A Worden gravimeter in use in the Himalayas, birthplace of the theory of isc The Earth's Shape and Gravity by G. D. GARLAND P E R G A M ON P R E SS NEW YORK · TORONTO · OXFORD SYDNEY · BRAUNSCHWEIG Pergamon Press Inc. Maxwell House, Fairview Park, Elmsford, N.Y. 10523 Pergamon of Canada Ltd. 207 Queen's Quay West, Toronto 117, Ontario Pergamon Press Ltd. Headington Hill Hall, Oxford Pergamon Press (Aust.) Pty. Ltd. Rushcutters Bay, Sydney, N.S.W. Vieweg & Sohn GmbH Burgplatz 1, Braunschweig Copyright © 1965 Pergamon Press Ltd, First edition 1965 Library of Congress Catalog Card No. 65-24227 Reprinted in the U.SA. 1970 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright holder. 1836/65 Preface MEASUREMENTS of the acceleration due to gravity form one of the geophysical methods for investigating the interior of the earth, and also one of the tools of geodesy for the determination of the earth's shape. Several recent developments have led to a great increase of activity in the field: the improvement of instruments, the use of satellites to indicate variations in gravity over the earth, and the availabiUty of high-speed computers to assist in the interpretation of the measurements, to name just three. There has been a demand for a book which would provide at least an intro­ duction to the modem work. The present volume forms one of a series on the earth sciences. It is intended for readers with some background in mathematics and physics, although for the reading of many sections this is not essential. In view of the rapid development of techniques, the author believes it best to emphasize physical principles rather than details of specific procedures. At the same time, several examples are given of geological problems in which gravity measurements have been of assistance. Certain material has been placed in appendixes. This includes an outline of potential theory, which is essential for an apprecia­ tion of the subject but is not always convenient to locate in modern books. Finally, an introduction is given to the analysis of satellite orbits, but this is not developed in detail because the mathematical background required would be considerably more extensive than for the remainder of the book. G. D. GARLAND Edmonton, Alberta vn CHAPTER 1 Gravity, Geophysics, Geodesy and Geology ONE of the most familiar facts about the earth is that a body released near it will fall with increasing velocity. The rate of increase of velocity is called the acceleration due to gravity, which Galileo showed to be the same for all bodies at a given point on earth. This is the most obvious example we have of a truly uni­ versal phenomenon, that of the mutual attraction of all masses. Newton formulated the principle of universal gravitation by deductions from Kepler's laws on the motions of the planets, showing that these laws were evidence of a force between each planet and the sun. The forces were shown to be proportional to the product of the masses involved, and inversely proportional to the square of the distance between them. In the case of a body on the earth, the force of attraction is determined by the product of the earth's mass and the mass of the body, and the distance between the body and the earth's centre. If the earth were a uniform, non-rotating sphere, the force on a body at a given distance would be everywhere the same, and there would be a single constant value of the acceleration due to gravity. In fact, our earth is non-uniform, non-spherical, and rotating, and all of these facts contribute to variations in the acceleration over its surface. Measurements and analyses of the variation in gravity (that is, the acceleration due to gravity) form a powerful branch of the science of geophysics, which is the investigation of the earth by the methods of physics. Those variations in gravity which are related to the departure of the earth from a spherical form are of particular interest in geodesy, the science of the earth's shape. Variations which reflect the non-uniform density of the earth can be used to infer the presence of structures beneath the surface, and are of 1 2 THE EARTH S SHAPE AND GRAVITY interest to the geologist. The aim of the present book is to provide a general background for the appreciation of both applications of gravity measurements. In geophysics, geodesy, and geology, there are other methods available for studying the shape and internal nature of the earth. Where possible, these methods are mentioned, and their relation to gravity studies are discussed. As one example, in the investi­ gation of the earth's crust, there is a considerable advantage to combining results obtained from seismology with gravity measure­ ments. On the other hand, limitations of space prevent a full discussion of other methods, especially in the case of geodesy, and it should not be thought that these methods are considered inferior. Fundamental Concepts Newton's law of gravitation for two particles of masses mj and m2, separated by a distance r, is or F = G - ^ ', (l.l) where G is a constant. Here, F is the force on either mass, and is directed along the line joining the masses. The numerical value of the constant G was not determined in Newton's lifetime, being first measured in the laboratory by Cavendish in 1798. During the latter half of the eighteenth century, a number of attempts were made to measure G by determining the attraction of large features, such as mountains. None of these gave a quantitatively satisfactory result, although the work of Maskelyne (1774) was among the best. The Cavendish apparatus, which is well known, made use of the fact (Appendix 1) that the attraction between spheres is the same as that between massive particles at their centres, and pro­ vided for the measurement of this force by the determination of the torque acting on a suspended beam. The Cavendish experi- GRAVITY, GEOPHYSICS, GEODESY AND GEOLOGY 3 ment has been repeated a number of times with improved appa­ ratus. We shall use, for numerical calculations, the value deter­ mined by Heyl(1930): G = 6-67 X 10-8c.g.s. units. (1.2) To