A Discussion of Hill's Method of Secular Perturbations and its Application to the Determination of the Zero-Rank Effects in Non-Singular Vectorial Elements of a Planetary Motion

Details A Discussion of Hill's Method of Secular Perturbations and its Application to the Determination of the Zero-Rank Effects in Non-Singular Vectorial Elements of a Planetary Motion A Discussion of Hill's Method of Secular Perturbations and its Application to the Determination of the Zero-Rank Effects in Non-Singular Vectorial Elements of a Planetary Motion

Mar 1970

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1970cemec...2...41m&link_type=abstract

Celestial Mechanics, Volume 2, Issue 1, pp.41-59

Statistics

Computation

4

Scientific paper

Knowledge of the perturbations of zero-rank is essential for the understanding of the behavior of a planetary or cometary orbit over a long interval of time. Recent investigations show that these zero-rank perturbations can cause large oscillations in both the shape and position of the orbit. At present we lack a complete analytical theory of these perturbations that can be applied to cases where either the eccentricity or inclination is large or has large oscillations. For this reason we here develop formulas for the numerical integration of the zero-rank effects, using a modified Hill's theory and suitable vectorial elements. The scalar elements of our theory are the two components of Hamilton's vector in a moving ideal reference frame and the three components of Gibb's rotation vector in an inertial system. The integration step can be taken to be several hundred years in the planetary or cometary case, and a few days in the case of a near-Earth space probe. We re-discuss Hill's method in modern symbolism and by applying the vectorial analysis in a pseudo-euclidean spaceM 3, we obtain a symmetrical computational scheme in terms of traces of dyadics inM 3. The method is inapplicable for two orbits too close together. In Hill's method the numerical difficulty caused by such proximity appears in the form of a small divisor, whereas in Halphen's method it appears as a slow convergence of a hypergeometric series. Thus, in Hill's method the difficulty can be watched more directly than in Halphen's method. The methods of numerical averaging have, at the present time, certain advantages over purely analytical methods. They can treat a large range of eccentricities and orbital inclinations. They can also treat the free ‘secular’ oscillations as well as the forced ones, and together with their mutual cross-effects. At the present time, no analytical theory can do this to the full extent.

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