1. An Iterative General Planetary Theory P. K. Seidelmann University of Virginia

2. Introduction Retrospective Experience from 1965-8 Computer Limitations have changed Experience of Others Numerical Integrations Some Challenges are the same Possibilities if done 45 years later

3. The Approach in 1965 Hansen’s Method Used by Hill and Clemence Iterative instead of by Taylor series expansion Computer Programs for Method Test against Hill & Clemence Controlled Minor Planet Integration as a Test

4. Regular Hansen’s Method Perturbations to Mean Anomaly Avoids small differences of large numbers W to determine all perturbations Osculating plane of the orbit Coordinates n 0  z,, and u Method dependent on order

5. Iterative Hansen’s Method Rigorous Equations used always Present theories could be used as input Fourier series manipulation subroutines multiplication, addition, integration, differentiation, Hanson’s bar operation, constants of integration Iterate into Improved Theories

6. Computer Challenges IBM 360 40 Computer Speed Storage Limitations Over night computer runs Must be interruptible for other computer users

7. Computer Implementation Terms contain: number, sine & cosine coefficients (double precision), power of t,11 integers for angles, integer for  8 words, 32 bytes or 256 bits Subroutines in Assembler language Accuracy testing for minimum values carried Testing based on mean motions present Variable accuracy testing through process A large amount of computing is necessary

8. Results Use of mean orbital plane added complicated terms, so osculating plane used Undisturbed mean anomaly better than disturbed a/Δ determined using Brown correction for each pass Commensurabilities require many terms in all series Convergence problems with commensurability terms which diverged Tolerances are difficult to set, secular terms need more accuracy Storage protection requires setting limits on number of terms

9. Comparison of general theory and numerical integration Disappointing Minor planet was highly perturbed Terms did not converge after 3 iterations Commensurable terms affected other terms Iterative method agreed with Clemence Mars, but perturbations were small Clemence theory did not fit the numerical integration well Newcomb encountered divergence of a term in his theory of Mars

10. Successes Matches with 1 st order Clemence’s Mars Theory Matches with 1 st order Hill’s Jupiter and Saturn Theories Some matches of 2 nd Iteration with 2 nd orders

11. Problems Expansions become very large Setting accuracy limits on individual terms Mixed secular terms Binomial expansions Lack of convergence after 3 rd iteration Long period terms contributing

12. Conclusions Iterative method results solely based on final pass Old theories can be used Only one method to be programmed Commensurabilities are a real problem With current computer speed and storage, can the problems be over come?

13. What is the Truth Observations Numerical Integrations Clemence and Hill Theories?

14. Experience in last 45 years Bretagnon Laskar Numerical Integrations Radar and improved observations

15. If Done Today An iterative Hansen Method? No computer speed and storage problems? Computer software improvements help? Remaining challenges Commensurabilities Binomial expansions Setting accuracy limits Convergence