investigate the relation between the constant of gravitation G and the acceleration due to gravity g, we consider a mass m on the surface of the earth, of mass M^, Then, neglecting effects of rotation and non-uniformity of shape and density, the force exerted on the mass is F = 0.3) where R is the earth's radius. If the mass were released, in a vacuum, it would accelerate toward the earth's centre with the acceleration g, where g = Flm GM, (1.4) The determination of g and G is thus sufficient to yield a value for the mass of the earth, or equivalently, the mean density of the earth. During the eighteenth century, attempts to measure G were often considered to be experiments for the determination of mean density. The results of Cavendish showed that the mean density was about 5-4 g/cm^, and this was one of the first proofs that the interior consists of material considerably denser than surface rock. Heyl's value for G corresponds to a value of mean density of 5-52 g/cm^. The value of g varies between 978 and 983 cm/sec^ over the earth's surface. For investigations of the shape, or of internal structures, it is necessary to measure variations of 0-001 cm/sec^ or less, and it is convenient to introduce a smaller unit of accelera­ tion. We designate 1 cm/sec^ as 1 gal (after Galileo), and 1 χ 10"^ cm/sec^ as 1 milligal (mgal). 4 THE EARTH'S SHAPE Amy GRAVITY Gravity Measurements and Reductions As will be shown in the next chapter, the measurement of gravity in absolute terms to an accuracy of 1 mgal is a difficult problem. Fortunately, for both geodetic and geophysical pur­ poses, the variation in g over the earth is the quantity usually required, and this can be measured more easily. The most direct application to geophysics of the absolute value is that given in equation (1.4), that is, the determination of the earth's mass. However, in other fields of physics, particularly in the setting up of standards of pressure, temperature, electric current, etc., the absolute value of g plays a very important role, and the geo- physicist is often approached for assistance in providing a value to use in a given laboratory. It has been mentioned that g varies over the earth for a number of reasons, and it might appear hopeless to separate these effects. However, it will be shown that we can treat, in turn, variations due to the shape of the sea-level surface, variations with height of the land, and variations due to concealed masses. For the study of the latter, the effect of the first two factors is removed from the observed values through a series of reductions. The quantities obtained are known as gravity anomalies, since they indicate the presence of anomalous conditions within the earth. The Potential and Equipotential Surfaces A mass in the presence of an attracting body has energy, by virtue of the attraction. This energy, known as potential energy, can be evaluated by considering the mass to have been brought from infinity, and calculating the work done on it in the process. This is done in detail in Appendix 1, where it is shown that the force of attraction can be obtained from the potential energy by differ­ entiation. As the potential energy is a scalar quantity, in contrast to the vector force, it is often convenient to describe the gravitational field in terms of it, rather than in terms of the force. In particular, we shall make use of the potential energy of a unit mass (usually with a change of sign), which is known simply as the potential of the field. GRAVITY, GEOPHYSICS, GEODESY AND GEOLOGY 5 A gravitational field can be represented by surfaces over which the potential is constant, known as equipotential surfaces. The force vectors are everywhere normal to these surfaces, so that there is no component of force along them. Thus, the surface of a liquid in a gravitational field coincides with an equipotential surface, and for this reason the potential is a quantity of great importance in the study of the shape of the sea-level surface of the earth. Mathematical functions which represent the potential have a number of remarkable properties. The most important of these are derived in Appendix 1. Some knowledge of these is essential for an appreciation of the chapters on methods of analysis of the gravitational field, but not, to the same extent, for those chapters dealing with the results of gravity surveys. CHAPTER 2 Gravity Measurements THE measurement of the acceleration due to gravity provides an interesting example of the difficulties which may be met in the attempt to measure a relatively simple physical quantity. An acceleration involves only the fundamental dimensions of length and time, but, as indicated before, the measurements must be sufficiently precise to show extremely small variations in the quantity. To measure g precisely at some point on earth, in absolute units, without reference to any other point, is in fact an experiment requiring the greatest care. On the other hand, it is now relatively straightforward to measure the differences in g from place to place. In theory, therefore, an absolute measurement is needed at only one place, as the value of gravity at all other places could be determined with reference to it. However, it is desirable to have several absolute determinations, made with different types of apparatus, so that intercomparisons can be made, to indicate the precision achieved. In this chapter, methods which have been used for absolute measurements will be described first, then instruments suitable for measuring differences in gravity will be discussed. Absolute Measurements Measurements of g in terms of length and time must be based on some physical equation which relates that quantity to times and distances which can be measured. The most direct approach is to time a freely-falling body over a known distance, but it is only recently that timing standards adequate for the precise measure­ ment of short time intervals have become available. Almost all of the older determinations made use of some form of pendulum, as the expression for the period of a pendulum involves gy and the 6