A study on dynamic processes at late stages in the formation of planetary systems in gas and dust disks

The discovered exoplanetary systems have highly diverse dynamic properties, which differ from those of the Solar System. A single model including planet migration effects and their gravitational interaction is used to investigate the features of dynamic processes that lead to the formation of giant-planet systems with different orbital characteristics. It is shown for a system of four giant planets similar to the Solar System how Type I migration could lead to all the planets being captured into resonant configurations. The resonant motion can continue for a long period of time after the transition to Type II migration and after the dissipation of the gas-and-dust disk. The three-planet system of GJ 876 is used to investigate the migration of the planets inward the orbit of the most massive planet and their capture into low-order resonant configurations under the conditions of Type II migration. A system similar to the exoplanetary system of HD 102272 is used to study the capture into high-order resonances followed by an increase in the orbit’s eccentricity.     

Formation of gas giant planets: core accretion models with fragmentation and planetary envelope

We have calculated formation of gas giant planets based on the standard core accretion model including effects of fragmentation and planetary envelope. The accretion process is found to proceed as follows. As a result of runaway growth of planetesimals with initial radii of ∼10 km, planetary embryos with a mass of ∼10²⁷ g (∼ Mars mass) are found to form in ∼10⁵ years at Jupiter's position (5.2 AU), assuming a large enough value of the surface density of solid material (25 g/cm²) in the accretion disk at that distance. Strong gravitational perturbations between the runaway planetary embryos and the remaining planetesimals cause the random velocities of the planetesimals to become large enough for collisions between small planetesimals to lead to their catastrophic disruption. This produces a large number of fragments. At the same time, the planetary embryos have envelopes, that reduce energies of fragments by gas drag and capture them. The large radius of the envelope increases the collision rate between them, resulting in rapid growth of the planetary embryos. By the combined effects of fragmentation and planetary envelope, the largest planetary embryo with 21M_⊕ forms at 5.2 AU in 3.8×10⁶ years. The planetary embryo is massive enough to start a rapid gas accretion and forms a gas giant planet.     

The formation and habitability of terrestrial planets in the presence of close-in giant planets

‘Hot jupiters,’ giant planets with orbits very close to their parent stars, are thought to form farther away and migrate inward via interactions with a massive gas disk. If a giant planet forms and migrates quickly, the planetesimal population has time to re-generate in the lifetime of the disk and terrestrial planets may form [P.J. Armitage, A reduced efficiency of terrestrial planet formation following giant planet migration, Astrophys. J. 582 (2003) L47-L50]. We present results of simulations of terrestrial planet formation in the presence of hot/warm jupiters, broadly defined as having orbital radii ?0.5 AU. We show that terrestrial planets similar to those in the Solar System can form around stars with hot/warm jupiters, and can have water contents equal to or higher than the Earth's. For small orbital radii of hot jupiters (e.g., 0.15, 0.25 AU) potentially habitable planets can form, but for semi-major axes of 0.5 AU or greater their formation is suppressed. We show that the presence of an outer giant planet such as Jupiter does not enhance the water content of the terrestrial planets, but rather decreases their formation and water delivery timescales. We speculate that asteroid belts may exist interior to the terrestrial planets in systems with close-in giant planets.     

A dynamical analysis of the 14 Herculis planetary system

Precision radial velocity measurements of the Sun-like dwarf 14 Herculis published by Naef et al., Butler et al. and Wittenmyer, Endl & Cochran reveal a Jovian planet in a 1760-d orbit and a trend indicating the second distant object. On the grounds of dynamical considerations, we test a hypothesis that the trend can be explained by the presence of an additional giant planet. We derive dynamical limits to the orbital parameters of the putative outer Jovian companion in an orbit within ∼13 au. In this case, the mutual interactions between the Jovian planets are important for the long-term stability of the system. The best self-consistent and stable Newtonian fit to an edge-on configuration of Jovian planets has the outer planet in 9-au orbit with a moderate eccentricity of ∼0.2 and confined to a zone spanned by the low-order mean motion resonances 5:1 and 6:1. This solution lies in a shallow minimum of (χ²_ν)^1/2 and persists over a wide range of the system inclination. Other stable configurations within 1σ confidence interval of the best fit are possible for the semimajor axis of the outer planet in the range of (6,13) au and the eccentricity in the range of (0, 0.3). The orbital inclination cannot yet be determined but when it decreases, both planetary masses approach ∼10 m _J and for i ∼ 30° the hierarchy of the masses is reversed.     

