The web of three-planet resonances in the outer Solar System: II. A source of orbital instability for Uranus and Neptune

The motion of the giant planets from Jupiter to Neptune is chaotic with Lyapunov time of approximately 10 Myr. A recent theory explains the presence of this chaos with three-planet mean-motion resonances, i.e. resonances among the orbital periods of at least three planets. We find that the distribution of these resonances with respect to the semi-major axes of all the planets is compatible with orbital instability. In particular, they overlap in a region of 10⁻³ AU with respect to the variation of the semi-major axes of Uranus and Neptune. Fictitious planetary systems with initial conditions in this region can undergo systematic variations of semi-major axes. The true Solar System is marginally in this region, and Uranus and Neptune undergo very slow systematic variations of semi-major axes with speed of order 10⁻⁴ AU/Gyr.     

Neptune migration model with one extra planet

We explore conventional Neptune migration model with one additional planet of mass at 0.1-2.0M_⊕. This planet inhabited in the 3:2 mean motion resonance with Neptune during planet migration epoch, and then escaped from the Kuiper belt when jovian planets parked near the present orbits. Adding this extra planet and assuming the primordial disk truncated at about 45 AU in the conventional Neptune migration model, it is able to explain the complex structure of the observed Kuiper belt better than the usual Neptune migration model did in several respects, which are the following. (1) High-inclination Plutinos with i?15-35° are produced. (2) Generating the excitation of the classical Kuiper belt objects, which have moderate eccentricities and inclinations. (3) Producing the larger ratio of Neptune’s 3:2 to 2:1 resonant particles, and the lower ratio of particles in the 3:2 resonance to those in the classical belt, which may be more consistent with observations. (4) Finally, several Neptune’s 5:2 resonant particles are obtained. However, numerical experiments imply that this model is a low-probability event. In addition to the low probability, two features produced by this model may be inconsistent with the observations. They are small number of low-inclination particles in the classical belt, and the production of a remnant population with near-circular and low-inclination orbit within . According to our present study, including one extra planet in the conventional Neptune migration model as the scenario we explored here may be unsuitable because of the low probability, and the two drawbacks mentioned above, although this model can explain better several features which is hard to produce by the conventional Neptune migration model. The issues of low-probability event and the lack of low-inclination KBOs in the classical belt are interesting and may be studied further under a more realistic consideration.     

The web of three-planet resonances in the outer Solar System

In this paper we numerically detect the web of three-planet resonances (i.e., resonances among mean anomalies, nodes and perihelia of three planets) with respect to the variation of the semi-major axis of Saturn and Jupiter, in a model including the planets from Jupiter to Neptune. The measure confirms the relevance of these resonances in the long-term evolution of the outer Solar System and provides a technique to identify some of the related coefficients.     

Giant planet migration through the action of disk torques and planet-planet scattering

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 a, 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 realizations of the simulations are performed in order to determine the distributions of the resulting orbital elements of the surviving planets; this paper presents the results of ∼8500 numerical experiments. Our goal is to study the physics of this particular migration mechanism and to test it against observations of extrasolar planets. The action of disk torques and planet-planet scattering results in a distribution of final orbital elements that fills the a-? plane, in rough agreement with the orbital elements of observed extrasolar planets. In addition to specifying the orbital elements, we characterize this migration mechanism by finding the percentages of ejected and accreted planets, the number of collisions, the dependence of outcomes on planetary masses, the time spent in 2:1 and 3:1 resonances, and the effects of the planetary IMF. We also determine the distribution of inclination angles of surviving planets and the distribution of ejection speeds for exiled planets.     

Transformation of Trojans into quasi-satellites during planetary migration and their subsequent close-encounters with the host planet

