After the two-body problem, the next more complicated system consists of three bodies. Let us call these bodies the Sun, planet and asteroid. Some further assumptions are made to keep the system as simple as possible. The word restricted means here that the mass of the asteroid is so small that it does not significantly affect the motion of the primaries (the Sun and the planet). The primaries move in circular orbits, and the asteroid is assumed to move in the same plane as the primaries. The perturbations due to the third body can be neglected and the positions of the primaries can be calculated analytically for all times. The problem is now to find the trajectory of the massless body.

The assumption about the mass of the asteroid is a little problematic. If the primaries affect the motion of the asteroid, it must, of course, affect their motions according to Newton's third law. The accuracy required determines whether the third body can actually be considered massless. Discarding Newton's third law has a side effect: total energy is no longer conserved. However, the energy conservation law can be replaced by another similar law.

Coordinate frames

When studying the restricted circular three body problem, the units are usually chosen in such a way that the properties of the system depend on a single parameter.

Classical orbit calculation in Newtonian mechanics has experienced a renaissance in recent decades. With the beginning of space flights there was suddenly a great practical need to calculate orbits with high accuracy. At the same time, advances in computer technology have improved the speed of orbit calculations enormously.

These advances have also made it possible to study the gravitational three-body problem with new rigour. The solutions of this problem go beyond the practicalities of space flight into the area of modern astrophysics. They include problems in the Solar System, in the stellar systems of our Galaxy as well as in other galaxies. The present book has been written with the astrophysical applications in mind.

The book is based on two courses which have been taught by us: Celestial Mechanics and Astrodynamics. The former course includes approximately Chapters 2–5 of the book, with some material from later chapters. It is a rather standard introduction to the subject which forms the necessary background to modern topics. The celestial mechanics course has been developed in the University of Helsinki by one of us (H. K.) over about two decades. The remainder of the book is based on the astrodynamics course which arose subsequently in the University of Turku. Much of the material in the course is new in the sense that it has not been presented at a textbook level previously.

The formalism of the previous chapters used a rather arbitrary coordinate system. In the Hamiltonian formalism the coordinates are chosen in quite a different way to reflect more deeply the dynamical properties of the system. In this chapter we derive the Hamiltonian equations of motion. The results of this chapter are later needed mainly to derive some standard results that are the starting point for further studies. The same results can also be obtained in a more traditionalway, but the Hamiltonian approach makes the calculations considerably shorter and more straightforward.

Hamiltonian mechanics and its applications to mechanics in general are explained more extensively in many books on theoretical mechanics. This chapter is based mainly on Goldstein (1950).

Generalised coordinates

We have this far used ordinary Euclidean rectangular coordinates to describe positions and velocities of the objects. They are purely geometric quantities that describe the system in a very simple and understandable way. However, they do not tell us anything about the dynamic properties of the system nor do they utilise any specific features of the system.We now want to find a different kind of description in terms of quantities which do not have these problems.

Motions of bodies may be constrained in various ways. For example, two points of a solid body must always be at the same distance from each other.

Three-body scattering

Three-body scattering is a process where a third body comes from a large distance in a hyperbolic orbit and interacts with a binary. The interaction may result in a capture of the third body into the vicinity of the binary. Then we say that a resonance (a long lasting state, as in atomic physics) has formed. We expect that the resonance will finally end with an escape of one of the bodies. The other alternatives are an exchange where the interaction leads to an immediate expulsion of one of the binary members, or a flyby when the third body immediately leaves the scene of the close interaction with the binary. These processes will be discussed in turn, in the following sections. Here the basic theoretical groundwork is formulated, using the results from Section 7.1. At very high energies the three bodies may fly apart separately; then the process is called ionisation.

Sometimes a different definition of exchange is used: whenever one of the original binary members is ejected, the process is called exchange. It may happen immediately (prompt exchange) or after an intermediate resonance (resonance exchange). We do not follow this wider definition but define exchange as prompt exchange.

The calculation of the scattering process is performed in two steps: (1) the probability that the third body meets the binary is calculated, and (2) the probability that the binary gains or loses a given amount of energy in the interaction with the third body is estimated.

Binary black holes in centres of galaxies

Galactic nuclei are regions of high star densities as well as sites of very massive black holes, at least in many galaxies. For example, the central black hole in our own Galaxy is thought to be about 2 × 106M⊙ while the giant elliptical galaxy M87 possesses a dark central body of 3 × 109M⊙. The observed masses of these central objects are typically 1.2 × 10–3 times the mass of the spheroidal stellar component of their host galaxies (Merritt and Ferrarese 2001).

It is quite likely that there are also supermassive binary black holes in the centres of some galaxies, based both on theoretical (Saslaw et al. 1974, Begelman et al. 1980) and observational grounds (Komberg 1967, Sillanpää et al. 1988, Lehto and Valtonen 1996). They result most likely from mergers of galaxies. While the stars and gas of the two merged galaxies intermingle and form a new single galaxy, the central black holes remain separate for a long time, perhaps as long as the Hubble time (Milosavljevic and Merritt 2001). In the currently popular cold dark matter (CDM) model of cosmology it is believed that merging of galaxies is a common process (e.g. Frenk et al. 1988). Therefore binary black holes must also be common. How common they are depends on the interaction of the binary with the surrounding stars and gas clouds.

Complete analytical solutions are not available for systems with more than two bodies. However, it is possible to describe three-body orbits by approximate methods when the system is hierarchical, i.e. if there is a clearly defined binary and a third body which stays separate from the binary. These methods may be validated by comparison with numerical orbit integrations. Then we may take exact two-body orbits as a first approximation, and the effects of other bodies and other disturbances are taken into account as small forces which make the true trajectory deviate from this reference orbit.

Whenanalysing perturbations we have to make some approximations that depend on the form of the perturbing force. Thus perturbation theory is a collection of various methods applicable in different situations rather than a single theory. In this chapter we will study a classical method that applies to the usual orbital elements. Another method will be discussed in the next chapter.

The problem which we consider by using this method is the long term evolution of a binary orbit when it is perturbed by a distant companion. This applies especially to triple stars and to the stability of planetary orbits around binary members.

Osculating elements

Consider the motion of a planet in a heliocentric xyz-frame. At the moment t = t0 the planet is at (x0, y0, z0).

About the three-body problem

The three-body problem arises in many different contexts in nature. This book deals with the classical three-body problem, the problem of motion of three celestial bodies under their mutual gravitational attraction. It is an old problem and logically follows from the two-body problem which was solved by Newton in his Principia in 1687. Newton also considered the three-body problem in connection with the motion of the Moon under the influences of the Sun and the Earth, the consequences of which included a headache.

There are good reasons to study the three-body gravitational problem. The motion of the Earth and other planets around the Sun is not strictly a two-body problem. The gravitational pull by another planet constitutes an extra force which tries to steer the planet off its elliptical path. One may even worry, as scientists did in the eighteenth century, whether the extra force might change the orbital course of the Earth entirely and make it fall into the Sun or escape to cold outer space. This was a legitimate worry at the time when the Earth was thought to be only a few thousand years old, and all possible combinations of planetary influences on the orbit of the Earth had not yet had time to occur.

Another serious question was the influence of the Moon on the motion of the Earth.

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).

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.