zotero/storage/R7CL66XZ/.zotero-ft-cache

Michel Capderou
Handbook of Satellite Orbits
From Kepler to GPS

Handbook of Satellite Orbits

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 publication 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. Permissions 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 publication, 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.

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Foreword
Since the dawn of the space age with the launch of Sputnik 1 and Explorer 1, orbital mechanics 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 orbital 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 discipline 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 interesting 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 derivation 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, geopotential and perturbation methods connects theory to physical measurements and observables. The chapter on Orbit 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 occultation, 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 determination, 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 gravity assist that enables missions like Voyager Grand Tour, Galileo, Cassini-
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Huygens and tours of their satellites. 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 planetary 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 discipline.
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 commercial, 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 succeeded 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 functioning 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.
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• 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 orbital elements, 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 is often used by 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 representations 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 observation, 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.

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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 somewhat 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.

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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, the Ecole 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, webmaster 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

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ÁÜ ÛÒ
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 atmosphere 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 representations 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.

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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
2 Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1 Earth Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.1 Diﬀerent Deﬁnitions of Latitude . . . . . . . . . . . . . . . . . . . . 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
3 Geopotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.1 Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.2 Review of Work and Potential . . . . . . . . . . . . . . . . . . . . . . 54
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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
4 Keplerian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Mean Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Mean Anomaly M . . . . 108 4.4.2 Expression t = t(θ) and the True Anomaly v . . . . . . . . . . 108 4.4.3 Expression t = t(r) and the Eccentric Anomaly E . . . . . . 110 4.4.4 Relating the Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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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 ) and E(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

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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

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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 Geostationary Orbit . . . . . . . . . . . . . . . . . . . . 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

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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 Circular Orbit . . . . . . . . . . . . . . . . . . . 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

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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

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11.8 11.9 11.10

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 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 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 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 Latitudes 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

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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
14.10 14.11

Historical Note: The First Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 698 14.9.1 Transit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 14.9.2 The Soviet System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 Appendix: GPS and Tectonic Plates . . . . . . . . . . . . . . . . . . . . . . . . 702 Appendix: GPS and Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 14.11.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 14.11.2 Proper Time Diﬀerence . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 14.11.3 Eﬀect Due to Orbital Eccentricity . . . . . . . . . . . . . . . . . . . 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 15.10 15.11

Spatiotemporal and Angular Sampling . . . . . . . . . . . . . . . . . . . . . . . 790 15.9.1 Examples of Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 15.9.2 Sun Glint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792 Natural Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 15.10.1 Phobos and Deimos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 15.10.2 Space Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 15.10.3 View and Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 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 16.10

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 Historical Note: Kepler and the Solar System . . . . . . . . . . . . . . . . . 881

Index of Astronomia Nova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891

Chapter 1
Geometry of the Ellipse

We will be concerned with ellipses in two diﬀerent contexts:
• The orbit of a satellite around the Earth (or the orbit of a planet around the Sun) is an ellipse.
• A cross-section of the planet Earth containing the polar axis is an ellipse.
Naturally, in both cases, this is just a ﬁrst approximation. However, a preliminary study of this geometric object will prove useful before going into greater detail.
In the ﬁrst case, the ellipse will be viewed as a geometrical object localised by its focus and speciﬁed by its eccentricity e. In the second, it will arise rather as a ﬂattened circle, localised by its center and characterised by the degree of ﬂattening f .
Ellipses can be deﬁned in many diﬀerent ways, as we shall now see. It is indeed a rich geometrical object.
1.1 Deﬁnition and Properties
1.1.1 Conic Sections
Consider a cone C, with apex S, and a plane P which does not pass through S. Let P be the plane parallel to P which does pass through S. There are three possible cases:
• If P lies outside the cone, the intersection of P and C is an ellipse. • If P is tangent to the cone, the intersection of P and C is a parabola. • If P lies within the cone, the intersection of P and C is a hyperbola.
In ancient times, these conic sections were viewed as a geometrical curiosity.

M. Capderou, Handbook of Satellite Orbits: From Kepler to GPS,

1

DOI 10.1007/978-3-319-03416-4 1,

© Springer International Publishing Switzerland 2014

2

Chapter 1. Geometry of the Ellipse

When he wanted to explain the orbit of Mars around the Sun, Kepler was the ﬁrst to rediscover the notion of ellipse that had been so carefully enunciated by the Greek geometers. Later on, with the help of Newton’s theory, astronomers were able to show that the trajectory of a body subject to a gravitational force could be not only an ellipse, but also a parabola or a hyperbola. Kepler’s ellipse was thus replaced by the whole family of conic sections.
We begin by describing the many diﬀerent ways of understanding conic sections.
Knowledge of this particular curve goes back at least to Ancient Greece. Menaechmus1 and Apollonius of Perga2 deﬁned the ellipse as one of the family of conic sections.3
1.1.2 Deﬁnition and Properties of the Ellipse
In this book, we shall be interested almost exclusively in the ellipse, because of all the conic sections, only this one constitutes a periodic trajectory.
The ellipse has many properties. We may thus choose one as the deﬁnition and deduce the others from it by means of rather straightforward demonstration. In general, the ﬁrst of the following is usually taken as the deﬁnition, while the others are considered as consequences.
Deﬁnition
[1] The ellipse is the locus of points M in the plane such that the sum of the distances M F and M F to two ﬁxed points F and F , called the foci, is constant.4

Å Ò ÕÑÓ 1Menaechmus (

, −375 to −325), was a Greek mathematician and member of

Plato’s Academy. He is credited by Eratosthenes with the discovery of the conic sections,

objects which arose in the so-called Delian problem, also known as doubling the cube (given a cube of volume a3, construct another of volume 2a3). Menaechmus sought two numbers

x and y such that a/x = x/y = y/2a, which leads to x3 = 2a3, by ﬁnding the intersections

ÔÓÐÐôÒ Ó È Ö Ó of a parabola (y = x2/a) and a hyperbola (y = 2a2/x). 2Apollonius of Perga, a city in Pamphylia in Anatolia (

, −262 to

−180) was a Greek geometer and astronomer and a student of Euclid. He wrote an eight

volume work entitled Conics, part of which has come down to us directly in the Greek

version, and part through the Arab translation. He studied the possible intersections of a

cone by an arbitrary plane, classifying them into the three types of conic section which still

carry the names he attributed to them.

ÛÒÓ ¸ ÓÙ 3The word “conic” comes from the Greek kˆonikos, an adjective derived from kˆonos (

˜

), meaning “pine cone”, the fruit of certain conifers.

4A direct application of this deﬁnition can be used to draw an ellipse in what is sometimes

known as the gardener’s method. Two stakes are stuck in the ground, some distance d apart.

A non-stretch string of length > d is attached with one end at each stake. If we then hold

a third stake against the string and move it in such a way that the string is always taut,

this stake will trace out an ellipse with focal points at the two ﬁxed stakes. The major axis

is equal to and the eccentricity is d/ .

1.1 Deﬁnition and Properties

3

Main Properties
[2a] The ellipse is an aﬃne transformation of a circle. [2b] The projection of a circle on a plane is an ellipse. [3] The ellipse is the locus of points M such that the ratio M F/d is a constant q, q < 1, where F is a ﬁxed point, the focus, and d is the distance between M and a ﬁxed straight line called the directrix. [4] The ellipse is obtained from the intersection of a cone and a plane in the case speciﬁed above.
Other Properties
Here are a few more properties that we shall not prove here. [5] The ellipse is the locus of the centers of circles passing through a ﬁxed point F and tangent to a ﬁxed circle, the director circle, with center F . [6] A light ray passing through one focus will reﬂect on the ellipse and pass through the second focus.5 [7] If the ends of a line segment of ﬁxed length run along two perpendicular straight lines, an arbitrary point on the segment will describe an ellipse and the envelope of the segment is then an astroid.

1.1.3 Applications of the Deﬁnition
Ellipse as a Geometrical Object
From the deﬁnition [1], the ellipse has a center6 of symmetry O, which we shall call the center of the ellipse. Taking O as the origin for an orthonormal Cartesian frame of reference (see Fig. 1.1), the various points of interest have the following coordinates:

OF =

c 0

,

OF =

−c 0

,

OA =

a 0

,

OB =

0 b

.

We use the standard terminology:

a = OA , semi-major axis of the ellipse , b = OB , semi-minor axis of the ellipse , c = OF = OF , focal distance .

b≤a,

5The term was introduced into scientiﬁc Latin (New Latin) by Kepler in 1603, from the

Latin word focus, i, which means “a place where ﬁre is made”. A pencil of light rays passing

through one focus will converge at the other, and could start a ﬁre there. This is indeed the

explanation for the name “focus” (see Fig. 1.11).

6The word “center” comes from Old French centre, which comes from the Latin centrum,

Ø ÒØÖÓÒ¸ ÓÙ ÒØ Û i, meaning “compass point”, “center of a circle”, or “midpoint of an ensemble”. The word

derives from the Greek kentron,

, a “goad”, from the verb

, meaning

“to prick or sting”. This is a rather rare example of a technical country word which became

scientiﬁc and later assumed a very general meaning. The Latin word spread to all the Latin

languages and many non-Latin ones, too.

4

Chapter 1. Geometry of the Ellipse

y

Y

B
M G

q

Ax

FЈ

O

HF

X

e = 0.65

Fig. 1.1 : Ellipse and principal circle, showing the notation used for the Cartesian coordinate system (axes Ox, Oy) and the polar coordinate system (r = F M , angle θ).

Since c is always strictly less than a, we can always deﬁne a number e such that

c = ea , where 0 ≤ e < 1 .

(1.1)

The real number e is the eccentricity of the ellipse.
Equation of the Ellipse in Cartesian Coordinates Centred on O
Consider an orthonormal Cartesian frame of reference (O; x, y), centered on O. The axis Ox is the straight line F F , the focal axis. Let M (x, y) be a point on the ellipse and set

r = FM , r = F M .

Deﬁnition [1] applied to M at A yields

r + r = 2a ,

(1.2)

and when M is at B, we obtain b2 = a2 − c2 = a2(1 − e2) .

(1.3)

1.1 Deﬁnition and Properties

5

The lengths r and r are expressed in Cartesian coordinates through their squares:

r2 = (x − c)2 + y2 = (x − ea)2 + y2 ,

(1.4)

r 2 = (x + c)2 + y2 = (x + ea)2 + y2 ,

(1.5)

and by the diﬀerence

r 2 − r2 = 4cx = 4eax .

(1.6)

Using (1.2), we obtain

r 2 − r2

r −r=

= 2ex ,

r +r

and ﬁnally the following expressions for r and r , whose sum and diﬀerence are as given:

r = a − ex ,

(1.7)

r = a + ex .

(1.8)

Note that these expressions are very simple and do not involve any square roots.
Using (1.4) and (1.7), we then have
r2 = (x − ea)2 + y2 = (a − ex)2 ,

x2 + e2a2 + y2 = a2 − e2x2 ,

x2(1 − e2) + y2 = a2 − e2a2 ,

whence

x2

y2

a2 + a2(1 − e2) = 1 ,

(1.9)

or again

x2 y2 a2 + b2 = 1 ,

(1.10)

which clearly reveal the role played by the semi-minor and semi-major axes.

6

Chapter 1. Geometry of the Ellipse

Equation of the Ellipse in Polar Coordinates Centred on F
Consider ﬁrst the orthonormal Cartesian frame (F ; X, Y ), centered on F , with axis F X identiﬁed with Ox and F Y parallel to Oy. We then have the following relation between the abscissæ:

X = x − ea .

For a point M (X, Y ) of the ellipse, (1.7) becomes r = a(1 − e2) − eX .

We then deﬁne the quantity

p = a(1 − e2) ,

(1.11)

known as the parameter or semi-latus rectum of the ellipse. It corresponds to the distance between the focus F and the point of intersection of the ellipse with the axis F Y (which is the straight line with equation X = 0). In Fig. 1.1, p = F G. Note that b is the geometric mean of a and p :

b2 = ap .

(1.12)

The equation for the ellipse can once again be written very simply as

r = p − eX .

(1.13)

Consider polar coordinates centered on F (see Fig. 1.1). The coordinates r and θ of an arbitrary point M are deﬁned by

r = FM = FM ,

(1.14)

θ = (F X, F M ) .

(1.15)

The abscissa of M in Cartesian coordinates is then X = r cos θ and (1.13) yields

p

r=

.

1 + e cos θ

(1.16)

The equation for the ellipse in polar coordinates is thus

a(1 − e2)

r(θ) =

.

1 + e cos θ

(1.17)

1.1 Deﬁnition and Properties

7

Note on Conic Sections

If we always deﬁne the parameter p to be the distance between the focus

F and the point G at which the conic section intersects the perpendicular to

the focal axis passing through F , the relation (1.16) can be used to deﬁne all

the conic sections,7 with the eccentricity discriminating between them:

⎧ ⎨ 0 e < 1 , for the ellipse ,

⎩

e e

= >

1 1

, ,

for the parabola , for the hyperbola .

Note that, for e = 0, the ellipse is a circle, and if e ≥ 1, the conic section extends to inﬁnity.

1.1.4 Demonstrating the Main Properties
Here we deduce the main properties of the ellipse. Deﬁnition [1] =⇒ Property [2a]
An aﬃne transformation or aﬃnity with axis Δ, direction δ, and coeﬃcient k (k = 0) is the point transformation which maps any point P in the plane

7Appolonius used the following three terms to characterise conic sections, inspired by

the language of the Pythagoreans:

é ÐÐ Ý ¸ Û • Ellipse ( é Ô Ö ÓÐ ¸ • Parabola (

) means “lacking” or “omitted”. ˜ ) describes the action of throwing, and hence suggests a map-

é Ô Ö ÓÐ ¸ ping, or comparison.

• Hyperbola (

˜ ) describes the action of throwing higher, hence invokes the

idea of an excess.

Ð Ô Ò The ﬁrst term comes from , meaning “in”, and the verb

, meaning “to leave or ne-

ÐÐ Ò glect”. The last two derive from the verb

, meaning “to throw”. The word “parabola”

should not be associated with the idea of throwing something in the sense of throwing a

projectile. This link between “parabola” and “throw” was described by Galileo, but was

unknown to the Ancient Greeks. It is interesting to note that, taken in this order, the three

terms “lacking”, “comparable”, and “in excess” associated with the conic sections corre-

spond to the values of the eccentricity as compared to unity. In fact, Apollonius introduced

the length p, the parameter of the ellipse, and calculated the areas of squares and rectangles

based on this length. With modern notation, taking the focal axis as the axis Ox, with Oy

perpendicular to it and the origin O at the apex of the conic section, we obtain the following

relation for the parabola:

y2 = 2px ,

which means that the area of a square of side y is equal to that of a rectangle of dimensions 2p × x. For the two other conic sections, we have:
• For the ellipse, y2 = 2px − (p/a)x2 . • For the hyperbola, y2 = 2px + (p/a)x2 .
In comparison with the parabola, it is thus the quantity (p/a)x2 which lacks for the ellipse and which is in excess for the hyperbola. The name for the conic sections was introduced by Kepler, Desargues, and Descartes, ﬁrst in Latin, then in the modern European languages.

8

Chapter 1. Geometry of the Ellipse

to the point P by the following construction: the straight line P P , parallel to δ, intersects Δ at H, where

HP = kHP .

Consider a circle of radius a, centered on O, and described by

x2 y2 a2 + a2 = 1 .

(1.18)

When applied to this circle, the aﬃne transformation of axis Ox and direction Oy (see Kepler’s drawing in Fig. 1.2 and also Fig. 1.3) transforms (1.18) to (1.10), provided that the aﬃne ratio is taken to be

b k= =

1 − e2 .

a

(1.19)

The circle of radius a deﬁned by (1.18) is called the principal circle of the ellipse of semi-major axis a, itself deﬁned by (1.10).
The area A of the ellipse can be deduced from that of the circle:

A = πab .

(1.20)

Property [2a] =⇒ Property [2b]
Property [2b] follows immediately from [2a]. Consider a circle of radius a and center O, lying in the plane P1. Now project it onto the plane P2, which

Fig. 1.2 : At the end of his book Astronomia Nova, Kepler devoted his Chap. LIX Elements of Geometry to the properties of the ellipse. On p. 286, he explains how the circle can be transformed into an ellipse by an aﬃnity.

k = 0.661 f = 0.339

1.1 Deﬁnition and Properties

9

y P PЈ

O

H

x

e = 0.750

Fig. 1.3 : Ellipse and principal circle. The ellipse is obtained from the principal circle by an aﬃne transformation with axis Ox, direction Oy, and coeﬃcient k.

makes a dihedral angle α with P1 and passes through O. Let Ox be the axis of intersection of the two planes and Oy perpendicular to it and lying in the plane P2. For the points of the circle in P1 which are projected into P2, the x coordinates remain the same, while the y coordinates are modiﬁed by a multiplicative coeﬃcient equal to cos α. The equation of the image in P2 is thus

x2

y2

a2 + (a cos α)2 = 1 ,

which corresponds to an aﬃne transformation with coeﬃcient k = cos α. This object is thus an ellipse.

