Embedded in the gaseous protoplanetary disk, Jupiter and Saturn naturally become trapped in 3:2 resonance and migrate outward. This serves as the basis of the Grand Tack model. However, previous hydrodynamical simulations were restricted to isotherma
l disks, with moderate aspect ratio and viscosity. Here we simulate the orbital evolution of the gas giants in disks with viscous heating and radiative cooling. We find that Jupiter and Saturn migrate outward in 3:2 resonance in modest-mass ($M_{disk} approx M_{MMSN}$, where MMSN is the minimum-mass solar nebula) disks with viscous stress parameter $alpha$ between $10^{-3}$ and $10^{-2} $. In disks with relatively low-mass ($M_{disk} lesssim M_{MMSN}$) , Jupiter and Saturn get captured in 2:1 resonance and can even migrate outward in low-viscosity disks ($alpha le 10^{-4}$). Such disks have a very small aspect ratio ($hsim 0.02-0.03$) that favors outward migration after capture in 2:1 resonance, as confirmed by isothermal runs which resulted in a similar outcome for $h sim 0.02$ and $alpha le 10^{-4}$. We also performed N-body runs of the outer Solar System starting from the results of our hydrodynamical simulations and including 2-3 ice giants. After dispersal of the gaseous disk, a Nice model instability starting with Jupiter and Saturn in 2:1 resonance results in good Solar Systems analogs. We conclude that in a cold Solar Nebula, the 2:1 resonance between Jupiter and Saturn can lead to outward migration of the system, and this may represent an alternative scenario for the evolution of the Solar System.
Planets close to their stars are thought to form farther out and migrate inward due to angular momentum exchange with gaseous protoplanetary disks. This process can produce systems of planets in co-orbital (Trojan or 1:1) resonance, in which two plan
ets share the same orbit, usually separated by 60 degrees. Co-orbital systems are detectable among the planetary systems found by the Kepler mission either directly or by transit timing variations. However, no co-orbital systems have been found within the thousands of Kepler planets and candidates. Here we study the orbital evolution of co-orbital planets embedded in a protoplanetary disk using a grid-based hydrodynamics code. We show that pairs of similar-mass planets in co-orbital resonance are disrupted during large-scale orbital migration. Destabilization occurs when one or both planets is near the critical mass needed to open a gap in the gaseous disk. A confined gap is opened that spans the 60 degree azimuthal separation between planets. This alters the torques imparted by the disk on each planet -- pushing the leading planet outward and the trailing planet inward -- and disrupts the resonance. The mechanism applies to systems in which the two planets masses differ by a factor of two or less. In a simple flared disk model the critical mass for gap opening varies from a few Earth masses at the inner edge of the disk to 1 Saturn-mass at 5 AU. A pair of co-orbital planets with masses in this range that migrates will enter a region where the planets are at the gap-opening limit. At that point the resonance is disrupted. We therefore predict an absence of planets on co-orbital configurations with masses in the super-Earth to Saturn mass range with similar masses.
Earth-mass bodies are expected to undergo Type I migration directed either inward or outward depending on the thermodynamical state of the protoplanetary disc. Zones of convergent migration exist where the Type I torque cancels out. We study the evol
ution of multiple protoplanets of a few Earth masses embedded in a non-isothermal protoplanetary disc. The protoplanets are located in the vicinity of a convergence zone located at the transition between two different opacity regimes. Inside the convergence zone, Type I migration is directed outward and outside the zone migration is directed inward. We used a grid-based hydrodynamical code that includes radiative effects. We performed simulations varying the initial number of embryos and tested the effect of including stochastic forces to mimic the effects resulting from turbulence. We also performed N-body runs calibrated on hydrodynamical calculations to follow the evolution on Myr timescales. For a small number of initial embryos (N = 5-7) and in the absence of stochastic forcing, the population of protoplanets migrates convergently toward the zero-torque radius and forms a stable resonant chain that protects embryos from close encounters. In systems with a larger initial number of embryos, or in which stochastic forces were included, these resonant configurations are disrupted. This in turn leads to the growth of larger cores via a phase of giant impacts, after which the system settles to a new stable resonant configuration. Giant planets cores with masses of 10 Earth masses formed in about half of the simulations with initial protoplanet masses of m_p = 3 Earth masses but in only 15% of simulations with m_p = 1 Earth mass. This suggests that if ~2-3 Earth mass protoplanets can form in less than ~1 Myr, convergent migration and giant collisions can grow giant planet cores at Type I migration convergence zones.
Low-mass objects embedded in isothermal protoplanetary discs are known to suffer rapid inward Type I migration. In non-isothermal discs, recent work has shown that a decreasing radial profile of the disc entropy can lead to a strong positive corotati
on torque which can significantly slow down or reverse Type I migration in laminar viscous disc models. It is not clear however how this picture changes in turbulent disc models. The aim of this study is to examine the impact of turbulence on the torque experienced by a low-mass planet embedded in a non-isothermal protoplanetary disc. We particularly focus on the role of turbulence on the corotation torque whose amplitude depends on the efficiency of diffusion processes in the planets horseshoe region. We performed 2D numerical simulations using a grid-based hydrodynamical code in which turbulence is modelled as stochastic forcing. In order to provide estimations for the viscous and thermal diffusion coefficients as a function of the amplitude of turbulence, we first set up non-isothermal disc models for different values of the amplitude of the turbulent forcing. We then include a low-mass planet and determine the evolution of its running time-averaged torque. We show that in non-isothermal discs, the entropy-related corotation torque can indeed remain unsaturated in the presence of turbulence. For turbulence amplitudes that do not strongly affect the disc temperature profile, we find that the running time-averaged torque experienced by an embedded protoplanet is in fairly good agreement with laminar disc models with appropriate values for the thermal and viscous diffusion coefficients. In discs with turbulence driven by stochastic forcing, the corotation torque therefore behaves similarly as in laminar viscous discs and can be responsible for significantly slowing down or reversing Type I migration.
We investigate the evolution of a system of two super-Earths with masses < 4 Earth masses embedded in a turbulent protoplanetary disk. The aim is to examine whether or not resonant trapping can occur and be maintained in presence of turbulence and ho
w this depends on the amplitude of the stochastic density fluctuations in the disk. We have performed 2D numerical simulations using a grid-based hydrodynamical code in which turbulence is modelled as stochastic forcing. We assume that the outermost planet is initially located just outside the position of the 3:2 mean motion resonance (MMR) with the inner one and we study the dependance of the resonance stability with the amplitude of the stochastic forcing. For systems of two equal-mass planets we find that in disk models with an effective viscous stress parameter {alpha} 10^{-3}, damping effects due to type I migration can counteract the effects of diffusion of the resonant angles, in such a way that the 3:2 MMR can possibly remain stable over the disk lifetime. For systems of super-Earths with mass ratio q=m_i/m_o < 1/2, where m_i (m_o) is the mass of the innermost (outermost) planet, the 3:2 MMR is broken in turbulent disks with effective viscous stresses 2x10^{-4}< {alpha}< 10^{-3} but the planets become locked in stronger p+1:p resonances, with p increasing as the value for {alpha} increases. For {alpha}> 2x10^{-3}, the evolution can eventually involve temporary capture in a 8:7 commensurability but no stable MMR is formed. Our results suggest that for values of the viscous stress parameter typical to those generated by MHD turbulence, MMRs between two super-Earths are likely to be disrupted by stochastic density fluctuations. For lower levels of turbulence however, as is the case in presence of a dead-zone, resonant trapping can be maintained in systems with moderate values of the planet mass ratio.
