ORBITAL MECHANICSWord下载.docx
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ORBITAL MECHANICSWord下载.docx
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∙MotionsofPlanetsandSatellites
∙LaunchofaSpaceVehicle
∙PositioninanEllipticalOrbit
∙OrbitPerturbations
∙OrbitManeuvers
∙EscapeVelocity
Orbitalmechanics,alsocalledflightmechanics,isthestudyofthemotionsofartificialsatellitesandspacevehiclesmovingundertheinfluenceofforcessuchasgravity,atmosphericdrag,thrust,etc.Orbitalmechanicsisamodernoffshootofcelestialmechanicswhichisthestudyofthemotionsofnaturalcelestialbodiessuchasthemoonandplanets.Therootoforbitalmechanicscanbetracedbacktothe17thcenturywhenmathematicianIsaacNewton(1642-1727)putforwardhislawsofmotionandformulatedhislawofuniversalgravitation.Theengineeringapplicationsoforbitalmechanicsincludeascenttrajectories,reentryandlanding,rendezvouscomputations,andlunarandinterplanetarytrajectories.
OrbitalElements
Tomathematicallydescribeanorbitonemustdefinesixquantities,calledorbitalelements.Theyare
∙Semi-MajorAxis,a
∙Eccentricity,e
∙Inclination,i
∙ArgumentofPeriapsis,
∙TimeofPeriapsisPassage,T
∙LongitudeofAscendingNode,
Anorbitingsatellitefollowsanovalshapedpathknownasanellipsewiththebodybeingorbited,calledtheprimary,locatedatoneoftwopointscalledfoci.Anellipseisdefinedtobeacurvewiththefollowingproperty:
foreachpointonanellipse,thesumofitsdistancesfromtwofixedpoints,calledfoci,isconstant(seefiguretoright).Thelongestandshortestlinesthatcanbedrawnthroughthecenterofanellipsearecalledthemajoraxisandminoraxis,respectively.Thesemi-majoraxisisone-halfofthemajoraxisandrepresentsasatellite'
smeandistancefromitsprimary.Eccentricityisthedistancebetweenthefocidividedbythelengthofthemajoraxisandisanumberbetweenzeroandone.Aneccentricityofzeroindicatesacircle.
Inclinationistheangulardistancebetweenasatellite'
sorbitalplaneandtheequatorofitsprimary(ortheeclipticplaneinthecaseofheliocentric,orsuncentered,orbits).Aninclinationofzerodegreesindicatesanorbitabouttheprimary'
sequatorinthesamedirectionastheprimary'
srotation,adirectioncalledprograde(ordirect).Aninclinationof90degreesindicatesapolarorbit.Aninclinationof180degreesindicatesaretrogradeequatorialorbit.Aretrogradeorbitisoneinwhichasatellitemovesinadirectionoppositetotherotationofitsprimary.
Periapsisisthepointinanorbitclosesttotheprimary.Theoppositeofperiapsis,thefarthestpointinanorbit,iscalledapoapsis.Periapsisandapoapsisareusuallymodifiedtoapplytothebodybeingorbited,suchasperihelionandaphelionfortheSun,perigeeandapogeeforEarth,perijoveandapojoveforJupiter,periluneandapolunefortheMoon,etc.Theargumentofperiapsisistheangulardistancebetweentheascendingnodeandthepointofperiapsis(seefigurebelow).Thetimeofperiapsispassageisthetimeinwhichasatellitemovesthroughitspointofperiapsis.
Nodesarethepointswhereanorbitcrossesaplane,suchasasatellitecrossingtheEarth'
sequatorialplane.Ifthesatellitecrossestheplanegoingfromsouthtonorth,thenodeistheascendingnode;
ifmovingfromnorthtosouth,itisthedescendingnode.Thelongitudeoftheascendingnodeisthenode'
scelestiallongitude.CelestiallongitudeisanalogoustolongitudeonEarthandismeasuredindegreescounter-clockwisefromzerowithzerolongitudebeinginthedirectionofthevernalequinox.
Ingeneral,threeobservationsofanobjectinorbitarerequiredtocalculatethesixorbitalelements.Twootherquantitiesoftenusedtodescribeorbitsareperiodandtrueanomaly.Period,P,isthelengthoftimerequiredforasatellitetocompleteoneorbit.Trueanomaly,v,istheangulardistanceofapointinanorbitpastthepointofperiapsis,measuredindegrees.
