The dynamical architecture and habitable zones of the quintuplet planetary system 55 Cancri

We perform numerical simulations to study the secular orbital evolution and dynamical structure of the quintuplet planetary system 55 Cancri with the self-consistent orbital solutions by Fischer and coworkers. In the simulations, we show that this sys-tem can be stable for at least 108 yr. In addition, we extensively investigate the planetary configuration of four outer companions with one terrestrial planet in the wide region of 0.790 AU ≤ a ≤ 5.900 AU to examine the existence of potential asteroid structure and Habitable Zones (HZs). We show that there are unstable regions for orbits about 4:1, 3:1 and 5:2 mean motion resonances (MMRs) of the outermost planet in the system, and sev-eral stable orbits can remain at 3:2 and 1:1 MMRs, which resembles the asteroid belt in the solar system. From a dynamical viewpoint, proper HZ candidates for the existence of more potential terrestrial planets reside in the wide area between 1.0 AU and 2.3 AU with relatively low eccentricities.     

The Effect of Tidal Interaction with a Gas Disk on Formation of Terrestrial Planets

We have performed N-body simulation on final accretion stage of terrestrial planets, including the effect of damping of eccentricity and inclination caused by tidal interaction with a remnant gas disk. As a result of runway and oligarchic accretion, about 20 Mars-sized protoplanets would be formed in nearly circular orbits with orbital separation of several to ten Hill radius. The orbits of the protoplanets would be eventually destabilized by long-term mutual gravity and/or secular resonance of giant gaseous planets. The protoplanets would coalesce with each other to form terrestrial planets through the orbital crossing. Previous N-body simulations, however, showed that the final eccentricities of planets are around 0.1, which are about 10 times higher than the present eccentricities of Earth and Venus. The obtained high eccentricities are the remnant of orbital crossing. We included the effect of eccentricity damping caused by gravitational interaction with disk gas as a drag force (“gravitational drag”) and carried out N-body simulation of accretion of protoplanets. We start with 15 protoplanets with 0.2M⊕ and integrate the orbits for 10⁷ years, which is consistent with the observationally inferred disk lifetime (in some runs, we start with 30 protoplanets with 0.1M⊕). In most runs, the damping time scale, which is equivalent to the strength of the drag force, is kept constant throughout each run in order to clarify the effects of the damping. We found that the planets' final mass, spatial distribution, and eccentricities depend on the damping time scale. If the damping time scale for a 0.2M⊕ mass planet at 1 AU is longer than 10⁸ years, planets grow to Earth's size, but the final eccentricities are too high as in gas-free cases. If it is shorter than 10⁶ years, the eccentricities of the protoplanets cannot be pumped up, resulting in not enough orbital crossing to make Earth-sized planets. Small planets with low eccentricities are formed with small orbital separation. On the other hand, if it is between 10⁶ and 10⁸ years, which may correspond to a mostly depleted disk (0.01-0.1% of surface density of the minimum mass model), some protoplanets can grow to about the size of Earth and Venus, and the eccentricities of such surviving planets can be diminished within the disk lifetime. Furthermore, in innermost and outermost regions in the same system, we often find planets with smaller size and larger eccentricities too, which could be analogous to Mars and Mercury. This is partly because the gravitational drag is less effective for smaller mass planets, and partly due to the “edge effect,” which means the innermost and outermost planets tend to remain without collision. We also carried out several runs with time-dependent drag force according to depletion of a gas disk. In these runs, we used exponential decay model with e-folding time of 3×10⁶ years. The orbits of protoplanets are stablized by the eccentricity damping in the early time. When disk surface density decays to ?1% of the minimum mass disk model, the damping force is no longer strong enough to inhibit the increase of the eccentricity by distant perturbations among protoplanets so that the orbital crossing starts. In this disk decay model, a gas disk with 10⁻⁴-10⁻³ times the minimum mass model still remains after the orbital crossing and accretional events, which is enough to damp the eccentricities of the Earth-sized planets to the order of 0.01. Using these results, we discuss a possible scenario for the last stage of terrestrial planet formation.     

