1.3 Theequinoctialsiderealframe (I, J, K),theeclipticsynodicframe (i, j, k),andEarthcentred celestialmeridian.

nodesisan ascendingnode,wheretheapparentmotionofthesunasseenfromEarth(called the apparentSun)occursfromthesouthtothenorthoftheequator.Thishappensatthe vernal equinox,occurringeveryyeararoundMarch21.The descendingnode oftheapparentsunisat the autumnalequinox,whichtakesplacearoundSeptember22.Sincethevernalequinoxpoints inaspecificdirectionfromthecentreofEarth,itcanbeusedtoorientoneoftheaxesofthe siderealframe,astheaxis I inFig.1.3.Anotheraxisofthesiderealframecanbetakentobe normaltoeithertheeclipticortheequatorialplane(axis K inFig.1.3),andthethirdaxiscan bechosentobeperpendiculartothefirsttwo(axis J inFig.1.3).

TherateofrotationofEarthonitsownaxis(normaltotheequatorialplane)isfromthewest totheeast,andcanbemeasuredinasiderealreferenceframeorientedwiththevernalequinox direction.Thisrateiscalledthe siderealrotationrate,andwouldbethetruerotationrateof Earthifthevernalequinoxwereaconstantdirection.A siderealday istheperiodofrotation ofEarthmeasuredfromthevernalequinox.Ifthesunisusedfortimingtherotationalrateof Earth,theperiodfromnoontonoonisa meansolarday (m.s.d.)of24-hourduration.However, themeansolardayisnotthetruerotationalrateofEarthbecauseofEarth’sorbitaroundthe sun,whichalsotakesplacefromthewesttotheeast.Tocalculatethesiderealdayfromthe meansolarday,acorrectionmustbeappliedbyaddingtheaveragerateatwhichEarthorbits thesun.The tropicalyear istheperiodofEarth’sorbitaroundthesunmeasuredfromonevernal equinoxtothenext,andequals365.242meansolardays.Thisimpliesthatthemeanapparent sunisslightlylessthanonedegreeperday(360∘ ∕365.242).Suchacorrectiongivesthesidereal

dayasthefollowing:

or23hr.,56min.,4.0904s.

Unfortunately,thevernalequinoxisnotaconstantdirectionbecauseoftheslow precession ofEarth’saxis(thustheequatorialplane)causedbythegravitationalinfluenceofthesunand themoon(calledthe luni-solarattraction).Whenaspinningrigidbody,suchasEarth,isacted uponbyanexternaltorque,suchasduetothegravityofthesunandthemoon,itsspinaxis undergoesacomplexrotationcalled“precession”and“nutation”,whichwillbeexplainedin detailinChapter11.Thisrotationoftheequatorialplanecausesthetwoequinoxestoshift towardsthewest,andisthuscalledthe precessionoftheequinoxes.Theperiodoftheprecessionisabout25772yr.,whichimpliesthatthesiderealdaydiffersonlyslightlyfromthetrue rotationalperiodofEarth.Italsomeansthatanequinoctialsiderealreferenceframe,suchas theframe (I, J, K) inFig.1.3,rotatesveryslowlyagainstabackgroundofdistantstars.Hence thevernalequinox(andtheequinoctialsiderealreferenceframe)canbeapproximatedtobe thefixedreferencesformostspaceflightapplications.However,foralongflighttimeofseveralyears’duration,thecalculationsmustbebroughttoacommonreferenceataspecifictime (calledan epoch1 )byapplyingthenecessarycorrections,whichtakeintoaccounttheslow movementofthevernalequinoxtowardsthewest.Theequinoxisgivenforvariousepochsby the InternationalEarthRotationandReferenceSystemsService (IERS)intermsofthelongitudeoftheequinoxmeasuredfroma celestialmeridian (seeFig.1.3).Theinclinationof Earth’sspinaxisfromthenormaltotheeclipticiscalledthe obliquityoftheecliptic (Fig.1.3), andalsovarieswithtimeduetothe nutation causedbytheluni-solarattraction.(Theprecessionandnutation,discussedindetailinChapter11,causeEarth’sspinaxistorotatewithtime duetotheluni-solarattraction.)Thevalueoftheobliquityoftheeclipticinthecurrentepoch ismeasuredbyIERStobeabout23∘ 26’21”.TheperiodofnutationofEarth’sspinaxisis about41000yr.,whichisconsiderablylongerthantheperiodofitsprecession.TheprecessionandnutationareexplainedinChapter11whenconsideringtherotationofarigidbody (suchasEarth).

