We investigate the stability of prograde versus retrograde planets in circular binary systems using numerical simulations. We show that retrograde planets are stable up to distances closer to the perturber than prograde planets. We develop an analyti
cal model to compute the prograde and retrograde mean motion resonances locations and separatrices. We show that instability is due to single resonance forcing, or caused by nearby resonances overlap. We validate our results regarding the role of single resonances and resonances overlap on orbit stability, by computing surfaces of section of the CR3BP. We conclude that the observed enhanced stability of retrograde planets with respect to prograde planets is due to essential differences between the phase-space topology of retrograde versus prograde resonances (at p/q mean motion ratio, prograde resonance is of order p - q while retrograde resonance is of order p + q).
The HD,196885 system is composed of a binary star and a planet orbiting the primary. The orbit of the binary is fully constrained by astrometry, but for the planet the inclination with respect to the plane of the sky and the longitude of the node are
unknown. Here we perform a full analysis of the HD,196885 system by exploring the two free parameters of the planet and choosing different sets of angular variables. We find that the most likely configurations for the planet is either nearly coplanar orbits (prograde and retrograde), or highly inclined orbits near the Lidov-Kozai equilibrium points, i = 44^{circ} or i = 137^{circ} . Among coplanar orbits, the retrograde ones appear to be less chaotic, while for the orbits near the Lidov-Kozai equilibria, those around omega= 270^{circ} are more reliable, where omega_k is the argument of pericenter of the planets orbit with respect to the binarys orbit. From the observers point of view (plane of the sky) stable areas are restricted to (I1, Omega_1) sim (65^{circ}, 80^{circ}), (65^{circ},260^{circ}), (115^{circ},80^{circ}), and (115^{circ},260^{circ}), where I1 is the inclination of the planet and Omega_1 is the longitude of ascending node.
Constructing dynamical maps from the filtered output of numerical integrations, we analyze the structure of the $ u_odot$ secular resonance for fictitious irregular satellites in retrograde orbits. This commensurability is associated to the secular a
ngle $theta = varpi - varpi_odot$, where $varpi$ is the longitude of pericenter of the satellite and $varpi_odot$ corresponds to the (fixed) planetocentric orbit of the Sun. Our study is performed in the restricted three-body problem, where the satellites are considered as massless particles around a massive planet and perturbed by the Sun. Depending on the initial conditions, the resonance presents a diversity of possible resonant modes, including librations of $theta$ around zero (as found for Sinope and Pasiphae) or 180 degrees, as well as asymmetric librations (e.g. Narvi). Symmetric modes are present in all giant planets, although each regime appears restricted to certain values of the satellite inclination. Asymmetric solutions, on the other hand, seem absent around Neptune due to its almost circular heliocentric orbit. Simulating the effects of a smooth orbital migration on the satellite, we find that the resonance lock is preserved as long as the induced change in semimajor axis is much slower compared to the period of the resonant angle (adiabatic limit). However, the librational mode may vary during the process, switching between symmetric and asymmetric oscillations. Finally, we present a simple scaling transformation that allows to estimate the resonant structure around any giant planet from the results calculated around a single primary mass.
