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tokens: by stochastic kicks as seen by the torque fluctuating about zero, resulting in small inward and outward migration. The bottom panel is in a constant β disc, where it is clear that despite stochastic kicks being present, the torque, on average, is slightly negative, resulting in inward migration. Figure 14. Total z -component of the torque exerted by the disc material on a 1 M ⊕ planet is shown on the left-hand panels. The migration tracks are shown on the right-hand panels. The top five panels are simulations modelled with a variable β , whilst the bottom panel is one of the simulations modelled with a constant β . The migration is dominated by stochastic kicks, which results in random outward or inward motion, as seen in the torques oscillating about zero. 5.2 Comparison with previous work We compare our results to those of Baruteau et al. ( 2011 ) and Malik et al. ( 2015 ), who considered giant planet migration in self-gravitating discs. There are a couple of differences in this work. They perform their simulations using a 2D grid-based code, whereas the simulations in this work are performed using a 3D SPH code. The initial conditions of the disc are also slightly different. Their disc surface density and temperature decreased as R −2 and R −1 , respectively, whereas in this work, the disc surface density and temperature decreases as R −1 and R −0.5 , respectively. To ensure that the different initial conditions and codes used do not influence the results, we initially performed a set of simulations with a constant value of β = 15 to compare with Baruteau et al. ( 2011 ) and Malik et al. ( 2015 ). Despite the differences, we agree with their findings that using a constant β results in the giant planets rapidly migrating towards the inner edge of the disc. However, the crucial development in our work is how the disc is cooled: Whilst we also utilize the β -cooling approach, we allow β to vary with radius to mimic a more realistic disc such that our discs remain gravitationally stable in the inner regions, and the spiral structure only forms in the outer regions. The planets are able to slow down their rapid inward migration before reaching the inner disc edge. Using radiative transfer, Stamatellos Inutsuka ( 2018 ) migrate 1 M Jup planets in self-gravitating discs using a 3D SPH code. The initial conditions in their discs are more similar to ours: the same initial disc mass and surface density profile. Their usage of a realistic treatment of the disc thermodynamics meant that like in the simulations presented here, the planets are evolved in discs that are only gravitationally unstable in the outer regions. Using realistic thermodynamics to allow the gas to control its cooling/heating based on its properties, the thermodynamics of the circumstellar and circumplanetary disc is better captured. This enhances the mass growth of the planet causing the planets to grow to beyond the brown dwarf limit. This resulted in their planets opening up