ApartfromtheprecessionandthenutationofEarth’sspinaxis,thereisalsoaprecessionof theeclipticcausedbythegravitationalattractionoftheotherplanets.Thisisamuchsmaller variationintheequinoxes(about100timessmallerthanthatcausedbyluni-solarattraction).

Sincethevernalequinoxmovesslightlywestwardeveryyear,thetropicalyearisnotthe trueperiodofrevolutionofEarthinitsorbitaroundthesun.Thetrueperiodofrevolutionis the siderealyear,whichismeasuredbytimingthepassageofEarthagainstthebackgroundof distantstars,andequals365.25636meansolardays.Thusatropicalyearisshorterthanthe actualyearby20hr.,40min.,and42.24s.

1 Anepochisamomentintimeusedasareferencepointforatime-varyingastronomicalquantity,suchastheorbital elementsspecifyingtheshapeandtheplaneofanorbit,thedirectionofthespinaxisofabody,thecoordinatesof importantcelestialobjects,etc.

1.2.2CelestialFrame

Foramotiontakingplaceinsidethesolarsystem,anytwostars(exceptthesun)appeartobe fixedforthedurationofthemotion.Hence,areferenceframeconstructedoutofthreemutually perpendicularaxes,eachofwhicharepointingtowardsdifferentdistantstars,wouldappearto befixedinspace,andcanserveasasiderealreferenceframe.Areferenceframefixedrelative todistantstarsistermeda celestialreferenceframe.Forexample,therateofrotationofEarth aboutitsownaxiscanbemeasuredbyanobserverstandingastridetheNorthPolebytiming therateatwhichastraightlinejoiningEarthtoadistantstar,calledacelestialmeridian(see Fig.1.3),appearstorotate.ThisrategivesthetruerotationaltimeperiodofEarth,calledthe stellarday,whichismeasuredbyIERStobe23hr.,56min.,4.0989s.Hence,thesiderealday isshorterthanthestellardaybyabout8.5 × 10 3 s.

1.2.3SynodicFrame

Whentwoobjectsorbitoneanotheratnearlyconstantratesonafixedplane,areferenceframe canbedefinedbytwoofitsaxesontheplaneofrotationandrotatingattheconstantrate,and thethirdaxisnormaltotheplane.Sucharotatingreferenceframeiscalleda synodicframe.An exampleofasynodicframeisthe eclipticframe,whichisareferenceframeconstructedoutof theeclipticplane,suchastheframe (i, j, k) inFig.1.3.Themotionofanobjectmeasuredrelative toasynodicframemustbecorrectedbyavectorsubtractionofthemotionoftheframeitself, asexemplifiedbythecalculationofthesiderealdayfromtheobservedrotationintheecliptic frame.Theeclipticframehasbeenusedasareferencesincetheearliestdaysofastronomical observations.Thedivisionofthecircleinto360∘ aroseoutoftheapparentmotionofthesun perday,whichsubtendsanarcofonediameterevery12hourswhenseenfromEarth.Since themoon’sapparentdiameterfromEarthisroughlythesameasthatofthesun,theeclipsesof thesunandthemoonareobservedintheecliptic(thusthename).However,sincethemoon’s orbitalplanearoundEarthistilted ±5 1∘ relativetotheecliptic,theeclipseshappenonlyalong theintersection(i.e.,thelineofnodes)ofthetwoplanes.

TheEarth-moonlineprovidesanothersynodicreferenceframeforspaceflight.The Earthandthemoondescribecoplanarcirclesaboutthecommoncentreofmass(calledthe barycentre)every27.32meansolardaysrelativetothevernalequinox(calleda sidereal month).Thisrotationalperiodappearsinthesynodicframetobe29.53meansolardays(a synodicmonth)fromonenewmoontothenext,whichisobtainedfromthesiderealmonthby subtractingtherateofrevolutionofEarth-moonsystemaroundthesun.

