## A i mpi ai a6pu t m afi ai a6Pi p t

Next, we expand a, and p, in powers of the masses m a and m p, which are small relative to that of the Sun, so that the elements vary slowly. Thus, we write a,(t) = £ ajk(t)mJamkp, pt(t) = £ ftk(t)mimkp, j,k = 0 j,k = 0

where a 1ik(t ) and p/k(t ) are functions of time to be determined. These expressions then canbe substituted into Eqn (9.17), and the coefficients of like powers of m a and m ¡ equated. We obtain aj° = ¡3i1 = 0, for any j,

¡i = Vi («1 ,¡1 ,...,p6 , q, and so on. To carry the analysis up to the first order in the planetary masses, therefore, we can integrate simply the right-hand sides of the equations in (9.17), treating the elements as constants. Moreover, the problems for the two planets decouple, and when considering the perturbations of the planet m a one only needs to consider the Keplerian orbit of the perturbing planet, m p.

The six elements of a Keplerian orbit can be taken as a semi-major axis ^ longitude of ascending node e eccentricity rn longitude of perihelion

, inclination t time of perihelion passage where a and e determine the shape of the orbit,,, and m fix the orientation of the orbit in space, and t determines the position of a planet on its orbit given the actual time t. Before Laplace, it was customary to use the aphelion as the reference for the line of apsides, but Laplace realized that, in order to allow comets to fit into the same theory, it was more appropriate to use the perihelion. An alternative to t is to use e = m — «t for the sixth element, where « is the mean motion. This is known as the 'mean longitude at epoch' and, since the mean longitude is simply m + n(t — t), e is the mean longitude of the planet at t = 0.

Trigonometric series can be used to express the differential equation for an element of the planet m a (to the first order of the planetary masses) in the form in which j and k are any integers, the coefficients Pjk depend on the elements a, e, and, for both planets, and A jk (a complicated function of the elements) is independent of t. The quantities na and np are the mean motions of the two planets (assumed constant).

When this is integrated, two types of term will result, as well as the constant of integration. First, if jna + knp = 0 (and in particular if j = k = 0) the term in the sum is independent of t and we will get a term proportional to t in the solution. Otherwise, periodic terms will result and so the final solution will be of the form a = Co + cit + ^ Pjk sin(jna t + knß t + Ajk) jna + knß

where it is understood that pairs of integers for which the denominator in the final term vanishes are not included in the summation. The magnitude of the coefficients Pjk decreases rapidly as | j + k| increases and, since na and can never be known exactly anyway, it is reasonable to assume that jna + kn$ is never zero except when j = k = 0. The coefficient of the term proportional to t is thus simply P00, and this term is what Lagrange and Laplace referred to as the secular variation in the element. Since the coefficients Pjk depend only on the elements a, e, and i for the two planets, the secular variation is a function of the orbits but not of the actual positions of the planets.

As far as Laplace was concerned, the successful determination of the secular inequalities affecting planetary motion was the key to progress, and the first major result he presented concerned the secular variations in the elements of a planet's orbit (attempts by Euler and Lagrange in this direction had led to contradictory results). Laplace focused his attention initially on the variation of the semi-major axis, which is related to the mean motion through Newton's modification of Kepler's third law Eqn (8.9). In a detailed analysis, he expanded the coefficient Poo up to third order in the eccentricities and inclinations of the orbits and showed that, to this degree of approximation, P00 = 0. Laplace thus concluded that the mutual interaction between two planets could not result in an acceleration in the mean motion of either of them which was independent of their relative position and, hence, the semi-major axes of the orbits could not be subject to a continual increase or decrease. Laplace suggested that perhaps the observed anomalies in the mean motions of Jupiter and Saturn were due to interactions with comets. What he did not think of at this time was that they might be oscillations, dependent on the positions of the planets, with very long periods.

The situation with the other elements was different. It had been accepted widely since Kepler's Rudolphine Tables that the nodal and apsidal lines of planetary orbits were subject to secular variation, and Euler had concluded from his perturbation analysis that both these elements and the inclinations and eccentricities of the orbits changed over long periods of time. In other words, it was to be expected that the coefficient c1 in the solution for these elements was nonzero. However, in 1774, Lagrange presented a memoir in which he

Lagrange subsequently produced a proof that c = 0, which was not restricted to small values of the eccentricities and inclinations (see Wilson (1985), pp. 198-205).

showed - much as he had done earlier for the problem of 'arcs of circles'- that the secular inequalities actually were periodic, the terms proportional to t in the solution simply representing developments of a periodic function in the eccentricities and inclinations of the orbits. As was so often the case, the key step involved choosing the most appropriate independent variables for the problem under consideration, and Lagrange managed to derive first-order linear differential equations for the secular variations of the nodes and inclinations. For example, in the case of Jupiter and Saturn, he found that the period of the oscillations in both the nodes and inclinations was 51150 years. Both Lagrange and Laplace realized that the same approach could be brought to bear on the secular variations in eccentricity and aphelion, with the same conclusions. Thus, it appeared that all apparently secular phenomena caused by the mutual interactions of two planets were, in fact, slow oscillations.

