Planetary systems are born in the disks of gas, dust and rocky fragments that surround newly formed stars. Solid content assembles into ever-larger rocky fragments that eventually become planetary embryos. These then continue their growth by accreting leftover material in the disc. Concurrently, tidal effects in the disc cause a radial drift in the embryo orbits, a process known as migration [1][2][3][4] . Fast inward migration is predicted by theory for embryos smaller than three to five Earth masses [5][6][7] . With only inward migration, these embryos can only rarely become giant planets located at Earth's distance from the Sun and beyond 8,9 , in contrast with observations 10 . Here we report that asymmetries in the temperature rise associated with accreting infalling material 11, 12 produce a force (which gives rise to an effect that we call "heating torque") that counteracts inward migration. This provides a channel for the formation of giant planets 8 and also explains the strong planet-metallicity correlation found between the incidence of giant planets and the heavy-element abundance of the host stars 13, 14 .We solve the equations governing the disc hydrodynamics in combination with the equations of radiative transfer. Planets have an angular momentum that increases with their orbital radius. In the case of a nearly circular orbit, the rate of change of angular momentum, or torque, gives the migration rate. Our calculations are performed in three dimensions, yielding a reliable value for the net torque, from which the direction and rate of migration are inferred.Our fiducial computation is one in which a rocky core with 3 Earth masses is located at a distance comparable to that of Jupiter from the Sun and is being bombarded by solid material at a rate that doubles its mass in 100 thousand years. We assume that the gravitational energy of the infalling solid material is transformed entirely into heat and ultimately radiated by the planet 11 . A second computation is performed with the same set up, but without the planet's radiation, in order to distinguish the effects of the heating torque from other torques. We find that the heating torque (defined as the torque difference between cases with accretion turned respectively on and off) has a positive sign (figure 1), which enables it to counteract the effect of the standard, negative torque.The latter includes all torque components of the non-heating case, and is always negative for small 2 mass embryos (typically smaller than 5 M ⊕ , where M ⊕ is the Earth's mass). Thus, the effect of the heating torque is to either slow down the inward migration, cancel it, or reverse its direction.The most important factors governing the strength of the heating torque and thus, the direction of migration, are the accretion rate of the embryo, its mass and the opacity of the disc. For our fiducial values of opacity, disc structure and embryo mass, we find that outward migration occurs for accretion rates corresponding to a mass doubling time less than approximately 60 t...

We present an asymptotically and unconditionally stable numerical method to account for the momentum transfer between multiple species. Momentum is conserved to machine precision. This implies that the asymptotic equilibrium corresponds to the velocity of the center of mass. Aimed at studying dust dynamics, we implement this numerical method in the publicly available code FARGO3D. To validate our implementation, we develop a test suite for an arbitrary number of species, based on analytical or exact solutions of problems related to perfect damping, damped sound waves, shocks, local and global gas-dust radial drift in a disk and, linear streaming instability. In particular, we obtain first-order, steady-state solutions for the radial drift of multiple dust species in protoplanetary disks, in which the pressure gradient is not necessarily small. We additionally present non-linear shearing-box simulations of the streaming instability and compare them with previous results obtained with Lagrangian particles. We successfully validate our implementation by recovering the solutions from the test suite to second-and first-order accuracy in space and time, respectively. From this, we conclude that our scheme is suitable, and very robust, to study the self-consistent dynamics of several fluids. In particular, it can be used for solving the collisions between gas and dust in protoplanetary disks, with any degree of coupling.

Aerodynamic theory predicts that dust grains in protoplanetary disks will drift radially inward on comparatively short timescales. In this context, it has long been known that the presence of a gap opened by a planet can alter the dust dynamics significantly. In this paper, we carry out a systematic study employing long-term numerical simulations aimed at characterizing the critical particle-size for retention outside a gap as a function of particle size and for various key parameters defining the protoplanetary disk model. To this end, we perform multifluid hydrodynamical simulations in two dimensions, including different dust species, which we treat as pressureless fluids. We initialize the dust outside of the planet's orbit and study under which conditions dust grains are able to cross the gap carved by the planet. In agreement with previous work, we find that the permeability of the gap depends both on dust dynamical properties and the gas disk structure: while small dust follows the viscously accreting gas through the gap, dust grains approaching a critical size are progressively filtered out. Moreover, we introduce and compute a depletion factor that enables us to quantify the way in which higher viscosity, smaller planet mass, or a more massive disk can shift this critical size to larger values. Our results indicate that gap-opening planets may act to deplete the inner reaches of protoplanetary disks of large dust grains -potentially limiting the accretion of solids onto forming terrestrial planets.

The streaming instability is thought to play a central role in the early stages of planet formation by enabling the efficient bypass of a number of barriers hindering the formation of planetesimals. We present the first study exploring the efficiency of the linear streaming instability when a particle-size distribution is considered. We find that, for a given dust-to-gas mass ratio, the multi-species streaming instability grows on timescales much longer than those expected when only one dust species is involved. In particular, distributions that contain close-to-order-unity dust-to-gas mass ratios lead to unstable modes that can grow on timescales comparable, or larger, with those of secular instabilities. We anticipate that processes leading to particle segregation and/or concentration can create favourable conditions for the instability to grow fast. Our findings may have important implications for a large number of processes in protoplanetary disks that rely on the streaming instability as usually envisioned for a unique dust species. Our results suggest that the growth rates of other resonant-draginstabilities may also decrease considerably when multiple species are considered.