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Extra info for Advances in Computers, Vol. 3
Normally b(t) will have to be obtained numerically since the solution y(t) will not be known analytically. 6) leads t o circular motion for which an explicit solution is known. The solution is given by y ( t ) = (cos t, -sin 2, sin t, cos t ) and the vector b(t) by (sin t, cos t , -cos t , sin t). 7) is then h(t) = (t sin t , t cos t, - t cos t, t sin t). The accumulated truncation error e(t) a t any time t is thus seen to oscillate with a period of 27r and t o grow linearly with time. Its numerical value can be obtained from the formula e(t) = -ch5h(t), where h is the iiitegration step size.
18) was obtained with extremely high precision. On the other hand experiments by Thomas using a machine with a 12-digit word length and based on Cowell’s method led to optimistic conclusions. Because of the possible application of this approach to the satellite lifrtimc problem, further investigation of its capabilities appear warranted. 4. General Perturbation Methods The large majority of vehicles launched over the past three years have been placed into orbits within 1000 miles of the earth. The major forces SATELLITE ORBIT TRAJECTORIES 29 affecting the motion of such satellites are drag and the oblateness of the earth.
For the perturbed orbit, the results provided by the standard are correct to a t least seven significant figures, an accuracy more than adequate for this study. Single precision, floating point programs for the Cowell, Herrick, and Encke methods were then run on an IBM 7090 and compared with the double precision standard. Great care was used to insure that the physical constants and the initial conditions were identical in all programs. The following table gives the method of integration used, the local truncation error criterion, the number of integration steps required, the computing SATELLITE ORBIT TRAJECTORIES 43 TABLE11.