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Handbook of From Kepler to GPS Michel Capderou

Handbook of Satellite Orbits

From Kepler to GPS

Translated by Stephen Lyle

Foreword by Charles Elachi, Director, NASA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA

123 Michel Capderou Universite´ Pierre et Marie Curie Paris, France

ISBN 978-3-319-03415-7 ISBN 978-3-319-03416-4 (eBook) DOI 10.1007/978-3-319-03416-4 Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014930341

© Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this pub- lication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permis- sions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publica- tion, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com) Foreword

Since the dawn of the space age with the launch of and Ex- plorer 1, became a major discipline in space exploration. This book reflects many years of research and teaching in this field by Michel Capderou. It is a comprehensive and modern treatment of the theory of or- bital mechanics, its application, and current day samples of how it is used in the field. In that sense, it is not just a textbook for classroom-style lectures; it is truly a handbook for practitioners. It is full of fascinating historical information and references that intrigue the readers to follow the anecdotes and details on how this particular disci- pline evolved from the collective genius of giants in mathematics, physics and astronomy such as Tycho Brahe, Kepler, Newton, Galileo, Lagrange, Laplace, Gauss, Poincar´e and Einstein. This story telling not only makes reading in- teresting but also challenges the readers to understand the fundamentals used by these giants before the advent of computers. Most classroom-style textbook would skip intermediate steps in the deriva- tion of equations or refer the readers to the original papers or textbooks. This book provides sufficient intermediate steps so readers with basic freshman mathematics can follow the logical steps. Its treatment of geodesy, geopoten- tial and methods connects theory to physical measurements and observables. The chapter on and Mission is unique in that it provides a comprehensive survey of how theory is applied to real-life missions. It connects this discipline to science and inspires the reader to appreciate how a satellite orbit provides a special vantage point for conducting scientific measurements. Orbital mechanics is not just about getting into space, but it is integral to the measurement technique such as altimetry, radar topography, radio occul- tation, interferometry and gravity field through radiometric observables. The comprehensive treatment on designing an orbit for systematic ground track control and target point visibility is unique. In the past, practitioners had to conduct a literature search and examine multiple publications. Its treatment on GPS begs the reader to further explore the world of precision orbit deter- mination, timing and terrestrial reference frame. The book is sprinkled with stories of much innovative use of “tricks” in orbital mechanics such as frozen orbit, sun-synchronicity, aerobraking, libration point and Lissajous orbits, and that enables missions like Voyager Grand Tour, Galileo, Cassini-

V VI Foreword

Huygens and tours of their . This book has superb illustrations and graphics enhanced by colorful photographs. Since the flight of Explorer 1, JPL prides itself in pioneering techniques in orbital mechanics and its applications to carry out NASA’s mission in space-borne observation of our Earth; in fly-by, orbiting and landing of plan- etary bodies and their satellites; in astronomical telescopes that can observe our galaxy and the early Universe. We continue to recruit the best and the brightest graduates in this discipline from universities around the world, who understand not only the physics and mathematics of orbital mechanics but also its applications of real-life missions. The Handbook of Satellite Orbits: From Kepler to GPS is exactly what is needed for all graduates of this disci- pline. Michel Capderou’s book is an essential treatise in orbital mechanics for all students, lecturers and practitioners in this field, as well as other aerospace systems engineers.

Charles Elachi Director NASA Jet Propulsion Laboratory California Institute of Technology Pasadena, CA, USA Preface

Of all the fields of modern science and technology, space exploration is the one that most clearly displays the following fundamental contrast: on the one hand, its theoretical basis is underpinned by long-established, historically tested and almost immutable, one might even say timeless, principles; on the other, the whole field of space science is undergoing meteoric technological evolution, with exponential growth, bringing with it a broad mix of commer- cial, political and ideological considerations. And so we have come from Kepler to GPS. Regarding the “immutable foundations”, we know that the notion of geopotential or the solution of Lagrange’s equations is no easy matter. We just hope that, with teaching experience among the ingredients, we have suc- ceeded in presenting these issues in a sufficiently clear and interesting way. To illustrate unbridled technological progress, we supply a wealth of examples. The book falls into six main parts: • The first part, consisting of Chaps. 1–3, is devoted to geodesy. We begin with the ellipse and its geometrical properties and work our way to the Earth’s gravitational potential and the geoid. • The second part, Chaps. 4–8, focuses on the motion of the satellite, working from the ideal, Keplerian case to the real, perturbed case. • The third part, Chaps. 9–11, takes us into the actual running and func- tioning of satellites, discussing their missions and the ways that orbits are designed to fulfil those missions. We consider some novel issues, such as the constant of Sun-synchronicity kh and, for recurrent satellites, the constant κ and the index Φ. Abundant illustrations are provided, always relating to past, present or future space programmes.

