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.     

The integrable cases of the planetary three-body problem at first-order resonance

This paper investigates the stability of the motion in the averaged planar general three-body problem in the case of first-order resonance. The equations of the averaged motion of bodies near the resonance surface is obtained and is analytically integrated by quadratures. The stability of the averaged motion is analytically investigated in relation to the semi-major axes, the eccentricities and the resonance phases. An autonomous second-order equation is obtained for the deviation of semiaxes from the resonance surface. This equation has an energy integral and is analytically integrated by quadratures. The quasi-periodic dependence on time with two-frequency basis of the averaged motion of bodies is found. The basic frequencies are analytically calculated. With the help of the mean functionals calculated along integral curves of the averaged problem the new analytic first integrals are constructed with coefficients periodic in time. The analytic conditions of librations of resonance phases are obtained.     

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.     

Non-integrability of the three-body problem

We consider the planar problem of three bodies which attract mutually with the force proportional to a certain negative integer power of the distance between the bodies. We show that such generalisation of the gravitational three-body problem is not integrable in the Liouville sense.     

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.     

The main mean motion commensurabilities in the planar circular and elliptic problem

We study the motion of asteroids in the main mean motion commensurabilities in the frame of the planar restricted three-body problem. No assumption is made about the size of the eccentricity of the asteroid. At small to moderate eccentricity, we recover existing results (shape of the phase space and location of secondary resonances). We also provide global pictures of the dynamics in the region of secondary resonances. At high eccentricity, the phase space portraits of the integrable part of the Hamiltonian show new families of stable orbits for the 3:2 and 2:1 cases and the secular resonances ₅ and ₆ are located.     

Numerical results to the Sitnikov-problem

We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e _primaries 0.66) and also a range of initial conditions for the distance of the 3 ^rd body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities.     

On the double averaged three-body problem

The Hamiltonian of three point masses is averaged over fast variablel and ll (mean anomalies) The problem is non-planar and it is assumed that two of the bodies form a close pair (stellar three-body problem). Only terms up to the order of (a/á)⁴ are taken into account in the Hamiltonian, wherea andá are the corresponding semi-major axes. Employing the method of elimination of the nodes, the problem may be reduced to one degree of freedom. Assuming in addition that the angular momentum of the close binary is much smaller than the angular momentum of the motion of the binary around a third body, we were able to solve the equation for the eccentricity changes in terms of the Jacobian elliptic functions.     

Spin axis evolution of two interacting bodies

We consider the solid-solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boué and Laskar [Boué, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but not trivially. We describe the general features of the motions and we provide explicit analytical approximations for the solutions. We demonstrate that the general solution of the secular system can be decomposed as a uniform precession around the total angular momentum and a periodic symmetric orbit in the precessing frame. More generally, we show that for a general n-body system of rigid bodies in gravitational interaction, the regular quasiperiodic solutions can be decomposed into a uniform precession around the total angular momentum, and a quasiperiodic motion with one frequency less in the precessing frame.     

Short Time Lyapunov Indicators in the Restricted Three-Body Problem

We applied the method of the short time Lyapunov indicators to the planar circular and to the planar elliptic restricted three-body problem in order to study the structure of the phase space in some selected regions. In the circular case we computed the short-time averages of the stretching numbers to distinguish between regular and chaotic domains of the phase space. The results obtained in this way are in good agreement with the corresponding Poincaré's surface of sections. Using the short time Lyapunov indicators we determined the detailed structure of the phase space in the semi-major axis-eccentricity plane of the test particle both in the circular and in the elliptic restricted problem (in the latter case for some values of the eccentricities of the primaries) and we studied the main features of the phase space.This revised version was published online in October 2005 with corrections to the Cover Date.     

Motion of dust in mean motion resonances with planets

Effect of stellar electromagnetic radiation on the motion of spherical dust particle in mean motion orbital resonances with a planet is investigated. Planar circular restricted three-body problem with the Poynting–Robertson (P–R) effect yields monotonic secular evolution of eccentricity when the particle is trapped in the resonance. Planar elliptic restricted three-body problem with the P–R effect enables nonmonotonous secular evolution of eccentricity and the evolution of eccentricity is qualitatively consistent with the published results for the complicated case of interaction of electromagnetic radiation with nonspherical dust grain. Thus, it is sufficient to allow either nonzero eccentricity of the planet or nonsphericity of the grain and the orbital evolutions in the resonances are qualitatively equal for the two cases. This holds both for exterior and interior mean motion orbital resonances. Evolutions of argument of perihelion in the planar circular and elliptical restricted three-body problems are shown. Numerical integrations show that an analytic expression for the secular time derivative of the particle’s argument of perihelion does not exist, if only dependence on semimajor axis, eccentricity and argument of perihelion is admitted. Connection between the shift of perihelion and oscillations in secular eccentricity is presented for the planar elliptic restricted three-body problem with the P–R effect. Period of the oscillations corresponds to the period of one revolution of perihelion. Change of optical properties of the spherical grain with the heliocentric distance is also considered. The change of the optical properties: (i) does not have any significant influence on the secular evolution of eccentricity, (ii) causes that the shift of perihelion is mainly in the same direction/orientation as the particle motion around the Sun. The statements hold both for circular and noncircular planetary orbits.     

