Celestial Dynamics E-Book

Rudolf Dvorak

• Herausgeber: Wiley-VCH
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2013

Seitenzahl: 416

Beschreibung

Written by an internationally renowned expert author and researcher, this monograph fills the need for a book conveying the sophisticated tools needed to calculate exo-planet motion and interplanetary space flight. It is unique in considering the critical problems of dynamics and stability, making use of the software Mathematica, including supplements for practical use of the formulae. A must-have for astronomers and applied mathematicians alike.

Leseprobe

Contents

Preface

1 Introduction: the Challenge of Science

2 Hamiltonian Mechanics

2.1 Hamilton’s Equations from Hamiltonian Principle

2.2 Poisson Brackets

2.3 Canonical Transformations

2.4 Hamilton–Jacobi Theory

2.5 Action-Angle Variables

3 Numerical and Analytical Tools

3.1 Mappings

3.2 Lie-Series Numerical Integration

3.3 Chaos Indicators

3.4 Perturbation Theory

4 The Stability Problem

4.1 Review on Different Concepts of Stability

4.2 Integrable Systems

4.3 Nearly Integrable Systems

4.4 Resonance Dynamics

4.5 KAM Theorem

4.6 Nekhoroshev Theorem

4.7 The Froeschlé–Guzzo–Lega Hamiltonian

5 The Two-Body Problem

5.1 From Newton to Kepler

5.2 Unperturbed Kepler Motion

5.3 Classification of Orbits: Ellipses, Hyperbolae and Parabolae

5.4 Kepler Equation

5.5 Complex Description

5.6 Motion in Space and the Keplerian Elements

5.7 Astronomical Determination of the Gravitational Constant

5.8 Solution of the Kepler Equation

6 The Restricted Three-Body Problem

6.1 Set-Up and Formulation

6.2 Equilibria of the System

6.3 Motion Close to L4 and L5

6.4 Motion Close to L1, L2, L3

6.5 Potential and the Zero Velocity Curves

6.6 Spatial Restricted Three-Body Problem

6.7 Tisserand Criterion

6.8 Elliptic Restricted Three-Body Problem

6.9 Dissipative Restricted Three-Body Problem

7 The Sitnikov Problem

7.1 Circular Case: the MacMillan Problem

7.2 Motion of the Planet off the z-Axes

7.3 Elliptic Case

7.4 The Vrabec Mapping

7.5 General Sitnikov Problem

8 Planetary Theory

8.1 Planetary Perturbation Theory

8.2 Equations of Motion for n Bodies

8.3 Lagrange Equations of the Planetary n-Body Problem

8.4 The Perturbing Function in Elliptic Orbital Elements

8.5 Explicit First-Order Planetary Theory for the Osculating Elements

8.6 Small Divisors

8.7 Long-Term Evolution of Our Planetary System

9 Resonances

9.1 Mean Motion Resonances in Our Planetary System

9.2 Method of Laplace–Lagrange

9.3 Secular Resonances

9.4 Three-Body Resonances

10 Lunar Theory

10.1 Hill’s Lunar Theory

10.2 Classical Lunar Theory

10.3 Principal Inequalities

11 Concluding Remarks

Appendix A Important Persons in the Field

Appendix B Formulae

B.1 Hansen Coefficients

B.2 Laplace Coefficients

B.3 Bessel Functions

B.4 Expansions in the Two-Body Problem

Acknowledgement

References

Index

Related Titles

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The Authors

Prof. Dr. Rudolf Dvorak

University of Vienna

Department of Astronomy

Tuerkenschanzstrasse 17

1180 Vienna

Austria

Dr. Christoph Lhotka

University of Namur

Department of Mathematics

Rempart de la Vierge 8

5000 Namur

Belgium

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Dedicated to Caroline, Sophia, Stephanie, and Barbara

