Introduction

In tests of the weak principle of equivalence, exact calculations of the attractions of masses are not necessary, but they are essential in experiments to test the inverse square law and to measure the gravitational constant. In fact, the calculation of the gravitational attraction of laboratory masses is usually not at all simple, because the dimensions of the masses are comparable with the separations between them, so that neither the test mass nor the attracting mass can be treated as a point object. In the following sections we discuss the gravitational attractions of laboratory masses with various common geometrical shapes. We present the results in terms of the gravitational efficiency, that is, the ratio of the gravitational attraction of a laboratory mass at a certain separation to that of a point mass with the same mass and separation. Furthermore, the precision demanded in measurements of separations of masses, the most difficult measurements in the determination of G and the test of gravitational law, depends on the geometry of the masses. These effects can have a strong influence on the conduct and final results of an experiment and it is essential to discuss in detail the calculation of potentials and attractions before going on to describe experiments.

Masses of three forms are often used in the laboratory: spheres, cylinders and rectangular prisms. The formula for the gravitational attraction of a sphere is well known and simple, but in practice it is not possible to manufacture an ideal sphere, the practical problem is usually how the real precision of manufacture affects the results; cylinders and prisms can be made very precisely but calculating the attraction is difficult.

Introduction

Although the weak principle of equivalence has been verified for ordinary macroscopic matter to very high precision, two questions remain open:

1. Is the principle valid for antimatter? Although indirect evidence from virtual antimatter in nuclei and short-lived antiparticles suggests that antimatter may have normal gravitational properties, no direct tests of the validity of the weak principle of equivalence for antimatter have been made.

2. Is the principle valid for microparticles? As the test bodies in macroscopic experiments are formed of neutrons, protons and electrons bound in nuclei, there is no doubt about the validity of the weak principle of equivalence for bound particles. However, the possibility of the principle of equivalence being violated for free particles should be studied.

Two main features characterize laboratory tests of the weak principle of equivalence for free elementary particles, both the consequence of their small masses. (1) When forces on substantial masses of bulk material are compared, a null experiment based on comparing different test bodies of two kinds of material can be devised. That is not possible for microscopic particles, and the gravitational accelerations have to be measured directly and subsequently compared with the acceleration of ordinary bulk matter to obtain the Eötvös coefficient. (2) The gravitational forces are very weak, even in the field of the Earth (which is the strongest attractive field), and so the accuracy of any experiment is very poor compared with Eötvös-type experiments using bulk masses.

We have not dealt in this book with all possible experiments on gravitation that have been or could be carried out in the laboratory, whether on the ground or in a space vehicle, but have concentrated on those on which most work has been done and from which most results have been obtained. That is because we have been concerned more with questions of experimental design and technique rather than with the bearing of the results on theories of gravitation. Something was said of that in the Introduction and we simply call attention again to recent reviews such as those of Cook (1987b), Will (1987) and others in the Newton Tercentary review of Hawking & Israel (1987). We have restricted our accounts to the weak principle of equivalence, the inverse square law and the measurement of the constant of gravitation partly because in numbers of results they dominate the subject, but more importantly because, having been so frequently and thoroughly studied, it seems that all the significant issues of experimental method and design are brought out when they are considered.

It was observed in the conclusion of the last chapter on the constant of gravitation, that the definition and calculation of the entire attraction upon a detector such as a torsion balance is no simple matter, and that applies equally to experiments on the inverse square law, as may be shown by the details of the calculations that were necessary in the experiments of Chen et al., (1984).

The launch of Sputnik 1 on 4 October 1957 was a traumatic event for the USA and much of the western world. For years there had been an unspoken assumption that the Russians were dark and backward people, and that all new initiatives in science and technology occurred, almost as a natural law, in ‘the West’. Disbelief was widespread. ‘What I say is truth, and truth is what I say’, that popular saying of the 1980s, had its adherents in the 1950s too, and they assured the world that Sputnik 1 was just propaganda and was not really in orbit at all.

My view of the event was different. For several years we had been showing in theory how ballistic rockets could be turned into satellite launchers by adding a small upper stage to produce the necessary extra velocity. The USSR had launched an intercontinental rocket in August 1957, and little extra velocity would be needed to attain orbit. So it would be quite easy for the USSR to launch a small satellite like Sputnik 1, which was a sphere 58 cm in diameter of mass 84 kg with four long aerials (Fig. 2.1). The real surprise was the final-stage rocket that accompanied Sputnik 1 into orbit. The rocket appeared much brighter than the pole star as it crossed the night sky, and seemed likely to be at least 20 m long, far larger than anything contemplated in our paper-studies of satellites: the final-stage rocket for our reconnaissance satellite was less than 5 m long.

