Satellite Orbits: Models, Methods and Applications

381 Pages·2000·4.347 MB·English

by  Dr. Oliver Montenbruck, Dr. Eberhard Gill (auth.)

Astrodynamical Constants
Quantity Value References and Remarks
Time
MJD(J2000) 51544.5 IAU 1976 (Seidelrnann 1992)
fl—TA I 32. 184 s MU 1991 (Seidelmann 1992)
GPS—TAI —19s Ilofmann-Wellenhofet at. (1997)
Universal
C 299 792 458 rn/s IAU 1976 (Seidelmann 1992)
0 6.67310—20 km3/(kg Cohen & Taylor 1987
Earth
U Me 398 600.44l5km3/s2 .IGM-3
J2 0.00108263 JGM-3
Re 6378. 137 km WGS-84 (NIMA 1997)
f
1/298.257223563 WGS-84 (NIMA 1997)
WED 0.72921 Moritz 1980
Sun
GM® 1.3271244001810" km3/s2 DE405 (Standish 1998)
AU 149597 870.691 km 1DE405 (Standish 1998)
km Seidelrnann 1992
P® 4.56010—6 N/rn2 IERS 1996 (McCarthy 1996)
Moon
0MM 4902.80 ikm3/s2 DE4QS (Standish 1998)
aM 384 400 km Seidelrnann 1992
1738 km Seidelrnann 1992
Satellites
TOEC 42164km 23h56m04s orbital period
VOEC 3.075 km/s
Tops 26561 km 1h58ni02s orbital period
vcps 3.874 km/s
TLEO 6678.. .7878km 300... 1500 km altitude
7.726.7.113km/s
VLEO
Underlined numbers indicate a rounding of the original value to the given number of digits. DE405
constants refer to the TDB time system.
Montenbruck Gill
Satellite Orbits
Oliver Montenbruck Eberhard Gill
•
Satellite Orbits
Models, Methods and Applications
With 97 Figures
Including 10 Color Figures,
47 Tables, and a CD-ROM
Springer
Dr. Oliver Montenbruck
Dr. Eberhard Gill
Deutsches Zentrum für Luft- und Raumfahrt (DLR) e.V.
Oberpfaffenhofen
Postfach 1116
82230 WeBling, Germany
Email: [email protected]
Cover Figure: Designed for a mission time of two years; on duty for eight years. Built by Dornier
Satellitensysteme GmbH, the German X-ray satellite Rosat was an ongoing success story. ©DSS
Library of Congress Control Number: 00038815
Corrected 3rd Printing 2005
1St Edition 2000
Additional material to this book can be downloaded from http://extras.springer.com.
ISBN 978-3-642-63547-2 ISBN 978-3-642-58351-3 (eBook)
DOl 10.1007/978-3-642-58351-3
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© Springer-Verlag Berlin Heidelberg 2000
Originally published by Springer-Verlag Berlin Heidelberg 2000
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Preface
SatelliteOrbits–Models,Methods,andApplicationshasbeenwrittenasacompre-
hensivetextbookthatguidesthereaderthroughthetheoryandpracticeofsatellite
orbit prediction and determination. Starting from the basic principles of orbital
mechanics,itcoverselaborateforcemodelsaswellasprecisemethodsofsatellite
tracking and their mathematical treatment. A multitude of numerical algorithms
used in present-day satellite trajectory computation is described in detail, with
properfocusonnumericalintegrationandparameterestimation.Thewiderangeof
levelsprovidedrendersthebooksuitableforanadvancedundergraduateorgradu-
atecourseonspaceflightmechanics,uptoaprofessionalreference innavigation,
geodesy and space science. Furthermore, we hope that it is considered useful by
theincreasingnumberofsatelliteengineersandoperatorstryingtoobtainadeeper
understandingofflightdynamics.
