Mercury's orbit is, of course, perturbed by the other planets. These perturbations can be divided into two types (see Eqn (9.18), p. 330): a secular term proportional to the time t, and a sum of periodic terms. Moreover, to first-order in the planetary masses, we can consider the orbit of the perturbing planet as a Keplerian ellipse with fixed elements. If, in Eqn (9.24), we retain only those parts of the disturbing function R that give rise to secular perturbations, then we find that the variations in the elements m and e of Mercury's orbit (the longitude of perihelion and the eccentricity, respectively) are given (to lowest order in the eccentricities) by:3

— = nm'Be' sin(m — a>'), dt in which e' and are the corresponding elements of the perturbing body, n is the mean motion of Mercury, m' the disturbing mass, and A and B are constants depending on the ratio of the semi-major axes of Mercury and the perturbing planet.

From these equations, Leverrier could calculate the effect of all the other members of the Solar System on the perihelion and eccentricity of Mercury separately. For the perihelion advance he found a total contribution of just i a 4

under 530" per century (about 11 per orbit) made up as shown in Table 12.1.

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