Excitation of planetary inclinations via dynamical bifurcations driven by misaligned disks

Open Access

Fig. 4

Values of f(ψ_1⋆; ψ_⋆0, ϵ_⋆d) defined by Eq. (23) as functions of ψ_1⋆ and ϵ_⋆d for ψ_⋆0 = 70°. The red and blue curves represent the critical ϵ_⋆d that trigger the bifurcations B₁ and B₂, respectively, which are characterized by f = 0 and df/dψ_1⋆ = 0. The two bifurcations are further distinguished by the sign of $d^{2} f / d ψ_{1 ⋆}^{2}$ $\[\mathrm{d}^{2} f / \mathrm{d} \psi_{1 \star}^{2}\]$ : B₁ (<0) and B₂ (>0).

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