Theoretical models of planetary system formation: mass vs. semi-major axis

Context. Planet formation models have been developed during the past years to try to reproduce what has been observed of both the solar system and the extrasolar planets. Some of these models have partially succeeded, but they focus on massive planets and, for the sake of simplicity, exclude planets belonging to planetary systems. However, more and more planets are now found in planetary systems. This tendency, which is a result of radial velocity, transit, and direct imaging surveys, seems to be even more pronounced for low-mass planets. These new observations require improving planet formation models, including new physics, and considering the formation of systems. Aims: In a recent series of papers, we have presented some improvements in the physics of our models, focussing in particular on the internal structure of forming planets, and on the computation of the excitation state of planetesimals and their resulting accretion rate. In this paper, we focus on the concurrent effect of the formation of more than one planet in the same protoplanetary disc and show the effect, in terms of architecture and composition of this multiplicity. Methods: We used an N-body calculation including collision detection to compute the orbital evolution of a planetary system. Moreover, we describe the effect of competition for accretion of gas and solids, as well as the effect of gravitational interactions between planets. Results: We show that the masses and semi-major axes of planets are modified by both the effect of competition and gravitational interactions. We also present the effect of the assumed number of forming planets in the same system (a free parameter of the model), as well as the effect of the inclination and eccentricity damping. We find that the fraction of ejected planets increases from nearly 0 to 8% as we change the number of embryos we seed the system with from 2 to 20 planetary embryos. Moreover, our calculations show that, when considering planets more massive than ~5 M⊕, simulations with 10 or 20 planetary embryos statistically give the same results in terms of mass function and period distribution.

Planet formation models: the interplay with the planetesimal disc

Context. According to the sequential accretion model (or core-nucleated accretion model), giant planet formation is based first on the formation of a solid core which, when massive enough, can gravitationally bind gas from the nebula to form the envelope. The most critical part of the model is the formation time of the core: to trigger the accretion of gas, the core has to grow up to several Earth masses before the gas component of the protoplanetary disc dissipates. Aims: We calculate planetary formation models including a detailed description of the dynamics of the planetesimal disc, taking into account both gas drag and excitation of forming planets. Methods: We computed the formation of planets, considering the oligarchic regime for the growth of the solid core. Embryos growing in the disc stir their neighbour planetesimals, exciting their relative velocities, which makes accretion more difficult. Here we introduce a more realistic treatment for the evolution of planetesimals' relative velocities, which directly impact on the formation timescale. For this, we computed the excitation state of planetesimals, as a result of stirring by forming planets, and gas-solid interactions. Results: We find that the formation of giant planets is favoured by the accretion of small planetesimals, as their random velocities are more easily damped by the gas drag of the nebula. Moreover, the capture radius of a protoplanet with a (tiny) envelope is also larger for small planetesimals. However, planets migrate as a result of disc-planet angular momentum exchange, with important consequences for their survival: due to the slow growth of a protoplanet in the oligarchic regime, rapid inward type I migration has important implications on intermediate-mass planets that have not yet started their runaway accretion phase of gas. Most of these planets are lost in the central star. Surviving planets have masses either below 10 M⊕ or above several Jupiter masses. Conclusions: To form giant planets before the dissipation of the disc, small planetesimals (~0.1 km) have to be the major contributors of the solid accretion process. However, the combination of oligarchic growth and fast inward migration leads to the absence of intermediate-mass planets. Other processes must therefore be at work to explain the population of extrasolar planets that are presently known.

Gravitational slopes, geomorphology, and material strengths of the nucleus of comet 67P/Churyumov-Gerasimenko from OSIRIS observations

Aims. We study the link between gravitational slopes and the surface morphology on the nucleus of comet 67P/Churyumov-Gerasimenko and provide constraints on the mechanical properties of the cometary material (tensile, shear, and compressive strengths). Methods. We computed the gravitational slopes for five regions on the nucleus that are representative of the different morphologies observed on the surface (Imhotep, Ash, Seth, Hathor, and Agilkia), using two shape models computed from OSIRIS images by the stereo-photoclinometry (SPC) and stereo-photogrammetry (SPG) techniques. We estimated the tensile, shear, and compressive strengths using different surface morphologies (overhangs, collapsed structures, boulders, cliffs, and Philae's footprint) and mechanical considerations. Results. The different regions show a similar general pattern in terms of the relation between gravitational slopes and terrain morphology: i) low-slope terrains (0-20 degrees) are covered by a fine material and contain a few large (>10 m) and isolated boulders; ii) intermediate-slope terrains (20-45 degrees) are mainly fallen consolidated materials and debris fields, with numerous intermediate-size boulders from <1m to 10m for the majority of them; and iii) high-slope terrains (45-90 degrees) are cliffs that expose a consolidated material and do not show boulders or fine materials. The best range for the tensile strength of overhangs is 3-15 Pa (upper limit of 150 Pa), 4-30 Pa for the shear strength of fine surface materials and boulders, and 30-150 Pa for the compressive strength of overhangs (upper limit of 1500 Pa). The strength-to-gravity ratio is similar for 67P and weak rocks on Earth. As a result of the low compressive strength, the interior of the nucleus may have been compressed sufficiently to initiate diagenesis, which could have contributed to the formation of layers. Our value for the tensile strength is comparable to that of dust aggregates formed by gravitational instability and tends to favor a formation of comets by the accrection of pebbles at low velocities.

Theoretical models of planetary system formation II. Post-formation evolution

Aims. We extend the results of planetary formation synthesis by computing the long-term evolution of synthetic systems from the clearing of the gas disk into the dynamical evolution phase. Methods. We use the symplectic integrator SyMBA to numerically integrate the orbits of planets for 100 Myr, using populations from previous studies as initial conditions. Results. We show that within the populations studied, mass and semimajor axis distributions experience only minor changes from post-formation evolution. We also show that, depending upon their initial distribution, planetary eccentricities can statistically increase or decrease as a result of gravitational interactions. We find that planetary masses and orbital spacings provided by planet formation models do not result in eccentricity distributions comparable to observed exoplanet eccentricities, requiring other phenomena, such as stellar fly-bys, to account for observed eccentricities.