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tokens: magnitude smaller than the size of the thermal disturbance. In all cases, the back torque exerted on the planet by the perturbed disc was evaluated using the softening length ε 2 . We have found that thermal torques thus evaluated do not depend on the softening length, and that there are only minute differences in the perturbation of the density field arising from heat release. All runs with larger mass planets as well as those focused on the study of the cold torque were performed using a softening length ε 2 . 2.3 Code and mesh domain To numerically solve equations ( 1 )–( 5 ), we use the hydrodynamic public code fargo3d 1 (Benítez-Llambay Masset 2016 ) with orbital advection enabled (Masset 2000 ). We have implemented the energy diffusion equation as an additional source step corresponding to the following differential equation: where is given by equation ( 4 ). In our simulations we adopt parameters for the pressure length-scale and for the thermal diffusivity that are typical of those found at a few astronomical units in protoplanetary discs (e.g. Benítez-Llambay et al. 2015 ; Bitsch et al. 2015 ; Lega et al. 2015 ). As a consequence the size of the thermal disturbance λ c is significantly smaller than the pressure length-scale H (Masset 2017 ). This is in contrast with the work of Hankla et al. ( 2020 ), who considered a thermal disturbance comparable in size to the pressure length-scale, which allowed them to satisfy largely the scale separation between the distance x p of the planet to corotation, and λ c , i.e. they have x p ≪ λ c ≲ H , whereas in our case we have x p ≲ λ c ≪ H . Namely, we take the following values in all our runs: where Ω p is the planet’s frequency. At the planet’s location we have Resolving this length-scale over a few zones while simulating a full disc would have a prohibitive computational cost, especially for parameter space explorations such as those presented here. For this reason we restrict our computational domain in the three coordinates. In the r -direction we cover the interval [ r min , r max ] = [0.9 r p , 1.1 r p ], in the θ-direction we use (hence we simulate only one hemisphere of the disc and use reflecting boundary conditions at the mid-plane). Finally our azimuthal extent is . The cell numbers in each direction are ( N r , N θ , N φ ) = (880, 64, 1536). Recast in pressure length-scale our box size is (4 H, H , 7 H ), while its resolution is (2.3 × 10 −4 r p , 7.8 × 10 −4 r p , 2.3 × 10 −4 r p ). To avoid reflections on the radial boundaries of our computational box, we use damping boundary conditions as in de Val-Borro et al. ( 2006 ), the width of the inner damping ring being 3 × 10 −2 r p and that of the outer ring being 3.5 × 10 −2 r p , the damping time-scale at the edge of each damping ring being 1/20th of the local orbital period. The size of the box considered is too small to properly capture the Lindblad and corotation torques. However, when our focus is on the heating torque, we