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Keplerian EllipsesThe physics of the gravitational two-body problem

Keplerian EllipsesThe physics of the gravitational two-body problem

Bruce Cameron ReedAlma College, Alma, MI, USA

Morgan & Claypool Publishers

Copyright ª 2019 Morgan & Claypool Publishers

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ISBN 978-1-64327-470-6 (ebook)ISBN 978-1-64327-467-6 (print)ISBN 978-1-64327-468-3 (mobi)

DOI 10.1088/2053-2571/ab0303

Version: 20190501

IOP Concise PhysicsISSN 2053-2571 (online)ISSN 2054-7307 (print)

A Morgan & Claypool publication as part of IOP Concise PhysicsPublished by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA,94901, USA

IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

This work is dedicated to Laurie.

Again.

Contents

Preface ix

Acknowledgments xiii

Author biography xiv

1 Spherical coordinates—a review 1-1

1.1 Fundamental definitions 1-1

1.2 Spherical coordinate unit vectors 1-1

1.3 Time-derivatives of spherical coordinate unit vectors 1-3

1.4 Some useful integrals 1-4

2 Dynamical quantities in spherical coordinates 2-1

2.1 Position, velocity, acceleration, angular momentum,torque, and energy

2-1

2.2 Uniform circular motion: a specific case of the acceleration formula 2-3

3 Central forces 3-1

3.1 The reduced mass 3-1

3.2 Central force dynamics: the potential 3-3

3.3 Why an inverse-square law? 3-6

3.4 Central force dynamics: conservation of angular momentum 3-8

3.5 Central force dynamics: integrals of the motion 3-10

3.6 Central force dynamics: acceleration in terms of the azimuthal angle 3-11

4 The ellipse 4-1

4.1 The ellipse in Cartesian and polar coordinates 4-1

4.2 Area of an ellipse 4-4

4.3 Area as a vector cross-product, and Kepler’s second law 4-4

4.4 How did Kepler plot the orbits? 4-6

5 Elliptical orbits and the inverse-square law: geometry meetsphysics

5-1

5.1 Proof by assuming an elliptical orbit: angular momentum 5-1

5.2 Velocity, the vis-viva equation, and energy 5-3

5.3 Proof of elliptical orbits by direct integration 5-4

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5.4 Kepler’s third law 5-7

5.5 The time–angle equation 5-8

5.6 Example: an Earth-orbiting spy satellite 5-10

6 Kepler’s equation: anomalies true, eccentric, and mean 6-1

7 Some sundry results 7-1

7.1 Average distance of a planet from the Sun 7-1

7.2 Determining initial launch conditions 7-3

7.3 A brief lesson in unit conversions 7-5

7.4 Orientation of Earth’s orbit 7-6

7.5 Some final words 7-10

Bibliography 8-1

Glossary of Symbols 9-1

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Preface

Celestial mechanics is the crown jewel of Newtonian physics. Building on theobservational and theoretical work of Galileo, Tycho Brahe, Copernicus, Keplerand others, Isaac Newton deployed his hypothesized law of universal gravitationand his newly-developed calculus to show why the orbits of the Moon and planetshave the shapes they do: Kepler’s three laws of planetary motion (figures P.1 andP.2). This was a stunning development in human intellectual history. Contrivedsystems of equants, deferents, and epicycles gave way to a single elliptical orbit foreach planet (table P.1). This new approach facilitated predictions of celestialphenomena to unprecedented levels of accuracy, and the realization that these orbitsreflected an underlying universal physical law whose validity extended from thefirmament of the Earth to the remotest cosmos marked the birth of modern science.

This book presents a self-contained treatment of the Newton/Kepler two-bodyorbit problem at the level of an advanced undergraduate physics student.

Any student of physics or astronomy will surely encounter Kepler’s three laws ofplanetary motion in a dynamics or astrophysics class. Proofs of these laws are someof the ultimate products of this book, so let’s summarize them right away, expressedin modern parlance:

Kepler’s first law: Each planet orbits the Sun on an elliptical path, with the Sunat one focus of the ellipse.

Kepler’s second law: A line from the Sun to a planet sweeps out equal areas inequal times.

Kepler’s third law: The square of a planet’s orbital period is directly propor-tional to the cube of the semi-major axis of its elliptical orbit.