Preface

The idea of writing a book about astrodynamics primarily for students and colleagues wishing to learn about and perhaps to work in this interesting field of astronomy came into my mind (RD) already some twenty years ago. In those days my concept was a different one than the book in your hands: based on a very special topic equally interesting for nonlinear dynamics and astrodynamics (see the book of J.Moser1) the so-called Sitnikov Problem I wanted to present all the classical ideas of perturbation theory and connect them to the modern tools of chaos theory. Years went by without such a book being written! But then, some ten years ago, I was fortunate enough to have a brilliant young student (CL) working especially on modern development in nonlinear dynamics who extensively made use of the tools of computer algebra. After his PhD he then moved as a postdoc to two famous Mathematics Departments (Rome and Namur under the direction of two world known colleagues, Profs. Alessandra Celletti, respectively Anne Lemaitre). Immediately after his PhD, still in Vienna, I (RD) got an offer by Wiley to write a book on the subject I have been working on for 40 years. Being aware of this opportunity I invited my former student to write a book with me which, on one hand, deals with the modern tools of nonlinear dynamics and, on the other hand, introduces to the classical methods used since more than two centuries with great success. We realize that this is a difficult task because recently two excellent books concerning this subject appeared written by Prof. S. Ferraz-Mello2) and by Prof. A. Celletti3). Nevertheless we hope to combine with our book different aspects of Celestial Mechanics in an understandable way for interested students and colleagues in Physics and Astronomy, and to succeed in infecting the reader with our enthusiasm for the subject.

Vienne and Namur, December 2012

Rudolf Dvorak and Christoph Lhotka

1) Stable and Random Motion in Dynamical Systems.

2) Canonical Perturbation Theories.

3) Stability and Chaos in Celestial Mechanics.

1

Introduction: the Challenge of Science

Science is one of the main challenges of mankind: it is not only to follow the traces of nature and to try to understand why things are there and how they ‘function’ – it is also to help humans to live in better conditions. Nevertheless we need to be cautious since our history shows that the technical development is gradually destroying our mother planet Earth. One may understand science as one side of a coin, where art is on the other side, a wonderful possibility of using the human spirit to produce things which are not present in nature and are created just for the sake of beauty. Both are the outcome of a development of nature from the primitive forms of life to human beings, the only creatures being able to think about their own existence. However we begin to discover more and more ‘intellectual abilities’ even in the ‘animal kingdom’, for example elephants recognizing themselves in a mirror, different species using tools for preparing their food and their place to live. Still, there seems to be a development in the human brain, far beyond the capacity of any animal or supercomputer, that allows us to be creative in a dual sense, in art AND science.

Tracing back the development of astronomy we need to go back thousands of years into the past, when our ancestors were looking into the sky during daytime following the path of the sun and during night time when the scintillating stars were visible in the dark sky. The Moon is the only ‘body’ sometimes visible during daytime and sometimes during the night, its shape changed while it apparently was also ‘eating’ the Sun for up to several minutes. Although the view of the world changed with time – and even now it is changing with new exciting discoveries like the dark matter and dark energy – the main point of view was a geocentric picture of the universe. In principle it did not change because mankind today still continuously follows an aspect of being in the center of the universe and the only intelligence capable of questioning their own existence.

Definitely the Sun, the Moon and the stars were subjects of religious worship, unapproachable and unalterable, but at the same time priests (and astronomers) tried to predict the celestial phenomena and to use them for measuring time and for making a calendar for these civilized nations in China, India, Central America, in Babylon and in Egypt and later also in Greece. Relatively early they could distinguish between the planets, moving in the sky in somewhat interesting ‘orbits’ and the stars, which never change their positions with respect to each other. Of special importance was to predict eclipses of the Sun and the Moon, which were often regarded to be able to alter the destiny of mankind. It seems that the highest development reached was by the Babylonians with big influence on the Greek astronomy and later on the Arabic astronomers which then was the basis for the European view of the world up to the middle ages, which was not always favorable1) for the progress.

Even in these early days of the dawn of science the description of the motion of the Sun, the Moon and the planets asked for a knowledge of determining positions on the sky relative to a never changing background of the stars. In this sense measurements of positions of celestial objects – astrometry – was born; their results serve as basis for the topic of the book on astrodynamics. In the old civilizations the motions of the Sun, the Moon and the planets were necessary to determine with as high precision as possible for the purpose of computing a calendar based on periodic events. The day is automatically the shortest measure for time, but the problem is that the length of the day from sunrise to sunset is changing. But again the long period where the length of the day repeats can well be determined and defines the year. Between the day and the year the motion of the Moon defines another time period, the month, which was also determined with quite a good precision from the ancient astronomers.

