TIME-STEP FOR COMPUTATION OF ORBITS BY NUMERICAL INTEGRATION 181 On the Time-Step to be Used for the Computation of Orbits by Numerical Integration

W. J. Eckert [1] has adapted Cowell's method of numerical integration to the determination of orbits on punched card machines. Eckert, Brouwer, and Clémence [2] have used Cowell's method on a large-scale computer. We shall assume that Cowell's method as modified by Eckert is the method of numerical integration best suited for the determination of orbits on large-scale computers. What we are concerned with in this paper is how to make use of this method in such a way that we do the least work (have the fewest arithmetic operations) per unit of advance in the time. Our results apply also to the computation of orbits on ordinary desk computers, but additional factors, such as the work involved in transcribing, must be considered. In Cowell's method as modified by Eckert the attractions and the central differences of the attractions which are required at the "new time-step" are obtained from the values at previous time-steps by an extrapolation process. This extrapolation process may be shown to be equivalent to extrapolating the attractions ahead in the time using Newton's backward difference interpolation formula, so that Cowell's integration formula may be written in terms of backward differences instead of central differences. Let xn\ i = 1, 2, 3, be rectangular coordinates and let X„' be the attractions at time / = nAt. Then we obtain for x'B+i in terms of backward differences,