A study of the planetary secular perturbations

We compare the classical method and Gauss' method for deriving secular inequalities and find the latter to be more accurate, especially in cases where the orbital eccentricities and inclination are moderate or large. Based on Gauss' method, we derive some practical algorithms and then Investigate numerically the orbital evolution of the nine major planets over the last 2,100,000 years or so, taking into account all their secular perturbations. The main results are shown in TABLE 3, alongside with Stockwell's results for easy comparison. Detailed variations of the elements are plotted in Figs. 1 and 2.     

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.     

Expansion of the planetary Hamiltonian function

We expand the planetary Hamiltonian function with its two parts, the principal and the indirect, up to the seventh order in the planetary masses. We adopt the Jacobi-Radau system of origins. The expansiion is valid for any number of planets.     

Expansion of the Hamiltonian for the planetary problem into a Poisson series in the heliocentric reference frame

An expansion of the Hamiltonian for the N-planet problem into a Poisson series using a system of modified (complex) Poincare´ canonical elements in the heliocentric coordinate system is constructed. The Lagrangian and Hamiltonian formalisms are used. The first terms in the expansions of the principal and complementary parts of the disturbing function are presented. Estimates of the number of terms in the presented expansions have been obtained through numerical experiments. A comparison with the results of other authors is made.     

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.     

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.     

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.     

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.     

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 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.     

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.     

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.     

Stability of the planetary three-body problem

We present a direct method for the expansion of the planetary Hamiltonian in Poincaré canonical elliptic variables with its effective implementation in computer algebra. This method allows us to demonstrate the existence of simplifications occurring in the analytical expression of the Hamiltonian coefficients. All the coefficients depending on the ratio of the semi major axis can thus be expressed in a concise and canonical form.     

Exoplanetary systems: The role of an equilibrium at high mutual inclination in shaping the global behavior of the 3-D secular planetary three-body problem

On the basis of a high-order (order 12) expansion of the perturbative potential in powers of the eccentricities and the inclinations, we analyze the secular interactions of two non-coplanar planets which are not in mean-motion resonance. The model is based on the planetary three-body problem which can be reduced to two degrees of freedom by the well-known elimination of the nodes [Jacobi, C.G.J., 1842. Astron. Nachr. XX, 81-102]. We introduce non-singular canonical variables which bring forward the symmetries of the problem. The main dynamical features depend on the location and stability of the equilibria which are easily found with our analytical model. We find that there exists an equilibrium when both eccentricities are zero. When the mutual inclination is small, this equilibrium is stable, but for larger mutual inclination it becomes unstable, generating a large chaotic zone and, by bifurcation, two regular regions, the so-called Kozai resonances. This analytical study which depends on only two parameters (the ratio of the semi-major axes and the mass ratio of the planets) makes possible a large survey of the problem and enables us to identify and quantify its main dynamical features, periodic orbits, regular and chaotic zones, etc. The results of our analytical model are illustrated and confirmed by numerical integrations.     

Sirtf: Capabilities for the study of planetary systems

The Space Infrared Telescope Facility, to be launched into a near-Earth heliocentric orbit in the year 2001, will open broad new vistas for the study, at infrared wavelengths, of the objects in the Solar System and planetary systems around other stars. This paper focuses on the study of Kuiper-belt comets and circumstellar planetary debris disks.Paper presented at the Conference onPlanetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A.     

On the stability of circumbinary planetary systems

The dynamics of circumbinary planetary systems (the systems in which the planets orbit a central binary) with a small binary mass ratio discovered to date is considered. The domains of chaotic motion have been revealed in the “pericentric distance–eccentricity” plane of initial conditions for the planetary orbits through numerical experiments. Based on an analytical criterion for the chaoticity of planetary orbits in binary star systems, we have constructed theoretical curves that describe the global boundary of the chaotic zone around the central binary for each of the systems. In addition, based on Mardling’s theory describing the separate resonance “teeth” (corresponding to integer resonances between the orbital periods of a planet and the binary), we have constructed the local boundaries of chaos. Both theoretical models are shown to describe adequately the boundaries of chaos on the numerically constructed stability diagrams, suggesting that these theories are efficient in providing analytical criteria for the chaoticity of planetary orbits.