Deﬁnition [1] =⇒ Property [3]

If we write r in (1.7) in the form

r = a − ex = e

a e

−

x

,

(1.21)

this brings in the quantity a/e − x, which is the distance from the point M , with abscissa x, to the straight line parallel to Oy and with abscissa a/e.

10

Chapter 1. Geometry of the Ellipse

y

DЈ AЈ

FЈ

B

M

J

O

F

AD

x

BЈ

e = 0.750

Fig. 1.4 : Ellipse and directrixes. The ellipse is the locus of points M such that the distance F M to the focus and the distance to a straight line denoted by d = M J have constant ratio e.
Hence, with r = F M and a/e − x = M J (see Fig. 1.4), it can be shown that the points of the ellipse are such that the ratio F M/M J is constant and in fact equal to e :
FM =e.
MJ Note that, if D is the intersection of the directrix with Ox, we have
OF · OD = OA2 = a2 .
Deﬁnition [1] =⇒ Property [4]
The equivalence between the deﬁnition of a conic by focus and directrix and its deﬁnition as the intersection of a plane and a cone was not shown until the nineteenth century. Here we consider the ellipse and give the proof of Dandelin’s theorem.8
Consider a cone with apex S and axis Sx. Consider also a plane P which cuts the cone and such that the plane parallel to P passing through S lies
8This theorem, proven in 1822, is also called the Dandelin–Qu´etelet theorem, after the two Belgian mathematicians Germinal Dandelin (1794–1847) and Adolphe Qu´etelet (1796– 1841).

1.1 Deﬁnition and Properties

11

S

C
FЈ AЈ CЈ

D O
A F
DЈ

OЈ

x

Fig. 1.5 : Plane representation, in the plane T perpendicular to the intersecting plane P, of the cross-section of the cone of axis Sx by the plane P, represented here by the straight line AA .

outside the cone. Let T be the plane perpendicular to P and containing Sx. The intersection of this plane T (called the plane of the page) with the cone comprises the two generators SA and SA, while its intersection with the plane P is the straight line AA .
Consider ﬁrst the view in the plane T , as shown in Fig. 1.5. Among the circles tangent to the three sides of the triangle SAA , we consider the two circles with centers lying on Sx. O is the center of the inscribed circle and the points of contact between the circle and the triangle are F on AA and C and D on SA and SA, respectively. Likewise, let O be the center of the escribed circle, and let the points of contact of this circle with the triangle be F on AA and C and D on SA and SA , respectively. Segments CD and C D are then parallel, each being perpendicular to the axis Sx.
We now rotate the ﬁgure about the axis Sx, but keeping the straight line AA ﬁxed: SA and SA generate the cone, while the circles centered at O and O generate two spheres inscribed within the cone. By translation, the straight line AA generates the plane P perpendicular to T (see Fig. 1.6). The spheres O and O are tangent to the plane P at F and F , and they are tangent to the cone at two parallel circles of diameter CD and C D , respectively.
Let M be a point on the conic section, i.e., the intersection of the given cone and the plane P. SM cuts the circles CD and C D at J and J , and

12

Chapter 1. Geometry of the Ellipse

S

C

D

JO

A F

FЈ M
AЈ

IЈ CЈ

DЈ

JЈ OЈ

x

Fig. 1.6 : Three-dimensional representation of the cone of axis Sx and the intersecting plane P. The curve of intersection goes through the points A, A , and M .

at these points, the generator SM is tangent to the spheres centered at O and O . Since M F and M F are tangent to the same spheres, respectively, we have:
• For the sphere centered at O, M F = M J. • For the sphere centered at O , M F = M J .
Now M J + M J = JJ , which is constant and equal to both CC and DD . Hence,
M F + M F = constant ,
and M describes an ellipse with foci F and F . The plane P cuts the parallel planes containing the circles CD and C D
in straight lines perpendicular to the plane T , passing through I and I , respectively. It can be shown that these two straight lines, perpendicular to F F , are the directrixes of the ellipse.

1.1 Deﬁnition and Properties

13

1.1.5 Eccentricity and Flattening

From (1.3), the eccentricity e can be expressed in the form

e2

=

a2 − b2 a2

.

(1.22)

The relation between a and b can also be given in terms of the ﬂattening f , which is deﬁned by

a−b

f=

.

a

Note that the aﬃne ratio k deﬁned by (1.19) is k = 1 − f . We immediately deduce the relationship between e and f :

(1.23)

b2 a2

=

1 − e2

=

(1 − f )2

.

This leads to the explicit relations

(1.24)

f = 1 − 1 − e2 ,

(1.25)

e = f (2 − f ) .

(1.26)

Figure 1.7 shows the transformation of an ellipse with ﬁxed semi-major axis a when e is increased in regular steps of 0.1. When instead we increase f in the same steps, the behaviour of the ellipse is very diﬀerent, as shown in Fig. 1.8.
Gauss introduced the angle of eccentricity , deﬁned by

e = sin .

(1.27)

We may then write b = a cos ,

c = a sin ,

p = a cos2 .

Later we shall see how certain formulas can be made more elegant with the help of this auxiliary variable.

Low Eccentricities

There are many situations in which an ellipse is very close to its principal circle. In the case e 1, (1.24) gives

e2 2f ,

(1.28)

whence the ﬂattening varies approximately as the square of the eccentricity.

14

Chapter 1. Geometry of the Ellipse

0.0
0.6 0.7 0.8
0.9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 1.7 : Ellipses with the same semi-major axis but eccentricity e varying from 0.0 to 0.9, in constant steps of 0.1. The corresponding eccentricity e is indicated on one of the two foci and on one end of the semi-minor axis. Note that the foci are regularly spaced.

e

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.0

0.1 0.2 0.3 0.4

Fig. 1.8 : Ellipses with the same semi-major axis but ﬂattening f varying from 0.0 to 0.9, in constant steps of 0.1. The corresponding ﬂattening f is indicated on one of the two foci and on one end of the semi-minor axis. Note that the ends of the semi-minor axes are regularly spaced.

f

1.1 Deﬁnition and Properties

15

Example 1.1 Eccentricity and ﬂattening for the orbit of Mars and Kepler’s doubts.

The elliptic orbit of Mars is characterised by a = 1.52366 a.u. (astronomical units) and e = 0.093412. The orbit looks almost circular:
√ b/a = 1 − e2 = 0.099564 = 0.99782 =⇒ f = 0.0022 ,
and the relative diﬀerence between a and b is 0.22 %. However, the distance between the center of the almost circular ellipse and the focus (the Sun) is signiﬁcant:
c = ea = 0.14233 a.u.
This means that the distance from Mars to the Sun varies between 1.38133 a.u. at perihelion and 1.66599 a.u. at aphelion.
In 1600, Kepler thus formulated what was to be his ﬁrst law (in 1609): the orbit of Mars is circular and the Sun is not at its center. Kepler used the notion of ellipse from 1603. He later wrote:
I originally assumed that the orbits of the planets were perfect circles. This mistake cost me all the more time and eﬀort in that it was supported by the authority of all the philosophers and was indeed quite plausible from a metaphysical point of view.
Figure 1.9 is one of Kepler’s drawings, taken from Astronomia Nova, in which he shows the orbits of Mercury, Venus, the Earth, and Mars (relative to the Sun). They are almost circular.9 However, the orbit of Mars is quite clearly oﬀ center, as is Mercury’s. See also the historical note on Kepler and the planet Mars at the end of Chap. 15.

Example 1.2 Calculating the lengths b and c for an ellipse speciﬁed by a and e in the case of the almost circular orbit of the satellite MetOp-A.

The ellipse representing the orbit of the satellite MetOp-A is speciﬁed by its semi-major axis a = 7,195,606 m and its eccentricity e = 0.0011655. With this value of the eccentricity, we obtain
1 − e2 = 1 − 1.3584 × 10−6 = 1 − 0.6792 × 10−6 ,

9To illustrate this property, V.I. Arnold suggests the following experiment in one of his books: Drop a droplet of tea close to the center of the cup. The waves will come together at the symmetric point. This is due to the deﬁnition of the foci of the ellipse, which implies that waves coming from one focus of the ellipse will converge at the other.

16

Chapter 1. Geometry of the Ellipse

Fig. 1.9 : Positions of Mars in a heliocentric frame. The planet is observed at four diﬀerent dates separated by time intervals that are multiples of the sidereal period of revolution (687 days). Mars is then in the same position relative to the stars. Kepler calculated the positions of the Earth at these dates and deduced the eccentricity of the Earth orbit. He was thus able to reconstruct the orbits of all the planets. The orbit of Mars appears almost circular but signiﬁcantly oﬀ center. The orbits of Mercury, Venus, and the Earth (with the Moon) are also shown. They look circular, but the Sun is not exactly at the center. It is interesting to note that the printers of the day were averse to completely white regions in drawings, so the empty parts of illustrations were systematically adorned with little ﬂowers. J. Kepler: Astronomia Nova, Chap. XXVII, p. 149.

1.2 Applications and Other Characteristics

17

and hence a − b = f a = 4.9 m. We thus obtain the following values for a, b, and c :
a = 7,195,606 m , b = 7,195,601 m , c = ea = 8,386 m .
In conclusion, the orbit is almost circular since the diﬀerence between a and b is only 5 m for a total of 7,200 km. However, the value of c, which measures the distance between the center of the ellipse and the focus is 8.4 km, which would be signiﬁcant in any calculation of the altitude of the satellite.

1.2 Applications and Other Characteristics

1.2.1 Arc Length of an Ellipse

The aﬃne transformation of the circle (O; a) (see Fig. 1.10) with axis Ox, orthogonal direction, and ratio b/a gives the ellipse (O; a, b). The coordinates of N on the circle and M on the ellipse can be expressed in terms of the polar angle10 u, centered on O :

ON =

a cos u a sin u

,

OM =

a cos u b sin u

.

(1.29)

y N
M

uψ O

Fig. 1.10 : Ellipse and

principal circle, show-

A

ing the angles u and ψ

x

to the center and the

arc AM of the ellipse.

e = 0.75
10This angular parameter u corresponds to the parametric (or reduced) latitude u in geodesy or Kepler’s eccentric anomaly E in astronomy, as we shall see later on.

18

Chapter 1. Geometry of the Ellipse

Arc Length
To obtain the arc length of an ellipse, measured from an arbitrary origin, we ﬁrst calculate the element ds of the curvilinear abscissa in terms of dx and dy :
ds2 = (a2 sin2 u + b2 cos2 u) du2 = a2(1 − e2 cos2 u) du2 .

The arc length of the ellipse between A (u = 0) and M (u = α) is then obtained by integration:

α

s(α) = a

1 − e2 cos2 u du .

0

(1.30)

The value of s(α) cannot be expressed in terms of simple analytic functions. This integral11 is an incomplete elliptic integral of the second kind. It can be
obtained as the limit of an inﬁnite series.

Perimeter of the Ellipse

The perimeter L of an ellipse is obtained as a complete elliptic integral of the second kind:

π/2

L = 4a

1 − e2 cos2 u du ,

0

(1.31)

which has no simple analytic representation. Expanding in powers of the eccentricity, we obtain Lambert’s formula:

L = 2πa 1 −

1 2 e2 −

1×3

2 e4 −

1×3×5

2 e6 +···

. (1.32)

2 1 2×4 3 2×4×6 5

For this elliptic integral, there are several approximate formulas with varying degrees of complexity and which become more accurate as the eccentricity e decreases. The simplest approximate result is the one provided by Euler:

L π 2(a2 + b2) ,

(1.33)

or again,

L

π

3

(a

+

b)

−

√ ab

.

2

(1.34)

11Such integrals are classiﬁed into three kinds. Those of the second kind have the form

φ

E(φ, k) =

1 − k2 sin2 ϑ dϑ ,

0

with parameter k such that 0 < k2 < 1. These elliptic integrals are said to be incomplete when φ is arbitrary. When φ = π/2, one speaks of complete elliptic integrals.

1.2 Applications and Other Characteristics

19

For greater accuracy, we may use the formulas given by Ramanujan12:

L π 3(a + b) − (a + 3b)(3a + b) ,

or again,

L π(a + b) 1 +

√3h

a−b 2

, where h =

.

10 + 4 − 3h

a+b

To ﬁrst order in f , all these formulas are equivalent to

f L π(a + b) = 2πa 1 −
2

e2 2πa 1 − ,
4

(1.35)

which amounts to considering the ellipse as a circle with radius equal to the arithmetic mean of the two semi-axes of the ellipse.13

1.2.2 Radius of an Ellipse

The radius of an ellipse is the distance from an arbitrary point M on the ellipse to its center O. This radius is denoted Rψ, since it is not constant but varies with ψ. The polar coordinates centered on O (see Fig. 1.10) are

Rψ = OM , ψ = (Ox, OM ) .

The Cartesian coordinates of M , viz.,

OM =

x = Rψ cos ψ y = Rψ sin ψ

,

(1.36)

give, using (1.10),

Rψ2

cos2 ψ sin2 ψ a2 + b2

=1.

12Srinivaˆsa Aiyangˆar Rˆamˆanujan (1887–1920) was a highly original Indian mathematician, considered as a genius in his ﬁeld. He taught himself mathematics using a compendium of 6,000 theorems, most of which were given without proof. From the age of 17, and in particular during his time in England between 1913 and 1919, he established hundreds of formulas which he noted down without proof in his notebooks. His intuition and memory were astonishing. He worked on number theory, elliptic integrals, Bernoulli numbers, and so on. Ramanujan established the values of π and e using continued fractions, series, or extremely concise formulas.
13In the words of Kepler: “Any elliptic circumference is very close to the arithmetic mean between the circle of longest diameter and the circle of shortest diameter.” Astronomia Nova, Chap. LIX, p. 287. He considered only the elliptic orbits of the known bodies.

20

Chapter 1. Geometry of the Ellipse

Then, in terms of the ﬂattening f , the radius Rψ of the ellipse is

Rψ =

a ,

cos2

ψ

+

sin2 ψ (1 − f )2

and in terms of the eccentricity e, Rψ =

√ a 1 − e2
. 1 − e2 cos2 ψ

For small ﬂattening (f 1), we have

Rψ

a 1 − f sin2 ψ , or Rψ

a 1 − e2 sin2 ψ . 2

(1.37) (1.38) (1.39)

1.2.3 Radius of Curvature of an Ellipse

Any curve can be approximated near one of its points (provided it is not a cusp), over an inﬁnitesimal interval, by a circle known as the osculating circle. Its radius ρ is the radius of curvature. The locus of the centers of these osculating circles is called the evolute of the curve. Equivalently, we may deﬁne the evolute of a plane curve as the envelope of all the normals to this curve (see Fig. 1.11). It can be shown that, at a given point, the osculating circle is unique.
Consider an ellipse speciﬁed by the angular parameter u. Let M be a point on the ellipse and M the corresponding point on the evolute. This means that the circle of center M and radius ρ = M M is tangent to the ellipse at M . The vector M M is normal to the ellipse at M .
The normal at point M , with parameter u, is written in the following way in terms of the coordinates X and Y :

aX − bY − c2 = 0 . cos u sin u

(1.40)

The parametric equation of the evolute is obtained by determining the two unknowns X and Y using two equations: the equation of the normal given by (1.40) and the derivative of this equation with respect to u. These can be written

aX sin u − bY cos u = c2 sin u cos u ,

aX cos u + bY sin u = c2(cos2 u − sin2 u) ,

1.2 Applications and Other Characteristics

21

FЈ f = 0.200

M MЈ F
e = 0.600

Fig. 1.11 : Radius of curvature of an ellipse. M is an arbitrary point on an ellipse with foci F and F . M is the center of the osculating circle of the ellipse at M . When M moves around the ellipse, M describes the evolute of the ellipse. This diagram illustrates another property of the ellipse: M M bisects the angle F M F , so any ray of light passing through F will pass through F after reﬂection on the ellipse.

and we obtain

aX = c2 cos3 u , bY = c2 sin3 u .

