A semi-numerical expansion of the averaged disturbing function for some very-high-eccentricity orbits

A note on the construction of expansions of the disturbing function valid for any eccentricity and on some applications.     

The convergence domain of the Laplacian expansion of the disturbing function

A computer-assisted reformulation of Sundman's determination of the the domain of absolute convergence of the Laplacian expansion fo the disturbing function is given. Sundman's results are extended to the cases of librating perihelions and a convergence criterion is established for the case of mutually inclined orbits.     

Convergence of Newton's iteration for the expansion of the planetary disturbing function

A method is suggested for choosing the first approximation in Newton's iterations to expand the planetary disturbing function. The method ensures convergence of the process for any planetary orbits. An estimation is given for the number of iterations depending on a given accuracy of calculation.     

On a global expansion of the disturbing function in the planar elliptic restricted three-body problem

Starting with a simple Taylor-based expansion of the inverse of the distance between two bodies, we are able to obtain a series expansion of the disturbing function of the three-body problem (planar elliptic case) which is valid for all points of the phase space outside the immediate vicinity of the collision points. In particular, the expansion is valid for very high values of the eccentricity of the perturbed body. Furthermore, in the case of an interior mean-motion resonant configuration, the above-mentioned expression is easily averaged with respect to the synodic period, yielding once again a global expansion of (R) valid for very high eccentricities.Comparisons between these results and the numerically computed exact function are presented for various resonances and values of the eccentricity. Maximum errors are determined in each case and their origin is established. Lastly, we discuss the applicability of the present expansion to practical problems.     

Satellite orbits perturbed by direct solar radiation pressure: General expansion of the disturbing function

An expression is derived for the solar radiation pressure disturbing function on an Earth satellite orbit which takes into account the variation of the solar radiation flux with distance from the Sun's centre and the absorption of radiation by the satellite. This expression is then expanded in terms of the Keplerian elements of the satellite and solar orbits using Kaula's method. The Kaula inclination functions are replaced by an equivalent set of modified Allan inclination functions.The resulting expression reduces to the form commonly used in solar radiation pressure perturbation studies (e.g. Aksnes, 1976), when certain terms are neglected. If, as happens quite often in practice, a satellite's orbit is in near-resonsnce with certain of these neglected terms, these near-resonant terms can cause changes in the satellite's orbital elements comparable to those produced by the largest term in Aksnes's expression. A new expression for the solar radiation pressure disturbing function expansion is suggested for use in future studies of satellite orbits perturbed by solar radiation pressure.     

Expansion of the planetary disturbing function

Some methods are described for the expansion of the disturbing function in planetary theory. One method uses the classical binomial expansion theorem or a successive approximation process derived from it. Another method is a direct application of the Laplace series expansions. For both methods it is proposed to first prepare the series to be manipulated by a scaling operation. These methods can be applied either in a literal or in a numerical form, or any combination of both, but they are especially designed for use on a large scale digital computer with standard Poisson series programs. No usage is made of Newcomb operators or derivatives of Laplace coefficients.     

A new expansion of planetary disturbing function and applications to interior,co-orbital and exterior resonances with planets

In this study,a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and...     

A note on the tidal expansion of the orbit of Mercury

Under the assumption that the secular decrease in the sidereal mean motion of Mercury is due to the Sun-Mercury tidal torque, the expansion of the orbit of Mercury and the tidal phase-lag angle of the tidal bulge are estimated.     

A numerical investigation of secular terms of the planetary disturbing function

The secular terms of the planetary disturbing function are given, after elimination of short period terms by von Zeipel's transformation. The adequacy of this expansion up to terms of eighth order in the inclination and eccentricity is investigated by numerical processes, as a function of the Keplerian elementsa, e andi. The eccentricityé of the outer planet, is taken equal to zero. It is concluded that for values ofi which are not small the inclusion of additional terms in the expression for the disturbing function, results to drastic changes in its values, while larger values ofe do not have an equaly large effect on the disturbing function.     

Expansion of the first-order planetary disturbing function by Smart's method

We expand the principal part of the planetary disturbing function, by Smart's method, using Taylor's theorem. In our expansion we neglect terms of degree higher than the fourth in the eccentricities and tangents of the inclinations.Now at the JPL Pasadena, California.     

