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Motions of Celestial Bodies: Computer Simulations

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Motions of Celestial Bodies: Computer Simulations Educational Software MOTIONS OF CELESTIAL BODIES: COMPUTER SIMULATIONS Eugene I. Butikov Department of Physics St. Petersburg State University St. Petersburg, Russia 2014 ii Contents Preface vii 1 Introduction: Getting Started 1 1.1 List of the Simulation Programs . 1 1.2 How to Operate the Simulation Programs . 2 1.3 Keplerian Motions in Celestial Mechanics . 3 1.4 Numerical and Analytical Methods . 6 I Review of the Simulations 9 2 Kepler’s Laws 11 2.1 Kepler’s First Law . 11 2.2 Kepler’s Second Law . 16 2.3 Kepler’s Third Law . 19 2.4 The Approximate Nature of Kepler’s Laws . 23 3 Hodograph of the Velocity Vector 27 3.1 Hodograph of the Velocity for Closed Orbits . 27 3.2 Hodograph of the Velocity for Open Orbits . 29 4 Satellites and Missiles 33 4.1 Families of Keplerian Orbits . 34 4.1.1 Various Directions of the Initial Velocities . 34 4.1.2 Equal Magnitudes of the Initial Velocities . 38 4.1.3 Different Magnitudes of the Initial Velocities . 40 4.2 Evolution of an Orbit in the Atmosphere . 43 4.2.1 Evolution of an Elongated Elliptical Orbit . 43 4.2.2 Late Stage of the Evolution and Aerodynamical Paradox . 45 4.2.3 Air Density over the Earth . 47 5 Active Maneuvers in Space Orbits 51 5.1 How to Operate the Program . 51 5.2 Space Flights and Orbital Maneuvers . 54 5.2.1 Designing a Space Flight . 54 iii iv CONTENTS 5.2.2 Way Back from Space to the Earth . 55 5.3 Relative Orbital Motion . 60 5.3.1 Motion of a Small Body Ejected from the Orbital Station . 60 5.3.2 Numerical Estimations . 62 5.3.3 Secular Component of the Relative Motion . 63 5.4 Space Probe and Relative Motion . 65 5.4.1 Space Probes in Inner Orbits . 65 5.4.2 Space Probes in Outer Orbits . 68 5.5 Interplanetary Flights . 71 6 Precession of an Equatorial Orbit 75 7 Binary Star—the Two-Body Problem 81 8 Three-Body Systems 87 8.1 The Restricted Three-Body Problem . 87 8.2 Managing the Program “Planet with a Satellite” . 88 8.3 Satellites of the Planet that Orbits a Star . 89 8.4 Exact Particular Solutions . 94 8.4.1 A System with Equal Masses of Heavy Bodies . 94 8.4.2 Satellites at the Triangular Libration Points . 97 8.4.3 The Collinear Libration Points . 99 8.5 Over the Back Side of the Moon . 104 8.6 Lunar Perturbations of a Satellite’s Orbit . 106 8.7 To a Distant Planet and Back . 108 8.8 Comets—Interplanetary Vagabonds . 112 8.9 A Double Star with a Planet . 115 9 Many-Body Systems in Celestial Mechanics 121 9.1 Planetary System—a Many-Body Problem . 121 9.2 A Model of the Solar System . 124 9.2.1 Kinematics of the Planetary Motion . 124 9.2.2 Kinematics of the Inferior Planets . 127 9.3 Hypothetical Planetary Systems . 128 9.4 Multiple Stars . 131 9.5 Exact Solutions to the Many-Body Problem . 134 9.5.1 A Star with Two Planets of Equal Masses . 134 9.5.2 A “Round Dance” of Identical Planets . 136 9.5.3 Keplerian Motions in Equilateral Configurations . 138 9.5.4 Remarkable Three-Body Motion Along Figure Eight . 141 II The Simulated Phenomena 145 10 Phenomena and Concepts 147 10.1 Newton’s Law of Gravitation . 147 CONTENTS v 10.2 Energy in the Gravitational Field . 149 10.3 Circular Velocity and Escape Velocity . 150 10.4 Geometric Properties of Orbits . 152 10.5 Parameters of the Orbits . 154 10.6 Satellite in the Atmosphere . 157 10.7 Trajectories of a Landing Module . 160 10.8 A Space Probe . 162 10.9 Space Rendezvous . 165 10.10Kepler’s Laws and the Solar System . 168 10.11The Three-Body Problem . 169 11 Theoretical Background 173 11.1 Angular Momentum and Areal Velocity . 173 11.2 Derivation of Kepler’s First Law . 175 11.3 Kepler’s Third Law . 178 11.4 Hodograph of the Velocity Vector . 180 11.5 Another Derivation of Kepler’s First Law . 183 11.6 Orbits with Equal Energies . 185 11.6.1 The Envelope Surface for the Family of Orbits . 185 11.6.2 Applications of the Envelope Surface . 189 11.7 Relative Orbital Motion . 191 11.8 Gravitational Field of a Distorted Planet . 195 11.8.1 A Planet with Additional Masses at the Poles . 196 11.8.2 A Planet with an Equatorial Bulge . 197 11.9 The Two-Body Problem . 198 11.9.1 Reduced Mass and Relative Motion . 198 11.9.2 An Alternative Approach to the Two-Body Problem . 200 11.10 Exact Solutions to Three-Body Problem . 201 11.11 Non-Restricted Three-Body Problem . 207 11.11.1 A Star with Two Identical Planets . 207 11.11.2 Three Bodies in the Equilateral Configuration . 209 11.12 Sphere of Gravitational Action . 211 11.13 The Oceanic Tides . 214 11.13.1 The Origin of Tidal Forces: an Elementary Approach . 215 11.13.2 Tidal Forces at an Arbitrary Point Near the Earth . 