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tokens: only occur with a libration period, T , in years given by with and H θθ = − R θθ , which can be calculated numerically, and both H II , H θθ are evaluated at the stable equilibrium point. Fig. 1 Level curves of a fictitious Hamiltonian. O indicates a stable equilibrium point at ( θ s = 60°, I 0 = 2) and X an unstable equilibrium point at ( θ u = 240°, I 0 = 2). The arrows show the total resonance width, which is 2( I sep − I 0 ), where I sep = 2.44. To obtain the width of the resonance in terms of Δ I , we follow the same reasoning as in Gallardo (2020) . Figure 1 shows that the resonance half-width is equal to the difference between I 0 and I sep , where I sep is defined by the separatrix such that where θ s and θ u are the stable and unstable equilibrium points. When we assume that the domain of the resonance is symmetric with respect to the nominal position, I 0 , the total width is twice Δ I . In some cases, asymmetrieswith respect to I = I 0 are present and a precise examination of the Hamiltonian level curves is required to obtain the exact width. In general, however, because Δ H = H ( θ s , I sep ) − H ( θ s , I 0 ), we can approximate Evaluating the derivatives at the stable equilibrium point ( θ s , I 0 ) and using Eq. ( 14 ), we have Using Eq. ( 6 ), the left-hand side is while is given by Eq. ( 13 ), then We recall that Δ I is in fact Δ I 1 . Strength is commonly defined as SR = ⟨ R ⟩− R min ( Gallardo 2006 ), which can be associated with the coefficient multiplying cos ( σ ) when R is a well-behaved cosine-like function, but in general, R could be very different from a simple cosine. In these situations, SR therefore is a reliable estimate of the sensitivity of R with σ . The strength is a global property of the resonance, as are the libration center and period. It is an indicator of the dynamical relevance of the resonance, and it generates different Δ a on each planet. The width in terms of Δ a 1 is obtained from the definition of I 1 in Eq. ( 4 ) and Δ a 2 is obtained from I 2 = constant and is evaluated at the center of the resonance. We obtain the half-widths in au: The higher the planetary mass m i , the narrower the relative width Δ a i ∕ a i . The resonance is the same dynamical mechanism that affects the two planets, but stronger effects are present in the planet with the lower mass. It is important to stress that Δ a i are given interms of Poincaré elements, but for planetary systems, the computed widths Δ a i in Poincaré elements arevery similar to the widths in astrocentric elements. Moreover, the widths obtained by the formulae above are correct as long as Δ a ≪ a . If large widths are obtained using these formulae, it is convenient to recalculate the resonance limits ( I max , I min ) by a numerical procedure that considers the separatrix obtained from the Hamiltonian given by Eq. ( 6 ). In this circumstance, it is possible to observe asymmetries between the two limits of the resonance around the exact (nominal)