1.2.4JulianDate

Insteadofthe calendaryear of365meansolardays,thetropicalyearof365.242meansolar days,andthesiderealyearof365.25636meansolardays,itismuchmoreconvenienttousea Julianyear of365.25meansolardays,whichavoidstheadditionofleapyearsincarryingout astronomicalcalculations.A Juliandaynumber (JDN )isdefinedtobethecontinuouscount ofthenumberofmeansolardayselapsedsince12:00noon universaltime (UT)onJanuary1, 4713BC.Universaltimereferstothetimetakenas12:00noonwhenthesunisdirectlyoverthe Greenwichmeridian(whichisdefinedtobezerolongitude).TheJuliandaynumber0isassigned

tothedaystartingatthattimeontheJulianprolepticcalendar.The Juliandate ofageneraltime instantisexpressedastheJDNplusthefractionofthe24-hourdayelapsedsincethepreceding noonUT.JuliandatesarethusexpressedasaJuliandaynumberplusadecimalfraction.For example,theJuliandatefor10:00a.m.UTonApril21,2020,isgivenbyJ2458960.91667, andtheJDNis2458960.Epochsarelistedinephemerischartsandnauticalalmanacsaccording totheirJuliandates.HenceaJuliandateservesasacommontimemeasureforastronautical calculationsinvolvingtwoeventsseparatedintime.

ComputationoftheJuliandate(JD)fromaGregoriancalendardateiscomplicateddueto thethreecalendarcyclesusedtoproducetheJuliancalendar,namelythesolar,thelunar,and theindictioncyclesof28,19,and15yearperiods,respectively(Seidelmann,1992).Aproduct ofthesegivesthe Julianperiod of7980years.TheJulianperiodbeginsfrom4713BC,which ischosentobethefirstyearofsolar,lunar,andindictioncyclesbeginningtogether.Thenext epochwhenthethreecyclesbegintogetherwillhappenatnoonUTonJanuary1,3268.The followingconversionformulafortheJDN,truncatedtothelastinteger,usesthenumberingof themonthsfromJanuarytoDecemberas M = 1, 2, … , 12;theGregoriancalendaryearsare numberedsuchthattheyear1BCistheyearzero, Y = 0,(i.e.,2BCis Y =−1,4713BCis Y =−4712,etc.);andthedaynumber, D,isthelastcompleteddayofthemonthuptonoonUT:

ThisformulacalculatestheJDNfor09:25a.m.UTonJune25,1975,bytaking Y = 1975, M = 6, D = 24,andyieldsthelasttruncatedintegervalueas JDN = 2442589.Thenthetime elapsedfromnoonUTonJune24to09:25a.m.UTonJune25isaddedasafractiontogive thefollowingJuliandate:

JD = JDN + 12 + 9 + 25∕60 24 = 2442589.892361 .

AnepochintheJuliandateisdesignatedwiththeprefix J ,andthesuffixbeingtheclosest Gregoriancalendardate.Forexample, J 2000refersto12:00noonUTonJanuary1,2000,and hastheJuliandateof2451545.Similarly,theepoch J 1900,whichoccursexactly100Julian years before 12:00noonUTonJanuary1,2000,mustreferto12noonUTonJanuary0,1900; henceitsdateintheGregoriancalendarisDecember31,1899,anditsJuliandateis2415020. Thedifferenceintheepochs J 2000and J 1900istherefore2451545 2415020 = 36525mean solardays(whichisexactly100Julianyears).

SinceJuliandaynumberswiththeepoch J 4712canbecomeverylarge,itisoftenconvenient tousealaterepochforcomputing JD.Epochscanbechosenwithsimpler JDN figures,such as12:00hr.UTonNovember16,1858,whichhas JDN = 2400000.ThenJuliandatescanbe convertedtothisepochbyreplacing JD with JD 2400000.Forexample,theJuliandatefor 09:25a.m.UT,June25,1975,convertedtotheepochofNov.16,1858,is JD = 42589.892361. Fortheconsistencyofdata,allmodernastronomicalcalculationsarereducedtotheepoch, J 2000,byinternationalagreement.ThismeansthatalltheJuliandatesmustbeconvertedto thisepochbyreplacing JD with JD 2451545.

