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Table A.1
Summary of the main notations used throughout the article.
| Name | Expression | Description | Reference |
|---|---|---|---|
| rj | Heliocentric coordinate position | Laskar (1991) | |
 |
Heliocentric coordinate momentum | Laskar (1991) | |
| Λj |  |
Circular angular momentum | (5) |
| λj | Mean longitude | (5) | |
| Cj |  |
Planet j AMD | (5) |
| ϖj | Longitude of the periapsis | (5) | |
| xj |  |
Complex Poincaré coordinate | (6) |
| C | C1 + C2 + C3 | Total AMD | (7) |
| G | Λ1 + Λ2 + Λ3 − C | Total angular momentum | (7) |
| nj |  |
Mean motion | (12) |
| ε | Dimensionless parameter related to the planet to star mass ratio | ||
 |
Keplerian part of the Hamiltonian | (8) | |
 |
Planet interactions Hamiltonian | (9) | |
| εχ1 | First-order averaging generating Hamiltonian | (16) | |
| νij | Pi∕Pj | Period ratio for planet i and j | (19) |
| η |  |
Resonance locator | (21) |
| ν |  |
Generalised period ratio separation | |
| θres | pλ1 − (p + q)λ2 + qλ3 | Zeroth order three-planet resonant angle | (24) |
| Θ |  |
Resonant action | (25) |
| Γ |  |
Scaling parameter | (25) |
| ϒ | Λ1 + Λ2 + Λ3 | Circular angular momentum | (25) |
| αij |  |
Semi-major axis ratio | |
 |
Laplace coefficients | (B.1) | |
| ε2 Rpq | Resonant coefficient | (42), (47) | |
| δij | 1-αij | Planet orbital spacing | (43) |
| δ |  |
Generalised orbital spacing | (45) |
 |
Coefficient of the second-order development of the Keplerian part | (49) | |
| ωpq |  |
Small oscillation frequency around the resonance | (52) |
| A | 3.47 | Numerical factor appearing in ωpq | |
| εM | Relevant mass ratio for the problem | (53) | |
| Δηpq | Resonance width in period ratio space | (55) | |
| ρp+q | (p + q)Δηpq | Density of the subnetwork of resonances with index p + q | (57) |
| ρtot | Total density of the zeroth order three-planet MMR network | (58) | |
| δov | Generalised orbital spacing such that the MMR cover the full space | (59) | |
| kov | Minimum index such that the resonances with lower index locally cover the period ratio space | (63) | |
| ξov | kovδ | (65) | |
| Dp+q |  |
Diffusion coefficient linked to the resonance p, q | (62) |
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