The intention in this introduction is not to give a course on the – very interesting – history of astronomy2), but to describe the big steps forward in the development of the description of the motion of celestial bodies caused by outstanding discoveries and of technical developments. We thus skip the developments of the astronomical knowledge up to the discoveries of the noncircular orbits of the planets around the Sun by Johannes Kepler3), the discoveries of Galileo Galilei4) and the brilliant discovery of the universal law of gravitation by Isaac Newton5).

Up to the early twentieth century the idea was that all events in nature may be described properly because we are living in a fully deterministic world as it was the thinking of Laplace6)

“Une intelligence qui, à un instant donné, connâtrait toutes les forces dont la nature est animé et la situation respective des êtres qui la compose embrasserait dans la même formule les mouvements des plus grands corps de l’univers et ceux du plus léger atome; rien ne serait incertain pour elle, et l’avenir, comme le passé, serait présent à ses yeux.”

Pierre Simon de Laplace,

Essai philosophique sur les probabilités, Bacheliers, Paris 1840.

In the concept of determinism it is believed that it is possible to describe nature with the basic laws of physics and the aid of appropriate mathematical tools (here we need to mention excellent mathematicians and astronomers of these periods like Leonard Euler (1707–1783), Lagrange (1736–1813), d’Alembert (1717–1783) and Carl Friedrich Gauss (1777–1855) without being complete in this list).

And just at the end of the nineteenth century and beginning of the twentieth century the work of Henri Poincaré (1854–1912) led to a new understanding of describing nature: in his Méthodes Nouvelles de la Mécanique céleste he already described a phenomenon which some 50 years later entered into the scientific mind as CHAOS; in contrary to the idea of Laplace it is NOT possible to describe nature in a deterministic way. About this time also from the theories of specific and general relativity by Albert Einstein (1879–1955) and quantum theory7) we learned that we have to accept the fact that nature can be described only statistically.

Three breakthroughs in the last century originated in new challenges for science in general and for astronomy and astrodynamics in particular: the rôle of the chaos described above, the fast development of computers opened up new frontiers, and the launch of the first artificial satellites by the Russians in 1957.8)

Although there had been made extremely interesting discoveries in astronomy during the last decades like the existence of dark matter and the surprisingly ‘fact’ of an accelerating expansion of our universe due to dark energy, we believe that for the human race the discovery of extrasolar planets is the most exciting one and a real breakthrough for scientific thinking. This first discovery of a planet around a main sequence star some almost twenty years ago was a real positive ‘shock’ for astronomers.

The idea of other worlds besides our own is an idea which goes back to the Greek philosopher Epicure and later in the middle ages as Giordano Bruno expressed his fantastic idea of other worlds This space we declare to be infinite…In it are an infinityof worlds of the same kind as our own. He believed in the Copernican theory that the Earth and other planets orbit the Sun. In addition he was convinced that the stars are similar to our Sun, and that they are accompanied by inhabited planets. Giordano Bruno was burned in 1600 in Rome primarily because of his pantheistic ideas.

Ever since that first discovery of an extrasolar planet astronomers work hard to confirm the existence of extrasolar planets and today more than 700 exoplanets are known and the number of discoveries is growing fast with finer and better tools of detecting them around other stars. Although most of the planets discovered seem to be ‘alone’, we have to assume that other planets are present in such systems because from the theory of formation of planets it is rather improbable that they form alone. To understand the diversity of the of the planets and especially the architecture of multiplanetary systems is one big task to be answered by astrodynamics.

The concept of our book is the following: In the first chapters we introduce the basic mathematical tools to understand the complexity of the subject we are dealing with in celestial mechanics. The next chapters are dedicated to classical problems in celestial mechanics from the two-body problem to the motion of the Moon. We briefly outline the content of each chapter:

We start by describing Hamiltonian mechanics, which is the basis of all later subjects. We explain the concept of canonical transformations, what are action angle variables and so on. We then move on to describe the different methods of solving the complex problems: we shortly explain the principle of mappings with some simple examples, introduce an efficient tool – the Lie-integrator – for solving differential equations and demonstrate how analytical methods may be used for solutions of problems in nonlinear dynamics in general. A central section is devoted to the problem of the stability of motions where we also present the theories of Arnold–Kolmogorov–Moser and of Nekhoroshev.