We use numerical integrations to investigate the dynamical evolution of resonant Trojan and quasi-satellite companions during the late stages of migration of the giant planets Jupiter, Saturn, Uranus, and Neptune. Our migration simulations begin with Jupiter and Saturn on orbits already well separated from their mutual 2:1 mean-motion resonance. Neptune and Uranus are decoupled from each other and have orbital eccentricities damped to near their current values. From this point we adopt a planet migration model in which the migration speed decreases exponentially with a characteristic timescale τ (the e-folding time). We perform a series of numerical simulations, each involving the migrating giant planets plus test particle Trojans and quasi-satellites. We find that the libration frequencies of Trojans are similar to those of quasi-satellites. This similarity enables a dynamical exchange of objects back and forth between the Trojan and quasi-satellite resonances during planetary migration. This exchange is facilitated by secondary resonances that arise whenever there is more than one migrating planet. For example, secondary resonances may occur when the circulation frequencies, f, of critical arguments for the Uranus-Neptune 2:1 mean-motion near-resonance are commensurate with harmonics of the libration frequency of the critical argument for the Trojan and quasi-satellite 1:1 mean-motion resonance . Furthermore, under the influence of these secondary resonances quasi-satellites can have their libration amplitudes enlarged until they undergo a close-encounter with their host planet and escape from the resonance. High-resolution simulations of this escape process reveal that ≈80% of jovian quasi-satellites experience one or more close-encounters within Jupiter’s Hill radius (R_H) as they are forced out of the quasi-satellite resonance. As many as ≈20% come within R_H/4 and ≈2.5% come within R_H/10. Close-encounters of escaping quasi-satellites occur near or even below the 2-body escape velocity from the host planet. Finally, the exchange and escape of Trojans and quasi-satellites continues to as late as 6-9τ in some simulations. By this time the dynamical evolution of the planets is strongly dominated by distant gravitational perturbations between the planets rather than the migration force. This suggests that exchange and escape of Trojans and quasi-satellites may be a contemporary process associated with the present-day near-resonant configuration of some of the giant planets in our Solar System.     

A hybrid scenario for gas giant planet formation in rings

The core-accretion mechanism for gas giant formation may be too slow to create all observed gas giant planets during reasonable gas disk lifetimes, but it has yet to be firmly established that the disk instability model can produce permanent bound gaseous protoplanets under realistic conditions. Based on our recent simulations of gravitational instabilities in disks around young stars, we suggest that, even if instabilities due to disk self-gravity do not produce gaseous protoplanets directly, they may create persistent dense rings that are conducive to accelerated growth of gas giants through core accretion. The rings occur at and near the boundary between stable and unstable regions of the disk and appear to be produced by resonances with discrete spiral modes on the unstable side.     

Neptune’s rotational period suggested by the extraordinary stability of two features

The interior rotation and motions in giant planets have generally been probed only at radio wavelengths from spacecraft near the planet, except for Jupiter’s radio emission detectable from Earth. Here I suggest that Neptune’s interior can be indirectly probed at visible wavelength by tracking 10 features that are connected with a stationary latitudinal speed pattern of 7 m/s amplitude. All 10 features remained aligned at the same longitude throughout the Voyager observation period in 1989. Two of them, the South Polar Wave and South Polar Feature, have been observed from Earth for ∼20 years, but their extraordinary rotational stability was never recognized. They probably pinpoint Neptune’s rotational period (15.9663 ± 0.0002 h), one of the largest improvements in 346 years of measuring the giant planets’ rotations. The previous best estimate of Neptune’s rotational period (16.108 ± 0.006 h) was based on Voyager 2 radio data (Lecacheux, A., Zarka, P., Desch, M.D., Evans, D.R. [1993]. Geophys. Res. Lett. 20, 2711-2714). The new result suggests an upward revision of the mass of Neptune’s core. This finding may also question the accepted value of Uranus’ rotational period. The first reliable wind measurements within 15° of Neptune’s South Pole, based on tracking four features in Voyager images, show a 300 m/s eastward jet peaking near 76° South, while the area within 4° of the South Pole seems to be rotationally locked to the interior. These new observations of the stationary features and winds could address the long-standing question about the depth of the atmospheric circulation and may allow some constraints on convection currents in Neptune’s interior.     

Detecting the signatures of Uranus and Neptune

With more than 15 years since the first radial velocity discovery of a planet orbiting a Sun-like star, the time baseline for radial velocity surveys is now extending out beyond the orbit of Jupiter analogs. The sensitivity to exoplanet orbital periods beyond that of Saturn orbital radii however is still beyond our reach such that very few clues regarding the prevalence of ice giants orbiting solar analogs are available to us. Here we simulate the radial velocity, transit, and photometric phase amplitude signatures of the Solar System giant planets, in particular Uranus and Neptune, and assess their detectability. We scale these results for application to monitoring low-mass stars and compare the relative detection prospects with other potential methods, such as astrometry and imaging. These results quantitatively show how many of the existing techniques are suitable for the detection of ice giants beyond the snow line for late-type stars and the challenges that lie ahead for the detection true Uranus/Neptune analogs around solar-type stars.     

