Modified equinoctial elements are introduced which are suitable for perturbation analysis of all kinds of orbit. Equations of motion in Lagrangian and Gaussian forms are derived. Identities connecting the partial derivatives of the disturbing function with respect to equinoctial elements are established. Numerical comparisons of the evolution of a perturbed, highly eccentric, elliptic orbit analysed in equinoctial elements and by Cowell's method show satisfactory agreement.

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A Set of Modified Equinoctial Orbit Elements

The results of a simultaneous solution for the orbital elements of Moon and planets are given and their derivation is discussed. A modern Cowell integrator is used for orbit computations, and least-squares fits are made to some 40000 optical observations taken since 1913. The model includes relativistic terms, the leading zonal harmonics of Earth and Moon, the precession of the lunar equator, and the tidal couple between Earth and Moon. The tidal term in the Moon's mean longitude is found to be −19″±4″ per century squared. The solution also yields an extrapolation of the atomic time scale back to 1912.5. At that time, the difference between atomic and ephemeris time is about 6±2 s. Lunar declinations observed by the Washington transit circles, after receiving limb corrections and thus with respect to the center of Watts' reference sphere, are smaller than computed values by 0″.33±0″.01. It is found that solar oblateness cannot quite be determined with optical data covering about 50 yr, butJ2 is unlikely to be much larger than 10−5. The advance of Mercury's perihelion is verified to within our resolution of 2″ per century to match that predicted by Einstein.

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New orbital elements for Moon and planets

A SURVEY of orbit element sets was conducted as part of a research and technology development task [1] which sought to develop feedback control laws and prototype tools for computing feasible, low-thrust transfers in multibody environments, where at least two celestial bodies exert significant gravitational influence on the spacecraft. One approach considered in this research effort was to use two-body element sets and their associated variational equations with the multibody effects modeled as perturbations. The goal of the survey was to identify an orbit element set that is the most natural to use in developing feedback control laws. This survey identified 22 candidate orbit element sets plus variations defined in terms of Euler angles, Euler parameters, functions of classical elements, quaternions, set-III elements, fast or slow variables, or canonical variables. The list of candidates is provided in this paper, together with a brief description of their attributes.

Survey of Orbit Element Sets

Non-singular orbital elements for special perturbations in the two-body problem

Using the rectangular equations of motion for the restricted three-body problem a comparison is made of the Encke and Cowell methods of integration. Each set of differential equations is integrated using Taylor series expansions where the coefficients of the powers of time are determined by recurrence relationships. It is shown that for fairly highly eccentric orbits in which the perturbing force is less than one thousandth of the two-body force the Encke method achieves a considerable saving in machine time. This is also true for almost circular orbits when low or moderate accuracy is required. When very high accuracy is required, however, the Cowell method is faster unless the perturbing force is less than 10−6 of the two-body force. There is little difference in the accuracy of the two methods, the Cowell method being slightly more accurate when a low or moderate accuracy criterion is imposed.

C. J. Cohen

Papers

A nonsingular set of orbit elements