Secular orbital dynamics of the innermost exoplanet of the υ -Andromedæ system

We introduce a quasi-periodic restricted Hamiltonian to describe the secular motion of a small-mass planet in a multi-planetary system. In particular, we refer to the motion of υ -And b which is the innermost planet among those discovered in the extrasolar system orbiting around the υ -Andromedæ A star. We preassign the orbits of the Super-Jupiter exoplanets υ -And c and υ -And d in a stable configuration. The Fourier decompositions of their secular motions are reconstructed by using the well-known technique of the (so-called) frequency analysis and are injected in the equations describing the orbital dynamics of υ -And b under the gravitational effects exerted by those two external exoplanets (that are expected to be major ones in such an extrasolar system). Therefore, we end up with a Hamiltonian model having 2 + 3 / 2 degrees of freedom; its validity is confirmed by the comparison with several numerical integrations of the complete four-body problem. Furthermore, the model is enriched by taking into account also the effects due to the relativistic corrections on the secular motion of the innermost exoplanet. We focus on the problem of the stability of υ -And b as a function of the parameters that mostly impact on its orbit, that are the initial values of its inclination and the longitude of its node (as they are measured with respect to the plane of the sky). In particular, we study the evolution of its eccentricity, which is crucial to exclude orbital configurations with high probability of (quasi)collision with the central star in the long-time evolution of the system. Moreover, we also introduce a normal form approach, that is based on the complete average of our restricted model with respect to the angles describing the secular motions of the major exoplanets. Therefore, our Hamiltonian model is further reduced to a system with 2 degrees of freedom, which is integrable because it admits a constant of motion that is related to the total angular momentum. This allows us to very quickly preselect the domains of stability for υ -And b , with respect to the set of the initial orbital configurations that are compatible with the observations.