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(GMAT) Mathematical Specifications DRAFT

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(GMAT) Mathematical Specifications DRAFT Draft for Release R2013a General Mission Analysis Tool (GMAT) Mathematical Specifications DRAFT April 4, 2013 NASA Goddard Space Flight Center Greenbelt RD Greenbelt, MD 20771 Draft for Release R2013a Contents 1 Time 11 1.1 TimeSystems....................................... ...... 11 1.1.1 Atomic Time: TAI and A.1 . .. 11 1.1.2 UniversalTime:UTCandUT1. .. .. .. .. .. .. .. .. .. .. .. .... 12 1.1.3 DynamicTime:TT,TDBandTCB .. .. .. .. .. .. .. .. .. .. .. .. 12 1.2 TimeFormats....................................... ...... 13 1.2.1 Julian Date and Modified Julian Date . .... 13 1.2.2 GregorianDate.................................... .... 14 1.3 Conclusions........................................ ...... 15 2 Coordinate Systems 17 2.1 GeneralCoordinateSystemTransformations . .............. 17 2.2 Pseudo-RotatingCoordinateSystems . ............. 19 2.3 ITRFandICRF ...................................... ..... 21 2.4 TransformationfromICRTtoMJ2000Eq . ........... 22 2.5 The InertialSystemandFK5Reduction . ... 22 FJ2k 2.5.1 OverviewofFK5Reduction. .. .. .. .. .. .. .. .. .. .. .. .. ..... 22 2.5.2 Precession Calculations . ..... 25 2.5.3 Nutation Calculations . .... 25 2.5.4 Sidereal Time Calculations . .... 27 2.5.5 Polar Motion Calculations . .... 27 ˙ 2.6 Deriving RJ2k,i and RJ2k,i forVariousCoordinateSystems . 28 2.6.1 EquatorSystem ................................... .... 28 2.6.2 MJ2000 Ecliptic (MJ2000Ec) . ..... 29 2.6.3 TrueofEpochEquator(TOEEq) . ...... 29 2.6.4 MeanofEpochEquator(MOEEq) . ...... 30 2.6.5 TrueofDateEquator(TODEq) . ...... 30 2.6.6 MeanofDateEquator(MODEq). ...... 31 2.6.7 Mean of Date Ecliptic (MODEc) . .... 31 2.6.8 True of Date Ecliptic (TODEc) . ..... 31 2.6.9 TopocentricHorizon ................................ ..... 31 2.6.10 Celestial Body Fixed . ..... 33 2.6.11 BodyInertial .................................... ..... 34 2.6.12 ObjectReferenced ............................... ....... 34 2.6.13 Geocentric Solar Ecliptic (GSE) . ...... 36 2.6.14 Geocentric Solar Magnetic (GSM) . ....... 36 2.6.15 Body-SpinSunCoordinates . ...... 37 2.7 Appendix 1: Derivatives of ObjectReferenced Unit Vectors . ................. 38 2.7.1 Basic Rotation Matrices . ..... 40 3 Calculation Objects 45 3.1 SpacecraftStateRepresentations . .............. 45 3.1.1 Definitions ........................................ .. 45 2 Draft for Release R2013a CONTENTS 3 3.1.2 Cartesian State to Keplerian Elements . ........ 46 3.1.3 Keplerian Elements to CartesianState . ........ 49 3.1.4 Equinoctial Elements to Cartesian State . ......... 51 3.1.5 Cartesian State to Equinoctial Elements . ......... 52 3.1.6 Cartesian State to SphericalAZFPA State . ......... 53 3.1.7 SphericalAZFPA State to Cartesian State . ......... 54 3.1.8 Cartesian State to SphericalRADEC State . ......... 55 3.1.9 SphericalRADEC State to Cartesian State . ......... 55 3.1.10 Keplerian or Cartesian, to Modified Keplerian Elements . .......... 56 3.1.11 Modified Keplerian Elements to Keplerian Elements . ........ 56 3.2 SimpleParameters ................................... ....... 57 3.2.1 A1Gregorian ...................................... ... 57 3.2.2 A1ModJulian....................................... .. 57 3.2.3 Altitude.......................................... .. 57 3.2.4 AOP............................................. 58 3.2.5 Apoapsis ......................................... .. 59 3.2.6 AZI ............................................. 59 3.2.7 BdotTandBdotR ................................... ... 59 3.2.8 BetaAngle........................................ ... 62 3.2.9 BVectorAngleandBVectorMag. ....... 62 3.2.10 C3Energy....................................... .... 62 3.2.11DEC............................................ .. 62 3.2.12DECV........................................... .. 63 3.2.13ECC............................................ .. 63 3.2.14FPA............................................ .. 63 3.2.15EA............................................. .. 63 3.2.16 Energy......................................... .... 64 3.2.17HMAG ........................................... 64 3.2.18 HX,HY,andHZ ..................................... .. 64 3.2.19HA............................................. .. 64 3.2.20INC ............................................ .. 65 3.2.21 Latitude........................................ .... 65 3.2.22 Longitude....................................... .... 66 3.2.23LST ............................................ .. 66 3.2.24MA ............................................. 67 3.2.25MHA ............................................ 68 3.2.26MM ............................................. 68 3.2.27 OrbitPeriod...................................... .... 