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 tra
nsition. 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 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 disks 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.
We evaluate the horseshoe drag exerted on a low-mass planet embedded in a gaseous disk, assuming the disks 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 flo
w 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 potentials 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.
