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2 Motions of Celestial Bodies Computer simulations

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4 Motions of Celestial Bodies Computer simulations Eugene Butikov Saint Petersburg State University, Russia IOP Publishing, Bristol, UK

5 ª IOP Publishing Ltd 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organisations. Permission to make use of IOP Publishing content other than as set out above may be sought at permissions@iop.org. Eugene Butikov has asserted his right to be identified as author of this work in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act ISBN ISBN ISBN (ebook) (print) (mobi) DOI / Version: IOP Expanding Physics ISSN (on-line) ISSN (print) British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Published by IOP Publishing, wholly owned by The Institute of Physics, London IOP Publishing, Temple Circus, Temple Way, Bristol, BS1 6HG, UK US Office: IOP Publishing, Inc., 190 North Independence Mall West, Suite 601, Philadelphia, PA 19106, USA

6 Contents Preface Author biography ix xi 1 Introduction: getting started List of the simulation programs How to operate the simulation programs Keplerian motions in celestial mechanics Numerical and analytical methods 1-6 Part I Review of the simulations Kepler s laws Kepler s first law 2-3 Questions and problems Kepler s second law 2-8 Questions and problems Kepler s third law 2-11 Questions and problems The approximate nature of Kepler s laws 2-15 Questions and problems Hodograph of the velocity vector for Keplerian motion Hodograph of the velocity for closed orbits Hodograph of the velocity for open orbits 3-3 Questions and problems Orbits of satellites and trajectories of missiles Families of Keplerian orbits Orbits with various directions of the initial velocities Satellites with equal magnitudes of the initial velocities Orbits of satellites launched in one direction with different magnitudes of the initial velocities 4-7 Questions and problems Evolution of an orbit in the atmosphere Evolution of an elongated elliptical orbit 4-11 v

7 Motions of Celestial Bodies Late stage of the evolution and the aerodynamical paradox Air density over the Earth 4-14 Questions and problems Active maneuvers in space orbits How to operate the program Space flights and orbital maneuvers Designing a space flight The way back from space to the Earth 5-5 Questions and problems Relative motion of bodies in space orbits The motion of a small body ejected from the orbital station Numerical estimates The secular component of the relative motion Space probes and relative motion Space probes in inner orbits Space probes in outer orbits 5-17 Questions and problems Rendezvous in space and interplanetary flights 5-20 Questions and problems Precession of an equatorial orbit 6-1 Questions and problems Binary stars the two-body problem 7-1 Questions and problems Three-body systems The restricted three-body problem Managing the program Planet with a Satellite Satellites of a planet that orbits a star Exact particular solutions to the three-body problem A system with equal masses of heavy bodies Satellites at the triangular libration points The collinear libration points A space flight over the back side of the Moon 8-18 vi

8 Motions of Celestial Bodies 8.6 Lunar perturbations of a satellite s orbit A space voyage to a distant planet and back Comets interplanetary vagabonds A double star with a planet 8-28 Questions and problems Many-body systems in celestial mechanics Planetary system a many-body problem A model of the Solar System The kinematics of planetary motion Kinematics of the inferior planets Hypothetical planetary systems and heavenly catastrophes Multiple stars Exact particular solutions to the many-body problem A star with two planets of equal mass A round dance of identical planets Keplerian motions in equilateral configurations A remarkable three-body motion along a figure-eight 9-20 Questions and problems 9-21 Part II The simulated phenomena Phenomena and concepts in celestial mechanics an introductory approach Newton s law of universal gravitation Potential energy of a body in the Newtonian gravitational field Circular velocity and escape velocity Geometric properties of Keplerian orbits Initial conditions and parameters of Keplerian orbits A satellite in the atmosphere Trajectories of a landing module A space probe Space rendezvous Kepler s laws and the solar system An approximate approach to the restricted three-body problem Questions for further thought vii

9 Motions of Celestial Bodies 11 Theoretical background Angular momentum and areal velocity Dynamical derivation of Kepler s first law Kepler s third law A hodograph of the velocity vector for Keplerian motion Another derivation of Kepler s first law A family of orbits with equal energies and a common initial point The envelope surface for the family of orbits Applications of the envelope surface Relative orbital motion The gravitational field of a distorted planet A planet with additional masses at the poles A planet with an equatorial bulge The two-body problem Reduced mass and relative motion An alternative approach to the two-body problem Exact particular solutions to the three-body problem The non-restricted three-body problem A star with two identical planets The regular Keplerian motions of three different bodies in the equilateral configuration The sphere of gravitational action The oceanic tides The origin of tidal forces: an elementary approach Tidal forces at an arbitrary point near the Earth The horizontal and vertical components of the tidal force The static distortion of the water surface Tidal forces on the rotating Earth The potential function for tidal forces The natural wave and the driving tidal forces The tides as forced oscillations of the ocean A mathematical description of the forced oscillations Real-world complications The evolution of orbital motions and spins of celestial bodies induced by tidal forces 12 Glossary 12-1 viii

