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A bin with mass $m$ loses mass at approximately $m{R}_{\mathrm{cc}}$, so changing an order-one fraction takes $m/(m{R}_{\mathrm{cc}})=1/R_{\mathrm{cc}}$. The catastrophic planetesimal collision rate must be evaluated using the contemporaneous projectile population. It supplies a relaxation scale, not an exact equilibration time for an arbitrary fragment kernel.

Let $L_k={\dot{m}}_k^{-c}$ be the mass processed by catastrophic impacts in logarithmic bin $k$. Steady mass conservation requires the same downward mass flux across every interior size threshold. A scale-independent fragment redistribution function makes the transfer kernel depend only on the number of logarithmic bin steps. Consequently a self-similar interior steady collisional cascade has $L_k=L$, independent of size: a constant processed mass per bin supplies the constant flux, with the same kernel-dependent proportionality at every bin. This is the constant mass flux in a collisional cascade result, away from injection and removal cutoffs. Size-independent fragmentation does not eliminate boundary waves at the very ends of a finite distribution.

For equal-density planetesimals, body mass is proportional to $D^3$ and a fixed logarithmic bin contains a number proportional to $Dn(D)$. The mass per logarithmic size bin is therefore $m_{\log }\propto D^{4-\alpha }$. With the given catastrophic planetesimal collision rate, $Q_D^{\ast }\propto D^b$ givesConstant mass flux in a collisional cascade sets the exponent to zero, yielding the strength-dependent steady collisional-cascade slopeThe time to change an order-one fraction of a bin's mass is $m_k/(m_k{R}_k)=1/R_k$. Its collisional-cascade relaxation time is thus $t_c(D)\sim 1/R(D,Q_D^{\ast })$. The contemporaneous projectile population must be used when a different part of the distribution has already evolved.

For the primordial differential-number index ${\alpha }_0=7/2$, the collisional-cascade relaxation time isIt has asymptotic logarithmic slope $1/12$ in the strength regime and $7/4$ in the gravity regime. It increases monotonically: its logarithmic derivative is $1/2+(5/6)d\log Q_D^{\ast }/d\log D$, ranging from $1/12$ to $7/4$. In particular the minimum of the catastrophic disruption threshold at $D_w$ is not a minimum of the collision time. The strength-gravity disruption transition obeysSmall sizes start evolving first. Before $t\sim t_{c0}(D_{\min })$, the whole distribution is nearly primordial. At intermediate times, a transition diameter $D_t$ with $t_c(D_t)\sim t$ separates processed small sizes from mostly primordial larger sizes. The steady strength-regime index is ${\alpha }_s=41/11$, while the gravity-regime index is ${\alpha }_g=3$. Thus a plot of mass per logarithmic size bin has slopes $3/11$ and $1$ in the two evolved regimes, compared with the primordial slope $1/2$. Once $D_t\gg D_w$, both steady slopes appear below the remaining primordial tail. The transition moves to larger sizes, and the normalization eventually decays as the largest bodies are depleted. Sharp joined power laws are a schematic description; detailed kernels produce smooth transitions and possible waves.

For the uniform depletion, the binary-collision evolution operator $F$ is quadratic in all bin masses: $F(cm)=c^{2}F(m)$. Let $m(t)$ be the undepleted trajectory and suppose $m_d(T)=m(T)/f$. Differentiating verifies the exact scaling within this fixed-kernel collision model,Immediately after depletion the shape is unchanged and the catastrophic planetesimal collision rates are reduced by $f$. Further evolution therefore proceeds $f$ times more slowly on the original trajectory. The processed-to-primordial transition initially remains at $D_t$, then advances on the stretched collision clock.

A collision-only model starting with the smaller initial normalization $m(0)/f$ has trajectory $\tilde{m}(u)=m(u/f)/f$. Matching the observed depleted distribution requires $u/f=T+(t-T)/f$. Hence the collisional age after uniform dynamical depletion isThe lower observed normalization would make it appear that slow collisions had taken much longer to produce the existing break. Shape and normalization alone cannot determine $f$ without independent information about age, initial mass, or depletion history.

