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Register a new userDynamical corotation torques on low-mass planets
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Sijme-Jan Paardekooper
Publication date
2014
fields
Physics
and research's language is
English
Authors
Sijme-Jan Paardekooper
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We study torques on migrating low-mass planets in locally isothermal discs. Previous work on low-mass planets generally kept the planet on a fixed orbit, after which the torque on the planet was measured. In addition to these static torques, when the planet is allowed to migrate it experiences dynamical torques, which are proportional to the migration rate and whose sign depends on the background vortensity gradient. We show that in discs a few times more massive than the Minimum Mass Solar Nebula, these dynamical torques can have a profound impact on planet migration. Inward migration can be slowed down significantly, and if static torques lead to outward migration, dynamical torques can take over, taking the planet beyond zero-torque lines set by saturation of the corotation torque in a runaway fashion. This means the region in non-isothermal discs where outward migration is possible can be larger than what would be concluded from static torques alone.
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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.
Using linear perturbation theory, we investigate the torque exerted on a low-mass planet embedded in a gaseous protoplanetary disc with finite thermal diffusivity. When the planet does not release energy into the ambient disc, the main effect of thermal diffusion is the softening of the enthalpy peak near the planet, which results in the appearance of two cold and dense lobes on either side of the orbit, of size smaller than the thickness of the disc. The lobes exert torques of opposite sign on the planet, each comparable in magnitude to the one-sided Lindblad torque. When the planet is offset from corotation, the lobes are asymmetric and the planet experiences a net torque, the `cold thermal torque, which has a magnitude that depends on the relative value of the distance to corotation to the size of the lobes $simsqrt{chi/Omega_p}$, $chi$ being the thermal diffusivity and $Omega_p$ the orbital frequency. We believe that this effect corresponds to the phenomenon named `cold finger recently reported in numerical simulations, and we argue that it constitutes the dominant mode of migration of sub-Earth-mass objects. When the planet is luminous, the heat released into the ambient disc results in an additional disturbance that takes the form of hot, low-density lobes. They give a torque, named heating torque in previous work, that has an expression similar, but of opposite sign, to the cold thermal torque.
We present the results of hydrodynamical simulations of the orbital evolution of planets undergoing runaway gas accretion in radiative discs. We consider accreting disc models with constant mass flux through the disc, and where radiative cooling balances the effect of viscous heating and stellar irradiation. We assume that 20-30 $M_oplus$ giant planet cores are formed in the region where viscous heating dominates and migrate outward under the action of a strong corotation torque. In the case where gas accretion is neglected, we find evidence for strong dynamical torques in accreting discs with accretion rates ${dot M}gtrsim 7times 10^{-8} ;M_odot/yr$. Their main effect is to increase outward migration rates by a factor of $sim 2$ typically. In the presence of gas accretion, however, runaway outward migration is observed with the planet passing through the zero-torque radius and the transition between the viscous heating and stellar heating dominated regimes. The ability for an accreting planet to enter a fast migration regime is found to depend strongly on the planet growth rate, but can occur for values of the mass flux through the disc of ${dot M}gtrsim 5times 10^{-8} ;M_odot/yr$. We find that an episode of runaway outward migration can cause an accreting planet formed in the 5-10 AU region to temporarily orbit at star-planet separations as large as $sim$60-70 AU. However, increase in the amplitude of the Lindblad torque associated with planet growth plus change in the streamline topology near the planet systematically cause the direction of migration to be reversed. Our results indicate that a planet can reach large orbital distances under the combined effect of dynamical torques and gas accretion, but an alternative mechanism is required to explain the presence of massive planets on wide orbits.
The solid material of protoplanetary discs forms an asymmetric pattern around a low-mass planet (M_p<=10M_Earth) due to the combined effect of dust-gas interaction and the gravitational attraction of the planet. Recently, it has been shown that although the total solid mass is negligible compared to that of gas in protoplanetary discs, a positive torque can be emerged by a certain size solid species. The torque magnitude can overcome that of gas which may result in outward planetary migration. In this study, we show that the accretion of solid species by the planet strengthens the magnitude of solid torque being either positive or negative. We run two-dimensional, high-resolution (1.5Kx3K) global hydrodynamic simulations of an embedded low-mass planet in a protoplanetary disc. The solid material is handled as a pressureless fluid. Strong accretion of well-coupled solid species by a M_p<0.3M_Earth protoplanet results in the formation of such a strongly asymmetric solid pattern close to the planet that the positive solid torque can overcome that of gas by two times. However, the accretion of solids in the pebble regime results in increased magnitude negative torque felt by protoplanets and strengthened positive torque for Earth-mass planets. For M_p>=3M_Earth planets the magnitude of the solid torque is positive, however, independent of the accretion strength investigated. We conclude that the migration of solid accreting planets can be substantially departed from the canonical type-I prediction.
We introduce a general mathematical framework to model the internal transport of angular momentum in a star hosting a close-in planetary/stellar companion. By assuming that the tidal and rotational distortions are small and that the deposit/extraction of angular momentum induced by stellar winds and tidal torques are redistributed solely by an effective eddy-viscosity that depends on the radial coordinate, we can formulate the model in a completely analytic way. It allows us to compute simultaneously the evolution of the orbit of the companion and of the spin and the radial differential rotation of the star. An illustrative application to the case of an F-type main-sequence star hosting a hot Jupiter is presented. The general relevance of our model to test more sophisticated numerical dynamical models and to study the internal rotation profile of exoplanet hosts, submitted to the combined effects of tides and stellar winds, by means of asteroseismology are discussed.
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