Dynamical models to explain observations with SPHERE in planetary systems with double debris belts

Context. A large number of systems harboring a debris diskshow evidence for a double belt architecture. One hypothesis for explaining the gap between the debris belts in these disks is the presence of one or more planetsdynamically carving it. For this reason these disks represent prime targets for searching planetsusing direct imaging instruments, like the Spectro-Polarimetric High-constrast ExoplanetResearch (SPHERE) at the Very Large Telescope. Aim. The goal of this work is to investigate this scenario in systems harboring debris disksdivided into two components, placed, respectively, in the inner and outer parts of the system. All the targets in the sample were observed with the SPHERE instrument, which performs high-contrast direct imaging, during the SHINE guaranteed time observations. Positions of the inner and outer belts were estimated by spectral energy distributionfitting of the infrared excessesor, when available, from resolved images of the disk. Very few planetshave been observed so far in debris disksgaps 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 planetsable to open the observed gaps and comparing such parameters with detection limits obtained with SPHERE. Methods. The relation between the gap and the planetis due to the chaotic zone neighboring the orbit of the planet. The radial extent of this zone depends on the mass ratio between the planetand the star, on the semi-major axis, and on the eccentricityof the planet, and it can be estimated analytically. We first tested the different analytical predictions using a numerical tool for the detection of chaotic behavior and then selected the best formula for estimating a planet’s physical and dynamical properties required to open the observed gap. We then apply the formalism to the case of one single planeton a circular or eccentricorbit. We then consider multi- planetary systems: two and three equal-mass planetson circular orbitsand two equal-mass planetson eccentricorbits in a packed configuration. As a final step, we compare each couple of values ( M p , a p ), derived from the dynamical analysis of single and multiple planetary models, with the detection limits obtained with SPHERE. Results. For one single planeton a circular orbitwe obtain conclusive results that allow us to exclude such a hypothesis since in most cases this configuration requires massive planetswhich should have been detected by our observations. Unsatisfactory is also the case of one single planeton an eccentricorbit for which we obtained high masses and/or eccentricitieswhich are still at odds with observations. Introducing multi planetary architectures is encouraging because for the case of three packed equal-mass planetson circular orbitswe obtain quite low masses for the perturbing planetswhich would remain undetected by our SPHERE observations. The case of two equal-mass planetson eccentricorbits is also of interest since it suggests the possible presence of planetswith masses lower than the detection limits and with moderate eccentricity. Our results show that the apparent lack of planetsin gaps between double belts could be explained by the presence of a system of two or more planetspossibly of low mass and on eccentricorbits whose sizes are below the present detection limits.