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 scaleheight 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 per cent of the numerical simulations.

At a distance of 1.295 parsecs, 1 the red-dwarf Proxima Centauri (α Centauri C, GL 551, HIP 70890, or simply Proxima) is the Sun's closest stellar neighbour and one of the best studied low-mass stars. It has an effective temperature of only ∼ 3050 K, a luminosity of ∼0.1 per cent solar, a measured radius of 0.14 R ⊙ 2 and a mass of about 12 per cent the mass of the Sun. Although Proxima is considered a moderately active star, its rotation period is ∼ 83 days, 3 and its quiescent activity levels and X-ray luminosity 4 are comparable to the Sun's. New observations reveal the presence of a small planet orbiting Proxima with a minimum mass of 1.3 Earth masses and an orbital period of ∼11.2 days. Its orbital semi-major axis is ∼ 0.05 AU, with an equilibrium temperature in the range where water could be liquid on its surface. 5 The results presented here consist of the analysis of previously obtained Doppler measurements (pre-2016 data), and the confirmation of a signal in a specifically designed follow-up campaign in 2016. The Doppler data comes from two precision radial velocity instruments, both at the European Southern Observatory (ESO): the High Accuracy Radial velocity Planet Searcher (HARPS) and the Ultraviolet and Visual Echelle Spectrograph (UVES). HARPS is a high-resolution stabilized echelle spectrometer installed at the ESO 3.6m telescope (La Silla observatory, Chile), and is calibrated in wavelength using hollow cathode lamps. HARPS has demonstrated radial velocity measurements at ∼1 ms −1 precision over time-scales of years, 6 including on low-mass stars. 7 All HARPS spectra were extracted and calibrated with the standard ESO Data Reduction Software, and radial velocities were measured using a least-squares template matching technique. 7 HARPS data is separated into two datasets. The first set includes all data obtained before 2016 by several programmes (HARPS pre-2016 work, and its value is then used to assess the false-alarm probability (or FAP) of the detection. 14 A FAP below 1% is considered suggestive of periodic variability, and anything below 0.1% is considered to be a significant detection. In the Bayesian framework, signals are first searched using a specialized sampling method 16 that enables exploration of multiple local maxima of the posterior density (the result of this process are the gray lines in Figure 1), and significances are then assessed by obtaining the ratios of evidences of models. If the evidence ratio exceeds some threshold (e.g. B 1 /B 0 > 10 3 ), then the model in the numerator (with one planet) is favoured against the model in the denominator (no planet).A well isolated peak at ∼11.2 days was recovered when analyzing all the night averages in the pre-2016 datasets (Figure 1, panel a). Despite the significance of the signal, the analysis of pre-2016 subsets produced slightly different periods depending on the noise assumptions and which subsets were considered. Confirmation or refutation of this signal at 11.2 days was the main driver for proposing the HARPS PRD campaign. T...

We study the torque on low‐mass planets embedded in protoplanetary discs in the two‐dimensional approximation, incorporating non‐isothermal effects. We couple linear estimates of the Lindblad (or wave) torque to a simple, but non‐linear, model of adiabatic corotation torques (or horseshoe drag), resulting in a simple formula that governs type I migration in non‐isothermal discs. This formula should apply in optically thick regions of the disc, where viscous and thermal diffusion act to keep the horseshoe drag unsaturated. We check this formula against numerical hydrodynamical simulations, using three independent numerical methods, and find good agreement.

We perform numerical simulations of a disc-planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analyzed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5% for the Eulerian simulations while the SPH results predict the correct shape of the gap although have less resolution in the low density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10%. The spread is larger in the tidal torques acting on the planet which agree within a factor 2 at the end of the simulation. In the Neptune case the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.Comment: 32 pages, accepted for publication in MNRA

Aims. We investigate the effect of including a proper energy balance on the interaction of a low-mass planet with a protoplanetary disk. Methods. We use a three-dimensional version of the RODEO method to perform hydrodynamical simulations including the energy equation. Radiation is included in the flux-limited diffusion approach.Results. The sign of the torque is sensitive to the ability of the disk to radiate away the energy generated in the immediate surroundings of the planet. In the case of high opacity, corresponding to the dense inner regions of protoplanetary disks, migration is directed outward, instead of the usual inward migration that was found in locally isothermal disks. For low values of the opacity we recover inward migration and show that torques originating in the coorbital region are responsible for the change in migration direction.