A Satellite Footprint Visualisation Tool Master-Thesis Îi× ËaØ by Patrick Daum Department of Communication Systems September 2005 Dissertation submitted in partial fulfillment of the MSc in Satellite Communications and Space Environment Supervisors: Dr. J.A. Wild and Dr. A.J. Kavanagh . Lancaster University attn.: P.Daum 44581 Castrop-Rauxel GERMANY Phone: (+49) 2367-98044 EMail: [email protected] www.vissat.net.ms MATLAB r is a registered trademark of The MathWorks Inc. For production information, please contact: The MathWorks Inc. 3 Apple Hill Drive Natick, MA 01760-2098, USA Phone: (508) 647-7000 EMail: [email protected] www.mathworks.com c ENSO . Abstract This thesis concerns the development of a numerical software tool for propagating and visualising orbits of artificial bodies orbiting the Earth and the calculation of magnetic conjunctions between these objects and the ground. The developed i× ËaØ toolbox Î enables the users to visualise the trajectories of satellites under different parameters and to estimate the visibility of a satellite to maintain efficient satellite ground communication links. The trajectory predictions enables the user to visualise the magnetic conjunctions for “space weather” studies by plotting the northern and southern footprint of the magnetic field line. The satellite’s motion results from the analysis of all forces which have an effect on the trajectory of the satellite. The resultant of all forces acting on a satellite is pro- portional to its acceleration, so it is possible to describe the trajectory of a satellite in the central force field it is acting in. In this case the main acting force is the gravi- tational force. This force is very variable in the near Earth space due to the effect of the Earth’s flattening and is therefore very variably marked; it could be characterised by the zonal harmonics. The equations to describe the motion of a satellite arises from Kepler’s empirically laws and Newton’s mathematical description afterwards. The major focus here lies on Newton’s law of universal gravitation and Newtons second law of motion. These laws leads to a description of the orbital motion by using differential equations, which are only possible to solve analytically when an ideal gravitational force field could be assumed but the Earth is known to neither have an even mass distribution nor be a perfect sphere. Therefore in order to get a precise description of the orbital motion the physical laws have to be expanded. This is done by using the Simplified General Perturbation Model (SGP4/SDP4) which was developed by the North American Aerospace Defense Command (NO- RAD) and implementing the methods, described in this model, in MATLAB r . The determination of the magnetic conjunction is performed by Tsyganenko’s magnetic field model (Tsy’03). By means of this model and the numerical representation in MATLAB r the magnetic connection between the satellite and the ground station can be determined. i× ËaØ All these models are summarised in three basic functions of Î in order to facilitate a fast numerical calculation of the given equations of motion and field descriptions, so it is possible to receive accurate results which agree with the theory in the context of computed precision. i× ËaØ The toolbox Î , the online documentation and all functions can be down- loaded from the following website hØØÔ:»»ÛÛÛºÚi××aغÒeغÑ× Also there is an attached CD-ROM with all functions and models used in the thesis. Contents 1 Introduction 1 2 Satellite Orbit Dynamics and Kinematics 3 2.1 CoordinateSystems............................ 3 2.1.1 EarthCenteredInertial-ECI . 4 2.1.2 EarthCenteredEarthFixed-ECEF . 4 2.1.3 TopocentricHorizonCoordinateSystem . 4 2.1.4 OrbitCoordinateSystem. 5 2.2 Newton’slaws............................... 5 2.2.1 GeneralConsiderations. 5 2.2.2 TwobodyProblem ........................ 6 2.3 Kepler’slaws ............................... 10 2.3.1 Determinationoftheorbitshape . 10 2.3.2 ConservationofEnergy. 10 2.3.3 Timederivationoftheorbit . 12 2.3.4 KeplerianElements. 15 2.4 NORADOrbitPropagation . 19 2.4.1 NORADTLE........................... 19 2.4.2 SimplifiedGeneralPerturbationModel . 21 2.4.2.1 PerturbationsduetoGeopotential . 21 2.4.2.2 Perturbations due to the Sun and the Moon . 22 2.4.3 Simplified General Perturbation Model Version 4 . .. 22 2.4.3.1 Perturbationsdue to Atmospheric Drag . 23 2.4.3.2 Perturbationsdue to Solar Radiation . 23 2.5 SatelliteOrbits .............................. 23 2.5.1 SynchronousOrbits. 24 2.5.1.1 EarthSynchronous . 24 2.5.1.2 SunSynchronous . 24 2.5.2 PolarOrbits............................ 25 2.5.3 RecurrentOrbit. .... ... .... ... ... .... ... 25 2.6 Summary ................................. 25 i 3 Magnetic Coupling Model 26 3.1 EarthMagneticField. 