Formation of the giant planets

Observational constraints on interior models of the giant planets indicate that these planets were all much hotter when they formed and they all have rock and/or ice cores of ten to thirty earth masses. These cores are probably soluble in the envelopes above, especially in Jupiter and Saturn, and are therefore likely to be primordial. They persist despite the continual upward mixing by thermally driven convection throughout the age of the solar system, because of the inefficiency of double-diffusive convection. Thus, these planets most probably formed by the hydrodynamic collapse of a gaseous envelope onto a core rather than by direct instability of the gaseous solar nebula. Recent calculations by Mizuno (1980, Prog. Theor. Phys.64, 544) show that this formation mechanism may explain the similarity of giant planet core masses. Problems remain however, and no current model is entirely satisfactory in explaining the properties of the giant planets and simultaneously satisfying the terrestrial planet constraints. Satellite systematics and protoplanetary disk nebulae are also discussed and related to formation conditions.     

Models of the giant planets

Models of the giant planets were constructed based on the assumption that the hydrogen to helium ratio is solar in these planets. This assumption, together with arguments about the condensation sequence in the primitive solar nebula, yields models with a central core of rock and possibly ice surrounded by an envelope of hydrogen, helium, methane, ammonia, and water. These last three volatiles may be individually enhanced due to condensation at the period of core formation. Jupiter was found to have a core of about 40 earth masses and a water enhancement in the atmosphere of about 7.5 times the solar value. Saturn was found to have a core of 20 earth masses and a water enhancement in the atmosphere of about 25 times the solar value. Rock plus ice constitute 75–85% of the mass of Uranus and Neptune. Temperatures in the interiors of these planets are probably above the melting points, if there is an adiabatic relation throughout the interiors. Some aspects of the sensitivities of these results to uncertainties in rotational flattening are discussed.     

Numerical simulation of the final stages of terrestrial planet formation

Three representative numerical simulations of the growth of the terrestrial planets by accretion of large protoplanets are presented. The mass and relative-velocity distributions of the bodies in these simulations are free to evolve simultaneously in response to close gravitational encounters and occasional collisions between bodies. The collisions between bodies, therefore, arise in a natural way and the assumption of expressions for the relative velocity distribution and the gravitational collision cross section is unnecessary. These simulations indicate that the growth of bodies with final masses approaching those of Venus and the Earth is possible, at least for the case of a two-dimensional system. Simulations assuming an initial uniform distribution of orbital eccentricities on the interval from 0 to e_max are found to produce final states containing too many bodies with masses which are too small when e_max < 0.10, while simulations with e_max > 0.20 result in too many catastrophic collisions between bodies thus preventing rapid accretion of planetary-size bodies. The e_max = 0.15 simulation ends with a state surprisingly similar to that of the present terrestrial planets and, therefore, provides a rough estimate of the range of radial sampling to be expected for the terrestrial planets.     

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.     

Aeronomy of extra-solar giant planets at small orbital distances

One-dimensional aeronomical calculations of the atmospheric structure of extra-solar giant planets in orbits with semi-major axes from 0.01 to 0.1 AU show that the thermospheres are heated to over 10,000 K by the EUV flux from the central star. The high temperatures cause the atmosphere to escape rapidly, implying that the upper thermosphere is cooled primarily by adiabatic expansion. The lower thermosphere is cooled primarily by radiative emissions from H⁺₃, created by photoionization of H₂ and subsequent ion chemistry. Thermal decomposition of H₂ causes an abrupt change in the composition, from molecular to atomic, near the base of the thermosphere. The composition of the upper thermosphere is determined by the balance between photoionization, advection, and H⁺ recombination. Molecular diffusion and thermal conduction are of minor importance, in part because of large atmospheric scale heights. The energy-limited atmospheric escape rate is approximately proportional to the stellar EUV flux. Although escape rates are large, the atmospheres are stable over time scales of billions of years.     

Love numbers of the giant planets

We calculate the Love numbers k_n for n = 2 to 10, and determine the “gravitational noise” from tides. The new values k₂ for Jupiter, Saturn, and Uranus yield new estimates for the planetary dissipation functions: $\text{Q}{}_{\mathrm{<mtext>J</mtext>}}\text{? 2.5 × 10}{}^4$, $\text{Q}{}_{\mathrm{<mtext>S</mtext>}}\text{? 1.4 × 10}{}^4\text{, Q}{}_{\mathrm{<mtext>U</mtext>}}\text{? 5 × 10}{}^3$.     