It was in 1953 that the metamorphosis of missiles into satellites began. One important new start was the prospect of rockets for upper-atmosphere research. The impetus came from a group of scientists belonging to the Royal Society's Gassiot Committee, particularly Professor Harrie Massey of University College London, and Professor David Bates of the Queen's University, Belfast. The existence of the Gassiot Committee was an extraordinary stroke of luck for space science, as I came to realize much later. The Royal Society covers all science, and until 1935 the one exception to this rule was the Gassiot Committee, the Society's only specialized ‘inhouse’ committee: it had been formed in 1871, to oversee Kew Observatory, and was expanded during the Second World War to cover atmospheric physics in general. The Gassiot Committee was vitally important for two reasons: first, it was a preconstructed official pathway into space; second, the Royal Society was fully committed from the outset, thus making respectable a subject dismissed by many as ‘utter bilge’.

The Gassiot Committee organized an Anglo-American conference on rocket exploration of the upper atmosphere, at Oxford in August 1953, and this can now be seen as the first British step on the ladder into space which we climbed for nearly twenty years. I cannot remember much about the meeting, except that it was held in a dark medieval lecture-room, lit by a few light bulbs with dusty white shades: it seemed paradoxical that these new ventures into space were being planned in such antiquated surroundings.

A rocket fired up the north face of the Eiger towards the summit might serve as a suitable simile for the worldly aspects of my career in science. From 1957 until about 1970 the upward thrust was strong, and the rocket seemed on course for the stratosphere. During the 1970s the propellant seemed to burn out and the momentum decreased. About 1980 the rocket came to rest on a rather precarious shelf, halfway up the cliff: there was a danger of being pushed off into free fall; on the other hand, the position was a commanding one, from which good work might be done. As it turned out, the danger was averted and the decade was most productive.

In 1980 the researches based on orbit analysis seemed to be in good health. The Earth Satellite Research Unit at Aston University, under Dr Brookes, had moved to a spacious modern building at St Peter's College, Saltley, and the prediction service was transferred from the Appleton Laboratory to ESRU in July, because the Appleton Laboratory was being moved and merged with the Rutherford Laboratory. (Pierre Neirinck retired from Appleton but continued as a keen analyst of satellites.) In September 1980, when a meeting of visual observers was held at St Peter's College, ESRU was thriving, with four staff members working on predictions, four more as Hewitt camera observers, and a strong research team that included Philip Moore and two recently-appointed Research Fellows, Graham Swinerd and Bill Boulton, both working on orbit analysis and popularly known as the heavenly twins.

The clouds had obediently unfolded to reveal that ‘chariot of fire’ over the Caribbean on 14 April 1958; but the descent of Sputnik 2 left us without any satellites to predict. The first US satellite, the pencil-shaped Explorer 1, had been launched on 1 February; the grapefruit-like Vanguard 1 followed on 17 March; and Explorer 3 on 26 March. But these three satellites were small and faint, and, with orbits inclined at less than 35° to the equator, they were far to the south and nearly always below the horizon for observers in Britain.

During this welcome respite there was time, on 22 April, for a visit to Herstmonceux, where the moated castle was worlds apart from the hotchpotch of rather ugly buildings at the RAE. (The contrast always startled me, even in later years.) Thus began a secure and friendly cooperation with the Royal Greenwich Observatory that flourished for more than thirty years, with benefit to both sides. The road back from historic Herstmonceux ran through Piltdown, a name redolent of even older times – or so it was thought until the Piltdown Man was exposed as bogus.

The hiatus in prediction did not last long, for Sputnik 3 was launched on 15 May, which was presciently marked in 1958 diaries as Ascension Day. We heard about the launch just before noon, and early that afternoon sent out the first set of predictions, which proved accurate to half a minute.

Retiring from paid employment at the RAE in May 1988 proved to be the prelude to two years of unpaid attendance part-time, clearing up the loose ends. During the last twenty of my forty years at the RAE I was able to adopt the most efficient procedure of filing away all working papers and reports received, by subject, in filing cabinets that remained in place and just increased in numbers. No time was wasted in going through them on throwing-away sprees. It was ‘onward undaunted’ continuously, with everything undisturbed, in the same office. All that had to end in 1988. In a long and traumatic series of evening massacres at home, I ploughed through the 25,000 neatly-filed letters in my ‘general correspondence’ and threw away about 98%. My wife with great forbearance allowed the smallest bedroom in the house to be lined with shelves and converted into an archive room, to store the papers needed for this book, and some of my books and reports on space topics. A second round of massacres is now in prospect among those archives …

At the RAE, meanwhile, I was obliged to vacate my office in Q134 Building in May 1988, and took a suitcase-full of selected papers each day across to a new office in R14 Building, reducing the bulk to a mere six filing cabinets. The Table of satellites was taken over by Doreen Walker and Alan Winterbottom: Geoffrey Perry, uniquely knowledgeable in current space activities, continued to supply the basic data under contract.

