On the expansion of the secular part of the perturbing function of mutual attraction in the satellite systems of planets

We propose a special representation for the secular part of the perturbing function describing the mutual attraction of satellites. In contrast to the known representations, it has a single analytical form for any ratio between the semimajor axes of the perturbed and perturbing satellites. The resulting expression is a partial sum of a power series with respect to the small eccentricities and planet-equatorial inclinations of the satellites’ orbits. This sum includes terms up to and including the fourth degree with respect to these small parameters. The proposed expansion is compared with one of the known expansions for the secular part of the perturbing function.     

Equilibria in the secular, non-co-planar two-planet problem

We investigate the secular dynamics of a planetary system composed of the parent star and two massive planets in mutually inclined orbits. The dynamics are investigated in wide ranges of semimajor axes ratios (0.1–0.667) and planetary masses ratios (0.25–2), as well as in the whole permitted ranges of the energy and total angular momentum. The secular model is constructed by semi-analytic averaging of the three-body system. We focus on equilibria of the secular Hamiltonian (periodic solutions of the full system) and we analyze their stability. We attempt to classify families of these solutions in terms of the angular momentum integral. We identified new equilibria, yet unknown in the literature. Our results are general and may be applied to a wide class of three-body systems, including configurations with a star and brown dwarfs and substellar objects. We also describe some technical aspects of the seminumerical averaging. The HD 12661 planetary system is investigated as an example configuration.     

A method for averaging the perturbing function in Hill’s problem

A modified method for averaging the perturbing function in Hill’s problem is suggested. The averaging is performed in the revolution period of the satellite over the mean anomaly of its motion with a full allowance for a variation in the position of the perturbing body. At its fixed position, the semimajor axis of the satellite orbit during the revolution of the satellite is constant in view of the evolution equations, while the remaining orbital elements undergo secular and long-period perturbations. Therefore, when the motion of the perturbing body is taken into account, the semimajor axis of the satellite orbit undergoes the strongest perturbations. The suggested approach generalizes the averaging method in which only the linear (in time) term is included in the perturbing function. This method requires no expansion in powers of time. The described method is illustrated by calculating the perturbations of the semimajor axes for two distant satellites of Saturn, S/2000 S 1 and S/2000 S5. An approximate analytic solution is compared with the results of numerical integration of the averaged system of equations of motion for these satellites.     

On the habitability of the OGLE-2006-BLG-109L planetary system

We investigate the dynamics of putative Earth-mass planets in the habitable zone (HZ) of the extrasolar planetary system OGLE-2006-BLG-109L, a close analogue of the Solar system. Our work is inspired by the work of Malhotra & Minton. Using the linear Laplace–Lagrange theory, they identified a strong secular resonance that may excite large eccentricity of orbits in the HZ. However, due to uncertain or unconstrained orbital parameters, the subsystem of Jupiters may be found in a dynamically active region of the phase space spanned by low-order mean-motion resonances. To generalize this secular model, we construct a semi-analytical averaging method in terms of the restricted problem. The secular orbits of large planets are approximated by numerically averaged osculating elements. They are used to calculate the mean orbits of terrestrial planets by means of a high-order analytic secular theory developed in our previous works. We found regions in the parameter space of the problem in which stable, quasi-circular orbits in the HZ are permitted. The excitation of eccentricity in the HZ strongly depends on the apsidal angle of jovian orbits. For some combinations of that angle, eccentricities and semimajor axes consistent with the observations, a terrestrial planet may survive in low eccentric orbits. We also study the effect of post-Newtonian gravity correction on the innermost secular resonance.     

Computing secular motion under slowly rotating quadratic perturbation

We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described.     

General presentation of secular part of the perturbing function of mutual attraction in the satellite system of a planet

A special presentation of secular part of the perturbing function of mutual attraction in the satellite system is presented. In contrast to known ones, it is given in general analytical form for any ratio between semimajor axes for perturbed and perturbing satellites.     

A semi-analytical model of the Galilean satellites’ dynamics

The Galilean satellites’ dynamics has been studied extensively during the last century. In the past it was common to use analytical expansions in order to get simple models to integrate, but with the new generation of computers it became prevalent the numerical integration of very sophisticated and almost complete equations of motion. In this article we aim to describe the resonant and secular motion of the Galilean satellites through a Hamiltonian, depending on the slow angles only, obtained with an analytical expansion of the perturbing functions and an averaging operation. In order to have a model as near as possible to the actual dynamics, we added perturbations and we considered terms that in similar studies of the past were neglected, such as the terms involving the inclinations and the Sun’s perturbation. Moreover, we added the tidal dissipation into the equations, in order to investigate how well the model captures the evolution of the system.     

The construction of a fifth-order Jupiter-Saturn motion: Analytical theory

We construct a fifth-order with respect to masses Jupiter-Saturn secular theory by Hori-Lie canonical technique. The J-S Hamiltonian includes both parts of the perturbing function. The influence of the 2:5 critical terms is taken into consideration. The Jacobi-Radau system of origins is adopted and the theory is expressed in terms of the Poincaré canonical variables.     