VII VIII Preface

• The fourth part, Chaps. 12 and 13, considers the instruments carried aboard the satellite from a geometrical point of view. We begin with the different ways of observing the Earth from a satellite, then move on to sampling, i.e. the conditions under which a given point on the Earth can view the satellite, considering the viewing angle and frequency of visibility. • We then devote the whole of Chap. 14 to GPS. This navigation system, entirely satellite-based, appeals to almost all branches of modern physics. • In the final part, Chaps. 15 and 16, we leave the confines of our own planet to apply all these theories first to Mars, then to the other planets of the Solar System, and even to the natural satellites of those planets, around which artificial satellites may gravitate.  The orbit and sampling software Ixion forms the backbone of this book. We first developed it as a teaching tool for an M.Sc. in climatology and space observation, and also in the research context, as an aid to understanding issues of , satellite–pixel–Sun configuration, and so on, which arise when processing the data transmitted to us by our satellites. But once the accuracy of Ixion had been proven in the context of real data, by the confrontation with pixels, one might say, we extended it to all types of orbit and included some didactic features that would make it accessible and useful to a broader audience. The software Ixion has since been used for preliminary studies of orbital strategy, as it is known, which serve to match orbital elements in the best possible way to the physical phenomena we need to observe. Among the orbits studied in this context, we cite the French–Indian satellite Megha-Tropiques and the planned Mars missions Premier-07 and MEMO. Ixion isoftenusedby our colleagues for calibration and validation campaigns in the field, as for the satellites Calipso, MetOp-A and -B, Megha-Tropiques, and others. Ixion/Web is the part of Ixion that is now accessible online. Our mapping software Atlas has been coupled with Ixion to produce graphical representa- tions of orbits and their ground tracks. We hope the maps it produces will be pleasant and useful to the reader. They should provide a refreshing change to the deeply saddening lack of cartographic imagination and the striking flatness of the projections generally used in this field. We have selected many examples among experiments that are familiar to us, such as the CERES and ScaRaB instruments and the Megha-Tropiques satellite. They may appear to be over-represented in the book, but perhaps it is better to stick to the things we know best! Because this book focuses on space mechanics and the geometry of obser- vation, with all its concomitants, such as spatiotemporal sampling, there is no consideration at all of the satellite as a technological object. There is not one word about the launch vehicles or the functioning of the onboard instruments, apart from the geometric aspect of the swath in the latter case, since this is directly relevant to our purpose. Preface IX

By concentrating in this way on the orbits, we have made every effort to prove or at least explain all the formulas used. This may look some- what austere, so we have tried to brighten things up with plenty of examples and illustrations. The examples will show the reader some rather unexpected orbits, while the photographs will demonstrate the level of accuracy achieved today in images acquired by satellite-borne instruments. To liven up the whole discussion, we have also included many references to historical aspects, even presenting several pages of the books that founded celestial mechanics and discussing some of those early results which leave us in admiration. And so we have come from Kepler to GPS. With that, let me wish you a good trip into space ... and into time. X Preface

Acknowledgements

This work was accomplished in the context of my teaching at the University of Pierre and Marie Curie (UPMC, Paris) and my research at the Laboratoire de m´et´eorologie dynamique (LMD), whose director, Vincent Cass´e, I would like to thank. The LMD is a research unit depending on the Centre national de la recherche scientifique (CNRS) and four institutes of higher education, namely, the UPMC, the Ecole Polytechnique,theEcole normale sup´erieure (ENS) and the Ecole nationale des Ponts et chauss´ees (ENPC). I am particularly grateful to Jacques Lefr`ere, Fran¸cois Forget and Florent Deleflie for their comments and criticisms during the preparation of this book. Their advice was invaluable and I extend my warmest gratitude to them. I would also like to thank R´emy Roca, Olivier Chomette, Patrick Raberanto and the whole team at Megha-Tropiques, as well as Karim Ramage, webmas- ter for Ixion/Web, my colleagues at the university and the CNRS, Fran¸cois Barlier, Pierre Briole, Nicole Capitaine, Xavier Collilieux, Michel Desbois, Albert Hertzog, Robert Kandel, Richard Kerner, Richard Marchand, Patrick Rocher, Jerˆome Sirven and Aymeric Spiga. I am grateful for the trust shown to me by the publisher Springer, and in particular by Nathalie Huilleret in Paris and Harry Blom and Jennifer Satten in New York. And last but not least, my thanks to Stephen Lyle who translated this book: bravo et merci !

