Dynamical models to explain double debris belts systems and comparison with SPHERE observations

A large number of systems harboring a debris disk show evidence for a double belt architecture, as also observed in the solar system. We observed several tens of these systems with SPHERE as for example HR8799. One likely hypothesis for explaining the gap between the belts is the presence of one or more planets dynamically carving it. This work aims to investigate this scenario in systems harboring two components debris disks. All the targets in the sample were observed with the SPHERE instrument which performs high-contrast direct imaging. Positions of the inner and outer belts were estimated by SED fitting of the infrared excesses or, when available, from resolved images of the disk. Very few planets have been observed so far in debris disks gaps and we intended to test if such non-detections depend on the observational limits of the present instruments. This aim is achieved by deriving theoretical predictions of masses, eccentricities and semi-major axes of planets able to open the observed gaps and comparing such parameters with detection limits obtained with SPHERE. The relation between the gap and the planet is due to the chaotic zone around the orbit of the planet. The radial extent of this zone depends on the mass ratio between the planet and the star, on the semi-major axis and on the eccentricity of the planet and it can be estimated analytically. We apply the formalism to the case of one planet on a circular or eccentric orbit. We then consider multi-planetary systems: 2 and 3 equal-mass planets on circular orbits and 2 equal-mass planets on eccentric orbits in a packed configuration. We then compare each couple of values (M,a), derived from the dynamical analysis of single and multiple planetary models, with the detection limits obtained with SPHERE. Our results show that the apparent lack of planets in gaps between double belts could be explained by the presence of a system of two or more planets possibly of low mass and on an eccentric orbits whose sizes are below the present detection limits