Keplerian solutions

236 Pages·2008·6.044 MB·English

by  Guochang Xu

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Liping,Jia,Yuxi,PanDandYan
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Preface
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Thepurposeofthisreferenceandhandbookistodescribea ndtoderivetheanalytic
solutionsoftheequationsofsatellitemotionperturbedbyDextraterrestrialandgeopo-
tentialdisturbancesofthesecondorder.Theequationsofsatellitemotionperturbed
by extraterrestrial disturbances are solved by means of discretization and approx-
E
imated potential function as well as Gaussian equations. The equations perturbed
bygeopotentialdisturbancesaresolvedbysymbolicmathematicaloperations.The
traditionalproblemofsingularityinthesolutionsTissolvedbyso-calledsingularity-
freeorbittheory.Simplifieddisturbedequationsofmotionareproposedtosimplify
the solutions. Applications of the theory forCanalytic orbit determination are also
discussed.Indeed,thisisthefirstbooksincethesatelliteera,whichdescribessys-
tematicallytheorbittheorywithanalyticalsolutions,withrespecttoallofextrater-
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restrialandgeopotentialdisturbancesofthesecondorder,andthesolutionsarefree
ofsingularity.Basedonsuchatheory,thealgorithmsoforbitdeterminationcanbe
renewed;deeperinsightintothephysicRsofdisturbancesbecomespossible;theway
toavarietyofnewapplicationsandrefinementsisopened.
My primary knowledge of the orbit theory came from my education of mathe-
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maticswhilestudyingphysicsandtheoreticalmechanics(1981).Myfirstpractical
experience with orbit came from the research activity at the Technical University
(TU) Berlin on orbit correctioOns of the satellite altimetry data (1988–1992). The
extensiveexperienceonorbitcamefromtheGPS/Galileosoftwaredevelopmentfor
orbitdeterminationandgeopotentialmappingattheGFZ(2001–2004). Thetradi-
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tional adjustment model of the solar radiation used in numerical orbit determina-
tionisinvestigatedandconsiderednotreasonablephysically;andanewadjustment
modelisproposedintNheusermanualoftheMulti-FunctionalGPS/Galileosoftware
(MFGsoft)(Xu,2004),whichisalsoreportedinthe2ndeditionofthebookGPS–
Theory, Algorithms and Applications (Xu, 2007). Indeed, one of the ways to ob-
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tainthesolutionsoftheextraterrestrialdisturbancesofthesatellitemotionisfound
during that investigation. However, it has not been realised until two scientists,
Dr.XiaochunLuandDr.XiaohuiLioftheNationalTimeServiceCenter(NTSC)
inXi’an,cametovisitandtocooperatewithmeatGFZ.Wediscussedthevirtual
navigationsystemandtriedtosolvethestabilityproblemofthe3-Dpositioningof
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tem.Byconsideringwhatissignificantintheoryand,whatismoreimportantthan
ournumericalstudy,theideaofsolvingthedisturbedequationsofmotionwasob-
tained,andthesolutionsoftheextraterrestrialdisturbancesoftheequationofsFatel-
litemotionwerefound.Becauseoftheimportanceofthegeopotentialdisturbances,
greateffortswerethenmadetoderivetherelatedsolutions.Thereafter,altOernative
solutions of the extraterrestrial disturbances were found by using different means
(besidesthediscretization,alsoapproximatedpotentialfunctionandGaussiandis-
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turbedequations).Tosimplifythesolutions,thesimplifieddisturbedequationswere
proposed. Tosolvetheproblem ofsingularity,thesingularity-freetheorywasalso
developed.