Palaiseau, France Michel Capderou

Preface XI ÁÜÛÒ

The figure from Greek mythology known as Ixion was not, if the truth be told, a particularly savoury one. The King of the Lapiths, he behaved in a decidedly reprehensible manner on the day of his wedding, causing his future father-in-law to fall into a burning pit so that he would not have to pay the dowry. This act was considered the ultimate crime, for it broke all the rules of hospitality, and indeed, it was reproved by all the gods but one. The only deity who would agree to purify Ixion for the murder was Zeus, a connaisseur when it came to perjury and other misdemeanours. Zeus even felt some compassion for this strong-minded king, inviting him to Olympus and offering him hospitality. As an exceptional sign of friendship, he bade him drink ambrosia, which made him immortal. Ixion admired Zeus’ antics and escapades and, encouraged by the atmo- sphere of familiarity in the Olympian realm, began to covet Zeus’ own wife Hera. But this was where he overstepped the mark! The king of the gods cried out: “A little respect for one’s host!” As a punishment, he bound him to a fiery wheel which whirled him forever through the skies. As he had been made immortal, the poor fellow must still be spinning around up there. One may thus consider Ixion as the first of all artificial satellites, and this is therefore the name we have chosen for our software.

Ixion is the orbitography and sampling software that forms the basis for this book. Since 2010, many of the features of this software have been put online in collaboration with Karim Ramage at the Institut Pierre-Simon Laplace (IPSL). This software Ixion/Web is perfectly operational. The orbital elements of the satellites are updated daily (NORAD data) and the calculation of the satellite trajectory is thus fully accurate. For past dates, the automatically archived NORAD data are used. The ground track of the satellite and instrumental swath are indicated on Google Maps, providing much detail and easy consultation. Graphic repre- sentations can also be obtained on standard maps, with a choice of over a hundred cartographic projections using the software Atlas. Another option is 3D visualisation of the orbit with Google Earth. At a given location, sampling tables indicate the times (day and time) of satellite overpasses for the whole month, specifying conditions of viewing such as sight angle and solar configuration. Statistical tables provide global data for the whole Earth. XII Preface

Ixion/Web also gives orbits and sampling for the planet Mars. Satellite ground tracks are represented on Google Map Mars or conventional maps. Ground tracks of satellites orbiting other planets (Venus, Mercury, etc.) or the Moon can also be represented. Apart from being fully operational, Ixion/Web has a clear pedagogical interest as an aid to understanding satellite motion in different frames, i.e. Galilean or moving with the Earth. We give here four examples of graphical representations: a 3D representation of the orbit of the satellite LAGEOS-1, a close-up of the orbital ground track of Jason-2, an orbital ground track of Molniya-3-50 and an orbital ground track of Terra with swath. http://climserv.ipsl.polytechnique.fr/ixion.html Contents

1 Geometry of the Ellipse ...... 1 1.1 Definition and Properties ...... 1 1.1.1 Conic Sections ...... 1 1.1.2 Definition and Properties of the Ellipse ...... 2 1.1.3 Applications of the Definition ...... 3 1.1.4 Demonstrating the Main Properties ...... 7 1.1.5 Eccentricity and Flattening ...... 13 1.2 Applications and Other Characteristics ...... 17 1.2.1 Arc Length of an Ellipse ...... 17 1.2.2 Radius of an Ellipse ...... 19 1.2.3 Radius of Curvature of an Ellipse ...... 20

2Geodesy...... 25 2.1 Earth Ellipsoid ...... 25 2.1.1 Different Definitions of ...... 25 2.1.2 Cartesian Coordinates: Great Normal ...... 31 2.1.3 Radius of Curvature ...... 32 2.1.4 Radius of the Ellipse ...... 32 2.1.5 Degrees of Latitude and Longitude ...... 33 2.1.6 Meridian Arc Length ...... 35 2.2 Altitude Relative to the Ellipsoid ...... 37 2.2.1 Definition of Geodetic Altitude and Nadir ...... 37 2.2.2 Latitude Related to Geodetic Altitude ...... 37 2.2.3 Determining the Geodetic Altitude and Nadir ...... 39 2.3 A Little History ...... 42 2.3.1 Before the Enlightenment ...... 42 2.3.2 A French Affair ...... 44 2.3.3 Dynamical Geodesy ...... 51

3Geopotential...... 53 3.1 Some Preliminaries ...... 53 3.1.1 Reference Systems ...... 53 3.1.2 Review of Work and Potential ...... 54 XIII XIV Contents

