A Discussion Of Halphen S Method For Secular Perturbations And Its Application To The Determination Of Long Range Effects In The Motions Of Celestial Bodies

The long range effects caused by the moon and the sun are of primary importance in establishing the stability of highly eccentric satellite orbits. At present no complete analytical theory exists which can treat such orbits. It is shown here that Halphen's method of treating secular planetary effects can, by means of step-by-step integration, also be used to determine long range lunar effects in the motions of artificial satellites. Halphen's method permits the numerical integration of long range lunar effects can be treated by averaging the disturbing function over the orbit of the satellite. Halphen's method is applicable to the determination of long range ("secular") effects in the motion of minor planets over the interval of hundreds of thousands of years. We assume that no sharp commensurability between mean motions of the disturbed and disturbing bodies does exist. A complete theory of Halphen's method is presented in modern symbols. Goursat transformations and a summability process are applied to speed the convergence of series which appear in the theory.

This paper discusses some applications of Halphen's method for determining the long range (secular) perturbations for planetary and cometary type orbits. This method of treating secular planetary effects has been suggested by Musen for the determination of the long range perturbations due to the moon for artificial satellite orbits. Two FORTRAN II computer programs incorporating this method are described and representative results are presented. The comparison of results obtained here with those obtained by applying other methods demonstrates the adequacy of this method for minor planets and appropriate artificial satellites. It shows also that considerable saving in computer time can be made in the study of artificial satellite orbits when the short period perturbations are not of interest by using a program based on Halphen's method rather than one based on the use of an unaveraged disturbing function. The program for artificial satellites is given.

This paper discusses some applications of Halphen's method for determining the long range (secular) perturbations for planetary and cometary type orbits. This method of treating secular planetary effects has been suggested by Musen for the determination of the long range perturbations due to the moon for artificial satellite orbits. Two FORTRAN II computer programs incorporating this method are described and representative results are presented. The comparison of results obtained here with those obtained by applying other methods demonstrates the adequacy of this method for minor planets and appropriate artificial satellites. It shows also that considerable saving in computer time can be made in the study of artificial satellite orbits when the short period perturbations are not of interest by using a program based on Halphen's method rather than one based on the use of an unaveraged disturbing function. The program for artificial satellites is given.