Thermal evolution of the three-dimensional simulation. Left: time evolution of the spherically averaged entropy. The lines are spaced logarithmically in time and the last four lines overlap, indicating the emergence of a radial entropy profile that is constant in time. Right: relative temperature deviation to the mean temperature in the steady state at three distances to the protoplanetary core. The trend line slopes of (− 0.84 ± 4.21) × 10⁻⁶% Ω_K at 0.1 R_Hill, (− 4.21 ± 1.12) × 10⁻⁶% Ω_K at 0.3 R_Hill, and (− 2.28 ± 0.31) × 10⁻⁵% Ω_K at 1 R_Hill are consistent with a true steady state.