First order planetary perturbations with elliptic functions

Details First order planetary perturbations with elliptic functions First order planetary perturbations with elliptic functions

Sep 1987

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1987cemec..40..367w&link_type=abstract

Celestial Mechanics (ISSN 0008-8714), vol. 40, no. 3-4, 1987, p. 367-391. Research supported by the University of South Florida.

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Differential Equations, Elliptic Functions, Orbit Perturbation, Planetary Orbits, Computational Astrophysics, Equations Of Motion, Jacobi Integral, Quadratures, Resonant Frequencies

Scientific paper

The differential equations of planetary theory are solved analytically to first order for the two-dimensional case, using only Jacobian elliptic functions and the elliptic integrals of the first and second kind. This choice of functions leads to several new features potentially of importance for planetary theory. The first of these is that the solutions do not require the expansion of the reciprocal of the distance between two planets, even for those variables which depend on two angular arguments. A second result is that the solution is free from small divisors with the exception of two special resonances. In fact, not only are the solutions for resonant orbits free from small divisors, the perturbations for all variablles are expressible in closed form. A subset of the resonant orbits maintains this form and in addition has the remarkable feature that the first order perturbations are purely periodic; they contain no secular terms. A solution for the 1:3 resonance case is given as an example.

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