68 3.2.28 PercentShadow .................................. ...... 68 3.2.29RA............................................. .. 70 3.2.30RAV............................................ .. 70 3.2.31RAAN........................................... .. 71 3.2.32 RadApo......................................... ... 71 3.2.33 RadPer ......................................... ... 71 3.2.34 RLAandDLA ...................................... .. 71 3.2.35RMAG ........................................... 72 3.2.36 SemilatusRectum.................................. ..... 72 3.2.37SMA............................................ .. 72 3.2.38TA............................................. .. 73 3.2.39 TAIModJulian ..................................... ... 73 3.2.40 TTModJulian...................................... ... 74 3.2.41 TTGregorian ..................................... .... 74 3.2.42 UmbraandPenumbra ............................... ..... 74 3.2.43 UTCModJulian..................................... ... 75 Draft for Release R2013a 4 CONTENTS 3.2.44 VelApoapsis...................................... .... 75 3.2.45 VelPeriapsis . .... 75 3.2.46VMAG ........................................... 76 3.3 OtherCalculations ................................... ...... 76 3.3.1 MAtoTA.......................................... 76 3.3.2 EAtoTA.......................................... 77 3.3.3 HAtoTA.......................................... 77 3.4 LibrationPoints ..................................... ...... 77 3.5 Barycenter ....................................... ....... 80 3.6 GroundStationModel ................................ ........ 81 4 Dynamics Modelling 83 4.1 OrbitDynamics...................................... ...... 83 4.1.1 Orbital Equations of Motion . ..... 83 4.1.2 Coordinate Systems for Integration of the Equations of Motion............. 84 4.1.3 Orbit Variational Equations and the State Transition Matrix . ........... 84 4.1.4 Multiple Spacecraft Propgation and Coupled Propagation of the Equations of Motion 85 4.2 ForceModelling...................................... ...... 85 4.2.1 n-BodyPointMassGravity ................................ 85 4.2.2 Non-SphericalGravity .............................. ..... 87 4.2.3 AtmosphericDrag .................................. .... 91 4.2.4 Solar Radiation Pressure . ...... 91 4.2.5 Relativistic Corrections . ..... 92 4.2.6 SpacecraftThrust ................................ ...... 92 4.3 Attitude.......................................... ...... 95 4.3.1 AttitudePropagation ............................... ..... 96 4.3.2 Attitude Parameterizations and Conversions . ........... 104 4.4 SpacecraftModel.................................. ......... 113 4.4.1 RFHardwareModels ................................. 113 4.4.2 ThrusterModels ................................... 113 4.4.3 TankModels ....................................... 118 4.4.4 MassProperties ................................... 129 4.5 EnvironmentModels .................................. ....... 131 4.5.1 Ephemerides ...................................... 131 4.5.2 AtmosphericDensity................................ ..... 133 4.6 FluxandGeomagneticIndex ............................ ........ 133 4.6.1 File Overview . 133 4.6.2 Historical(Observed)data. ........ 134 4.6.3 Near Term Daily Predictions . 134 4.6.4 Long Term Monthly Predictions . ..... 134 4.6.5 TankModels ....................................... 140 5 Numerical Integrators 151 5.1 Runge-KuttaIntegrators. ............ 151 5.1.1 Constructor& DestructorDocumentation . ........... 152 5.2 Prince-DormandIntegrators. ............ 152 5.3 AdamsBashforthMoulton. .. .. .. .. .. .. .. .. .. .. .. .. ......... 152 5.4 Bulirsch-Stoer..................................... ........ 153 5.5 Stopping Condition Algorithm . ....... 153 5.6 IntegratorCoefficients .............................. .......... 153 6 Measurement Modeling 157 6.1 GeneralFormoftheMeasurementModel . ............ 157 6.1.1 Ideal Observables: Geometry, Coordinate Systems, and Notation ............ 157 Draft for Release R2013a CONTENTS 5 6.1.2 OneWayRangeExample ............................... 159 6.2 Light-Time Solution . ...... 160 6.2.1 One-WayLightTime.................................. 160 6.2.2 Partial Derivatives of the One-Way Light Time . ........ 160 6.3 ComputedValueofTwo-WayRange . ......... 162 6.3.1 Overview ......................................... 162 6.3.2 NASA Ground Network (STDN) and Universal Space Network (USN) ......... 163 6.3.3 NASASpaceNetwork(TDRSS) . ..... 163 6.3.4 NASADeepSpaceNetwork(DSN) . ...... 165 6.4 ComputedValueofAveragedTwo-WayDoppler . ........... 166 6.5 NASADeepSpaceNetwork(DSN) . ......... 167 6.6 Computed Values of Optical Angles Observables . ............ 167 6.7 GeometricMeasurements . .. .. .. .. .. .. .. .. .. .. .. .. ......... 169 6.7.1 GeometricRange..................................
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