On the corotation torque in a radiatively inefficient disk

We consider the angular momentum exchange at the corotation resonance between a two-dimensional gaseous disk and a uniformly rotating external potential, assuming that the disk flow is adiabatic. We first consider the linear case for an isolated resonance, for which we give an expression of the corotation torque that involves the pressure perturbation, and which reduces to the usual dependence on the vortensity gradient in the limit of a cold disk. Although this expression requires the solution of the hydrodynamic equations, it provides some insight into the dynamics of the corotation region. In the general case, we find an additional dependence on the entropy gradient at corotation. This dependence is associated to the advection of entropy perturbations. These are not associated to pressure perturbations. They remain confined to the corotation region, where they yield a singular contribution to the corotation torque. In a second part, we check our torque expression by means of customized two-dimensional hydrodynamical simulations. In a third part, we contemplate the case of a planet embedded in a Keplerian disk, assumed to be adiabatic. We find an excess of corotation torque that scales with the entropy gradient, and we check that the contribution of the entropy perturbation to the torque is in agreement with the expression obtained from the linear analysis. We finally discuss some implications of the corotation torque expression for the migration of low mass planets in the regions of protoplanetary disks where the flow is radiatively inefficient on the timescale of the horseshoe U-turns.Comment: 36 pages, 19 figures, accepted for publication in Ap

Inertial waves in a differentially rotating spherical shell

We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in E^{1/3} with the Ekman number E, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with E^{1/2} as Ekman boundary layers. The behavior of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.Comment: 31 pages, 16 figures, accepted for publication in Journal of Fluid Mechanic

Gas and dust hydrodynamical simulations of massive lopsided transition discs - I. Gas distribution

Motivated by lopsided structures observed in some massive transition discs, we have carried out 2D numerical simulations to study vortex structure in massive discs, including the effects of disc self-gravity and the indirect force which is due to the displacement of the central star from the barycenter of the system by the lopsided structure. When only the indirect force is included, we confirm the finding by Mittal & Chiang (2015) that the vortex becomes stronger and can be more than two pressure scale heights wide, as long as the disc-to-star mass ratio is >1%. Such wide vortices can excite strong density waves in the disc and therefore migrate inwards rapidly. However, when disc self-gravity is also considered in simulations, self-gravity plays a more prominent role on the vortex structure. We confirm that when the disc Toomre Q parameter is smaller than pi/(2h), where h is the disc's aspect ratio, the vortices are significantly weakened and their inward migration slows down dramatically. Most importantly, when the disc is massive enough (e.g. Q~3), we find that the lopsided gas structure orbits around the star at a speed significantly slower than the local Keplerian speed. This sub-Keplerian pattern speed can lead to the concentration of dust particles at a radius beyond the lopsided gas structure (as shown in Paper II). Overall, disc self-gravity regulates the vortex structure in massive discs and the radial shift between the gas and dust distributions in vortices within massive discs may be probed by future observations.Comment: 10 pages, 7 figures, accepted for publication in MNRA

On the horseshoe drag of a low-mass planet. II Migration in adiabatic disks

We evaluate the horseshoe drag exerted on a low-mass planet embedded in a gaseous disk, assuming the disk's flow in the coorbital region to be adiabatic. We restrict this analysis to the case of a planet on a circular orbit, and we assume a steady flow in the corotating frame. We also assume that the corotational flow upstream of the U-turns is unperturbed, so that we discard saturation effects. In addition to the classical expression for the horseshoe drag in barotropic disks, which features the vortensity gradient across corotation, we find an additional term which scales with the entropy gradient, and whose amplitude depends on the perturbed pressure at the stagnation point of the horseshoe separatrices. This additional torque is exerted by evanescent waves launched at the horseshoe separatrices, as a consequence of an asymmetry of the horseshoe region. It has a steep dependence on the potential's softening length, suggesting that the effect can be extremely strong in the three dimensional case. We describe the main properties of the coorbital region (the production of vortensity during the U-turns, the appearance of vorticity sheets at the downstream separatrices, and the pressure response), and we give torque expressions suitable to this regime of migration. Side results include a weak, negative feed back on migration, due to the dependence of the location of the stagnation point on the migration rate, and a mild enhancement of the vortensity related torque at large entropy gradient.Comment: Accepted for publication in Ap

Disk-planets interactions and the diversity of period ratios in Kepler's multi-planetary systems