In short, the coordinates of M and M can be written in the form

OM =

a cos u b sin u

,

OM

=

⎛ ⎜⎝

a2 − b2 a
a2 − b2

⎞
cos3 u ⎟⎠ sin3 u

.

b

(1.41)

The evolute of the ellipse shown in Fig. 1.12 (upper) is the curve A B C D with four cusps. It is the aﬃne transformation with axis Ox, direction orthogonal, and ratio −b/a, of the astroid with equation

A cos3 u A sin3 u

,

where A = a2 − b2 = c2 = ae2 .

a

a

The components of the vector M M can be calculated from (1.41):

⎛ M M =⎜⎜⎝

a sin2 u + b2 cos2 u a
b cos2 u + a2 sin2 u b

cos

⎞ u⎟⎟⎠ =

(a2

sin2

u

+

b2

cos2

⎛ cos

u)⎝

a sin

u⎞ u⎠

.

sin u

b

(1.42)

22

Chapter 1. Geometry of the Ellipse

J
C
f = 0.165
K
f = 0.005 f = 0.046 f = 0.134 f = 0.286 f = 0.564

B

DЈ

CЈ O

AЈ

BЈ

D

I

A
e = 0.550
L
e = 0.100 e = 0.300 e = 0.500 e = 0.700 e = 0.900

Fig. 1.12 : Relationship between an ellipse and its evolute. Upper: The straight line joining A and B , centers of curvature for A and B, respectively, passes through I and is perpendicular to the diagonal JL of the rectangle IJKL, a rectangle with sides 2a and 2b, center O, escribed on the ellipse. Lower: The same construction for diﬀerent values of the eccentricity between e = 0.10 and e = 0.90, in steps of 0.20. The semi-major axis remains constant.

We deduce the value of the radius of curvature ρ = M M :

(a2 sin2 u + b2 cos2 u)3/2

ρ=

.

ab

(1.43)

1.2 Applications and Other Characteristics

23

We note the particular values for the points A and B :

⎛ a2 − b2 ⎞

⎛ b2 ⎞

u = 0 =⇒ OA =

a 0

, OA = ⎝

a

⎠ , AA=⎝ a ⎠ ,

0

0

⎛

⎞

⎛⎞

π u=

=⇒ OB =

2

0 b

,

0 OB = ⎝ b2 − a2 ⎠ ,

0 B B = ⎝ a2 ⎠ .

b

b

The radius of curvature varies between two extreme values:

• maximal for B and D, with ρ = a2/b , • minimal for A and C, with ρ = b2/a .

Let IJKL be the rectangle with center O, sides 2a and 2b, in which the ellipse is inscribed (see Fig. 1.12 upper). Using the coordinates of the various points calculated above, we obtain the following relations:

BI b =,
BB a

OA b =,
OB a

JL =

+2a −2b

,

which show that:
• The straight line A B passes through I. • The diagonal of the rectangle which does not pass through I is perpendic-
ular to A B .
Figure 1.12 (lower) shows this property for a range of eccentricities.

Chapter 2
Geodesy

2.1 Earth Ellipsoid

2.1.1 Diﬀerent Deﬁnitions of Latitude
Spherical Coordinates
Consider an orthonormal Cartesian frame (O; x, y, z). A point M in space can be identiﬁed by the three spherical coordinates r, ψ, λ, deﬁned in the following way. We select one axis Oz and project M to M on the plane xOy perpendicular to Oz passing through O. We then set

r = OM , ψ = (OM , OM ) , λ = (Ox, OM ) ,

where the ranges of the three coordinates are

r ∈ [0, ∞) ,

ππ ψ ∈ − ,+ ,
22

λ ∈ [0, 2π) .

The Cartesian coordinates of the point M can be expressed as follows in terms

of the spherical coordinates:

⎛

⎞

x = r cos ψ cos λ

OM = ⎝ y = r cos ψ sin λ ⎠ .

(2.1)

z = r sin ψ .

Now consider a sphere of radius R, centered on O. If this sphere is a representation of the (spherical) Earth, the geometric quantities deﬁned above have geographic signiﬁcance:
• Oz is the polar axis and xOy is the equatorial plane. • The distance r = R + h is often speciﬁed by the altitude h. • ψ is the latitude and λ is the longitude.

M. Capderou, Handbook of Satellite Orbits: From Kepler to GPS,

25

DOI 10.1007/978-3-319-03416-4 2,

© Springer International Publishing Switzerland 2014

26

Chapter 2. Geodesy

Fig. 2.1 : British stamp issued in June 1984 to commemorate the 100th anniversary of the adoption of the Greenwich Meridian as Longitude Zero for the Earth. Created by Sedley Place Design.

Latitude and longitude are quite generally deﬁned here in spherical coordinates. Circles passing through both poles are meridians (loci of points of constant longitude). Circles parallel to the equatorial plane are called parallels (loci of points at constant latitude), and the parallel in the equatorial plane is the equator. The meridians are great circles, while all the parallels apart from the equator are small circles.1
We apply the following conventions:
• The polar axis is oriented from the South Pole to the North Pole. • The meridian plane xOz serving as origin is called the prime meridian,
zero meridian, or (see Fig. 2.1) Greenwich meridian.2 • Longitudes are measured in the right-handed trigonometric sense from 0◦
to 360◦ or from −180◦ to +180◦, which gives values [+E/−W], i.e., positive for the east and negative for the west.
1If they intersect at all, a sphere and a plane intersect in a circle. If the plane passes through the center of the sphere, the circle in question is a great circle, otherwise it is a small circle.
2When it was oﬃcially chosen as prime meridian at the International Meridian Conference, held in Washington in 1884, the meridian passing through the Royal Observatory in Greenwich was already being used in most shipping charts (the British delegate declared at the time that, in terms of sheer tonnage, 72 % of world shipping trade was using charts based on Greenwich). On the other hand, when it came to land charts, there was a multitude of diﬀerent zero meridians. Naturally, France used the Paris meridian set down by Cassini, which passed through the Paris Observatory. Italy referred to the meridian in Rome, Russia to the one in Pulkovo, and so on. The meridian which passes through El Hierro Island (Isla de el Hierro), the westernmost island of the Canaries, and hence the westernmost point known to Europe before the discovery of the Americas, had the advantage that it gave only positive longitudes at the time. It had often been used in Europe in the seventeenth and eighteenth centuries. Even in the nineteenth century it was still to be found on several maps from central Europe.

2.1 Earth Ellipsoid

27

Ellipsoid
Delambre Airy Everest Bessel Clarke I Clarke II Hayford International Krassowsky
AIG67 WGS72 GRS80 WGS84
GEM-T2 EGM96 GRIM5 EIGEN EGM2008

Year
1810 1830 1830 1840 1866 1880 1924 1942
1967 1972 1980 1984
1990 1996 2000 2008 2008

a (m)
6, 375, 653 6, 376, 542 6, 377, 206.4 6, 377, 397.16 6, 378, 206.4 6, 378, 249.2 6, 378, 388.2 6, 378, 245
6, 378, 160 6, 378, 135 6, 378, 137 6, 378, 137
6, 378, 137 6, 378, 136.30 6, 378, 136.46 6, 378, 136.46 6, 378, 136.46

1/f (dimensionless)
334 299.3 300.8017 299.1528 294.9787 293.4660 297 293.3
298.2471 298.26 298.257222101 298.257223563
298.257 298.25765 298.25765 298.25765 298.25765

Table 2.1 : Reference ellipsoids used in geodesy (with year of application). Changing estimates of the values of the semi-major axis a and the ﬂattening f . The ellipsoids are divided into three chronological groups: (a) now historical ellipsoids, (b) ellipsoids from the satellite era, and (c) ellipsoids relating to geopotential models.

• Latitudes are measured in the right-handed trigonometric sense from−90◦ to +90◦, which gives values [+N/−S], positive for the north and negative for the south.
Coordinates on the Ellipsoid
Let us now model the Earth as an ellipsoid of revolution, with the polar axis as the axis of revolution. Consider a point M on the surface of the ellipsoid (the notion of altitude relative to this ellipsoid will be considered later). The intersection of an arbitrary meridian plane with the ellipsoid will be an ellipse with center O. We set a = OA and b = OB (see Fig. 2.2), where A is a point on the equator and B is the North Pole. The plane xOz is the meridian plane.
The semi-major axis a is the equatorial radius Re and the semi-minor axis b is the polar radius Rp. This ellipse, and hence also the Earth ellipsoid, is thus determined by two quantities. We choose a and the ﬂattening f (see Table 2.1).
From the symmetry of revolution, the point M can be identiﬁed by the longitude speciﬁed on the sphere, but for the latitude, we must think again. Indeed, the latitude was obtained historically by measuring angles between the directions of carefully selected stars and the local horizontal plane (or

28

Chapter 2. Geodesy

z

B N

J

M

u

ψ

j

xЈ

O

H

A

MЈ I

T

x

e = 0.55 zЈ

Latitude Symbol Reference direction

Angle

Geocentric ψ Geodetic ϕ Geographic ϕ Parametric u

Center of the Earth Normal to the ellipsoid Normal to the geoid Center of the Earth

ψ = (OA, OM ) ϕ = (OA, IM) ϕ = (OA, [plumb line at M ]) u = (OA, ON )

Fig. 2.2 : Diﬀerent deﬁnitions of latitude at a point M .

the direction of a plumb line, which is perpendicular to it). In Fig. 2.2, the latitude of the point M corresponds to the angle between the sightline to the north (say, the pole star, sightline M N , parallel to the polar axis OB) and the tangent to the ellipse at M . This angle is equal to ϕ = (OA, IM ), angle between the normal to the ellipse and the equatorial radius, called the geodetic latitude.3
The geographic latitude, also called the astronomical latitude, which is measured “in the ﬁeld”, takes the plumb line as reference, rather than the normal to any theoretical ellipsoid of reference. The plumb line hangs perpendicu-

3The word “geodesy” comes from the New Latin geodesia, as attested in the sixteenth

é ¸ century. This in turn came from the Greek, and in particular from the preﬁx geo-, ˜

é ¸ ØÓ ˜ , meaning “the Earth” or “the country”, and -desy, ´

´ , meaning “share”, in

the sense of equal shares distributed at mealtimes.

2.1 Earth Ellipsoid

29

lar to the equipotential surface represented by the local geoid. The diﬀerence between geodetic and geographical latitude, known as the deviation from the vertical, is at most 3 s of arc in regions where the geoid is particularly “uneven”. This diﬀerence must be taken into account for certain very accurate measurements that will not be considered in this book.
The geocentric latitude ψ at the point M is deﬁned relative to the center of the Earth: ψ = (OA, OM ). This angle is used in particular to specify satellite positions. The various deﬁnitions of latitude are summarised in Fig. 2.2.
To ﬁnd the relation between ψ and ϕ, we introduce yet another latitude with a purely geometric role, namely the parametric latitude u (see Fig. 2.2). Consider the principal circle of the ellipse. The parallel to Oz passing through M cuts the principal circle at N , a point used to deﬁne the parametric latitude by u = (OA, ON ).
The point M is obtained from N by an aﬃne transformation with axis Ox, direction Oz, and ratio b/a. We thus have

b tan ψ = tan u .
a

(2.2)

The aﬃne transformation conserves contact: it transforms the tangent to the

circle at N to the tangent to the ellipse at M . These tangents cut the axis Ox

at the same point T . The angle (T O, T M ) is complementary to ϕ and the

angle (T O, T N ) is complementary to u. This gives

a tan ϕ = tan u .
b

(2.3)

We deduce the relation between the two latitudes from (2.2) and (2.3), using f or e :

tan ψ tan ψ tan ϕ = (1 − f )2 = 1 − e2 .

(2.4)

These three latitudes always have the same sign at any given location. Their absolute values are ordered as follows:

|ϕ| |u| |ψ| ,

with equality only at the equator or the poles. For small values of the ﬂattening (f 1), (2.4) yields

tan ϕ (1 + 2f ) tan ψ .

Setting δϕ = ϕ − ψ and expanding tan(ϕ − ψ), we obtain

tan(ϕ

−

ψ)

=

tan ϕ − tan ψ 1 + tan2 ψ

2f tan ψ cos2 ψ = f sin 2ψ .

With these small-angle approximations, the diﬀerence between the angles becomes

δϕ f sin 2ϕ .

(2.5)

30

Chapter 2. Geodesy

Its maximum value δϕ0, reached when ϕ = ±π/4, latitude 45◦ North or South,

is equal to

⎧ ⎨ f = 0.0035281 (in radians) ,

δϕ0

=

⎩

f f

× ×

180/π = 0.19210 3600 × 180/π = 692

(in degrees) , (in arc sec) .

(2.6)

Example 2.1 For the planet Earth, calculate the diﬀerence between the equatorial and polar radii, and the maximal diﬀerence of latitude δϕ0.
Consider the values of the Earth ellipsoid WGS84 (World Geodetic System 1984, revised in 2004, and updated by EGM96) given in Table 2.1. The values required here are Re = a, Rp = a(1 − f ), with f = 0.00335281:
Re = 6, 378, 137.000 m , Rp = 6, 356, 752.314 m . The diﬀerence between the two radii is
δR = Re − Rp = f Re = 21.285 km . For the maximal diﬀerence in the latitude values, we obtain
δϕ0 = f = 3.35 mrad = 0.19◦ = 11 32 = 692 , which represents a diﬀerence δ 0 in the ﬁeld given by
δ 0 Reδϕ0 f Re = 21.3 km . Note therefore that, as long as f is much smaller than unity, the values of δR and δ 0 are the same, because they are equal to f Re.

From (2.4), we obtain

tan(ϕ − ψ)

=

fo sin 2ϕ 1 + fo cos 2ϕ

with

fo

=

e2 2 − e2

.

A more accurate expansion gives

(2.7)

ϕ

−

ψ

=

fo

sin

2ϕ

−

fo2 2

sin 4ϕ

+

fo3 3

sin 6ϕ

+

...

,

ϕ

−

ψ

=

fo

sin

2ψ

+

fo2 2

sin 4ψ

+

fo3 3

sin

6ψ

+

...

.

For the Earth (WGS84):

(2.8) (2.9)

δϕ = 692.72622 sin 2ϕ − 1.16324 sin 4ϕ + 0.00260 sin 6ϕ − 0.00001 sin 8ϕ , (2.10)

with δϕ = ϕ − ψ in arc sec.

2.1 Earth Ellipsoid

31

2.1.2 Cartesian Coordinates: Great Normal

In a Cartesian frame (O; x, z), it is straightforward to express the coordinates of M in terms of the latitudes u or ψ, as in (1.29) or (1.36). But the geodetic latitude ϕ has the greater practical value. We use (2.3) to express the relation between u and ψ, while bringing in the eccentricity:

1 cos2

u

=

1 + tan2 u

=

1

+ (1 − e2) tan2 ϕ

=

1 cos2 ϕ

− e2 tan2 ϕ

=

1 cos2 ϕ (1

− e2 sin2 ϕ)

,

whence

a cos u = a cos ϕ 1 − e2 sin2 ϕ −1/2 .

Furthermore,

sin u = tan u cos u =

b tan ϕ cos ϕ

1 − e2 sin2 ϕ

−1/2

,

a

whence, using the relation b2/a = a(1 − e2),

b sin u = a(1 − e2) sin ϕ 1 − e2 sin2 ϕ −1/2 .

Equation (1.29) thus gives

⎛

OM

=

⎜⎜⎜⎜⎝

x z

= =

a a

cos ϕ

⎞

(1 − e2 (1 − e2)

sin2 ϕ) sin ϕ

⎟⎟⎟⎟⎠

.

(1 − e2 sin2 ϕ)

(2.11)

The normal to the ellipse at M cuts the axis z Oz at I. In geodesy, the length IM is sometimes called the great normal (see Fig. 2.2). It is denoted by N .
Since x = IM cos ϕ, we deduce the value of N from (2.11):

a

N=

,

1 − e2 sin2 ϕ

(2.12)

and the plane Cartesian coordinates of M can be written in the form

OM =

x = N cos ϕ z = N (1 − e2) sin ϕ

.

(2.13)

We can now obtain the coordinates of I, where OI = z − N sin ϕ :

OI =

0 −N e2 sin ϕ

.

(2.14)

32

Chapter 2. Geodesy

The coordinates (geodetic latitude ϕ, longitude λ) of M on the ellipsoid are

therefore

⎛

⎞

x = N cos ϕ cos λ

OM = ⎝ y = N cos ϕ sin λ ⎠ .

(2.15)

z = N (1 − e2) sin ϕ

2.1.3 Radius of Curvature

The center of the radius of curvature M lies between I and M . We use (1.43) to calculate ρ as a function of ϕ. To begin with,

a2

sin2

u

+

b2

cos2

u

=

a2(1

−

e2

cos2

u)

=

a2 1

1 − e2 − e2 sin2

ϕ

,

whence we obtain ρ(ϕ) as

1 − e2 ρ = a (1 − e2 sin2 ϕ)3/2 ,

(2.16)

which can also be written

ρ

=

N

1 − e2 1 − e2 sin2 ϕ

.

(2.17)

For small values of the ﬂattening, (2.16) becomes

ρ 1 + 3 sin2 ϕ − 1 e2 .

a

2

(2.18)

In this case, the radius of curvature is equal to the equatorial radius, whatever the value of e, for two values of the latitude, viz., ϕ = ± arcsin 2/3 :

ρ = a ⇐⇒ ϕ 54.7◦N or 54.7◦S .

