Periodic orbits in the 1:2:3 resonant chain and their impact on the orbital dynamics of the Kepler-51 planetary system

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Table 3

Configurations of families of symmetric periodic orbits in the 1:2:3 resonant chain.

Family	ϖ1₁	ϖ2₂	ϖ3₃	M₁	M₂	M₃	θ₁	θ₂	θ₃	θ₄
S₁	0	π	0	0	π	0	0	π	0	π	0
												0.0647425	0.0026077	0.0
	0	π	π	0	π	π	0	π	π	π	0
												0.0991335	0.0	0.0006738
	0	0	π	0	0	π	0	0	π	0	0
												0.2391378	0.0	0.0581071
	0	π	π	0	π	π	0	π	π	π	0
												0.0	0.2666913	0.4032193
	π	π	π	π	π	π	π	π	π	π	0

S₂	0	0	π	0	0	π	0	0	π	0	0
												0.3948695	0.3015711	0.0
	0	0	0	0	0	0	0	0	0	0	0

S₃	π	0	π	π	0	0	π	0	0	π	π

S₄	π	0	0	π	0	0	π	0	0	0	0

S₅	π	π	0	π	π	0	π	π	0	π	0

S₆	0	π	0	0	0	π	0	π	π	0	π

S₇	0	0	π	0	0	π	0	0	π	0	0
												0.0	0.0682943	0.2815644
	π	0	π	π	0	π	π	0	π	0	0
												0.1066329	0.0	0.343046
	π	π	π	π	π	π	π	π	π	π	0

S₈	0	0	π	0	0	π	0	0	π	0	0
												0.0	0.1101254	0.488658
	π	0	π	π	0	π	π	0	π	0	0
												0.1764644	0.0	0.5815609
	π	π	π	π	π	π	π	π	π	π	0

S₉	0	π	0	0	π	0	0	π	0	π	0

Notes. Transitions to different configurations as the families, S_i (i = 1…9), of symmetric periodic orbits in the 1:2:3 resonant chain (presented in Fig. 2) evolve from the origin of the axes (circular family) up to high eccentricity values. The transitions along the families S₇ and S₈ begin from the configuration where e₁ > 0 and are monitored as they evolve thereafter. The eccentricity values represent the periodic orbits where a change of a configuration takes place along the family.

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