This chapter introduces the basic concepts of Newtonian mechanics. We will emphasize the areas which are most useful in the three-body problem, and also familiarise ourselves with a system of units and scaling laws. The calculation of orbits using Newton's laws is a central theme of this book, and therefore a brief introduction to the methods follows. It is not the purpose of this work to teach the latest orbit calculation techniques; therefore only a brief introduction is given. Finally, we discuss the connection of Newtonian mechanics to chaos. It may come as a surprise that the introduction of just one more body to the well behaved two-body system brings about a chaotic, unpredictable dynamical system. This was realised by Poincaré well before the concept of deterministic chaos became a popular topic.

Newton's laws

We begin with the fundamental laws of mechanics, as given by Newton in his Principia in 1687, although in a more modern form.

First law If there are no external forces, an object will maintain its state of motion, i.e. it will stay at rest or continue rectilinear motion at constant velocity.

Second law The rate of change of the momentum of an object is proportional to the applied force.

Three-body systems tend to be unstable. Often they are only short-lived stages in the evolution of a dynamical system. Typically a body comes from a large distance, meets a binary, and escapes again far away. The meeting could be a distant flyby or a close encounter with one of the binary members. Both types of events are important and will be studied in turn. Here we will study only the latter situation, and limit ourselves to the case where the third body is of low mass in comparison with the binary. The general scattering problem is left to Chapters 8 and 10. As in the two-body problem, the transfer of the third body from one hyperbolic relative orbit to another is called scattering.

Scattering of small fast bodies from a binary

The restricted circular three-body problem deals with the motion of a ‘massless’ body in the gravitational field of a zero eccentricity binary. What we will now discuss is a similar problem, namely the motion of a low mass body in the binary field. In our problem the low mass body arrives from a large distance with a high speed, scatters from the binary and flies away. The problem is similar to the twobody scattering of Section 3.13.We present the discussion of three-body scattering following Gould (1991).

Most computer screens and image displays in use are 8-bit devices. This means that the displays can represent data projected on them with 28 = 256 different greyscale levels or data values of resolution. These greyscale levels can represent numeric values from 0 to 255 and it is common to only have about 200 levels actually available to the image display for representing data values with the remaining 50 or so values reserved for graphical overlays, annotation, etc. If displaying in color (actually pseudo-color), then one has available about 200 separate colors, each with a possible grey value of 0–255, or the famous “16 million possible colors” listed in many computer ads (see below).

On the display, the color black is represented by a value of zero (or in color by a value of zero for each of the three color guns, red (R), green (G), and blue (B)). White has R = G = B = 255, and various grey levels are produced by a combination of R = G = B = N, where N is a value from 0 to 255. Colors are made by having R ≠ G ≠ B or any combination thereof in which all three color guns are not operated at the same intensity. A pure color, say blue, is made with R = G = 0 and B = 255 and so on.

Escapes in a bound three-body system

When three self-gravitating bodies are placed inside a small volume, the three-body system becomes unstable. Sooner or later one of the bodies leaves the volume and the two other bodies form a binary system. By recoil, the binary also leaves the original volume and escapes in the opposite direction from the single body. This instability is not at all obvious and the breakup of the bound three-body system was established as a general evolutionary path only after extensive computer simulations in the late 1960s and early 1970s. As mentioned in Chapter 1, there are exceptions to this but generally they do not represent much of the initial value space.

The breakup may be permanent in which case we say that the third body has escaped from the binary. However, sometimes the third-body motion is slowed down sufficiently that the third body returns and a vigorous three-body interaction resumes again. Then the breakup stage is called an ejection. We start by studying escape orbits, and will come to ejections in Section 8.3.

These orbit calculations and later ones have shown that the orbit behaviour of a three-body system is essentially chaotic. The chaoticity can be shown, for example, as follows. Take a given three-body configuration with position vectors r1, r2 and r3 and velocity vectors ṙ1, ṙ2 and ṙ3 for the three bodies labelled 1, 2 and 3.

