Table 2
Parameters of several representative compact multi-planet systems.
| Parameter | R⋆ (R⊙) | P⋆ (day) | J2 (10−4) | mi (m⊕) | a1 (au) | δa (RH) | ϵ⋆p,1 | ![$\[\psi_{\star 0, \text {crit}}^{P}\]$](/articles/aa/full_html/2026/01/aa55897-25/aa55897-25-eq117.png) (°) |
|---|---|---|---|---|---|---|---|---|
| Case I | 2.0 | 3.0 | 7.97 | m1 − m3 = 5 | 0.05 | 25 | 8.12 | 48.1 |
| Case II | 2.0 | 5.0 | 2.87 | m1 − m3 = 5 | 0.05 | 20 | 1.56 | 64.5 |
| Case III | 2.0 | 5.0 | 2.87 | m1 − m3 = 5 | 0.1 | 27 | 0.92 | − |
| Case IV | 2.0 | 5.0 | 2.87 | m1 = 5, m2 = m3 = 10 | 0.1 | 22 | 0.38 | − |
| Case V | 1.5 | 5.0 | 1.21 | m1 − m3 = 5 | 0.1 | 18 | 0.07 | − |
| Case VI | 1.5 | 5.0 | 1.21 | m1 = m2 = 5, m3 = m4 = 20 | 0.1 | 18 | 0.06 | − |
Notes. The number of planets in each system is indicated by the column mi. The separations between adjacent planets δa are expressed in terms of the Hill radius, RH. The critical obliquity ![$\[\psi_{\star 0, \text {crit}}^{P}\]$](/articles/aa/full_html/2026/01/aa55897-25/aa55897-25-eq118.png)
is calculated according to ϵ⋆p,1, utilizing the bifurcation conditions Eqs. (37) for hierarchical systems.
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