The idea for thisbook emerged when we realized thatdocumentationon the
methods,modelsandtoolsoforbitdeterminationwaseitherspreadovernumerous
technicalandscientificpublications,orhiddeninsoftwaredescriptionsthatarenot,
ingeneral,accessibletoawidercommunity.Havingworkedformanyyearsinthe
fieldofspaceflightdynamicsandsatelliteoperations,wetriedtokeepinclosetouch
withquestionsandproblemsthatariseduringdailywork,andtostressthepractical
aspectsoforbitdetermination.Nevertheless,ourinterestintheunderlyingphysics
motivatedustopresenttopicsfromfirstprinciples,andmakethebookmuchmore
thanjustacookbookonspacecrafttrajectorycomputation.
Withtheavailabilityofpowerfulongroundandonboardcomputers,aswellas
increasingdemandsforprecision,theneedforanalyticalperturbationtheorieshas
almostbeenreplacedbyapurelynumericaltreatmentoftheequationsofmotion.
Wethereforefocusonmodelsandmethodsthatcanbeappliedwithinanumerical
reconstructionofthesatelliteorbitanditsforecast.Asaconsequence,topicslike
orbitdesign,long-termorbitevolutionandorbitaldecayarenotaddressedspecifi-
cally,althoughtherequiredfundamentalsareprovided.Geodesicsatellitemissions,
ontheotherhand,havereachedanunprecedentedlevelofpositionaccuracywitha
needforverycomplexforceandmeasurementmodels,whichcouldnotalwaysbe
coveredinfulldetail.Inanycase,referencestobackgroundinformationaregiven,
soastoallowthereadereasyaccesstothesespecificareas.
Eachchapterincludesexercisesatvaryinglevelsofcomplexity,whichaimat
an additional practice of the presented material, or address supplementarytopics
ofpracticalinterest.Wherepossible,wehavetriedtofocusonproblemsthathigh-
VI Preface
lighttheunderlyingphysicalsmodelsoralgorithmicmethods,ratherthanrelying
on purely numerical reference examples. In most cases, the exercises include a
comprehensivedescriptionofthesuggestedsolution,aswellasthenumericalre-
sults.These are either deriveddirectlyfrom equationsgiveninthe text,or based
onsamplecomputerprograms.
The CD-ROM that was provided with previous printings of this edition has
beenreplacedbyazip-archivemadeavailableonSpringer’sExtraMaterialsserver
http://extra.springer.com/. Thisarchivecontains theC++sourcecodeofall
sample programs and applications, as well as relevant data files. The software is
builtaroundapowerfulspaceflightdynamicslibrary,whichislikewiseprovidedas
sourcecode.Forthesakeofsimplicitywehaverestrictedthelibrarytobasicmod-
els,butemphasizedtransparentprogrammingandin-codedocumentation.This,in
turn,allowsforanimmediateunderstandingofthecode,andpavesthewayforeasy
softwareextensionsbytheuser.Freeuseoftheentiresoftwarepackageincluding
therightformodificationsisgrantedfornon-commercialpurposes.Readers,stu-
dents and lecturers are, therefore, encouraged to apply it in further studies, and
todevelopnewapplications.Weassumethatthereaderisfamiliarwithcomputer
programming,buteveninexperiencedreadersshouldbeabletousethelibraryfunc-
tionsasblack boxes.Allsourcecode iswritteninC++, nowadaysa widelyused
programminglanguageandonewhichisreadilyavailableonavarietyofdifferent
platformsandoperatingsystems.
We would liketo thank Springer-Verlag for their cordial cooperationand in-
terestduringtheprocessofpublishingthisbook.Ourthanksarealsoduetoallour
friendsandcolleagues,who,withtheirideasandadvice,andtheirhelpincorrect-
ing the manuscriptand in testingthe programs, have playedan importantrole in
thesuccessfulcompletionofthisbook.Realmissiondatasetsfortheapplication
programshavekindlybeenprovidedbytheGPS/METprojectandtheFlightDy-
namicsAnalysisBranchoftheGoddardSpaceFlightCenter.Numerousagencies
andindividualshavecontributedimagesfortheintroductionofthisbook,whichis
gratefullyacknowledged.