Only now we discuss the two-body problem with the Kepler laws and the famous transcendental Kepler-equation and some methods of solving this simple looking equation. We also make use of a complex notation which makes the two-body problem even in three dimensions easy to describe in closed form. An important step from the integrable Kepler-problem to the problem where 3 bodies (one of them regarded as massless) are involved is described in the chapter of the restricted three-body problem. The equations of motion, in a rotating coordinate system, are well adapted to find a concept of qualitative description of stable motion via the zero-velocity curves. Because the basics in the motion of the Moon can be described in a dynamical system with a massless Earth companion we then turn to give insights into the Lunar motion.

A kind of intermezzo is dedicated to the Sitnikov problem where we both (RD and CL) worked a lot during the last years. Analytical approaches are presented together with results of extensive numerical integrations. The richness of motion in its different adaption from the circular and the elliptic problem (two equally massive bodies and a massless third body) is shown in many details up to the self-similarity in the representation of the Poincaré surfaces of section.

We move on in the next part to develop the main ideas of the planetary theories, where, after giving a simple example, we deal with first and second-order perturbations, and the role of small divisors. Mean motion resonances, secular resonances and three-body resonances are the topic of the next chapter, where we also describe many recent results about the interesting motions of a group of asteroids, the Trojans. Then we give an introduction into the Lunar theory based on the former results of the restricted three-body problem and discuss the main ‘inequalities’ of the motion of the Moon based on the perturbing function.

Finally, to conclude we list books dealing with Celestial Mechanics from the time of Poincaré on and briefly say some words about their content. This very personal choice is far from being complete and we are aware of the fact that many important contributions are not mentioned; we apologize for that.

Enjoy our book!

1) Think of the dragging influence of Aristotle with respect to science: he had in his thinking no concepts of mass and force. He had a basic conception of speed and temperature, but no quantitative understanding of them, which was partly due to the absence of basic experimental devices, like clocks and thermometers.

2) There are excellent books on the subject on the market like ‘Early Physics and Astronomy: A Historical Introduction’ by Olaf Pedersen [1].

3) (1571–1630) derived the surprising fact that Mars is in an elliptic orbit based on the excellent observations by Tycho Brahe (1546–1601). And he then formulated his three laws concerning the motion of the two body problem (see details in Chapter 5).

4) (1564–1642) who observed with a telescope the motion of the 4 large Moons of Jupiter, the Galilean satellites, which form a ‘planetary system’ themselves on a smaller scale.

5) (1642–1717) in his magnum opus ‘Philosophia Naturalis Principia Mathematica’ (1678) where he described the fundamental law of gravitation.

6) (1749–1827), “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.” [Laplace, A Philosophical Essay, New York, 1902, p. 4]

7) Here we need to mention besides Einstein Max Planck (1858–1947), Erwin Schrödinger (1887–1961), Niels Bohr (1885–1962), Wolfgang Pauli (1900–1958) and Werner Heisenberg (1901–1976) as some of the main founders of quantum physics.

8) In this year on the fourth of October the Soviet Union Sputnik 1 was the first man made satellite in orbit around the Earth. Only 4 years after the Soviet cosmonaut Yuri Gagarin on board of Vostok 1, was the first human in space orbiting the Earth. In fact already in 1903 the Russian Konstantin Tsiolkovsky in his main work (The Exploration of Cosmic Space by Means of Reaction Devices) established the basis for the later space missions with rockets. The endpoint (beginning) of a race between the Americans and the Soviets ended with the landing on the Moon of the first human in 1969. On board of Apollo 11, which landed on the Moon the American astronauts Neil Armstrong and Edwin Aldrin, Jr., acceded the Moons surface: That’s one small step for man – one – giant leap for mankind.

2

Hamiltonian Mechanics

In this chapter we introduce the basic concepts and formalism of Hamiltonian mechanics which are needed to present the contents of this book. The following short exposition is based on [2], see [3, 4] for more details on this interesting subject.

(2.1)

The notation was introduced by William Rowan Hamilton, and named after the French(-Italian) mathematician Joseph Louis Lagrange. In classical or Lagrangian mechanics the function L is defined as the difference

with