3.2 Gravitational Potential and Field ...... 56 3.2.1 Gravity ...... 56 3.2.2 Gauss’ Theorem ...... 57 3.2.3 Gravity and Weight ...... 60 3.3 Calculating the Geopotential ...... 61 3.3.1 Potential Element ...... 61 3.3.2 Obtaining the Potential by Integration ...... 62 3.3.3 Spherical Harmonics ...... 64 3.3.4 Second Degree Expansion of the Potential ...... 65 3.3.5 Expanding the Potential to Higher Degrees ...... 68 3.4 Weight Field and Potential for the Ellipsoid ...... 71 3.4.1 Calculating the Field and Potential ...... 71 3.4.2 Weight Field at the Earth’s Surface ...... 73 3.4.3 Clairaut’s Formula ...... 75 3.4.4 Somigliana’s Formula ...... 78 3.5 Geoid ...... 79 3.5.1 Gravity Anomalies ...... 79 3.5.2 Satellites and Geodesy ...... 79 3.5.3 Development of Geopotential Models ...... 83 3.5.4 Evaluation of the Geocentric Gravitational Constant .... 87 3.6 Appendix: Terrestrial Reference Systems ...... 88 3.6.1 Celestial Reference System ...... 88 3.6.2 Terrestrial Reference System ...... 89 3.7 Appendix: Summary of Legendre Functions ...... 92

4KeplerianMotion...... 95 4.1 Central Acceleration ...... 95 4.1.1 General Acceleration ...... 95 4.1.2 Properties of Central Acceleration ...... 96 4.1.3 Motion with Central Acceleration ...... 97 4.2 Newtonian Acceleration ...... 99 4.2.1 Equation for the Trajectory ...... 99 4.2.2 Types of Trajectory ...... 100 4.3 Trajectory and Period for Keplerian Motion ...... 102 4.3.1 Definition of Keplerian Motion ...... 102 4.3.2 Periodic Trajectories ...... 102 4.3.3 Period and ...... 106 4.3.4 Relation Between Position and Time ...... 107 4.4 Time as a Function of Position: The Three Anomalies ...... 107 4.4.1 Expression for the Time and the M ....108 4.4.2 Expression t = t(θ) and the v ...... 108 4.4.3 Expression t = t(r) and the E ...... 110 4.4.4 Relating the Anomalies ...... 112 Contents XV

4.5 Position as a Function of Time: Kepler’s Problem ...... 115 4.5.1 Methods for Solving Kepler’s Problem ...... 115 4.5.2 Solution by Numerical Iteration ...... 115 4.5.3 Other Methods of Solution ...... 120 4.6 Representation of Anomalies ...... 122 4.6.1 Representation of Anomalies v(M)andE(M) ...... 122 4.6.2 Equation of Center ...... 124 4.6.3 Summary of Anomalies ...... 127 4.7 First Integrals of the Motion ...... 136 4.7.1 Conservation Laws ...... 136 4.7.2 Note on Energy ...... 139 4.8 Historical Note on Universal Attraction ...... 141 4.8.1 Kepler’s Laws ...... 141 4.8.2 Newton and the Law of Universal Attraction ...... 145

5 Satellite in Keplerian Orbit ...... 149 5.1 Two-Body Problem ...... 149 5.2 Orbital Elements ...... 151 5.2.1 Specifying the Satellite Orbit in Space ...... 151 5.2.2 Keplerian Elements ...... 154 5.2.3 Adapted Orbital Elements ...... 154 5.3 Keplerian Period ...... 156 5.4 Appendix: Rotation of a Solid—Euler and Cardan Angles ...... 159

6 Satellite in Real (Perturbed) Orbit ...... 163 6.1 Perturbing Forces ...... 163 6.1.1 From Ideal to Real Orbits ...... 163 6.1.2 Order of Magnitude of Perturbing Forces ...... 164 6.1.3 Potential ...... 164 6.1.4 Perturbations and Altitude of a Satellite ...... 165 6.2 Perturbative Methods: Presentation ...... 171 6.2.1 Orbit Propagation: Numerical and Analytical Methods . . 171 6.2.2 Basic Principles ...... 174 6.2.3 Lagrange Brackets ...... 176 6.2.4 Properties of the Lagrange Bracket ...... 178 6.3 Perturbative Method: Solution ...... 179 6.3.1 Calculating the Coordinates ...... 179 6.3.2 Calculating the Lagrange Brackets ...... 182 6.3.3 Lagrange Equations ...... 183 6.3.4 Metric and Angular Orbital Elements ...... 186 6.3.5 Poorly Defined Parameters...... 187 6.3.6 Delaunay Elements ...... 187 XVI Contents

6.4 Perturbative Method: Results for the Geopotential up to J2 .....189

6.4.1 Expression for the Perturbative Potential up to J2 .....189 6.4.2 Variation of the Orbital Elements ...... 192 6.5 Perturbative Method: Results for General Case ...... 195 6.5.1 Geopotential up to Jn ...... 195 6.5.2 Full Geopotential ...... 201 6.5.3 Other Forces Deriving from a Potential ...... 202 6.5.4 Perturbative Forces not Derived from a Potential ...... 202 6.5.5 Different Definitions of the Period ...... 203 6.6 Appendix: Atmospheric Drag ...... 204 6.6.1 Description of the Earth’s Atmosphere ...... 204 6.6.2 Density of the Atmosphere ...... 205 6.6.3 Models of the Atmosphere ...... 206 6.6.4 Calculation of Atmospheric Drag: The Notion of ΔV . . . 207 6.6.5 Effect of Drag on the Orbit ...... 210 6.6.6 Simplified Calculations for an Eccentric Orbit: Air Braking ...... 210 6.7 Historical Note: First Determinations of the Harmonics Jn ...... 213