Kepler published his first two laws in his Astronomia Nova (New Astronomy) in1609, deductions based on Tycho Brahe’s detailed observations of the motions of

Figure P.1. (Left) Johannes Kepler (1751–1630). Source: https://upload.wikimedia.org/wikipedia/commons/d/de/JKepler.png. (Right) Sir Isaac Newton (1643–1727). Source: https://commons.wikimedia.org/wiki/Isaac_Newton#/media/File:Sir_Isaac_Newton_(1643–1727).jpg.

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Mars. He published the third law in 1619 in hisHarmonices Mundi (The Harmony ofthe World). Orbital mechanics has been picked apart for over 400 years (over 2000 ifyou count Ptolemy); the physics sections of school libraries are rife with dynamicstexts, and extensive discussions of Kelperian mechanics can be found online.Why, then, did I prepare this book?

17501500 1600 1700

Copernicus Galileo Newton

Kepler

Tycho

Figure P.2. Timeline of the lives of some of the founders of modern astronomy. Copernicus (1473–1543),Tycho Brahe (1546–1601), Galileo (1564–1642), Kepler (1571–1630) and Newton (1643–1727).

Table P.1. Solar-system orbital data.

Object Semi-major axis (AU) Period (years) Eccentricity Mass (Earth = 1)

Sun 332 946Mercury 0.3871 0.2408 0.2056 0.0553Venus 0.7233 0.6152 0.0068 0.8150Earth 1.0000 1.0000 0.0167 1.0000Mars 1.5237 1.8808 0.0934 0.1075Jupiter 5.2029 11.863 0.0484 317.83Saturn 9.537 29.447 0.0539 95.159Uranus 19.189 84.02 0.0473 14.50Neptune 30.0699 164.79 0.0086 17.20Ceres 2.7658 4.600 0.078 < 0.001Pluto 39.482 248.021 0.2488 0.002Haumea 43.34 285.4 0.189 < 0.001Makemake 45.79 309.9 0.159 < 0.001Eris 67.67 557.2 0.4418 0.003Sedna 525.6 ∼12 050 0.855 < 0.001Halley’s comet 17.834 75.3 0.967 < 0.001

Ceres was traditionally termed an asteroid; it is now considered to be a dwarf planet. Pluto, Haumea,Makemake, Eris, and Sedna are ‘Plutoid’-class dwarf planets, essentially round objects orbiting the Sun atdistances greater than that of Neptune. Semi-major axes are in terms of astronomical units (AUs), the meanEarth/Sun distance, and periods are in years.AU = 1.495 98 × 1011 m.One year = 365.25 days = 31 557 600 s = 3.156 × 107 s.MEarth = 5.972 × 1024 kg.MSun = 1.988 × 1030 kg.Radius of Earth = 6371 km.Newtonian gravitational constant: G = 6.674 × 10−11 m3 kg−1 s−2

Data adopted from https://www.princeton.edu/∼willman/planetary_systems/Sol/ and various online sources.

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When students encounter orbital mechanics, it is usually as a unit of a much moreextensive dynamics class that will include numerous topics such as resisted motion,rotating reference frames, damped and driven vibrational motions, Lagrangian andHamiltonian dynamics, least-action principles, and perhaps even relativity. In such avast docket of material to be crammed into one or two semesters—all worthy andcontaining much powerful physics—planetary orbits becomes just one more topic tosqueeze into the syllabus. It is difficult to do this beautiful topic justice in suchcircumstances. This book grew out of my conviction that students deserve, canbenefit from, and would, I hope, enjoy a brief, affordable, self-contained treatmentof the essential physics of Keplerian elliptical orbits. When you have finished readingthis book I hope that you will keep it as a reference/refresher.

My goal has been to prepare a volume that starts with fundamental concepts suchas position, velocity, acceleration, force, energy, momentum, and angular momen-tum (all expressed in spherical coordinates!), pairs them with Newton’s gravitationallaw, and shows you how elliptical orbits and Kepler’s laws result, all while setting upvery general expressions for exploring relationships between distances, angles, andtimes for such orbits. The derivations presented in this book are not complete inevery detail, but are given in enough detail that you should be able to fill in the gaps.And do fill in the gaps: think of them as exercises. It will be excellent algebra practiceand help reinforce the concepts involved.

