Compact Planetary Systems Perturbed by an Inclined Companion. I. Vectorial Representation of the Secular Model

The non-resonant secular dynamics of compact planetary systems are modeled by a perturbing function that is usually expanded in eccentricity and absolute inclination with respect to the invariant plane. Here, the expressions are given in a vectorial form which naturally leads to an expansion in eccentricity and mutual inclination. The two approaches are equivalent in most cases, but the vectorial one is specially designed for those cases where an entire quasi-coplanar system tilts to a large degree. Moreover, the vectorial expressions of the Hamiltonian and of the equations of motion are slightly simpler than those given in terms of the usual elliptical elements. We also provide the secular perturbing function in vectorial form expanded in semi-major axis ratio allowing for arbitrary eccentricities and inclinations. The interaction between the equatorial bulge of a central star and its planets is also provided, as is the relativistic periapse precession of any planet induced by the central star. We illustrate the use of this representation to follow the secular oscillations of the terrestrial planets of the solar system and for Kozai cycles which may take place in exoplanetary systems.