Evolution of the semi-major axis and eccentricity of Io with respect to the integration time-variable for different acceleration factors α. The time is given in a logarithmic scale. The duration of each integration is set so that it represents 1 Gyr of physical time t used in Sect. 3. Up to α = 10⁵, a change of α only amounts to a linear contraction of the integration time (i.e., the curves overlap when viewed with respect to the physical time t).