Table Of ContentIntroduction 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
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and by CRC Press
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© 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)
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are used only for identification and explanation without intent to infringe.
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(cid:4) 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(cid:4) 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(cid:4) Celestial Mechanics 83
3.1 THE PRINCIPLE OF LEAST ACTION 84
vii
viii (cid:4) 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’s1st LawasDiscoveredbyKeplerHimself
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(cid: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
Thisbookcollectsmyexperiencesafterteachinganintroductoryundergradu-
ate 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
arefirstobservedcarefullythenmodeledandfinallycalculatedandestimated
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...
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