Delivery of icy planetesimals to inner planets in the Proxima Centauri planetary system

The estimates of the delivery of icy planetesimals from the feeding zone of Proxima Centauri c (with mass equal to 7m_E, m_E is the mass of the Earth) to inner planets b and d were made. They included the studies of the total mass of planetesimals in the feeding zone of planet c and the probabilities of collisions of such planetesimals with inner planets. This total mass could be about 10–15m_E. It was estimated based on studies of the ratio of the mass of planetesimals ejected into hyperbolic orbits to the mass of planetesimals collided with forming planet c. At integration of the motion of planetesimals, the gravitational influence of planets c and b and the star was taken into account. In most series of calculations, planetesimals collided with planets were excluded from integrations. Based on estimates of the mass of planetesimals ejected into hyperbolic orbits, it was concluded that during the growth of the mass of planet c the semi-major axis of its orbit could decrease by at least a factor of 1.5. Depending on possible gravitational scattering due to mutual encounters of planetesimals, the total mass of material delivered by planetesimals from the feeding zone of planet c to planet b was estimated to be between 0.002m_E and 0.015m_E. Probably, the amount of water delivered to Proxima Centauri b exceeded the mass of water in Earth's oceans. The amount of material delivered to planet d could be a little less than that delivered to planet b.     

Atmospheric thermal structures of the giant planets

Thermal models of planetary atmospheres can be calculated from assumptions of the energy budget of the atmosphere and from the knowledge of the effective temperature of the studied planet. On the other hand, the retrieval of the thermal atmospheric profiles from infrared measurements by means of the numerical inversion of the radiative transfer equation presents the advantages of not requiring such assumptions. The extent of the atmospheric range which can then be sounded is examined and the vertical resolution of the inferred profiles is discussed. Comparisons of thermal models and retrieved thermal profiles are made for the four giant planets. The retrieved profiles lead to brightness temperature spectra which fit all the available infrared measurements fairly well for Jupiter and Saturn but only part of them for Uranus and Neptune. The values of the planetary effective temperatures calculated from the retrieved profiles show that Jupiter, Saturn, and Neptune have strong internal heating sources while Uranus probably has a very small or null one.     

Capture of planetesimals by the giant planets

We investigate the possibility of gravitational capture of planetesimals as temporary or permanent satellites of Uranus and Neptune during the process of planetary growth. The capture mechanism is based in the enhancement of the Hill's sphere of action not only due to the mass acquired by the planet, but also by the variation of the planet-Sun distance as a consequence of the scattering of planetesimals by the planets of the outer solar system. Our calculations indicate that satellite capture was very important, specially during the first stages of the accretion process, contributing in a significant way to the planetary growth.     

A dynamo driven by zonal jets at the upper surface: Applications to giant planets

We present a dynamo mechanism arising from the presence of barotropically unstable zonal jet currents in a rotating spherical shell. The shear instability of the zonal flow develops in the form of a global Rossby mode, whose azimuthal wavenumber depends on the width of the zonal jets. We obtain self-sustained magnetic fields at magnetic Reynolds numbers greater than 10³. We show that the propagation of the Rossby waves is crucial for dynamo action. The amplitude of the axisymmetric poloidal magnetic field depends on the wavenumber of the Rossby mode, and hence on the width of the zonal jets. We discuss the plausibility of this dynamo mechanism for generating the magnetic field of the giant planets. Our results suggest a possible link between the topology of the magnetic field and the profile of the zonal winds observed at the surface of the giant planets. For narrow Jupiter-like jets, the poloidal magnetic field is dominated by an axial dipole whereas for wide Neptune-like jets, the axisymmetric poloidal field is weak.     

Calculations of the evolution of the giant planets

Evolutionary calculations are presented for spherically symmetric protoplanetary configurations with a homogeneous solar composition and with masses of 10^?3, 1.5 × 10^?3, 2.85 × 10^?4, and 4.2 × 10^?4M_⊙. Recent improvements in equation-of-state and opacity calculations are incorporated. Sequences start as subcondensations in the solar nebula with densities of ～10^?10 to 10^?11 g cm^?3, evolve through a hydrostatic phase lasting 10⁵ to 10⁷ years, undergo dynamic collapse due to dissociation of molecular hydrogen, and regain hydrostatic equilibrium with densities ～1 g cm^?3. The nature of the objects at the onset of the final phase of cooling and contraction is discussed and compared with previous calculations.     

Resonance capture and the evolution of the planets

The present nearly resonant orbital periods of the planets are explained in terms of past two-body resonance capture of planetesimals in the solar nebula. Planetary formation then occurs sequentially starting with Jupiter for the outer planets and Venus for the inner planets and propagates outward due to two-body orbital resonances. It might now be possible to reconstruct the evolutionary history of the planets from their nearly commensurable orbital periods and, hence, provide an explanation for the Titius-Bode law.