Secular Dynamics of Self-gravitating Debris Discs
Debris discs are the leftovers from the epoch of planetary formation, akin to the Solar System's asteroid and Kuiper belts. They preserve a record of the physical processes at work throughout the system's age, not least planet--disc interactions. Deciphering this record has become a central focus of modern celestial mechanics, as it may provide unique insights into the formation and evolution of exoplanetary systems, in addition to indirectly detecting planets based on the structures (e.g., gaps and spiral arms) they imprint on debris discs. However, the problem is compounded by two interrelated issues. First, most existing studies of planet—debris disc interactions do not account for the (self-)gravitational effects of debris discs. That is, debris discs are usually treated as a collection of massless particles, subject only to the gravity of external massive companions. This is partly related to the second issue pertaining to the theoretical and computational challenges involved in modelling the long-term (secular) gravitational effects of discs in general. This dissertation aims to partly address these two problems. In the first part of the dissertation, I provide a comparative analysis of the two existing methods for the analytical computation of the secular disturbing function due to a flat disc: softened and non-softened. I first show that the various methods which are based on softening the Newtonian point-mass potential, in the appropriate limit, may converge to the expected non-softened behaviour exactly, approximately, or not converge at all. I then explain this variety of outcomes by formulating a generalised Laplace--Lagrange theory for flat discs which is applicable for a wide variety of softening prescriptions. Finally, I study the conditions that must be obeyed by any discretised numerical treatment of self-gravitating discs to accurately reproduce the expected gravitational potential, finding that a fine numerical sampling is required. In the second part of the dissertation, I explore the secular dynamical interaction between an eccentric planet and a massive, external debris disc. I first employ a simplified model accounting only for the axisymmetric component of the disc gravity to probe the ensuing basic dynamical effects. I find that even when the disc is less massive than the planet, the system may feature secular resonances which in turn can sculpt a gap within the debris disc through excitation of planetesimal eccentricities. I then characterise the properties of the secular resonances and illustrate how the planet--disc parameters can be constrained for a given system hosting a gapped disc. Then, using a numerical method developed based on the first part of the dissertation, I examine the case where the full gravity of the debris disc is accounted for, including both axi- and non-axisymmetric components. In this case, I find new dynamical phenomena whereby the secular resonances cause e.g. the planetary orbit to circularise in time, as well as to launch a one-armed spiral arm beyond the gap. As an example, I apply these results to known gapped debris discs, and show that they could be sculpted by a single interior planet/companion, provided the discs have enough mass. In summary, then, the contributions of this dissertation are three-fold: to better understand the methods for quantifying the gravitational potential of discs in general; to propose a new mechanism for sculpting gaps in debris discs; and more generally, to pave the way for advancing the theory of debris disc modelling by including the effects of disc gravity.