218 11.13.3 Horizontal and Vertical Components of the Tidal Force . 219 11.13.4 The Static Distortion of the Water Surface . 221 11.13.5 Tidal Forces on the Rotating Earth . 222 11.13.6 The Potential Function for Tidal Forces . 223 11.13.7 The Natural Wave and the Driving Tidal Forces . 225 11.13.8 The Tides as Forced Oscillations of the Ocean . 226 11.13.9 Mathematical Description of the Forced Oscillations . 227 11.13.10 Real-World Complications . 229 11.13.11 The Evolution of Orbital Motions . 231 Glossary 237 vi CONTENTS Index 243 Preface The textbook MOTIONS OF CELESTIAL BODIES: COMPUTER SIMULATIONS together with the accompanying award winning educational software package PLAN- ETS AND SATELLITES is intended to help students learn and understand the funda- mental concepts and the laws of physics as they apply to the fascinating world of the motions of natural and artificial celestial bodies. In this wonderful space laboratory all phenomena are observed in their purest form, without numerous complications that are inevitable in an ordinary earth laboratory. It is the understanding of the founda- tions of classical and modern physics that form the primary aim of the book while their application to the real world celestial mechanics is rather illustrative and incidental. The textbook relies heavily on the software package PLANETS AND SATEL- LITES which includes several highly interactive computer programs presenting a set of exciting computer-simulated experiments. The programs of the package provide students and their instructors with a powerful tool which enables them to investigate basic concepts and phenomena that are difficult to imagine and study in an abstract conventional manner. The textbook with the software package PLANETS AND SATELLITES is devel- oped as an exploration-oriented complement to various physics courses. It can be help- ful to a wide range of students, from those in introductory physics to those in advanced courses. With this textbook and the software, the students can learn the basic princi- ples and concepts of classical dynamics, and the application of these principles to the motions of various celestial bodies—stars, planets, comets, natural and artificial satel- lites, manned and automatic space vehicles. The simulation programs make visible the beauty and aesthetics of the mathematics and of the fundamental laws of physics in their application to the motions of celestial bodies. The software package PLANETS AND SATELLITES allows the students to con- struct and investigate a model of the solar system, or to create an imaginary planetary system on their own—complete with the star, planets, moons, comets, asteroids, and satellites. Contemporary interactive media provides students with a powerful means to visualize the evolution of such a planetary system, and to explore the orbital mo- tions governed by the gravitational forces. The simulations bring to life many abstract concepts of classical dynamics. Interactive work with PLANETS AND SATELLITES helps students understand phenomena better. Students can work at a pace they can enjoy, varying parameters of the simulated systems and repeating several times the most interesting experiments on their own. The experience based on student’s own actions results in deeper under- vii viii PREFACE standing than the mere reception of someone else’s knowledge. No doubt that for a great majority of human beings a visual impression is much more intensive and perma- nent than a heard or read one. With some of the suggested programs, students have an opportunity to perform interesting mini-research projects in physics and astronomy. Computer simulations in PLANETS AND SATELLITES enable students to see clearly how the systems that obey simple and precise physical laws behave, sometimes in unexpected and even irregular, chaotic ways. Although designed as a desk-top lab- oratory for individual interactive work, the software also provides the instructor with powerful demonstration tools to accompany lectures in mechanics and general physics. The structure of the programs allows students to study the subject at different levels of difficulty, depending on the time available and on the mathematical complexity of the course. Part I (Chapters 2 – 9) of the textbook contains a description of the simulation pro- grams and their possibilities, and explains how the programs are operated. It suggests experiments that demonstrate typical examples of behavior of the simulated systems. Part I of the textbook is aimed at building physical intuition. Understanding the un- derlying concepts of physics is given precedence over using formulae in calculations.
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