tokens: coupling, resonance, and weak inner planet coupling. We elaborate on these in the following subsections. Figure 2. RMS values of the eccentricities of the planets in a fiducial two-planet system under the influence of an eccentric perturber. The perturber has initial eccentricity e p = 0.1 and has an orbit co-planar with the inner system. The other parameters are the same as Fig. 1 . The solid coloured points represent the results of numerical integrations using secular equations, while the solid coloured curves are calculated using linearized theory; the dashed coloured lines show the forced eccentricity (equation 26 ). The dashed vertical line corresponds to ε 12 = 1. 2.3.1 Resonance When ω 1 ≃ ω 2 (note that this is approximately equivalent to the condition ε 12 ∼ 1), a potential resonance feature arises where large eccentricities can be excited in the inner planets, even for small e p . If we take ω 1 = ω 2 , equations ( 34 ) and ( 35 ) becomes We see that the eccentricity of planet 1 is boosted while the eccentricity of planet 2 is dampened near the resonance. The resonance feature is most pronounced when L 2 ≫ L 1 . To illustrate this, let L 1 = 0, then the forced eccentricity on the inner planet becomes If both the inner planets have zero initial eccentricity, then their eccentricity evolution is given by In this limiting case, according to the linear theory, at ε 12 ≃ 1 the eccentricity of the inner planet can become arbitrarily large, even for small initial values of e p . In reality, the linear theory breaks down as e 1 becomes too large, and higher order terms will keep e 1 to a modest value. An illustration of the resonance behaviour can be seen in Fig. 2 . Systems with larger ratios of m 2 / m 1 tend to exhibit pronounced resonance features, whereas for systems with more comparable masses, the feature is notably reduced. 2.3.2 Strongly and weakly coupled regime In the case where the mutual precession rates of the inner planets dominate over the influence of the perturber (i.e. ω 12 ≫ ω 1 p and ω 21 ≫ ω 2 p ), the general expressions ( 34 ) and ( 35 ) can be significantly simplified; we refer to this as the strongly coupled regime. In this regime, the two planets attain very similar forced eccentricities (), and as a result equations ( 34 ) and ( 35 ) are dominated by their first terms. Explicitly, in this regime the RMS eccentricities are approximately given by where α 12 = ( a 1 / a 2 ) and β 12 = β( a 1 / a 2 ) (see equation 8 ). In the other limiting case (i.e. weak coupling), when the precession rates of the inner planets driven by the perturber dominate the their mutual precession rates (i.e. ν 1 p ≫ ω 12 and ν 2 p ≫ ω 21 ), the terms of order ν 12 /ω 1 p and ν 21 /ω 2 p in equations ( 34 ) and ( 35 ) can be dropped, and the final inner planets’ RMS eccentricities are again given by . In this case, the planets precess independently of one another, and their eccentricities are given by equations ( 11 ) and ( 12 ), i.e. Note
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