(2.19)

2.1.4 Radius of the Ellipse

The “natural” variable to use for the radius of the ellipse, denoted Rψ, is the geocentric latitude ψ, as we saw previously [see (1.38) on p. 20]. However, if we wish to express this radius as a function of the geodetic latitude ϕ, the resulting formula is more involved:

Rψ(ϕ) = a

(1 − e2)2 sin2 ϕ + cos2 ϕ =N
1 − e2 sin2 ϕ

(1 − e2)2 sin2 ϕ + cos2 ϕ . (2.20)

Note the relation obtained with the projection of OM on the axis Ox :

Rψ cos ψ = N cos ϕ .

(2.21)

2.1 Earth Ellipsoid

33

6410

6400 N
6390

6380

Radius (km)

6370

6360

Rψ

6350
r
6340

6330

EARTH 0

10

20

30

40

50

60

Latitude [N/S] (deg)

70

80 90

Fig. 2.3 : Diﬀerent radii relating to the Earth ellipsoid: radius of curvature ρ in the meridian plane, the great normal N , and the radius of the ellipsoid Rψ.

2.1.5 Degrees of Latitude and Longitude
Distance Along a Meridian or a Parallel
Let us return to the Earth ellipsoid and evaluate the distance corresponding to an inﬁnitesimal increase in the latitude (or the longitude) along a meridian (along a parallel). Along a meridian, a change dϕ in geodetic latitude will correspond to an elliptical arc dLM, identiﬁed at this point with the arc of the osculating circle, or radius ρM = M M [see Fig. 2.2 and (2.16) or (2.17)]:

dLM = ρM dϕ , with ρM = ρ .

(2.22)

Along a parallel, a change dλ in longitude will correspond to a circular arc dLP in a plane perpendicular to the polar axis, of radius ρP = JM [see Fig. 2.2 and (2.12)]:

dLP = ρP dλ , with ρP = N cos ϕ .

(2.23)

Figure 2.3 shows the changes in the various radii as a function of the latitude, whether it be a radius of curvature, like ρ (for ρM) or N (for ρP/ cos ϕ), or the

34

Chapter 2. Geodesy

Latitude
ϕ 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

1◦ latitude
ΔLM 110.574 110.583 110.608 110.649 110.704 110.773 110.852 110.941 111.035 111.132 111.229 111.324 111.412 111.493 111.562 111.618 111.660 111.685 111.694

−→
ΔLP/ cos ϕ 111.319 111.322 111.331 111.344 111.363 111.386 111.413 111.442 111.474 111.506 111.539 111.570 111.600 111.627 111.650 111.669 111.683 111.691 111.694

1◦ longitude
ΔLP 111.319 110.899 109.639 107.550 104.647 100.950
96.486 91.288 85.394 78.847 71.696 63.994 55.800 47.176 38.187 28.902 19.393
9.735 0.000

Arc
L(ϕ) 0.000 552.885 1,105.855 1,658.990 2,212.366 2,766.054 3,320.114 3,874.593 4,429.529 4,984.944 5,540.847 6,097.230 6,654.073 7,211.339 7,768.981 8,326.938 8,885.140 9,443.509 10,001.966

Historical
Lh(ϕ) 0.000 553.074 1,106.223 1,659.520 2,213.032 2,766.823 3,320.946 3,875.444 4,430.349 4,985.683 5,541.451 6,097.648 6,654.255 7,211.241 7,768.561 8,326.162 8,883.982 9,441.951 10,000.000

Table 2.2 : One degree of latitude and longitude as a function of latitude ϕ. Reference
ellipsoid: WGS84. Length ΔLM in km on the meridian for Δϕ = 1 degree of latitude. Length ΔLP in km on the parallel for Δλ = 1◦ of longitude. Length L of the meridian arc in km, measured from the equator. A historical value of the arc length Lh is also given, using the reference ellipsoid of Delambre and M´echain which served to deﬁne
the standard metre.

radius of the ellipse, like Rψ(ϕ). Note also the ranges of these variations for the Earth, between the equator and the pole (all monotonic in [0, π/2]):
• ρ varies between b2/a and a2/b, i.e., between 6, 335.439 and 6, 399.594 km. • N varies between a and a2/b, i.e., between 6, 378.137 and 6, 399.594 km. • Rψ varies between a and b, i.e., between 6, 378.137 and 6, 356.752 km.
Change of One Degree
It is perhaps more meaningful to replace the inﬁnitesimal change of angle by a change of one degree,4 which remains small compared with the whole circumference. Table 2.2 gives the value ΔLM for the length along the meridian of an arc of 1◦ around a central value ϕ of the geodetic latitude. Likewise, ΔLP
4In the navy, the nautical mile is deﬁned as the distance equivalent to 1 of arc of latitude, with the relation 1 nautical mile = 40 × 106/(360 × 60) = 1 851.851 m. The second of arc is equivalent to 1 = 1 nm/60 = 30.864 m. The speed is obtained directly in nautical miles per hour by spacing knots every 15.432 m (equivalent to 0.5 of arc) and measuring the rate at which the knots go by for a period of 30 s (or 0.5 min).

2.1 Earth Ellipsoid

35

C
f = 0.200

B DЈ
CЈ AЈ
BЈ D

A
e = 0.600

Fig. 2.4 : Length of 1◦ of latitude at the surface of the ellipsoid as a function of the latitude. Diﬀerent distances correspond to equal angles, and the eﬀect increases toward the poles. Angular diﬀerence in geodetic latitude represented in the ﬁgure: 5◦.

is the length along the parallel ϕ of an arc of 1◦ longitude. It is interesting to compare ΔLP/ cos ϕ with ΔLM (Fig. 2.4).

2.1.6 Meridian Arc Length
Arc Length of an Ellipse
In the general case, the length of arc of the ellipse, which we denote by L(ϕ), is calculated as an incomplete elliptic integral of the second kind, as in (1.30). Such integrals are usually carried out with the help of expansions using the Wallis integrals.
For a small value of the ﬂattening, Levallois notes that direct integration gives results that are just as accurate and which converge more quickly. Consider an element dLM of the meridian arc:
dLM = ρ dϕ = a(1 − e2)(1 − e2 sin2 ϕ)−3/2 dϕ .

This can be expanded as follows:

dLM

a(1 − e2) 1 + 3 e2 sin2 ϕ + 15 e4 sin4 ϕ + 105 e6 sin6 ϕ + · · ·

2

8

48

dϕ ,

36

Chapter 2. Geodesy

noting that terms in e8 and beyond, neglected here, contribute less than 1 mm in the arc length of the Earth meridian. The sine functions are linearised using the Moivre formula:
2 sin2 ϕ = 1 − cos 2ϕ , 8 sin4 ϕ = 3 − 4 cos 2ϕ + cos 4ϕ ,

32 sin6 ϕ = 10 − 15 cos 2ϕ + 6 cos 4ϕ − cos 6ϕ .

We thus obtain an expression for dLM that can be integrated term by term from 0 up to the latitude ϕ :

ϕ
L(ϕ) = dLM .
0
The meridian arc length L(ϕ) is then

(2.24)

L(ϕ) a(1 − e2)A(ϕ) ,

(2.25)

where

A(ϕ)

=

A0ϕ

−

3 8

A2

e2

sin 2ϕ

+

15 256

A4e4

sin

4ϕ

−

35 3072

A6e6

sin

6ϕ

and

A0

=

1

+

3 e2 4

+

45 e4 64

+

175 e6 256

,

A2

=

1

+

5 e2 4

+

175 e4 128

,

A4

=

1

+

7 e2 4

,

A6 = 1 .

A length of great historical importance was the quarter meridian, i.e., the arc length of the meridian from the equator to the pole:

L

π 2

=

π a(1
2

−

e2)A0

.

(2.26)

Expanding to second order in e, this yields

Lπ 2

π

e2

a 1−

2

4

π a

1− f

πa+b

=

,

2

2 22

(2.27)

already given in (1.35). For the Earth,

e2 1/150 6.7 × 10−3 , e4 1/22500 4.4 × 10−5 .

Lengths of the Meridian and the Equator

The full length of the meridian LM going right around the globe (the ellipsoid) and the full length of the equator LP/equator are

LM = 2πa(1 − e2)A0 ,

LP/equator = 2πa .

Numerical values are given in Table 2.3.

2.2 Altitude Relative to the Ellipsoid

37

Ellipsoid

Length of the Length of the meridian (m) equator (m)

Delambre and M´echain (creation of the metre)

40, 000, 000

40, 059, 944

WGS84 (and current ellipsoids)

40, 007, 864

40, 075, 016

Table 2.3 : Lengths of the meridian and the equator. Historical values (deﬁnition of the metre) and current values, in metres.

2.2 Altitude Relative to the Ellipsoid

2.2.1 Deﬁnition of Geodetic Altitude and Nadir

Consider a point S above the ellipsoid at a distance r from the point O

at the center of the ellipsoid (see Fig. 2.5). Its longitude is λ. If S is identiﬁed

by the geocentric latitude ψ = (Ox, OS), its Cartesian coordinates are, as

obtained in (2.1),

⎛

⎞

x = r cos ψ cos λ

OS = ⎝ y = r cos ψ sin λ ⎠ .

(2.28)

z = r sin ψ

The point T is the intersection of OS and the ellipsoid. If S is identiﬁed by

the geodetic latitude ϕ = (Ox, IS), its Cartesian coordinates are, adapting

(2.15) and denoting the great normal by N = IS,

⎛

⎞

x = (N + h) cos ϕ cos λ

OS = ⎝ y = (N + h) cos ϕ sin λ ⎠ ,

(2.29)

z = N (1 − e2) + h sin ϕ

where h = SN is called the geodetic altitude or ellipsoidal height, that is, the distance between the point S and the base N of the normal to the ellipsoid.
In the terminology of space mechanics, S represents a satellite, O the center of attraction (the center of the Earth), T the ground track (or geocentric ground track ), and N the nadir 5 (or geodetic ground track, or subsatellite point ).

2.2.2 Latitude Related to Geodetic Altitude
The angles used to determine these quantities are shown in Fig. 2.5. Note that, although ψ and ϕ are both related to the point S, these angles do not refer to the same point on the surface of the ellipsoid.

5The nadir is the direction given by the vertical, but in the downward direction. The
opposite direction is the zenith. The word “nadir” comes from the Arabic na¯d. ir, from the root of the verb “to look towards”.

38

Chapter 2. Geodesy

z S

T N

ψ JЈ j

x

O

J

I IЈ

Symbol
ψ ϕT ψN ϕ

Latitude
Geocentric, of T Geodetic, of T Geocentric, of N Geodetic, of N

Latitude
Geocentric, of S – – Geodetic, of S

Angle
ψ = (Ox, OS) ϕT = (Ox, I T ) ψN = (Ox, ON ) ϕ = (Ox, IS)

Fig. 2.5 : Representation of the geodetic latitude ϕ and the geocentric latitude ψ of the point S. Shown are the ground track T and the nadir N , together with the geodetic altitude or ellipsoidal height SN .

The point S is considered to be perfectly determined in space by its Cartesian coordinates (x, y, z), or equivalently by its geocentric spherical coordinates (r, ψ, λ) [see (2.28)]. Regarding its geodetic coordinates (h, ϕ, λ), only the longitude is easily obtained:

y λ = arctan ,

with sign(λ) = sign(y) , for [+E/−W] .

x

(2.30)

It is more diﬃcult to obtain h and ϕ, bearing in mind that they are related. Let us begin by examining the relationship between the geocentric and geodetic latitudes of S. Equations (2.28) and (2.29) give

r cos ψ = (N + h) cos ϕ , r sin ψ = N (1 − e2) + h sin ϕ ,

2.2 Altitude Relative to the Ellipsoid

39

whence

tan ψ = 1 − N e2 ,

tan ϕ

N +h

(2.31)

or again,

e2

−1

tan ϕ = 1 −

tan ψ .

1 + h/N

(2.32)

This immediately shows the limiting values of ϕ :
• For h = 0, tan ϕ = (1 − e2)−1 tan ψ =⇒ ϕ = ϕT . • For h → ∞, tan ϕ = tan ψ =⇒ ϕ = ψ .
The geodetic latitude of S lies between the geodetic latitude of the ground track T and its geocentric latitude.

2.2.3 Determining the Geodetic Altitude and Nadir
The coordinates h and ϕ can be obtained iteratively, by approximation, or directly. We consider each in turn. Iterative Method
Let P be the projection of OS on the equatorial plane:

P = x2 + y2 = (N + h) cos ϕ ,

(2.33)

whence the geodetic altitude h is given by

h= P −N . cos ϕ

(2.34)

The projection of OS on the polar axis Oz is z = P tan ψ, and using (2.31), we have

z = 1−

N

e2 tan ϕ ,

P

N +h

and hence,

z

ϕ = arctan

1−

e2

−1
.

P

1 + h/N

(2.35)

We then proceed as shown in Table 2.4. Convergence is very fast. In fact, two or three iterations give the result to high accuracy.

40

Chapter 2. Geodesy

◦0◦ ◦1◦ ◦2◦
3

•4•

•5•

•6•
•7• 8 ◦9◦

Ellipsoid: a = Re , e2 = f (2 − f )

Data: x, y, z =⇒ P = x2 + y2 , ψ = arctan(z/P )

Initialisation: ϕ1 = ψ

Start loop i = 1, . . . , n

Ni = a 1 − e2 sin2 ϕi −1/2

hi

=

P cos ϕi

−

Ni

z ϕi = arctan P

1−

e2

−1

1 + hi/Ni

Test ϕi −→ 3 or 8

End loop

Results

Table 2.4 : Iterative method for obtaining the height h and latitude ϕ of the nadir (subsatellite point).

Approximation Method

In the triangle ON S shown in Fig. 2.5, the angle at O (equal to ψ−ψN ) and the angle at S (equal to ϕ − ψ) are both small, in fact, always less than 0.19◦
for the Earth ellipsoid. This justiﬁes trigonometric approximations leading to
the following formulas:

f

f2

η

ϕ = ψ + η sin 2ψ + η2

1− 4

sin 4ψ ,

(2.36)

h

1 − cos 2ψ f 2

η 1 − cos 4ψ

= (η − 1) + f

+ 1−

,

Re

2

η

4

4

(2.37)

using the reduced distance η deﬁned by

r η= ,
a

(2.38)

where r is the distance OS and a the semi-major axis of the ellipsoid,6 here a = Re.

Direct Method

The Borkowski algorithm can be used to obtain the values of ϕ and h directly. It exploits the fact that this problem reduces to a fourth order polynomial equation. This is rather diﬃcult to solve, and indeed, much more involved than the two preceding methods, while the gain in accuracy is almost negligible. The three methods agree to within 10−4 degrees for the angles.

6Note that η = 1 only corresponds to h = 0 in the equatorial plane for ψ = 0. If we set η = 1 with ψ = π/2 in (2.37), we obtain h/Re = f , or h = Re − Rp. At the poles, zero altitude h = 0 corresponds to η = 1 − f .

2.2 Altitude Relative to the Ellipsoid

41

Some Remarks Concerning Altitude and Latitude

For a given point S, we deﬁne the diﬀerence in latitude δϕ by

δϕ = δϕ(h, ψ) = ϕ − ψ .

(2.39)

We deﬁne also the diﬀerence in altitude:

δh = δh(h, ψ) = h − h ,

(2.40)

where h = ST is the geocentric altitude. Since the tangent to the ellipse lies outside the ellipse, h is always greater than h (h is the smallest distance between S and the ellipsoid).
With r = OS = OT + T S = Rψ + h , the relative distance η is equal to (Rψ + h )/Re. Using (1.39), we deduce that

h = η − Rψ = (η − 1) + f 1 − cos 2ψ + o(f 2) .

Re

Re

2

(2.41)

Comparing (2.39) and (2.36), then (2.41) and (2.37), we see that δϕ is a small quantity, proportional to f , while δh is another small quantity, proportional to f 2. In the triangle SN T shown in Fig. 2.5, the arc N T is almost proportional to the angle at S (equal to δϕ), hence proportional to f , whereas the diﬀerence between the sides SN and ST depends, if we write down the distances, on
the square of f . Figure 2.6 shows the dependence of the latitude diﬀerence δϕ = δϕ(h, ψ)
on the latitude (varying from 0◦ to 90◦) and the altitude (varying from 0 to
inﬁnity).

Example 2.2 Calculating the diﬀerences δϕ and δh, as deﬁned above, for two satellites, one in low Earth orbit (LEO) and the other in medium Earth orbit (MEO).