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Afterpublishingmybook,GPS–Theory,AlgorithmsandApplications,in2003,
Ididnotwanttoeverwriteanotherscientificbookbecausethisprocesstookmore
thantwoyearsextremehardwork.However,ImustfinishthisPbookbecausesome
of the scientists have contributed their lifetime to the theoretical solutions of the
geopotential disturbances of the equation of satellite mot ion and now the results
are here. The solutions of the extraterrestrial disturbancDes of the orbit motion are
of extreme importance for practice, but they are rarely investigated because they
arehighlycomplex.Fromthetheory,aspecialconfusionrelatedtothesolarradia-
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tionfromthepurenumericalorbitdeterminationhasbeencleared.Manyinteresting
applicationswillfollowsoon.Tomaketheprocessofwritingeasy,asmallportion
ofthebasiccontentsofmyGPSbookispartlymoTdifiedandimportedorrearranged
andused.
The book includes ten chapters. After a bCrief introduction, the coordinate and
timesystemsaredescribedinthesecondchapter.Thethirdchapterisdedicatedto
the Keplerian satellite orbits – the orbits of the satellite under the attraction of the
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centralforceoftheEarth.
Thefourthchapter dealswithperturbationsoftheorbit.Perturbedequations of
satellitemotionarederived.PerturbatioRnforcesofthesatellitemotionarediscussed
indetail,includingtheperturbationsoftheEarth’sgravitationalfield,Earthtideand
oceantide,thesun,themoonandplanets,solarradiationpressure,andatmospheric
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drag,aswellascoordinateperturbation.
The fifth chapter covers the analytic solution ofC¯ perturbation, including the
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completeformulasofthelongOterm,andlongandshortperiodicterms.Thederiva-
tionalsogivesthealgorithmandmodeloftheorbitcorrection.Thesolutionsofother
geopotentialdisturbancesofhigherorderanddegreearedescribedinthesixthchap-
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ter.Asexamples,solutionsofdisturbancesofC ,D andD aregiven.General
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solutionsofdisturbanceofD arederived.Symbolicoperationsoftwareforderiv-
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ingsolutionsofgeopoNtentialdisturbancesofanyorderanddegreesaredesignedand
used.
The seventh chapter covers the solutions of extraterrestrial disturbances such
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as solar radiation pressure, atmospheric drag and the disturbances of the sun, the
moonandplanets.Theprincipleandstrategythatleadtothesolutionaredescribed.
Thesolutionsarederivedviadiscretizationandapproximatedpotentialfunctionas
wellasGaussianperturbedequationsofmotion.Simplifieddisturbedequationsare
Preface ix
proposedandusedpartly.Theephemerisofthesun,themoonandplanetsaregiven
forpracticaluse.
The eighth chapter is dedicated to numerical orbit determination, includinFg its
principle,thealgebraicsolutionsofthevariationequations,andthenumericalinte-
grationandinterpolationalgorithms,aswellastherelatedderivatives. O
Theninthchapterdescribestheprincipleofanalyticalorbitdeterminationbased
ontheproposednewsolutions.Realtimeabilityandpropertiesoftheanalyticorbit
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solutionsarediscussed.
The final chapter includes algorithms that lead to singularity-free orbit theory
andtheequationsofmotioninnon-inertialframeaswellasdiscussionsconcerning
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thefurtherdevelopmentoftheorbittheoryanditsapplicationsaswellascomments
onsomeremainingproblems.
ThebookhasbeensubjectedtoanindividualreviewofchapPtersandsectionsand
ageneralreview.IamgratefultoreviewersProf.MarkusRothacherofGFZ,Prof.
DieterLelgemannofTUBerlin,Prof.YuanxiYangoftheIn stituteofSurveyingand
Mapping(ISM)inXi’an,Dr.JianfengGuoofInformatioDnEngineeringUniversity
(IEU)inZhengzhou,Prof.XuhaiYangofNTSCinXi’an,Dr.JunpingChenofGFZ.
AgrammaticalcheckoftechnicalEnglishwritinghasbeenperformedbySpringer
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Heidelberg.
