Introduction

The dynamical properties of a planet depend on the way in which the density varies with radius, and seismological properties depend also on the way in which the elastic moduli and elastic dissipation vary with radius. The data we have for the Earth are sufficiently complete that the variations of density and elastic moduli with radius can be derived from them and we are then presented with the problem of inferring the mineralogical and chemical composition consistent with them. When, as for the other planets, seismic data are lacking, we must proceed in a different way and derive the variation of density from a postulated composition, asking if it leads to the observed mass and moment of inertia. In either case, we must know how the density depends on pressure, temperature and composition, for all these vary with radius, and when we discuss the Earth and the Moon, for which we have seismic data, we must also examine the dependence of the elastic moduli upon the three variables. Some idea of the problems that arise, of the theoretical principles, of possible experimental methods, and of the systematics of equations of state of minerals has already been given in Chapter 1, and it is the aim of this chapter to give a more extensive and systematic account.

The wonders of the heavens

The planets have been a subject of wonder to man from earliest recorded times. Their very name, the Wandering Ones, recalls the fact that their apparent positions in the sky change continually, in contrast to the fixed stars. Greek astronomers, Ptolemy particularly, had shown how the motions of the planets, the Sun and the Moon could be accounted for if they were all supposed to move around a stationary Earth, and in mediaeval times an elaborate cosmology was created, at its most allegorical, evocative and poetic in the Paradiso of Dante. The men of the Renaissance overthrew these ideas but provided fresh cause for wonder in their place. Placed in motion around the Sun by Copernicus, their paths observed with care by Kepler, the planets led Newton to his ideas of universal gravitation. Galileo, his telescope to his eye, showed that they had discs of definite size and that Jupiter had moons, the Medicean satellites, which formed a system like the planets themselves.

The discoveries of the seventeenth century settled notions of the planets for three centuries, but within that framework a most extra-ordinary flowering of the intellect attended the working out of the ideas of Newton. Closer and closer observation showed ever more intricate departures of the paths of the planets from the simple ellipses of Kepler, and each was accounted for by ever subtler applications of mechanics as the consequence of the gravitational pull of each planet upon its fellows.

Introduction

The models of the planets which have been adopted so far depend explicitly on the assumption that the planet is in the hydrostatic state, so that the density is a function only of radial distance (in a generalized sense when the planet is flattened by spin). That may be an appropriate first assumption to provide a starting point for further developments, but it is clearly not adequate: the gravity fields of the Earth, the Moon and Mars contain harmonic components that would be absent if the internal state were hydrostatic; the irregular surface features of the terrestrial planets are inconsistent with strict hydrostatic equilibrium; and the structure seen in the atmosphere of Jupiter reveals internal motions, if only superficial. A density distribution not in hydrostatic equilibrium requires a stress system to support it that departs from the simple normal pressure to which hydrostatic equilibrium corresponds. Such a stress system may be developed in two ways: statically, through strains of the planet, or dynamically, through movements of the material. According to which mode is effective, so the planet may be considered to be cold or hot (though, as has already been argued, no planet is hot in relation to the effect on the equation of state). If the planet is cold, the materials within it have high strengths, can support large stresses and so maintain statically non-hydrostatic distributions of density. If the planet is hot, then parts will be molten, as is the core of the Earth, or will be sufficiently hot to creep steadily under applied stress.

Introduction

Mars, Venus and Mercury form with the Earth and the Moon a group of rather similar bodies. By comparison with the giant planets on the one hand and the small satellites on the other, the sizes lie in a relatively restricted range, while the mean densities are higher than those of most other bodies in the solar system. It is natural to think that their compositions are similar and that the structures of Mars, Venus and Mercury might be inferred from what is known of the Earth and the Moon.

Seismological data are, of course, not available for any of the planets other than the Earth, so that the structures of the terrestrial planets must be derived from the dynamical data, together with such inferences as may be drawn from the magnetic and electrical properties, together with analogies with the Earth and the Moon.

Unfortunately, the dynamical data themselves are less informative for Mars, Venus and Mercury than they are for the Earth and the Moon or for the major planets. The solar precession of Mars has not so far been observed and, in consequence, the moment of inertia cannot be derived from the value of J2 without making the assumption of hydrostatic equilibrium. Yet it is clear that Mars is not in hydrostatic equilibrium. The theory of the errors likely to be committed by making the assumption of hydrostatic equilibrium was given in Chapter 3 and subsequently in this chapter (section 6.6) it will be applied to Mars.

