The importance of the dynamical corotation torque for the migration of low-mass planets -- 1D analytical prescriptions verified by 2D hydrodynamical simulations

Recent developments suggested that planet formation occurs in regions of the discs with low turbulent viscosity. There, the dynamical corotation torque is thought to play an important role by slowing down type I migration. We aim to provide a simple analytical prescription for the dynamical corotation torque for use in 1D global models of planet formation and evolution, and assess the importance of the dynamical corotation torque for the migration of low-mass planets in low-viscosity discs. We propose simple prescriptions for calculating in 1D the time evolution of the vortensities of the librating and orbit-crossing flows around a low-mass planet, which both enter the analytical expression for the dynamical corotation torque. One of our prescriptions involves a memory timescale for the librating flow, and 2D hydrodynamical simulations of disc-planet interactions are used to assess the memory timescale and validate our model. The orbital evolution of a low-mass planet is calculated by 1D simulations where the dynamical corotation torque features our prescriptions for the vortensities of the librating and orbit-crossing flows, and by 2D hydrodynamical simulations of disc-planet interactions, assuming locally isothermal discs. We find very good agreement between the 1D and 2D simulations for a wide parameter space, whether the dynamical corotation torque slows down or accelerates inward migration. We provide maps showing how much the dynamical corotation torque reduces the classical type I migration torque as a function of planet mass and orbital distance. The reduction is about 50\% for a 10 Earth-mass planet at 10 au in a young disc with surface density profile in $r^{-1/2}$ and alpha viscosity of $10^{-4}$. In discs with low turbulent viscosity, the dynamical corotation torque should be taken into account in global models as it can strongly slow down type I migration.