Determination of secular terms in general planetary theory

A method to calculate secular terms of the two parts of the planetary disturbing function— when it is expressed in terms of the true anomalies or the eccentric anomalies instead of the mean anomalies - is described. Also an alternative method is outlined.     

Critical inclinations in planetary problems

The general conception of the critical inclinations and eccentricities for theN-planet problem is introduced. The connection of this conception with the existence and stability of particular solutions is established. In the restricted circular problem of three bodies the existence of the critical inclinations is proved for any values of the ratio of semiaxes . The asymptotic behaviour of the critical inclinations as 1 is investigated.

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A determination of secular terms in first-order planetary theory

The secular terms of the first-order planetary Hamiltonian is determined, by two methods, in terms of the variables of H. Poincaré, neglecting powers higher than the second in the eccentricity-inclination.     

A secular theory of coplanar, non-resonant planetary system

We present the secular theory of coplanar N -planet system, in the absence of mean motion resonances between the planets. This theory relies on the averaging of a perturbation to the two-body problem over the mean longitudes. We expand the perturbing Hamiltonian in Taylor series with respect to the ratios of semimajor axes which are considered as small parameters, without direct restrictions on the eccentricities. Next, we average out the resulting series term by term. This is possible thanks to a particular but in fact quite elementary choice of the integration variables. It makes it possible to avoid Fourier expansions of the perturbing Hamiltonian. We derive high-order expansions of the averaged secular Hamiltonian (here, up to the order of 24) with respect to the semimajor axes ratio. The resulting secular theory is a generalization of the octupole theory. The analytical results are compared with the results of numerical (i.e. practically exact) averaging. We estimate the convergence radius of the derived expansions, and we propose a further improvement of the algorithm. As a particular application of the method, we consider the secular dynamics of three-planet coplanar system. We focus on stationary solutions in the HD 37124 planetary system.     

Second order secular planetary theory for the four major planets

We generalize our results of a second order Jupiter-Saturn planetary theory to be applicable for the case of the four major planets.We use the Von Zeipel method and we neglect powers higher than the third with respect to the eccentricities and sines of the inclinations in our expansions. We consider the critical terms as the only periodic terms.     

Holomorphy of coordinates with respect to eccentricities in the planetary three-body problem

The boundaries of the domains of holomorphy of the coordinates of unperturbed elliptic motion with respect to the eccentricities of planetary orbits are determined for the cases when any of the five anomalies of one of the planets-eccentric, true, tangential, or one of two mutual anomalies suggested by M.F. Subbotin—is used as an independent variable. The resulting equations are a generalization of the known equations for the boundaries of the domains of the holomorphy of coordinates for the cases when the time is the independent variable and determine the bisymmetric ovals, whose size and shape depend on the eccentricities and on the ratio of the planetary mean motions. The largest domains of holomorphy are obtained when the tangential anomaly or one of the Subbotin mutual anomalies is used. A function was found that conformally maps the domain of holomorphy to the unit disk. It was demonstrated that the application of any anomaly of the outer planet as the independent variable can result in a significant shrinking of the domain of the holomorphy of the coordinates of the inner planet, so that the analytic continuation of the initial power series with the center at the origin of the coordinates of a complex plane becomes impossible.     

Implementation of an explicit numerical method to the study of stability of asteroids with large eccentricities and inclinations

In this paper, we consider the secular variations in the restricted three bodies problem by implementing an explicit numerical technique for studying stability of equilibrium solutions. In the present case, the three bodies are the Sun, the Jupiter and an asteroid. Our results arise from studying some maximum stable orbital elements corresponding to the stable equilibrium solutions of the particular problem.     

Solar nebula dispersal and the stability of the planetary system: I. Scanning secular resonance theory

Secular resonances in the early solar system are studied in an effort to establish constraints on the time scale and/or method of solar nebula dispersal. Simplified nebula models and dispersal routines are employed to approximate changes in an assumed axisymmetric nebula potential. These changes, in turn, drive an evolutionary sequence of Laplace-Lagrange solutions for the secular variations of the solar system. A general feature of these sequences is a sweep of one or more giant planet resonances through the inner solar system. Their effect is rate dependent; in the linearized models considered, characteristic dispersal times ≤O(10^4?5 years) are required to avoid the generation of terrestrial eccentricities and inclinations in excess of observed values. These times are short compared to typical estimates of the accretion time scales [i.e., ～O(10^7?9 years)] and may provide an important boundary condition for developing models of nebula dispersal and solar system formation in general.     

A way to generate a model of secular perturbations for planetary orbits

The reciprocal distance between two material points that rotate around a central body in nonintersecting orbits is expanded and the results are presented. The expansion is obtained accurate to the tenth order with respect to small parameters: the eccentricities and sine of the orbital inclination angle. The result is the basis of the averaging operation of the perturbation function in the system of eight major planets in the solar system, and of the numerical integration of the averaged equations of motion. The averaged Hamiltonian contains the terms whose period of variation is greater than 200 years. Forty eight equation of first order are numerically integrated with increments of 100 years for two intervals from the beginning of the Christian era: 25 million years forward and 25 million years backward over time. To present the results of calculation, the website (URL: http://vadimchazov.narod.ru/secequat.htm) was developed, where the initial codes, executable program modules, the results of calculations presented in graphical form, text files with initial conditions, tables for expanding the reciprocal distance between two material points, and the tables with the results of expansion of the perturbation function for eight major planets of the solar system are presented.     