Finally, count all impacts exceeding the strength threshold $Q_s=Q_a{D}^{-1/2}$, including impacts which also disperse the target. This is the source's inclusive rubblising convention; a bound-remnant rubblising collision alone would exclude the dispersing impacts. In the evolved gravity-regime projectile approximation, ${\alpha }_g=3$, so the given threshold-rate law gives the rubblising-to-dispersal collision-rate ratioAt the collisional front $D_t$, the age is of order one catastrophic collision time. Thus the expected number of strength-shattering impacts has the large-$D_t/D_w$ scalingThis is the requested scaling. It is not an exact unit-coefficient identity from the stated assumptions: using the true minimum diameter gives $R_{\mathrm{rub}}/R_{\mathrm{cc}}\simeq 3^{-2/3}(D_t/D_w{)}^{4/3}$. The actual cumulative number is ${\int }_0^T{R}_{\mathrm{rub}}(D_t,t)dt$, not automatically $T{R}_{\mathrm{rub}}(D_t,T)$, and its coefficient depends on evolution of the projectile population. The $4/3$ estimate also assumes the relevant projectiles sample the gravity-regime slope; sampling the strength or primordial slope changes the exponent. The steady-cascade framework and evolving transitions are developed by Wyatt, Clarke and Booth.

Neglect gravitational focusing and assume a fixed well-mixed volume $V$, $\alpha >1$, and $D_{\min }\ll D_{\mathrm{cc}}\ll D\ll D_{\max }$. For the small impactors that dominate the rate, the geometric collision cross-section is approximately $\pi D^2/4$. ThusEvaluating and neglecting the upper-end term gives the catastrophic planetesimal collision rateUsing part (ii), its size dependence isIf the destructive cutoff is near an endpoint or comparable to $D$, retain the full integral and the cross-section $\pi (D+d{)}^2/4$. The simple power law then need not apply.

Let the belt contain $n(D)dD=C{D}^{-\alpha }dD$ bodies. Its total mass fixesFor equal mass density, an impactor of diameter $D$ supplies specific impact energyThus catastrophic disruption requires $D\ge D_{\mathrm{cc}}=X_c{D}_t$, whereThe impactor number density per diameter is $C{D}^{-\alpha }/V$. Neglecting gravitational focusing, the geometric collision cross-section is $\pi (D_t+D{)}^2/4$, so the exact catastrophic planetesimal collision rate in this model isEquivalently, with $A=\pi C{v}_{\mathrm{rel}}/(4V)$,If $D_{\min }\ll X_c{D}_t\ll D_t\ll D_{\max }$, the steep distribution makes impactors just above $X_c{D}_t$ dominate and their collision cross-section is approximately $\pi D_t^2/4$. HenceThis approximation also assumes a common $v_{\mathrm{rel}}$ and size-independent catastrophic disruption threshold $Q_D^{\ast }$.

For equal mass density, the specific impact energy isThus the smallest catastrophic impactor has $D_{\mathrm{im}}=X_{c}D$, whereIntegrating the collision rate givesFor $X_c\ll 1$, the last term is the dominant lower-limit contribution, so the catastrophic planetesimal collision rate is

Cratering impactors satisfy $D_{\mathrm{im}}<X_{c}D$. For $X_c\ll 1$, they also satisfy $D_{\mathrm{im}}\ll D$, so $R_{\mathrm{col}}\simeq A{D}^2{D}_{\mathrm{im}}^{-\alpha }$. The fractional mass-loss rate is thenUsing the catastrophic planetesimal collision rate found above,

With size-independent collision speed and catastrophic disruption threshold $Q_D^{\ast }\propto D^b$, the catastrophic planetesimal collision rate scales as $R\propto D^{3-\alpha +b(1-\alpha )/3}$. Multiplying by mass per logarithmic size bin gives processed mass proportional to $D^{7-2\alpha +b(1-\alpha )/3}$. Constant mass flux in a collisional cascade sets the exponent to zero, yielding the displayed differential-number index when $b\ne -6$. This is the zero-velocity-exponent case of a 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.