The Kepler mission is dramatically increasing the number of planets known in multi-planetary systems. Many adjacent planets have orbital period ratios near resonant values, with a tendency to be larger than required for exact first-order mean-motion resonances. This intriguing feature has been shown to be a natural outcome of orbital circularization of resonant planetary pairs due to star-planet tidal interactions. However, this feature holds in multi-planetary systems with periods longer than ten days, for which tidal circularization is unlikely to provide efficient divergent evolution of the planets orbits. Gravitational interactions between planets and their parent protoplanetary disk may instead provide efficient divergent evolution. For a planet pair embedded in a disk, we show that interactions between a planet and the wake of its companion can reverse convergent migration, and significantly increase the period ratio from a near-resonant value. Divergent evolution due to wake-planet interactions is particularly efficient when at least one of the planets opens a partial gap around its orbit. This mechanism could help account for the diversity of period ratios in Kepler's multiple systems comprising super-Earth to sub-jovian planets with periods greater than about ten days. Diversity is also expected for planet pairs massive enough to merge their gap. The efficiency of wake-planet interactions is then much reduced, but convergent migration may stall with a variety of period ratios depending on the density structure in the common gap. This is illustrated for the Kepler-46 system, for which we reproduce the period ratio of Kepler-46b and c.Comment: 15 pages, 11 figures, accepted for publication in Ap

Planet--planet scattering in circumstellar gas disks

Hydrodynamical simulations of two giant planets embedded in a gaseous disk have shown that in case of a smooth convergent migration they end up trapped into a mean motion resonance. These findings have led to the conviction that the onset of dynamical instability causing close encounters between the planets can occur only after the dissipation of the gas when the eccentricity damping is over. We show that a system of three giant planets may undergo planet-planet scattering when the gaseous disk, with density values comparable to that of the Minimum Mass Solar Nebula, is still interacting with the planets. The hydrodynamical code FARGO--2D--1D is used to model the evolution ofthe disk and planets, modified to properly handle close encounters between the massive bodies. Our simulations predict a variety of different outcomes of the scattering phase which includes orbital exchange, planet merging and scattering of a planet in a hyperbolic orbit. This implies thatthe final fate of a multiplanet system under the action of the disk torques is not necessarily a packed resonant configuration.Comment: Astronomy and Astrophysics Letters, in pres

Tidal inertial waves in the differentially rotating convective envelopes of low-mass stars - I. Free oscillation modes

Star-planet tidal interactions may result in the excitation of inertial waves in the convective region of stars. In low-mass stars, their dissipation plays a prominent role in the long-term orbital evolution of short-period planets. Turbulent convection can sustain differential rotation in their envelope, with an equatorial acceleration (as in the Sun) or deceleration, which can modify the waves' propagation properties. We explore in this first paper the general propagation properties of free linear inertial waves in a differentially rotating homogeneous fluid inside a spherical shell. We assume that the angular velocity background flow depends on the latitudinal coordinate only, close to what is expected in the external convective envelope of low-mass stars. We use i) an analytical approach in the inviscid case to get the dispersion relation, from which we compute the characteristic trajectories along which energy propagates. This allows us to study the existence of attractor cycles and infer the different families of inertial modes; ii) high-resolution numerical calculations based on a spectral method for the viscous problem. We find that modes that propagate in the whole shell (D modes) behave the same way as with solid-body rotation. However, another family of inertial modes exists (DT modes), which can propagate only in a restricted part of the convective zone. Our study shows that they are less common than D modes and that the characteristic rays and shear layers often focus towards a wedge - or point-like attractor. More importantly, we find that for non-axisymmetric oscillation modes, shear layers may cross a corotation resonance with a local accumulation of kinetic energy. Their damping rate scales very differently from what we obtain for standard D modes and we show an example where it is independent of viscosity (Ekman number) in the astrophysical regime in which it is small.Comment: 17 pages, 15 figures, accepted for publication in A&

A torque formula for non-isothermal Type I planetary migration - II. Effects of diffusion

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.Comment: 12 pages, 8 figures, accepted for publication in MNRA

Saturated torque formula for planetary migration in viscous disks with thermal diffusion: recipe for protoplanet population synthesis

We provide torque formulae for low mass planets undergoing type I migration in gaseous disks. These torque formulae put special emphasis on the horseshoe drag, which is prone to saturation: the asymptotic value reached by the horseshoe drag depends on a balance between coorbital dynamics (which tends to cancel out or saturate the torque) and diffusive processes (which tend to restore the unperturbed disk profiles, thereby desaturating the torque). We entertain here the question of this asymptotic value, and we derive torque formulae which give the total torque as a function of the disk's viscosity and thermal diffusivity. The horseshoe drag features two components: one which scales with the vortensity gradient, and one which scales with the entropy gradient, and which constitutes the most promising candidate for halting inward type I migration. Our analysis, which is complemented by numerical simulations, recovers characteristics already noted by numericists, namely that the viscous timescale across the horseshoe region must be shorter than the libration time in order to avoid saturation, and that, provided this condition is satisfied, the entropy related part of the horseshoe drag remains large if the thermal timescale is shorter than the libration time. Side results include a study of the Lindblad torque as a function of thermal diffusivity, and a contribution to the corotation torque arising from vortensity viscously created at the contact discontinuities that appear at the horseshoe separatrices. For the convenience of the reader mostly interested in the torque formulae, section 8 is self-contained.Comment: Affiliation details changed. Fixed equation numbering issue. Biblio info adde