Dust torques for realistic dust size distributions
Previous studies have shown that a population of dust particles with a fixed Stokes number can exert a substantial torque on a low-mass planet embedded in a protoplanetary disk, modifying its migration rate. We aim to characterize the dust torque on a low-mass planet for a realistic distribution of dust grain sizes. We performed 2D hydrodynamical simulations of planet-disk interactions using the PLUTO code, with the addition of Lagrangian superparticles representing dust dynamics. We apply an energy-based criterion to exclude the particles that are gravitationally bound to the planet, to prevent circumplanetary flow to contaminate the torque measurements. We find that the dust torque is dominated by the largest grains in the size distribution and is highly sensitive to the maximum grain size. For typical disk conditions, the torque becomes positive for marginally coupled particles ($\mathrm{St} \gtrsim 10^{-2}$) and can exceed the gas torque in the presence of cm-sized pebbles, leading to outward migration of low-mass planets. Unlike previous studies, the turbulent dust diffusion has a negligible influence on the torque over the explored range of $\alpha = 10^{-4}$ to $3\times 10^{-3}$. The dominant contribution arises from within the planetary Hill sphere, highlighting the need for high spatial resolution and accurate integration of particle trajectories. We derive a scaling law for the dust torque as a function of the maximum grain size and the planetary mass, suitable for implementation in population synthesis models.