The full set of gas giant structures: I: On the origin of planetary masses and the planetary initial mass function

Context

Current planet search programs are detecting extrasolar planets at a rate of 60 planets per year. These planets show more diverse properties than was expected.

Aims

We try to get an overview of possible gas giant (proto-) planets for a full range of orbital periods and stellar masses. This allows the prediction of the full range of possible planetary properties which might be discovered in the near future.

Methods

We calculate the purely hydrostatic structure of the envelopes of proto-planets that are embedded in protoplanetary disks for all conceivable locations: combinations of different planetesimal accretion rates, host star masses, and orbital separations. At each location all hydrostatic equilibrium solutions to the planetary structure equations are determined by variation of core mass and pressure over many orders of magnitude. For each location we analyze the distribution of planetary masses.

Results

We get a wide spectrum of core-envelope structures. However, practically all calculated proto-planets are in the planetary mass range. Furthermore, the planet masses show a characteristic bimodal, sometimes trimodal, distribution. For the first time, we identify three physical processes that are responsible for the three characteristic planet masses: self-gravity in the Hill sphere, compact objects, and a region of very low adiabatic pressure gradient in the hydrogen equation of state. Using these processes, we can explain the dependence of the characteristic masses on the planet’s location: orbital period, host star mass, and planetesimal accretion rate (luminosity). The characteristic mass caused by the self-gravity effect at close proximity to the host star is typically one Neptune mass, thus producing the so-called hot Neptunes.

Conclusions

Our results suggest that hot Jupiters with orbital period less than 64 days (the exact location of the boundary depends on stellar type and accretion rate) have quite distinct properties which we expect to be reflected in a different mass distribution of these planets when compared to the “normal” planetary population. We use our theoretical survey to produce an upper mass limit for embedded planets: the maximum embedded equilibrium mass (MEEM). This naturally explains the lack of high mass planets between 3 and 64 days orbital period.     

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 effects of metallicity and grain growth and settling on the early evolution of gaseous protoplanets

Giant protoplanets formed by gravitational instability in the outer regions of circumstellar disks go through an early phase of quasi-static contraction during which radii are large (∼1 AU) and internal temperatures are low (<2000 K). The main source of opacity in these objects is dust grains. We investigate two problems involving the effect of opacity on the evolution of isolated, non-accreting planets of 3, 5, and 7 M_J. First, we pick three different overall metallicities for the planet and simply scale the opacity accordingly. We show that higher metallicity results in slower contraction as a result of higher opacity. It is found that the pre-collapse time scale is proportional to the metallicity. In this scenario, survival of giant planets formed by gravitational instability is predicted to be more likely around low-metallicity stars, since they evolve to the point of collapse to small size on shorter time scales. But metal-rich planets, as a result of longer contraction times, have the best opportunity to capture planetesimals and form heavy-element cores. Second, we investigate the effects of opacity reduction as a result of grain growth and settling, for the same three planetary masses and for three different values of overall metallicity. When these processes are included, the pre-collapse time scale is found to be of order 1000 years for the three masses, significantly shorter than the time scale calculated without these effects. In this case the time scale is found to be relatively insensitive to planetary mass and composition. However, the effects of planetary rotation and accretion of gas and dust, which could increase the timescale, are not included in the calculation. The short time scale we find would preclude metal enrichment by planetesimal capture, as well as heavy-element core formation, over a large range of planetary masses and metallicities.     

Tidal interactions and disruptions of giant planets on highly eccentric orbits

We calculate the evolution of planets undergoing a strong tidal encounter using smoothed particle hydrodynamics (SPH), for a range of periastron separations. We find that outside the Roche limit, the evolution of the planet is well-described by the standard model of linear, non-radial, adiabatic oscillations. If the planet passes within the Roche limit at periastron, however, mass can be stripped from it, but in no case do we find enough energy transferred to the planet to lead to complete disruption. In light of the three new extrasolar planets discovered with periods shorter than two days, we argue that the shortest-period cases observed in the period-mass relation may be explained by a model whereby planets undergo strong tidal encounters with stars, after either being scattered by dynamical interactions into highly eccentric orbits, or tidally captured from nearly parabolic orbits. Although this scenario does provide a natural explanation for the edge found for planets at twice the Roche limit, it does not explain how such planets will survive the inevitable expansion that results from energy injection during tidal circularization.     