May2000andAugust2012 OliverMontenbruckandEberhardGill
Contents
1 AroundtheWorldinaHundredMinutes. . . . . . . . . . . . . . . . 1
1.1 APortfolioofSatelliteOrbits . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Low-EarthOrbits . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 OrbitsofRemoteSensingSatellites . . . . . . . . . . . . . . 3
1.1.3 GeostationaryOrbits . . . . . . . . . . . . . . . . . . . . . . 4
1.1.4 HighlyEllipticalOrbits . . . . . . . . . . . . . . . . . . . . . 6
1.1.5 Constellations. . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 NavigatinginSpace . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 TrackingSystems . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 AMatterofEffort . . . . . . . . . . . . . . . . . . . . . . . 10
2 IntroductoryAstrodynamics . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 GeneralPropertiesoftheTwo-BodyProblem . . . . . . . . . . . . 16
2.1.1 PlaneMotionandtheLawofAreas . . . . . . . . . . . . . . 16
2.1.2 TheFormoftheOrbit . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 TheEnergyIntegral . . . . . . . . . . . . . . . . . . . . . . 19
2.2 PredictionofUnperturbedSatelliteOrbits . . . . . . . . . . . . . . 22
2.2.1 Kepler’sEquationandtheTimeDependenceofMotion . . . . 22
2.2.2 SolvingKepler’sEquation . . . . . . . . . . . . . . . . . . . 23
2.2.3 TheOrbitinSpace . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 OrbitalElementsfromPositionandVelocity . . . . . . . . . 28
2.2.5 Non-SingularElements . . . . . . . . . . . . . . . . . . . . . 29
2.3 Ground-BasedSatelliteObservations . . . . . . . . . . . . . . . . . 32
2.3.1 SatelliteGroundTracks . . . . . . . . . . . . . . . . . . . . 32
2.3.2 SatelliteMotionintheLocalTangentCoordinateSystem . . . 36
2.4 PreliminaryOrbitDetermination . . . . . . . . . . . . . . . . . . . 39
2.4.1 OrbitDeterminationfromTwoPositionVectors . . . . . . . . 40
2.4.2 OrbitDeterminationfromThreeSetsofAngles . . . . . . . . 43
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 ForceModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Geopotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 ExpansioninSphericalHarmonics . . . . . . . . . . . . . . . 56
3.2.2 SomeSpecialGeopotentialCoefficients . . . . . . . . . . . . 59
3.2.3 GravityModels . . . . . . . . . . . . . . . . . . . . . . . . . 61
VIII Contents
3.2.4 Recursions . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.5 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 SunandMoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.1 PerturbingAcceleration . . . . . . . . . . . . . . . . . . . . 69
3.3.2 Low-PrecisionSolarandLunarCoordinates . . . . . . . . . . 70
3.3.3 ChebyshevApproximation . . . . . . . . . . . . . . . . . . . 73
3.3.4 JPLEphemerides . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 SolarRadiationPressure . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.1 EclipseConditions . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.2 ShadowFunction . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5 AtmosphericDrag . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.1 TheUpperAtmosphere . . . . . . . . . . . . . . . . . . . . . 86
3.5.2 TheHarris–PriesterDensityModel . . . . . . . . . . . . . . 89
3.5.3 TheJacchia1971DensityModel. . . . . . . . . . . . . . . . 91
3.5.4 AComparisonofUpperAtmosphereDensityModels. . . . . 98
3.5.5 PredictionofSolarandGeomagneticIndices . . . . . . . . . 102
3.6 ThrustForces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.7 PrecisionModeling . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.7.1 EarthRadiationPressure . . . . . . . . . . . . . . . . . . . . 107