1.3ClassificationofSpaceMissions

Spacecraftareclassifiedaccordingtotheirmissions.AlargemajorityofspacecraftorbitEarth asartificialsatellitesforobservation,mapping,thermalandradioimaging,navigation,scientific experimentation,andtelecommunicationspurposes.Thesesatellitesareclassifiedaccordingto theshapesandsizesoftheirorbits.Aspacecraftorbitingacentralbodyataltitudessmaller thanthemeanradius, r0 ,ofthebody, z < r0 ,istermeda low-orbitingspacecraft.Examples ofsuchspacecraftforEarth(r0 = 6378.14km)arethe low-Earthorbit (LEO)satellites,which orbittheplanetinnearlycircularorbitsof200 ≤ z ≤ 2000km.OrbitalperiodsofLEOsatellites rangefrom90to127min.,andaremainlyusedforEarthobservation,photoreconnaissance, resourcemapping,andspecialsensingandscientificmissions.The InternationalSpaceStation isamannedLEOspacecraftwithanearlycircularorbitofmeanaltitude, z = 400km.There arehundredsofactiveLEOsatellitesinorbitatanygiventime,launchedbyvariousnationsfor civilandmilitaryapplications.

A medium-Earthorbit (MEO)satellitehasaperiodofabout12hours.Examplesofsuch spacecraftarethe GlobalPositioningSystem (GPS)navigationalsatellitesincircularorbits ofaltitudesabout20,000km,and Molniya telecommunicationssatellitesofRussiainhighly eccentricellipticalorbitsinclinedat63.435∘ relativetoEarth’sequatorialplane.

ThehighestaltitudeofEarthsatellitesisforthoseinthe geosynchronousequatorialorbit (GEO),whichisacircularorbitintheequatorialplaneofaperiodexactlymatchingasidereal day,i.e.,23hr.,56min.,4.0904s.Thistranslatesintoanaltitudeof z = 35786.03km.Sincethe orbitalfrequencyofaGEOsatelliteequalstherateofrotationofEarthonitsaxis,suchasatellite returnstothesamepointabovetheequatoraftereachsiderealday,therebyappearingtobe stationarytoanobserverontheground.Hence,aGEOsatelliteisusedasatelecommunications relayplatformforsignalsbetweenanytwogroundstationsdirectlyinthelineofsightofthe satellite.DuetothehighaltitudeoftheGEOsatellite,abroadcoverageofsignalsisprovidedto thereceivingstationsontheground,andisthebasisofmoderntelevisionbroadcastsandmobile telephonecommunications.

Asmallnumberofspacecraftareputintohighlyspecializedlunar,interplanetary,andasteroid/cometaryinterceptorbitsfortheexplorationofthesolarsystem.Duetothetypicallylarge distancesinvolvedintheirmissions,whichmightincludethetimespentbeyondtheline-of-sight ofEarth,suchspacecraftmustbefullyautonomousintermsoftheirbasicoperations.Thespacecraftwhicharesenttoexploretheouterplanets(suchasNASA’s Voyager1 and Voyager2, Cassini, Galileo,and NewHorizons)mustalsohaveanonboardelectricalpowersourcefor chargingtheirbatteries,duetotheunavailabilityofeffectivesolarpower(thesunistoodimat suchlargedistances).

Exercises

1.UsingthefollowingexponentialatmospheremodelforEarthwiththescaleheight, H = 6.7 km,andbasedensity, ��0 = 1.752kg∕m3 ,calculatetheatmosphericdensityatthealtitude, z = 150km:

ComparetheresultwiththatgiveninTable1.1.

2.CalculatetheJuliandatefor3:30p.m.UTonOctober15,2007,referringtothe J 2000epoch.

3.Whatistheexacttimedifferencebetweentwoeventshappeningat11:05a.m.onJuly28, 1993,and8:31p.m.onNovember3,2005,respectively?

References

TewariA2006. AtmosphericandSpaceFlightDynamics.Birkhäuser,Boston. SeidelmannKP(ed.)1992. ExplanatorySupplementtotheAstronomicalAlmanac.UniversityScienceBooks,Sausalito, CA.