This book comprises seven chapters, a bibliography and a glossary of mathe-matical symbols. Since gravitation is a ‘central’ force, that is, one which acts directlyalong the line joining the centers-of-mass of two objects, the naturally convenientcoordinate framework within which to analyze orbital mechanics is the sphericalpolar coordinate system. Chapter 1 is devoted to a brief review of sphericalcoordinates and the properties of their corresponding unit vectors, and also lists afew useful integrals. Chapter 2 develops expressions for position, velocity, accel-eration, angular momentum, torque, and energy in spherical coordinates. Sincethese expressions do not assume any particular force law (that comes later), they arevery general. Chapter 3 introduces the idea of a central force in spherical-coordinateform; shows how the mutual motion of two bodies under the action of a central forcecan be reduced to considering that of a single ‘reduced’mass which is imagined to beorbiting another mass which serves as the origin of the coordinate system; showswhy angular momentum must be conserved within such a system; sets up verygeneral integrals for expressing relationships between time, angle, and radialposition for such motions; and shows how Newton’s second law F = ma can betransformed into an expression that relates rates of change of radial position to theangular position of the orbiting body as opposed to rates of change with respect totime. All of this can be done without yet specifying the particular (i.e. gravitational)form of the central force.

It is due to Johannes Kepler that we know that planets orbit the Sun in ellipticalpaths with the Sun at one focus of the ellipse. Chapter 4 reviews some of theproperties of ellipses, and shows how the polar-form equation for an ellipse, which isso handy for analyzing orbits, can be derived from the fundamental Cartesianequation for an ellipse. This chapter also presents a proof of Kepler’s second law by

Keplerian Ellipses

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examining the area of an ellipse, and briefly describes how Kepler actually wentabout plotting planetary orbits. This sets the stage for chapter 5, where it is shown bytwo methods that ellipses satisfy the force equation set up in chapter 3 in the case ofNewton’s law of gravitation and that Kepler’s three laws result. Chapter 6 is devotedto a derivation of ‘Kepler’s equation’, a compact although not analytically solvableexpression which relates the position of a planet in its orbit to time. Chapter 7collects some sundry results. The bibliography provides a brief list of sources forreaders who wish to pursue more detailed study. The glossary gives brief definitionsof the many mathematical symbols used throughout the text.

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Acknowledgments

This work grew out of my notes for various classes that I have taught over manyyears at both Alma College in Michigan and Saint Mary’s University in Halifax,Nova Scotia. Thanks are due my many students who pointed out confusing oroverly-elaborate derivations, thus helping me to distill orbital mechanics to itsessence. My interest in this material began during my own student days long ago atthe University of Waterloo and Queen’s University, and I am grateful to my manyexcellent professors at those institutions for their superb classes and encouragement.Thanks are also due Susanne Filler and Joel Claypool of Morgan and ClaypoolPublishers for agreeing to this project and encouraging me to see it through. And toLaurie, again.

Johannes Kepler set out to discover India and found America. It is an event repeatedover and again in the quest for knowledge. But the result is indifferent to the motive. Afact once discovered, leads to an existence of its own, and enters into relations withother facts of which their discoverers never dreamt. Apollonius of Perga discovered thelaws of the useless curves which emerge when a plane intersects a cone at variousangles: these curves proved, centuries later, to represent the paths followed by planets,comets, rockets, and satellites.

Arthur Koestler 1959 The Sleepwalkers (New York: Grosset and Dunlap) p 398.

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Author biography

Bruce Cameron Reed

Bruce Cameron Reed is the Charles A Dana Professor of PhysicsEmeritus at Alma College, Alma, Michigan. He holds a PhD inPhysics from the University of Waterloo in Canada. In addition toa quantum mechanics text and four books on the ManhattanProject (including the IOP Concise Physics volumes Atomic Bomb:The Story of the Manhattan Project and The Manhattan Project: AVery Brief Introduction to the Physics of Nuclear Weapons), he has

published over 150 papers in peer-reviewed scientific journals on research in thefields of astronomy, data analysis, quantum physics, mathematics, nuclear physics,the history of physics, and the physics of nuclear weapons. In 2009 he was elected aFellow of the American Physical Society ‘For his contributions to the history of boththe physics and the development of nuclear weapons in the Manhattan Project.’After a 35 year teaching career spent in both Canada and the United States, he isnow semi-retired and living in Nova Scotia with his wife Laurie and a variablenumber of cats, and where he continues to teach part-time and keeps busy withvarious publication projects.

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