Planet-planet scattering is the leading mechanism to explain the large eccentricities of the observed exoplanet population. However, scattering has not been considered important to the production of pairs of planets in mean motion resonances (MMRs). We present results from a large number of numerical simulations of dynamical instabilities in 3-planet systems. We show that MMRs arise naturally in about five percent of cases. The most common resonances we populate are the 2:1 and 3:1 MMRs, although a wide variety of MMRs can occur, including high-order MMRs (up to eleventh order). MMRs are generated preferentially in systems with uneven mass distributions: the smallest planet is typically ejected after a series of close encounters, leaving the remaining, more massive planets in resonance. The distribution of resonant planets is consistent with the phase-space density of resonant orbits, meaning that planets are randomly thrown into MMRs rather than being slowly pulled into them. It may be possible to distinguish between MMRs created by scattering vs. convergent migration in a gaseous disk by considering planetary mass ratios: resonant pairs of planets beyond ~1 AU with more massive outer planets are likely to have formed by scattering. In addition, scattering may be responsible for pairs of planets in high-order MMRs (3:1 and higher) that are not easily populated by migration. The frequency of MMRs from scattering is comparable to the expected survival rate of MMRs in turbulent disks. Thus, planet-planet scattering is likely to be a major contributor to the population of resonant planets.
A migrating planet can capture planetesimals into mean motion resonances. However, resonant trapping can be prevented when the drift or migration rate is sufficiently high. Using a simple Hamiltonian system for first and second order resonances, we explore how the capture probability depends on the order of the resonance, drift rate and initial particle eccentricity. We present scaling factors as a function of the planet mass and resonance strength to estimate the planetary migration rate above which the capture probability drops to less than 1/2. Applying our framework to multiple extra solar…
With the help of the Laplace-Lagrange solution of the secular perturbation theory in a double-planet system, we study the occurrence and the stability of apsidal secular resonance between the two planets. The explicit criteria to predict whether two planets are in apsidal resonance is derived. The pobability of two planets falling in apsidal resonance is given in the initial element space. We verify the criteria with numerical integrations for the HD12661 system, and find they give good predictions except at the boundary of the criteria or when the planet eccentricities are too large. The nonl…
We investigate the hypothesis that interactions between a giant planet and the disk from which it forms promote eccentricity growth. These interactions are concentrated at discrete Lindblad and corotation resonances. Interactions at principal Lindblad resonances cause the planet's orbit to migrate and open a gap in the disk if the planet is sufficiently massive. Those at first order Lindblad and corotation resonances change the planet's orbital eccentricity. Eccentricity is excited by interactions at external Lindblad resonances which are located on the opposite side of corotation from the pla…
We investigate the secular resonances for massless small bodies and Earth-like planets in several planetary systems. We further compare the results with those of Solar System. For example, in the GJ 876 planetary system, we show that the secular resonances $ν_1$ and $ν_2$ (respectively, resulting from the inner and outer giant planets) can excite the eccentricities of the Earth-like planets with orbits 0.21 AU $\leq a <$ 0.50 AU and eject them out of the system in a short timescale. However, in a dynamical sense, the potential zones for the existence of Earth-like planets are in the area 0.50 …
We explore the possibility that extrasolar planets might be found in the 1:1 mean-motion resonance. There are a variety of stable co-orbtial configurations, and we specifically examine three different versions of the 1:1 resonance. These include tadpole and horseshoe type orbits, as well as a more exotic configuration which occurs when one planet has a highly eccentric orbit while the other planet moves on a nearly circular orbit. We show that pairs of planets in 1:1 resonance yield characteristic radial velocity signatures which are not prone to the sin(i) degeneracy. Indeed, Keplerian fits t…
The 2:1 orbital resonances of the GJ 876 system can be easily established by the differential planet migration due to planet-nebula interaction. Significant eccentricity damping is required to produce the observed orbital eccentricities. The geometry of the GJ 876 resonance configuration differs from that of the Io-Europa pair, and this difference is due to the magnitudes of the eccentricities involved. We show that a large variation in the configuration of 2:1 and 3:1 resonances and, in particular, asymmetric librations can be expected among future discoveries.
