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tokens: our calculations on a detailed level, we show below that the envisioned picture generally holds. We note that orbital excitation by sweeping secular resonances has previously been studied within the context of the solar system (see, e.g., Ward et al. 1976 ; Ward 1981 ; Nagasawa Ida 2000 ; Nagasawa et al. 2002 ; Minton Malhotra 2011 ; Agnor Lin 2012 ). In contrast with our work, these studies do not observe capture and instead report impulsive excitation of the orbital parameters upon encounter. However, these analyses are consistently based on a second order (Lagrange–Laplace) expansion of the Hamiltonian and therefore cannot properly model the establishment of secular commensurabilities, because homoclinic curves do not exist in phase space within this framework, disallowing a resonant domain to even be defined. A counter example is provided by the N -body simulations of Nagasawa et al. ( 2005 ), where long-term capture into first-order secular eccentricity resonances is observed. A final point worthy of attention is that as the host star spins down, the assumed angular momentum hierarchy (Equation ( 2 )) breaks down. Therefore, the assumption that the stellar spin axis remains fixed in inertial space does not formally apply toward the end of the evolutionary track of δ . If m is taken to approach a typical hot-Jupiter-like mass, however, it can be shown that by the time the angular momentum hierarchy is severely violated, the value of δ is already very close to that given by Equation ( 19 ). Accordingly, this complication does not pose a practical limitation. 3.1.2. Eccentricity Evolution Thus far, we have neglected orbital eccentricities in order to obtain an integrable Hamiltonian. Recalling that eccentricity and inclination terms are decoupled to second order in either quantity (Murray Dermott 1999 ), this assumption is justified as long as the eccentricities remain small. However, the ultimate result of adiabatic capture into a secular inclination resonance raises the question of whether or not similar dynamics can occur in the degree of freedom associated with the outer orbit’s eccentricity. Fortunately, very similar techniques to those already employed above can be used to attack this problem. In direct analogy to Equation ( 8 ), the inner orbit’s longitude of perihelion will not remain stationary in time, and will instead precess at a constant rate: π = μ t . In contrast with nodal regression, however, the rate of apsidal precession comprises numerous contributions (Ragozzine Wolf 2009 ):where quantities with the subscript p refer to the planet, and c is the speed of light. For simplicity, in the above expressions we have assumed that the planetary obliquity is null (and therefore only contributes to apsidal precession and not nodal regression), the rotation rate is synchronous with the orbital frequency (Hut 1981 ), and that eccentricities are sufficiently low to warrant the retention of only leading-order terms in e (see Kopal 1978