2 Dynamics

Dynamics isthestudyofanobjectinmotion,andpertainstoachangeinthepositionand orientationoftheobjectasafunctionoftime.Thischapterintroducesthebasicprinciplesof dynamics,whicharelaterappliedtothemotionofavehicleinthespace.

2.1NotationandBasics

Thevectorsandmatricesaredenotedthroughoutthisbookinboldface,whereasscalarquantitiesareindicatedinnormalfont.Theelementsofeachvectorarearrangedinacolumn.The Euclideannorm (or magnitude)ofathree-dimensionalvector, A =(Ax , Ay , Az )T ,isdenotedas follows:

Allthevariablesrepresentingthemotionofaspacecraftarechangingwith time, t .The overdotsrepresentthetimederivatives,e.g.,d

.Thetime derivativeofavector A,whichischangingbothinitsmagnitudeanditsdirection,requiresan explanation.

Thetimederivativeofavector, A,whichischangingbothinmagnitudeanddirectioncanbe resolvedintwomutuallyperpendiculardirections–onealongtheoriginaldirectionof A,and theothernormaltoitontheplaneoftherotationof A.Theinstantaneous angularvelocity, ��, of A denotesthevectorrateofchangeinthedirection,whereas A istherateofchangeinits magnitude.Bydefinition, �� × A isnormaltothedirectionoftheunitvector, A∕A,andliesin theinstantaneousplaneofrotationnormalto ��.Therotationof A isindicatedbythe right-hand rule,wherethethumbpointsalong ��,andthecurledfingersshowtheinstantaneousdirection

ofrotation,1 �� × A.Thetimederivativeof A isthereforeexpressedasfollows:

wheretheterm A∕A representsaunitvectorintheoriginaldirectionof A,and �� × A isthe changenormalto A causedbyitsrotation.Equation(2.2)willbereferredtoasthe chainrule ofvectordifferentiationinthisbook.

Similarly,thesecondtimederivativeof A isgivenbytheapplicationofthechainruleto differentiate A asfollows:

ApplyingEq.(2.1)tothetimederivativeoftheangularvelocity, ��,wehavethefollowing expressionforthe angularacceleration of A:

where �� istheinstantaneousangularvelocityatwhichthevector �� ischangingitsdirection. Hence,thesecondtimederivativeof A isexpressedasfollows:

Thebracketedtermontheright-handsideofEq.(2.5)isparallelto A,whilethesecondterm ontheright-handsideisperpendiculartoboth A and ��.Thelasttermontheright-handsideof Eq.(2.5)denotestheeffectofatime-varyingaxisofrotationof A.

2.2PlaneKinematics

Asaspecialcase,considerthemotionofapoint, P,inafixedplanedescribedbythe radius vector, r,whichischangingintime.Thevector r isdrawnfromafixedpoint, o,ontheplane, tothemovingpoint, P,andhencedenotestheinstantaneousradiusofthemovingpointfrom o. Theinstantaneousrotationofthevector r isdescribedbytheangularvelocity, �� = ��k,which isfixedinthedirectiongivenbytheunitvector k,normaltotheplaneofmotion.Thuswehave thefollowinginEq.(2.4): �� = 0 .

1 Areferenceframe, (i, j, k),consistingofthreemutuallyperpendicularaxes, i, j,and k,istermeda right-handedframe ifitsatisfiestheright-handruleofvectormultiplicationofthefirsttwoaxesinthepropersequence,toproducethethird axis: i × j = k

Thenet velocity ofthepoint, P,isdefinedtobethetimederivativeoftheradiusvector, r,which isexpressedasfollowsaccordingtothechainruleofvectordifferentiation:

andconsistsofthe radialvelocity component, r ∕r ,andthe circumferentialvelocity component, ∣ �� × r ∣.Similarly,whenthechainruleisappliedtothevelocity, v,theresultisthenet acceleration ofthemovingpoint, P,whichisdefinedtobethetimederivativeof v,orthesecond timederivativeof r.Inthisspecialcaseoftheradiusvector, r,alwayslyingonafixedplane, itsangularvelocityvector, ��,isalwaysperpendiculartothegivenplane(hencethedirection k = ��∕�� isconstant),butcanhaveatime-varyingmagnitude, ��.Hence,Eq.(2.4)yieldsthe followingexpressionforthetimederivativeof ��:

