CONSTANTS OF NATURE

Natural units

We measure distances in units such as meters and light years, intervals of time in units such as seconds and years, and masses in units such as grams and kilograms. There is nothing sacred about these units, which are determined by our history, environment, and physiology. If we communicate with beings in another planetary system and inform them that something has a size of so many meters, an age of so many seconds, and a mass of so many kilograms, they will not understand because their units of measurement are undoubtedly different. But they will understand if we say the size is so many times that of a hydrogen atom, the age is so many times that of a certain atomic period, and the mass is so many times that of a hydrogen atom, simply because their atoms are the same as ours (if they were not, it would be an incoherent universe, incomprehensible, and we might not be able to communicate with them). The basic uniformity of the universe provides us all with the same set of natural units of measurement.

WHAT ARE COSMOLOGICAL HORIZONS?

Horizons

We look out in space and back in time and do not see the galaxies stretching away endlessly to an infinite distance in an infinite past. Instead, we look out a finite distance and see only things within the “observable universe.” Like the sea-watching folk in Robert Frost's poem, we “cannot look out far” and “cannot look in deep.”

The observable universe is normally only a portion of the whole universe. We are at the center of our observable universe; its distant boundary acts as a cosmic horizon beyond which lie things that cannot be observed. Observers in other galaxies are located at the centers of their observable universes that are also bounded by horizons. A person on a ship far from land, who sees the sea stretching away to a horizon, is at the center of an “observable sea.” People on other ships are at the centers of their own observable seas that are bounded by horizons. Despite this analogy the horizons of the universe are not as simple as the horizons of the sea.

GRAVITATIONAL COLLAPSE

Stars are luminous globes of gas in which the inward pull of gravity matches the outward push of pressure. The nuclear energy released in the interior at high temperature is radiated from the surface at low temperature and this low-temperature radiation sustains the chemistry of planetary life.

But to each star comes a day of reckoning. Its central reservoir of hydrogen approaches exhaustion and the star begins to die. The tireless pull of gravity causes the central regions to contract to higher densities and temperatures, and as a consequence the outer regions swell up and the star becomes a red giant. A star like the Sun then evolves into a white dwarf in which most of its matter is compressed into a sphere roughly the size of Earth. Many stars end as white dwarfs, slowly cooling, supported internally against gravity by the pressure of electron waves (as in ordinary metals).

More massive stars do not give up the game so easily. Gravity is stronger in these stars and their central regions continue to contract to even higher densities and temperatures, thus enabling them to draw on the last reserves of nuclear energy. These stars become luminous giants squandering energy at a prodigious rate. Soon their reserves of nuclear energy are exhausted. Only gravitational energy remains with its fatal price of continual contraction.

THE PRIMEVAL ATOM

The universe expands, and naturally we conclude that in the past the universe was in a more condensed state than at present. If we journeyed back in time we would expect to see the universe get steadily denser. Ultimately, we would arrive at the very high-density state popularly called the “big bang.” This conclusion seems unavoidable. It might be a mistake, however, to forget entirely the many debates among cosmologists concerning the reality of a big bang beginning. Eddington was firmly against the idea of a universe that begins in a dense state, and many persons – particularly those who were drawn to science by Eddington's popular works – have felt disinclined to set his views aside lightly. The steady–state theory of an expanding universe, proposed in the late 1940s, attracted many who were united in their dislike of the big bang idea, and even now, as the 20th century closes, a few cosmologists continue to think that a big bang interpretation of the observations is mistaken.

What do we mean by the expression “big bang?” The actual singularity of maximum density at the origin of time? Or an early period in cosmic history? If the latter, how long a period?

Introduction

The first ring system to be observed in the solar system was discovered around Saturn by Galileo in 1610. Unsure of the nature of the phenomenon he had observed, he originally interpreted the ring ansae as two moons, one on each side of the planet. In a Latin anagram sent to fellow scientists he announced, “I have observed the most distant planet to have a triple form”. Galileo was surprised to find that the phenomenon had disappeared by 1612, only to reappear again soon afterwards. Huygens (1659) correctly attributed the varying appearance as being due to the different views of a thin disk of material surrounding Saturn. It was Maxwell (1859) who provided a mathematical proof that the rings could not be solid; they had to be composed of individual particles orbiting the planet.

The rings of Uranus were detected serendipitously in March 1977 by astronomers observing an occultation of a star by the planet. The Voyager spacecraft detected a faint ring around Jupiter (Smith et al. 1979a), and occultations of stars by Neptune led to the discovery of the ring arcs of Neptune, subsequently shown to be the optically thicker parts of a faint ring system. The flybys of the outer planets by the Voyager spacecraft and the continuing ground- and space-based observations of the ring systems have provided evidence of a wide variety of dynamical phenomena, which provide an ideal testing ground for some of the concepts covered in this book.

We are living in a new age of discovery. The major voyages of exploration in the fifteenth and sixteenth centuries have modern parallels in the interplanetary spacecraft missions that have “discovered” our solar system. The data from these spacecraft combined with ground-based observations have revealed a solar system that is more than a collection of planets, satellites, asteroids, comets, and dust distributed in some arbitrary fashion: It has an intricate dynamical structure, which can be largely understood by the application of a simple inverse square law of force to its constituent bodies. To understand the dynamical structure and evolution of the solar system we must therefore understand the qualitative and quantitative effects of the universal law of gravitation.

We consider solar system dynamics to be the application of the techniques of celestial mechanics to solve real problems in planetary science. There are several classical texts on celestial mechanics and many are still in use today. These include the books by Plummer (1918), Brown & Shook (1933), Brouwer & Clemence, (1961) and, more recently Danby (1988). The books by Hagihara (1970, 1972a,b, 1974a,b, 1975a,b, 1976a,b) are authoritative works of reference but make little attempt to convey understanding.

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.