The morphology of the epsilon Eridani dust ring is reproduced by a numerical simulation of dust particles captured into the 5:3 and 3:2 exterior mean-motion resonances with a 0.3 eccentricity 10^-4 solar mass planet at periastron at a semi-major axis of 40 AU. The morphology will differ when the planet is at aphelion, in about 140 years. Moderate eccentricity planets in outer extra-solar systems will cause observable variations in the morphology of associated dusty rings.
(Abridged) A diversity of 2:1 resonance configurations can be expected in extrasolar planetary systems, and their geometry can provide information about the origin of the resonances. Assembly during planet formation by the differential migration of planets due to planet-disk interaction is one scenario for the origin of mean-motion resonances in extrasolar planetary systems. The stable 2:1 resonance configurations that can be reached by differential migration of planets with constant masses and initially coplanar and nearly circular orbits are (1) anti-symmetric configurations with the mean-mo…
The formation of resonant planets pairs in exoplanetary systems involves planetary migration inside the protoplanetary disc : an inwards migrating outer planet captures in Mean Motion Resonance an inner planet. During the migration of the resonant pair of planets, the eccentricities are expected to rise excessively, if no damping mechanism is applied on the inner planet. We express the required damping action to match the observations, and we show that the inner disk can play this role. This result applies for instance to the system GJ 876 : we reproduce the observed orbital elements through a…
The nonlinear stability domain of Lagrange's celebrated 1772 solution of a three-body problem is obtained numerically as a function of the masses of the bodies and the common eccentricity of their Keplerian orbits. This domain shows that this solution may be realized in extra-solar planetary systems similar to those that have been discovered recently with two Jupiter-size planets orbiting a solar-size star. For an exact 1:1 resonance, the Doppler shift variation in the emitted light would be the same as for stars which have only a single planetary companion. But it is more likely that in actua…
We analyze the possible relationship between the current orbital elements fits of known exoplanets in the 2/1 mean-motion resonance and the expected orbital configuration due to migration. It is found that, as long as the orbital decay was sufficiently slow to be approximated by an adiabatic process, all captured planets should be in apsidal corotations. In other words, they should show a simultaneous libration of both the resonant angle and the difference in longitudes of pericenter. We present a complete set of corotational solutions for the 2/1 commensurability, including previously known s…
A restricted planar circular three-body system, consisting of the Sun and two planets, is studied as a simple model for a planetary system. The mass of the inner planet is considered to be larger and the system is assumed to be moving in a uniform interplanetary medium with constant density. Numerical integrations of this system indicate a resonance capture when the dynamical friction of the interplanetary medium is taken into account. As a result of this resonance trapping, the ratio of orbital periods of the two planets becomes nearly commensurate and the eccentricity and semimajor axis of t…
Multi-planet systems detected until now are in most cases characterized by hot-Jupiters close to their central star as well as high eccentricities. As a consequence, from a dynamical point of view, compact multi-planetary systems form a variety of the general N-body problem (with N >= 3), whose solutions are not necessarily known. Extrasolar planets are up to now found in prograde (i.e. direct) orbital motions about their host star and often in mean-motion resonances (MMR). In the present paper, we investigate a theoretical alternative suitable for the stability of compact multi-planetary syst…
Orbits of known extrasolar planets that are located outside the tidal circularization regions of their parent stars are often substantially eccentric. By contrast, planetary orbits in our Solar System are approximately circular, reflecting planet formation within a nearly axisymmetric, circumsolar disk. We propose that orbital eccentricities may be generated by divergent orbital migration of two planets in a viscously accreting circumstellar disk. The migration is divergent in the sense that the ratio of the orbital period of the outer planet to that of the inner planet grows. As the period ra…
We revisit the "dynamical shakeup" model of Solar System terrestrial planet formation, wherein the whole process is driven by the sweeping of Jupiter's secular resonance as the gas disk is removed. Using a large number of 0.5 Gyr-long N-body simulations, we investigate the different outcomes produced by such a scenario. We confirm that in contrast to existing models, secular resonance sweeping combined with tidal damping by the disk gas can reproduce the low eccentricities and inclinations, and high radial mass concentration, of the Solar System terrestrial planets. At the same time, this also…