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.     

Asteroid long-periodic perturbations: The second order Hamiltonian

The Hamiltonian of the second order with respect to the disturbing mass, as defined in the higher order-higher degree theory of asteroid secular perturbations by Yuasa (1973), is expressed in the heliocentric, ecliptic coordinate system. Errors found in the original paper with terms coming from the principal part of the disturbing function are removed, and corrected values of the coefficients are computed. The importance of second-order perturbations and the improvement in the accuracy of proper element determination, achieved by using the newly-obtained coefficients, are demonstrated. Finally, a table of the secular frequencies as functions of the semimajor axis is given, and compared with the analogous one by Kozai (1979).     

Global dynamics of the Gliese 876 planetary system

The Gliese 876 planetary system consists of two Jupiter-like planets having a nearly commensurate 2:1 orbital periods ratio. Because the semimajor axes of the planets are very small (of the order 0.1 au and 0.2 au, respectively), and the eccentricity of the inner companion is ≃0.3, the mutual perturbations are extremely large. However, many authors claim the long-term orbital stability of the system, at least over 500 Myr for initial conditions found by Rivera & Lissauer. Results of investigations of a migration of initially separated planets into the close 2:1 mean motion resonance lock from Lee & Peale also support the conclusion that the system should be stable for the lifetime of the parent star. Initial conditions of the system, found from non-linear N -body fits by Laughlin & Chambers and Rivera & Lissauer, to the radial velocity curve, formally allow for a variety of orbital configurations of the GJ 876 system, e.g. coplanar, with planetary inclinations in the range [≃30°, 90°], and with relative inclinations of orbital planes as high as 80°. Our work is devoted to the stability investigation of the systems originating from the fitted initial conditions. We study neighbourhoods of these initial states in the orbital parameter space. We found estimations of the 2:1 mean motion resonance width and dynamical limitations on the planetary masses. We also obtain a global representation of the domains of the orbital parameters space in which initial conditions leading to stable evolutions can be found. Our results can be useful in localization of the best, stable fits to the observational data. In our investigations we use the MEGNO technique (the Mean Exponential Growth factor of Nearby Orbits) invented by Cincotta & Simó. It allows us to distinguish efficiently and precisely between chaotic and regular behaviour of a planetary system.     

The Dynamical Evolution of the Atens

In this investigation the orbits of 21 Atens (semimajor axes smaller than the Earth) are studied with the aid of numerical integrations over the time interval of one million years. The dynamical model was a 6-body Solar System, where the perturbations of Uranus and Neptune were ignored, and where Mercury's mass was added to the Sun's mass. Thus mean motion resonances, secular resonances and the Kozai resonance were fully taken into account. The evolution of the semimajor axes shows the typical step function like pattern which we know also from comets although some Atens have a very fuzzy development of the orbital elements, and some of them stay in a mean motion resonance for very long time. The evolution from Atens to Apollos (with semimajor axes larger than the Earth) and vice versa is also a phenomenon which we could observe. The main goal was the study of encounters of the Atens with the Earth and Venus. We found out that Venus encounters occur somewhat more often than Earth encounters (approximately one within the distance Earth - Moon every 40000 years with Venus, one every 50000 years with the Earth). This revised version was published online in July 2006 with corrections to the Cover Date.     

The stabilising mechanism of the HD 82943 planetary system

We carry out simulations to investigate the dynamics of the HD 82943 planetary system with two resonant Jupiter-like planets, and to reveal possible stabilizing mechanism for the system. By following different coplanar configurations in the neighborhood of the best-fit orbits, we find that all the stable cases are involved in the 2:1 mean motion resonance and that the alignment of the periastra of the two planets also plays important part in the secular orbital evolution, indicating that these two kinds of mechanisms could be responsible for the dynamics of the system under study.     

Secular evolution of the orbits of hypothetical satellites of Uranus

The problem of the secular perturbations of the orbit of a test satellite with a negligible mass caused by the joint influence of the oblateness of the central planet and the attraction by its most massive (or main) satellites and the Sun is considered. In contrast to the previous studies of this problem, an analytical expression for the full averaged perturbing function has been derived for an arbitrary orbital inclination of the test satellite. A numerical method has been used to solve the evolution system at arbitrary values of the constant parameters and initial elements. The behavior of some set of orbits in the region of an approximately equal influence of the perturbing factors under consideration has been studied for the satellite system of Uranus on time scales of the order of tens of thousands of years. The key role of the Lidov–Kozai effect for a qualitative explanation of the absence of small bodies in nearly circular equatorial orbits with semimajor axes exceeding ~1.8 million km has been revealed.     

