Constraining planetesimal stirring: how sharp are debris disc edges?

The dust production in debris discs by grinding collisions of planetesimals requires their orbits to be stirred. However, stirring levels remain largely unconstrained, and consequently the stirring mechanisms as well. This work shows how the sharpness of the outer edge of discs can be used to constrain the stirring levels. Namely, the sharper the edge the lower the eccentricity dispersion must be. For a Rayleigh distribution of eccentricities (e), I find that the disc surface density near the outer edge can be parametrized as tanh [(r_max - r)/l_out], where r_max approximates the maximum semimajor axis and l_out defines the edge smoothness. If the semimajor axis distribution has sharp edges e_rms is roughly 1.2l_out/r_max or e_rms = 0.77l_out/r_max if semimajor axes have diffused due to self-stirring. This model is fitted to Atacama Large Millimeter/submillimeter Array data of five wide discs: HD 107146, HD 92945, HD 206893, AU Mic, and HR 8799. The results show that HD 107146, HD 92945, and AU Mic have the sharpest outer edges, corresponding to e_rms values of 0.121 ± 0.05, $0.15^{+0.07}_{-0.05}$, and 0.10 ± 0.02 if their discs are self-stirred, suggesting the presence of Pluto-sized objects embedded in the disc. Although these stirring values are larger than typically assumed, the radial stirring of HD 92945 is in good agreement with its vertical stirring constrained by the disc height. HD 206893 and HR 8799, on the other hand, have smooth outer edges that are indicative of scattered discs since both systems have massive inner companions.