Formation of Gaps in Self-gravitating Debris Disks by Secular Resonance in a Single-planet System. II. Towards a Self-consistent Model

High-resolution observations of several debris disks reveal structures such as gaps and spirals, suggestive of gravitational perturbations induced by underlying planets. Most existing studies of planet--debris disk interactions ignore the gravity of the disk, treating it as a reservoir of massless planetesimals. In this paper, we continue our investigation into the long-term interaction between a single eccentric planet and an external, massive debris disk. Building upon our previous work, here we consider not only the axisymmetric component of the disk's gravitational potential, but also the non-axisymmetric torque that the disk exerts on the planet (ignoring for now only the non-axisymmetric component of the disk self-gravity). To this goal, we develop and test a semi-analytic `$N$-ring' framework that is based on a generalized (softened) version of the classical Laplace--Lagrange secular theory. Using this tool, we demonstrate that even when the disk is less massive than the planet, not only can a secular resonance be established within the disk that leads to the formation of a wide gap, but that the very same resonance also damps the planetary eccentricity $e_p$ via a process known as resonant friction. The resulting gap is initially non-axisymmetric (akin to those observed in HD 92945 and HD 206893), but evolves to become more axisymmetric (similar to that in HD 107146) as $e_p(t)\rightarrow0$ with time. We also develop analytic understanding of these findings, finding good quantitative agreement with the outcomes of the $N$-ring calculations. Our results may be used to infer both the dynamical masses of (gapped) debris disks and the dynamical history of the planets interior to them, as we exemplify for HD 206893.