The Kepler missions discovery of a number of circumbinary planets orbiting close (a_p < 1.1 au) to the stellar binary raises questions as to how these planets could have formed given the intense gravitational perturbations the dual stars impart on the disk. The gas component of circumbinary protoplanetary disks is perturbed in a similar manner to the solid, planetesimal dominated counterpart, although the mechanism by which disk eccentricity originates differs. This is the first work of a series that aims to investigate the conditions for planet formation in circumbinary protoplanetary disks. We present a number of hydrodynamical simulations that explore the response of gas disks around two observed binary systems: Kepler-16 and Kepler-34. We probe the importance of disk viscosity, aspect-ratio, inner boundary condition, initial surface density gradient, and self-gravity on the dynamical evolution of the disk, as well as its quasi steady-state profile. We find there is a strong influence of binary type on the mean disk eccentricity, e_d, leading to e_d = 0.02 - 0.08 for Kepler-16 and e_d = 0.10 - 0.15 in Kepler-34. The value of alpha-viscosity has little influence on the disk, but we find a strong increase in mean disk eccentricity with increasing aspect-ratio due to wave propagation effects. The choice of inner boundary condition only has a small effect on the surface density and eccentricity of the disk. Our primary finding is that including disk self-gravity has little impact on the evolution or final state of the disk for disks with masses less than 12.5 times that of the minimum-mass solar nebula. This finding contrasts with the results of self-gravity relevance in circumprimary disks, where its inclusion is found to be an important factor in describing the disk evolution.
We perform one of the first studies into the nonlinear evolution of tidally excited inertial waves in a uniformly rotating fluid body, exploring a simplified model of the fluid envelope of a planet (or the convective envelope of a solar-type star) subject to the gravitational tidal perturbations of an orbiting companion. Our model contains a perfectly rigid spherical core, which is surrounded by an envelope of incompressible uniform density fluid. The corresponding linear problem was studied in previous papers which this work extends into the nonlinear regime, at moderate Ekman numbers (the ratio of viscous to Coriolis accelerations). By performing high-resolution numerical simulations, using a combination of pseudo-spectral and spectral element methods, we investigate the effects of nonlinearities, which lead to time-dependence of the flow and the corresponding dissipation rate. Angular momentum is deposited non-uniformly, leading to the generation of significant differential rotation in the initially uniformly rotating fluid, i.e. the body does not evolve towards synchronism as a simple solid body rotator. This differential rotation modifies the properties of tidally excited inertial waves, changes the dissipative properties of the flow, and eventually becomes unstable to a secondary shear instability provided that the Ekman number is sufficiently small. Our main result is that the inclusion of nonlinearities eventually modifies the flow and the resulting dissipation from what linear calculations would predict, which has important implications for tidal dissipation in fluid bodies. We finally discuss some limitations of our simplified model, and propose avenues for future research to better understand the tidal evolution of rotating planets and stars.
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 corotation 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.
Discs in binaries have a complex behavior because of the perturbations of the companion star. Planet formation in binary-star systems both depend on the companion star parameters and on the properties of the circumstellar disc. An eccentric disc may increase the impact velocity of planetesimals and therefore jeopardize the accumulation process. We model the evolution of discs in close binaries including the effects of self-gravity and adopting different prescriptions to model the discs radiative properties. We focus on the dynamical properties and evolutionary tracks of the discs. We use the hydrodynamical code FARGO and we include in the energy equation heating and cooling effects. Radiative discs have a lower disc eccentricity compared to locally isothermal discs with same temperature profile. As a consequence, we do not observe the formation of an internal elliptical low density region as in locally isothermal disc models. However, the disc eccentricity depends on the disc mass through the opacities. Akin to locally isothermal disc models, self-gravity forces the discs longitude of pericenter to librate about a fixed orientation with respect to the binary apsidal line ($pi$). The discs radiative properties play an important role in the evolution of discs in binaries. A radiative disc has an overall shape and internal structure that are significantly different compared to a locally isothermal disc with same temperature profile. This is an important finding both for describing the evolutionary track of the disc during its progressive mass loss, and for planet formation since the internal structure of the disc is relevant for planetesimals growth in binary systems. The non-symmetrical distribution of mass in these discs causes large eccentricities for planetesimals that may affect their growth.
