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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.

Write the differential power-law size distribution as $n(D)=K{D}^{-\alpha }$ and take all bodies to have the same bulk density. In a fixed belt volume, $K$ is proportional to $M_{\mathrm{tot}}$ when the size cutoffs and the shape of the distribution are fixed, becauseFor $\alpha \ne 4$ the integral is $(D_{\max }^{4-\alpha }-D_{\min }^{4-\alpha })/(4-\alpha )$; at $\alpha =4$ it is a logarithm. Thus no assumption that the mass is always dominated by the largest bodies is needed for this normalization step.

A size-independent catastrophic disruption threshold and collision speed imply a fixed minimum projectile-to-target diameter ratio $\xi$, since $\frac{1}{2}{m}_p{v}^2/m_t\sim Q_D^{\ast }$ gives $\xi \sim (2Q_D^{\ast }/v^2{)}^{1/3}$. With negligible gravitational focusing, a target of size $D$ has a catastrophic collision rate proportional toFor $\alpha >3$ and targets well inside the cutoffs, the dimensionless integral converges at the upper end and is size independent. ConsequentlyHere $A$ contains the belt geometry, speed, density, cutoffs and disruption parameters. Close to a cutoff, or if $\xi D<D_{\min }$, this scaling needs modification. A finite belt without replenishment can only be in quasi-steady state over times short compared with depletion of its largest reservoir.

The mass in a logarithmic size interval is $m_{\ln D}\propto D^{4}n(D)\propto D^{4-\alpha }$. For a scale-independent fragment redistribution function, steady collisional gain and loss transfer a constant mass flux down the cascade: each logarithmic interval processes the same mass per unit time, away from boundaries. Equivalently, a normalized translation-invariant redistribution kernel acting on logarithmic bins admits constant processed mass as its steady solution. Thusmust be size independent. Therefore $\alpha =7/2$, the Dohnanyi collisional cascade, and $t_c=(A/M_{\mathrm{tot}})D^{1/2}$.

The Yarkovsky effect is recoil from anisotropic thermal radiation. Finite thermal inertia shifts the hottest region away from the instantaneous substellar point. The resulting recoil has a tangential component and produces a secular change of semimajor axis; the diurnal component can drift in either direction depending on spin, and the seasonal component generally drifts inward. Migration into a dynamical escape region can remove a body from the belt. For large bodies, the intercepted luminosity scales as $D^2$ and inertia as $D^3$, giving a recoil acceleration roughly proportional to $D^{-1}$, hence a removal time growing as $D$. Very small bodies become nearly isothermal when heat penetrates the whole body; the anisotropy decreases and the removal time again grows. The stated phenomenological lawencodes these limits. Its minimum occurs at $D_0=\sqrt{C/B}$, with $t_{y,\min }=2\sqrt{BC}$. This minimum in absolute removal time differs from the minimum relative to the collision time.

For the collision-only cascade, form the ratioDifferentiation gives the minimum at $D_{\ast }=\sqrt{3C/B}=\sqrt{3}{D}_0$. At this diameter,Thus $t_y<t_c$ for some sizes if and only if $M_{\mathrm{tot}}<M_y$ for an unrestricted size interval. In a finite belt, that interval must also overlap $[D_{\min },D_{\max }]$; the displayed inequality alone is necessary but not sufficient if all favorable sizes lie outside the belt. On logarithmic axes $t_y$ has slopes $-1$ and $+1$, while $t_c$ is a line of slope $1/2$. Increasing belt mass shifts the collision line downward.

For $M_{\mathrm{tot}}\gg M_y$, collisions dominate at all sizes. The belt retains $n_b(D)\propto D^{-7/2}$. Let $T_{\mathrm{dyn}}$ be the size-independent residence time of objects after Yarkovsky removal. Their steady number distribution isHence $n_{\mathrm{out}}\propto D^{-5/2}$ for $D\ll D_0$ and $n_{\mathrm{out}}\propto D^{-9/2}$ for $D\gg D_0$, with a smooth change around $D_0$. These are differential number slopes, not cumulative-number or mass-per-logarithmic-bin slopes.

For $M_{\mathrm{tot}}\ll M_y$, first quantify the nominal crossings obtained by extending the collision-only cascade:The two roots surround $D_{\ast }$, and in the well-separated limit they areThey mark where the undepleted distribution first becomes susceptible to Yarkovsky removal. The upper crossing remains the leading estimate for the onset of depletion, since larger bodies still constitute a collision-dominated reservoir. The lower crossing will be shifted by the depletion itself, as described below.

In the removal-dominated band, the fragment number injection spectrum is $S_N(D)=J{D}^{-7/2}$ under the stated fragment redistribution function assumption. Balancing production against escape givesThus the belt slopes in the depleted band are $-5/2$ above $D_0$ and $-9/2$ below $D_0$; the removed population has slope $-7/2$ across that band. The steep small-size side recovers towards a secondary collision-dominated cascade as escape becomes inefficient. It does not keep the $-9/2$ slope to zero size.

For a useful quantified sketch, approximate the collision integral by its scale-free local dependence and match adjacent asymptotic branches. Let $n_b=K{D}^{-7/2}$ above $D_{+}$, and write its collision rate as $(M_{\mathrm{tot}}/A)D^{-1/2}$. Matching at $D_{+}$ gives $J\simeq K/(B{D}_{+})$. On the small-size side of the removal band,Its collision rate scales as $(M_{\mathrm{tot}}/A)(D_0^2/D_{+})D^{-3/2}$, rather than the undepleted $D^{-1/2}$ law. Equating this rate to $D/C$ gives the corrected lower transitionThe equality is an order-of-magnitude matching law: the collision integral has different dimensionless coefficients for slopes $7/2$ and $9/2$, and projectiles of size $\xi D$ spread transitions over a finite range. The powers follow from the stated scale-free model; an exact numerical lower crossing requires the collision kernel and fragment normalization. In particular, simply retaining $D_{-,0}$ as the true lower crossing silently treats the depleted projectile abundance as unchanged.

Below $D_{-}$, the secondary Dohnanyi collisional cascade has $n_b\simeq K_{-}{D}^{-7/2}$ with matching normalizationThe four belt branches, in descending diameter, are thereforeThe corresponding Yarkovsky removal population has differential slopes $-9/2$, $-7/2$, $-7/2$, $-5/2$, respectively: division by $t_y$ cancels the bend at $D_0$ within the removal-dominated band. Its genuine changes of slope occur at $D_{+}$ and $D_{-}$. Multiplication by $T_{\mathrm{dyn}}$ changes only its normalization. If one is using the preliminary fixed-projectile approximation instead, the same four asymptotic slopes apply, with $D_{-,0}$ as its nominal lower breakpoint; that approximation omits the feedback just quantified.

All sketches presume the relevant breakpoints lie between the size cutoffs, enough time to establish the asserted steady portions, and a large-body reservoir feeding fragments. Otherwise only the branches within the actual size interval appear. Constant collision speed, size-independent strength, a self-similar redistribution law, and a size-independent external residence time are essential: changing them changes the exponents. Boundary waves and nonlocal collisions round the sharp branch joins in the sketch.