For the low Earth orbit (LEO), we choose the satellite SPOT-5 (average height at the equator 822.3 km, inclination i = 98.670◦). Results are given for latitudes in steps of 15◦, starting from 0◦, the extreme latitude being |ψ| = 180◦ − i = 81.330◦. We give the three angles ψ (geocentric latitude
of the satellite S), ϕT (geodetic latitude of its ground track T ), ϕ (geodetic latitude of the nadir N and hence of the satellite S), and also the diﬀerence |δϕ|. In addition, h is the geodetic altitude and h the geocentric altitude. The quantity δh is at most a few metres.
We give the corresponding results for a medium Earth orbit (MEO) satel-
lite, choosing one component of the Navstar/GPS constellation, namely, the
satellite Navstar-2RM-6 (average altitude at the equator 20, 182.3 km, inclination i = 55.284◦). The extreme latitude is in this case |ψ| = i = 55.284◦.
All the results are displayed in Table 2.5. Technical data on satellite orbits: calculations using Ixion with NORAD initialisation. SPOT-5, Revolution
34006, Date 2008-11-24. Navstar-2RM-6 [PRN 07], Revolution 510, Date 2008-
11-22.

42

Chapter 2. Geodesy

Difference: Geodetic Lat. - Geocentric Lat.

(arcminute) (n. mi.) 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0

(km)

27

26
25 Altitude (km)

24

23

22

21

0

20

400

19

800

18

1250

17

16

15

2500

14

13

12

5000

11

10

9

8

10000

7

6

5

20000

4

3

40000

2 1

80000

0

infinity

10

20

30

40

50

60

70

80

90

Latitude (degree)

Fig. 2.6 : Latitudinal dependence of δϕ = ϕ − ψ, the diﬀerence between the geodetic latitude ϕ and the geocentric latitude ψ, for a point S representing a satellite. Each curve corresponds to a speciﬁc altitude of S, indicated on the right opposite the highest point of the curve. The value of δϕ is given in arc minutes (ordinate on the left) and in kilometres corresponding to the distance on the ground (right). Recall that 1 of latitude = 1 nautical mile of distance. The variable on the abscissa is indicated as “latitude” because it is impossible to distinguish the two latitudes for this coordinate. For each altitude, the maximum is attained at latitude 45◦.

2.3 A Little History
2.3.1 Before the Enlightenment
If humanity had applied the principle of Saint Thomas—seeing is believing—we would have had to wait for Gagarin before we could say “the Earth is round”. But fortunately for the human intellect, this fact has been known for a long time now, and no satellites were involved in discovering or checking it. However, as we shall see, it is mainly thanks to satellites that we have been able to reﬁne, to a very high level of accuracy, our knowledge of the true shape of the Earth, and indeed, the shapes of other planets in the Solar System.
The oldest description of the real world that has come down to us is to be found in Homer’s Odyssey. The aoidos describes Ulysses’ return journey, with his 10 years of wandering in the Mediterranean. The whole trip can be reconstructed to establish a geographical map of the world as it was perceived

2.3 A Little History

43

S

ψ

ϕT

ϕ

|δϕ|

(degrees) (degrees) (degrees) (degrees)

L

0.000

0.000

0.000

0.000

14.963 15.059 15.048

0.085

29.957 30.124 30.105

0.148

44.992 45.185 45.162

0.170

59.987 60.153 60.134

0.147

74.905 75.002 74.991

0.086

81.330 81.387 81.381

0.051

−0.022 −0.022 −0.022

0.000

−81.330 −81.387 −81.381

0.051

h (km) 822.011 823.341 827.187 832.510 837.849 841.777 842.796 822.631 843.647

h (km) 822.011 823.341 827.190 832.514 837.851 841.778 842.796 822.631 843.647

δh (km) 0.000 0.000 0.003 0.004 0.002 0.001 0.000 0.000 0.000

M

0.000

0.000

0.000

14.965 15.061 14.988

29.874 30.040 29.914

44.981 45.174 45.027

55.284 55.464 55.327

−0.027 −0.027 −0.027

−55.284 −55.464 −55.327

0.000 0.023 0.040 0.046 0.043 0.000 0.043

20,240.459 20,235.031 20,226.316 20,211.941 20,184.322 20,124.195 20,208.988

20,240.459 20,235.039 20,226.338 20,211.969 20,184.346 20,124.195 20,209.010

0.000 0.008 0.022 0.027 0.023 0.000 0.023

Table 2.5 : Geodetic latitude ϕ and geodetic altitude h for two satellites (S), one in an LEO orbit (L), like SPOT-5, and the other in an MEO orbit (M), like Navstar/GPS. Comparison with geocentric quantities ψ and h . Refer to Fig. 2.5 for notation.

in ancient times: a ﬂat disk, surrounded by the World Ocean, a great river encircling the world.7
Later on, the ﬁrst suggestion that the Earth might be round also emanated from Greece. Philosophical theories, as put forward by Aristotle, had clearly incorporated the fact that, when a boat disappears at the horizon—ﬁrst the hull, then the sail—it is because the sea surface is not ﬂat but rounded, while this was corroborated by scientiﬁc theories supported by measurement, as presented by Eratosthenes with his comparative observation of the noonday sun in Alexandria and Syene. As far as we know by juggling with the length units of the day, the value found by this geometer–mathematician–astronomer for the Earth’s circumference was rather accurate.
From a geographical point of view, the Middle Ages represent a dark period. In fact, very dark. The Hereford mappa mundi shows that, by 1300, the perception of the world had barely evolved since the time of Homer—except for one signiﬁcant change: Jerusalem had replaced Delphi as the center of the world. The so-called T and O maps, from orbis terrarum, represent the land mass as a T-shape, surrounded by an ocean O. These mappæ mundi reﬂect above all the overwhelming obscurantism of the age, founded on and serving the prevailing religion.

Ó Ð Ó ¸ ÛÒ é Ð Ù ¸ ÙÓ 7At the exact center of this circular world was the Temple of Apollo in Delphi. The

name Delphi,

´ ˜ in Greek, is closely related to the word

´ ´ , which

means “womb”.

44

Chapter 2. Geodesy

And then there was light. Or rather, the stage was lit again by the likes of Copernicus, Kepler, and Galileo. The Earth became round again, but this time it lost its place at the center of the Universe.
2.3.2 A French Aﬀair
In its early stages, from 1666 (the foundation of the Acad´emie des Sciences, by Colbert, under Louis XIV) up until around 1810, geodesy was a French aﬀair, brilliantly developed by the learned assembly and its appendage, the Paris Observatory. There are really three stages in this fruitful development:
(a) measurement of the Earth’s radius, (b) measurement of the ﬂattening of the Earth, (c) deﬁnition of the metre.
Picard’s Measurement of the Earth
Father Jean Picard8 carried out the ﬁrst scientiﬁcally serious measurement of 1◦ of latitude in order to establish the radius of the Earth, which he assumed at the time to be spherical. With precision instruments, he used the method of triangulation9 invented by the Dutch astronomer and mathematician Snellius. Picard measured (1669–1671) 1◦ on the meridian from Paris to Amiens (the so-called Paris meridian). More precisely, he established 13 triangles between Malvoisine to the south and Sourdon to the north, covering 1◦22 , with Villejuif–Juvisy as baseline.10

8Jean–F´elix Picard (1620–1683) was a French astronomer and geodesist. He invented the sighting telescope with crosswires, which allowed him to carry out very accurate surveying, and in particular, levelling. Having determined the Earth’s radius, as explained in the main text (Mesure de la Terre, 1671), he immediately communicated his result to Newton, who was thus able to check the relation between the accelerations and the squares of the distances, and thereby obtained a clear conﬁrmation of his universal theory of gravitation. In another area, Picard was the ﬁrst to carry out systematic measurements of the diameter of the solar disk. He observed its variations and sought the connection with climate change on Earth. His series of measurements between 1666 and 1682 was continued by La Hire from 1683 to 1718.
9Given a straight line distance to be measured in the ﬁeld, the idea is to consider a chain of adjacent triangles along the straight line. The apices of the triangles are church towers or other features that are clearly visible from afar. The angles of these triangles are then measured, and if the length of one side is known, the lengths of the other two sides can be deduced by trigonometry. This yields the required distance. The side that is actually measured is called the baseline.
10In Picard’s own words: “The cobbled road from the mill in Ville–Juive to the pavillion in Juvisy, a straight line with no signiﬁcant unevenness, was considered ideal as baseline for this undertaking.” Its length was very carefully measured in both directions by juxtaposition of toises. Today, this 11 km section of the D7 (previously the N7), which goes under the landing strips at Orly airport, is still very straight, apart from one or two recent urban adjustments. Each end is commemorated by a pyramidal marker stone.

2.3 A Little History

45

Fig. 2.7 : Sighting and measurement instruments used for triangulation. Colour plate (p. 16) taken from La Mesure de la Terre by J. Picard, Paris, 1671.
He found a value of 57,060 toises for 1◦ along the meridian, which corresponds to 6,372 km for the radius of the Earth. This is remarkably accurate, within 0.1 % of the exact value. This is certainly due in part to the quality of the measurements (see Fig. 2.7), but also to the fact that the region measured is close to latitude 50◦, where the radius of curvature happens to be practically equal to the radius of the Earth, as can be seen from (2.19).

46

Chapter 2. Geodesy

Fig. 2.8 : Map of France (by Maraldi and Cassini de Thury) 1744. Detail of frontispiece. This map shows the rays observed in building up the primary triangulation network. Cartographic projection: Cassini projection.
From Sphere to Ellipsoid In 1672, the astronomer J. Richer was sent to French Guiana.11 He noticed
that the mechanical clock he had brought with him, and which had been
11As the radius of the Earth had been known since the previous year, the Academy sent Jean Richer to Cayenne to observe the parallax of Mars (the angle subtended by the Earth’s diameter as viewed from Mars) in a joint eﬀort with Picard, who had remained in Paris. By thus measuring the distance to Mars, Kepler’s third law allows one to deduce the sizes of the planetary orbits. So the length scale of the whole Solar System was at this point underpinned by the Villejuif–Juvisy baseline!

2.3 A Little History

47

scrupulously adjusted to “beat seconds” at the Paris Observatory, was losing 2 min every day (the expression battre la seconde in French indicates that the period is 2 s). He attributed this failing to a reduction in weight, and for him this could have only one explanation: the equator was further from the center of the Earth than Paris. He thus suggested that the Earth was ﬂattened.
Shortly afterwards, Newton and Huygens independently showed, in 1687 and 1690, that if the Earth’s innards were more or less ﬂuid, its daily rotation ought to transform the sphere into an ellipsoid, ﬂattened along the polar axis. Newton obtained a value of f = 1/230 by considering a uniform distribution of matter within the ellipsoid, while Huygens obtained f = 1/576 by assuming that the mass of the Earth was concentrated in a central core.
In 1668, J.D. Cassini12 observed the ﬂattening phenomenon on Jupiter, which is indeed considerably ﬂattened along the polar axis (f = 1/18), and he measured it later for Saturn.
To measure the ﬂattening eﬀect on Earth, the French Academy of Sciences decided in 1683 to extend the Picard meridian to the north as far as Dunkerque and the south to Collioure, whence it would extend right across France. This

12The name Cassini will come up several times during this section. Indeed, there was a genuine dynasty of outstanding astronomers, often numbered with Roman numerals like the crowned heads of Europe:
• Gian Domenico Cassini (1625–1712), known to the French as Jean Dominique Cassini, or Cassini I, was actually of Italian origin (from Nice). He soon became famous for his work on geodesy, and especially on astronomy, with his very accurate observations of the planets Mars and Jupiter. He set up tables of the Galilean moons of Jupiter, a fundamental step in the determination of longitudes, because the eclipses of these satellites constitute instantaneous signals for an Earth-based observer. So it was really the “transfer of the century” when Louis XIV called Cassini to Paris in 1669 to entrust him with the foundation and the running of the Paris Observatory. In 1679, the observatory began to publish La Connaissance des Temps, a publication that is still alive today and which lists the positions of the heavenly bodies to the greatest possible accuracy. Cassini continued his observation of the moons of Jupiter, work which allowed Olau¨s R¨omer to show that the speed of light was not inﬁnite. He also improved observations of the Moon and Saturn. This was the Cassini who gave his name to the Cassini Division of Saturn’s rings and also to the Cassini space probe, designed to explore Saturn and its environment.
• Jacques Cassini (1677–1756), or Cassini II, was the son of Jean Dominique. He pursued his father and Picard’s geodetic measurements, but the publication of De la grandeur et de la ﬁgure de la Terre (1722), in which he made a mistake over the ﬂattening of the Earth, was later to reduce his scientiﬁc status. This was the Cassini who instigated the scientiﬁc dispute between the Cassini dynasty and Newton.
• C´esar Franc¸ois Cassini de Thury (1714–1784), or Cassini III, was the son of Jacques. After assisting his father with his geodetic measurements, he devoted himself to cartography. See Fig. 2.8. In 1750, Louis XV asked him to map the whole of the kingdom. This is the Cassini who is remembered for the Cassini projection and the Cassini map of France.
• Jacques Dominique Cassini (1748–1845), or Cassini IV, was the son of C´esar Fran¸cois. He was to ﬁnally publish the map of France in 1790.
From 1669 to 1793, the Cassinis ran the Paris Observatory, either oﬃcially or unoﬃcially, each son succeeding his father.

48

Chapter 2. Geodesy

work was carried out from 1700 to 1718 under the supervision of J. Cassini, Maraldi, and La Hire, see Fig. 2.8. Measurements implied a degree of latitude that was longer in the south than in the north. This in turn suggested an ellipsoid that was stretched along the polar axis, with f = −1/95 (f is negative if b > a), the opposite of what is shown in Fig. 2.4.
This was the beginning of a 20 year feud between supporters of Newton and those of the Cassinis: was the Earth shaped like an apple or a lemon?13
Between the north and the south of France, the change in latitude is not signiﬁcant enough to be able to obtain a reliable result. On the recommendation of the French Academy of Sciences, the Count of Maurepas, Secretary of State for the Marine, sent two expeditions to the geodetic limits of the world, one polar, the other equinoctial:
• P.L.M. de Maupertuis,14 A.C. Clairaut, and A. Celsius measured 1◦ in Lapland, on the frontier between Sweden and Finland, in 1736 and 1737.
• L. Godin, C.M. de la Condamine, and L. Bouguer measured 1◦ in Peru (in a region that has now become part of Ecuador), from 1736 to 1744, under very diﬃcult conditions.

At the same time, in 1739 and 1740, Cassini de Thury and Father La Caille went back to Picard’s meridian, and then from 1750 to 1754, La Caille went to the Cape of Good Hope and measured 1◦ in the southern hemisphere.
We can exploit these results to work out the ﬂattening (see Table 2.6). Using (2.16), the ratio of two radii of curvature, ρ1 at ϕ1 and ρ2 at ϕ2, yields

ρ1 = ρ2

1 − e2 sin2 ϕ2 1 − e2 sin2 ϕ1

3/2
.

Expanding to ﬁrst order in e2, we obtain

ρ1 ρ2

=

1−

3 4

e2

(cos

2ϕ1

− cos 2ϕ2)

.

Since the ratio of the radii of curvature is equivalent to the ratio of the measurements ΔLM for 1◦, and since e2 is almost equal to 2f , this means that

f = 2 1 − ΔLM1/ΔLM2 . 3 cos 2ϕ1 − cos 2ϕ2

(2.42)

13Traditionally, the contrast is illustrated by: mandarin or lemon? There seems to be an anachronism here, since the word “mandarin” did not appear in French until 1773.
14Pierre Louis Moreau de Maupertuis (1698–1759), a French physicist, led the expedition to Lapland. In so doing, he earnt these two graceful lines from Voltaire:
Vous allˆates v´eriﬁer en ces lieux pleins d’ennui Ce que Newton connut sans sortir de chez lui.
[In this soulless landscape you are sure to construe What Newton in his college lodgings always knew.] While there is probably no connection with this typical irony, Maupertuis subsequently published, in 1744, his famous Principe de moindre action (Principle of least action).

2.3 A Little History

49

Name of degree
Lapland Paris Peru Cape

Latitude ϕ
+66◦ 20 +49◦ 29 −01◦ 30 −33◦ 18

ΔLM (toises)
57, 438 57, 074 56, 746 57, 037

ΔLM (km)
111.949 111.239 110.600 111.167

Table 2.6 : Results found for the degree of latitude ΔLM between 1736 and 1754, for diﬀerent values of the average latitude ϕ, in degrees [+N/−S]. The value of ΔLM is given in toises, together with the modern equivalent in kilometres.

If we calculate the ﬂattening from the measurements (ΔLM1, ϕ1) in Peru and (ΔLM2, ϕ2) in Paris, we obtain

ΔLM1 = 0.99425 , ΔLM2

cos 2ϕ1 − cos 2ϕ2 = 1.15506 ,

=⇒ f = 2 5.4769 × 10−3 = 3.31696 × 10−3 = 1 .