Accretion of terrestrial planets from oligarchs in a turbulent disk

We have investigated the final accretion stage of terrestrial planets from Mars-mass protoplanets that formed through oligarchic growth in a disk comparable to the minimum mass solar nebula (MMSN), through N-body simulation including random torques exerted by disk turbulence due to Magneto-Rotational Instability. For the torques, we used the semi-analytical formula developed by Laughlin et al. [Laughlin, G., Steinacker, A., Adams, F.C., 2004. Astrophys. J. 608, 489-496]. The damping of orbital eccentricities (in all runs) and type-I migration (in some runs) due to the tidal interactions with disk gas is also included. Without any effect of disk gas, Earth-mass planets are formed in terrestrial planet regions in a disk comparable to MMSN but with too large orbital eccentricities to be consistent with the present eccentricities of Earth and Venus in our Solar System. With the eccentricity damping caused by the tidal interaction with a remnant gas disk, Earth-mass planets with eccentricities consistent with those of Earth and Venus are formed in a limited range of disk gas surface density (∼10⁻⁴ times MMSN). However, in this case, on average, too many (?6) planets remain in terrestrial planet regions, because the damping leads to isolation between the planets. We have carried out a series of N-body simulations including the random torques with different disk surface density and strength of turbulence. We found that the orbital eccentricities pumped up by the turbulent torques and associated random walks in semimajor axes tend to delay isolation of planets, resulting in more coagulation of planets. The eccentricities are still damped after planets become isolated. As a result, the number of final planets decreases with increase in strength of the turbulence, while Earth-mass planets with small eccentricities are still formed. In the case of relatively strong turbulence, the number of final planets are 4-5 at 0.5-2 AU, which is more consistent with Solar System, for relatively wide range of disk gas surface density (∼¹⁰⁻⁴-¹⁰⁻² times MMSN).     

Eccentricities of Planets in Binary Systems

The most puzzling property of the extrasolar planets discovered by recent radial velocity surveys is their high orbital eccentricities, which are very difficult to explain within our current theoretical paradigm for planet formation. Current data reveal that at least 25% of these planets, including some with particularly high eccentricities, are orbiting a component of a binary star system. The presence of a distant companion can cause significant secular perturbations in the orbit of a planet. At high relative inclinations, large-amplitude, periodic eccentricity perturbations can occur. These are known as “Kozai cycles” and their amplitude is purely dependent on the relative orbital inclination. Assuming that every planet host star also has a (possibly unseen, e.g., substellar) distant companion, with reasonable distributions of orbital parameters and masses, we determine the resulting eccentricity distribution of planets and compare it to observations? We find that perturbations from a binary companion always appear to produce an excess of planets with both very high (?0.6) and very low (e ? 0.1) eccentricities. The paucity of near-circular orbits in the observed sample implies that at least one additional mechanism must be increasing eccentricities. On the other hand, the overproduction of very high eccentricities observed in our models could be combined with plausible circularization mechanisms (e.g., friction from residual gas) to create more planets with intermediate eccentricities (e? 0.1–0.6).     

Eccentric Extrasolar Planets: The Jumping Jupiter Model

Most extrasolar planets discovered to date are more massive than Jupiter, in surprisingly small orbits (semimajor axes less than 3 AU). Many of these have significant orbital eccentricities. Such orbits may be the product of dynamical interactions in multiplanet systems. We examine outcomes of such evolution in systems of three Jupiter-mass planets around a solar-mass star by integration of their orbits in three dimensions. Such systems are unstable for a broad range of initial conditions, with mutual perturbations leading to crossing orbits and close encounters. The time scale for instability to develop depends on the initial orbital spacing; some configurations become chaotic after delays exceeding 10⁸ y. The most common outcome of gravitational scattering by close encounters is hyperbolic ejection of one planet. Of the two survivors, one is moved closer to the star and the other is left in a distant orbit; for systems with equal-mass planets, there is no correlation between initial and final orbital positions. Both survivors may have significant eccentricities, and the mutual inclination of their orbits can be large. The inner survivor's semimajor axis is usually about half that of the innermost starting orbit. Gravitational scattering alone cannot produce the observed excess of “hot Jupiters” in close circular orbits. However, those scattered planets with large eccentricities and small periastron distances may become circularized if tidal dissipation is effective. Most stars with a massive planet in an eccentric orbit should have at least one additional planet of comparable mass in a more distant orbit.     