Expansions of the derivatives of the disturbing function in planetary problems

A method is suggested to develop literal expansions of derivatives of the disturbing function especially for the case of large values of the major axis ratio . The series remain convergent as well if =1, unless the eccentricities vanish at the same time. The treatment holds true in the case when usual analytical expansions are not valid, that is if the orbits have points equidistant from the primary. The general case is considered too, the intersecting orbits being included.     

The influence of the great inequality on the secular disturbing function of the planetary system

This paper derives the contributionF ₂ ^* by the great inequality to the secular disturbing function of the principal planets. Andoyer's expansion of the planetary disturbing function and von Zeipel's method of eliminating the periodic terms is employed; thereby, the corrected secular disturbing function for the planetary system is derived. An earlier solution suggested by Hill is based on Leverrier's equations for the variation of elements of Jupiter and Saturn and on the semi-empirical adjustment of the coefficients in the secular disturbing function. Nowadays there are several modern methods of eliminating periodic terms from the Hamiltonian and deriving a purely secular disturbing function. Von Zeipel's method is especially suitable. The conclusion is drawn that the canonicity of the equations for the secular variation of the heliocentric elements can be preserved if there be retained, in the secular disturbing function, terms only of the second and fourth order relative to the eccentricity and inclinations.The Krylov-Bogolubov method is suggested for eliminating periodic terms, if it is desired to include the secular perturbations of the fifth and higher order in the heliocentric elements. The additional part of the secular disturbing functionF ₂ ^* derived in this paper can be included in existing theories of the secular effects of principal planets. A better approach would be to preserve the homogeneity of the theory and rederive all the secular perturbations of principal planets using Andoyer's symbolism, including the part produced by the great inequality.     

Expansion of the first order planetary disturbing function by Smart's method

In this part we expand the indirect part of the planetary perturbing function by Smart's method, via Taylor's theorem. We neglect, in our expansion, terms of degree higher than the fourth with regard to the eccentricities and tangents of the inclinations.     

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula’s expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane’s expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time. We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long-term motion of the highly elliptical and critically-inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.     

An expansion of the Voigt function

The Voigt function and its derivatives are represented by means of series in Hermite polynomials. The equations obtained can be used both for numerical calculations of these functions and for analytical research.Translated from Astrofizika, Vol. 39, No. 4, pp. 619–625, October–December, 1996.     

Expansion of the disturbing function for high eccentricity and large amplitude of libration

Following some ideas, developed by Woltjer (1928), Message (1989), Yokoyama (1988, 1989) and Duriez (1990) an expansion of the disturbing function is given for high values of the eccentricity and large amplitude of libration. The classical expansion can be obtained as a particular case of the present model. Several asteroids with high eccentricity and large amplitude of libration are tested and the results are much better than those obtained from the classical theory.     

Minor planet short periodic perturbations: The indirect part of the disturbing function

Leverrier's development of the indirect part of the disturbing function has been extended to include terms up to degree 4 in eccentricity and inclination; the resulting series has been expressed with respect to a fixed plane, and in a computer readable form (a list of integers). Tests have been performed for the relative significance of the terms of degrees 2, 3 and 4, and estimates have been obtained for the accuracy of the short periodic perturbations of a minor planet, and of the corresponding mean orbital elements. It was found that: (i) even in extreme cases, the indirect part of the disturbing function gives rise to very small short periodic perturbations; (ii) bodies of very high eccentricity/inclination and those close to mean motion resonances are most significantly affected; (iii) indirect perturbations for minor planets can be computed up to the degree 2 terms only, without any significant loss of accuracy; and (iv) higher degree indirect perturbations appear to be important only for their contribution to the long periodic effects of higher order (with respect to the perturbing mass).     

A note on the averaging method

When time-averaged equations are used to discuss the secular behavior of dynamical systems, the action-angle variables conjugate to the action variables of the unperturbed motion of the system should be chosen as dependent variables; otherwise, the results are not correct.Presented at the Division of Dynamical Astronomy Meeting of the American Astronoming Society, Chapel Hill, N.C., U.S.A., December 4, 1976.     

A note on the linear stellar model

For a purely gaseous self-gravitating stellar configuration with linear matter density distribution the total power generated by nuclear reactions is considered. The analytic connection between physical parameters of the macroscopic and microscopic levels of the stellar equilibrium configuration is revealed.     

A note on the Hori-Lie perturbation technique

In this paper we describe how to avoid the introduction of Hori's pseudo time in general planetary theory in the case of two planets.