Resonance sticking in the scattered disk

We investigate the dynamical evolution of trans-neptunian objects (TNOs) in typical scattered disk orbits (scattered TNOs) by performing simulations using several thousand particles lying initially on Neptune-encountering orbits. We explore the role of resonance sticking in the scattered disk, a phenomenon characterized by multiple temporary resonance captures (‘resonances’ refers to external mean motion resonances with Neptune, which can be described in the form r:s, where the arguments r and s are integers). First, all scattered TNOs evolve through intermittent temporary resonance capture events and gravitational scattering by Neptune. Each scattered TNO experiences tens to hundreds of resonance captures over a period of 4 Gyr, which represents about 38% of the object's lifetime (mean value). Second, resonance sticking plays an important role at semimajor axes , where the great majority of such captures occurred. It is noteworthy that the stickiest (i.e., dominant) resonances in the scattered disk are located within this distance range and are those possessing the lowest argument s. This was evinced by r:1, r:2 and r:3 resonances, which played the greatest role during resonance sticking evolution, often leading to captures in several of their neighboring resonances. Finally, the timescales and likelihood of temporary resonance captures are roughly proportional to resonance strength. The dominance of low s resonances is also related to the latter. In sum, resonance sticking has an important impact on the evolution of scattered TNOs, contributing significantly to the longevity of these objects.     

The occurrence of high-order resonances and Kozai mechanism in the scattered disk

By means of numerical methods we explore the relevance of the high-order exterior mean motion resonances (MMR) with Neptune that a scattered disk object (SDO) can experience in its diffusion to the Oort cloud. Using a numerical method for estimate the strength of these resonances we show that high-eccentricity or high-inclination resonant orbits should have evident dynamical effects. We investigate the properties of the Kozai mechanism (KM) for non-resonant SDO's and the conditions that generate the KM inside a MMR associated with substantial changes in eccentricity and inclination. We found that the KM inside a MMR is typical for SDO's with Pluto-like or greater inclinations and is generated by the oscillation of ω inside the mixed (e,i) resonant terms of the disturbing function. A SDO diffusing to the Oort cloud should experience temporary captures in MMR, preferably of the type 1:N, and when evolving inside a MMR and experiencing the KM it can reach regions where the strength of the resonance drops and consequently there is a possibility of being decoupled from the resonance generating by this way a long-lived high-perihelion scattered disk object (HPSDO).     

Secular apsidal configuration of non-resonant exoplanetary systems

Using a high-order (order 12) expansion of the perturbative potential in powers of eccentricities [Libert, A.-S., Henrard, J., 2005. Celest. Mech. Dynam. Astron. 93, 187-200], we study the secular effects of two coplanar planets which are not in mean motion resonances. The main results concern eccentricity variations, oscillation amplitude of the angular difference of the apsidal lines (Δ?) and frequency of such an oscillation. We show that this analytical approach describes correctly the behaviour of most of the exosystems and underlines the known limitations of the linear Laplace-Lagrange theory. Apsidal configuration of υ Andromedae, HD 168443, HD 169830, HD 38529, HD 74156 and HD 12661 are examined. We also point out the great sensitivity of the υ Andromedae system to the initial values (e₁(0), e₂(0) or Δ?(0)).     

Origin of scattered disk resonant TNOs: Evidence for an ancient excited Kuiper belt of 50 AU radius

Resonance occupation of trans-neptunian objects (TNOs) in the scattered disk (>48 AU) was investigated by integrating the orbits of 85 observed members for 4 Gyr. Twenty seven TNOs were locked in the 9:4, 16:7, 7:3, 12:5, 5:2, 8:3, 3:1, 4:1, 11:2, and 27:4 resonances. We then explored mechanisms for the origin of the resonant structure in the scattered disk, in particular the long-term 9:4, 5:2, and 8:3 resonant TNOs (median 4 Gyr), by performing large scale simulations involving Neptune scattering and planetary migration over an initially excited planetesimals disk (wide range of eccentricities and inclinations). To explain the formation of Gyr-resident populations in such distant resonances, our results suggest the existence of a primordial planetesimal disk of at least 45-50 AU radius that suffered a dynamical perturbation leading to 0.1-0.3 or greater eccentricities and a range of inclinations up to ∼20° during early stages of the Solar System history, before planetary migration.     

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.     

