Velocity-dependent collisional cascade

For catastrophic disruption threshold $Q_D^{\ast }=Q_b{D}^b$, encounter speed $v_{\mathrm{rel}}=v_p{D}^p$, fixed belt volume, and small impactors, the catastrophic projectile diameter scales as $D_{\mathrm{cc}}\propto D^{1+(b-2p)/3}$. If $\alpha >1$, small destructive impactors dominate the catastrophic planetesimal collision rate, givingMass per logarithmic bin scales as $D^{4-\alpha }$. Constant mass flux in a collisional cascade then givesSize-independent speed and strength recover the Dohnanyi collisional cascade slope $7/2$. A size-dependent belt thickness changes this derivation. The broader coupled size-and-velocity problem is studied by Pan and Schlichting.