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.

General relativity is the flagship of applied mathematics. Although from its inception this has been regarded as an extraordinarily difficult theory, it is in fact the simplest theory to consummate the union of special relativity and Newtonian gravity. Einstein's ‘popular articles’ set a high standard which is now emulated by many in the range of introductory textbooks. Having mastered one of these the new reader is recommended to move next to one of the more specialized monographs, e.g. Chandrasekhar, 1983, Kramer et al., 1980, before considering review anthologies such as Einstein (centenary), Hawking and Israel, 1979, Held, 1980 and Newton (tercentenary), Hawking and Israel, 1987. As plausible gravitational wave detectors come on line in the next decade (or two) interest will focus on gravitational radiation from isolated sources, e.g., a collapsing star or a binary system including one, and I have therefore chosen to concentrate in this book on the theoretical background to this topic.

The material for the first three chapters is based on my lecture courses for graduate students. The first chapter of this book presents an account of local differential geometry for the benefit of the beginner and as a reminder of notation for more experienced readers. Chapter 2 is devoted to two-component spinors which give a representation of the Lorentz group appropriate for the description of gravitational radiation. (The relationship to the more common Dirac four-component spinors is discussed in an appendix.) Far from an isolated gravitating object one might expect spacetime to become asymptotically Minkowskian, so that the description of the gravitational field would be especially simple.