Introduction

The possibility that, at the pressures encountered in the planets, materials ordinarily non-metallic at low pressures might transform into metals has been discussed for more than forty years. Two main ideas have been considered: one, that metal silicates, such as olivine, might become metallic at pressures developed in the core of the Earth, and the other, that hydrogen, helium and other light elements might transform to metals at pressures encountered in the major planets. Sufficient is now known about changes of density in metallic transformations under high pressure to be sure that the jump of density between the mantle and the core of the Earth is too great to be explained by such a transformation and in the preceding chapters on the terrestrial planets it has been assumed that the difference between the core and mantle of the Earth is one of composition (see also Anderson, 1977). It is otherwise with Jupiter and Saturn. The mean densities of those planets are too low for them to be composed of anything but hydrogen, helium and other materials of low atomic number, and the likelihood of a metallic transformation of hydrogen in particular is crucial to a discussion of their internal structures. One of the first studies of the metallic transformation in hydrogen (Kronig, de Boer and Korringa, 1946) was prompted by the idea of Kuhn and Rittman (1941) that the inability of the core of the Earth to support shear waves might be because it was of solar composition, that is, mainly of hydrogen, and by the subsequent suggestion of van der Waals that at core pressures the hydrogen might be metallic.

The lesser objects of the solar system

So far in our studies, no notice has been taken of the smaller bodies in the solar system, i.e. Pluto, the asteroids and the satellites of Mars and the major planets, for their properties are but poorly known on the whole and it is not very rewarding to apply to them the type of analysis that was applied in the foregoing parts of this book to the greater objects. Yet, in considering the solar system as a whole, their existence and such information as we have of them cannot be ignored.

Pluto, the outermost known planet, is in a highly eccentric orbit highly inclined to the ecliptic, and, in consequence, although it comes on occasion within the orbit of Neptune, it never approaches Neptune closely, and detailed studies have shown the outer solar system to be stable. Pluto is a very small object as seen from the Earth. Its diameter is estimated from the brightness and supposed reflectivity. The latter has recently been redetermined from infra-red spectroscopy and the diameter of Pluto is consequently now estimated to lie between 2800 and 3300 km (Cruickshank, Pilcher and Morrison, 1976). The mass of Pluto was originally estimated from the perturbations of the orbits of Uranus and Neptune, but a satellite has now been detected (Christy and Harrington, 1978) with a period of 6.4 d, from which the mass of Pluto is estimated to be about 0.002 times that of the Earth (Meadows, 1980).

Introduction

The only observed mechanical data we have for any planet are the mass and moment of inertia, and infinite sets of models can be constructed consistent with such pairs of data. The sets of models are not, however, unbounded, and, further, certain models are in some sense more probable than others. It is the purpose of this appendix to set out the bounds on two particular models and to give some most probable models. The models considered are: that of two zones, each of constant density, and that in which the density is determined by hydrostatic compression alone. The terrestrial planets may be modelled by the former, and the major planets by the latter. Neither model can represent the complexities of actual planets but, given only two data, no more elaborate model is justified. Guided by the constitution of the Earth, and by such seismic data as are available for the Moon, it is natural to choose the two-zone model as an approximation to the structures of the terrestrial planets. In this model, the maximum pressure is such that changes of density under self-compression are less than differences of density arising from differences of chemical composition or crystal structure in different parts of the planet. Thus, a model comprising two zones of different density is chosen as a basis for study of the terrestrial planets.

Introduction

The Earth, as will appear, is not typical of the planets. It is the largest of the inner planets, it is the only one on which active tectonic development of the surface appears to be going on at present and, so far as we know, it has the most complex structure. Yet it is the only one which can be studied in detail; from it we may derive empirically equations of state of the materials of the inner planets; and the methods that have been used to study the structure of the Earth are those we should like to use, but are inhibited from using by the difficulties of observing, in the investigation of the other planets. For these reasons it is helpful to preface an account of the methods used to study the planets and of the results that have been obtained with a review of the way in which the Earth is examined and what has been discovered.

Our knowledge of the internal structure of the Earth comes by two routes. In the first place the mass, size and density of the Earth provide a rough idea of the overall composition and of the central pressure, while the value of the moment of inertia shows that the density increases strongly towards the centre. Naturally a wide range of models could be constructed to fit just three facts and so it is necessary to turn, in the second place, to seismology to provide more detailed information.

