Self-Stirring of Debris Discs by Planetesimals Formed by Pebble Concentration

When a protoplanetary disc loses gas, it leaves behind planets and one or more planetesimal belts. The belts get dynamically excited, either by planets ('planet stirring') or by embedded big planetesimals ('self-stirring'). Collisions between planetesimals become destructive and start to produce dust, creating an observable debris disc. Following Kenyon & Bromley (2008), it is often assumed that self-stirring starts to operate as soon as the first ~1000 km-sized embedded 'Plutos' have formed. However, state-of-the-art pebble concentration models robustly predict planetesimals between a few km and ~200 km in size to form in protoplanetary discs rapidly, before then slowly growing into Pluto-sized bodies. We show that the timescale, on which these planetesimals excite the disc sufficiently for fragmentation, is shorter than the formation timescale of Plutos. Using an analytic model based on the Ida & Makino (1993) theory, backed up by numerical integrations, we find the excitation timescale to be $T_{\rm excite} \approx 100 x_{\rm m}^{-1} M_*^{-3/2} a^3$ Myr, where $x_{\rm m}$ is the total mass of a protoplanetary disc progenitor in the units of the Minimum-Mass Solar Nebula, $a$ its radius in the units of 100 AU, and $M_*$ is the stellar mass in solar masses. These results are applied to a set of 23 debris discs that have been well resolved with ALMA or SMA. We find that the majority of these discs are consistent with being self-stirred. However, three large discs around young early-type stars do require planets as stirrers. These are 49 Cet, HD 95086, and HR 8799, of which the latter two are already known to have planets.