First-order theory of satellite attitude motion application to HIPPARCOS

This work aims at finding an analytic solution corresponding to the attitude evolution in space of a satellite submitted to disturbing torques. This paper presents a basic frame applicable to any perturbed rotation satellite, and a method of resolution leading to a formal solution which is given here to the first order. Thus, the main problem is the slow rotation of a body with three unequal axes of inertia, essentially submitted to a dominant solar radiation pressure torque, with the axis pointing far away from a position of equilibrium. The comparison of the results with a numerical integration based upon a HIPPARCOS model is convincing.     

Progress in general planetary theory

Celestial Mechanics and Dynamical Astronomy - When on searches for a planetary theory valid over 1 million years, one can leave in the solution the short period terms whose amplitude are small, and...     

Onset of secular chaos in planetary systems: period doubling and strange attractors

As a result of resonance overlap, planetary systems can exhibit chaotic motion. Planetary chaos has been studied extensively in the Hamiltonian framework, however, the presence of chaotic motion in systems where dissipative effects are important, has not been thoroughly investigated. Here, we study the onset of stochastic motion in presence of dissipation, in the context of classical perturbation theory, and show that planetary systems approach chaos via a period-doubling route as dissipation is gradually reduced. Furthermore, we demonstrate that chaotic strange attractors can exist in mildly damped systems. The results presented here are of interest for understanding the early dynamical evolution of chaotic planetary systems, as they may have transitioned to chaos from a quasi-periodic state, dominated by dissipative interactions with the birth nebula.     

New approach to determining planetary perturbations in lunar theory

The technique of the general planetary theory has been proposed for constructing a theory of motion of the Moon. This method enables us to elaborate the consistent theory of motion of the principal planets and the Moon, which is of particular importance for determining planetary perturbations in lunar motion. As an initial approximation for lunar motion, an intermediate orbit generalizing the Hill's variational curve has been built. This orbit includes all solar and planetary inequalities independent of eccentricities and inclinations of the Moon, Sun and planets. In calculating this orbit, the motion of the principal planets in quasi-periodic intermediate orbits has been taken into account. This solution was produced with the aid of the Universal Poissonian Processor (UPP) elaborated in the Institute for Theoretical Astronomy (Leningrad).Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

Synthetic secular theories of the planetary orbits: Regular and chaotic behavior

Modern computer technology allows dynamical astronomers to investigate the long term stability of real systems as thoroughly as ever. However, the process is not straightforward and new problems need to be solved. This work deals with only one such problem: the construction-from the numerical integration- of a secular perturbation theory that is able to describe the dynamical behavior of the system. The discussion refers to the outer planets and is based on the knowledge acquired by the author during her participation in project LONGSTOP. A digital filter is used in order to reduce the output and eliminate short periodic terms. Filtering uncovers long term variations in the semimajor axes. From the filtered output a secular perturbation theory is constructed in the assumption that the solution is regular, as secular perturbation theories can only be constructed for regular solutions. If we succeed, this means that the solution is indeed regular for the computed span of time; if not-and this can be established in a rigorous way-it has to be concluded a posteriori that the solution is not regular. The LONGSTOP 1A and 1B integrations show well that as the timespan of the integration increases it is possible to detect the non-regular behavior of the solution. This happens in the eccentricity of Saturn at the 10^–4 level.     

Secular perturbations in general planetary theory

Based on a general planetary theory, the secular perturbations in the motion of the eight major planets (excluding Pluto) have been derived in polynomial form. The results are presented in the tables. The linear terms of second order with respect to the planetary masses and the nonlinear terms of first order up to the fifth (and partly seventh) degree with respect to eccentricities and inclinations were taken into account in the right-hand members of the secular system. Calculations were carried out by computer with the use of a system that performed analytic operations on power series with complex coefficients.

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General planetary theory in elliptic functions

It is shown that the first-order general planetary theory, i.e. the theory without secular terms, developed in (Brumberg and Chapront, 1973) may be re-constructed and presented by the series in powers of the eccentricity and inclination variables with the closed form coefficients expressed in terms of elliptic functions. The intermediate solution of the zero degree in eccentricities and inclinations has been given explicitly with the aid of elliptic functions and the Hansen type quadratures with trigonometric function kernels. In determining the first and higher degree terms in eccentricities and inclinations one meets the Hansen type quadratures with elliptic function kernels. The secular evolution is described by the autonomous polynomial differential system.     

The main problem of artificial satellite theory for small and moderate eccentricities

Perturbation techniques based on Lie transforms as suggested by Deprit were used as the theoretical foundation for programming the analytical solution of the Main Problem in Satellite Theory (all gravitational harmonics being zero exceptJ ₂). The collection of formulas necessary and sufficient to construct an ephemeris is given in the exposition. Short and long period displacements, as well as the secular terms, have been obtained up to the third order inJ ₂ as power series of the eccentricity. They result from two successive completely canonical transformations which it has been found convenient not to compose into a unique transformation. Division by the eccentricity appears nowhere in the developments-neither explicitly nor implicitly. The determination of the constants of motion from the initial conditions has been given an elementary solution that is both complete and explicit without being iterative. The program was developed by Rom from MAO's package of subroutines forMechanizedAlgebraicOperations. Reliability tests have been run in two instances: in-track errors for ANNA 1B are only 20 cm after 210 days in orbit, while for RELAY II, they are 2.4 m, even after 350 days in orbit.     

Rotation of asteroids and planetary axial rotation theory

A theory (Barricelli, 1972) developed for the interpretation of planetary axial rotations is here applied to an interpretation of the axial rotations of major asteroids. The interpretation is based on the assumption that also asteroids can have satellite-systems, which may influence the axial rotation of the respective primaries. The reason why smaller asteroids tend to have slower axial rotation than the major ones as an average is discussed. Predictions of the theory can be tested by space-craft exploration of asteroids.