Introduction

So far we have considered all objects as being point masses with no physical dimensions. Since this is evidently not the case for real bodies, we must now consider the effects of the application of universal gravitation to the matter that forms the bodies of the solar system. A tide is raised on one body by another because of the effect of the gravitational gradient or the variation of the gravitational force across the body. For example, if we consider the tide raised on a planet by an orbiting satellite, the force experienced by the side of the planet facing the satellite is stronger than that experienced by the far side of the planet. Since none of the bodies that make up the solar system is perfectly rigid, there will be a distortion that gives rise to a tidal bulge.

The magnitude of the tidal bulge on a body is determined in part by its internal density distribution and thus, in principle, a measurement of the tidal amplitude could lead to a determination of the internal structure. Such measurements are not possible for any of the planets in the solar system other than the Earth.

Introduction

In the last chapter we saw how the disturbing function can be expanded in an infinite series where the individual terms can be classified as secular, resonant, or short period, according to the given physical problem. We have already stated in Sect. 3 that the N-body problem (for N ≥ 3) is nonintegrable. However, in this chapter we will show how, with suitable approximations, it is possible to find an analytical solution to a particular form of the N-body problem that can be applied to the motion of solar system bodies. We can do this by considering the effects of the purely secular terms in the disturbing function for a system of N masses orbiting a central body. The resulting theory can be applied to satellites orbiting a planet, or planets orbiting the Sun, and then used to study the motion of small objects orbiting in either of these systems. This is the subject of secular perturbation theory.

Secular Perturbations for Two Planets

Consider the motion of two planets of mass m1 and m2 moving under their mutual gravitational effects and the attraction of a point-mass central body of mass mc where m1 ≪ mc and m2 ≪ mc. Let R1 and R2 be the disturbing functions describing the perturbations on the orbit of the masses m1 and m2 respectively, where R1 and R2 are functions of the standard osculating orbital elements of both bodies.

Introduction

This appendix contains lists of important astronomical constants, information about the use of the Julian date, orbital data, and physical properties of the known planets and satellites, as well as limited information about some of the minor bodies that make up the solar system.

The data are taken from a number of sources including The Astronomical Almanac for the Year 1995 (HMSO, 1994), The Explanatory Supplement to the Astronomical Almanac (Seidelmann, 1992), and the article by Yoder (1995) and the references therein. The data from the first of these publications is reproduced with permission from HMSO. Other sources of data are indicated in the appropriate sections.

Astronomical Constants

In 1976 the International Astronomical Union (IAU) defined a system of astronomical constants. The IAU system has units of length (the astronomical unit), mass (the mass of the Sun), and time (the day). If the units of length, mass, and time are the astronomical units of these quantities then the astronomical unit of length is the length for which the Gaussian gravitational constant k has the value 0.01720209895. In effect, if the gravitational constant G is expressed in the astronomical units of length, mass, and time then k2 = G. Some of the 1976 IAU constants are given in Table A1.1.

Introduction

It is a laudable human pursuit to try to perceive order out of the apparent randomness of nature; science is, after all, an attempt to make sense of the world around us. Moving against the background of the “fixed” stars, the regularity of the Moon and planets demanded a dynamical explanation.

The history of astronomy is the history of a growing awareness of our position (or lack of it) in the universe. Observing, exploring, and ultimately understanding our solar system is the first step towards understanding the rest of the universe. The key discovery in this process was Newton's formulation of the universal law of gravitation; this made sense of the orbits of planets, satellites, and comets, and their future motion could be predicted: The Newtonian universe was a deterministic system. The Voyager missions increased our knowledge of the outer solar system by several orders of magnitude, and yet they would not have been possible without knowledge of Newton's laws and their consequences. However, advances in mathematics and computer technology have now revealed that, even though our system is deterministic, it is not necessarily predictable. The study of nonlinear dynamics has revealed a solar system even more intricately structured than Newton could have imagined.

Introduction

In the last chapter,we considered the effect of tides raised on a satellite by a planet where we assumed that the satellite was in a synchronous spin state (i.e., that the rotational period of the satellite was equal to its orbital period). As mentioned in Sect. 1.6, most of the major natural satellites in the solar system are observed to be rotating in the synchronous state. How did this situation arise and what determines the spin–orbit state of a given satellite or planet? In this chapter, we start by further examining the effects of a tidal torque on a satellite's rotation. This analysis reveals why, for example, in order to maintain its synchronous spin–orbit resonance, the Moon must have a permanent quadrupole moment. The consequences of this extra torque on the system are then examined and this leads to a general approach to the concept of spin–orbit resonance in the solar system. The origin and stability of these resonances are also discussed.

Tidal Despinning

Consider the case of a satellite orbiting a planet in an elliptical orbit. Those parts of the orbit in which the satellite's spin rate, which we denote by + n, is less (or greater) than its angular velocity or the rate of change of its true anomaly, are shown in Fig. 5.1a.

Introduction

We saw in Chapter 6 how resonant effects arise in the small divisor problem when we considered the motion of an asteroid whose orbital period was a simple fraction of Jupiter's period. The naïve theory predicted that, as the ratio of mean motions approached the exact resonant value, the small divisor approached zero and large-amplitude variations in the elements would result. In this chapter we examine the theory of resonance in more detail. Starting from simple geometrical and physical approaches, we go on to show how the simple model breaks down. In order to understand the basic dynamics of resonance we start by using the pendulum approach valid for resonant phenomena in the asteroid belt. We then give a complete and detailed model of resonance using a Hamiltonian approach. Throughout this chapterwedevelop a variety of approaches to handle the problem of orbit–orbit resonance in the solar system and beyond.

