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Panoramic Rings

Credit: NASA/JPL/Space Science Institute

Here we provide resources in the form of Mathematica and Python code to assist the reader in understanding the content of the book.

Mathematica code is provided below in the form of Mathematica notebooks. The notebooks were created using Mathematica 13.2. Each notebook is self-contained. The notebooks are provided as-is, and the code can be modified freely if you have a licensed copy of Mathematica. You can also view and interact with (but not edit) Mathematica notebooks by using the free software, Wolfram Player. Mathematica code was written by Carl Murray unless stated otherwise.

Python code is provided below in the form of Jupyter notebooks. The notebooks were created using JupyterLab, running the Python 3 kernel. Each notebook is self-contained, with the required dependencies and packages shown at the top of each notebook file. The notebooks are provided as-is, and the code can be modified freely, as required. Python code was written by Nick Cooper.

Panoramic Rings
Chapter 1

Chapter One

1. Structure of the Solar System

1.1 Introduction

1.2 The belief in number

1.3 Kepler's laws of planetary motion

1.4 Newton's universal law of gravitation

1.5 The Titius-Bode 'law'

1.6 Resonance in the Solar System

1.7 The preference for commensurability

1.8 Recent developments

1.9 Exercises

Panoramic Rings
Chapter 2
Panoramic Rings

Chapter Three

3. The Restricted Three-Body Problem

3.1 Introduction

3.2 Equations of motion

3.3 ​The Jacobi integral​

3.4 The Tisserand relation

3.5 Lagrangian equilibrium points​

3.6 Location of equilibrium points​​

3.7 Stability of equilibrium points

3.8 Motion near L4 and L5

​3.9 Tadpole and horseshoe orbits​

3.10 Orbits and zero velocity curves

3.11 Trojan asteroids and satellites

3.12 Janus and Epimetheus​

3.13 Hill's equations​

3.14 The effects of drag​

3.15 Exercises

Chapter 3
Panoramic Rings

Chapter Four

4. Tides, Rotation and Shape

4.1 Introduction

4.2 The tidal bulge

4.3 ​Potential theory

4.4 Tidal deformation

4.5 Rotational deformation

4.6 The Darwin-Radau relation

4.7 Shapes and internal structures of satellites​

4.8 The Roche zone​

​4.9 Tidal torques​

4.10 Satellite tides

4.11 Tidal heating of Io

4.12 Tides on Titan​​

4.13 Tidal evolution​

4.14 The double synchronous state

4.15 Exercises

Chapter 4
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Panoramic Rings

Chapter Six

6. The Disturbing Function

6.1 Introduction

6.2 The disturbing function

6.3 ​Expansion using Legendre polynomials

6.4 Literal expansion in orbital elements​

6.5 Literal expansion to second order

6.6 Terms associated with a specific argument

6.7 Use of the disturbing function

6.8 Lagrange's planetary equations

6.9 Classification of arguments in the disturbing function

6.10 Sample calculations of the averaged disturbing function

6.11 The effect of planetary oblateness

6.12 Exercises

Chapter 6
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Chapter Seven

7. Secular Perturbations​

7.1 Introduction

7.2 Secular perturbations for two planets

7.3 ​Jupiter and Saturn

7.4 Free and forced elements

7.5 Jupiter, Saturn and a test particle

7.6 Gauss's averaging method

7.7 Generalised secular perturbations

7.8 Secular theory for the solar system

7.9 Generalised free and forced elements​

7.10 Hirayama families and the IRAS dust bands

7.11 Secular resonances

7.12 Higher order secular theory

7.13 Exercises

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Chapter Eight

8. Resonant Perturbations​

8.1 Introduction

8.2 The geometry of resonance

8.3 ​The physics of resonance

8.4 Variation of orbital elements

8.5 Resonance in the circular restricted three-body problem

8.6 The pendulum model

8.7 Libration width

8.8 The Hamiltonian approach

8.9 The 2:1 resonance​

  • Figures 8.16-18: Mathematica notebook

  • 8.9.1 Exact resonance

  • 8.9.2 Medium amplitude libration

  • 8.9.3 Large amplitude libration

  • 8.9.4 Apocentric libration

  • 8.9.5 Internal circulation

  • 8.9.6 External circulation

  • 8.9.7 Other types of motion

  • 8.9.8 Comparison with analytical theory

8.10 The 3:1 and 7:4 resonances​

8.11 Additional resonances and resonance splitting

8.12 Resonant encounters

8.13 The dynamics of capture and evolution in resonance​

8.14 Two-body resonances in the solar system

  • 8.14.1 The Titan-Hyperion resonance

  • 8.14.2 The Mimas-Tethys resonance

8.15 Resonant encounters in satellite systems​

8.16 Three-body resonance​

8.17 The Laplace resonance

8.18 Secular and resonant motion​

8.19 LONGSTOP Uranus

8.20 Pulsar planets

8.21 Exercises

Chapter 8
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Chapter Nine

9. Chaos and Long-Term Evolution

9.1 Introduction

9.2 Sensitive dependence on initial conditions

​9.3 ​Regular and chaotic orbits

9.4 Chaos in the circular restricted problem

9.5 Algebraic mappings

9.6 Separatrices and resonance overlap

9.7 The rotation of Hyperion

​9.8 The Kirkwood gaps

9.9 The Neptune-Pluto system​

9.10 The stability of the Solar System

9.11 Exercises

Chapter 9
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Chapter Ten

10. Planetary Rings

10.1 Introduction

10.2 Planetary ring systems

  • Figure 10.1: Mathematica notebook

  • 10.2.1 The rings of Jupiter

  • 10.2.2 The rings of Saturn

  • 10.2.3 The rings of Uranus

  • 10.2.4 The rings of Neptune

  • 10.2.5 Rings and satellites

​10.3 ​Resonances in the rings

10.4 Density waves and bending waves

10.5 Narrow rings and sharp edges

10.6 The Encke gap and Pan​

10.7 The F ring of Saturn

​10.8 The Adams ring of Neptune

10.9 The evolution of rings

10.10 The Earth's dust ring

10.11 Exercises

Chapter 10
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