The Three-Body problem is a famously intractable aspect of Newtonian mechanics. The demand for highly accurate predictions of one instance of the problem, lunar motion, led to practical approximate solutions of great complexity, constiuted by trigonometric series with hundreds of terms. Such considerations meant there was demand for high speed machine computation from astronomers during the earliest stages of computer development.
One early innovator in this regard was Wallace J. Eckert, a Columbia University professor of astronomer and IBM researcher. When IBM unveiled its first large electronic machine the Selective Sequence Electronic Calculator (SSEC) in 1948 Eckert chose the starting problem as a more accurate calculation of Lunar positions based on the work of his mentor E. W. Brown. Here the speed of electronic computing was used to make practical arithmetic previously to onerous in time and effort. In the 1950s and 60s Eckert would seek to improve the underlying equations for Lunar motion to achieve new levels of accuracy. This involved two distinct efforts involving two different complex analytic solutions to the problem, both involving solutions proposed in the 19th century, that would make extensive use of computer capabilities of the day.
In my paper I will discuss Eckert's work, his choice of techniques in lunar theory, the role of computers and connect his work with other developments in celestial mechanics and mathematics. Eckert's work was the state of the art in his day and formed the nucleus of the trajectories of the Moon NASA used in their lunar missions. However, NASA's lunar missions also demonstrated the limits of Eckert's techniques and new numerical integrations of the lunar motion would be developed to meet the demands for accuracy of space exploration. Eckert had himself used numerical integration in planetary astronomy but opted for more traditional analytic techniques in his lunar work. Despite its status as applied these developments tested mathematical issues such as the convergence of series solutions and appropriate techniques for solving large systems of linear equations.
My research draws on the published record, Eckert's papers and interviews with some of the participants.