The Orbital Architecture of Exoplanetary Systems

The great diversity of extrasolar planetary systems has challenged our understanding of how planets form. During the formation process their orbits are modified while the protoplanetary disk is present. After its dispersal orbits may also be modified as a result of mutual gravitational interactions leading to their currently observed configurations in the longer term. A number of potentially significant phenomena have been identified. These include radial migration of solids in the protoplanetary disk, radial migration of protoplanetary cores produced by disk-planet interaction and how it can be halted by protoplanet traps, formation of resonant systems and subsystems, and gravitational interactions among planets or between a planet and an external stellar companion. These interactions may cause excitation of orbital inclinations and eccentricities which in the latter case may attain values close to unity. When the eccentricity approaches unity, tidal interaction with the central star could lead to orbital circularization and a close orbiting Hot Jupiter, providing a competitive process to direct migration through the disk or in-situ formation. Long-term dynamical instability may also account for the relatively small number of observed compact systems of super-Earths and Neptune class planets that have attained and subsequently maintained linked commensurabilities in the long term.

Formation of planetary systems by pebble accretion and migration

Super-Earths – planets with sizes between the Earth and Neptune – are found in tighter orbits than that of the Earth around more than one third of main sequence stars. It has been proposed that super-Earths are scaled-up terrestrial planets that also formed similarly, through mutual accretion of planetary embryos, but in discs much denser than the solar protoplanetary disc. We argue instead that terrestrial planets and super-Earths have two clearly distinct formation pathways that are regulated by the pebble reservoir of the disc. Through numerical integrations, which combine pebble accretion and N-body gravity between embryos, we show that a difference of a factor of two in the pebble mass flux is enough to change the evolution from the terrestrial to the super-Earth growth mode. If the pebble mass flux is small, then the initial embryos within the ice line grow slowly and do not migrate substantially, resulting in a widely spaced population of approximately Mars-mass embryos when the gas disc dissipates. Subsequently, without gas being present, the embryos become unstable due to mutual gravitational interactions and a small number of terrestrial planets are formed by mutual collisions. The final terrestrial planets are at most five Earth masses. Instead, if the pebble mass flux is high, then the initial embryos within the ice line rapidly become sufficiently massive to migrate through the gas disc. Embryos concentrate at the inner edge of the disc and growth accelerates through mutual merging. This leads to the formation of a system of closely spaced super-Earths in the five to twenty Earth-mass range, bounded by the pebble isolation mass. Generally, instabilities of these super-Earth systems after the disappearance of the gas disc trigger additional merging events and dislodge the system from resonant chains. Therefore, the key difference between the two growth modes is whether embryos grow fast enough to undergo significant migration. The terrestrial growth mode produces small rocky planets on wider orbits like those in the solar system whereas the super-Earth growth mode produces planets in short-period orbits inside 1 AU, with masses larger than the Earth that should be surrounded by a primordial H/He atmosphere, unless subsequently lost by stellar irradiation. The pebble flux – which controls the transition between the two growth modes – may be regulated by the initial reservoir of solids in the disc or the presence of more distant giant planets that can halt the radial flow of pebbles.

Element history of the Laplace resonance: a dynamical approach

Context. We consider the three-body mean motion resonance defined by the Jovian moons Io, Europa, and Ganymede, which is commonly known as the Laplace resonance. In terms of the moons’ mean longitudes λ1 (Io), λ2 (Europa), and λ3 (Ganymede), this resonance is described by the librating argument φL ≡ λ1 − 3λ2 + 2λ3 ≈ 180°, which is the sum of φ12 ≡ λ1 − 2λ2 + ϖ2 ≈ 180° and φ23 ≡ λ2 − 2λ3 + ϖ2 ≈ 0°, where ϖ2 denotes Europa’s longitude of perijove. Aims. In particular, we construct approximate models for the evolution of the librating argument φL over the period of 100 yr, focusing on its principal amplitude and frequency, and on the observed mean motion combinations n1 − 2n2 and n2 − 2n3 associated with the quasi-resonant interactions above. Methods. First, we numerically propagated the Cartesian equations of motion of the Jovian system for the period under examination, and by comparing the results with a suitable set of ephemerides, we derived the main dynamical effects on the target quantities. Using these effects, we built an alternative Hamiltonian formulation and used the normal forms theory to precisely locate the resonance and to semi-analytically compute its main amplitude and frequency. Results. From the Cartesian model we observe that on the timescale considered and with ephemerides as initial conditions, both φL and the diagnostics n1 − 2n2 and n2 − 2n3 are well approximated by considering the mutual gravitational interactions of Jupiter and the Galilean moons (including Callisto), and the effect of Jupiter’s J2 harmonic. Under the same initial conditions, the Hamiltonian formulation in which Callisto and J2 are reduced to their secular contributions achieves larger errors for the quantities above, particularly for φL. By introducing appropriate resonant variables, we show that these errors can be reduced by moving in a certain action-angle phase plane, which in turn implies the necessity of a tradeoff in the selection of the initial conditions. Conclusions. In addition to being a good starting point for a deeper understanding of the Laplace resonance, the models and methods described are easily generalizable to different types of multi-body mean motion resonances. Thus, they are also prime tools for studying the dynamics of extrasolar systems.

Slippery evidence on masses in the Local Group

Once it is agreed that not all mass gives a significant contribution to light or any other emission, then one must rely on the dynamics of the visible objects to determine the total gravity field. It is clearly impossible to do this without subsidiary hypotheses. Here we shall assume that all members of the Local Group began together in the Big Bang, and that their dynamics have been governed by their mutual gravitational interactions since the system first achieved a size of some 200 kpc. This must have been some 109 years after the Big Bang.