Table 2
Properties of the ACR solutions in the low-order three-planet resonant chain.
| Chain | Degeneracy | φ0 | ACRs | Centre |
|---|---|---|---|---|
| 1:2:3 | (1,1,1,1) | (1, −4, 3) | 1S | 180 |
| 1:2:4 | (1,1,1,1) | (1, −3, 2) | 1S | 180 |
| 2:3:4 | (2,2,2,1) | (1, −3, 2) | 2A | ±82 |
| 2:3:5 | (1,1,1,1) | (4, −9, 5) | 1S | 180 |
| 2:3:6 | (2,2,2,1) | (1, 1, −2) | 2A | ±90 |
| 3:4:5 | (1,1,1,1) | (3, −8, 5) | 1S | 180 |
| 3:4:6 | (3,3,3,1) | (1, −2, 1) | 1S+2A | 180, ±51 |
| 3:4:8 | (1,1,1,1) | (3, −5, 2) | 1S | 180 |
| 3:5:7 | (1,1,1,1) | (3, −10, 7) | 1S | 0 |
| 3:6:8 | (1,1,1,1) | (1, −5, 4) | 1S | 180 |
| 4:6:9 | (1,1,1,1) | (2,-5, 3) | 1S | 180 |
| 12:15:20 | (4,4,4,1) | (1, −2, 1) | 4A | ±135, ±45 |
Notes. The resonant chain is given in the first column. The degeneracies (see Appendix A) of the resonant angles (θ12, Δϖ12, θ23, Δϖ23) defined as in Eq. (9) are in the second column. The third column gives the Laplace angle φ0 = l1λ1 + l2λ2 + l3λ3 by the coefficients (l1, l2, l3). The number of ACR families we found are given in the fourth column, with suffixes ‘S’ and ‘A’ indicating ‘symmetric’ and ‘asymmetric’, respectively. The libration centre(s) of φ0 are in the fifth column.
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