Self-gravity in thin-disc simulations of protoplanetary discs: smoothing length rectified and generalised to bi-fluids

Abstract

To mimic protoplanetary discs (PPDs) evolution, 2D simulations with self-gravity must introduce a softening prescription of the gravitational potential. When the disc is only made of gas the smoothing length is proportional to the gas scale height. On the other hand when a dust component is included, the question arises as whether the smoothing length approach can still be used to quantify not only the dust self-gravity but also its gravitational interaction with gas. We identified grey areas in the standard smoothing length formalism for computing self-gravity in PPDs uniquely made of gas. We revisit the smoothing length approach which is then generalised to two phases when the dust component can be considered as a pressureless fluid. Analytical developments are used to approximate the vertically averaged self-gravity when the smoothing length is not assumed to be constant but rather a spatial function. We obtained an analytical expression for the space varying smoothing length, which strongly improves the accuracy of the self-gravity computation. For the first time, this method is generalised to address bi-fluid interactions in a PPD: two additional smoothing lengths are proposed for featuring an isolated dusty disc and gas-dust self-gravity interactions. We checked that our method remains compatible with standard fast Fourier transform algorithms and evaluated computational costs. Our space varying smoothing length permits (i) to solve the contradictions inherent to a constant smoothing length hypothesis, (ii) to fit accurately the 3D vertically averaged self-gravity and (iii) is applicable to a bi-fluid description of PPDs with the use of two additional smoothing lengths. Such results are crucial to enable realistic 2D numerical simulations accounting for self-gravity and are important to deepen our understanding of planetesimals formation and type I migration.