The vertical structure of debris discs and the role of disc gravity: a primer using a simplified model

Debris discs provide valuable insights into the formation and evolution of exoplanetary systems. Their structures are commonly attributed to planetary perturbations, serving as probes of as-yet-undetected planets. However, most studies of planet-debris disc interactions ignore the disc’s gravity, treating it as a collection of mass-less planetesimals. We develop a simplified analytical model as a primer to investigate how the vertical structure of a back-reacting debris disc responds to secular perturbations from an inner, inclined planet. Considering the disc’s axisymmetric potential, we identify two dynamical regimes: planetdominated and disc-dominated, which may coexist, separated by a secular-inclination resonance. In the planet-dominated regime (M_d /m_p ≪ 1), we recover the classical result: a transient warp propagates outward until the disc settles into a box-like structure centred around the planetary orbit’s initial inclination I_p(0), with a distance-independent aspect ratio H(R) ≈ I_p(0). In contrast, in the disc-dominated regime (M_d /m_p ⪰ 1), the disc exhibits dynamical rigidity, remaining thin and misaligned, with significantly suppressed inclinations and a sharply declining aspect ratio, H(R) ∝ I_p(0)R^−7/2. In the intermediate regime (M_d /m_p ≲ 1), the system exhibits a secular-inclination resonance, leading to long-lived, warp-like structures and a bimodal inclination distribution, containing both dynamically hot and cold populations. We provide analytic formulae describing these effects as a function of system parameters. We also find that the vertical density profile is intrinsically non-Gaussian and recommend fitting observations with non-zero slopes of H(R). Our results may be used to infer planetary parameters and debris disc masses based on observed warps and scale heights, as demonstrated for HD 110058 and β Pic.