The Earth's Density. K.E. Bullen M.A., Sc.D., Hon.D.Se., F.R.S. University of Sydney CHAPMAN AND HALL LONDON

Transcription

1 THE EARTH'S DENSITY

2 The Earth's Density K.E. Bullen M.A., Sc.D., Hon.D.Se., F.R.S. University of Sydney LONDON CHAPMAN AND HALL

3 First published 1975 by Chapman and Hall Ltd 11 New Fetter Lane London EC4P 4EE 1975 K.E. Bullen Soficover reprint o/the hardcover 1st edition 1975 Typeset by E. W.C. Wilkins Ltd, London and Northampton All rights reserved. No part of this book may be reprinted, or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without permission in writing from the publisher. Distributed in the U.S.A. by Halsted Press, a Division of John Wiley & Sons, Inc., New York Library of Congress Catalog Card Number ISBN-13: e-isbn-13: DOl: /

4 Contents Preface page xi 1. Noted ancient investigations Size of the Earth Contributions of Newton and contemporaries The period after Newton 10 References Determination of the mean density of the Earth Bouguer's experiments in Peru The Schiehallion experiment Other pendulum and plumb-line experiments The Michell-Cavendish experiment Other small-scale experiments 18 References Spherical harmonics Solutions of Laplace's equation Legendre polynomials Integrals of products of spherical harmonics Tesseral harmonics Expansions in spherical harmonics Spherical harmomcs and Earth oscillation theory 27 References Theory of the Earth's gravitational attraction General theorems on gravitational attraction Attractions due to some particular mass distributions MacCullagh's formula Representation of the Earth's surface Attraction due to spheroidal Earth model 39 References The figure and moment of inertia of the Earth The geopotential function 44 v

5 CONTENTS 5.2 Forms of surfaces of equal density inside rotating Earth model Relations involving a and h Clairaut's equation on the internal variation of Estimation of the moment of inertia of the Earth Numerical results on the hydrostatic theory Use of artificial satellites International reference systems Ellipticities of internal surfaces of constant density 58 References Early models of the Earth's density variation Earth models Clairaut's equation and the density problem The Legendre-Laplace density law Background theory in density determination Other early model density laws Numerical results for early models Some further nineteenth century results Early evidence on the Earth's rigidity Early twentieth century models 83 References Representation of elasticity in the Earth Stress Strain and rotation Model stress-strain relations Stress-strain relations for the Earth Interpretation of coefficients in perfect elasticity Strength of a material The terms 'fluid' and 'solid' Pressure and finite strain Thermodynamic considerations 101 References Seismic wave transmission Earthquakes and other sources of seismic waves Equations of motion of seismic disturbances Bodily seismic waves Scalar and vector potentials Surface seismic waves Refraction and reflex ion of bodily seismic waves 123 References 127 VI

6 CONTENTS 9. First approximation to seismic P and S distributions in the Earth Seismic rays Effect of the Earth's ellipticity on seismic travel times Normal and abnormal seismic velocity variation Bodily seismic phases Evolution of travel-time tables Derivation of P and S velocity distributions in the Earth 145 References Earth models of type A Introductory theory of density variation in the Earth Historical background on the Earth's internal layering The regions A, B, C, D, E, F and G Density near the Earth's surface Early evidence on inhomogeneity insitle the upper mantle Minimum central density Earth models of type A Corrections for temperature and inhomogeneity Critique of A-type models 179 References Evidence on compressibility in the Earth Compression in the Earth Variation of incompressibility in homogeneous regions Some further implications of finite-strain theory Compressibility-pressure hypothesis Theory for inhomogeneous regions Degrees of inhomogeneity in particular regions of the Earth Solidity of the inner core Earth models of type B 219 References Some second approximations P and S velocities in the mantle Structure of the outer core Structure of the inner core Radius of the Earth's core The Earth's central density Further evidence bearing on rigidity in lower core Improved B-type models 251 References 255 Vll

7 CONTENTS 13. Evidence from seismic surface waves Underlying principles in applying surface-wave data More complex model structures Direct observation of phase velocities Allowance for Earth's curvature and gravity Evidence on crustal structure Evidence on mantle structure Surface waves and density variation Further remarks 282 References Evidence from free Earth oscillations Free oscillations of a dynamical system Approach to the theory of Earch oscillations Equations of motion of an oscillating Earth model Solving the equations of motion Observational data Early inferences from free Earth oscillation data The modelhb Other models using free Earth oscillation data Oscillation evidence on solidity of inner core 317 References Miscellaneous developments Equations of state, and related equations, for the Earth's interior Some miscellaneous Earth models Monte Carlo techniques The general problem of 'inverting' observation data Density and seismic wave amplitudes Implications of wave-scattering investigations Deviations from spherical symmetry Changes in gravitational constant 345 References Optimum and standard Earth models General requirements of Earth models Consequence of non-uniqueness Approaches to the optimum model problem Progress towards an optimum Earth model The problem of a standard Earth model 363 References 365 Vlll