IwishtosincerelythankProf.MarkusRothacherforhissupportandtrustduring
myresearchactivitiesatGFZ.Dr.Ju¨rgenKuscheTisthankedforhisencouragement
and help. Dr. Christoph Reigber is thanked for granting me special freedom of re-
search. My grateful thanks go to Dr. XiaochCun Lu and Dr. Xiaohui Li of NTSC
in Xi’an. Their visit to and cooperation at the GFZ have led to the derivations of
thekeycontentsofthisbook.Dr.JiangfengGuoofIEUinZhengzhou followeda
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partofmyderivationandcheckedforthecorrectness.VolkerGrundofGFZhelped
me greatly by assisting in the application of software tools, which is another key
tothesolutionofgeopotentialdisturbaRnces.QianxinWangofGFZhelpedtocheck
a part of the formula typing. Dr. Jinghui Liu of the educational department of the
ChineseEmbassyinBerlin,Prof.YuanxiYangofISMinXi’an,Prof.HepingSun
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of the Institute of Geodesy and Geophysics (IGG) in Wuhan and Prof. Qin Zhang
of ChangAn University in Xi’an are thanked for their friendly support during my
scientificactivitiesinChina.TOheChineseAcademyofSciencesisthankedforthe
OutstandingOverseasChineseScholarsFund,whichsupportedgreatlymanyvalu-
ablescientificactivitiesevenoutsideChina.
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Duringthiswork,manyvaluablediscussionshavebeenheldwithmanyscientists
and friends.My special thanks go to Dr.Luisa Bastos of the Astronomical Obser-
vatory of University PNorto, Dr. Rene Forsberg of Danish National Space Center,
Prof. Jo¨rg Reinking of Oldenburg University of Applied Sciences, Prof. Jikun Ou
andProf.YunbinYuanofIGGinWuhan,Prof.WuChenofHongKongPolytech-
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nic University, Prof. Yunzhong Shen of Tongji University in Shanghai, Dr. Yanx-
iong Liu of the First Oceanic Institute in Qingdao, Prof. Jiancheng Li of Wuhan
University, Prof. Ta-Kang Yeh of the ChingYun University of Taiwan, Dr. Ju¨rgen
Neumeyer,Dr.FranzBarthelmes,andDr.SvetozarPetrovicofGFZ,Dr.UweMeyer
ofGeoZentrumHannover,Dr.LudgerTimmenofUniversityHannover,Dr.Xiong
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Li of Hugro Inc. Houston, Dr. Daniela Morujao of Lisbon University, Prof. Klaus
HehlofTechnicalUniversityofAppliedSciencesBerlin,etc.
IalsowishtosincerelythankAngelikaSvarovskyandHartmutPflugofGFFZfor
their kind help. I am also grateful to Dr. Chris Bendall of Springer Heidelberg for
hisvaluableadvice. O
MywifeLiping,sonJiaanddaughtersYuxi,PanandYanarethankedfortheir
constantsupportandunderstanding,aswellasfortheirhelp.
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October2007 GuochangXu
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Contents
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1 Introduction.................................. ................. 1
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2 CoordinateandTimeSystems ................................... 5
2.1 GeocentricEarth-FixedCoordinateSystems .................... 5
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2.2 CoordinateSystemTransformations........................... 8
2.3 LocalCoordinateSystem .................................... 9
2.4 Earth-CentredInertialCoordinateSystTem ...................... 11
2.5 IAU2000Framework....................................... 15
2.6 GeocentricEclipticInertialCoordinCateSystem.................. 19
2.7 SatelliteFixedCoordinateSystem............................. 19
2.8 TimeSystems.............................................. 22
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3 KeplerianOrbits............................................... 25
3.1 KeplerianMotion ........R.................................. 25
3.2 SatelliteMotionintheOrbitalPlane........................... 28
3.3 KeplerianEquation ......................................... 30
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3.4 StateVectoroftheSatellite .................................. 33
4 PerturbationsontheOrbits ..................................... 37
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4.1 PerturbedEquationofSatelliteMotion......................... 37
4.1.1 LagrangianPerturbedEquationofSatelliteMotion ........ 38
4.1.2 GaussiaCnPerturbedEquationofSatelliteMotion.......... 40
4.2 PerturbationForcesofSatelliteMotion ........................ 44
4.2.1 PerturbationoftheEarth’sGravitationalField ............ 44
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4.2.2 PerturbationoftheSunandtheMoonaswellasPlanets.... 48
4.2.3 EarthTideandOceanTidePerturbations ................ 49
