## R 4 r VR116

where R is the disturbing function, considered small, and expanded in terms of small parameters. If R = 0, we have the standard two-body problem with (setting G = 1 for simplicity) 4, the sum of the masses of the bodies, and r, the position of one of the bodies with respect to the other. In the simplest case, the perturbation comes from one other body, the position of which is r' and the mass of which is m, for example, and then, from Eqn (9.3),

In the problems of planetary motion, R is small because m is small relative to 4, whereas in the lunar theory, R is small because the two terms in parentheses -representing the effect of the Sun on the Moon and on the Earth, respectively -are very similar due to the smallness of the quantity r /r'.

There are two distinct approaches to perturbation theory: absolute perturbations, and variation of orbital parameters. In the former, one seeks equations for the small variations in the coordinates about some approximate orbit, whereas in the latter, one seeks equations that determine how the elements of the orbit vary with time. Variation of orbital parameters is particularly useful for discovering r'3

long-period inequalities such as that which exists between Jupiter and Saturn, and is appropriate more generally in planetary theory. In the early nineteenth century, it was the planetary theory that received the most attention from theoreticians, but in the latter part of the century there was a shift toward the lunar theory. This stemmed partly from the difficulty astronomers found in reconciling theory and observation for the Moon, and partly it reflected a change in the nature of mathematical astronomy. In 1800, celestial mechanics was pursued for one purpose only, i.e. to provide the theory necessary to produce accurate tables with which to predict future positions of heavenly bodies. By 1900, another competing goal had emerged as mathematicians began to analyse the equations of celestial mechanics in their own right to see what general conclusions could be drawn about the dynamics of the Solar System. The mathematical tools needed to do this came to the fore in attempts to understand the motion of the Moon.

In the early nineteenth century, many people, including Johann Karl Burckhardt, Philippe Comte de Pontecoulant, and Leverrier, extended Laplace's general approach by expanding the disturbing function to higher and higher powers in the eccentricity and inclination. The algebra becomes horrendous, and some turned to direct integration of the equations of motion ('mechanical quadrature'). Significant theoretical progress was made by Peter Hansen, who succeeded Encke as the director of the observatory at Seeberg when the latter moved to Berlin in 1825. The standard procedure for analysing planetary perturbations resulted in long-period inequalities that had to be applied to the mean longitude and short period variations in the true longitude. The two effects thus had to be calculated separately. Hansen's theory improved matters by treating both effects as variations in mean longitude. The method was based on an unperturbed orbit that was an ellipse of fixed dimensions located in the moving plane of the actual orbit and with a perigee the longitude of which was a linear function of time. The theory is difficult, but the series that result are convergent more rapidly than in previous methods and, as a result, it can be applied to orbits of greater eccentricity and inclination.10 Hansen applied his technique to the Jupiter-Saturn problem and to the lunar theory.

Following his success with Neptune, Leverrier embarked on an ambitious programme to produce extremely accurate tables for all the planets. Between 1858 and 1861, he published theories for the four inner planets, Mercury, Venus, Earth, and Mars; all represented significant improvements over existing tables. The case of Venus, however, illustrates clearly the problems inherent in the use of perturbation theory. Laplace previously had calculated perturbations up to

10 Details can be found in Brouwer and Clemence (1961), Chapter XIV.

e3, i3, but Airy discovered the fifth-order 239-year perturbation due to the near resonance in the mean motions of the Earth and Venus (13nE - 8nV ^ 1/239 revolutions per year). The amplitude of this inequality in the longitude of Venus is about 3 ", which compares with 1" .4 for the maximum amplitude of the e3, i3 terms, and 0". 1 for the fourth-order terms.

Jupiter, Saturn, Uranus, and Neptune present a more difficult challenge but, nevertheless, Leverrier succeeded in producing accurate theories. In the case of Neptune, the work was completed by A. J.-B. Gaillot a month after Leverrier's death. Leverrier's planetary theories are a magnificent testimony to the legacy of Laplace. They represent, with some minor modifications, the motions of the planets over hundreds of years with errors of at most a few arc-seconds. And yet they are somewhat unsatisfactory. The masses of many of the planets were not known with any real precision, and the last step in Leverrier's construction of each planetary theory was to tweak these masses slightly so as to minimize the discrepancies between theory and observation. Unfortunately, this meant that the same planet sometimes was accorded a different mass in each of the separate theories.

The first to develop a consistent set of planetary theories was the American astronomer, Simon Newcomb.11 Newcomb made significant technical improvements to the underlying Laplacian theory, which reduced the labours involved in computing perturbations to high order in e and i. Tables based on his theory for the four inner planets appeared in 1898. In the same year, Hill produced tables for the outer planets based on Hansen's method. The planetary theories of Newcomb and Hill are marginally superior to those of Leverrier, and were not improved upon until the advent of the electronic computer.12

There was still a significant problem in the theory for Mercury, which has so far been glossed over. In order to get theory and observation to agree, Leverrier found that he had to alter significantly the theoretical value for the rate at which the perihelion of Mercury moved round the Sun. Newcomb was confronted by the same problem, but could not explain the discrepancy. At the time, this may have appeared a rather small problem but, as it turned out, its resolution had profound consequences. This is the subject of Chapter 12.

Newcomb was involved in the first significant astronomical discovery made by American astronomers, who in the second half of the nineteenth century

11 Newcomb's early career is described in Norberg (1978).

Unlike Newcomb and Hill, Leverrier was not influenced by Hansen - some copies of the latter's work presented to Leverrier remained with their pages uncut!, (see Williams (1945), who examines carefully the differences between Leverrier's and Newcomb's tables). Newcomb described Hansen as the 'greatest living master of celestial mechanics since Laplace' (Roseveare (1982), p. 53).

were beginning to compete with their European counterparts. The architect of the discovery, though, was an amateur astronomer who had spent time working as a 'calculator' for first Benjamin Apthorp Gould and then Benjamin Peirce at the US Coast Survey. Seth Chandler established, by reducing thousands of extant observations, that the latitude of a point on the surface of the Earth (defined in terms of the declination of the celestial pole) undergoes a small but measurable variation.

It had been known since Euler's work on the dynamics of rigid bodies that, if the Earth rotated about an axis that was inclined slightly to its polar axis, then it would wobble, and the period of such an oscillation had been computed to be approximately 10 months. People searched in vain for just such an effect to explain small but persistent errors in their observations. Chandler was no theoretician and did not set out to find a variation with a 10-month period. His phenomenal abilities at reducing data led him in 1891 to the discovery that there was a variation in latitude, but that its period was not the theoretical value of 306 days - it was 427 days. Newcomb made Chandler's discovery believable when he pointed out that the Earth was not perfectly rigid as the Eulerian theory supposed, and that the fluidity of the ocean and the elasticity of the solid earth would both serve to lengthen the period of the 'Chandler wobble'. Chandler went on to discover that the variation in latitude actually was rather more complicated, with the superpositionof two oscillations, one withaperiod of427 days

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