The fact that mutual interaction between planets appeared to lead to oscillations in the semi-major axes, inclinations and eccentricities of their orbits, appeared to suggest that the Solar System was a stable entity, in that the positions of planets would remain within certain fixed limits for all time. However, actual calculations for specific planets required knowledge of their relative masses and, except for those planets with satellites, these were unknown. From the mid 1770s onwards, the search for a 'proof' of the stability of the Solar System became one of the main objectives of celestial mechanics. Lagrange continued to devise improvements to the techniques by which planetary perturbations could be determined, but he remained frustrated in his attempts to resolve the great inequality of Jupiter and Saturn. This resolution was achieved by Laplace who, following 10 years of silence on the matter, announced his findings in 1785.

All the previous results on the Jupiter-Saturn problem suggested that the changes that were observed in the mean motions of these two planets were not caused by their mutual interaction. However, Laplace came to realize that, notwithstanding all the previous work, gravitational interaction had to be the cause. From the conservation of energy relation Eqn (9.16), and on the basis that the planetary masses are much smaller than that of the Sun, he derived the result m m' m"

where f is a constant and m, m', m", ... and a, a, a", ... are, respectively, the masses and semi-major axes of the planets. The contributions from Jupiter and Saturn were much larger than from any of the other planets and thus a

77 The full theory followed a few months later in La théorie de Jupiter et de Saturne (1786).

small increase in the mean motion of Jupiter, Sn, would lead to a corresponding decrease in that of Saturn, Sn', and vice versa. Numerical calculations revealed that

Sn' ~ —2.33 Sn, so that Saturn should be decelerating at about 3 times the rate at which Jupiter was accelerating, and this appeared to agree with observation:

It is therefore very probable that the observed variations in the movements of Jupiter and Saturn are an effect of their mutual action, and since it is established that this action can produce no inequality that either increases constantly or is of very long period and independent of the situation of the planets, and since it can only cause inequalities dependent on their mutual configuration, it is natural to think that there exists in their theory a considerable inequality of this kind, of which the period is very long.

Being convinced of the cause was one thing, but finding the effect was another. Laplace took a lead from Lagrange's work in which he had shown that, in the interaction between two orbiting bodies, the effect of small divisors would lead to problems if the mean motions of the bodies, n and n' say, were commensurable, i.e. if jn = kn' for some integers j and k. Now, in the case of Jupiter and Saturn it happens that 2n « 5n' and so Laplace looked for terms in the analysis that contained the factor sin(5n' — 2n)t, even though he knew that these would be multiplied by cubes of small quantities (since 5 — 2 = 3). With hindsight, knowing that the period of such an oscillation is 360°/(5n' — 2n) « 900 years, and that the sought-after inequality is periodic with a period of about 900 years, this is a fairly obvious thing to do, but Laplace had no prior knowledge of the form of what actually he was looking for. However, once he had hit upon the right idea it did not take him long to resolve the issue. Showing great technical ability in following a single aim through a mass of complex calculations, Laplace obtained 20' for the amplitude of the inequality of Jupiter in longitude and 46' 50 ' for that of Saturn, values he subsequently improved upon but which already compare well with the more accurate values given on p. 310 above.

The familiar pattern had repeated itself once more. An observed phenomenon had appeared out of line with Newton's theory of gravitation, but the fault had, as in all previous cases, turned out to be due to the incomplete understanding of the differential equations describing the motion rather than with the equations themselves. Laplace later wrote:

Laplace La memoire sur les inegalites seculaires des planetes et des satellites (1785). Quoted from Wilson (1985), p. 229.

The irregularities of the two planets appeared formerly to be inexplicable by the law of universal gravitation - they now form one of its most striking proofs. Such has been the fate of [Newton's] brilliant discovery, that each difficulty which has arisen has become for it a new triumph, a circumstance which is the surest

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