Introduction

Depending on one’s point of view, the realization that solutions to even simple deterministic dynamical systems could produce highly irregular – chaotic – behavior happened 40 years ago with the publication of Edward Lorenz’ seminal paper “Deterministic nonperiodic flow” or probably more than 100 years ago with Poincaré’s study of complicated orbits in three-body problems of classical Hamiltonian mechanics. Each study indicated the prevalence of complex orbits in classical state space when only a few degrees of freedom were involved. Each study was an unpleasant surprise to physical scientists, and Poincaré’s work was roundly ignored for more than half a century, while Lorenz’ results were reported in a geosciences journal read by a relatively small group of atmospheric scientists.

Each result, one on the celestial mechanics of Hamiltonian systems and the other on a severe approximation to the dissipative fluid dynamics of convection, had no place in the mainstream pursuits of the day. This was in remarkable contrast to the development of the wave equation for nonrelativistic quantum theory, or the crystal structure of DNA. Both of these were at the core of widely identified important problems and were developments for which a huge body of scientists was prepared. Scientists were not even looking in the right direction when chaotic behavior in deterministic systems was found.

e-Science and Licklider

It is no coincidence that it was at CERN, the particle-physics accelerator laboratory in Geneva, that Tim Berners-Lee invented the World Wide Web. Given the distributed nature of the multi-institute collaborations required for modern particle-physics experiments, the particle-physics community desperately needed a tool for exchanging information. After a slow start, their community enthusiastically adopted the Web for information exchange within their experimental collaborations – the first Web site in the USA was at the Stanford Linear Accelerator Center. Since its beginnings in the early 1990s, the Web has taken by storm not only the entire scientific world but also the worlds of business and recreation. Now, just a decade later, scientists need to develop capabilities for collaboration that go far beyond those of the Web. Besides being able to access information from different sites they want to be able to use remote computing resources, to integrate, federate, and analyze information from many disparate and distributed data resources, and to access and control remote experimental equipment. The ability to access, move, manipulate, and mine data is the central requirement of these new collaborative-science applications – be they data held in a file or database repositories, data generated by accelerators or telescopes, or data gathered from mobile sensor networks.

At the end of the 1990s, John Taylor became Director General of Research Councils at the Office of Science and Technology (OST) in the UK – roughly equivalent to Director of the National Science Foundation (NSF) in the USA. Before his appointment to the OST, Taylor had been Director of HP Laboratories in Europe and HP as a company have long had a vision of computing and IT resources as a “utility.” Rather than purchase expensive IT infrastructure outright, users in the future would be able to pay for IT services as they require them, in the same way as we use the conventional utilities such as electricity, gas, and water. In putting together a bid to government for an increase in science funding, Taylor realized that many areas of science could benefit from a common IT infrastructure to support multidisciplinary and distributed collaborations. He therefore articulated a vision for this type of collaborative science and introduced the term “e-Science”:

Introduction

The great discoveries in physics and the technological breakthroughs in the twentieth century have completely revolutionized astronomy – the observational study of the physical Universe beyond Earth and its theoretical understanding. These great discoveries included special relativity, general relativity, quantum mechanics, atomic structure, and nuclear structure, together with the elementary particles and their unified interactions. The technological developments of the twentieth century which had the greatest impact on observational astronomy included microelectronics, microdetectors, computers, and space-age technologies. They allowed astronomical observations deep into space with unprecedented resolution and sensitivity. The New Physics, together with these observations, led by the end of the twentieth century to an amazing understanding of an extremely complex Universe that contains more than 1021 stars in more than 100 billion galaxies with enormous variety, diverse environments, and complex evolutions. Nevertheless, astronomy, one of the oldest sciences, is still one of the most rapidly developing. This is because many fundamental questions related to the origin of our physical Universe, to its contents, to its laws, and to the existence of life in it are still unanswered. They may be answered as science progresses, new technologies for high-resolution observations are exploited, and new fundamental theories are developed and tested. In this chapter, we give a brief account of our present knowledge of the physical Universe, our current understanding of it, and our major observational endeavors to widen this knowledge and understanding.

Advances in observational astronomy

Until the invention of the optical telescope for military purposes at the beginning of the seventeenth century, astronomical observations were made with the naked eye. The Universe observable from planet Earth included only five other planets – Mercury, Venus, Mars, Jupiter, and Saturn – orbiting the Sun and a few thousand more distant stars. The invention of the telescope dramatically increased the horizon of the observable Universe, the number of observable stars, and the resolving power of observations.