26 3.2 Tsyganenko’sMagneticFieldModel. 28 3.3 FieldLineTracing ............................ 30 3.4 Summary ................................. 32 i× Ëaص 4 Program/Toolbox ´Î 34 4.1 StructureandImplementation . 34 4.1.1 ReadNORADTLE........................ 36 4.1.2 SGP4 and SDP4 implementation (MEX Function orbit).... 37 4.1.3 Satellite Position Footprint (SatPOS) .............. 40 4.1.4 Ground Station Tracking (SatTRACK).............. 44 4.1.5 Magnetic Footprint (MagPOS) .................. 47 4.2 GraphicalUserInterface . 49 4.2.1 Operationandfunctionality . 51 4.2.1.1 SatelliteInputData . 51 4.2.1.2 TimeSettings. 52 4.2.1.3 MagnetosphericParameters . 53 4.2.1.4 SimulationandView . 53 4.3 PlotFunctions............................... 54 4.3.1 Plot on 2D Map (plot2dmap) .................. 54 4.3.2 Plot on Blue Marble Maps (plotonmarble) .......... 54 4.3.3 Plot Skymap (skymap)...................... 55 4.4 Summary ................................. 56 5 Conclusion and Future Work 58 References 61 A Equations, Tables, Explanations 65 A.1 Geopotential................................ 65 A.2 EarthAtmosphere ............................ 67 A.3 Time-VariantKeplerElements. 67 A.4 DetailedSGPModel ........................... 69 ii A.4.1 SGPModel ............................ 69 A.4.2 SGP4Model............................ 70 A.4.3 SDP4Model............................ 74 A.4.4 BoundariesoftheNORADmodels . 75 A.5 AdditionalAccelerations . 77 A.6 Earth’sMagneticFieldAdditions . 77 A.6.1 QuasiSchmidtnormalised . 77 B MATLAB 79 B.1 Explanation: MATLAB r /MEX ...................... 79 B.2 norad2kep.m ............................... 79 B.3 orbit.m .................................. 81 B.4 SatPOS.m ................................. 99 B.5 SatTRACK.m ................................100 B.6 MagPOS.m .................................102 B.7 Validation .................................104 C GEOPACK 104 List of Figures 2.1 Sputnik1 ................................. 3 2.2 Earth Centered Inertial CoordinateSystem . .... 4 2.3 TopocentricHorizonCoordinateSystem . .. 4 2.4 OrbitCoordinateSystem. 5 2.5 TwoBodyProblem............................ 6 2.6 PlaneMotion ............................... 8 2.7 ConicSections............................... 9 2.8 KeplerianOrbit.............................. 11 2.9 MotionalIntegral,AngularMomentum . 12 2.10 IllustrationoftheEccentricAnomaly . .... 13 2.11 KeplerianElements . 15 2.12 KeplerianOrbitwithPerturbations . .. 21 iii 2.13 EarthSynchronousOrbits . 24 2.14RecurrentOrbit.............................. 25 3.1 Three Dimensional Structure of the Earth’s Magnetic Field...... 27 3.2 Earth’s Dipole Field under different outer Conditions . ....... 28 3.3 Magnetosphere .............................. 29 3.4 Tsyganenko’sGeomagneticFieldModel. .. 31 3.5 SimpleShapesoftheEarth’smagneticfield . .. 33 4.1 ECItoGeodeticConversion . 41 4.2 GeodeticPositionFootprints. 42 4.3 ThreeDimensionalSimulatedTrajectory . ... 43 4.4 Skymap .................................. 44 4.5 GroundStation .............................. 45 4.6 SkymapplotstotrackSatellites . 46 4.7 MagneticConjunctionFootprint. 49 4.8 GraphicalUserInterface . 50 4.9 SatelliteData ............................... 51 4.10 OsculatingElementPerturbation . .. 52 4.11TimeSetting ............................... 53 4.12 MagnetosphericSetting. 53 4.13 GeodeticPositionFootprintonMarble . ... 55 5.1 TrackingDevice.............................. 60 A.1 Geopotential................................ 66 A.2 Nutation.................................. 76 A.3 Accelerations on Earth orbiting Satellites . ...... 77 List of Tables 2.1 KeplerianElements............................ 15 4.1 ArraystructureofTLE. 37 4.2 Assignment Structure between MATLAB r andC/C++ . 39 B.1 GODDARDSSCLocatorValues . .104 iv List of mathematical abbreviations a Semimajor Axis A Cross Section Area A~ Integration Constant for the Orbit plane b Semiminor Axis B∗ Ballistic Coefficient cw Drag Coefficient C~ Plane Vector for the Orbit e Eccentricity ex Unity Vector in x Direction of the given Reference Frame ey Unity Vector in y Direction of the given Reference Frame ez Unity Vector in z Direction of the given Reference Frame E Eccentric Anomaly G Gravitational Constant h Energy Constant for Central Force Fields i Inclination k Number of Revolutions per Day L Angular Momentum ms Mass of the Satellite M Mean Anomaly ME Mass of the Earth n Mean Angular Velocity ν True Anomaly ν˙ Angular Velocity of the Satellite ω Argument of Perigee Ω Right Ascension of Ascending Node (RAAN) p Ellipse Parameter q0 Density of the Atmosphere ~r Direction Vector ~r˙ Velocity ~r¨ Acceleration ¨ ~rDrag Acceleration due to Atmospheric Drag ¨ ~rsolar Acceleration due to Solar Perturbations t0 Epoch Time or Time of Perigee T Period of the