The paper introduces a modified Laplace-Lagrange formula, Eq. (10), that adds the Heppenheimer eccentricity factor to handle eccentric companion stars, and extends it to multiple perturbers via superposition.
Secular Orbit Evolution in Systems with a Strong External Perturber - A Simple and Accurate Model
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abstract
We present a semi-analytical correction to the seminal solution for the secular motion of a planet's orbit under gravitational influence of an external perturber derived by Heppenheimer (1978). A comparison between analytical predictions and numerical simulations allows us to determine corrective factors for the secular frequency and forced eccentricity in the co-planar restricted three-body problem. The correction is given in the form of a polynomial function of the system's parameters that can be applied to first-order forced eccentricity and secular frequency estimates. The resulting secular equations are simple, straight forward to use and improve the fidelity of Heppenheimer's solution well beyond higher-order models. The quality and convergence of the corrected secular equations are tested for a wide range of parameters and limits of its applicability are given.
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Analytical estimates of secular frequencies for binary star systems
The paper introduces a modified Laplace-Lagrange formula, Eq. (10), that adds the Heppenheimer eccentricity factor to handle eccentric companion stars, and extends it to multiple perturbers via superposition.