3.7.2 EarthTides . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.7.3 RelativisticEffects . . . . . . . . . . . . . . . . . . . . . . . 110
3.7.4 EmpiricalForces . . . . . . . . . . . . . . . . . . . . . . . . 112
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4 NumericalIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1 Runge–KuttaMethods. . . . . . . . . . . . . . . . . . . . . . . . . 118
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.1.2 GeneralRunge–KuttaFormulas . . . . . . . . . . . . . . . . 120
4.1.3 StepsizeControl . . . . . . . . . . . . . . . . . . . . . . . . 121
4.1.4 Runge–Kutta–NyströmMethods . . . . . . . . . . . . . . . . 123
4.1.5 ContinuousMethods . . . . . . . . . . . . . . . . . . . . . . 127
4.1.6 ComparisonofRunge–KuttaMethods . . . . . . . . . . . . . 129
4.2 MultistepMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.2.2 Adams–BashforthMethods . . . . . . . . . . . . . . . . . . 134
4.2.3 Adams–MoultonandPredictor–CorrectorMethods . . . . . . 136
4.2.4 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.2.5 VariableOrderandStepsizeMethods . . . . . . . . . . . . . 141
4.2.6 StoermerandCowellMethods . . . . . . . . . . . . . . . . . 143
4.2.7 Gauss–JacksonorSecondSumMethods. . . . . . . . . . . . 145
4.2.8 ComparisonofMultistepMethods . . . . . . . . . . . . . . . 146
4.3 ExtrapolationMethods . . . . . . . . . . . . . . . . . . . . . . . . 147
4.3.1 TheMid-PointRule . . . . . . . . . . . . . . . . . . . . . . 147
4.3.2 Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Contents IX
4.3.3 ComparisonofExtrapolationMethods . . . . . . . . . . . . . 150
4.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5 TimeandReferenceSystems . . . . . . . . . . . . . . . . . . . . . . . 157
5.1 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1.1 EphemerisTime . . . . . . . . . . . . . . . . . . . . . . . . 160
5.1.2 AtomicTime . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1.3 RelativisticTimeScales . . . . . . . . . . . . . . . . . . . . 162
5.1.4 SiderealTimeandUniversalTime . . . . . . . . . . . . . . . 165
5.2 CelestialandTerrestrialReferenceSystems . . . . . . . . . . . . . 169
5.3 PrecessionandNutation . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3.1 LunisolarTorquesandtheMotionoftheEarth’sRotationAxis 172
5.3.2 CoordinateChangesduetoPrecession . . . . . . . . . . . . . 174
5.3.3 Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.4 EarthRotationandPolarMotion . . . . . . . . . . . . . . . . . . . 181
5.4.1 RotationAbouttheCelestialEphemerisPole . . . . . . . . . 181
5.4.2 FreeEulerianPrecession . . . . . . . . . . . . . . . . . . . . 182
5.4.3 ObservationandExtrapolationofPolarMotion . . . . . . . . 183
5.4.4 TransformationtotheInternationalReferencePole . . . . . . 185
5.5 GeodeticDatums . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6 SatelliteTrackingandObservationModels. . . . . . . . . . . . . . . 193
6.1 TrackingSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.1.1 RadarTracking . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.1.2 LaserTracking . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.1.3 TheGlobalPositioningSystem . . . . . . . . . . . . . . . . 203
6.2 TrackingDataModels . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.2.1 TransmitterandReceiverMotion . . . . . . . . . . . . . . . 208
6.2.2 AngleMeasurements . . . . . . . . . . . . . . . . . . . . . . 209
6.2.3 RangeMeasurements . . . . . . . . . . . . . . . . . . . . . . 213
6.2.4 DopplerMeasurements . . . . . . . . . . . . . . . . . . . . . 215
