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 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.     

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.     

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.     

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.     

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 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.     

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.     

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.     

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.     

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.     

Convergence of the expansions of the disturbing functions in the planar three-body planetary problem

In the framework of the planar three-body planetary problem, conditions are found for the absolute convergence of the expansions of the disturbing functions in powers of the eccentricities, with coefficients represented by trigonometric polynomials with respect to the mean, eccentric, or true anomaly of the inner planet. It is found that using the eccentric or true anomaly as the independent variable instead of the mean anomaly (or time) extends the holomorphy domain of the principal part of the perturbation functions. The expansions of the second parts converge in open bicircles, which admit values of the eccentricity of the inner planet in excess of the Laplace limit.     

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.     

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...     

The influence of stellar evolution on planetary nebula formation

A discussion of how the fact that the central star evolves while the planetary nebula is forming affects the formation of the nebula. Numerical models point to a number of effects, such as more efficient shaping in the early phases and the formation of a surrounding shell. It is also found that for identical initial density distributions, quickly evolving (more massive) stars will form bipolar PNs and more slowly evolving stars will form elliptical PNs.Physics Dept. UMIST Manchester UK     

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.