Collisional cascade

A collisional cascade transfers mass from large bodies to progressively smaller fragments. A steady cascade has a size distribution whose mass flux through logarithmic size bins is independent of size.

A simplified collisional cascade tracks large objects $S$ and small fragments $F$. A fragment can disrupt a large object and multiply, while fragment–fragment collisions remove material. With $\dot{S}=-\kappa SF$ and $\dot{F}=\alpha SF-\beta F^2$, elimination of time gives a linear differential equation, $dF/dS-(\beta /\kappa )F/S=-\alpha /\kappa$. It has solution $F=C{S}^k-(\alpha /\kappa )S/(1-k)$ for $k=\beta /\kappa \ne 1$. Its maximum occurs at $F=(\alpha /\beta )S$; early exponential growth does not imply permanent exponential growth after the large-object reservoir is depleted.

If a collision operator is homogeneous of degree two in its bin masses, reducing every mass by $f$ slows the subsequent evolution by $f$. If $m(t)$ is the undepleted trajectory and depletion occurs at $T$, the post-depletion trajectory is $m_d(t)=m(T+(t-T)/f)/f$. Compared with a collision-only model starting with normalization $m(0)/f$, the same observed distribution has inferred age $fT+(t-T)$. The distribution alone cannot distinguish the depletion factor from unknown initial normalization and age.

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.

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.

Once escape depletes a diameter interval, collision rates must be recomputed from the depleted projectile population. For a local power law $n(D)=K{D}^{-\alpha }$ and self-similar catastrophic-projectile limits, the rate scales as $K{D}^{3-\alpha }$. Retaining the original rate after changing $K$ or $\alpha$ can give incorrect transition sizes.

When a loss mechanism becomes slow again at very small diameters, a depleted intermediate band can feed a lower-normalization collisional cascade. For size-independent strength the asymptotic collision-dominated slope returns to $-7/2$. In the local two-limit Yarkovsky removal model, matching gives $D_{-}\sim D_0^{8/5}{D}_{+}^{-3/5}$ and normalization ratio $K_{-}/K\sim (D_0/D_{+}{)}^{2/5}$. Coefficients depend on the collision kernel; a sharp join is only an asymptotic sketch.

This function specifies the fraction of disrupted mass assigned to each fragment size interval. A self-similar law depends on fragment-to-parent size ratios, rather than an absolute size. In logarithmic bins its translation invariance gives the constant processed mass flux of the interior steady collisional cascade. In a removal-dominated interval, it instead sets the fragment source spectrum.

If fragments are injected with number spectrum $S_N(D)\propto D^{-{\alpha }_r}$ and escape faster than they collide, steady balance gives $n(D)=S_N(D)t_y(D)$. Thus $t_y\propto D^{\gamma }$ changes the differential number index to ${\alpha }_r-\gamma$. A removed population with size-independent subsequent lifetime retains the source index ${\alpha }_r$.

In a steady collisional cascade supplied at large sizes and depleted only at small sizes, mass conservation makes the downward mass flux across every intermediate size threshold equal. With self-similar fragmentation and comparable transfer between logarithmic bins, the mass removed per bin per unit time is consequently independent of size. Nonlocal or size-dependent fragmentation, additional sources, and proximity to the cutoffs can invalidate the simplified bin-by-bin relation.

A differential power-law size distribution has $n(D)dD=k{D}^{-\alpha }dD$ objects with sizes between $D$ and $D+dD$.

For equal-density spherical bodies with differential number distribution $n(D)\propto D^{-\alpha }$, a fixed logarithmic bin has width $dD=Dd\log D$. Multiplying the body mass proportional to $D^3$ by $n(D)dD$ gives $m_{\log }\propto D^{4-\alpha }$. Distinguishing differential-number and logarithmic-mass slopes is essential in collisional cascade plots.

For spherical fragments of common mass density $\rho$, with $n(D)=K{D}^{-\alpha }$ between specified diameter cutoffs, mass normalization of a power-law size distribution and integration of the geometric cross-section giveIndeed $m=(\pi \rho K/6)\int D^{3-\alpha }dD$ and ${\sigma }_{\mathrm{tot}}=(\pi K/4)\int D^{2-\alpha }dD$; eliminating $K$ proves the formula. If both cutoff ratios raised to $\alpha -3$ and $4-\alpha$ are small, respectively, this reduces to $3m(4-\alpha )D_{\min }^{3-\alpha }{D}_{\max }^{\alpha -4}/[2\rho (\alpha -3)]$. Approaching either endpoint requires the corresponding logarithmic integral.

If a spherical parent and its fragments have the same mass density, substitute $m=\pi \rho D_c^3/6$ into the endpoint-dominated cross-section of a power-law fragment population. The diameter obeysThis inversion requires specified fragment cutoffs. If $D_{\max }=\eta D_c$ instead, solving that dependence changes the power of $D_c$ to $\alpha -1$. It is useful for estimating the parent comet needed to supply an observed debris disk, subject to the assumed fragment model.

For spherical planetesimals with mass density $\rho$ and total-number distribution $n(D)=K{D}^{-\alpha }$ between $D_{\min }$ and $D_{\max }$,Thus $K=6M_{\mathrm{tot}}(4-\alpha )/[\pi \rho (D_{\max }^{4-\alpha }-D_{\min }^{4-\alpha })]$ for $\alpha \ne 4$. At $\alpha =4$, use $K=6M_{\mathrm{tot}}/[\pi \rho \log (D_{\max }/D_{\min })]$. Divide $n(D)$ by belt volume to obtain the number-density distribution used in a planetesimal collision rate.

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.

The ideal steady Dohnanyi cascade assumes self-similar fragmentation, size-independent collision speeds, and a size-independent catastrophic disruption threshold. Its differential size distribution is $n(D)\propto D^{-7/2}$.

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.

For a common projectile power-law size distribution, the rate of impacts exceeding a threshold $Q$ is proportional to $Q^{(1-\alpha )/3}$. Taking the strength-only threshold $Q_s=Q_a{D}^{-a}$ rather than the full catastrophic disruption threshold gives the displayed ratio. Here the strength-shattering rate counts all impacts above $Q_s$, including dispersive ones; subtracting $R_{mcc}$ gives the bound-remnant-only rubblising collision rate. In a gravity-dominated cascade with $\alpha =3$, $a=1/2$, and $b=3/2$, it scales as $D^{4/3}$. Using the actual strength-gravity disruption transition minimum diameter gives coefficient $3^{-2/3}$ in the asymptotic rate ratio; treating that diameter as the equal-contribution crossover silently loses this factor.

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.

A cratering collision has $Q<Q_D^{\ast }$ and removes part of a target without catastrophically dispersing it.