The full set of published radial velocity data (52 measurements from Keck + 58 ones from ELODIE + 17 ones from CORALIE) for the star HD37124 is analysed. Two families of dynamically stable high-eccentricity orbital solutions for the planetary system are found. In the first one, the outer planets c and d are trapped in the 2/1 mean-motion resonance. The second family of solutions corresponds to the 5/2 mean-motion resonance between these planets. In both families, the planets are locked in (or close to) an apsidal corotation resonance. In the case of the 2/1 MMR, it is an asymmetric apsidal cor…
We present new results related to the coupled evolution of a two giant planet system embedded in a protoplanetary disk, in which a Saturn mass protoplanet is trapped in an outer mean motion resonance with a Jupiter mass protoplanet. The gaps opened in the disk by the two planets overlap, therefore the two planet system exchanges angular momentum with the disk at the inner planet's Inner Lindblad Resonances (ILRs) and at the outer planet Outer Lindblad Resonances (OLRs). Since the torques are proportional to the square of the planet masses, and since the inner planet is about three times more m…
The recent radial velocity Keck data for GJ876 in Laughlin et al. (2005) is shown to be in good agreement, apart from a 6 m/s scatter, with a theoretical calculation (Nauenberg, 2002) based on orbital parameters from a fit to the earlier Keck data. The time variation of the periods of the inner and outer planets, which are locked in a near 2:1 resonance, are evaluated, and their mean values, Pi and Po, are shown to satisfy closely the relation Po/Pi=2+Po/P, where P is the period for the retrograde precesion of the common mean periastron of these planets.
(Abridged) In planetary systems with two or more giant planets, dynamical instabilities can lead to collisions or ejections through strong planet--planet scattering. Previous studies for simple initial configurations with two equal-mass planets revealed some discrepancies between the results of numerical simulations and the observed orbital elements of extrasolar planets. Here, we show that simulations with two unequal mass planets starting on nearly circular orbits predict a reduced frequency of collisions and a broader range of final eccentricities. The two-planet scattering model can easily…
Mean-motion resonances (MMRs) are likely to play an important role both during and after the lifetime of a protostellar gas disk. We study the dynamical evolution and stability of planetary systems containing two giant planets on circular orbits near a 2:1 resonance and closer. We find that by having the outer planet migrate inward, the two planets can capture into either the 2:1, 5:3, or 3:2 MMR. We use direct numerical integrations of ~1000 systems in which the planets are initially locked into one of these resonances and allowed to evolve for up to ~10^7 yr. We find that the final eccentric…
This paper considers the effects of turbulence on mean motion resonances in extrasolar planetary systems and predicts that systems rarely survive in a resonant configuration. A growing number of systems are reported to be in resonance, which is thought to arise from the planet migration process. If planets are brought together and moved inward through torques produced by circumstellar disks, then disk turbulence can act to prevent planets from staying in a resonant configuration. This paper studies this process through numerical simulations and via analytic model equations, where both approach…
Planets in extrasolar systems tend to interact such that their orbits lie near a boundary between apsidal libration and circulation, a "separatrix", with one eccentricity periodically reaching near-zero. One explanation, applied to the Upsilon Andromedae system, assumed three original planets on circular orbits. One is ejected, leaving the other two with near-separatrix behavior. We test that model by integrating hundreds of hypothetical, unstable planetary systems that eject a planet. We find that the probability that the remaining planets exhibit near-separatrix motion is small (< 5% compare…
Using numerical methods we thoroughly investigate the dynamical stability in the region between the two planets found in HD 74156. The two planets with minimum masses 1.56 M_JUP (HD 74156b) and 7.5 M_JUP (HD 74156c), semimajor axes 0.276 AU and 3.47 AU move on quite eccentric orbits (e=0.649 and 0.395). There is a region between 0.7 and 1.4 AU which may host additional planets which we checked via numerical integrations using different dynamical models. Besides the orbital evolution of several thousands of massless regarded planets in a three-dimensional restricted 4-body problem (host star, t…