6.7.1 First Satellite Determination of J2 ...... 213

6.7.2 First Satellite Determination of J3 ...... 214

6.7.3 First Determinations of Jn up to J14 ...... 215 6.8 Historical Note: Success in Calculating Perturbations ...... 215 6.8.1 The Delayed Return of Halley’s Comet ...... 215 6.8.2 The Discovery of Neptune by Le Verrier ...... 216 6.8.3 Advance of the Perihelion of Mercury ...... 217 6.9 Astronomical Note: Perturbations and the Solar System ...... 220 6.9.1 Stability of the Solar System ...... 220 6.9.2 Precession of the Equinoxes ...... 223 6.9.3 The Earth as a Satellite ...... 224 6.10 Appendix: Astronomical Constants ...... 227 6.10.1 Systems of Units ...... 227 6.10.2 Astronomical Constants ...... 228 6.10.3 Time Scales ...... 228 6.11 Appendix: Gravitational Sphere of Influence ...... 231 6.11.1 Attraction of the Sun and Earth ...... 231 6.11.2 Determining the Sphere of Influence ...... 233 6.12 Appendix: Lagrange Points ...... 234 6.12.1 Restricted Three-Body Problem ...... 234

6.12.2 Simplified Study of Points L1 and L2 ...... 234 6.12.3 Lagrange Points and Sphere of Influence ...... 236 6.12.4 The Five Lagrange Points ...... 237 6.12.5 Lagrange Points in Astronomy ...... 238 6.12.6 Artificial Satellites at Lagrange Points ...... 239 Contents XVII

6.13 Appendix: Spherical Trigonometry ...... 241 6.13.1 Gauss’ Relations ...... 241 6.13.2 Fifteen Relations for the Spherical Triangle ...... 243

7 Motion of Orbit, Earth and Sun ...... 245 7.1 Motion of the Orbit ...... 245 7.1.1 Secular Variations: Simplified Case ...... 245 7.1.2 Secular Variations up to Degree 4 ...... 250 7.1.3 Secular Variations up to Degree n ...... 252 7.1.4 Removing Precessional Motion ...... 252 7.1.5 Effective Calculation of Period and Altitude ...... 256 7.2 Motion of the Earth ...... 259 7.2.1 Motion of the Earth About the Sun ...... 260 7.2.2 Motion of the Earth About the Polar Axis ...... 261 7.2.3 Motion of the Poles ...... 263 7.2.4 Motion of the Orbit and Earth ...... 264 7.3 Apparent Motion of the Sun ...... 266 7.3.1 Celestial Sphere and Coordinates ...... 266 7.3.2 Hour Angle ...... 267 7.3.3 Equation of Time ...... 267 7.3.4 Solar Time ...... 272 7.3.5 Declination ...... 273 7.3.6 Julian Day, Julian Date ...... 274 7.4 Geosynchronicity ...... 276 7.4.1 Definition ...... 276 7.4.2 Calculating the Orbit ...... 277 7.4.3 Geostationary Satellites ...... 278 7.4.4 Drift of the ...... 280 7.4.5 Stationkeeping ...... 282 7.4.6 Geosynchronous Satellites with Highly Eccentric Orbits . 289 7.5 Sun-Synchronicity ...... 291 7.5.1 Definition ...... 291 7.5.2 Constant of Sun-Synchronicity ...... 291 7.5.3 Calculating the Orbit: Circular Case ...... 292 7.5.4 Calculating the Orbit: Elliptical Case ...... 296 7.5.5 Sun-Synchronous Satellites ...... 298 7.5.6 Orbit Maintenance ...... 299

8 Ground Track of a Satellite ...... 301 8.1 Position of Satellite on Its Orbit ...... 301 8.1.1 Using Euler Angles to Describe Satellite Motion ...... 301 8.1.2 Position of Satellite in Cartesian Coordinates ...... 304 8.1.3 Position of Satellite in Spherical Coordinates ...... 305 XVIII Contents