WhentheseresultsaresubstitutedintoEq.(2.3),thefollowingexpressionfortheacceleration ofthepoint, P,isobtained:

Thenetaccelerationofthepoint, P,paralleltotheinstantaneousradiusvector, r,isidentified fromEq.(2.8)tobethefollowing:

Thedirectionoftheterm �� ×(�� × r)=−��2 r isalwaystowardstheinstantaneouscentreof rotation(i.e.,along r∕r ).Theotherradialaccelerationterm, ̈ r r∕r ,iscausedbytheinstantaneouschangeintheradius, r ,andispositiveinthedirectionoftheincreasingradius(i.e.,away fromtheinstantaneouscentreofrotation).

Thecomponentofaccelerationalongthevector �� × r inEq.(2.8)isperpendiculartoboth r and ��,andisgivenby

Intermsofthe polarcoordinates, (r ,�� ),wehave �� = ̇ �� k;hencethemotionisresolvedin twomutuallyperpendiculardirections,(r∕r , i�� ),where i�� isaunitvectoralongthedirectionof increasing �� (calledthe circumferentialdirection),definedby

Thustherotatingframe, (r∕r , i�� , k),constitutesaright-handedtriad.Inthisrotatingcoordinate frame,themotionofthepoint, P,isrepresentedasfollows:

ItisclearfromEq.(2.13)thatintherotatingcoordinatesystem, (r∕r , i�� , k),theacceleration alongtheinstantaneousradiusvector, r,isgivenby ( ̈ r r �� 2 ) r r

andconsistsoftheaccelerationtowardstheinstantaneouscentreofrotation, r �� 2 r∕r ,aswellas thatawayfromtheinstantaneouscentre, ̈ r r∕r .Oftheaccelerationnormaltotheinstantaneous radiusvector r,theterm2 ̇ r ̇ �� i�� iscausedbyachangeoftheradiusintherotatingcoordinate frame, (r∕r , i�� , k),whereastheotherterm, r �� i�� ,isduetothevariationoftheangularvelocityof rotation, �� = �� ,inthesamerotatingframe.

Analternativerepresentationofthemotionofthepoint P isvia Cartesiancoordinates, (x, y), measuredinareferenceframewhoseaxesarefixedinspace.Letusconsider (i, j, k) assuch afixed,right-handedcoordinatesystemwith i × j = k,and (i, j) beingtheconstantplaneof rotation.Theradiusvectoranditstimederivativesinthefixedframearethengivenby

Ingeneral,atimevariationoftheradiusvector, r,givesrisetoaradialacceleration, r,which isresolvedinafixedcoordinateframe, (i, j, k),withoutresortingtoanyrotationalacceleration terms.Suchacoordinateframewhoseaxesarefixedinspaceistermedan inertialreference frame,andtheaccelerationmeasuredbysuchaframeistermedtheinertial(or“true”)acceleration.Theinertialacceleration, r,canbethoughtofasbeingdirectedtowards(orawayfrom) aninstantaneouscentreofrotation,whichitselfcouldbeamovingpoint.Forexample,apoint movingalonganarcofaconstantradius, r = √x2 + y2 ,ataconstantangularrate, ��,hasits accelerationdirectedtowardsthearc’scentre, r =−��2 r.

2.3Newton’sLaws

In1687Newtongavehisthreefamouslawsofmotion,whicharevalidforthemotionofall objects(unlesstheyaremovingatspeedscomparabletothespeedoflight).Statedbriefly,they arethefollowing:

(i)Anobjectcontinuestomoveinastraightlineataconstant velocity,unlessacteduponby a force appliedtoitbyanotherobject.

(ii)Thetimerateofchangeofthevelocity(calledthe acceleration)ofanobjectisdirectly proportionaltotheforceappliedtotheobject.Theconstantofproportionalityisaproperty oftheobject,calledthe mass

(iii)Ifanobject, A,appliesaforceonanotherobject, B,then B appliesaforceon A ofthesame magnitude,butoppositeindirectiontothatappliedby A.