Our understanding of the Solar System has been revolutionized over the past decade by the finding that the orbits of the planets are inherently chaotic. In extreme cases, chaotic motions can change the relative positions of the planets around stars, and even eject a planet from a system. Moreover, the spin axis of a planet-Earth's spin axis regulates our seasons-may evolve chaotically, with adverse effects on the climates of otherwise biologically interesting planets. Some of the recently discovered extrasolar planetary systems contain multiple planets, and it is likely that some of these are …
We investigate the evolution of two-planet systems embedded in a protoplanetary disc, which are composed of a Jupiter-mass planet plus another body located further out in the disc. We consider outermost planets with masses ranging from 10 earth masses to 1 M_J. We also examine the case of outermost bodies with masses < 10 earth masses (M_E). Differential migration of the planets due to disc torques leads to different evolution outcomes depending on the mass of the outer protoplanet. For planets with mass < 3.5 M_E the type II migration rate of the giant exceeds the type I migration rate of the…
We calculate the linear torque exerted by a planet on a circular orbit on a disc containing a toroidal magnetic field. All fluid perturbations are singular at the so--called magnetic resonances, where the Doppler shifted frequency of the perturbation matches that of a slow MHD wave propagating along the field line. These lie on both sides of the corotation radius. Waves propagate outside the Lindblad resonances, and also in a restricted region around the magnetic resonances. The magnetic resonances contribute to a significant global torque which, like the Lindblad torque, is negative (positive…
We show that interaction with a gas disk may produce young planetary systems with closely-spaced orbits, stabilized by mean-motion resonances between neighbors. On longer timescales, after the gas is gone, interaction with a remnant planetesimal disk tends to pull these configurations apart, eventually inducing dynamical instability. We show that this can lead to a variety of outcomes; some cases resemble the Solar System, while others end up with high-eccentricity orbits reminiscent of the observed exoplanets. A similar mechanism has been previously suggested as the cause of the lunar Late He…
A new era of directly imaged extrasolar planets has produced a three-planet system (Marois et al. 2008), where the masses of the planets have been estimated by untested cooling models. We point out that the nominal circular, face-on orbits of the planets lead to a dynamical instability in ~10^5 yr, a factor of at least 100 shorter than the estimated age of the star. Reduced planetary masses produce stability only for unreasonably small planets (<~2 M_ Jup). Relaxing the face-on assumption, but still requiring circular orbits while fitting the observed positions, makes the instability time even…
Observations in the past decade have revealed extrasolar planets with a wide range of orbital semimajor axes and eccentricities. Based on the present understanding of planet formation via core accretion and oligarchic growth, we expect that giant planets often form in closely packed configurations. While the protoplanets are embedded in a protoplanetary gas disk, dissipation can prevent eccentricity growth and suppress instabilities from becoming manifest. However, once the disk dissipates, eccentricities can grow rapidly, leading to close encounters between planets. Strong planet--planet grav…
Following the general numerical analysis of Melita and Woolfson (1996), I showed in a recent paper that a restricted, planar, circular planetary system consisting of Sun, Jupiter and Saturn would be captured in a near (2:1) resonance when one would allow for frictional dissipation due to interplanetary medium (Haghighipour, 1998). In order to analytically explain this resonance phenomenon, the method of partial averaging near a resonance was utilized and the dynamics of the first-order partially averaged system at resonance was studied. Although in this manner, the finding that resonance lock …
The orbits of the outer two known planets orbiting Upsilon Andromedae are remarkably eccentric. Planet C possesses an orbital eccentricity of e1 = 0.253. For the more distant planet D, e2 = 0.308. Previous dynamical analyses strongly suggest that the two orbits are nearly co-planar and are trapped in an apsidal resonance in which the difference between their longitudes of periastron undergoes a bounded oscillation about 0 degrees. Here we elucidate the origin of these large eccentricities and of the apsidal alignment. Resonant interactions between a remnant circumstellar disk of gas lying exte…
The resonance perturbation of planetary ring by distant satellite is considered. The gravity interaction between rings particles is taken into account. The gravitation interaction between particles in rings can change shape of non-circular rings, so it is differ from keplerian ellipse. The shift between simple mean motion resonances and parametric resonance zones is detected. This shift depends on ring properties.