TypesOfOrbits
ForaspacecrafttoachieveEarthorbit,itmustbelaunchedtoanelevationabovetheEarth'
satmosphereandacceleratedtoorbitalvelocity.Themostenergyefficientorbit,thatisonethatrequirestheleastamountofpropellant,isadirectlowinclinationorbit.Toachievesuchanorbit,aspacecraftislaunchedinaneastwarddirectionfromasiteneartheEarth'
sequator.TheadvantagebeingthattherotationalspeedoftheEarthcontributestothespacecraft'
sfinalorbitalspeed.AttheUnitedStates'
launchsiteinCapeCanaveral(28.5degreesnorthlatitude)adueeastlaunchresultsina"
freeride"
of1,471km/h(914mph).Launchingaspacecraftinadirectionotherthaneast,orfromasitefarfromtheequator,resultsinanorbitofhigherinclination.HighinclinationorbitsarelessabletotakeadvantageoftheinitialspeedprovidedbytheEarth'
srotation,thusthelaunchvehiclemustprovideagreaterpart,orall,oftheenergyrequiredtoattainorbitalvelocity.Althoughhighinclinationorbitsarelessenergyefficient,theydohaveadvantagesoverequatorialorbitsforcertainapplications.Belowwedescribeseveraltypesoforbitsandtheadvantagesofeach:
Geosynchronousorbits(GEO)arecircularorbitsaroundtheEarthhavingaperiodof24hours.Ageosynchronousorbitwithaninclinationofzerodegreesiscalledageostationaryorbit.AspacecraftinageostationaryorbitappearstohangmotionlessaboveonepositionontheEarth'
sequator.Forthisreason,theyareidealforsometypesofcommunicationandmeteorologicalsatellites.Aspacecraftinaninclinedgeosynchronousorbitwillappeartofollowaregularfigure-8patternintheskyonceeveryorbit.Toattaingeosynchronousorbit,aspacecraftisfirstlaunchedintoanellipticalorbitwithanapogeeof35,786km(22,236miles)calledageosynchronoustransferorbit(GTO).Theorbitisthencircularizedbyfiringthespacecraft'
sengineatapogee.
Polarorbits(PO)areorbitswithaninclinationof90degrees.Polarorbitsareusefulforsatellitesthatcarryoutmappingand/orsurveillanceoperationsbecauseastheplanetrotatesthespacecrafthasaccesstovirtuallyeverypointontheplanet'
ssurface.
Walkingorbits:
Anorbitingsatelliteissubjectedtoagreatmanygravitationalinfluences.First,planetsarenotperfectlysphericalandtheyhaveslightlyunevenmassdistribution.Thesefluctuationshaveaneffectonaspacecraft'
strajectory.Also,thesun,moon,andplanetscontributeagravitationalinfluenceonanorbitingsatellite.Withproperplanningitispossibletodesignanorbitwhichtakesadvantageoftheseinfluencestoinduceaprecessioninthesatellite'
sorbitalplane.Theresultingorbitiscalledawalkingorbit,orprecessingorbit.
Sunsynchronousorbits(SSO)arewalkingorbitswhoseorbitalplaneprecesseswiththesameperiodastheplanet'
ssolarorbitperiod.Insuchanorbit,asatellitecrossesperiapsisataboutthesamelocaltimeeveryorbit.Thisisusefulifasatelliteiscarryinginstrumentswhichdependonacertainangleofsolarilluminationontheplanet'
ssurface.Inordertomaintainanexactsynchronoustiming,itmaybenecessarytoconductoccasionalpropulsivemaneuverstoadjusttheorbit.
MolniyaorbitsarehighlyeccentricEarthorbitswithperiodsofapproximately12hours(2revolutionsperday).Theorbitalinclinationischosensotherateofchangeofperigeeiszero,thusbothapogeeandperigeecanbemaintainedoverfixedlatitudes.Thisconditionoccursatinclinationsof63.4degreesand116.6degrees.Fortheseorbitstheargumentofperigeeistypicallyplacedinthesouthernhemisphere,sothesatelliteremainsabovethenorthernhemispherenearapogeeforapproximately11hoursperorbit.Thisorientationcanprovidegoodgroundcoverageathighnorthernlatitudes.
Hohmanntransferorbitsareinterplanetarytrajectorieswhoseadvantageisthattheyconsumetheleastpossibleamountofpropellant.AHohmanntransferorbittoanouterplanet,suchasMars,isachievedbylaunchingaspacecraftandacceleratingitinthedirectionofEarth'
srevolutionaroundthesununtilitbreaksfreeoftheEarth'
sgravityandreachesavelocitywhichplacesitinasunorbitwithanaphelionequaltotheorbitoftheouterplanet.Uponreachingitsdestination,thespacecraftmustdeceleratesothattheplanet'
sgravitycancaptureitintoaplanetaryorbit.
Tosendaspacecrafttoaninnerplanet,suchasVenus,thespacecraftislaunchedandacceleratedinthedirectionoppositeofEarth'
srevolutionaroundthesun(i.e.decelerated)untilitachievesasunorbitwithaperihelionequaltotheorbitoftheinnerplanet.ItshouldbenotedthatthespacecraftcontinuestomoveinthesamedirectionasEarth,onlymoreslowly.
Toreachaplanetrequiresthatthespacecraftbeinsertedintoaninterplanetarytrajectoryatthecorrecttimesothatthespacecraftarrivesattheplanet'
sorbitwhentheplanetwillbeatthepointwherethespacecraftwillinterceptit.Thistaskiscomparabletoaquarterback"
leading"
hisreceiversothatthefootballandreceiverarriveatthesamepointatthesametime.Theintervaloftimeinwhichaspacecraftmustbelaunchedinordertocompleteitsmissioniscalledalaunchwindow.
Newton'
sLawsofMotionandUniversalGravitation
slawsofmotiondescribetherelationshipbetweenthemotionofaparticleandtheforcesactingonit.
Thefirstlawstatesthatifnoforcesareacting,abodyatrestwillremainatrest,andabodyinmotionwillremaininmotioninastraightline.Thus,ifnoforcesareacting,thevelocity(bothmagnitudeanddirection)will
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