8.2 Ground Track of Satellite ...... 305 8.2.1 Equation for Ground Track ...... 305 8.2.2 Maximum Attained Latitude ...... 306 8.3 Ground Track of Satellite in ...... 307 8.3.1 Equation for Satellite Ground Track ...... 308 8.3.2 Equatorial Shift ...... 308 8.3.3 Apparent Inclination ...... 310 8.3.4 Angle Between Ground Track and a Meridian ...... 316 8.3.5 Velocity of a Satellite and Its Ground Track ...... 317 8.3.6 Eliminating Time from the Ground Track Equation ....320 8.4 Appendix: NORAD Orbital Elements ...... 322 8.4.1 NORAD: The Organisation ...... 322 8.4.2 Two-Line Element (TLE) Set Format ...... 323 8.4.3 Decoding the TLE ...... 323 8.4.4 Conditions of Use ...... 327 8.5 Appendix: Cartographic Projections ...... 329 8.5.1 Definitions and Properties ...... 329 8.5.2 Classifying Projections by Type or Aspect ...... 330 8.5.3 Description of Three Projections ...... 331

9 Orbit and Mission ...... 339 9.1 Classifying Orbit Types...... 339 9.2 Classifying Satellites by Mission ...... 340 9.2.1 The First Satellites ...... 341 9.2.2 Satellites for Geodesy ...... 344 9.2.3 Earth Environment Satellites ...... 348 9.2.4 Satellites for Meteorology and Climate Study ...... 357 9.2.5 Satellites for Remote-Sensing and Surveillance ...... 384 9.2.6 Oceanographic Satellites ...... 388 9.2.7 Navigation Satellites ...... 390 9.2.8 Communications Satellites ...... 391 9.2.9 Satellites for Astronomy and Astrophysics ...... 408 9.2.10 Satellites for Fundamental Physics ...... 421 9.2.11 Technological Satellites ...... 424 9.2.12 Satellites with Specific Military Missions ...... 425 9.2.13 Manned Satellites ...... 428 9.2.14 Non-Scientific Satellites ...... 429 9.3 Appendix: Delays in Scheduling Space Missions ...... 430

10 Orbit Relative to the Sun: Crossing Times and Eclipse ...... 433 10.1 Cycle with Respect to the Sun ...... 433 10.1.1 Crossing Time ...... 433

10.1.2 Calculating the Cycle CS ...... 434 Contents XIX

10.1.3 Cycle CS and Orbital Characteristics ...... 436 10.1.4 Cycle and Ascending Node Crossing Time ...... 448 10.2 Crossing Time for a Sun-Synchronous Satellite ...... 449 10.2.1 Passage at a Given Latitude ...... 449 10.2.2 Choice of Local Time at the Ascending Node ...... 454 10.2.3 Calculating the Drift in Local Crossing Time ...... 460 10.3 Angle Between Orbital Plane and Solar Direction ...... 464 10.3.1 Position of the Normal to the Orbital Plane ...... 464 10.3.2 Angle β ...... 465 10.4 Solar Eclipse for Circular Orbits ...... 466 10.4.1 Duration of Solar Eclipse ...... 469 10.4.2 Sun-Synchronous LEO Orbit ...... 470 10.4.3 Dawn–Dusk Sun-Synchronous LEO Orbit ...... 474 10.4.4 MEO Orbit ...... 478 10.4.5 GEO Orbit ...... 478 10.5 General Conditions for Solar Eclipse ...... 479 10.5.1 Establishing General Eclipse Conditions ...... 479 10.5.2 Criterion for Eclipse ...... 482

11 Orbit Relative to the Earth: Recurrence and Altitude ...... 487 11.1 Recurrence Constraint ...... 487 11.1.1 Definition of Recurrence ...... 487

11.1.2 Calculating the Recurrence Cycle CT ...... 488 11.1.3 Recurrence Triple ...... 490 11.2 Recurrence of Sun-Synchronous LEO Satellites ...... 491 11.2.1 Method for Obtaining Recurrence ...... 491 11.2.2 Recurrence Module ...... 491 11.2.3 Recurrence Diagram ...... 492 11.2.4 Recurrence Defined by the Recurrence Triple ...... 498 11.2.5 One-Day Recurrence Cycle ...... 506 11.3 Recurrence for Non-Sun-Synchronous LEO Satellites ...... 508 11.3.1 Obtaining the Recurrence Triple ...... 508 11.3.2 Recurrence, Altitude, and Inclination ...... 512 11.4 Recurrence for MEO and HEO Satellites ...... 514 11.5 Recurrence Grid ...... 516 11.5.1 Constructing the Recurrence Grid ...... 516 11.5.2 Grid Interval ...... 517 11.5.3 Recurrence Subcycle ...... 520 11.5.4 Reference Grids ...... 524 11.5.5 Grid Points for Recurrent Satellites ...... 526 11.6 Maintaining a Recurrent Satellite on Orbit ...... 533 11.7 Recurrence Index ...... 536 11.7.1 Definition of Recurrence Index ...... 536 XX Contents

