Analytical Partial Derivatives for the Moon's Mean Elements.

For the purpose of differential correction of the moon s mean orbital elements, partial derivatives of the moon s coordinates with respect to these elements are needed. Ordinary methods for forming these partials depend upon the assumption of Keplerian motion, which is too poor an approximation to be useful for the moon. Other numerical methods, such as integration of variational equations, require a great deal of computer time for the moon because of the small step size needed. Brown's theory of the motion of the moon is a literal, as well as a numerical, theory. Hence it is possible to analytically differentiate the expressions for the moon s coordinates to obtain the partial derivatives. This leads to compact expressions, generally only a few terms, which represent the partial derivatives to an accuracy of three significant figures adequate for the moon. The expressions treated in this paper are those for the moon's longitude N, latitude p, sine parallax sinir, range r, and range rate r. The expressions are for a geocentrTh moon. Topocentric corrections are small (but not negligible) for all of these except range rate, where they are very large. These will be developed in a later discussion. The six orbital elements selected to represent the moon's orbit are a, e, i, co, ~, L (semimajor axis, eccentricity, inclination, longitude of perigee, longitude of ascending node, and mean longitude). A table contains terms of the form C~~m~ (il +jl'+kF+rD), where C is the numeric coefficient, and 1,1', F, D are the fundamental arguments of the lunar theory. Only terms which can influence the third significant figure of the total derivative are retained. Expressions for the derivatives of the time rates of change of N, p, and sinir have also been developed to facilitate the conversion to osculating elements, if that is desired (e.g.,in a numerical integration). These are not presented here. Note that variations in semimajor axis, ~, are treated as scaling factors only. No corresponding variation in mean motion is allowed.