In 1961 a clear ocean of scientific research seemed to have opened up, ready to sail into and explore. The climate seemed set fair too. This optimism – fearing ‘nor sea rising, nor sky clouding’ – was justified by events: the 1960s proved to be a decade of fairly easy achievement, exploiting techniques already devised.

The RAE research on the upper atmosphere had so far been received in deafening silence by the Meteorological Office, which regarded anything at heights above about 20 km as rather ‘way out’ and of no interest to weather forecasters. This hardline attitude by meteorologists was slowly softening, and the Royal Meteorological Society invited me to give the Symons Memorial Lecture on 1 March 1961: the title was ‘Satellites and the Earth's outer atmosphere’, and I ranged more widely than in previous talks, discussing the history of ideas on the atmosphere and also venturing further outwards, above 1000 km height, into the exosphere and magnetosphere.

A month later came the most important scientific meeting I ever attended, the 1961 COSPAR Symposium at Florence. For this occasion we gathered all the data on air density for an updated picture of the variations with height, with solar activity and between day and night. Fig. 4.1 shows the graph of density versus height obtained from twenty-nine different satellites launched before 1961, as presented at Florence.

The morning was sunny and serene, the day was Monday 12 September 1948, and I was travelling by train to begin a new life working at the Royal Aircraft Establishment at Farnborough in Hampshire. As the steam-engine puffed along the last few miles from Guildford to the curiously-named North Camp station, I had no idea what was in store, never having ventured into Hampshire before (unnecessary travel had been frowned on during the Second World War). During the previous two years I had been working for a mathematics degree at Cambridge, and it was in the garden of the Cambridge Appointments Board in May that I was interviewed by two ‘Men from the Ministry’ and offered a post in the Guided Weapons Department at the RAE, as an alternative to three years of military service. My interviewers were very pleasant and persuasive, and the alternative was also persuasive: I accepted the post as a temporary Scientific Officer at the excellent salary of £340 a year, though with various deductions.

At first sight, the Royal Aircraft Establishment created a favourable impression, because I had seen nothing like it before. It covered about three square miles and seemed like a small town. Some of the buildings were rather scruffy, but some were quite presentable, and the built-up area was balanced by the extensive airfield. There were about 10,000 people working at the RAE then, and the whole place seemed to be buzzing with activity, the noisiest buzzing being produced by the frequent take-offs and landings of jet aircraft.

In 1970 a new world beckoned, the realm of resonance, with prospects of fresh and fertile fields of research. A satellite experiences resonance when longitudinal variations in gravity cause changes in the orbit that build up continually, day after day and month after month. Orbital changes that are basically very small then magnify themselves until they are large enough to be accurately determined: thus resonance creates a powerful technique for measuring the gravity field.

In earlier chapters the Earth's gravity has been taken to be composed of a series of zonal harmonics dependent only on latitude, and independent of longitude. This is an over-simplification, because in reality gravity varies with longitude: the variations are small, but detectable. The zonal harmonics discussed in previous chapters can be regarded as longitude-averaged, and each of them needs to be supplemented by a teeming family of harmonics that are dependent on longitude as well as latitude,‘tesseral harmonics’ as they are called, after the tesserae of varied shapes in a Roman mosaic floor.

The variation of a tesseral harmonic with longitude is specified by its order. A tesseral harmonic of order 15 gives rise to 15 undulations as you go round the equator (or any other line of latitude), as shown in Fig. 5.1. The symbol m is used to denote the order of a tesseral harmonic: it is helpful to think of m as specifying the variations between one meridian and another. (The zonal harmonics, being independent of longitude, are tesseral harmonics of order zero.)

This book is a personal account of the researches based on analysis of satellite orbits between 1957 and 1990 at the Royal Aircraft Establishment, Farnborough, work in which I played a leading role. The book is most definitely not an impartial history of the subject world-wide: contributions by other groups are mentioned only when necessary. Nor is the book an autobiography, though the science is punctuated – and perhaps enlivened – by some personal experiences.

A book of this kind, a hybrid of science and life, presents the author with many stylistic problems. I have ruthlessly gouged out as many ‘I's as possible, and have tried to avoid mentioning too many names (with apologies to all those who find themselves liquidated). I decided to use ‘we’ quite often: throughout the book we means ‘those of us at the RAE who were concerned with or working on the problem’. Individual names are mentioned too, of course, and often the we is defined by giving the authors of a paper in a note.

I have tried to make the book widely intelligible to readers without specialized knowledge. There is a light sprinkling of mathematical equations: but if you don't like them you can skip them without losing the thread.

Most spacecraft chatter continuously, sending back to the ground stations so much data that storage can be quite a problem. The satellites selected for orbit analysis, on the other hand, are usually dumb (and deaf and blind): but they can be seen from the ground as they cross the sky, and from the observations their orbits can be determined.