Locations and widths of first-order secular spin-orbit resonances for different masses of the hypothetical ancient satellite. In each panel, the mass labelled m_j is the minimum satellite mass in order to allow for Uranus to reach the v_j resonance and be tilted from 0° to 90° (see Table 2). The extent of all resonances is shown in pink and the centre of the v_j resonance is highlighted with a red curve. The approximate ridge line separating the close and far satellite regimes is shown by the dashed blue line (same as in Fig. 1). The black striped area is the region E₁ where the satellite’s classic Laplace plane is unstable. The red dot is the singular point S₁. Resonances are labelled by the frequency v_j of the forcing term (arrows).