10 Preface This textbook Motions of Celestial Bodies: Computer Simulations, together with the accompanying award-winning educational software package PLANETS AND SATELLITES, is intended to help students learn and understand the fundamental concepts and 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 the numerous complications that are inevitable in an ordinary Earth laboratory. It is the understanding of the foundations of classical and modern physics that form the primary aim of the book, while their application to real-world celestial mechanics is more illustrative and incidental. This textbook relies heavily on PLANETS AND SATELLITES, 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 a conventional abstract manner. This textbook along with PLANETS AND SATELLITES has been developed as an exploration-oriented complement to various physics courses. It can be helpful 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 principles and concepts of classical dynamics and the application of these principles to the motions of various celestial bodies stars, planets, comets, natural and artificial satellites, and manned and automatic space vehicles. The simulation programs make visible the beauty and aesthetics of the mathematics and the fundamental laws of physics in their application to the motions of celestial bodies. PLANETS AND SATELLITES allows the students to construct 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 motions 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 enjoy, varying the parameters of the simulated systems and repeating the most interesting experiments several times on their own. Experience based on students own actions results in deeper understanding than merely receiving someone elseʼs knowledge. Undoubtedly, for the great majority of humans, a visual experience is much more intensive and permanent than an aural or reading experience. With some of the suggested programs, students have an opportunity to perform interesting mini-research projects in physics and astronomy. ix

11 Motions of Celestial Bodies The computer simulations in PLANETS AND SATELLITES enable students to see clearly how systems that obey simple and precise physical laws behave, sometimes in unexpected and even irregular, chaotic ways. Although designed as a desk-top laboratory 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 programs and their possibilities, and explains how the programs are operated. It suggests experiments that demonstrate typical examples of the behavior of the simulated systems. Part I is aimed at building physical intuition. Understanding the underlying concepts of physics is given precedence over using formulas in calculations. Students who are not going to study the subject thoroughly may restrict themselves to only this part of the textbook. In order to remain simple, the text focuses on concepts while keeping mathematics to the necessary minimum. However, to answer some of the suggested questions and to solve the more difficult problems, students may also need to read the following parts. Once they master the basics, the structure of the textbook and software allows for increasingly complex investigations of the subject matter. Chapter 10 of part II presents a more detailed, though rather elementary, description of the fundamental concepts and the laws of physics underlying the simulated phenomena. Quantitative, mathematical formulation of the physical laws and their consequences provides a means to predict and explain the results of the simulation experiments and to calculate the parameters that must be entered in order to obtain the desired results. Chapter 11 of part II is much more sophisticated and is intended for an in-depth study of the subject. This highly mathematical chapter delves into the serious theoretical background for the computer-aided study of celestial mechanics and space dynamics. In the sections Questions and problems an asterisk (*) after the problem number is used to grade the problems by difficulty. Simple problems and questions have no asterisk. One asterisk means that the problem requires thought. Two- and especially three-asterisk problems are challenging in some way. The best learning takes place when you struggle with a tough problem. Eugene Butikov St Petersburg State University, Russia x

12 Author biography Eugene Butikov Eugene Butikov was born on 17 January 1940 in Yekaterinburg (formerly Sverdlovsk), Russia. He graduated from Saint Petersburg (formerly Leningrad) State University in 1962 (Department of Physics). All of his research and academic activity took place at Saint Petersburg State University and Saint Petersburg Research University of Fine Mechanics and Optics, where presently he is a full professor of general physics. He lectures on general physics, optics, the quantum theory of solids, and the theory of oscillations. He has written several textbooks and handbooks on physics which are used widely in Russia. His research work is associated with solid state physics (quantum theory of electronic paramagnetic resonance, theory of Josephson effects in weak superconductivity), theory of nonlinear oscillations. Several new complicated and even counterintuitive modes of regular and chaotic behavior have been discovered recently in parametrically excited simple nonlinear systems by virtue of his efforts with the help of computer simulations. He has succeeded in finding clear physical explanations for some of these modes, and in a theoretical determination of their boundaries in the parameter space of the system. During the last decades he has devoted much time and effort to developing (and using) interactive educational software for university-level physics students. His software is a kind of desk-top laboratory in which the computer is used as a universal research instrument for investigating mathematical models of physical systems rather than simply as a convenient means to search and display information. He has developed several packages of interactive simulation programs with detailed tutorials. These programs are published and distributed in Russia by the Physicon Company, and in the USA by the American Institute of Physics. With these packages, he has tried to change the traditional paradigm of teaching physics in Universities. Students do most of the work on their own, using detailed manuals with assignments, which prompt them to do particular activities to explore the use of the simulation programs. Their work on those assigned projects is a model of real scientific research. xi