Theconsequencesoftheselawsareprofound,astheygovernthemotionofallobjects(which aremovingquiteslowlywhencomparedtothespeedoflight2 ).Thefirstlawimpliesthatthere isno absoluteposition inspace,becausetwoobserversmovinginparallelstraightlinesata constantspeed, c,relativetooneanotherfindtwoeventsseparatedintimetotakeplaceat differentpositions.Suchobserverscanberegardedtobelocatedattheoriginsoftwodifferent referenceframes, oxyz and o′ x′ y′ z′ ,suchthattherelativemotiontakesplacealongtheparallel axes, ox,and o′ x′ .Thedistancestravelledbyamovingobjectduringthetime, t ,measuredin thetwoframesaredifferent,asgivenbythefollowing Galileantransformation:

Newton’slawsappliedtoamovingobjectareequallyvalidinthetworeferenceframes, oxyz and o′ x′ y′ z′ ;hencetheyarebothreferredtoas inertialreferenceframes.Anotherconsequence ofthefirstlawisthatitpostulatesanabsolutequantitycalledthe time,whichisthesameinall referenceframes,andisthereforeunaffectedbythemotion.Thesecondlawassignsaproperty calledthe mass toallmaterialobjects,whichcanbedeterminedbymeasuringtheforceapplied totheobjectandthecorrespondingaccelerationexperiencedbyit,whilethethirdlawdefines theforceappliedbytwoisolatedobjectsuponeachother.

2.4ParticleDynamics

A particle isdefinedtohaveafinitemassbutinfinitesimaldimensions,andisthereforeregarded tobeapointmass.Sinceaparticlehasnegligibledimensions,itspositioninspaceiscompletely determinedbythe radius vector, r,measuredfromafixedpoint, o,atanyinstantof time, t Thecomponentsof r areresolvedinaright-handed referenceframe withoriginat o,having threemutuallyperpendicularaxesdenotedbytheunitvectors i, j,and k,andaredefinedtobe theCartesianpositioncoordinates, (x, y, z),oftheparticlealongtherespectiveaxes, (i, j, k),as showninFig.2.1.The velocity, v,oftheparticleisdefinedtobethetimederivativeoftheradius vector,andgivenby

Ifthereferenceframe, (i, j, k),usedtomeasurethevelocityoftheparticleisatrest,thenthe componentsofthevelocity, v,resolvedalongtheaxesoftheframe, i, j,and k,aresimplythe timederivativesofthepositioncoordinates, x,̇y,and z,respectively.However,iftheorigin, o,of thereferenceframeitselfismovingwithavelocity, vo ,andtheframe, (i, j, k),isrotatingwithan

2 Thiscaveatmustbeaddedbecause,aspostulatedbyEinsteinin1905,Newton’slawsceasetoapplytoobjectstravelling closetothespeedoflight,whichisthesameinallreferenceframes.

Figure2.1 Thepositionvector, r,ofaparticleresolvedinaninertialreferenceframeusingCartesian coordinates,(x, y, z).

angularvelocity, ��,withrespecttoan inertialframe3 ,thenthevelocityofthemovingreference framemustbevectoriallyaddedtothatoftheparticleinordertoderivethenetvelocityofthe particleinthestationaryframeasfollows:

Thelasttermontheright-handsideofEq.(2.19)isthechangecausedbyrotatingaxes, (i, j, k), eachofwhichhavethesameangularvelocity, ��.Therelationshipbetweenthepositionand velocitydescribedbyavectordifferentialequation,eitherEq.(2.18)orEq.(2.19),istermedthe kinematics oftheparticle.

