Orbital Evolution of the Extrasolar Planetary Systems HD 39194, HD 141399, and HD 160691

Abstract

The averaged semi-analytical motion theory of second order in planetary masses for the four-planet problem has been constructed. The Hamiltonian and equations of motion are given in Jacobi coordinates and written in elements of the second Poincaré system. The eccentric and oblique Poincaré orbital elements are conserved in the equations of motion up to third order. The orbital evolution of the three-planet system HD 39194 and the four-planet systems HD 141399 and HD 160691 (μ Ara) is considered. The numerical integration of the equations of motion was carried out for a set of initial conditions, in which unknown orbital elements and orbital elements that are known from observations with some uncertainty were varied within admissible limits. The ranges of variation of the orbital elements are determined as a function of the initial conditions. The assumption that the observed planetary systems are stable can be used to exclude initial conditions leading to extreme growth in the orbital eccentricities and inclinations. Initial conditions for which the orbital elements remain small over the entire modeling interval are identified. A method that can be used to narrow the range of possible values of the unknown orbital elements and identify most probable values from the point of view of stability is shown.