ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: three regimes: the stable regime, the intermediate regime, and the unstable regime. In the stable regime, the Laplace–Lagrange rapid angular momentum exchange between the two inner planets dominates over the gravitational perturbations of the outer orbit. The left column of Fig. 3 depicts this evolution; the system in this column resides in the bottom left of the parameter space in Fig. 2 . The system remains stable for 10 Gyr of evolution and the orbits never come close to one another. The two inner planets’ inclinations oscillate together around the initial z -axis, due to the precession of the nodes (see below). In the intermediate region, systems might appear to be stable for a long time but the outer orbit perturbations become too dominant, causing instability of the inner system. We show the dynamics of this type of evolution in the middle panel of Fig. 3 . This system resides very close to the stability criterion plotted in Fig. 2 , inside the transition zone between e 3,c ( f LL,min ) and e 3,c ( f LL,max ). In the unstable regime, gravitational perturbations from the third planet cause high amplitude eccentricity excitation in the inner planets, causing orbits to cross. We show this behaviour in the right column 3 . This system lies in the top right of the parameter space depicted in Fig. 2 . Figure 3. Here are the three prominent types of dynamics from each region. From left to right, stable, intermediate, and unstable, for our nominal system. Shown in the top panels are the apocentres and the pericentres of the innermost planets ( m 1 in blue and m 2 in red). The middle panels show the inclinations of the inner planets ( i 1 in blue and i 2 in red). Note that this inclination is not with respect to the total angular momentum, but rather with respect to the initial angular momentum of the two inner planets . The bottom panels show, coloured in black, the difference between the longitude of ascending nodes of the two inner planets as cos (Ω 1 − Ω 2 ), and, in yellow, the difference between the longitude of the periapsis as cos (π 1 − π 2 ). These two parameters are present in equation ( 11 ). Recall that the nominal system has the following parameters: M = 1 M ⊙ , m 1 = m 2 = 1 M ⊕ , m 3 = 1 M J , a 1 = 1 au, a 3 = 100 au, and we set initially ω 1 = ω 2 = ω 3 = Ω 1 = Ω 2 = Ω 3 = 0, e 1 = e 2 = 0.001, i 1 = i 2 = 0.001 and i 3 = 45°. The only difference between each integrated system is the second planet’s separation and the third planet’s eccentricity. For the stable system we chose a 2 = 3 au and e 3 = 0.8. For the intermediate system we had a 2 = 3 au and e 3 = 0.9 (which placed it on the stability curve), and finally for the unstable system we had a 2 = 5.9 au and e 3 = 0.551. In the system’s reference frame (see Fig. 1 ), the inner planet’s inclinations are the angles between their respective angular momenta and the normal to the initial orbits. Thus, the inclination modulation shown in the stable system (the left column in Fig. 3