In many astrophysical problems involving discs (gaseous or particulate) orbiting a dominant central mass, gravitational potential of the disc plays an important dynamical role. Its impact on the motion of external objects, as well as on the dynamics of the disc itself, can usually be studied using secular approximation. This is often done using softened gravity to avoid singularities arising in calculation of the orbit-averaged potential -disturbing function -of a razor-thin disc using classical Laplace-Lagrange theory. We explore the performance of several softening formalisms proposed in the literature in reproducing the correct eccentricity dynamics in the disc potential. We identify softening models that, in the limit of zero softening, give results converging to the expected behavior exactly, approximately or not converging at all. We also develop a general framework for computing secular disturbing function given an arbitrary softening prescription for a rather general form of the interaction potential. Our results demonstrate that numerical treatments of the secular disc dynamics, representing the disc as a collection of N gravitationally interacting annuli, are rather demanding: for a given value of the (dimensionless) softening parameter, ς ≪ 1, accurate representation of eccentricity dynamics requires N ∼ Cς −χ ≫ 1, with C ∼ O(10), 1.5 χ 2. In discs with sharp edges a very small value of the softening parameter ς ( 10 −3 ) is required to correctly reproduce eccentricity dynamics near the disc boundaries; this finding is relevant for modelling planetary rings.

A relatively massive and moderately eccentric disk of trans-Neptunian objects (TNOs) can effectively counteract apse precession induced by the outer planets, and in the process shepherd highly eccentric members of its population into nearly-stationary configurations which are anti-aligned with the disk itself. We were sufficiently intrigued by this remarkable feature to embark on an extensive exploration of the full spatial dynamics sustained by the combined action of giant planets and a massive trans-Neptunian debris disk. In the process, we identified ranges of disk mass, eccentricity and precession rate which allow apse-clustered populations that faithfully reproduce key orbital properties of the much discussed TNO population. The shepherding disk hypothesis is to be sure complementary to any potential ninth member of the Solar System pantheon, and could obviate the need for it altogether. We discuss its essential ingredients in the context of Solar System formation and evolution, and argue for their naturalness in view of the growing body of observational and theoretical knowledge about self-gravitating disks around massive bodies, extra-solar debris disks included.

Spatially resolved images of debris disks frequently reveal complex morphologies such as gaps, spirals, and warps. Most existing models for explaining such morphologies focus on the role of massive perturbers (i.e., planets, stellar companions), ignoring the gravitational effects of the disk itself. Here we investigate the secular interaction between an eccentric planet and a massive, external debris disk using a simple analytical model. Our framework accounts for both the gravitational coupling between the disk and the planet, as well as the disk self-gravity—with the limitation that it ignores the non-axisymmetric component of the disk (self-)gravity. We find generally that even when the disk is less massive than the planet, the system may feature secular resonances within the disk (contrary to what may be naively expected), where planetesimal eccentricities get significantly excited. Given this outcome, we propose that double-ringed debris disks, such as those around HD 107146 and HD 92945, could be the result of secular resonances with a yet-undetected planet interior to the disk. We characterize the dependence of the properties of the secular resonances (i.e., locations, timescales, and widths) on the planet and disk parameters, finding that the mechanism is robust provided the disk is massive enough. As an example, we apply our results to HD 107146 and find that this mechanism readily produces ∼20 au wide non-axisymmetric gaps. Our results may be used to set constraints on the total mass of double-ringed debris disks. We demonstrate this for HD 206893, for which we infer a disk mass of ≈170M ⊕ by considering perturbations from the known brown dwarf companion.

The vertical structure of debris disks provides clues about their dynamical evolution and the collision rate of the unseen planetesimals. Thanks to the ever-increasing angular resolution of contemporary instruments and facilities, we are beginning to constrain the scale height of a handful of debris disks, either at near-infrared or millimeter wavelengths. Nonetheless, this is often done for individual targets only. We present here the geometric modeling of eight disks close to edge-on, all observed with the same instrument (SPHERE) and using the same mode (dual-beam polarimetric imaging). Motivated by the presence of CO gas in two out of the eight disks, we then investigate the impact that gas can have on the scale height by performing N-body simulations including gas drag and collisions. We show that gas can quickly alter the dynamics of particles (both in the radial and vertical directions), otherwise governed by gravity and radiation pressure. We find that, in the presence of gas, particles smaller than a few tens of microns can efficiently settle toward the midplane at the same time as they migrate outward beyond the birth ring. For second generation gas (Mgas ≤ 0.1 M⊕), the vertical settling should be best observed in scattered light images compared to observations at millimeter wavelengths. But if the gas has a primordial origin (Mgas ≥ 1 M⊕), the disk will appear very flat both at near-infrared and sub-mm wavelengths. Finally, far beyond the birth ring, our results suggest that the surface brightness profile can be as shallow as ∼−2.25.