Quasi-steady thermal regulation along a constant angular momentum migration track at fixed eccentricity, e = 0.7. Columns correspond to cooling indices β = 1, 4, and 8 in Equation (28). Colored curves show solutions of the thermal balance equation for three values of the dimensionless normalization A₀. In each column, the cooling normalization L_cool,0 is chosen so that the middle model, A₀ = 10⁻⁵, has t_e = 5 Gyr at a_f = 0.06 au; the dashed gray fixed-Δt comparison is normalized at the same point. Upper panels show the eccentricity damping time t_e, middle panels show the representative tidal quality factor Q_eff, and lower panels show the corresponding equilibrium temperature X = T_c/T_ref. Thermal regulation weakens the dependence of t_e on a_f relative to the standard constant time lag scaling, ﹩{t}_{e}\propto {a}_{f}^{8}﹩, by driving larger Q_eff at smaller circularization radii.