Progress in general planetary theory

Abstract

When on searches for a planetary theory valid over 1 million years, one can leave in the solution the short period terms whose amplitude are small, and compute only the long period terms which give the essential of the solution; but one must take into account the short period terms contribution in the computation of the long period terms. Among recent works on such theories, Bretagnon (1974) computed a long period solution for the 8 planets, including the contribution of all terms up to order 2 with respect to the masses and degree 3 in the eccentricity-inclination. He improved his solution later on by adding the effects of the relativity and the Moon, and by adjusting the main frequencies of the outer planets to the secular terms of his short period theory VSOP82 (Bretagnon 1984). Duriez (1979) computed an analytical solution for the 4 outer planets up to order 2 of the masses and degree 5 in the eccentricity-inclination (and even 7 for the Jupiter-Saturn couple). Nowadays, the great development of the Milankovitch astronomical theory of palaeoclimates, resulting from the improvements of geological datation procedures for sediments (Hays, Imbrie and Shackleton 1976), requests a very accurate knowledge of the Earth orbital elements over a very long span of time. We resumed the study of General Planetary Theory, focussing on the accuracy of the solution. The aim of our study is no longer to have the global features of the solution, but to obtain an accurate solution with known error bounds over 1 million years, in order to know how far we can extend the solution keeping a reasonable precision. In the first step of the work, we have developed a new method for computing the terms of the autonomous system which gives the secular variations of the 8 planets orbital elements. This system includes all the secular terms and the contribution of the short period terms up to order 2 of the masses and up to degree 5 in the eccentricity-inclination. The method is sufficiently efficient to allow us to kep all the 150 000 monomial terms in the autonomous system. We have also paid special attention to the precision of the computation and all the coefficients of the autonomous system are given with a relative precision of 10−6 (Laskar, 1984a). We then elaborated an algorithm to compute an analytical solution of the autonomous system, following the method of normalization given in Brumberg (1980). We thus convinced ourselves that it was very difficult, if not impossible to obtain a very accurate solution with this method in the case of our 8 planet system, nowithstanding the fact that if the degree 3 solution contains 25 000 terms, the degree 5 solution contains 3000 000 terms, and may not give a more accurate solution, due to the presence of small divisors which damage very much the solution. On the contrary, to integrate numerically the autonomous system is rather easy: the main periods are larger than 50 000 years, so we can choose a step of integration of about 1000 years. We thus made a numerical integration of the whole autonomous system over 1 million years using an Adams method with varying stepsize and tolerance of 10−12. To check the accuracy of the solution, we made an expansion of the solution at the origin in polynomial of the time, in order to compare it with the secular variations given by the highly accurate short period solution VSOP82 (Bretagnon 1982). As the constants of the two theories are the same, we can assume that the comparison of the coefficients of T gives a good estimate of the precision of the solution. We made also a comparison with Bretagnon (1984) long period theory. For the variable k=e cos $$\overline \omega $$ of the Earth, for example, we obtain a precision of 0.068×10−6rd/ky for our solution against 13.371×10−6rd/ky for the Bretagnon (1984) solution. If we take then the terms in T3, T4, T6 given by our general theory and add them to the secular terms of VSOP82, we reduce considerably the discrepancies VSOP82-DEl02 over the whole range of DEl02, that is 3000 years. In conclusion, our general planetary theory fulfils a double task: A complete exposition of the results is to be found in Laskar (1984b).