3 1.15506

301

The measurement at the Cape of Good Hope was not retained. Its value was overestimated because the vertical was aﬀected by the presence of mountains nearby.
The calculation of f using the Lapland degree gave results that diﬀered too much to be accepted by the scientiﬁc community (f = 1/123 and f = 1/207), whereupon the Swedish Academy opted for a new expedition to Lapland in 1801, which would give ΔLM = 57, 196 toises, i.e., 0.42 % less than the ﬁrst measurement.
The use of (2.42) to calculate f is extremely sensitive to the accuracy of the measurements. If we assume that the latitudes ϕ1 and ϕ2 are known with certainty and the degree in Paris ΔLM2 measured exactly, a relative error of 0.1 % in the degree in Peru ΔLM1 (or an error of just 57 toises per degree) would lead to an error of 17.3 % in f . We ﬁnd the following results: for ΔLM1 = 56, 803 toises, 1/f = 364.6, for ΔLM1 = 56, 689 toises, 1/f = 257.0.

50

Chapter 2. Geodesy

Fig. 2.9 : Commemorating the revolution: for all time, for all peoples.

As a result of all this state-of-the-art geodesy, France was the ﬁrst country in the world to draw up a highly accurate map of its own territory, known as the Cassini map.15
Deﬁnition of the Meter
A tous les temps, a` tous les peuples (for all time, for all peoples, see Fig. 2.9), in the universal spirit of the French Revolution which characterised what was happening in France at the time, there was born the idea of oﬀering humanity a single consistent system of physical units.
Since the length of a pendulum which “beats the second” is not independent of the latitude,16 the Assembly decided on 30 March 1791 that the unit of length would be one ten-millionth part of the Earth meridian, based on a proposal by a committee whose members were Borda, Lagrange, Laplace, Monge, and Condorcet. The astronomers Delambre and M´echain were requested to make accurate measurements of the meridian arc from Dunkerque to Barcelona, towns separated by “9 and 2 thirds degrees” on the Paris meridian, and both situated by the sea.
15This map, already mentioned earlier, used a scale of 1/86,400 (one line for 100 toises). It comprised 182 sheets to cover the whole kingdom. Cassini de Thury used a novel projection, now called the Cassini projection, in which he plotted lines perpendicular to the Paris meridian. These lines are not parallels, i.e., they are not the loci of points at constant latitude, but great circles (for a spherical Earth). The perpendicular line passing through the Paris Observatory goes from Granville to Strasbourg. This projection is the transverse aspect of the projection known in French as the plate-carr´ee (ﬂat square) projection.
16In a ﬁeld with acceleration due to gravity equal to g, the period T of a pendulum of length l is given by T = 2π l/g in SI units. With T = 2 s, this gives l = g/π2 numerically. As g varies between the equator and the pole, l varies from 0.991 to 0.996 m. We note that the metre was chosen close to the length of this pendulum, whereas a doubled metre would have been close to the toise.

2.3 A Little History

51

The process of triangulation was carried out17 from June 1792 to the end of 1798,18 with 115 triangles and two baselines, the Melun baseline (from Lieusaint to Melun) and the Perpignan baseline (from Salses to Vernet).
The result was proclaimed in June 1799. The quarter meridian, as calculated from these measurements, was
L π = 5, 130, 740 toises , 2
which was thus equal, by deﬁnition of the new unit of length, to L π = 10, 000, 000 m . 2
The oﬃcial conversion rate was thus ﬁxed at
1 Chˆatelet toise = 1.9490363 m .
The law instating the metre19 was signed on 10 December 1799 (19 frimaire An VIII in the revolutionary dating system).
Using Delambre and M´echain’s ellipsoid (a = 6, 375.738 m, 1/f = 334), which gives this deﬁnitive value for L(π/2), the last column of Table 2.2 shows the values of the meridian arc as a function of latitude. These can then be compared with current values. The relative error introduced by Delambre et M´echain is a mere 0.02 %, which attests to the high quality of these measurements.20
2.3.3 Dynamical Geodesy
Modern geodesy begins with Clairaut. With his Th´eorie de la ﬁgure de la Terre, tir´ee des principes de l’hydrostatique (Theory of the Earth’s shape, based on the principles of hydrostatics, 1743), he laid the foundations of dynamical geodesy: measurements of the acceleration due to gravity should be used, like the measurement of degrees of latitude, to determine the shape of the ellipsoid. The shape of the Earth depends on its rate of rotation along the polar axis and the distribution of mass within it.
17The angles were measured to an accuracy of 1 of arc using Borda’s repeating circle method, with instruments made by his assistant Lenoir.
18It was not a good time to be carrying out this kind of expedition. In the thick of the revolution, hauling strange-looking instruments to the tops of church towers or onto the battlements of castles was unnecessarily intriguing for the local populations. There were many unfortunate incidents, with material being sabotaged and surveyors arrested, among other things.
19Given the delays in completing the measurements, a provisional metre had been adopted on 1 August 1793. This metre would give L(π/2) = 5, 130, 430 toises.
20Once it had been determined relative to the Earth ellipsoid, the metre was then ﬁxed by the General Conference on Weights and Measures (Conf´erence G´en´erale des Poids et Mesures CGPM). In 1889 (the ﬁrst CGPM), the metre was deﬁned by the prototype deposited at the Archives de France. In 1960 (the 11th CGPM), it was deﬁned in terms of a particular wavelength of light emitted by krypton 86. Since 1983 (the 17th CGPM), the metre has been deﬁned relative to the speed of light (see Sect. 6.10).

52

Chapter 2. Geodesy

Fig. 2.10 : The Soviet Union remembers its conquest of space.

Lagrange invented the notion of gravitational potential and was no longer satisﬁed to deﬁne the Earth ellipsoid by its ﬂattening. In 1810, he expanded the gravitational potential of the Earth in spherical harmonics and the coeﬃcients of this expansion provide a much better representation for the imperfections in the Earth’s shape when compared with a sphere. The notion of the Earth’s shape was then replaced by the “geoid”. This is the equipotential surface which best ﬁts the mean level of the oceans.
On 4 October 1957, Sputnik-1 was put into orbit (see the commemorative stamp in Fig. 2.10), and this marked the beginning of a new era of space geodesy.

Chapter 3
Geopotential

3.1 Some Preliminaries

3.1.1 Reference Systems

Consider a reference frame centered on the Sun and with axes pointing
to distant (ﬁxed) stars. This is a Copernican frame, which we denote by 0. Any frame 1 in uniform translational motion relative to 0 is a Galilean frame. In this kind of frame, experiment shows that Newton’s second law1 is
perfectly satisﬁed:

d(mv)

F=

,

dt

(3.1)

where F is the force applied to a body of mass m and dv/dt is the acceleration

of the body.

Consider a frame with origin at the center of the Earth and axes parallel

to the axes of the frame 1. Strictly speaking, this is not a Galilean frame, because the motion of the Earth around the Sun is neither linear nor uniform.

1Isaac Newton (1643–1727) was an English mathematician, physicist, and astronomer. In 1687, he stated his three laws of motion in Philosophæ Naturalis Principia Mathematica: (1) the principle of inertia, (2) his famous second law, which says that, in a Galilean frame, the force is equal to mass times acceleration, and (3) the principle of action and reaction. It can be shown that (1) is a special case of (2) and that (3) can be deduced from (2). The fundamental second law (2) was not expressed in exactly this way by Newton. Combined with Kepler’s law of elliptical orbits, the second law can be used to derive Newton’s universal law of gravity [see (4.115)–(4.117)]. Newton’s work dominated the eighteenth century, in mathematics (analysis, solution of equations) and in physics, especially in optics, with the publication of Opticks. Regarding Newton’s date of birth, it is interesting to note that 25 December 1642 on the Julian calendar, which was still used in England at the time, corresponds to 4 January 1643 on the Gregorian calendar.

M. Capderou, Handbook of Satellite Orbits: From Kepler to GPS,

53

DOI 10.1007/978-3-319-03416-4 3,

© Springer International Publishing Switzerland 2014

54

Chapter 3. Geopotential

However, this motion is nevertheless slow, with one round trip per year, and above all, it is perfectly well known, so there is no diﬃculty in calculating the resulting apparent accelerations. In the following, this pseudo-Galilean frame
will be treated as a Galilean frame moving with the Earth. In this book, we shall use two frames with origin O at the center of the Earth:
• The Galilean (pseudo-Galilean) frame we have just deﬁned, ﬁxed relative to the orthonormal triad (O; xG, yG, zG), where OzG is the polar axis and OxG points in a ﬁxed direction in space, i.e., toward a distant star. This is called an Earth-centered inertial (ECI) frame, or Earth-centered spaceﬁxed (ECSF) frame.
• The terrestrial (non-Galilean) frame T, which is needed to describe ﬁxed points on the Earth, ﬁxed relative to the orthonormal triad (O; xT, yT, zT), where OzT = OzG is the polar axis, while OxT rotates with the Earth, remaining ﬁxed in the prime meridian through Greenwich. This is called the Earth-centered Earth-ﬁxed (ECEF) frame.
Theory and Practice
The frames ECI and ECEF are used to set up all the equations in this book from the standpoint of space mechanics. The deﬁnitions here are theoretical and didactic. In contrast, the practical and technical realisation of these frames, denoted respectively by ICRF and ITRF, is the work of astronomers and geodesists. They are explained brieﬂy in an appendix to this chapter (see Sect. 3.6).
3.1.2 Review of Work and Potential
Work, Force Field, Potential
The work done by a force F applied at a point M is the scalar quantity
dW = F ·dl ,
where dl is the displacement of the point of application of the force from M to M . In an orthonormal frame, with F (X, Y, Z) and dl (dx, dy, dz), the scalar product gives
dW = Xdx + Y dy + Zdz .
The total work done when the point of application of the force moves from A to B is
B
W = F ·dl .
A
A point M is said to be subjected to a force ﬁeld if, throughout its domain of application, a force F (X, Y, Z) can be associated with each position of the point M (x, y, z).

3.1 Some Preliminaries

55

If there is a function V (x, y, z) such that the components of the force F (X, Y, Z) can be expressed in the form

∂V X= ,
∂x

∂V Y= ,
∂y

∂V Z= ,
∂z

throughout the region of deﬁnition, the ﬁeld is said to derive from a potential . In this case, the force ﬁeld is said to be conservative. Using the vector operator gradient, denoted by grad or ∇ and deﬁned by dV = grad V ·dl, we can then write

F = grad V ,

whence dW becomes

dW = grad V ·dl ,

which represents the exact diﬀerential dV :

dW = dV .

Integrating between points A and B,
B
dW = WAB = V (B) − V (A) .
A
This shows that the work done by the force in going from A to B, denoted by WAB, depends only on the values of the potential V at the points A and B, and not on any of the intermediate values taken along the intervening path. The function V is only deﬁned up to an additive constant.
Equipotential Surface
An equipotential surface is a surface of the form

V (x, y, z) = constant .

This has the following properties:
• No work is done by displacement on a given equipotential surface, which shows that the component of the force tangent to the surface is zero. An equipotential surface is an equilibrium surface.
• For the same reason, i.e., the fact that F ·dl = 0, the force is normal to each equipotential surface.
• Two equipotential surfaces cannot intersect, otherwise work could be done without ever leaving one of the equipotential surfaces.

56

Chapter 3. Geopotential

Potential Energy
The potential energy U is deﬁned in the following way in terms of the work done by a conservative force ﬁeld on a moving point:
B
dW = WAB = U (A) − U (B) .
A
Hence, U = −V and the relationship with F is

F = −grad U .

(3.2)

This expression involving the potential energy can be used to deﬁne the mechanical energy E as the sum of the potential energy U and the kinetic energy T . Indeed, we can write dW in two diﬀerent ways:

dW = −d U ,

dW = F ·dl = m dv ·dl = d 1 mv2 = dT .

dt

2

Hence, dU + dT = 0. For an isolated system subjected to a conservative force ﬁeld, we thus establish conservation of mechanical energy:

E = T + U = constant .

(3.3)

We usually set U(∞) = 0. When E is not constant, energy is dissipated and the force is said to be
dissipative.

3.2 Gravitational Potential and Field

3.2.1 Gravity

The law of gravity, or universal law of gravitational attraction, established by Newton, states that two point bodies A and B, with masses M and m, respectively, will each exert an attractive force on the other that is proportional to their mass and inversely proportional to the square of the distance between them:

f A→B

=

−f B→A

=

−G

Mm r2

er

,

(3.4)

where f A→B is the force exerted by A on B and AB = r = rer. The gravitational constant G is not used as such in space mechanics. Instead, we use the speciﬁc gravitational constant μ, which is the product of G and the mass of the relevant attracting body:

μ = GM .

(3.5)

3.2 Gravitational Potential and Field

57

The relation (3.4) is symmetric. In order to distinguish the role of one of the two bodies, we may express the fact that body A, for example, creates a gravitational ﬁeld to which body B is subjected. This ﬁeld g is such that

f A→B = F = mg ,

or in terms of μ,

μ g = − r2 er .

There is a function U such that

F

=

−grad

U

=

−

∂U ∂r

er

,

(3.6)

This can be obtained by integrating over r :

U =−

F ·dr

=

μ −m

,

with U(∞) = 0 .

r

(3.7)

We then introduce the quantity U = −U/m. To sum up, U is the potential energy of the mass m in this force ﬁeld, the gravitational ﬁeld. U is then the gravitational potential produced by the mass M at distance r :

μ U= .
r

(3.8)

In astronomy and geodesy, the potential U is deﬁned like this so that the leading term in the potential, viz., μ/r, is positive [see (3.28)].

3.2.2 Gauss’ Theorem
In the last section, we obtained the ﬁeld and gravitational potential produced by a point body of mass M located at A and acting on another point body of mass m at a point B. For a continuous distribution of matter, we must then carry out an integral to obtain the force exerted on B. For a given conﬁguration, Gauss’ theorem gives the result directly without the need for such an integration.
Proof of Gauss’ Theorem
There are several ways to prove Gauss’ theorem.2 We shall use a method based on the idea of solid angle. We consider a closed surface S, enclosing a

2Carl Fiedrich Gauß (1777–1855) was a German astronomer, mathematician, and physicist. He was extremely precocious and interested in astronomy from an early age. He invented a method for calculating the orbital elements of the planets (see the note about Piazzi), then developed powerful methods for handling the problems of celestial mechanics, such as the theory of least squares, in his work Theoria motus corporum coelestium (1809). In mathematics, he invented congruences (modulo) and studied quadratic forms, er-

58

Chapter 3. Geopotential

volume τ . We can thus deﬁne an inside and an outside. We then consider a surface element dS with unit normal n pointing outwards. The ﬂux of an arbitrary vector g through this surface element is deﬁned as
dΦ = g·dS , with dS = n dS .
The total ﬂux of g out through the surface S is then given by the integral over the whole closed surface S, viz.,

Φ = g·dS .
S
We consider a surface S surrounding a distribution of masses: the diﬀerent points Ai are each attributed a mass Mi. The ﬁeld created by each mass Mi at a point B is then

gi

=

−GMi

AiB AiB3

,

where AiB is the vector from Ai to B and AiB its length. The ﬂux leaving S is

Φ = g·dS

S

=G

S

i

Mi

AiP AiP 3

·dS

=G

i

Mi

S

n · AiP AiP 3

dS

,

where P is a point running over S. Now,

n · AiP AiP 3

dS

=

dS cos αi AiP 2

=

dΣ AiP 2

=

dΩi

,

where αi is the angle between the normal and AiP and dΣ is the projection of dS on the plane perpendicular to AiP . The quantity dΩi is then the element of solid angle, represented by the inﬁnitesimal cone with apex Ai and base the surface element dS (or dΣ, which comes to the same thing).
The integration over dΩi is independent of the surface S. We thus take a sphere of center Ai and radius R. This gives

ror analysis (bell-shaped curve, 1821), regular polygons, conformal representations, spherical trigonometry, and the curvature of surfaces (1827). He revolutionised the ﬁeld of geodesy by introducing and developing novel methods, and he was not afraid to go out into the ﬁeld, e.g., setting up the cadastral survey of the Hanover region between 1817 and 1821. In physics, he carried out fundamental work on magnetism (Allgemeine Theorie des Erdgeomagnetismus, published in 1839), electricity (Gauss’ theorem), and optics (Gaussian optics). His contemporaries called him the Prince of Mathematicians. So who was the king?

3.2 Gravitational Potential and Field

59

dΣ 1

4πR2

Ωi = dΩi = R2 = R2 dΣ = R2 = 4π .

