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

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

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

The Dynamical Origin of the Multi-Planetary System HD45364

The recently discovered planetary system HD45364 which consists of a Jupiter and Saturn mass planet is very likely in a 3:2 mean motion resonance. The standard scenario to form planetary commensurabilities is convergent migration of two planets embedded in a protoplanetary disc. When the planets are initially separated by a period ratio larger than two, convergent migration will most likely lead to a very stable 2:1 resonance for moderate migration rates. To avoid this fate, formation of the planets close enough to prevent this resonance may be proposed. However, such a simultaneous formation of the planets within a small annulus, seems to be very unlikely. Rapid type III migration of the outer planet crossing the 2:1 resonance is one possible way around this problem. In this paper, we investigate this idea in detail. We present an estimate for the required convergent migration rate and confirm this with N-body and hydrodynamical simulations. If the dynamical history of the planetary system had a phase of rapid inward migration that forms a resonant configuration, we predict that the orbital parameters of the two planets are always very similar and hence should show evidence of that. We use the orbital parameters from our simulation to calculate a radial velocity curve and compare it to observations. Our model can explain the observational data as good as the previously reported fit. The eccentricities of both planets are considerably smaller and the libration pattern is different. Within a few years, it will be possible to observe the planet-planet interaction directly and thus distinguish between these different dynamical states.Comment: 9 pages, 9 figures - accepted for publication in Astronomy and Astrophysic

On the width and shape of the corotation region for low-mass planets

We study the coorbital flow for embedded, low mass planets. We provide a simple semi-analytic model for the corotation region, which is subsequently compared to high resolution numerical simulations. The model is used to derive an expression for the half-width of the horseshoe region, x_s, which in the limit of zero softening is given by x_s/r_p = 1.68(q/h)^(1/2), where q is the planet to central star mass ratio, h is the disc aspect ratio and r_p the orbital radius. This is in very good agreement with the same quantity measured from simulations. This result is used to show that horseshoe drag is about an order of magnitude larger than the linear corotation torque in the zero softening limit. Thus the horseshoe drag, the sign of which depends on the gradient of specific vorticity, is important for estimates of the total torque acting on the planet. We further show that phenomena, such as the Lindblad wakes, with a radial separation from corotation of ~ a pressure scale height H can affect x_s, even though for low-mass planets x_s << H. The effect is to distort streamlines and to reduce x_s through the action of a back pressure. This effect is reduced for smaller gravitational softening parameters and planets of higher mass, for which x_s becomes comparable to H.Comment: 15 pages, 11 figures, accepted for publication in MNRA