The equation of state here is deliberately simple, locally isothermal, but soundspeed varies in a prescribed power-law way with distance. c/vK is about the disk's vertical scale height to radius ratio at the distance to the planet.
All disk models have surface density at the planet's position Sigma=200 g/cm2, corresponding to the minimum solar nebula.
The migration timescale ta = a/|da/dt| and the eccentricity driving timescale te = e/|de/dt| correspond to the orbital distance of a planet equal a=5.2 AU.
Perhaps confusingly but consistently the Roche lobe of Jupiter is overplotted on all maps to indicate scale, independent of the actual planet's mass. This was done to have a uniform scale marker.