Second Generation Planet Formation in Post-AGB Discs: Testing the Role of Gravitational Instability

Post-asymptotic giant branch (post-AGB) binary stars are evolved systems that host circumbinary discs formed through mass loss during late stage binary interactions. Their structural, morphological, kinematic, and chemical similarities to planet-forming discs suggest that these systems may act as sites of "second generation" planet formation. In this study, we assess whether the disc instability mechanism -- a proposed pathway for rapid, giant planet formation in some protoplanetary discs - can operate in post-AGB discs; motivated by their short lifetimes. Using the Toomre criterion under well motivated assumptions for disc structure and size, mass, and thermal properties, we assess the conditions for gravitational instability. We first benchmark our analytical framework using well studied protoplanetary disc systems (including HL Tauri, Elias 2-27, GQ Lupi) before applying the same analysis to observed post-AGB discs. We find that post-AGB discs are generally gravitationally stable at present, due primarily to their low masses. Using viscous disk theory, we find that the discs were stable against collapse even in the past, when their masses were potentially higher. In contrast, several protoplanetary discs analysed in the same way show that they likely experienced gravitationally unstable phases early on. We also find that higher viscosity parameters are better aligned with expected post-AGB disc lifetimes. Finally, we revisit the planet formation scenario proposed for the post-common envelope system NN Ser, first carried out by Schleicher and Dreizler and we show that gravitational instability could be feasible under specific, high disc mass assumptions. Overall, our results provide the first systematic theoretical assessment of gravitational instability in post-AGB discs, demonstrating that this mechanism is unlikely to dominate second generation planet formation in these systems.