Sincethevelocityoftheparticlecouldbevaryingwithtime,the acceleration, a,oftheparticle isdefinedtobethetimederivativeofthevelocityvector,andisgivenby

withtheunderstandingthatthederivativesaretakenwithrespecttoastationaryreferenceframe. Ifthereferenceframeinwhichthepositionandvelocityoftheparticleareresolvedisitselfmovingsuchthatitsorigin, o,hasaninstantaneousvelocity, vo andaninstantaneousacceleration,

3 Rotationoftheframe, (i, j, k),referstotherotationofitsaxes.Since i, j,and k areunitvectors,theycannotchange inmagnitude.Hencethetimederivativeofeachaxisisjustduetoitsrotationbyanangularvelocity, ��.Thisangular velocityiscommontoallaxesbecausetheymustalwaysbemutuallyperpendicularandrighthanded.Therefore,we have

ao = dvo ∕dt ,anditsaxesarerotatingwithaninstantaneousangularvelocity, ��,allmeasuredin astationaryframe,thenthenetaccelerationoftheparticleisgivenby

Equation(2.21)isanalternativekinematicaldescriptionoftheparticle’smotion,andcanbe regardedasbeingequivalenttothatgivenbyEq.(2.19),whichhasbeendifferentiatedintime accordingtothechainrule.Equation(2.21)isusefulinfindingtheaccelerationoftheparticle fromthepositionandvelocitymeasuredinamovingreferenceframe.Thefirsttwotermsonthe right-handsideofEq.(2.21)representthenetaccelerationduetotheoriginofthemovingframe. Theterm2 ̇ r (�� × r∕r ) isthe Coriolisacceleration,and �� ×(�� × r) the centripetalacceleration oftheparticleinthemovingreferenceframe.Theterm �� × r istheeffectofthe angularacceleration ofthereferenceframe,whereas r (r∕r ) istheaccelerationduetoachangingmagnitude of r,andwouldbetheonlyaccelerationhadthereferenceframebeenstationary.

TheapplicationofNewton’ssecondlawtothemotionofaparticleofafixedmass, m,and acteduponbya force, f ,givesthefollowingimportantrelationship–calledthe kinetics –for thedeterminationoftheparticle’sacceleration:

The linearmomentum, p,oftheparticleisdefinedastheproductofitsmass, m,andvelocity, v:

Sincetheparticle’smassisconstant,thesecondlawofmotiongivenbyEq.(2.22)canalternativelybeexpressedasfollows:

whichgivesrisetotheprincipleoflinearmomentumconservationifnoforceisappliedtothe particle.

The angularmomentum, H,oftheparticleaboutapoint, o,isdefinedtobethevectorproduct oftheradiusvector, r,oftheparticlefrom o anditslinearmomentum, p:

ByvirtueofEq.(2.24),itisevidentthattheangularmomentumoftheparticleabout o canvary withtime,ifandonlyifa torque,definedby M = r × f ,actsontheparticleabout o:

Thisresultsintheprincipleofangularmomentumconservationifnotorqueactsontheparticle about o

The work doneonaparticlebyaforcewhilemovingfrompoint A topoint B isdefinedby thefollowingintegralofthescalarproductoftheforce, f ,andtheparticle’sdisplacement,dr:

TheapplicationofNewton’ssecondlawfortheconstantmassparticle,Eq.(2.22),resultsinthe followingexpressionfortheworkdone:

where vA and vB arethespeedsoftheparticleatthepoints A and B,respectively.Thusthenet workdoneonaparticleequalsthenetchangeinits kineticenergy,1∕2mv2

2.5The n-BodyProblem

Gravity,beingthepredominantforceinspaceflight,mustbeunderstoodbeforeconstructing anymodelforspaceflightdynamics.Considertwoparticlesofmasses, m1 and m2 ,whose instantaneouspositionsinaninertialframe, OXYZ,aredenotedbythevectors, R1 (t ) and R2 (t ), respectively.Therelativepositionofmass, m2 ,withrespecttothemass, m1 ,isgivenbythevector r21 = R2 R1 .ByNewton’slawofgravitation,thetwoparticlesapplyanequalandopposite attractiveforceoneachother,whichisdirectlyproportionaltotheproductofthetwomasses, andinverselyproportionaltothedistance, r21 =∣ R2 R1 ∣,betweenthem.Theequationsof motionofthetwoparticlesareexpressedasfollowsbyNewton’ssecondlawofmotion:

G beingthe universalgravitationalconstant.Addingthetwoequationsofmotionyieldsthe importantresultthatthe centreofmass ofthetwoparticlesisnon-accelerating:

isthepositionofthecentreofmass.Thisapproachcanbeextendedtoasystemof n particles, wherethe ith particlehasthefollowingequationofmotion: