Self-gravity in thin protoplanetary discs. I. The smoothing-length approximation versus the exact self-gravity Kernel

Planet-forming discs are increasingly found to harbour internal structures such as spiral arms. The origin and evolution of these is often associated with the disc's own gravitational force. When investigating discs using a 2D approximation, it is common to employ an ad hoc softening prescription for self-gravity. However, this approach ignores how the vertical structure of the disc is affected by the mass distribution of gas and dust. More significantly, it suppresses the Newtonian nature of gravity at short distances and does not respect Newton's third law. To overcome the inherent issues associated with approximate descriptions -- for instance, a Plummer potential -- we aim to derive an exact self-gravity kernel designed for hydrostatically supported thin discs, which moreover incorporates a potential dust fluid component embedded in the gas. We develop an analytical framework to derive an exact 2D self-gravity prescription suitable for modelling thin discs. The validity and consistency of the proposed kernel is then supported by analytical benchmarks and both 2D and 3D numerical tests. We derive the exact 2D self-gravity kernel valid for Gaussian-stratified thin discs. This kernel is built upon exponentially scaled modified Bessel functions and simultaneously adheres to all the expected features of Newtonian gravitation -- including point-wise symmetry, a smooth transition from light to massive discs, and a singularity for vanishing distances. Quite remarkably, the kernel displays a purely 2D nature at short distances, while transitioning to a fully 3D behaviour at longer distances. In contrast to other prescriptions found in the literature, it proves capable of leading to an additional, and previously unnoticed, source of gravitational runaway discernible only at infinitesimal distances. We finally note that our new prescription remains compatible with methods based on the fast Fourier transform, affording superior computational efficiency. Our exact kernel formulation overcomes substantial limitations inherent in the smoothing-length approach. It permits a novel, fully consistent treatment of self-gravity in Gaussian-stratified thin discs. The approach, which makes the usage of the Plummer potential obsolete, will prove useful for studying all common planet formation scenarios, which are often backed by 2D flat numerical simulations. Accordingly, in an accompanying paper we shall investigate how the occurrence of the gravitational instability is affected.