= Velocity-dependent collisional cascade
For <catastrophic disruption threshold> $Q_D^*=Q_bD^b$, encounter speed $v_{\rm rel}=v_pD^p$, fixed belt volume, and small impactors, the <catastrophic projectile diameter> scales as $D_{\rm cc}\propto D^{1+(b-2p)/3}$. If $\alpha>1$, small destructive impactors dominate the <catastrophic planetesimal collision rate>, giving
$$
R_{\rm cc}(D)\propto D^{2+p}D_{\rm cc}^{1-\alpha}.
$$
Mass per logarithmic bin scales as $D^{4-\alpha}$. <Constant mass flux in a collisional cascade> then gives
$$
\alpha=\frac{21+b+p}{6+b-2p}.
$$
Size-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 https://arxiv.org/abs/1111.0667[Pan and Schlichting].
Back to article page
