Self-consistent modelling of the dust component in protoplanetary and circumplanetary disks: the case of PDS 70

Context. Direct observations of young stellar objects are important to test established theories of planet formation. PDS 70 is one of the few cases where robust evidence favours the presence of two planetary mass companions inside the gap of the transition disk. Those planets are believed to be going through the last stages of accretion from the protoplanetary disk, a process likely mediated by a circumplanetary disk (CPD). Aims. We aim to develop a three-dimensional radiative transfer model for the dust component of the PDS 70 system which reproduces the system's global features observed at two different wavelengths: 855 mu m with the Atacama large millimeter/submillimeter array (ALMA) and 1.25 mu m with the Spectro-polarimetric high-contrast exoplanet research instrument at the Very large telescope (VLT/SPHERE). We use this model to investigate the physical properties of the planetary companion PDS 70 c and its potential circumplanetary disk. Methods. We base our modelling process on published ALMA and VLT/SPHERE observations of the dust continuum emission at 855 and 1.25 mu m, respectively. We selected initial values for the physical properties of the planet and CPD through appropriate assumptions about the nature and evolutionary stage of the object. We modified the properties of the protoplanetary disk iteratively until the predictions retrieved from the model were consistent with both data sets. Simulations were carried out with the three-dimensional radiative transfer code MCMax3D. Results. We provide a model that jointly explains the global features of the PDS 70 system seen in sub-millimetre and polarised-scattered light. Our model suggests that spatial segregation of dust grains is present in the protoplanetary disk. The sub-millimetre modelling of the PDS 70 c source favours the presence of an optically thick CPD and places an upper limit on its dust mass of 0.7 M-circle plus. Furthermore, analysis of the thermal structure of the CPD demonstrates that the planet luminosity is the dominant heating mechanism of dust grains inside 0.6 au from the planet, while heating by stellar photons dominates at larger planetocentric distances. Conclusions. A CPD surrounding the planetary-mass companion PDS 70 c is a plausible scenario to explain the substructure observed with ALMA. The heating feedback from the protoplanetary disk has an non-negligible effect on the equilibrium temperature of dust grains in the outskirts of the CPD. The connection between the CPD properties and the planet mass still depends on a series of key assumptions. Further observations with high spatial and spectral resolution also for the gas component of the CPD are required to break the degeneracy between the properties of the planet and the disk.