Fractal structures for the Jacobi Hamiltonian of restricted three-body problem

We study the dynamical chaos and integrable motion in the planar circular restricted three-body problem and determine the fractal dimension of the spiral strange repeller set of non-escaping orbits at different values of mass ratio of binary bodies and of Jacobi integral of motion. We find that the spiral fractal structure of the Poincaré section leads to a spiral density distribution of particles remaining in the system. We also show that the initial exponential drop of survival probability with time is followed by the algebraic decay related to the universal algebraic statistics of Poincaré recurrences in generic symplectic maps.     

Exchange orbits: an interesting case of co-orbital motion

In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist—at least—theoretically—two different types: the exchange-a orbits and the exchange-e orbits. The first one is the following: two celestial bodies are in orbit around a central body with almost the same semi-major axes on circular orbits. Because of the relatively small differences in semi-major axes they meet from time to time and exchange their semi-major axes. The inner one then moves outside the other planet and vice versa. The second configuration one is the following: two planets are moving on nearly the same orbit with respect to the semi-major axes, one on a circular orbit and the other one on an eccentric one. During their dynamical evolution they change the characteristics of the orbit, the circular one becomes an elliptic one whereas the elliptic one changes its shape to a circle. This ‘game’ repeats periodically. In this new study we extend the numerical computations for both of these exchange orbits to the three dimensional case and in another extension treat also the problem when these orbits are perturbed from a fourth body. Our results in form of graphs show quite well that for a large variety of initial conditions both configurations are stable and stay in these exchange orbits.     

Bifurcation points in the planar problem of three bodies

An essential parameter in the planar problem of three bodies is the product of the square of the angular momentum and of the total energy (c ² H). The role of this parameter, which may be called abifurcation parameter, in establishing regions of possible motions has been shown by Marchal and Saari (1975) and Zare (1976a). There exist critical values of this parameter below which exchange between bodies cannot occur. These critical values may be calledbifurcation points.This paper gives an analytical criterion to obtain these bifurcation points for any given masses of the participating bodies.     

Global bifurcation of planar and spatial periodic solutions in the restricted <Emphasis Type="Italic">n</Emphasis>-body problem

The paper deals with the study of a satellite attracted by n primary bodies, which form a relative equilibrium. We use orthogonal degree to prove global bifurcation of planar and spatial periodic solutions from the equilibria of the satellite. In particular, we analyze the restricted three body problem and the problem of a satellite attracted by the Maxwell’s ring relative equilibrium.     

On the 2/1 resonant planetary dynamics – periodic orbits and dynamical stability

The 2/1 resonant dynamics of a two-planet planar system is studied within the framework of the three-body problem by computing families of periodic orbits and their linear stability. The continuation of resonant periodic orbits from the restricted to the general problem is studied in a systematic way. Starting from the Keplerian unperturbed system, we obtain the resonant families of the circular restricted problem. Then, we find all the families of the resonant elliptic restricted three-body problem, which bifurcate from the circular model. All these families are continued to the general three-body problem, and in this way we can obtain a global picture of all the families of periodic orbits of a two-planet resonant system. The parametric continuation, within the framework of the general problem, takes place by varying the planetary mass ratio ρ. We obtain bifurcations which are caused either due to collisions of the families in the space of initial conditions or due to the vanishing of bifurcation points. Our study refers to the whole range of planetary mass ratio values  [ρ∈ (0, ∞)]  and, therefore we include the passage from external to internal resonances. Thus, we can obtain all possible stable configurations in a systematic way. As an application, we consider the dynamics of four known planetary systems at the 2/1 resonance and we examine if they are associated with a stable periodic orbit.     

The General Solution of the Planar Laplace Problem

The famous Laplace problem is the three-body, secular, planetary problem. Its plane version has the great theoretical advantage of being integrable (Ferraz-Mello, private correspondence, 2001) and Ferraz-Mello et al. (Chaotic World: from order to Disorder in Gravitational N-Body Systems, Kluwer Academic Publisher, 2004)). Nevertheless it remains a very complex problem with many singularities and many possibilities of collisions. Large eccentricities lead generally to large perturbations especially if the two planets have the same direction of revolution about their star.     

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.     

Periodic orbits of the general three-body problem for the Sun-Jupiter-Saturn system

Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the Sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 25 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem.