Inner edges of planetesimal belts: collisionally eroded or truncated?

The radial structure of debris discs can encode important information about their dynamical and collisional history. In this paper we present a 3-phase analytical model to analyse the collisional evolution of solids in debris discs, focusing on their joint radial and temporal dependence. Consistent with previous models, we find that as the largest planetesimals reach collisional equilibrium in the inner regions, the surface density of dust and solids becomes proportional to $\sim r^{2}$ within a certain critical radius. We present simple equations to estimate the critical radius and surface density of dust as a function of the maximum planetesimal size and initial surface density in solids (and vice versa). We apply this model to ALMA observations of 7 wide debris discs. We use both parametric and non-parametric modelling to test if their inner edges are shallow and consistent with collisional evolution. We find that 4 out of 7 have inner edges consistent with collisional evolution. Three of these would require small maximum planetesimal sizes below 10 km, with HR 8799's disc potentially lacking solids larger than a few centimeters. The remaining systems have inner edges that are much sharper, which requires maximum planetesimal sizes $\gtrsim10$ km. Their sharp inner edges suggest they could have been truncated by planets, which JWST could detect. In the context of our model, we find that the 7 discs require surface densities below a Minimum Mass Solar Nebula, avoiding the so-called disc mass problem. Finally, during the modelling of HD 107146 we discover that its wide gap is split into two narrower ones, which could be due to two low-mass planets formed within the disc.
Comment: Accepted for publication in MNRAS, 21 pages, 11 figures