Catastrophic disruption threshold ($Q_D^{\ast }$)

The catastrophic disruption threshold is the impact energy per unit target mass required both to shatter a body and to disperse enough fragments against self-gravity.

For $Q_D^{\ast }=Q_a{D}^{-a}+Q_b{D}^b$, $a,b>0$, differentiation gives the minimum at the displayed diameter. Equal strength and gravity contributions occur instead at $D_{\mathrm{eq}}=(Q_a/Q_b{)}^{1/(a+b)}=(b/a{)}^{1/(a+b)}{D}_w$. These two transition conventions coincide only for $a=b$.

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

A rubblising collision shatters a body but does not supply enough energy to disperse its fragments against self-gravity, leaving a gravitationally bound rubble pile.

Consider equal-density planetesimals with projectile differential number distribution $n(d)\propto d^{-\alpha }$, $\alpha >1$, and a fixed encounter speed. For projectiles small compared with a target of diameter $D$, the minimum projectile diameter needed to exceed specific impact energy $Q$ is proportional to $Q^{1/3}$. If this threshold is well below the upper projectile cutoff, integration over the projectile population gives a collision rate $R(>Q)\propto Q^{(1-\alpha )/3}$.

Write the catastrophic disruption threshold as $Q_D^{\ast }=Q_s+Q_g$, with strength term $Q_s=Q_a{D}^{-a}$ and gravity term $Q_g=Q_b{D}^b$. Let $R_{\mathrm{rub}}=R(>Q_s)$ count all strength-shattering impacts and $R_{\mathrm{cc}}=R(>Q_D^{\ast })$ count dispersive impacts. ThenWith this inclusive convention, the rate of rubblising collisions that leave a bound remnant is $R_{\mathrm{rub}}-R_{\mathrm{cc}}$, rather than $R_{\mathrm{rub}}$ alone.

For $a=1/2$, $b=3/2$ and $\alpha =3$, define $D_w$ as the actual minimum of $Q_D^{\ast }$ at the strength-gravity disruption transition. Differentiation gives $Q_b/Q_a=1/(3D_w^2)$, soin the gravity-dominated limit. Defining the crossover instead by $Q_s=Q_g$ changes the reference diameter and the apparent coefficient, but not the collision-rate scaling. Variable speeds, comparable projectile and target sizes, and finite-cutoff corrections require a different collision integral.

A catastrophic planetesimal collision has $Q\ge Q_D^{\ast }$ and disperses a substantial fraction of the target mass.

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 $v_{\mathrm{col}}^2\sim e_p^2(a/a_p)GM/a_p$. Comparing specific impact energy with a catastrophic disruption threshold gives a lower disruption radius $a/a_p\gtrsim Q_D^{\ast }{a}_p/(GM{e}_p^2)$, 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.

The catastrophic planetesimal collision rate is the collision rate integrated over impactors large enough to satisfy $Q\ge Q_D^{\ast }$.

The catastrophic collision mass-loss timescale is the target mass divided by the rate at which catastrophic impacts remove mass from its largest remnant.