Examples illustrating the four different types of past obliquity evolution of Saturn. Each graph shows a 4-Gyr numerical trajectory (black dots) computed for Titan’s nominal migration rate and for a given value of Saturn’s normalised polar moment of inertia specifiedin title. Today’s location of Saturn is represented by the big green spot; the vertical error bar corresponds to our full exploration interval of λ. The red curves show the centre of first-order secular spin-orbit resonances (Cassini state 2) and the coloured areas represent their widths (same as Fig. 1). The top large area is the resonance with ϕ₃ and the bottom thin area is the resonance with ϕ₁₉ (see Table 1). The separatrices of the ϕ₃ resonance are highlighted in blue. Going forwards in time, the trajectories go from bottom to top.