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Register a new userOn type-I migration near opacity transitions. A generalized Lindblad torque formula for planetary population synthesis
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F. S. Masset
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We give an expression for the Lindblad torque acting on a low-mass planet embedded in a protoplanetary disk that is valid even at locations where the surface density or temperature profile cannot be approximated by a power law, such as an opacity transition. At such locations, the Lindblad torque is known to suffer strong deviation from its standard value, with potentially important implications for type I migration, but the full treatment of the tidal interaction is cumbersome and not well suited to models of planetary population synthesis. The expression that we propose retains the simplicity of the standard Lindblad torque formula and gives results that accurately reproduce those of numerical simulations, even at locations where the disk temperature undergoes abrupt changes. Our study is conducted by means of customized numerical simulations in the low-mass regime, in locally isothermal disks, and compared to linear torque estimates obtained by summing fully analytic torque estimates at each Lindblad resonance. The functional dependence of our modified Lindblad torque expression is suggested by an estimate of the shift of the Lindblad resonances that mostly contribute to the torque, in a disk with sharp gradients of temperature or surface density, while the numerical coefficients of the new terms are adjusted to seek agreement with numerics. As side results, we find that the vortensity related corotation torque undergoes a boost at an opacity transition that can counteract migration, and we find evidence from numerical simulations that the linear corotation torque has a non-negligible dependency upon the temperature gradient, in a locally isothermal disk.
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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.
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 disks 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.
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.
Studies of planet migration derived from disc planet interactions began before the discovery of exoplanets. The potential importance of migration for determining orbital architectures being realised, the field received greater attention soon after the initial discoveries of exoplanets. Early studies based on very simple disc models indicated very fast migration times for low mass planets that raised questions about its relevance. However, more recent studies, made possible with improving resources, that considered improved physics and disc models revealed processes that could halt or reverse this migration. That in turn led to a focus on special regions in the disc where migration could be halted. In this way the migration of low mass planets could be reconciled with formation theories. In the case of giant planets which have a nonlinear interaction with the disc, the migration should be slower and coupled to the evolution of the disc. The latter needs to be considered more fully to make future progress in all cases. Here we are primarily concerned with processes where migration is connected with the presence of the protopolanetary disk. Migration may also be induced by disc-free gravitational interactions amongst planets or with binary companions. This is only briefly discussed here.
We present an analytical and numerical study of the orbital migration and resonance capture of fictitious two-planet systems with masses in the super-Earth range undergoing Type-I migration. We find that, depending on the flare index and proximity to the central star, the average value of the period ratio, $P_2/P_1$, between both planets may show a significant deviation with respect to the nominal value. For planets trapped in the 2:1 commensurability, offsets may reach values on the order of $0.1$ for orbital periods on the order of $1$ day, while systems in the 3:2 mean-motion resonance (MMR) show much smaller offsets for all values of the semimajor axis. These properties are in good agreement with the observed distribution of near-resonant exoplanets, independent of their detection method. We show that 2:1-resonant systems far from the star, such as HD82943 and HR8799, are characterized by very small resonant offsets, while higher values are typical of systems discovered by Kepler with orbital periods approximately a few days. Conversely, planetary systems in the vicinity of the 3:2 MMR show little offset with no significant dependence on the orbital distance. In conclusion, our results indicate that the distribution of Kepler planetary systems around the 2:1 and 3:2 MMR are consistent with resonant configurations obtained as a consequence of a smooth migration in a laminar flared disk, and no external forces are required to induce the observed offset or its dependence with the commensurability or orbital distance from the star.
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