Table 2
Minimum mass of the satellite in order for Uranus to reach the few closest resonances and be tilted from 0° to 90°.
| k | Identification | vk (″ yr−1) |  (Myr) |
mmin/M (×10−5) | mk/M (×10−5) |
|---|---|---|---|---|---|
| 15 | 𝒢7 − 𝒢8 + s7 | −0.58033 | 3021 | 35 | 36 |
| 3 | s8 | −0.69189 | 115 | 41 | 44 |
| 9 | 𝒢6 − 𝒢7 + s6 | −1.19906 | 519 | 71 | 79 |
| 7 | 𝒢5 − 𝒢7 + s7 | −1.84625 | 92 | 110 | 129 |
| 5 | − 𝒢5 + 𝒢6 + s6 | −2.35835 | 51 | 141 | 172 |
| 2 | s7 | −3.00557 | 4 | 179 | 233 |
| 8 | −𝒢7 + 𝒢8 + s8 | −3.11725 | 115 | 186 | 244 |
| 6 | −𝒢5 + 𝒢7 + s7 | −4.16482 | 40 | 248 | 368 |
Notes. k is the index of the resonant term in Uranus’s ζ series ordered by decreasing amplitude (see Table 1); 
is the libration period inside the resonance when the separatrix appears; mmin is the minimum mass of the satellite obtained analytically through Eq. (8); mk is the minimum mass of the satellite obtained semi-analytically using a more realistic model (see text). The terms are sorted here by increasing value of the frequency ∣vk∣ (i.e. by increasing value of mmin).
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