  We investigate the tidal interaction between a low-mass planet and a
self-gravitating protoplanetary disk, by means of two-dimensional hydrodynamic
simulations. We first show that considering a planet freely migrating in a disk
without self-gravity leads to a significant overestimate of the migration rate.
The overestimate can reach a factor of two for a disk having three times the
surface density of the minimum mass solar nebula. Unbiased drift rates may be
obtained only by considering a planet and a disk orbiting within the same
gravitational potential. In a second part, the disk self-gravity is taken into
account. We confirm that the disk gravity enhances the differential Lindblad
torque with respect to the situation where neither the planet nor the disk
feels the disk gravity. This enhancement only depends on the Toomre parameter
at the planet location. It is typically one order of magnitude smaller than the
spurious one induced by assuming a planet migrating in a disk without
self-gravity. We confirm that the torque enhancement due to the disk gravity
can be entirely accounted for by a shift of Lindblad resonances, and can be
reproduced by the use of an anisotropic pressure tensor. We do not find any
significant impact of the disk gravity on the corotation torque.