On the other hand, an external mass, i.e., with Ai outside S, produces a ﬁeld whose ﬂux through S is zero. Indeed, a cone with apex Ai standing on a surface element dS determines two opposing ﬂux elements whose total contribution
cancels, since dΦ is a scalar whose sign depends on the scalar product.
Finally, letting Mint = int Mi, the sum of the masses contained within the surface S, the ﬂux out of S is

Φ = −4πG Mi ,
int
and Gauss’ theorem can be stated as follows:

g·dS = −4πGMint .
S

(3.9)

For a continuous distribution of masses represented by the density ρ at each

point of space, Mint is given by

Mint =

ρ(r)dτ ,
V

where the triple integral extends over the whole volume V .

Calculating the Field Using Gauss’ Theorem

If the density depends only on r (the magnitude of r), i.e., if the mass distribution is spherically symmetric, the ﬁeld it produces will also have spherical symmetry:
r g(r) = g(r) .
r
It is then straightforward to calculate the ﬂux. For the surface S, we choose a sphere of radius r containing all the mass Mint. From the symmetry, the ﬁeld g must be orthogonal to S at every point. Remembering that r is constant over the whole surface S, we obtain

Φ = g·dS = g(r) r ·n dS = g(r) dS = 4π g(r) r2 .

S

S

r

S

Applying Gauss’ theorem (3.9), we ﬁnd that

4π g(r) r2 = −4πGMint ,

whence the gravitational ﬁeld g is given by

r g(r) = −GMint r3 .

(3.10)

We thus obtain the following very important result: the ﬁeld produced by a
spherically symmetric mass distribution is the same as would be produced by a point mass of the same value located at the center of the spherical distribution. This property results from the fact that the forces are central and go as r−2.

60

Chapter 3. Geopotential

Gravitational Field of the Earth

If we treat the Earth as spherical and assume that its density only depends

on the distance from the center O, then at a point outside or on the surface

of the Earth, and at distance r from O, the ﬁeld produced there is

r

μ

g(r) = −μ r3 = − r2 er ,

with μ = GM ,

(3.11)

where M is the total mass of the Earth. In this case, μ is called the geocentric

gravitational constant.

3.2.3 Gravity and Weight

If we assume once again that the Earth is spherically symmetric and in addition that it is not moving relative to a Galilean frame, the equipotential surfaces will be concentric spheres. But as Galileo pointed out, the Earth is spinning on its axis, so a point subjected to the force of gravity will also suﬀer an inertial force relative to a non-Galilean frame ﬁxed relative to the Earth. This is how the Earth’s shape was transformed and ﬂattened during its formation. Its outer envelope is an equipotential surface: its points are in equilibrium. If we ignore tides, currents, and winds, the ocean surface provides a faithful image of this equipotential surface,3 and it is generally taken as the zero altitude. This geoid naturally extends beneath the continents. The surface of a motionless lake4 also represents an equipotential surface, at another altitude, and the plumb line, which deﬁnes the local vertical, will be exactly perpendicular to this surface.
On such a surface, the potential U is constant. However, as mentioned above, the gravitational ﬁeld is not constant on this equilibrium surface. Indeed, it is stronger at the poles than at the equator, since Rp < Re, while the centrifugal acceleration is zero at the poles and maximal at the equator. We shall calculate the gravitational potential produced by a ﬂattened planet, then the weight potential which takes into account the Earth’s rotation. By integrating this potential, we will obtain the weight as a function of latitude.
Maupertuis was the ﬁrst to make a clear terminological distinction between gravity and weight, a distinction that was subsequently taken up by D’Alembert5 and Clairaut6:

3In 1742, MacLaurin showed that the ellipsoid of revolution spinning on its minor axis was the only geometrical shape that could meet requirements. Later, Poincar´e showed that, for much faster rotation, there were other possibilities, but they are not relevant to the planets.
4The water is in equilibrium so there is no reason why it should ﬂow from left to right, or from right to left!
5Jean le Rond d’Alembert (1717–1783) was a French mathematician, physicist, and philosopher. He published Recherche sur la pr´ecession des ´equinoxes et sur la nutation de l’axe de la Terre dans le syst`eme newtonien, in 1749. In 1743, he had stated the principle that carries his name in his Trait´e de dynamique. With Diderot, he wrote the Encyclop´edie.
6“I make here the same distinction as M. de Maupertuis (La Figure de la Terre d´etermin´ee, etc.) between weight and gravity. By weight, I understand the natural force

3.3 Calculating the Geopotential

61

• Gravity is the sum of the attractive eﬀects acting on a mass according to the universal law of gravitation.
• Weight is the resultant of gravity and the action of the centrifugal acceleration due to the Earth’s rotation.
In other words, gravity is the ﬁeld measured in the frame , while weight is the ﬁeld measured in the frame T. Any body on the Earth is subjected to weight, while a satellite in orbit around the Earth is subjected to gravity.

3.3 Calculating the Geopotential
3.3.1 Potential Element
The temporal variation in the terrestrial mass distribution (due to land and ocean tides and phenomena linked to internal geophysical processes) and the variation in the direction of the Earth’s axis of rotation (motion of the poles) are not taken into account here. We only consider the averaged eﬀect of these phenomena over a given period and calculate the static geopotential produced by a ﬁxed mass distribution (see Fig. 3.1).
Let O be the center of the Earth and (O; x, y, z) a coordinate system ﬁxed relative to the Earth, like T, where Oz is the polar axis and (xOy) the equatorial plane. Let S be a point outside the Earth (the satellite). Its position is speciﬁed by the three spherical coordinates r, λ, ψ [see (2.28)]. The angles λ and ψ represent the longitude and geocentric latitude of the point S. If T is a point inside the Earth, it can also be speciﬁed by its three spherical coordinates ρ, α, β, where ρ is the magnitude of OT , α the longitude, and β the geocentric latitude.

z S

T

O

y

x

Fig. 3.1 : Obtaining the gravitational potential at a point S. The volume element of mass dM contains the point T . In the integration, it runs over the whole volume of the Earth. Each such element produces a gravitational potential dU at S. The notations for the distances are r = OS, ρ = OT , and D = T S.
with which all bodies fall, and by gravity, the force with which the body would fall if the Earth’s rotation were not to alter its eﬀect and direction.” Clairaut, in his introduction to Th´eorie de la Figure de la Terre.

62

Chapter 3. Geopotential

We now have the standard relations giving components in Cartesian coor-

dinates:

⎛

⎞

OS

=

cos ψ cos λ ⎝ cos ψ sin λ ⎠

,

r

sin ψ

⎛

⎞

OT

cos β cos α = ⎝ cos β sin α ⎠

.

ρ

sin β

(3.12)

Let θ be the angle between the two radial vectors, viz.,

θ = (OS, OT ) ,

so that the distance D between the two points S and T is

D2 = T S 2 = r2 − 2rρ cos θ + ρ2 ,

and

ρ

ρ 2 1/2

D = D(T, S) = r 1 − 2 cos θ +

.

r

r

The scalar product OS · OT yields

cos θ = sin ψ sin β + cos ψ cos β cos(λ − α) .

The potential element dU produced at S by the mass element dM located at T , at a distance D from S, is given by (3.8) as

dμ dM

dU = = G .

D

D

(3.13)

3.3.2 Obtaining the Potential by Integration

The potential U we hope to calculate here is obtained by summing all the potential elements produced by the mass elements making up the mass distribution. The mass element dM is associated with the point T which ranges over the whole of the Earth:

U = U (S) =

dU = G

dM (T ) .

Earth

T ∈Earth D(T , S)

(3.14)

The expression for D arising in the calculation of the potential is given as a function of θ by

11 =
Dr

1

.

ρ

ρ2

1 − 2 cos θ +

r

r

(3.15)

This expression can be expanded in terms of Legendre polynomials (see the appendix at the end of the chapter). The expansion converges if ρ/r < 1.

3.3 Calculating the Geopotential

63

The calculation is thus valid if S remains strictly outside the sphere containing all the mass elements. We may then write

1 1∞ ρ l

= Dr

r Pl(cos θ) ,

l=0

(3.16)

where Pl is the l th Legendre polynomial (or Legendre polynomial of degree l). Replacing cos θ by its value in terms of spherical coordinates, the angles λ, ψ, α, and β, or more precisely ψ, β, and λ − α, we now use the Legendre addition formula:

Pl(cos θ) = Pl(sin ψ) · Pl(sin β)

+2

l

(l (l

− +

m)! m)!

Plm(sin

ψ)Plm(sin

β)

cos

m(λ

−

α)

,

m=1

where Plm are the associated Legendre functions. We thus obtain 1/D in terms of the six spherical coordinates. Substituting this expression into (3.16), then into (3.14), and using R to denote the equatorial radius Re = a of the Earth, we obtain:

dM (ρ, α, β) U (r, λ, ψ) = G
ρ α β D(r, λ, ψ, ρ, α, β)

1 =G
r

R 2π π/2

∞

ρ=0 α=0 β=−π/2 l=0

ρl r Pl(sin ψ)Pl(sin β)

l (l − m)!

+2

(l + m)! Plm(sin ψ) cos mλPlm(sin β) cos mα

m=1

l (l − m)!

+2

(l + m)! Plm(sin ψ) sin mλPlm(sin β) sin mα dM .

m=1

Finally, we obtain the expression for U in terms of the associated Legendre functions Plm and the coeﬃcients Clm and Slm :

μ∞ U (r, λ, ψ) =
r
l=0

Rl l

r

Clm cos mλ + Slm sin mλ Plm(sin ψ) ,

m=0

(3.17)

with μ = GM and M the mass of the Earth given by

R 2π π/2

M=

dM (ρ, α, β) ,

ρ=0 α=0 β=−π/2

and Clm and Slm the harmonic coeﬃcients of the geopotential of degree l and order m.

64

Chapter 3. Geopotential

In the expression (3.17), the terms for m = 0 refer to the Legendre polynomial Pl and the sum from m = 1 to m = l refers to the associated Legendre functions Plm. The coeﬃcients Clm and Slm are obtained by identifying the two formulas for U . There are two cases, depending on whether m is zero or not:

• Harmonic coeﬃcients for m = 0, Cl0 and Sl0 :

1 Cl0 = M Rl

R ρ=0

2π α=0

π/2
ρlPl(sin β)dM (ρ, α, β) ,
β=−π/2

Sl0 = 0 .

(3.18) (3.19)

The coeﬃcients Sl0 are always zero. • Harmonic coeﬃcients for m = 0, Clm and Slm :

2 (l − m)! Clm = M Rl (l + m)!

ρ

α

ρlPlm(sin β) cos mα dM
β

2 (l − m)! Slm = M Rl (l + m)!

ρ

α

ρlPlm(sin β) sin mα dM
β

(3.20) (3.21)

The function U (r, λ, ψ) representing the gravitational potential of the Earth is called the geopotential.
3.3.3 Spherical Harmonics
The potential U has been given as a linear combination of spherical functions Flm and Glm deﬁned by
Flm(λ, ψ) = Plm(sin ψ) cos mλ , Glm(λ, ψ) = Plm(sin ψ) sin mλ .
These can be considered as the real and imaginary parts of the functions Hlm, called spherical harmonics:
Hlm(λ, ψ) = eimλPlm(sin ψ) .
These functions have many mathematical properties (such as orthogonality) and there exists an extensive literature. In the present context, they can be used to give a graphical decomposition of the geopotential.
One can gain an idea of the way the spherical functions vary by plotting the points on the sphere where they vanish. To do so, the spherical harmonics are divided into three groups: the zonal harmonics, the sectorial harmonics, and the tesseral harmonics:

3.3 Calculating the Geopotential

65

• Zonal harmonics. These are obtained when m = 0. In this case,

Fl0 = Pl0(sin ψ) = Pl(sin ψ) , Gl0 = 0 , Hl0 = Pl(sin ψ) .

Hence, Hl0(λ, ψ) = Hl0(ψ) depends only on the latitude. Zonal harmonics have axial symmetry about the axis through the poles. In particular, they take into account the ﬂattening of the Earth. They divide the Earth up along the geographic parallels. • Sectorial harmonics. These are obtained when m = l. In this case,

Plm(sin ψ)

=

Pll(sin ψ)

=

(2l)! 2ll!

(cos2

ψ)l/2

.

This function of ψ is never zero, except at the poles. Hence, Hll is only zero for certain values of λ. The sectorial harmonics only vanish on the geographic meridians and one generally gives a picture of the sphere that looks like an orange separated into segments that meet at the poles. • Tesseral harmonics. These are obtained in all other cases. The zeros produce a kind of spherical chessboard pattern, marked out by the meridians and parallels.

Normalised Coeﬃcients
Geopotential models are generally expressed in terms of normalised coeﬃcients Cl∗m, while the coeﬃcients Clm used above are referred to as nonnormalised. The relation between Cl∗m and Clm is

Cl∗m =

(l + m)! (l − m)!(2l + 1)(2 − δ0m) Clm ,

(3.22)

where δ0m is the Kronecker symbol, equal to 1 if m = 0 or 0 if m = 0.

3.3.4 Second Degree Expansion of the Potential
To make use of these rather complex formulas, we begin by the simplest case, namely when the Earth is treated as an ellipsoid of revolution. This amounts to stopping the expansion at degree and order 2.
Theoretical Calculation of Coeﬃcients
If we expand the potential U given by (3.17) up to second degree, we obtain

66

Chapter 3. Geopotential

μ

U (r, λ, ψ) = r

C00P0(sin ψ)

R

+ r

C10P1(sin ψ) + (C11 cos λ + S11 sin λ)P11(sin ψ)

R2

+ r

C20P2(sin ψ) + (C21 cos λ + S21 sin λ)P21(sin ψ)

+ (C22 cos 2λ + S22 sin 2λ)P22(sin ψ) . (3.23)

The values of the ﬁrst few Legendre polynomials and functions for the argument sin β are as follows:
P0(sin β) = 1 , P1(sin β) = sin β , P2(sin β) = (3 sin2 β − 1)/2 ,
P11(sin β) = cos β , P21(sin β) = 3 sin β cos β , P22(sin β) = 3 cos2 β .
We can now calculate the harmonic coeﬃcients Clm and Slm using the four relations (3.18)–(3.21), adopting the spherical coordinates of the interior point T deﬁned by (3.12). The coordinates of the center of gravity of the Earth are (x0, y0, z0) and the components of the Earth’s inertia tensor7 are Ix, Ixy, and so on. The results are displayed in Table 3.1. Case of the Earth Ellipsoid
In the case of a solid Earth, the origin of the coordinate system for expanding the geopotential is taken at the center of the Earth. We then have x0 = y0 = z0 = 0, which implies that
C10 = 0 , C11 = 0 , S11 = 0 .
If the axis Oz passes through the center of inertia, we have Ixz = Iyz = 0, which implies that
C21 = 0 , S21 = 0 .
The most signiﬁcant inhomogeneity in the terrestrial mass distribution is due to the ﬂattening at the poles. The Earth is treated here as an ellipsoid of

7The moment of inertia Ix is deﬁned by Ix = (y2 + z2)dM , while the product
of inertial Ixy is deﬁned by Ixy = xy dM . In the literature, the moments of inertia
are often denoted by A = Ix, B = Iy, and C = Iz, whence (3.25) and (3.26) become J2 = (C − A)/M R2.

3.3 Calculating the Geopotential

67

1

C00 = M

dM (ρ, α, β) = 1
ραβ

1

C10 = M R

ρ

α

ρ sin β dM (ρ, α, β)
β

=1 MR

z dM = z0 R

1

C11 = M R

ρ

α

ρ cos β cos α dM (ρ, α, β)
β

=1 MR

x dM = x0 R

1

S11 = M R

ρ

α

ρ cos β sin α dM (ρ, α, β)
β

1 =
MR

y dM = y0 R

1 C20 = M R2

ρ

α

ρ2 3 sin2 β − 1 dM (ρ, α, β)

β

2

1 = 2M R2

3z2 − (x2 + y2 + z2) dM

1 =
2M R2

(x2 + z2) + (y2 + z2) − 2(x2 + y2) dM

=

1 2M R2 (Ix + Iy

− 2Iz)

1 C21 = 3M R2

ρ

α

3ρ2 sin β cos β cos α dM (ρ, α, β)
β

1 = M R2

1 xz dM = M R2 Ixz

1 S21 = 3M R2

ρ

α

3ρ2 sin β cos β sin α dM (ρ, α, β)
β

1 = M R2

1 yz dM = M R2 Iyz

1 C22 = 12M R2

ρ

α

3ρ2 cos2 β cos 2α dM (ρ, α, β)
β

1 =
4M R2

(x2

− y2) dM

=

1 4M R2 (Iy

− Ix)

1 S22 = 12M R2

ρ

α

3ρ2 cos2 β sin 2α dM (ρ, α, β)
β

1 = 2M R2

1 xy dM = 2M R2 Ixy

Table 3.1 : Harmonic coeﬃcients Clm and Slm of the geopotential of degree l and order m, up to l = 2, m = 2.