Orbital Evolution of Extrasolar Planets

The recent discovery of extrasolar planets and planetary systems has raised many new research problems for astronomers. It has become apparent that the newly discovered systems differ significantly from the Solar System. In particular, many massive planets of other stars, in contrast to Jupiter, have large orbital eccentricities. In the present paper, we investigate several dynamic implications of this finding. Numerical integration results show that the orbits of low-mass planets in such systems usually have large evolving eccentricities. If the motion remains regular and no close encounters occur, the orbital evolution can be described analytically by using secular perturbations of Laplace–Lagrange equations. In terms of the Lagrange variables, the trajectories are circles, and the semimajor axis remains constant. The loss of the regularity of motion is normally followed by a nonmonotone synchronous increase in the semimajor axis and eccentricity, and the orbit becomes similar to that of a large-period comet. Narrow resonance-related regions include more complex motions.     

Disc–Planet Interactions: Migration and Resonances in Extrasolar Planetary Systems

We consider orbital resonances in multiplanet systems. These are expected to arise during or just after formation in a gaseous disc. Disc–planet interaction naturally produces orbital migration and circularization through the action of tidal torques which in turn may lead to an orbital resonance. The mass and angular momentum content of the disc is likely to be comparable to that in the planets so that it is essential to fully incorporate the disc in the analysis.We study the orbital evolution of two planets locked in 2:1 commensurability through migration tidally induced by the disc using both analytic methods and numerical hydrodynamic simulations. The planets are assumed to orbit in an inner cavity containing at most only a small amount of disc material. Results are found to be sensitive to initial surface density profile, planet masses and disc parameters. The evolution may range between attaining and subsequently maintaining a resonance lock with two angles librating to divergent migration with no commensurability formed. In the former case eccentricities increase monotonically with time while the system undergoes inward migration. If the migration is halted by loss of the disc leaving the planets in a final configuration, there is likely to be a low probability of seeing resonant planets at small radii as well as a sensitive dependence on past history.We have also considered a multiplanet system in secular apsidal resonance. We consider the system as being in just one secular normal mode and include the effects of a gaseous disc. It is suggested that a normal mode may be selected by adding in some weak dissipative process in the disc and that it may remain, involving only the planets, when the disc is slowly removed.     

Influence of periodic orbits on the formation of giant planetary systems

The late-stage formation of giant planetary systems is rich in interesting dynamical mechanisms. Previous simulations of three giant planets initially on quasi-circular and quasi-coplanar orbits in the gas disc have shown that highly mutually inclined configurations can be formed, despite the strong eccentricity and inclination damping exerted by the disc. Much attention has been directed to inclination-type resonance, asking for large eccentricities to be acquired during the migration of the planets. Here we show that inclination excitation is also present at small to moderate eccentricities in two-planet systems that have previously experienced an ejection or a merging and are close to resonant commensurabilities at the end of the gas phase. We perform a dynamical analysis of these planetary systems, guided by the computation of planar families of periodic orbits and the bifurcation of families of spatial periodic orbits. We show that inclination excitation at small to moderate eccentricities can be produced by (temporary) capture in inclination-type resonance and the possible proximity of the non-coplanar systems to spatial periodic orbits contributes to maintaining their mutual inclination over long periods of time.     

The dynamics of two massive planets on inclined orbits

The significant orbital eccentricities of most giant extrasolar planets may have their origin in the gravitational dynamics of initially unstable multiple planet systems. In this work, we explore the dynamics of two close planets on inclined orbits through both analytical techniques and extensive numerical scattering experiments. We derive a criterion for two equal mass planets on circular inclined orbits to achieve Hill stability, and conclude that significant radial migration and eccentricity pumping of both planets occurs predominantly by 2:1 and 5:3 mean motion resonant interactions. Using Laplace-Lagrange secular theory, we obtain analytical secular solutions for the orbital inclinations and longitudes of ascending nodes, and use those solutions to distinguish between the secular and resonant dynamics which arise in numerical simulations. We also illustrate how encounter maps, typically used to trace the motion of massless particles, may be modified to reproduce the gross instability seen by the numerical integrations. Such a correlation suggests promising future use of such maps to model the dynamics of more coplanar massive planet systems.     

