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

Context. To mimic the evolution of protoplanetary discs (PPDs), 2D simulations that incorporate self-gravity must introduce a softening prescription of the gravitational potential. When the disc is only composed of gas, the smoothing length is proportional to the scale height of the gas. On the other hand, when a dust component is included, the question arises as to whether the smoothing length approach can still be used to not only quantify the dust self-gravity, but its gravitational interaction with gas as well.
Aims: We identified grey areas in the standard smoothing length formalism for computing self-gravity in PPDs entirely made up of gas. Our goal is to revisit the smoothing length approach, which can then be generalised to two phases, whereby the dust component may be considered as a pressureless fluid.
Methods: We used analytical developments to approximate the vertically averaged self-gravity when the smoothing length is not assumed to be constant, but by taking a spatial function instead.
Results: 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. On computational grounds, we prescribe the use of tapering functions for the purpose of avoiding numerical divergences. We also checked that our method continues to be compatible with standard fast Fourier transform algorithms and evaluated computational costs.
Conclusions: Our space-varying smoothing length allows us to: (i) solve the contradictions inherent in the constant smoothing length hypothesis; (ii) fit the 3D vertically averaged self-gravity with a high level of accuracy; and (iii) render it 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 that account for self-gravity and are essential to improving our understanding of planetesimal formation and type I migration.