Dynamical classification of trans-neptunian objects: Probing their origin, evolution, and interrelation

The orbital structure of trans-neptunian objects (TNOs) in the trans-neptunian belt (Edgeworth-Kuiper belt) and scattered disk provides important clues to understand the origin and evolution of the Solar System. To better characterize these populations, we performed computer simulations of currently observed objects using long-arc orbits and several thousands of clones. Our preliminary analysis identified 622 TNOs, and 65 non-resonant objects whose orbits penetrate that of at least one of the giant planets within 1 Myr (the centaurs). In addition, we identified 196 TNOs locked in resonances with Neptune, which, sorted by distance from the Sun, are 1:1 (Neptune trojans), 5:4, 4:3, 11:8, 3:2, 18:11, 5:3, 12:7, 19:11, 7:4, 9:5, 11:6, 2:1, 9:4, 16:7, 7:3, 12:5, 5:2, 8:3, 3:1, 4:1, 11:2, and 27:4. Kozai resonant TNOs are found inside the 3:2, 5:3, 7:4, and 2:1 resonances. We present detailed general features for the resonant populations (i.e., libration amplitude angles, libration centers, Kozai libration amplitudes, etc.). Taking together the simulations of Lykawka and Mukai [Lykawka, P.S., Mukai, T., 2007. Icarus 186, 331-341], an improved classification scheme is presented revealing five main classes: centaurs, resonant, scattered, detached and classical TNOs. Scattered and detached TNOs (non-resonant) have q (perihelion distance) <37 AU and q>40 AU, respectively. TNOs with 37 AU<q<40 AU occupy an intermediate region where both classes coexist. Thus, there are no clear boundaries between the scattered and detached regions. We also securely identified a total of 9 detached TNOs by using 4-5 Gyr orbital integrations. Classical objects are non-resonant TNOs usually divided into cold and hot populations. Their boundaries are as follows: cold classical TNOs (i?5°) are located at 37 AU<a<40 AU (q>37 AU) and 42 AU<a<47.5 AU (q>38 AU), and hot classical TNOs (i>5°) occupy orbits with 37 AU<a<47.5 AU (q>37 AU). However, a more firm classification is found with i>10° for hot classical TNOs. Lastly, we discuss some implications of our classification scheme comparing all TNOs with our model and other past models.     

Modeling the 3-D secular planetary three-body problem: Discussion on the outer υ Andromedae planetary system

The three-dimensional secular behavior of a system composed of a central star and two massive planets is modeled semi-analytically in the frame of the general three-body problem. The main dynamical features of the system are presented in geometrical pictures allowing us to investigate a large domain of the phase space of this problem without time-expensive numerical integrations of the equations of motion and without any restriction on the magnitude of the planetary eccentricities, inclinations and mutual distance. Several regimes of motion of the system are observed. With respect to the secular angle Δ?, possible motions are circulations, oscillations (around 0° and 180°), and high-eccentricity/inclination librations in secular resonances. With respect to the arguments of pericenter, ω₁ and ω₂, possible motions are direct circulation and high-inclination libration around ±90° in the Lidov-Kozai resonance. The regions of transition between domains of different regimes of motion are characterized by chaotic behavior. We apply the analysis to the case of the two outer planets of the υ Andromedae system, observed edge-on. The topology of the 3-D phase space of this system is investigated in detail by means of surfaces of section, periodic orbits and dynamical spectra, mapping techniques and numerical simulations. We obtain the general structure of the phase space, and the boundaries of the spatial secular stability. We find that this system is secularly stable in a large domain of eccentricities and inclinations.     

Evolution of the obliquities of the giant planets in encounters during migration

Tsiganis et al. [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461] have proposed that the current orbital architecture of the outer Solar System could have been established if it was initially compact and Jupiter and Saturn crossed the 2:1 orbital resonance by divergent migration. The crossing led to close encounters among the giant planets, but the orbital eccentricities and inclinations were damped to their current values by interactions with planetesimals. Brunini [Brunini, A., 2006. Nature 440, 1163-1165] has presented widely publicized numerical results showing that the close encounters led to the current obliquities of the giant planets. We present a simple analytic argument which shows that the change in the spin direction of a planet relative to an inertial frame during an encounter between the planets is very small and that the change in the obliquity (which is measured from the orbit normal) is due to the change in the orbital inclination. Since the inclinations are damped by planetesimal interactions on timescales much shorter than the timescales on which the spins precess due to the torques from the Sun, especially for Uranus and Neptune, the obliquities should return to small values if they are small before the encounters. We have performed simulations using the symplectic integrator SyMBA, modified to include spin evolution due to the torques from the Sun and mutual planetary interactions. Our numerical results are consistent with the analytic argument for no significant remnant obliquities.