Formation of terrestrial planets in a dissipating gas disk with Jupiter and Saturn

We have performed N-body simulations on final accretion stage of terrestrial planets, including the eccentricity and inclination damping effect due to tidal interaction with a gas disk. We investigated the dependence on a depletion time scale of the disk, and the effect of secular perturbations by Jupiter and Saturn. In the final stage, terrestrial planets are formed through coagulation of protoplanets of about the size of Mars. They would collide and grow in a decaying gas disk. Kominami and Ida [Icarus 157 (2002) 43-56] showed that it is plausible that Earth-sized, low-eccentricity planets are formed in a mostly depleted gas disk. In this paper, we investigate the formation of planets in a decaying gas disk with various depletion time scales, assuming disk surface density of gas component decays exponentially with time scale of τ_gas. Fifteen protoplanets with are initially distributed in the terrestrial planet regions. We found that Earth-sized planets with low eccentricities are formed, independent of initial gas surface density, when the condition (τ_cross+τ_growth)/2?τ_gas?τ_cross is satisfied, where τ_cross is the time scale for initial protoplanets to start orbit crossing in a gas-free case and τ_growth is the time scale for Earth-sized planets to accrete during the orbit crossing stage. In the cases satisfying the above condition, the final masses and eccentricities of the largest planets are consistent with those of Earth and Venus. However, four or five protoplanets with the initial mass remain. In the final stage of terrestrial planetary formation, it is likely that Jupiter and Saturn have already been formed. When Jupiter and Saturn are included, their secular perturbations pump up eccentricities of protoplanets and tend to reduce the number of final planets in the terrestrial planet regions. However, we found that the reduction is not significant. The perturbations also shorten τ_cross. If the eccentricities of Jupiter and Saturn are comparable to or larger than present values (∼0.05), τ_cross become too short to satisfy the above condition. As a result, eccentricities of the planets cannot be damped to the observed value of Earth and Venus. Hence, for the formation of terrestrial planets, it is preferable that the secular perturbations from Jupiter and Saturn do not have significant effect upon the evolution. Such situation may be reproduced by Jupiter and Saturn not being fully grown, or their eccentricities being smaller than the present values during the terrestrial planets' formation. However, in such cases, we need some other mechanism to eliminate the problem that numerous Mars-sized planets remain uncollided.     

Drag-induced resonant capture in a multiplanet scenario: An application to 55 Cancri A

The binary star 55 Cancri harbors at least five giant planets. The discovery of a possible dust disk around the primary of the system was announced in 1998 although it was later dismissed as an observational artifact induced by the presence of three background galaxies. However, the possible existence of an asteroid belt beyond 6 AU from the primary could not be excluded. The actual properties of a hypothetical outer asteroid belt around 55 Cancri A are likely to be the result of the gravitational influence of the outermost planets and gas drag during the late stages of the formation of the planetary system. Gaseous drag within a protoplanetary disk in a multiplanet context can induce resonant capture of solid material rather easily, triggering the formation of asteroid belts similar to our own transneptunian belt. In this paper we investigate this scenario using numerical simulations within the framework of the planar restricted four-body problem to find possible stable debris locations or additional planets around 55 Cancri A. Our results indicate that, beyond 55 Cancri d, only the 1:2 and 1:3 mean-motion resonances may be possible although the details of gas drag-induced resonant trapping depend strongly on the size of the particles, with larger particles being preferentially trapped farther away from the host star. For a primary of mass 0.94 M and i=53° we find solid particles submitted to confined chaos at 10.14 and 13.22 AU with average eccentricities 0.14 and 0.20, respectively. The argument of pericentre of the trapped particles is found to librate around π/2 with the pericentres of the two resonant structures nearly aligned. The resonance responsible for the inner ring effectively halts the migration of the particles. The outer structure resembles a rosette or separatrix orbit. Our calculations suggest that hypothetical planets or asteroid belts in these locations would be, in principle, dynamically stable.     

Eccentricity evolution of giant planet orbits due to circumstellar disk torques

The extrasolar planets discovered to date possess unexpected orbital elements. Most orbit their host stars with larger eccentricities and smaller semi-major axes than similarly sized planets in our own Solar System do. It is generally agreed that the interaction between giant planets and circumstellar disks (Type II migration) drives these planets inward to small radii, but the effect of these same disks on orbital eccentricity, ?, is controversial. Several recent analytic calculations suggest that disk-planet interactions can excite eccentricity, while numerical studies generally produce eccentricity damping. This paper addresses this controversy using a quasi-analytic approach, drawing on several preceding analytic studies. This work refines the current treatment of eccentricity evolution by removing several approximations from the calculation of disk torques. We encounter neither uniform damping nor uniform excitation of orbital eccentricity, but rather a function d?/dt that varies in both sign and magnitude depending on eccentricity and other Solar System properties. Most significantly, we find that for every combination of disk and planet properties investigated herein, corotation torques produce negative values of d?/dt for some range in ? within the interval [0.1, 0.5]. If corotation torques are saturated, this region of eccentricity damping disappears, and excitation occurs on a short timescale of less than 0.08 Myr. Thus, our study does not produce eccentricity excitation on a timescale of a few Myr—we obtain either eccentricity excitation on a short time scale, or eccentricity damping on a longer time scale. Finally, we discuss the implications of this result for producing the observed range in extrasolar planet eccentricity.     