6.2.5 GPSMeasurements . . . . . . . . . . . . . . . . . . . . . . . 217
6.3 MediaCorrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.3.1 InteractionofRadiationandAtmosphere . . . . . . . . . . . 219
6.3.2 TroposphericRefraction . . . . . . . . . . . . . . . . . . . . 221
6.3.3 IonosphericRefraction . . . . . . . . . . . . . . . . . . . . . 225
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.1 Two-BodyStateTransitionMatrix . . . . . . . . . . . . . . . . . . 235
7.1.1 Orbital-ElementsTransitionMatrix . . . . . . . . . . . . . . 235
7.1.2 Keplerian-to-CartesianPartialDerivatives . . . . . . . . . . . 236
7.1.3 Cartesian-to-KeplerianPartialDerivatives . . . . . . . . . . . 238
X Contents
7.1.4 TheStateTransitionMatrixandItsInverse . . . . . . . . . . 239
7.2 VariationalEquations . . . . . . . . . . . . . . . . . . . . . . . . . 240
7.2.1 TheDifferentialEquationoftheStateTransitionMatrix . . . 240
7.2.2 TheDifferentialEquationoftheSensitivityMatrix . . . . . . 241
7.2.3 FormandSolutionoftheVariationalEquations . . . . . . . . 241
7.2.4 TheInverseoftheStateTransitionMatrix . . . . . . . . . . . 243
7.3 PartialDerivativesoftheAcceleration . . . . . . . . . . . . . . . . 244
7.3.1 Geopotential . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.3.2 Point-MassPerturbations . . . . . . . . . . . . . . . . . . . . 247
7.3.3 SolarRadiationPressure . . . . . . . . . . . . . . . . . . . . 248
7.3.4 Drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.3.5 Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
7.4 PartialsoftheMeasurementswithRespecttotheStateVector . . . . 250
7.5 PartialswithRespecttoMeasurementModelParameters . . . . . . 252
7.6 DifferenceQuotientApproximations . . . . . . . . . . . . . . . . . 253
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
8 OrbitDeterminationandParameterEstimation . . . . . . . . . . . . 257
8.1 WeightedLeast-SquaresEstimation. . . . . . . . . . . . . . . . . . 258
8.1.1 LinearizationandNormalEquations . . . . . . . . . . . . . . 260
8.1.2 Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.1.3 StatisticalInterpretation . . . . . . . . . . . . . . . . . . . . 263
8.1.4 ConsiderParameters . . . . . . . . . . . . . . . . . . . . . . 265
8.1.5 EstimationwithAPrioriInformation . . . . . . . . . . . . . 266
8.2 NumericalSolutionofLeast-SquaresProblems . . . . . . . . . . . 268
8.2.1 QRFactorization . . . . . . . . . . . . . . . . . . . . . . . . 268
8.2.2 HouseholderTransformations . . . . . . . . . . . . . . . . . 270
8.2.3 GivensRotations . . . . . . . . . . . . . . . . . . . . . . . . 272
8.2.4 SingularValueDecomposition . . . . . . . . . . . . . . . . . 274
8.3 KalmanFiltering . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
8.3.1 RecursiveFormulationofLeast-SquaresEstimation. . . . . . 277
8.3.2 SequentialEstimation . . . . . . . . . . . . . . . . . . . . . 280
8.3.3 ExtendedKalmanFilter . . . . . . . . . . . . . . . . . . . . 282
8.3.4 FactorizationMethods . . . . . . . . . . . . . . . . . . . . . 283
8.3.5 ProcessNoise . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.4 ComparisonofBatchandSequentialEstimation . . . . . . . . . . . 286
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
9.1 OrbitDeterminationErrorAnalysis . . . . . . . . . . . . . . . . . . 293
9.1.1 ALinearizedOrbitModel . . . . . . . . . . . . . . . . . . . 294
9.1.2 ConsiderCovarianceAnalysis . . . . . . . . . . . . . . . . . 297
9.1.3 TheGEODAProgram . . . . . . . . . . . . . . . . . . . . . 299
9.1.4 CaseStudies . . . . . . . . . . . . . . . . . . . . . . . . . . 300
9.2 Real-TimeOrbitDetermination . . . . . . . . . . . . . . . . . . . . 303