The migration of low-mass planets is driven by the differential Lindblad torque and the corotation torque in non-magnetic viscous models of protoplanetary discs. The corotation torque has recently received detailed attention as it may slow down, stall, or reverse migration. In laminar viscous disc models, the long-term evolution of the corotation torque is intimately related to viscous and thermal diffusion processes in the planets horseshoe region. This paper examines the properties of the corotation torque in discs where MHD turbulence develops as a result of the magnetorotational instability, considering a weak initial toroidal magnetic field. We present results of 3D MHD simulations carried out with two different codes. Non-ideal MHD effects and the discs vertical stratification are neglected, and locally isothermal disc models are considered. The running time-averaged torque exerted by the disc on a fixed planet is evaluated in three disc models. We first present results with an inner disc cavity (planet trap). As in viscous disc models, the planet is found to experience a positive running time-averaged torque over several hundred orbits, which highlights the existence of an unsaturated corotation torque maintained in the long term in MHD turbulent discs. Two disc models with initial power-law density and temperature profiles are also adopted, in which the time-averaged torque is found to be in decent agreement with its counterpart in laminar viscous disc models with similar viscosity at the planet location. Detailed analysis of the averaged torque density distributions indicates that the differential Lindblad torque takes very similar values in MHD turbulent and laminar viscous discs, and there exists an unsaturated corotation torque in MHD turbulent discs. This analysis also reveals the existence of an additional corotation torque in weakly magnetized discs.
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 how 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.
The massive stars in the Galactic center inner arcsecond share analogous properties with the so-called Hot Jupiters. Most of these young stars have highly eccentric orbits, and were probably not formed in-situ. It has been proposed that these stars acquired their current orbits from the tidal disruption of compact massive binaries scattered toward the proximity of the central supermassive black hole. Assuming a binary star formed in a thin gaseous disk beyond 0.1 pc from the central object, we investigate the relevance of disk-satellite interactions to harden the binding energy of the binary, and to drive its inward migration. A massive, equal-mass binary star is found to become more tightly wound as it migrates inwards toward the central black hole. The migration timescale is very similar to that of a single-star satellite of the same mass. The binarys hardening is caused by the formation of spiral tails lagging the stars inside the binarys Hill radius. We show that the hardening timescale is mostly determined by the mass of gas inside the binarys Hill radius, and that it is much shorter than the migration timescale. We discuss some implications of the binarys hardening process. When the more massive (primary) components of close binaries eject most their mass through supernova explosion, their secondary stars may attain a range of eccentricities and inclinations. Such processes may provide an alternative unified scenario for the origin of the kinematic properties of the central cluster and S-stars in the Galactic center as well as the high velocity stars in the Galactic halo.
We study the effects of diffusion on the non-linear corotation torque, or horseshoe drag, in the two-dimensional limit, focusing on low-mass planets for which the width of the horseshoe region is much smaller than the scale height of the disc. In the absence of diffusion, the non-linear corotation torque saturates, leaving only the Lindblad torque. Diffusion of heat and momentum can act to sustain the corotation torque. In the limit of very strong diffusion, the linear corotation torque is recovered. For the case of thermal diffusion, this limit corresponds to having a locally isothermal equation of state. We present some simple models that are able to capture the dependence of the torque on diffusive processes to within 20% of the numerical simulations.
Numerical simulations of planets embedded in protoplanetary gaseous discs are a precious tool for studying the planetary migration ; however, some approximations have to be made. Most often, the selfgravity of the gas is neglected. In that case, it is not clear in the literature how the material inside the Roche lobe of the planet should be taken into account. Here, we want to address this issue by studying the influence of various methods so far used by different authors on the migration rate. We performed high-resolution numerical simulations of giant planets embedded in discs. We compared the migration rates with and without gas selfgravity, testing various ways of taking the circum-planetary disc (CPD) into account. Different methods lead to significantly different migration rates. Adding the mass of the CPD to the perturbing mass of the planet accelerates the migration. Excluding a part of the Hill sphere is a very touchy parameter that may lead to an artificial suppression of the type III, runaway migration. In fact, the CPD is smaller than the Hill sphere. We recommend excluding no more than a 0.6 Hill radius and using a smooth filter. Alternatively, the CPD can be given the acceleration felt by the planet from the rest of the protoplanetary disc. The gas inside the Roche lobe of the planet should be very carefully taken into account in numerical simulations without any selfgravity of the gas. The entire Hill sphere should not be excluded. The method used should be explicitly given. However, no method is equivalent to computing the full selfgravity of the gas.
We investigate the tidal interaction between a low-mass planet and a self-gravitating protoplanetary disk, by means of two-dimensional hydrodynamic simulations. We first show that considering a planet freely migrating in a disk without self-gravity leads to a significant overestimate of the migration rate. The overestimate can reach a factor of two for a disk having three times the surface density of the minimum mass solar nebula. Unbiased drift rates may be obtained only by considering a planet and a disk orbiting within the same gravitational potential. In a second part, the disk self-gravity is taken into account. We confirm that the disk gravity enhances the differential Lindblad torque with respect to the situation where neither the planet nor the disk feels the disk gravity. This enhancement only depends on the Toomre parameter at the planet location. It is typically one order of magnitude smaller than the spurious one induced by assuming a planet migrating in a disk without self-gravity. We confirm that the torque enhancement due to the disk gravity can be entirely accounted for by a shift of Lindblad resonances, and can be reproduced by the use of an anisotropic pressure tensor. We do not find any significant impact of the disk gravity on the corotation torque.