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.     

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.     

Effects of Planetary Migration on Natural Satellites of the Outer Planets

Numerous studies in the past few years have analyzed possible effects of planetary migration on the small bodies of the Solar System (mainly asteroids and KBOs), with the double aim of explaining certain dynamical structures in these systems, as well as placing limits on the magnitude of the radial migration of the planets. Here we undertake a similar aim, only this time concentrating on the dynamical stability of planetary satellites in a migration scenario. However, different from previous works, the strongest perturbations on satellite systems are not due to the secular variation of the semimajor axes of the planets, but from the planetesimals themselves. These perturbations result from close approaches between the planetesimals and satellites.We present results of several numerical simulations of the dynamical evolution of real and fictitious satellite systems around the outer planets, under the effects of multiple passages of a population of planetesimals representing the large-body component of a residual rocky disk. Assuming that this component dominated the total mass of the disk, our results show that the present systems of satellites of Uranus and Neptune do not seem to be compatible with a planetary migration larger than even one quarter that suggested by previous studies, unless these bodies were originated during the late stage of evaporation of the planetesimal disk. For larger variations of the semimajor axes of the planets, most of the satellites would either be ejected from the system or suffer mutual collisions due to excitation in their eccentricities. For the systems of Jupiter and Saturn, these perturbations are not so severe, and even large migrations do not introduce large instabilities.Nevertheless, even a small number of 1000-km planetesimals in the region may introduce significant excitation in the eccentricities and inclinations of satellites. Adequate values of this component may help explain the present dynamical distribution of distant satellites, including the highly peculiar orbit of Nereid.     

Secular Dynamics of Asteroids in the Inner Solar System

In the last three years we have carried out numerical and semi-analytical studies on the secular dynamical mechanisms in the region (semimajor axis a < 2 AU) where the NEA orbits evolve. Our numerical integrations (over a time span of a few Myr) have shown that: (i) the linear secular resonances with both the inner and the outer planets may play an important role in the dynamical evolution of NEAs; (ii) the apsidal secular resonance with Mars could provide an important dynamical transport mechanism by which asteroids in the Mars-crossing region eventually achieve Earth-crossing orbits; (iii) in this region, due to the interaction with the terrestrial planets, the Kozai resonance can occur at small inclinations, with the argument of perihelion ω librating around 0° or 180°, providing a temporary protection mechanism against close approaches to the planets. The location of the linear secular resonances in this zone has also been obtained by an automatic procedure using a semi-numerical method valid for all values of the inclinations and eccentricities of the small bodies, and also in the case of libration of the argument of perihelion. A map of the secular resonances in the (a, i) plane shows — in agreement with the numerical integrations — that all the resonances with the terrestrial and giant planets are present, and also that some of them overlap. Thus the way is now open to fully take into account secular resonances in modelling the dynamical evolution of NEAs. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Forced obliquities and moments of inertia of Ceres and Vesta

We examine models of secular variations in the orbit and spin poles of Ceres and Vesta, the two most massive bodies in the main asteroid belt. If the spin poles are fully damped, then the current values of obliquity, or angular separation between spin and orbit poles, are diagnostic of the moments of inertia and thus indicative of the extent of differentiation of these bodies. Using existing shape models and assuming uniform density, the present obliquity values are predicted to be 12.31° for Ceres and 15.66° for Vesta. Part of this difference is related to differing orbital inclinations; a more centrally condensed internal structure would yield more rapid spin pole precession, and larger obliquity. Time scales for tidal damping are expected to be rather long. However, at least for Vesta, current estimates of the spin pole location are consistent with its obliquity being fully damped. When the degree two gravity coefficients and spin pole orientations are determined by the Dawn spacecraft, it will allow accurate determination of the moments of inertia of these bodies, assuming the obliquities are damped.     

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.