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Register a new userA torque formula for non-isothermal Type I planetary migration - II. Effects of diffusion
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Sijme-Jan Paardekooper
Publication date
2010
fields
Physics
and research's language is
English
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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.
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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.
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.
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 evolution 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.
Context. Giant planets open gaps in their protoplanetary and subsequently suffer so-called type II migration. Schematically, planets are thought to be tightly locked within their surrounding disks, and forced to follow the viscous advection of gas onto the central star. This fundamental principle however has recently been questioned, as migrating planets were shown to decouple from the gas radial drift. Aims. In this framework, we question whether the traditionally used linear scaling of migration rate of a giant planet with the disks viscosity still holds. Additionally, we assess the role of orbit-crossing material as part of the decoupling mechanism. Methods. We have performed 2D (r, {theta}) numerical simulations of point-mass planets embedded in locally isothermal {alpha}-disks in steady-state accretion, with various values of {alpha}. Arbitrary planetary accretion rates were used as a means to diminish or nullify orbit-crossing flows. Results. We confirm that the migration rate of a gap-opening planet is indeed proportional to the disks viscosity, but is not equal to the gas drift speed in the unperturbed disk. We show that the role of gap-crossing flows is in fact negligible. Conclusions. From these observations, we propose a new paradigm for type II migration : a giant planet feels a torque from the disk that promotes its migration, while the gap profile relative to the planet is restored on a viscous timescale, thus limiting the planet migration rate to be proportional to the disks viscosity. Hence, in disks with low viscosity in the planet region, type II migration should still be very slow. Key words. protoplanetary disks; planet-disk interactions; planets and satellites: formation
In a further development of a deterministic planet-formation model (Ida & Lin 2004), we consider the effect of type-I migration of protoplanetary embryos due to their tidal interaction with their nascent disks. During the early embedded phase of protostellar disks, although embryos rapidly emerge in regions interior to the ice line, uninhibited type-I migration leads to their efficient self-clearing. But, embryos continue to form from residual planetesimals at increasingly large radii, repeatedly migrate inward, and provide a main channel of heavy element accretion onto their host stars. During the advanced stages of disk evolution (a few Myr), the gas surface density declines to values comparable to or smaller than that of the minimum mass nebula model and type-I migration is no longer an effective disruption mechanism for mars-mass embryos. Over wide ranges of initial disk surface densities and type-I migration efficiency, the surviving population of embryos interior to the ice line has a total mass several times that of the Earth. With this reservoir, there is an adequate inventory of residual embryos to subsequently assemble into rocky planets similar to those around the Sun. But, the onset of efficient gas accretion requires the emergence and retention of cores, more massive than a few M_earth, prior to the severe depletion of the disk gas. The formation probability of gas giant planets and hence the predicted mass and semimajor axis distributions of extrasolar gas giants are sensitively determined by the strength of type-I migration. We suggest that the observed fraction of solar-type stars with gas giant planets can be reproduced only if the actual type-I migration time scale is an order of magnitude longer than that deduced from linear theories.
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