Introduction to the Maths and Physics of the Solar System

Introduction to the Maths and Physics of the Solar System Introduction to the Maths and Physics of the Solar System Lucio Piccirillo First edition published 2020 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2020 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC International Standard Book Number: 978-0-367-02271-6 (Hardback) International Standard Book Number: 978-0-367-00254-1 (Paperback) Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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To Emma, Julia, Madelaine, Jacopo and Tommaso. This book is also dedicated to Mario Sortino, a.k.a. “Zio Mario”, for explaining to me the foundation of the scientific methods during long and extenuating mountain hikes during my teenager years. Finally, I would like to express my deepest affection and acknowledgement to a very special person, CB, for her lovely support in the final phases (the most stressful) of the writing. Thank you CB! Contents Foreword ix Preface xi Chapter 1 Basic Concepts 1 1.1 GEOMETRY 2 1.2 TRIGONOMETRY 6 1.3 CALCULUS 15 1.3.1 Functions 20 1.3.2 Infinity in Maths 20 1.3.3 Derivatives, Integrals and the Fundamental The- orem of Calculus 24 1.3.4 Inverse Trigonometric Functions 37 1.4 ERATOSTHENES’S FINAL CALCULATION 39 1.5 ARISTARCHUS’S CALCULATIONS 40 Chapter 2 Math and Physics Toolkit 49 2.1 VECTORS 49 2.1.1 Change of Coordinate Systems 52 2.1.2 Operations with Vectors 56 2.1.3 Differentials and Derivatives of Vectors 63 2.1.4 Polar and Cylindrical Coordinates 67 2.1.5 Vectors in Physics 70 2.1.6 Polar and Axial Vectors 70 2.2 NEWTON’S LAWS AND GRAVITY 74 2.3 THE CONCEPT OF MASS 79 Chapter 3 Celestial Mechanics 83 3.1 THE PRINCIPLE OF LEAST ACTION 84 vii viii Contents 3.1.1 Conservation Laws 89 3.1.2 Newtonian and Lagrangian Problem Solving 94 3.2 KEPLER’S LAWS 98 3.2.1 Theory of Conic Sections 100 3.2.2 Kepler’s 1st Law as Discovered by Kepler Himself in the Years 1600 - 1630 111 3.2.3 Kepler’s Problem: Geometrical Solution 115 3.2.4 Kepler’s Problem: Newton’s Solution Using Cal- culus 132 3.2.5 Kepler’s Problem: Solution Using Geometric Al- gebra with the Laplace-Runge-Lenz Vector 140 3.3 ENERGY AND ORBITS 145 3.4 THE UNIVERSAL LAW OF GRAVITATION: ONE VERY FAMOUS APPLE 148 3.4.1 Newton’s Shell Theorem Using Calculus 152 3.4.2 Newton’s Shell Theorem Using Geometry 158 3.4.3 Newton’s Shell Theorem Using Gauss’s Law 165 3.5 PLANET’S MOTION USING EULER-LAGRANGE EQUATIONS 173 Chapter 4 A Few Facts about the Solar System 177 4.1 GEOCENTRIC VERSUS HELIOCENTRIC 177 4.2 MOTION AND COORDINATES 185 4.3 THE ANALEMMA 196 4.4 TIDES IN THE SOLAR SYSTEM 200 4.5 ROCHE LIMIT 212 4.6 MEASURING THE SPEED OF LIGHT IN THE SO- LAR SYSTEM 213 Bibliography 221 Index 223 Foreword This book collects my experiences after teaching an introductory undergraduate course on the physics of the solar system at the University of Manchester. The course is oriented to first-year undergraduate students first semester, i.e. students just attending a university course for the first time. My philosophy of teaching has been always to teach trying to achieve several objectives. First, the course has to trigger interest in the student: this is relatively easy because space, planets, etc., are a natural interesting subject for practi- cally everybody. Who does not feel a sense of awe when thinking about the Universe? Second, students have the opportunity to learn how natural phenomena are first observed carefully then modeled and finally calculated and estimated more than just simply described. Every time there is a discovery in astro- physics, newspapers are full of sensational reports of phenomena described almost as mysteries of nature. Scientists are seen as “magicians” that myste- riously know what a black hole is or what space-time is. First year students are usually still in this mode of “acceptance” of qualitative explanations more than critical evaluation and quantitative estimation. Third, I wanted to write a book that is somehow “self-contained”, i.e. all the maths and physics contained are explained starting from basic concepts that any well-educated person can follow. I imagine my reader as a person who has the curiosity to understand quantitatively how scientists can make claims that apparently