4.2.4 USolarRadiationPressure .............................. 53
4.2.5 AtmosphericDrag ................................... 57
4.2.6 AdditionalPerturbations .............................. 60
4.2.7 OrderEstimationsofPerturbations ..................... 62
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5 SolutionsofC Perturbation.................................... 63
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5.1 C PerturbedEquationsofMotion............................ 63
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5.2 SolutionsofC PerturbedOrbit.............................F. 65
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5.3 PropertiesoftheSolutionsofC Perturbations ................. 78
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5.4 OrbitCorrection ........................................O... 79
6 SolutionsofGeopotentialPerturbations .......................... 83
6.1 PrincipleoftheDerivations ...........................O....... 83
6.2 SolutionsofC Perturbation................................. 87
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6.3 SolutionsofD Perturbations................................110
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6.4 SolutionsofD Perturbations................................125
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6.5 PropertiesoftheSolutionsofGeopotentialPerturbations .........139
6.6 SolutionsofGeopotentialPerturbationsof6×6OrPderandDegrees140
6.7 SolutionsofGeopotentialPerturbationsofl×mOrderandDegrees141

7 SolutionsofExtraterrestrialDisturbances ........................143
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7.1 KeyNotestotheProblems...................................143
7.2 SolutionsofDisturbanceofSolarRadiationPressure.............144
7.2.1 SolutionsviaGaussianPerturbedEEquations..............146
7.3 SolutionsofDisturbanceofAtmosphericDrag ..................156
7.3.1 SolutionsviaGaussianPerturbedEquations..............157
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7.4 SolutionsofDisturbanceoftheSun ...........................159
7.4.1 SolutionsviaGaussianPerturbedEquations..............162
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7.5 SolutionsofDisturbanceoftheMoon .........................162
7.6 SolutionsofDisturbanceofPlanets............................164
7.7 Summary ..................E...............................165
7.8 EphemerisoftheMoon,theSunandPlanets....................165
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8 NumericalOrbitDetermination .................................171
8.1 PrincipleofGPSPreciseOrbitDetermination...................171
8.1.1 AlgebraicSolutiRonoftheVariationEquation .............173
8.2 NumericalIntegrationandInterpolationAlgorithms..............174
8.2.1 Runge-KuttaAlgorithms ..............................175
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8.2.2 AdamsAlgorithms...................................179
8.2.3 CowellAlgorithms...................................182
8.2.4 MixedAClgorithmsandDiscussions .....................183
8.2.5 InterpolationAlgorithms ..............................184
8.3 Orbit-RelatedPartialDerivatives..............................186
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9 AnalyticOrbitDetermination ...................................197
9.1 PrincipleofAnalyticOrbitDetermination ......................197
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9.2 RealTimeAbility ..........................................199
9.3 PropertiesofAnalyticOrbitDetermination .....................200
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10 Singularity-FreeTheoryandDiscussions .........................203
10.1 Singularity-FreeOrbitTheory ................................203
10.1.1 ProblemofSingularityoftheSolutions.................F.203
10.1.2 DisturbedEquationsintheCaseofCircularOrbit .........204
10.1.3 DisturbedEquationsintheCaseofEquatorialOrbit ....O...204
10.1.4 Disturbed Equations in the Case of Circular
andEquatorialOrbit..................................205
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10.1.5 Singularity-FreeDisturbedEquationsofmotion ..........206
10.1.6 SimplifiedSingularity-FreeDisturbedEquationsofmotion .206
10.2 EquationsofMotioninNon-InertialFrame.....................207
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10.3 Discussions ...............................................208
Appendix1 P
IAU1980TheoryofNutation....................................211

References.........................................................215
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