The main problem of lunar theory solved by the method of Brown

Abstract

Brown's method for solving the main problem of lunar theory has been adapted for the computation by machine. The computations are carried out with the help of an algebraic processor called POLYPAK, which can manipulate power series in several real or complex variables. Brown's result have been recovered and refined first and the solution, in Cartesian coordinates to the sixth order, has been compared to the work of Eckert (see Gutzwiller, 1979). The solution has then been expanded to include most terms through order nine. This order is necessary to get an accuracy of 0.00001″ for the terms in longitude and latitude and of 0.000001″ for the terms in the sine parallax. A preliminary comparison with the theories of Chapront and Henrard (1980) indicates that the solution has an accuracy which is close to the one desired. For details see Schmidt (1980).

The next step in developing a complete lunar theory requires the computation of the partial derivatives of the solution with respect to the primary parameters. Since Brown's method gives a semianalytical solution only the derivative with respect ton, the mean motion of the Moon, is difficult to compute. It is possible to find this derivative with one quadrature from the other derivatives if one takes into account that the Jacobian has to be a symplectic matrix when a canonical set of primary parameters is used.

The mean motions of the perigee and node often exhibit the largest discrepancies among the different theories. Therefore it is not too surprising that also their derivatives show significant differences. It is hoped by providing another independent computation of the derivatives by a different method their accuracy can be improved.