Table 1
The 15 equilibria of the simplified Hamiltonian (25).
| # | 100 e1 | 100 e2 | 100 e3 | σ1 (°) | σ2 (°) | σ3 (°) | L−0.0345 | ξ(°) | Nature | Domain |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 4.449 | 7.490 | 5.878 | ±14.433 | ±22.572 | +92.014 | 6.461e–5 | ±59.760 | elliptic | δ ϵ ℝ |
| 2 | 0.168 | 0.165 | 0.093 | ∓119.18 | ±179.92 | ±4.6606 | 19.34e–5 | ±60.003 | δ-dependent | δ > 1.129 |
| 3 | 5.201 | 7.645 | 5.470 | ±14.216 | ∓15.688 | ±95.525 | 6.483e–5 | ±59.806 | elliptic | δ > 5.997 |
| 4 | 8.344 | 8.646 | 0.305 | 180 | 0 | 0 | 6.501e–5 | 180 | hyperbolic | δ ϵ ℝ |
| 5 | 7.939 | 7.180 | 5.922 | ∓151.76 | +20.944 | ±92.729 | 6.439e–5 | ±179.13 | hyperbolic | δ > 4.195 |
| 6 | 10.11 | 6.330 | 6.246 | ∓102.00 | ±177.92 | ±4.3287 | 7.200e–5 | ±62.538 | hyperbolic | δ > 1.129 |
| 7 | 6.009 | 8.721 | 1.696 | ±6.8311 | ∓1.8178 | ±86.449 | 6.518e–5 | ±59.710 | hyperbolic | δ > 5.999 |
| 8 | 0.164 | 0.165 | 0.079 | 0 | 180 | 0 | 19.34e–5 | 180 | hyperbolic | δ > 1.082 |
| 9 | 3.708 | 6.584 | 6.887 | 0 | 180 | 0 | 7.157e–5 | 180 | hyperbolic | δ > 1.082 |
Notes. The equilibria are found for the resonance chain 1:1:2 at δ = 7 using a Newton-Raphson method. Values given without decimal places are exact. The planetary masses are as in Fig. 1. Branch 2 is hyperbolic only for 5.548 ≤ δ ≤ 5.802 and elliptic elsewhere. The entry value of δ in the formal resonance (here 1.129) weakly depends on the planetary masses because of the normalisation by ΔGbif. Branch 1 is the only elliptic branch existing for all values of δ and it is the main branch introduced in Sect. 2.2.2. It corresponds to the only real solution of Eq. (35) when δ < 1. Branches 3, 5, and 7 do not exist at first order in eccentricity, while they exist at second order; hence, we cannot exclude that the complete Hamiltonian (3) has more libration centres, either because we did not discretise the phase space thinly enough to find them or because they do not exist at second order in eccentricity.
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