On corotation torques, horseshoe drag and the possibility of sustained stalled or outward protoplanetary migration

We study the torque oil low-mass protoplanets oil fixed circular orbits. embedded in I protoplanetary disc in the isothermal limit. We consider a wide range of sur-face density distributions including(I cases where the surface density increases smoothly outwards. We perform both linear disc response calculations and non-linear numerical simulations. We consider a large range of viscosities, including the inviscid limit. as well its a range of protoplanet mass ratios, with special emphasis oil the co-orbital region and the corotation torque acting between disc and protoplanet. For low-mass protoplanets and large viscosity, the corotation torque behaves as expected from linear theory. However. when the viscosity becomes small enough to enable horseshoe turns to occur, the linear corotation torque exists only temporarily after insertion of I planet into the disc, being replaced by the horseshoe drag first discussed by Ward. This happens after a time that is equal to the horseshoe libration period reduced by a factor amounting to about twice the disc aspect ratio. This torque scales with the radial gradient of specific vorticity, as does the linear torque, but we find it to be many times larger. If the viscosity is large enough for viscous diffusion across the co-orbital region to occur Within a libration period. we find that the horseshoe drag may be Sustained. It not. the corotation torque saturates leaving Only the linear Lindblad torques. As the magnitude Of file non-linear co-orbital torque (horseshoe drag) is always found to be larger than the linear torque, we find that the sign of the total torque InaV change even for mildly positive surface density gradients. In combination with a kinematic viscosity large enough to keel) the torque from saturating, Strong Sustained deviations front linear theory and outward or stalled migration may occur ill such Cases.