The planets, which have always been objects of wonder and curiosity to those with the opportunity or need to lift their eyes to the heavens, now in our times shine with new and strange lights revealed to us by the far seeing instruments carried upon space craft. The Moon, Mars, Venus and Mercury all bear on their surfaces the crater scars of innumerable meteorites that have fallen upon them from the beginning of the solar system. The Earth alone has an active surface that has obliterated those scars. The fluid surface of Jupiter is in constant and vigorous motion, driven by heat flowing out from the interior or, it may be, brought to it by the ultra-violet radiation from the Sun or by the solar wind. The Medicean satellites of Jupiter now' present to us strange and individual faces: would Galileo who first saw the mountains on the Moon or the spots on the Sun have been surprised by the eruption of sodium and sulphur from Io and the cloud of gas within which it moves, or by the strange stress patterns upon other of the satellites? Seeing these strange and varied faces of the planets, each apparently different from any other, who can forbear to ask, what bodies are these, how are they made up, that their appearances are so distinctive? Why are some active, and others apparently dead, some dry, and others thickly covered with atmosphere or ocean?

Introduction

Leaving aside the special case of the Moon, the properties of planets that can at present be determined are certain gross quantities descriptive of a planet as a whole; these are the size, the spin angular velocity, the mass and mean density, the moments of inertia and the coefficients in a spherical harmonic expansion of the gravitational potential, together with some features of the magnetic field and possibly electromagnetic induction in the planet. The Moon alone is open to the study of the variation of properties with depth by seismology. The investigation of the internal state of a planet depends on what can be inferred from the measured gross properties, and fails unless those properties can be measured with precision. Given only integral properties, a wide range of internal distributions of density is consistent with the data, but the more precisely the integral properties are known the more restricted the range of possible distributions.

The various dynamical properties of a planet are not independent, for all are determined by three factors: the spin, the chemical composition and the temperature. Suppose the spin acceleration at the surface at the equator (where it has its greatest value) to be small compared with the acceleration of the self-gravitational attraction. Then composition and temperature together determine the equation of state, the latter mainly by its control of the occurrence of any polymorphic phase changes.

Introduction

After the Earth, the Moon is much the best known body of the solar system. Almost all physical measurements that have been made on the Earth have also to some extent been made on the Moon. Artificial satellites have been placed in orbit about the Moon and have enabled the components of the gravitational potential to be estimated. The physical librations, the equivalent of the luni-solar precession of the Earth, have been observed, especially by laser ranging to the retroreflectors left on the Moon by Apollo astronauts. The Apollo astronauts took with them seismometers that have recorded impacts of meteorites and rockets on the surface and moonquakes within the Moon. The flow of heat through the surface of the Moon was measured.

The magnetic field of the Moon has been studied intensively, globally by satellites at a distance from the surface and in detail by others close to it, while the magnetization of rock samples brought back by the Apollo astronauts has been studied in the laboratory. In addition, electro-magnetic induction in the Moon has been studied. Thus, there is some prospect of being able to construct models of the interior of the Moon using much the same methods as are followed for the Earth, whereas there is at present no such prospect for any of the planets. However, there are major gaps in our knowledge of the Moon as compared with the Earth, and the principal one is that seismic data are comparatively very sparse because there are only four seismic stations on the Moon and all of them are on the same hemisphere and, furthermore, because free oscillations of the Moon have never been observed.

THIS BOOK is based on lectures given annually in the University of Cambridge and on a parallel course of instruction in Practical Astronomy at the Observatory. The recent changes in the almanacs have, in many respects, affected the position of the older textbooks as channels of information on current practice, and the present work is intended to fill the gap caused by modern developments. In addition to the time-honoured problems of Spherical Astronomy, the book contains the essential discussion of such important subjects as helio-graphic co-ordinates, proper motions, determination of position at sea, the use of photography in precise astronomical measurements and the orbits of binary stars, all or most of which have received little attention in works of this kind. In order to make certain subjects as complete as possible, I have not hesitated to cross the traditional frontiers of Spherical Astronomy. This is specially the case as regards the spectroscopic determination of radial velocity which is considered, the physical principles being assumed, in relation to such problems as solar parallax, the solar motion and the orbits of spectroscopic binary stars.

Throughout, only the simplest mathematical tools have been used and considerable attention has been paid to the diagrams illustrating the text. I have devoted the first chapter to the proofs and numerical applications of the formulae of spherical trigonometry which form the mathematical foundation of the subsequent chapters. Although other formulae have been given for reference, I have limited myself to the use of the basic formulae only.

A writer of a textbook on Spherical Astronomy cannot avoid a certain measure of detailed reference to the principal astronomical instruments and, accordingly, general descriptions of instruments have been given in the appropriate places, usually with a simple discussion of the chief errors which must be taken into account in actual observational work.

In numerical applications, the almanac for 1931 has been used.