Secular dynamics of the three-body problem: application to the υ Andromedae planetary system

The discovery of extra-solar planetary systems with multiple planets in highly eccentric orbits (∼0.1-0.6), in contrast with our own Solar System, makes classical secular perturbation analysis very limited. In this paper, we use a semi-numerical approach to study the secular behavior of a system composed of a central star and two massive planets in co-planar orbits. We show that the secular dynamics of this system can be described using only two parameters, the ratios of the semi-major axes and the planetary masses. The main dynamical features of the system are presented in geometrical pictures that allows us to investigate a large domain of the phase space of this three-body problem without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The topology of the phase space is also investigated in detail by means of spectral map techniques, which allow us to detect the separatrix of a non-linear secular apsidal resonance. Finally, the qualitative study is supplemented by direct numerical integrations. Three different regimes of secular motion with respect to the secular angle Δ? are possible: they are circulation, oscillation (around 0° and 180°), and high eccentricity libration in a non-linear secular resonance. The first two regimes are a continuous extension of the classical linear secular perturbation theory; the last is a new feature, hitherto unknown, in the secular dynamics of the three-body problem. We apply the analysis to the case of the two outer planets in the υ Andromedae system, and obtain its periodic and ordinary orbits, the general structure of its secular phase space, and the boundaries of its secular stability; we find that this system is secularly stable over a large domain of eccentricities. Applying this analysis to a wide range of planetary mass and semi-major axis ratios (centered about the υ Andromedae parameters), we find that apsidal oscillation dominates the secular phase space of the three-body coplanar system, and that the non-linear secular resonance is also a common feature.     

About averaging procedures in the problem of asteroid motion

A new mathematically correct approach to construct an averaging procedure for the motion of a massless body around the central body perturbed by fully interacting planets is developed and the errors of the standard solution are discussed. The new technique allows to combine the advantages of the Hamiltonian representation with the usage of standard osculating elements in combination with all the standard expansions of the perturbing functions. The main idea is to introduce new additional variables conjugate to all the standard elements and to work in a corresponding super phase space. In this way, the number of variables is doubled at first, but one has to deal with only one Hamiltonian. The artificially introduced variables disappear from the final averaged equations as well as from the transformation formulae connecting the osculating and the mean elements.     

Tidal Evolution of the Kepler Candidate Two-planet Systems

The tidal evolution of ten Kepler candidate two-planet systems is investigated by using the general secular perturbation theory, and then a general picture of tidal evolution for these systems is described. Taking the KOI 1239 system as an example, the tidal effect of the system is studied in detail, the results indicate that the dissipative term of the tidal effect causes the attenuation of planets’ orbital eccentricities, and it plays a dominant role in the process of orbital evolution, however, the conservative term of the tidal effect and the relativistic effect may damp the excitation of the eccentricity of the inner planet under the secular perturbation of the outer planet. In addition, the process of tidal evolution is also affected by both the initial eccentricity of the outer planet and the planet's tidal dissipation coeffcient. At the same time, the numerical simulation on the tidal evolution of the KOI 1239 system is also made, and the numerical results are consistent with those of general secular perturbation theory.     

Accurate analytical nutation series

In this letter we present the first accurate analytical nutation series, deduced from the Hamiltonian theory by the authors. They provide the highest accuracy ever obtained by any analytical nutation series, since the deviation in CEP (celestial ephemeris pole) offsets with respect to that of the IERS Conventions 1996 is kept below 1 mas in the time domain, in spite of still lacking ocanic corrections.     

The representation of the force functions of the Earth's and Moon's gravitation in ellipsoidal coordinates

This paper derives asymptotic expansions of ellipsoidal coordinates in Cartesian coordinates and an expansion in spherical harmonics of the dominant term for the solution of Laplace's equation corresponding to the gravitational force function for a two-dimensional finite body.On comparing the expansion of the dominant term derived here with known expansions of the force functions of the Earth's and Moon's gravitation the author obtains values for the semimajor axes and eccentricities of the singular ellipses of these bodies in terms of the second degree harmonic coefficientsc ₂₀ andc ₂₂.     

Secular dynamics of S-type planetary orbits in binary star systems: applicability domains of first- and second-order theories

We analyse the secular dynamics of planets on S-type coplanar orbits in tight binary systems, based on first- and second-order analytical models, and compare their predictions with full N-body simulations. The perturbation parameter adopted for the development of these models depends on the masses of the stars and on the semimajor axis ratio between the planet and the binary. We show that each model has both advantages and limitations. While the first-order analytical model is algebraically simple and easy to implement, it is only applicable in regions of the parameter space where the perturbations are sufficiently small. The second-order model, although more complex, has a larger range of validity and must be taken into account for dynamical studies of some real exoplanetary systems such as \(\gamma \) Cephei and HD 41004A. However, in some extreme cases, neither of these analytical models yields quantitatively correct results, requiring either higher-order theories or direct numerical simulations. Finally, we determine the limits of applicability of each analytical model in the parameter space of the system, giving an important visual aid to decode which secular theory should be adopted for any given planetary system in a close binary.