= Catastrophic projectile diameter
{title2=$D_{\rm cc}$}
For equal-density spherical <planetesimals>, an impactor much smaller than target diameter $D$ causes a <catastrophic planetesimal collision> when its <specific impact energy> reaches $Q_D^*$. Neglecting <gravitational focusing>,
$$
D_{\rm cc}=D\left(\frac{2Q_D^*}{v_{\rm rel}^2}\right)^{1/3}.
$$
When encounter speed is much smaller than the target's <escape velocity>, impact speed is approximately $v_{\rm esc}=D\sqrt{2\pi G\rho/3}$. For $Q_D^*=Q_bD^b$, the focusing-dominated threshold is $D_{\rm cc}=(3Q_b/\pi G\rho)^{1/3}D^{(1+b)/3}$. This assumes the disruption threshold uses impact energy per target mass and neglects energy-coupling efficiency variations.
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