Spherical meteoroids orbiting the Sun experience orbital decay due to the Poynting-Robertson effect and can become trapped in commensurability resonances with a planet. Detail conditions for trapping in the planar circular restricted three-body problem with action of solar electromagnetic radiation on spherical grain are presented. The results are given in terms of the radiation presure factor $β$. In general, the greater value of $β$, the shorter capture time in a resonance. Captures of particles may be even several times longer than it is presented in literature. Analytical formula for maxim…
In most extrasolar planetary systems, the present orbits of known giant planets admit the existence of stable terrestrial planets. Those same giant planets, however, have typically eccentric orbits that hint at violent early dynamics less benign for low mass planet formation. Under the assumption that massive planet eccentricities are the end point of gravitational scattering in multiple planet systems, we study the evolution of the building blocks of terrestrial planets during the scattering process. We find that typically, evolutionary sequences that result in a moderately eccentric giant pl…
We show that in a system of two planets initially in nearly circular orbits, an impulse perturbation that imparts a finite eccentricity to one planet's orbit causes the other planet's orbit to become eccentric as well, and also naturally results in a libration of their relative apsidal longitudes for a wide range of initial conditions. We suggest that such a mechanism may explain orbital eccentricities and apsidal resonance in some exo-planetary systems. The eccentricity impulse could be caused by the ejection of a planet from these systems, or by torques from a primordial gas disk. The amplit…
Most multi-planetary systems are characterized by hot-Jupiters close to their central star, moving on eccentric orbits. From a dynamical point of view, compact multi-planetary systems form a specific class of the general N-body problem (where N>=3). Moreover, extrasolar planets are found in prograde orbits about their host star, and often in mean motion resonances (MMR). In a first step, we study theoretically a new stabilizing mechanism suitable for compact two-planet systems. This mechanism involves counter-revolving orbits forming a retrograde MMR. In a second step, we investigate the feasi…
This paper continues previous work on the effects of turbulence on mean motion resonances in extrasolar planetary systems. Turbulence is expected to arise in the disks that form planets, and these fluctuations act to compromise resonant configurations. This paper extends previous work by considering how interactions between the planets and possible damping effects imposed by the disk affect the outcomes. These physical processes are studied using three approaches: numerical integrations of the 3-body problem with additional forcing due to turbulence, model equations that reduce the problem to …
Divergent migration of planets within a viscous circumstellar disk can engender resonance crossings and dramatic excitation of orbital eccentricities. We provide quantitative criteria for the viability of this mechanism. For the orbits of two bodies to diverge, a ring of viscous material must be shepherded between them. As the ring diffuses in radius by virtue of its intrinsic viscosity, the two planets are wedged further apart. The ring mass must be smaller than the planetary masses so that the crossing of an individual resonance lasts longer than the resonant libration period. At the same ti…
This paper presents a parametric study of giant planet migration through the combined action of disk torques and planet-planet scattering. The torques exerted on planets during Type II migration in circumstellar disks readily decrease the semi-major axes, whereas scattering between planets increases the orbital eccentricities. This paper presents a parametric exploration of the possible parameter space for this migration scenario using two (initial) planetary mass distributions and a range of values for the time scale of eccentricity damping (due to the disk). For each class of systems, many r…
A planet orbiting around a star in a binary system can be ejected if it lies too far from its host star. We find that instability boundaries first obtained in numerical studies can be explained by overlap between sub-resonances within mean-motion resonances (mostly of the j:1 type). Strong secular forcing from the companion displaces the centroids of different sub-resonances, producing large regions of resonance overlap. Planets lying within these overlapping regions experience chaotic diffusion, which in most cases leads to their eventual ejection. The overlap region extends to shorter-period…
We integrate the orbital solutions of the planets orbiting 55 Cnc. In the simulations, we find that not only three resonant arguments $θ_{1}=λ_{1}-3λ_{2}+2\tildeω_{1}$, $θ_{2}=λ_{1}-3λ_{2}+2\tildeω_{2}$ and $θ_{3}=λ_{1}-3λ_{2}+(\tildeω_{1}+\tildeω_{2})$ librate respectively, but the relative apsidal longitudes $Δω$ also librates about $250^{\circ}$ for millions of years. The results imply the existence of the 3:1 resonance and the apsidal resonance for the studied system. We emphasize that the mean motion resonance and apsidal locking can act as two important mechanisms of stabilizing the syst…