8 CONTENTS 17. Application to other planets and the Moon 17.1 Planetary observational data 17.2 Assumptions on the Earth's internal composition 17.3 Earth, Venus, Mars 17.4 Mercury 17.5 Moon 17.6 Jupiter and Saturn 17.7 Uranus and Neptune 17.8 Pluto 17.9 Further remarks References Index lx

9 Preface The book attempts to draw together the various strands of evidence that have led to present knowledge of the distribution of density throughout the interior of the Earth. Details are also given of other properties with which the density is closely linked, including pressure, compressibility and compression, rigidity, seismic velocities, Poisson's ratio and gravitational intensity. Questions of thermodynamics and chemical composition and phase enter discussions where they bear more or less sharply on the density determination; but the book does not purport to be a comprehensive text on the Earth's internal temperature distribution and composition. The density distributions of other planets are discussed. The quest for clues on the Earth's internal densities has long been an exciting one, and an aim of the book is to present a developing story which has fascinated the author over much of his worl(ing life. The early chapters refer to key developments from ancient times to around The later chapters recount, in greater detail, developments since The stage has now been reached where numerous published papers bearing on the Earth's density seem to do little more than fidget around the resolving power of long accumulated observational data. So the present seems an appropriate time to try to put some perspective into the story. Of course, the story will never be quite finished: in describing the interior of the Earth, there will always be extra decimal places to add as further significant evidence arises. I have sought to describe what have appeared to me to be the more important contributions among those that have caught my active interest during the past forty years. Thus the book inevitably reflects my tastes and interests: the subject now touches on so many fields that a wholly satisfying coverage could hardly be expected from a single author. So I apologize in advance to those whose important work may possibly have been overlooked. I mention also that my Xl

10 PREFACE approach has been mainly 'macroscopic': only brief reference is made to 'microscopic' approaches through, for example, lattice theory. Perhaps I should apologize also for making considerable reference to my own work. I have done this because: first, in a subject rather difficult to expound in all its intricacies, I felt I could contrive the best coherence by basing many of the developments on my own approaches, at least in the first approximation; secondly, I would like this book to help correct numerous recent distortions of detail in my past writings (a phenomenon which of course by no means afflicts only myself in this era of scientist population explosion); thirdly, since the book may be my last major effort on the subject, I have sought to make the account of my work as unambiguous as possible. I hope these reasons will help to counter any suggestion that I regard my contributions as more significant than they really are. The book will probably be found rather more cautious in its attitude to uncertainties than are many current writings. There is a strong tendency for modern writers (including some notable contributors) in the Earth sciences to be unduly black and white in their pronouncements - rather over-steady to 'prove' and 'disprove' and to declare the 'beliefs' and 'disbeliefs' of themselves and others in contexts where cautious assessments in terms of probability would be wiser. (This tendency is not confined to the Earth sciences.) Here, I have striven to avoid words such as 'proof', 'true', 'false', 'right', 'wrong', 'valid', 'invalid', except in formal deductive arguments. In inductive arguments, I have sought to 'infer', not 'deduce'; I have been at pains to distinguish between 'mathematical models' and 'facts', not only with density distributions and the like, but also with (so-called) physical 'laws'; and so on. Perhaps vainly, I cherish the hope that my pattern of writing may make a modest contribution towards improving the appreciation of some points of scientific inference that need to be specially heeded in geophysics. Because of complex interplay between different lines of evidence on the Earth's density, some degree of overlapping among several of the chapters has been unavoidable; I have tried to arrange the book so that this overlapping is minimal. The numerous cross-references are principally to help readers who may desire ready access to the subject. Many cross-references may be ignored in the first reading: their purpose is to help the readers who want them, not to distract those who don't. The book necessarily contains a fair quantity of seismological detail since important density findings have come through the help of seismology. But the book is not intended as a text on seismology. Intermittent reference is made to sections in the third edition of my 'Introduction to the Theory of Seismology' (Cambridge University XlI

11 PREFACE Press) - referred to as B - where additional seismological detail can be found. The second half of the book includes a considerable variety of numerical detail which, I hope, the reader will not find too oppressive. I felt that only by giving this detail could I fulfil my purpose of exhibiting in suitable perspective the consequences of varying approaches to a complete subject. By way of compensation, I hope that research workers, lecturers and students of solid-earth geophysics will find helpful the compilation of information given in the book. With such needs in mind, I have, further, taken special pains to provide an extended Index which should give rapid access to particular strands of the information. Most of the symbolism used is orthodox, but there are occasional deviations. For example, s is sometimes used instead of 5 for entropy in contexts where 5 is already used for other purposes. Most of the units are 5.1.; but I cannot bring myself to the point of expressing densities in kg/m3 with the consequent squandering of cyphers in a book where densities appear on nearly every page. lowe deep gratitude to Miss K. Yamamoto for her superb typing of an often difficult manuscript and her great patience in numerous re-typings needed to bring parts of the book up-to-date over the several years during which it was written. The book might possibly never have emerged had it not been for her great help. I also wish to thank my former colleagues Dr. A.P. Treweek for the information in Chapter 1 on ancient stadia, and Dr. R.A. Haddon for helpful comments on several seismological points. Sydney 29 June 1974 K.E. Bullen Xlll