Although there is an extensive range of literature on celestial mechanics, there is little devoted specifically to the theory of resonance. Useful reviews of the subject, particularly in the context of orbital evolution through resonance, have been given by Greenberg (1977), Peale (1986), and Malhotra (1988).

Introduction

In Chapter 2 we showed how the problem of the motion of two masses moving under their mutual gravitational attraction can be solved analytically and that the resulting motion is always confined to fixed geometrical paths that are closed in inertial space. We will now extend our analysis to consider the gravitational interaction of three bodies, paying particular attention to the problem in which the third body has negligible mass compared with the other two.

The simplicity and elusiveness of the three-body problem in its various forms have attracted the attention of mathematicians for centuries. Among the giants of mathematics who have tackled the problem and made important contributions are Euler, Lagrange, Laplace, Jacobi, Le Verrier, Hamilton, Poincaré, and Birkhoff. The books by Szebehely (1967) and Marchal (1990) provide authoritative coverage of the literature on the subject as well as derivations of the important results. Today the three-body problem is as enigmatic as ever and although much has been discovered already, the recent developments in nonlinear dynamics and the spur of new observations in the solar system have meant a resurgence of interest in the problem and the derivation of new results.

If two of the bodies in the problem move in circular, coplanar orbits about their common centre of mass and the mass of the third body is too small to affect the motion of the other two bodies, the problem of the motion of the third body is called the circular, restricted, three-body problem.

Introduction

In this book we have derived a number of equations of motion to study the rotational and orbital motion of solar system objects. These equations have described either conservative systems, such as the two- and three-body problems, or dissipative systems, such as the equations governing tidal evolution or the dynamical effects of drag forces. However, all have a common characteristic: They describe systems that are deterministic. This means that the current state of the system allows us to calculate its past and future state providing we know all the forces that are acting on it. In the case of the two-body problem we were able to solve the equations of motion and calculate the behaviour of the system at all past and future times. A complete analytical solution was not possible in the case of the three-body problem and we had to rely on numerical solutions if we wanted to follow the orbital evolution of a test particle. However, there was an implicit assumption that, given the initial state of the system, we should be able to calculate its future state by obtaining solutions of the equations of motion. Unfortunately this assumption is not valid for some of the systems we have investigated and this is because of the phenomenon called chaos.

Introduction

Star formation (SF) is a local phenomenon which must find its explanation in the stability and fragmentation of dense molecular clouds. Studies in our own Galaxy have focussed on the structure of dense proto-stellar cores and the chemical and thermal balance of star-forming regions. These studies lend indirect support to a Schmidt (1959) law, but emphasize the need to include explicitly the structure of the multi-phase ISM to model accurately the most important heating and cooling processes. A large unknown in these investigations is the role of feedback. Supernova explosions and stellar radiation associated with the process of SF influence the global physical structure of the interstellar gas which supports this process.

Introduction

The nature of the X-ray emission of early-type, prominent-bulge spirals has been the subject of an on-going controversy, which has sought to establish if and how much of this emission can be ascribed to thermal emission of an optically thin hot gaseous medium. This is an important issue, because if it can be established that the X-ray emission is dominated by gravity-bound gaseous halos, the X-ray data may be used to measure the mass of these galaxies (see review in Fabbiano 1989).

In what follows, I give a summary of the work on this subject, and point out future opportunities.

A Brief History of X-ray Studies of Early-type Spirals

With the clear exception of M31, most of the bulges of early-type spirals could not be studied in detail with X-ray observatories, starting with the Einstein Observatory, in the early ʾ80s, and including all the facilities in orbit and operational at this time.

Introduction

HST UV images of nearby galaxies presented by Maoz et al. (1996) and Barth et al. (1998), as well as analogous space-borne optical images of early-type galaxies discussed by Lauer et al. (1995) and Carollo et al. (1997) have shown that about 15% of imaged galaxies show evidence of unresolved central spikes.

In the following we discuss two ‘prototype’ galactic spheroids, NGC 2681 and NGC 4552, that we properly monitored with HST–which host UV-bright, unresolved spikes at their center. While the early-spiral (Sa) galaxy NGC 2681 shows a nonvariable unresolved cusp, the UV spike which became visible at the center of the Virgo Elliptical NGC 4552 is a UV flare caught in mid-action, presumably related to a transient accretion event onto a central supermassive black hole (Renzini et al. 1995; Cappellari et al. 1998).

Introduction

NGC 4698 is classified Sa by Sandage & Tammann (1981) and Sab(s) by de Vaucouleurs et al. (1991; RC3). Sandage & Bedke (1994; CAG) present NGC 4698 as an example of the early-to-intermediate Sa type since it is characterized by a large central bulge and tightly wound spiral arms. In addition to a remarkable geometrical decoupling between the bulge and the disk (whose apparent major axes appear oriented in an orthogonal way upon simple visual inspection of galaxy plates; see Panels 78, 79 and 87 in CAG), a spectrum taken along the minor axis of the disk shows the presence of a stellar velocity gradient which could be ascribed to the bulge.