11.7.2 Perfect or Imperfect Recurrence ...... 538 11.7.3 Applications of the Recurrence Index ...... 538 11.7.4 Recurrence Index and Orbital Characteristics ...... 540 11.8 Altitude Variations ...... 542 11.8.1 Altitude and Orbital Parameters ...... 543 11.8.2 Altitude During One Revolution ...... 546 11.8.3 Variation of the Altitude over a Long Period ...... 550 11.9 Frozen Orbits ...... 551 11.9.1 Definition of a Frozen Orbit ...... 551 11.9.2 Determining the Frozen Parameters ...... 551 11.9.3 Altitude of a Satellite on a Frozen Orbit ...... 554 11.10 Altitude and Atmospheric Drag ...... 558

12 View from the Satellite ...... 561 12.1 Swath of an Instrument ...... 561 12.1.1 Local Orbital Frame ...... 561 12.1.2 Scanning Modes ...... 562 12.2 Swath Viewing Geometry ...... 565 12.2.1 Definition of Angles ...... 565 12.2.2 Relations Between Angles ...... 567 12.2.3 Ground Swath ...... 567 12.2.4 Viewed and Latitude Overlap ...... 568 12.3 Pixel Distortion ...... 570 12.3.1 Calculating the Distortion Index ...... 570 12.3.2 Pixel Distortion for LEO Satellites ...... 572 12.3.3 Pixel Distortion for GEO Satellites ...... 574 12.4 Swath Track for an LEO Satellite ...... 576 12.4.1 Cross-track Swath ...... 576 12.4.2 Variable-Yaw Swath ...... 582 12.4.3 Conical Swath ...... 585 12.4.4 Ground Track Superposition ...... 592 12.5 View from a GEO Satellite ...... 593 12.5.1 Simplified Geometric Conditions ...... 593 12.5.2 Pixels and Geographic Coordinates Correspondence ....604

13 Spatiotemporal and Angular Sampling ...... 613 13.1 Satellite–Target Direction ...... 614 13.1.1 Line-of-Sight Direction of the Satellite ...... 614 13.1.2 Geostationary Satellites ...... 618 13.1.3 Local View and Sky Plots ...... 620 13.1.4 Visibility Window for LEO Satellites ...... 622 13.1.5 Visibility Window for HEO Satellites ...... 627 Contents XXI

13.2 Target–Sun Direction ...... 629 13.2.1 Solar Line of Sight ...... 629 13.2.2 Sunrise, Sunset, and Apparent Noon ...... 632 13.3 Sun–Target–Satellite Configuration ...... 633 13.3.1 Angles Describing the Sun–Target–Satellite Configuration ...... 633 13.3.2 Specular Reflection (Sun Glint) ...... 635 13.4 Monthly Sampling Tables ...... 639

14 Global Positioning Systems (GPS) ...... 653 14.1 Basic Principle of GPS ...... 653 14.1.1 Positioning in the Ideal Case ...... 653 14.1.2 Positioning in Real Situations ...... 654 14.1.3 Determining User Velocity ...... 658 14.1.4 Perturbation of Signal and Measurement ...... 660 14.1.5 Geometric Considerations and Measurement Accuracy . . 661 14.1.6 Position on Earth and Geographic Coordinates ...... 663 14.1.7 Differential GPS (DGPS) ...... 664 14.2 Navstar/GPS ...... 666 14.2.1 Setting up the System ...... 666 14.2.2 Space Segment ...... 667 14.2.3 Control Segment ...... 670 14.2.4 User Segment ...... 675 14.2.5 Local View ...... 676 14.2.6 Navstar/GPS and Other Systems ...... 676 14.3 Glonass...... 678 14.3.1 The Three Segments ...... 678 14.3.2 Local View and Visibility Table ...... 679 14.4 Galileo ...... 681 14.4.1 A European Project ...... 681 14.4.2 The Three Segments ...... 684 14.5 BeiDou NS ...... 684 14.5.1 The Three Segments ...... 687 14.5.2 Beidou-1 Experimental System ...... 688 14.6 Augmentation Systems ...... 691 14.7 Regional Systems ...... 694 14.7.1 IRNSS ...... 694 14.7.2 QZSS ...... 694 14.8 Non-positioning Uses of GPS ...... 696 14.8.1 Radio Occultation ...... 696 14.8.2 Studying the Troposphere via the Base Stations ...... 698 14.8.3 Other Applications ...... 698 XXII Contents

14.9 Historical Note: The First Systems ...... 698 14.9.1 ...... 698 14.9.2 The Soviet System ...... 701 14.10 Appendix: GPS and Tectonic Plates ...... 702 14.11 Appendix: GPS and Relativity ...... 705 14.11.1 Presentation ...... 705 14.11.2 Proper Time Difference ...... 705 14.11.3 Effect Due to ...... 713 14.11.4 Sagnac Effect ...... 715 14.11.5 Conclusion ...... 717

