Special perturbation theories: General principles
 

A different approach to the many-body problem is that of special perturbations, a method most workers in celestial mechanics before the days of high-speed computers shrank from using, since it involved the step-by-step numerical integration of the differential equations of motion of the bodies from the initial epoch to the epoch at which the bodies’ positions were desired. Each step consists essentially of calculating the bodies’ positions, velocities and mutual gravitational attractions at time t₁, say, their positions and velocities at time t₂, where (t₂ − t₁) is a small time interval, perhaps a few hours, perhaps a few days, depending upon the problem. The new positions give the new values of the mutual gravitational attractions, so that the calculation can be carried forward to a new time t₃. The great advantage of this method is that it is applicable to any problem involving any number of bodies. Nowadays, in both celestial mechanics and astrodynamics (the study of the trajectories followed by spacecraft, artificial satellites and interplanetary probes), special perturbations are applied to all sorts of problems, especially since many modern problems fall into regions in which general perturbation theories are absent. In astrodynamics, in particular, the solution is often required in a very short time: a high-speed computer suitably programmed will provide the answer. For example, in the case of a circumnavigation of the Moon by a spacecraft, the problem of calculating swiftly the orbit of the vehicle in the Earth–Moon gravitational field can be adequately treated only by special perturbations and a computer.
The main disadvantage of this method is that it rarely leads to any general formulas; in addition, though they may be of no interest to the worker, the bodies’ positions at all intermediate steps must be computed to arrive at the final configuration.
A further disadvantage is the accumulation of round-off error. During the numerical work for each step, a great many additions, subtractions, multiplications and divisions are carried out. Each is rounded off and this process is a source of error. If, for example, we worked to four significant figures and a number 1·2754 was, therefore, rounded to 1·275 then multiplied by two, the answer would be 2·550. If, however, we had retained five figures and multiplied by two, the answer would have been 2·5508. This number, rounded to four figures, is 2·551. This shows a difference of one in the fourth place when compared with the previous answer. Thus, in long calculation, where millions of operations are carried out, the number of significant figures gradually decreases until, by the end of the calculation, the answers may have lost any meaning.
Several long-term calculations of the positions of the bodies in the Solar System have been carried out by special perturbation methods. In recent years the positions of the five outer planets, Jupiter, Saturn, Uranus, Neptune and Pluto, have been computed over a period of 2 × 10⁸ years. Such investigations show that the orbits, at least within that length of time, are stable. The orbits of Pluto and Neptune seem to be locked together so that no close approach of these bodies to each other can take place. Jupiter and Saturn’s orbits also seem to be coupled, the two orbital planes rotating as if stuck together at a constant angle. Periodic disturbances occur, having cycles ranging from a few years to several million years.
Electronic computers, by writing suitable programmes for them, have also been used to prepare the general perturbation theories. The general perturbations are true analytical theories; the algebraic manipulations that formerly occupied years of a researcher’s life in carrying out are now done by the computer in hours, so that problems of a much higher order of complexity in celestial mechanics can now be tackled.

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