68

Chapter 3. Geopotential

revolution with axis Oz. In this case, the axial symmetry implies that Ixy = 0 and Ix = Iy, which in turn implies that

C22 = 0 , S22 = 0 .

(3.24)

The ﬂattening at the poles is expressed by the fact that Iz > Ix. Hence,

C20

=

1 M R2 (Ix

− Iz)

,

C20 < 0 .

(3.25)

When we expand the geopotential to second order and with the above assump-
tions, the only nonzero term (apart from the leading term C00 = 1) is thus the term C20 (which is negative). It is customary to introduce the coeﬃcients Jl deﬁned by8

Jl = −Cl0 .

(3.26)

The geopotential is then

μ U (r, λ, ψ) = U (r, ψ) =

1−

r

R 2 3 sin2 ψ − 1

r J2

2

,

(3.27)

with
J2 = 1.0826 × 10−3 .
This term is dimensionless, like all the coeﬃcients Clm and Slm. The value of the coeﬃcient J2 was known for a long time from geodetic considerations (see Sect. 3.4.3), and then to very high accuracy by studying the trajectories of the ﬁrst artiﬁcial satellites.

3.3.5 Expanding the Potential to Higher Degrees

For degrees higher than 2 and using the notation introduced above, the potential can be written

μ U (r, λ, ψ) =
r

∞
1−

R

r

l=1

∞l

+

l=1 m=1

l
JlPl(sin ψ)

(3.28)

Rl

r

Clm cos mλ + Slm sin mλ Plm(sin ψ) .

8In 1958, one of the pioneers of space geodesy, Desmond King-Hele, decided to attribute the letter J to this coeﬃcient, in homage to the British geodesist Sir Harold Jeﬀreys (1891– 1989).

3.3 Calculating the Geopotential

69

Jn = −Cn0
C00 J1
J2 J3 J4 J5 J6 J7 J8 J9 J10

Value [dimensionless]
1
0
+1,082.62652305×10−6 −2.53253531×10−6 −1.61997147×10−6 −0.22780140×10−6 +0.54066755×10−6 −0.36055772×10−6 −0.20402823×10−6 −0.12211470×10−6 −0.24439275×10−6

Table 3.2 : Harmonic coeﬃcients Jn for the geopotential, up to n = 10. Values taken from the EIGEN-6C2 model.

In the part between square brackets, there are three groups of terms:
• The ﬁrst comprises only the number 1, representing the central potential. • The second, with Jl and Pl, constitutes the contribution of the zonal har-
monics [see (3.45)]. • The third, involving Clm, Slm, and Plm, gives the contribution of the sec-
torial and tesseral harmonics.
These terms Jl, Clm, and Slm can only be known (except possibly for J2) by comparing the ellipsoid with the actual shape of the Earth, which is called the geoid.9 To do this, weight measurements can be carried out in situ, but the best approach today is to use precise observations of the motions of artiﬁcial satellites. These points will be discussed below.
For the Earth as it really is (dropping the ellipsoid approximation), the numerical values of Jl are given in Table 3.2 (but see also Table 3.3). These coeﬃcients are usually referred to in the literature as Jn terms. So for the geoid, the coeﬃcients C10 (or J1), C11, and S11 are zero, while the coeﬃcients C21 and S21 (∼ 10−9), C22, and S22 (∼ 10−6) are nonzero.
Regarding orders of magnitude, we see that the J2 term is about 103 times smaller than the leading term, but 103 times greater than the following coeﬃcients. To sum up, considering the expansion of the potential given by (3.28), we observe that (see Fig. 3.2):

9When geodesists realised that the shape of the Earth was not exactly ellipsoidal, they chose to call it the geoid (Listing in 1873), which is tautological: it is like saying that the Earth is Earth-shaped! One sometimes sees the word telluroid, a disharmonious product of Latin–Greek hybridisation that is just as tautological.

70

Chapter 3. Geopotential

Coeﬃcient GEM-T2

JGM-3

GRIM5-C1

GRIM5-S1

C2∗0 C3∗0 C4∗0 C5∗0 C6∗0 C7∗0 C8∗0 C9∗0 C1∗00 C2∗00 C9∗90

−484.1652998 0.9570331 0.5399078 0.0686883
−0.1496092 0.0900847 0.0483835 0.0284403 0.0549673 0.0199685

−484.165368 0.957171 0.539777 0.068659
−0.149672 0.090723 0.049118 0.027385 0.054130 0.018790

−484.16511551 0.95857491 0.53978784 0.06726760
−0.14984936 0.09301877 0.05039091 0.02628356 0.05101952 0.02340848
−0.00128836

−484.16511551 0.95857492 0.53978784 0.06720440
−0.14985240 0.09311367 0.05046451 0.02620763 0.05076191 0.02342817
−0.00001554

Coeﬃcient EGM96

EIGEN-CH03S EIGEN-6C2

C2∗0 C3∗0 C4∗0 C5∗0 C6∗0 C7∗0 C8∗0 C9∗0 C1∗00 C1∗10 C1∗20 C1∗30 C1∗40 C1∗50 C2∗00 C9∗90

−484.165371736 0.957254174 0.539873864 0.068532348
−0.149957995 0.090978937 0.049671167 0.027671430 0.052622249
−0.050961371 0.037725264 0.042298221
−0.024278650 0.001479101 0.022238461 0.001478118

−484.165562843 0.957477372 0.539923241 0.068584004
−0.149991332 0.090539419 0.049295631 0.028093014 0.053699211
−0.050765723 0.036209032 0.041543398
−0.022288877 0.002425544 0.021496270
−0.000779156

−484.165299956 0.957208401 0.539990490 0.068684705
−0.149954200 0.090513612 0.049484115 0.028015031 0.053330869
−0.507685657 0.036437330 0.041729879
−0.022669657 0.002192288 0.021558749 0.002263992

C2∗2

2.439143524

2.439311853

2.439355937

S2∗2

−1.400166837 −1.400342254 −1.400284583

C3∗1

2.029988822

2.030480649

2.030499314

S3∗1

0.248513159

0.248170920

0.248199233

C3∗3

0.721072657

0.721306788

0.721274250

S3∗3

1.414356270

1.414370341

1.414373139

Table 3.3 : Comparison between diﬀerent models. Normalised zonal coeﬃcients Cl∗0 and other normalised coeﬃcients Cl∗m and Sl∗m. All values should be multiplied by 10−6.

• The term of degree 0 is the leading term, causing the Keplerian motion (see below), in which the Earth is considered to be spherical and made up of homogeneous layers.

3.4 Weight Field and Potential for the Ellipsoid

71

J0

J2

J3,4,...

Fig. 3.2 : Changing perception of the shape of the Earth in geodesy. Left: Sphere, with C00 = −J0 = 1 and J1 = 0. Center: From the sphere to the ellipsoid of revolution, with J2 term indicating ﬂattening. Right: From the ellipsoid to the geoid, with Jn terms n ≥ 3.
• The term of degree 1, which would correspond to a shift in the center of mass of the Earth away from the geometrical center, is made to vanish by choice of the coordinate origin.10
• The term of degree 2 corresponds to the ﬂattening of the Earth when the latter is considered as an ellipsoid of revolution.
• The terms of degree 3 and higher cater for deviations between the geoid and the Earth ellipsoid.

3.4 Weight Field and Potential for the Ellipsoid
3.4.1 Calculating the Field and Potential
In order to investigate the weight ﬁeld at the surface of the Earth, one has to consider the gravitational force ﬁeld in a frame T moving with the Earth, rather than in the Galilean frame . To obtain the relations in T, in addition to the acceleration as calculated in , one must take into account the centrifugal acceleration ac due to the Earth’s rotation:
ac = − 2JM ,
where is the angular speed of the Earth’s rotation11 and J is the projection of M on the polar axis, which is the point of geodetic latitude ϕ (and geocentric

10For the great majority of geopotential models, C00 = 1 and exceptions, however, such as EIGEN-CHAMP03-S, with C00 = 1 10−9 or GRIM5-C1, with C00 = 1 − 1.14 × 10−10 and C1∗0 = 0.

aCn1d∗0C=1∗00=. T−h0e.r8e33a9re09s6o6m×e

11We use the notation only in this chapter. In subsequent chapters, we shall use the

notation Ω˙ T for this quantity, and we shall explain why when the time comes. The angular

speed is equal to one revolution per sideral day, or = 7.292115 × 10−5 rad s−1.

72

Chapter 3. Geopotential

z B J

MK

ψ

ϕ

xЈ

OQ

A

x

I

e = 0.60 zЈ
Fig. 3.3 : For a point M at the Earth’s surface with geodetic latitude ϕ and geocentric latitude ψ, we represent gravity, pointing toward O, and weight, pointing toward I, normal to the ellipsoid at M .

latitude ψ) at the Earth’s surface (see Fig. 3.3). The center of the Earth is O and we set r = OM . Therefore, JM = r cos ψ. The unit vector in the direction OM is denoted by er.
To simplify the notation, we set

g = g (gravity) , γ = g T (weight) . The rule for composition of accelerations is then

absolute acceleration

(g)

=

relative acceleration

(γ) +

centrifugal acceleration

(ac) .

We obtain

γ = g + 2JM .

(3.29)

The vector γ represents the weight. This is what deﬁnes the weight of a body at a given location. The weight is the vector sum of the gravity and the

3.4 Weight Field and Potential for the Ellipsoid

73

centrifugal acceleration. The vector g lies along OM . The angle between the
vector γ and g is very small, equal to ϕ − ψ. Its value, given by (2.5), is at most 0.19◦. We can thus write

g = −ger , γ = −γer .

Projecting the expression in (3.29) onto OM , we then have

−γer = (−g + 2r cos2 ψ)er .

(3.30)

Expressing the ﬁelds g and γ in terms of the respective potentials U and UT and integrating (3.30) with respect to r, we obtain

1 UT = U + 2

2r2 cos2 ψ .

Cutting oﬀ the expansion of U at the second order as in (3.27), i.e., treating the geoid as an ellipsoid, we obtain

μ UT(r, ψ) = r

1−

R r

2 3 sin2 ψ − 1

J2

2

2
+ r2 cos2 ψ . 2

(3.31)

The axial symmetry of the model appears through the absence of the variable λ (longitude) in the expression for the potential UT.

3.4.2 Weight Field at the Earth’s Surface

The weight ﬁeld γ is found by diﬀerentiating UT along the normal to the ellipsoid. To the same order of approximation as when we identify the directions of the vectors g and γ, we may consider the ﬁeld to be given by ∂UT/∂r. Its magnitude γ is then

μ γ(ψ) = r2 1 − 3

a 2 3 sin2 ψ − 1

r J2

2

−

2r cos2 ψ .

(3.32)

Replacing r by its value as a function of ψ, viz., r = Rψ(ψ) as given by (1.37), we obtain an expression for the magnitude γ(ψ) of the weight ﬁeld at the surface of the ellipsoid as a function of the latitude alone.
The dependence of the weight ﬁeld on the latitude is shown in Fig. 3.4. The latitude dependence of the gravitational ﬁeld is also shown. In SI units, the gravity g varies from 9.814 at the equator to 9.832 at the pole, due to the ﬂattening of the Earth, but this quantity is not directly measurable, because we cannot stop the Earth from rotating!

74

Chapter 3. Geopotential

9.840

9.830

Weight and gravitation (m.s-2)

9.820

9.810

9.800

9.790

9.780

9.770
0 EARTH

10

20

30

40

50

60

70

Latitude [N/S] (°)

80

90

Fig. 3.4 : Dependence of the weight (lower curve) on the latitude at the surface of the ellipsoid. The upper curve shows the theoretical latitude dependence, not directly measurable, of the gravitational ﬁeld. The diﬀerence between these two curves gives the value of the centrifugal acceleration.

The weight γ, measured experimentally,12 varies from 9.780 at the equator
to the same value 9.832 at the pole, because the variation caused by the Earth’s rotation, which is zero at the pole, adds algebraically to the variation of g.13

12ESA’s presentation of the satellite GOCE provides an interesting illustration of the

levels of accuracy attained in weight measurements at a given location:

Weight = 9.8

Mass of spherical Earth

9.81

Flattening and rotation

9.812

Mountain and oceanic rifts

9.8123

Internal mass distribution

9.81234 Major river dams

9.812345 Sea and land tides

9.8123456 Large buildings in the neighbourhood

13Let us indulge in a little science ﬁction! Imagine a planet just like the Earth, but

rotating faster, with angular speed . Let us calculate the weight at the equator, assuming

that the planet is spherical with radius R. From (3.32), we ﬁnd

μ γ = r2 −

2R

=

μ R2

1−

2R3 μ

μ = R2 (1 − ma) ,

where ma is deﬁned below by (3.34). This term represents the contribution of the centrifugal acceleration to the weight when gravity is taken as unity. For the Earth, ma = 1/288. If ma = 1, the weight is zero, and all bodies at the equator on the surface ﬁnd themselves

3.4 Weight Field and Potential for the Ellipsoid

75

This formula for the latitude dependence of the weight ﬁeld is quite accurate enough in many cases. However, if we require a more accurate formula in terms of the geodetic latitude and without approximation, we can use Somigliana’s formula discussed below.

3.4.3 Clairaut’s Formula

The J2 term in the expansion of the geopotential, which can be related to the diﬀerence in moments of inertia of the Earth about the polar axis and about an equatorial axis, as can be seen from (3.25) and (3.26), cannot be measured directly. But without waiting for the advent of the artiﬁcial satellite, it could be determined from geodetic considerations, exploiting the properties of equipotential surfaces.
Relation Between J2 and Flattening
Clairaut14 made the assumption that the Earth was in hydrostatic equilibrium in its rotation about its own axis. It follows that, for any point on the Earth’s surface, taken as an ellipsoid, the potential is constant. Let us choose a point at the pole (r = Rp = b) and a point at the equator (r = Re = a):

UT(r = a, ψ = 0) = UT(r = b, ψ = π/2) .

Equation (3.31) yields

μ a

1 1 + 2 J2

+

2
a2

=

μ

2

b

a2 1 − b2 J2

.

The quantities f and J2 are much smaller than 1. Neglecting small quantities to second order, the right-hand side becomes

μ (1
a

+

f)

1

−

J2(1

+

2f )

≈

μ (1 +
a

f

− J2)

,

in weightless conditions. For such a fast-spinning version of Earth, we thus have ( / )2 = 288, or ≈ 17 . With such an angular speed, the day lasts 17 times less than on the real Earth, i.e., a mere 84.5 min. Furthermore, this is equal to the period of rotation of a terrestrial satellite at zero altitude, as we shall see in (5.9) of Chap. 5.
14Alexis Claude Clairaut (1713–1765) was a French astronomer and mathematician. He entered the French Academy of Sciences at the age of eighteen, after astonishing the assembly by his investigation of geometric curves. He soon turned his attention to geodesy and celestial mechanics, publishing Th´eorie de la ﬁgure de la Terre tir´ee des principes de l’hydrostatique in 1743. This explored the diﬀerences in the acceleration due to the weight at the poles and the equator. He then studied the three-body problem and published his Th´eorie de la Lune in 1752. He was also one of the ﬁrst to investigate gravitational perturbations (see the historical note on the return of Halley’s comet in Sect. 6.8.1).

76

Chapter 3. Geopotential

which gives Clairaut’s ﬁrst equation:

21

J2

=

f 3

−

3 ma

,

(3.33)

where the dimensionless quantity

2a3 ma = μ

(3.34)

is easily found to be ma = 3.461 × 10−3. If we consider the ﬂattening to be given by f = 1/298.3, we obtain the value of J2 to ﬁrst order as

J2 = 1.0814 × 10−3 .

This is very close15 to the value of J2 given in Table 3.2.

Relation Between J2 and Weight
Historically, it was the quantity f that scientists sought to calculate. That is, they hoped to determine the ﬂattening without having to measure the Earth’s meridian. They thus had to ﬁnd some way of expressing J2, and this could be done by measuring γ, the acceleration due to the weight, at various points on the Earth’s surface. Now using (3.32), we can calculate γ = γe at the equator and γ = γp at the pole:

μ γe = a2
μ γp = b2

3 1 + 2 J2

−

2a ,

a2

μ

1 − 3 b2 J2 ≈ a2 (1 + 2f − 3J2) .

Neglecting small quantities to second order, the diﬀerence gives

μ γp − γe = a2

9 2f − 2 J2

+

2a .

Replacing γe by μ/a2 in the small terms, since g = μ/R2 to a ﬁrst approximation, we obtain

γp − γe γe

=

2f

−

9 2 J2

+ mg

,

15Carrying out the calculation to second order in the small quantities, we obtain

J2

=

2 f
3

−

1 3 mb

−

1f2 3

+

2 21 f mb

,

where mb = 2a2b/μ = ma(1 − f ). The numerical result is J2 = 1.082634 × 10−3, implying a relative error of 7 × 10−6 compared with the value of J2 given in the text.