  We study the disc planet interactions of low-mass protoplanets embedded in a
circumstellar disc. We extend the standard theory of planet migration from the
usual locally isothermal assumption to include non-barotropic effects, focusing
on the validity of linear theory. We compared solutions of the linear equations
with results from non-linear hydrodynamic simulations, where in both cases we
adopted a background entropy gradient and solved the energy equation. We show
that the migration behavior of embedded planets depends critically on the
background radial entropy gradient in the disc. The presence of such a gradient
not only changes the corotation torque on the planet, but also always
guarantees a departure from linear behavior, which gives a singular density
response at corotation, in the absence of thermal or viscous diffusion. A
negative entropy gradient tends to give rise to positive, non-linear corotation
torques apparently produced as material executes horseshoe turns at
approximately constant entropy. These torques have no counterpart in linear
theory, but can be strong enough to push the planet outwards until saturation
starts to occur after a horseshoe libration period. Increased thermal diffusion
acts to reduce these non-linear torques, but, at the same time, it can help to
prevent their saturation. In combination with a small kinematic viscosity that
is able to maintain a smooth density profile the positive torque could be
sustained.
