= Collision disruption radius under secular stirring
{title2=$a/a_p\gtrsim Q_D^*a_p/(GM e_p^2)$}
A phase-mixed population secularly forced by a distant eccentric <planet> has typical <orbital eccentricity> proportional to $e_p a/a_p$. Consequently a relative speed of order <orbital eccentricity> times Keplerian speed gives $v_{\rm col}^2\sim e_p^2(a/a_p)GM/a_p$. Comparing <specific impact energy> with a <catastrophic disruption threshold> gives a lower disruption radius $a/a_p\gtrsim Q_D^*a_p/(GM e_p^2)$, up to an order-one coefficient depending on velocity statistics and impact-energy convention. The <planet> <mass> sets the secular stirring time, but cancels from this energy scaling.
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