Physics is the science of matter – the stuff of the Universe around us, and of energy – the capacity of matter to act in different ways. Physics is the systematic study of how this matter and energy behave, the explanation of what this reveals, and the understanding it brings. A magnificent allegory of what a physicist does can be found in the Old Testament, the Book of Job, Chapter 28.

If our surroundings are seen as being built up of matter, much of Nature is ultimately physics, so physics underpins many other branches of science. It is difficult to be more ambitious than that. But as though such boldness were not enough of a challenge, new physics has gone on to reveal that matter and energy can exist in forms and behave in ways very different from those we know in everyday life. The goal becomes even more ambitious. Nature, and therefore physics, has become much wider than what we normally see around us.

Introduction

All scientists reading this would agree that scientific research in general, and Physics in particular, deserves to be pursued if only for the sake of understanding and enjoying the world in which we live. They share Albert Einstein’s opinion: “Why do we devise theories at all? The answer is simply: because we enjoy ‘comprehending’ . . . There exists a passion for comprehending, just as there exists a passion for music” [1]. At the same time, all scientists also believe that new basic knowledge will continue to bring concrete benefits to Society.

However, this intimate conviction, well-founded on past experience, is no longer sufficient. For about twenty years, politicians and the public have increasingly been asking scientists, and in particular physicists, to

1. better describe what they do and what they learn;

2. explain what advantages Society has gained and can expect to receive in the future from the funds allocated to fundamental research; and

3. organize the production and dissemination of fundamental research in such a way as to maximize its benefits to Society.

It is my opinion that in such an enterprise other scientists perform better than physicists, possibly (but not only) because the subject of their research helps them: advances in medicine and, more recently, in molecular biology are naturally close to human life and are thus easily perceived as “useful.”

Introduction

In the course of the twentieth century fundamental notions concerning the physical world were radically altered by a succession of unifying insights. In the early part of the century quantum theory provided the framework for understanding the structure of the atom, its nucleus, and the “elementary” particles of which it is made. This understanding, combined with the special theory of relativity, led, by the early 1970s, to a comprehensive account of three of the fundamental forces – the weak, strong, and electromagnetic forces – in the “Standard Model.” Meanwhile, Einstein’s 1915 general theory of relativity replaced Newton’s theory of gravity, leading to a unification of the force of gravity with spacetime geometry, and explanations of the evolution of the Universe from the time of the Big Bang and of weird astrophysical phenomena, such as black holes. Yet, with each advance ever-deeper questions were posed and remained unanswered. What determines the species of observed elementary particles, such as the electron, neutrinos, quarks, etc., and what principle leads to a unified description of all of them? How can quantum theory be consistent with general relativity? What is the meaning of space and time at extremely small distances, where quantum theory plays a dominant role? How did the Universe begin and how will it end?

An optimist should regard such unresolved puzzles as the seeds from which further insights will arise. This spirit of incompleteness is a distinguishing feature of this chapter. Here we will see that many, if not all, of these deep questions might be resolved by string theory. Although there is no single mathematical equation that summarizes string theory – indeed, it is not yet a complete theory – it has many intriguing interrelations with some of the most modern areas of mathematics. Its compelling qualities suggest that string theory has the potential to overcome key problems present in previous physical theories and to overcome them in a manner that is surprisingly simple and at the same time very novel.

Introduction

Cosmology is the study of the origin, evolution, composition, and structure of the Universe. As a scientific discipline cosmology began only in the twentieth century. Among the fundamental theoretical and observational developments that established the Big-Bang model were the general theory of relativity proposed by Albert Einstein in 1915, the development of the theory of an expanding relativistic cosmology by Alexander Friedmann in 1922 and Georges LemaÎtre in 1927, the observation of the expansion of the Universe by Edwin Hubble in 1929, the development of the theory of Big-Bang nucleosynthesis by Ralph Alpher, George Gamow, and Robert Herman in the early 1950s, and the discovery of the cosmic background radiation by Arno Penzias and Robert Wilson in 1964.

Traditionally, cosmology has been a data-starved science, but cosmology today is experiencing a fertile interplay between observation and theory. Precision measurements of the expansion rate of the Universe, the large-scale homogeneity and isotropy of the distribution of galaxies, the existence and high degree of isotropy of the 3-K cosmic microwave background radiation, and the abundances of the light elements support the basic picture of an expanding hot-Big-Bang Universe.