15 Satellites of Mars ...... 719 15.1 Presenting the Planet Mars ...... 719 15.1.1 Mars and Space Exploration ...... 719 15.1.2 Geography of Mars ...... 726 15.2 Geodetic and Astronomical Quantities ...... 731 15.2.1 Geodetic Data ...... 731 15.2.2 Astronomical Data ...... 732 15.2.3 Areocentric Longitude and Martian Day ...... 733 15.2.4 Declination ...... 739 15.2.5 Equation of Time ...... 741 15.3 Satellite in Real Orbit ...... 742 15.3.1 Satellite in Keplerian Orbit ...... 742 15.3.2 Perturbative Accelerations ...... 742 15.3.3 Secular Variation of Orbital Elements ...... 746 15.4 Different Orbits ...... 748 15.4.1 Areosynchronous Satellite ...... 748 15.4.2 Sun-Synchronous Satellite ...... 752 15.5 Ground Track of a Satellite ...... 754 15.5.1 Representing the Ground Track ...... 754 15.5.2 Apparent Inclination ...... 761 15.5.3 Velocity of Satellite in Circular Orbit ...... 762 15.6 Orbit Relative to the Sun: Crossing Times and Eclipse...... 763 15.6.1 Overpass Time for a Sun-Synchronous Satellite ...... 766 15.6.2 Eclipse Conditions ...... 767 15.7 Orbit Relative to Mars: Recurrence and Altitude ...... 773 15.7.1 Recurrence ...... 773 15.7.2 Altitude ...... 781 15.8 View from the Satellite ...... 782 15.8.1 Viewing Configuration and Pixel Distortion ...... 782 15.8.2 Swath Track for an LMO Satellite ...... 782 15.8.3 Image Acquisition and Apparent Inclination ...... 785 15.8.4 View from an SMO Satellite ...... 788 Contents XXIII

15.9 Spatiotemporal and Angular Sampling ...... 790 15.9.1 Examples of Sampling ...... 790 15.9.2 Sun Glint ...... 792 15.10 Natural Satellites ...... 795 15.10.1 Phobos and Deimos ...... 795 15.10.2 Space Exploration ...... 795 15.10.3 View and Sampling ...... 796 15.11 Historical Note: Kepler and the Planet Mars ...... 798 15.11.1 Calculating the Period of Revolution ...... 798 15.11.2 Other Calculations for the Earth and Mars ...... 802

16 Satellites of Other Celestial Bodies ...... 803 16.1 Planets of the Solar System ...... 804 16.1.1 Presenting the Planets ...... 804 16.1.2 Space Exploration of the Planets ...... 808 16.2 Geodetic and Astronomical Quantities for Planets ...... 816 16.2.1 Geodetic and Astronomical Data ...... 816 16.2.2 Satellite in Keplerian Orbit ...... 817 16.2.3 Geographical Maps ...... 821 16.3 Satellite of Planet in Real Orbit ...... 822 16.3.1 Perturbative Accelerations ...... 822 16.3.2 Classification of Satellites ...... 824 16.4 Ground Track for a Satellite of a Planet ...... 826 16.4.1 Satellites of Mercury ...... 827 16.4.2 Satellites of Venus ...... 830 16.4.3 Satellites of the Asteroid Eros ...... 834 16.4.4 Satellites of the Asteroids Vesta and Ceres ...... 838 16.4.5 Satellites of Giant Planets ...... 845 16.5 Natural Satellites in the Solar System ...... 852 16.5.1 Presentation of the Natural Satellites ...... 852 16.5.2 Space Exploration of Natural Satellites ...... 853 16.6 Geodetic and Astronomical Quantities for Natural Satellites .....853 16.6.1 Geodetic and Astronomical Data ...... 853 16.6.2 Satellite in Keplerian Orbit ...... 855 16.6.3 Geographical Maps ...... 855 16.7 Satellite of a Natural Satellite in Real Orbit ...... 856 16.7.1 Perturbative Accelerations ...... 856 16.7.2 Classification of Satellites ...... 857 16.8 Ground Track of a Satellite of a Natural Satellite ...... 861 16.8.1 Satellites of the Moon...... 861 16.8.2 Satellites of Europa and Ganymede ...... 872 16.8.3 Satellites of Titan ...... 873 16.8.4 Satellites of Triton ...... 877 XXIV Contents

16.9 Appendix: The Three Planetocentric Spheres ...... 877 16.9.1 Presenting the Three Spheres ...... 878 16.9.2 The Case of the Four Giant Planets ...... 879 16.10 Historical Note: Kepler and the Solar System ...... 881

Index of Astronomia Nova ...... 883

Bibliography ...... 885

Index ...... 891