A NEW METHOD TO DETERMINE THE DERIVATIVES OF THE LAPLACE COEFFICIENTS ÁRON SÜLI, BÁLINT ÉRDI and ANDRÁS PÁL Department of Astronomy, Eötvös University, H-1117 Budapest, Pázmány Péter st. 1/A, Hungary, e-mail: a.suli@astro.elte.hu (Received: 9 December 2002; revised: 8 September 2003; accepted: 15 September 2003) Abstract. The determination of the secular variations of the orbital elements of objects in N -body systems is based on the literal development of the perturbing function. The development makes use of the Laplace coefficients and their derivatives. In this paper a new method is described for the ana- lytical computation of the derivatives of the Laplace coefficients. It is an explicit formula in the sense that it only contains the Laplace coefficients and the parameter α on which the Laplace coefficients depend. The advantage of this method is that it is unnecessary to calculate all the derivatives up to the desired order. It is enough to calculate the Laplace coefficients. Easy coding is a further benefit of the method and it provides more accurate numerical results. The paper describes in detail the application of the method through an example and gives comparison with former methods. Key words: analytical methods, Laplace coefficients, N -body systems, perturbation theory 1. Introduction Many studies on the long-term variations of the orbital elements of objects in N - body systems have been carried out. The most natural example of an N -body system is the Solar System and on larger scale the Milky way. The problem of the secular variations of the orbital elements of the major planets was first tackled in 1781 by Lagrange and later by Laplace. The Laplace–Lagrange approach took into account only linear terms. Later on Hill (1897) was the first to introduce degree 3 and 5 of the eccentricities into the development of the perturbing function of the 2nd order with respect to the masses of Jupiter and Saturn. Brouwer and van Woerkom (1950) used the perturbing func- tion of Hill and incorporated it into their secular theory of the eight planets. This theory has been extensively used in the studies of motion of Solar System bodies. Duriez (1979) significantly improved the accuracy of the solution for the four outer major planets by using terms in the perturbing function up to the second order in the masses and degree 5 for the eccentricity–longitude of perihelion, and degree 3 for the inclination–longitude of node couples. Finally, Laskar (1984, 1985) developed a solution which represents, in general, the extension of that of Duriez to eight planets; his theory is of the second order in the masses and degree 5 for the eccentricities and inclinations, and it has been Celestial Mechanics and Dynamical Astronomy 88: 259–268, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.