On the horseshoe drag of a low-mass planet. I - Migration in isothermal disks

We investigate the unsaturated horseshoe drag exerted on a low-mass planet by an isothermal gaseous disk. In the globally isothermal case, we use a formal- ism, based on the use of a Bernoulli invariant, that takes into account pressure effects, and that extends the torque estimate to a region wider than the horse- shoe region. We find a result that is strictly identical to the standard horseshoe drag. This shows that the horseshoe drag accounts for the torque of the whole corotation region, and not only of the horseshoe region, thereby deserving to be called corotation torque. We find that evanescent waves launched downstream of the horseshoe U-turns by the perturbations of vortensity exert a feed-back on the upstream region, that render the horseshoe region asymmetric. This asymmetry scales with the vortensity gradient and with the disk's aspect ratio. It does not depend on the planetary mass, and it does not have any impact on the horseshoe drag. Since the horseshoe drag has a steep dependence on the width of the horseshoe region, we provide an adequate definition of the width that needs to be used in horseshoe drag estimates. We then consider the case of locally isothermal disks, in which the tempera- ture is constant in time but depends on the distance to the star. The horseshoe drag appears to be different from the case of a globally isothermal disk. The difference, which is due to the driving of vortensity in the vicinity of the planet, is intimately linked to the topology of the flow. We provide a descriptive inter- pretation of these effects, as well as a crude estimate of the dependency of the excess on the temperature gradient.Comment: Accepted for publication in Ap

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

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

Horseshoe drag in three-dimensional globally isothermal disks

We study the horseshoe dynamics of a low-mass planet in a three-dimensional, globally isothermal, inviscid disk. We find, as reported in previous work, that the boundaries of the horseshoe region (separatrix sheets) have cylindrical symmetry about the disk´s rotation axis. We interpret this feature as arising from the fact that the whole separatrix sheets have a unique value of Bernoulli´s constant, and that this constant does not depend on altitude, but only on the cylindrical radius, in barotropic disks. We next derive an expression for the torque exerted by the horseshoe region on the planet, or horseshoe drag. Potential vorticity is not materially conserved as in two-dimensional flows, but it obeys a slightly more general conservation law (Ertel´s theorem) that allows an expression for the horseshoe drag identical to the expression in a two-dimensional disk to be obtained. Our results are illustrated and validated by three-dimensional numerical simulations. The horseshoe region is found to be slightly narrower than previously extrapolated from two-dimensional analyses with a suitable softening length of the potential. We discuss the implications of our results for the saturation of the corotation torque, and the possible connection to the flow at the Bondi scale, which the present analysis does not resolve.Fil: Masset, F. S.. Universidad Nacional Autónoma de México; MéxicoFil: Benítez Llambay, Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Astronomía Teórica y Experimental. Universidad Nacional de Córdoba. Observatorio Astronómico de Córdoba. Instituto de Astronomía Teórica y Experimental; Argentin

On type-I migration near opacity transitions. A generalized Lindblad torque formula for planetary population synthesis

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.Comment: Appeared in special issue of "Celestial Mechanics and Dynamical Astronomy" on Extrasolar Planetary System

Low-mass planets in nearly inviscid disks: Numerical treatment

Embedded planets disturb the density structure of the ambient disk and gravitational back-reaction will induce possibly a change in the planet's orbital elements. The accurate determination of the forces acting on the planet requires careful numerical analysis. Recently, the validity of the often used fast orbital advection algorithm (FARGO) has been put into question, and special numerical resolution and stability requirements have been suggested. In this paper we study the process of planet-disk interaction for small mass planets of a few Earth masses, and reanalyze the numerical requirements to obtain converged and stable results. One focus lies on the applicability of the FARGO-algorithm. Additionally, we study the difference of two and three-dimensional simulations, compare global with local setups, as well as isothermal and adiabatic conditions. We study the influence of the planet on the disk through two- and three-dimensional hydrodynamical simulations. To strengthen our conclusions we perform a detailed numerical comparison where several upwind and Riemann-solver based codes are used with and without the FARGO-algorithm. With respect to the wake structure and the torque density acting on the planet we demonstrate that the FARGO-algorithm yields correct results, and that at a fraction of the regular cpu-time. We find that the resolution requirements for achieving convergent results in unshocked regions are rather modest and depend on the pressure scale height of the disk. By comparing the torque densities of 2D and 3D simulations we show that a suitable vertical averaging procedure for the force gives an excellent agreement between the two. We show that isothermal and adiabatic runs can differ considerably, even for adiabatic indices very close to unity.Comment: accepted by Astronomy & Astrophysic

Secular interactions between inclined planets and a gaseous disk

In a planetary system, a secular particle resonance occurs at a location where the precession rate of a test particle (e.g. an asteroid) matches the frequency of one of the precessional modes of the planetary system. We investigate the secular interactions of a system of mutually inclined planets with a gaseous protostellar disk that may contain a secular nodal particle resonance. We determine the normal modes of some mutually inclined planet-disk systems. The planets and disk interact gravitationally, and the disk is internally subject to the effects of gas pressure, self-gravity, and turbulent viscosity. The behavior of the disk at a secular resonance is radically different from that of a particle, owing mainly to the effects of gas pressure. The resonance is typically broadened by gas pressure to the extent that global effects, including large-scale warps, dominate. The standard resonant torque formula is invalid in this regime. Secular interactions cause a decay of the inclination at a rate that depends on the disk properties, including its mass, turbulent viscosity, and sound speed. For a Jupiter-mass planet embedded within a minimum-mass solar nebula having typical parameters, dissipation within the disk is sufficient to stabilize the system against tilt growth caused by mean-motion resonances.Comment: 30 pages, 6 figures, to be published in The Astrophysical Journa

Evolution of the eccentricity and inclination of low-mass planets subjected to thermal forces: a numerical study

By means of three dimensional, high resolution hydrodynamical simulations we study the orbital evolution of weakly eccentric or inclined low-mass protoplanets embedded in gaseous discs subject to thermal diffusion. We consider both non-luminous planets, and planets that also experience the radiative feedback from their own luminosity. We compare our results to previous analytical work, and find that thermal forces (the contribution to the disc's force arising from thermal effects) match those predicted by linear theory within $\sim 20$%. When the planet's luminosity exceeds a threshold found to be within $10$% of that predicted by linear theory, its eccentricity and inclination grow exponentially, whereas these quantities undergo a strong damping below this threshold. In this regime of low luminosity indeed, thermal diffusion cools the surroundings of the planet and allows gas to accumulate in its vicinity. It is the dynamics of this gas excess that contributes to damp eccentricity and inclination. The damping rates obtained can be up to $h^{-1}$ times larger than those due to the resonant interaction with the disc, where $h$ is the disc's aspect ratio. This suggests that models that incorporate planet-disc interactions using well-known formulae based on resonant wave-launching to describe the evolution of eccentricity and inclination underestimate the damping action of the disc on the eccentricity and inclination of low-mass planets by an order of magnitude.Comment: Accepted for publication in MNRA