are “magic” like, for example, stating that the age of the Earth is about 4.5 billion years. In this book, this person, with a little bit of effort, will be able to convince himself or herself that the claim is perfectly justified. Lastly, when I was a young high school student I always wanted to read about science but I also wanted to be “convinced” about the various scientific claims. I never found a book that was easy enough to describe all the maths and physics needed but complete enough to describe phenomena accurately. About 45 years ago I promised to myself that, if I was capable, I would have written such a book and here it is. Now I need a time machine to send it to myself... ix Preface This book is an attempt at having a self-contained treatment of some phenomena happening in the solar system. Self-contained because all the maths and physics needed are described starting from a few basic notions that every logical person should have no difficulty to accept and digest. Instead of a mere qualitative description of various phenomena, I tried to select a set of representative phenomena happening in our solar system and describe them quantitatively after a careful treatment of the maths and physics needed to understand. A correct title of this book should be: “How to Teach You Maths and Physics Concepts Using Interesting Facts about the Solar System”. The first chapter, for example, is designed to introduce all the geometry, trigonometry, calculus, and vector calculus needed to understand planetary orbits and more. I start the chapter by studying in detail Eratosthenes determination of the circumference of the Earth and I show how such a simple geometry problem contains in it a big number of assumptions. When determining an important angle through inverting the sine function instead of just stating that an angle α is such that its sine is 0.1256, I use this opportunity to introduce derivatives and series expansion of functions. The maths is obviously treated without the rigor usually associated with more complete and exhaustive textbooks that the reader is warmly invited to consult. My experience is that learning maths and physics using interesting prob- lems gives the students, and I hope more generally the readers of this book, additional enjoyment by showing how a variety of phenomena can be, and are, calculated when the theory that we have is a good theory. This approach has the risk that the book might be “too complex” for people who do not have a technically oriented mind and “too easy” for people that already know the basics. My writing is therefore always trying to excite anyone who already “knows” and allowing to understand who does not know “yet”. Who already “knows” might find, here and there, interesting new calculations and perhaps different views of already known facts. Those who, instead do not know enough, might still get the opportunity to try to understand how a physicist works: I only ask some efforts in following carefully my maths and physics explanations. I strongly believe that science is fun, simple maths used in physics is un- derstandable, and everybody equipped with patience, enthusiasm, and time xi xii Preface can enjoy the beauty of using maths to “predict” phenomena happening in the solar system. This book is my private acknowledgment of my hero, Galileo Galilei. He was the first to understand that “the book of nature is written in the language of mathematics”. CHAPTER 1 Basic Concepts CONTENTS 1.1 Geometry :::::::::::::::::::::::::::::::::::::::::::::::::: 2 1.2 Trigonometry :::::::::::::::::::::::::::::::::::::::::::::: 6 1.3 Calculus :::::::::::::::::::::::::::::::::::::::::::::::::::: 15 1.3.1 Functions ::::::::::::::::::::::::::::::::::::::::::: 20 1.3.2 Infinity in Maths ::::::::::::::::::::::::::::::::::: 20 1.3.3 Derivatives, Integrals and the Fundamental Theorem of Calculus :::::::::::::::::::::::::::::: 24 1.3.4 Inverse Trigonometric Functions ::::::::::::::::::: 37 1.4 Eratosthenes’s final calculation :::::::::::::::::::::::::::: 39 1.5 Aristarchus’s calculations :::::::::::::::::::::::::::::::::: 40 HERE are several elementary concepts that need to be properly re- viewed in order to fully understand the phenomena happening in our solarT system.

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