Excitation of planetary inclinations via dynamical bifurcations driven by misaligned disks

Open Access

Table 1

Parameters of several representative hierarchical systems.

Perturbation from an outer planetary companion

Parameter	R_⋆ (R_⊙)	P_⋆ (day)	J₂ (10⁻⁴)	a₁ (au)	m₂ (M_J)	a₂ (au)	ϵ_⋆p	$ψ_{⋆ 0, crit}^{P}$ $\[\psi_{\star 0, \text {crit}}^{P}\]$ (°)
Case I	2.0	5.0	2.87	0.1	1.0	2.0	41.7	45.3
Case II	1.5	5.0	1.21	0.1	1.0	2.0	9.9	47.5
Case III	1.5	5.0	1.21	0.1	1.0	1.0	1.24	71.8
Case IV	1.5	3.0	3.36	0.05	1.0	0.5	13.7	46.7
K2-290A(1)	2.5	3.0	13.1	0.0923	0.774	0.305	2.42	56.8
K2-290A(2)	2.0	3.0	6.70	0.0923	0.774	0.305	0.79	−
HD 3167	1.5	5.0	1.42	0.018	$\sum m_{j} / a_{j}^{3} = 41.05$ $\[\sum m_{j} / a_{j}^{3}=41.05\]$		158.6	44.8

Perturbation from the outer disk

Parameter	R_⋆ (R_⊙)	P_⋆ (day)	J₂ (10⁻⁴)	a₁ (au)	m_d (M_⊙)	r_c	ϵ_⋆p	$ψ_{⋆ 0, crit}^{P}$ $\[\psi_{\star 0, \text {crit}}^{P}\]$ (°)

Case 1	2.0	5.0	2.87	0.1	0.02	2	99.5	44.9
Case 2	1.5	5.0	1.21	0.1	0.02	1.5	13.3	46.7

Notes. The values of ϵ_⋆p and $ψ_{⋆ 0, crit}^{P}$ $\[\psi_{\star 0, \text {crit}}^{P}\]$ are calculated by using Eq. (35) and the bifurcation condition (Eq. (37)), respectively. If the perturbation is from the outer disk considering nonhomologous disk dissipation, ϵ_⋆p is calculated by ϵ_⋆p = ϕ_⋆,1/ϕ_dout,1, where ϕ_dout,1 is given by Eq. (19). The parameter J₂ is calculated through Eq. (39). Except R_⋆ and P_⋆, the other system parameters of the K2-290A and HD 3167 systems are taken from current observations (Hjorth et al. 2021; Bourrier et al. 2022). For HD 3167, the outer three planets collectively act as an outer perturbation, as discussed in Fu & Wang (2024). Except K2-290A with m_⋆ = 1.19M_⊙ and HD 3167 with m_⋆ = 0.852M_⊙, all other systems have stellar masses of m_⋆ = 1m_⊙. The disk’s outer truncation is set to be r_out = 50 au.

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