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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 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.

Assume the physical coefficients $Q_a,Q_b$ are positive. Differentiate the catastrophic disruption threshold:The bracket is strictly increasing from a negative value to positive infinity, so there is one critical point and it is the global minimum. In fact $Q_D^{\ast }$ tends to infinity at both endpoints $D\to 0$ and $D\to \infty$. The strength-gravity disruption transition is thereforeAt this diameter, $b{Q}_b{D}_w^b=a{Q}_a{D}_w^{-a}$, so $Q_w^{\ast }=[(a+b)/b]Q_a{D}_w^{-a}$. This minimizes the full catastrophic disruption threshold; it need not be where its two contributions are equal.

For equal-density spherical planetesimals, the specific impact energy per unit target mass is $\frac{1}{2}(D_{\mathrm{im}}/D_w{)}^3{v}_{\mathrm{rel}}^2$. Setting this to $Q_w^{\ast }$ gives the catastrophic projectile diameterHere $v_{\mathrm{rel}}$ is the impact speed, the threshold is defined per target mass, and energy-coupling changes are not included. If projectile and target densities differ, multiply the diameter by $({\rho }_{\mathrm{target}}/{\rho }_{\mathrm{projectile}}{)}^{1/3}$. For $a=1/2$, $b=3/2$,so collecting the powers givesThe result presumes that this projectile diameter is available in the population.

For the dust, $n(D)dD$ is the total number in a diameter interval, not a number per unit volume. The mass normalization of a power-law size distribution with dust mass $M/2$ givesThe approximation uses $(D_{\min }/D_{\max }{)}^{4-\alpha }\ll 1$. Merely having a small diameter ratio does not guarantee this accuracy if $\alpha$ approaches four, so retain the exact denominator in that case.

With $\gamma >0$, the stated Poynting–Robertson drag law integrates directly:Thus dust arrives in order of increasing diameter. This is burst transport under inverse-size radial drag, assuming an instantaneous release at $r_0$ and no intervening destruction or changes of diameter.

Dust that reaches the planet's orbital radius has several possible fates. It can pass without a close encounter and continue spiralling towards the star; undergo resonant trapping of dust in an exterior mean-motion resonance, where resonant torque temporarily balances the drag; be scattered onto a different stellar Kepler orbit or ejected; or collide with the planet and be accreted. Resonant trapping of dust can increase orbital eccentricity until escape or an encounter becomes possible. Temporary planet-bound capture can become permanent if drag or another dissipative process removes sufficient energy. Arrival at a radial circle alone does not guarantee a collision with the planet at the correct longitude.

A more massive planet has a larger Hill sphere, stronger resonant perturbations and stronger planetary scattering. Capture into a mean-motion resonance is favoured when drift across it is slow compared with its libration timescale. Since $|\dot{r}|\propto D^{-1}$ here, larger grains drift more slowly and are generally easier to trap; small fast-drifting grains more readily cross a resonance. Grain radiation pressure can also move resonance locations, though the prescribed nearly circular drift model omits that additional change. For direct impact, a larger physical radius increases the geometric cross-section, and gravitational focusing gives, in the two-body encounter approximation,A compact massive planet can nevertheless scatter or eject grains rather than hit them: its encounter region can be much larger than its physical radius. Slower dust migration permits more repeated encounter opportunities. These competing effects mean neither planet mass nor grain diameter alone specifies an accretion probability.

For the idealized complete-accretion calculation, put $T=t_p(D_{\max })$ and $\eta =D_{\min }/D_{\max }$. At time $t$, the arriving diameter is $D_t=D_{\max }t/T$ and $d{D}_t/dt=D_{\max }/T$. Hence the dust-burst accretion pulse follows from the arriving mass per diameter:Dropping the small lower-cutoff normalization correction gives the requested expressionThe rate is zero before the smallest grains arrive and after the largest arrive. Integrating the exact expression over its support returns $M/2$, as required by mass conservation. The approximate expression has that integral to the same lower-cutoff accuracy. This pulse assumes no resonant delay or scattering loss before accretion.

For the surface density of a size-sorted dust burst, take $0<r<r_0$ and writeAt fixed $t>0$, the size-radius map is monotone and $d{D}_r/dr=2r{D}_r/H_r$. Define $\Sigma (r,t)$ as the azimuthally averaged surface density, so the mass in an annulus is $2\pi r\Sigma dr$. Transforming the power-law size distribution into radial mass givesIt follows thatThe requested leading approximation omits $1-{\eta }^{4-\alpha }$ from the denominator. The surface density is zero where the inferred $D_r$ lies outside the original size interval. Equivalently, one can obtain it from the instantaneous inward mass flux divided by $2\pi r|\dot{r}(D_r)|$. At $t=0$ the idealized initial distribution is a delta-function ring at $r_0$; the formula above describes the subsequent radial spread. No uniform initial azimuthal distribution is required for this azimuthal average, but the local density of a clump would require its angular evolution. If the complete-absorption planet assumption is retained, this expression holds exterior to $r_p$ and $\Sigma =0$ interior to the planet.

At each fixed semimajor axis, let $B=b_{3/2}^2(\alpha )/b_{3/2}^1(\alpha )$ and $Z_f=B{e}_{\mathrm{pl}}$. The planet's longitude of pericentre is the zero of longitude. The secular perturbation equation becomes $\dot{z}=iA(z-Z_f)$. Integrating it with the initial condition $z(0)=0$ givesThe forced eccentricity vector is the constant real $Z_f$; the free eccentricity vector is $-Z_f{e}^{iAt}$. Consequently the complex eccentricity follows a circle in the Argand plane centered at $Z_f$, with radius $Z_f$, starting at the origin and moving counterclockwise. Its initial velocity is downward, $\dot{z}(0)=-iA{Z}_f$. The secular period is $2\pi /A$, andThe longitude of pericentre is undefined at the origin, but the complex eccentricity passes smoothly through it. At $At=\pi$, $z=2Z_f$ and the orbit is maximally eccentric and aligned with the planet.

Using the small-$\alpha$ Laplace coefficient expansion givesAlso $n=n_{\mathrm{pl}}{\alpha }^{-3/2}$ to leading order in the planet-to-star mass ratio. The secular forcing by a distant outer planet is thereforeA relative $O({\alpha }^2)$ correction to $A$ can accumulate a phase error at very long times; the leading-frequency formula is not a uniform-in-time expansion.

If $a_2=a_1(1+\delta )$ with $\delta =\delta a/a_1\ll 1$, then $Z_{f,2}/Z_{f,1}=1+\delta +O({\delta }^2)$ and $A_2/A_1=1+3\delta /2+O({\delta }^2)$. The outer planetesimal therefore has a slightly larger forced eccentricity circle and a shorter secular period. Its accumulated secular phase lead is $\Delta (At)\simeq 3A_{1}t\delta /2$. Initially both circles start at the origin, but their eccentricity vectors progressively lose alignment. Linearizing the exponential in $\delta$ additionally requires $|A_{1}t\delta |\ll 1$.

The orbit crossing is caused by this accumulating differential secular phase, even though individual orbital eccentricities remain small. To first order in orbital eccentricity, the radius of an orbit at fixed inertial longitude $\theta$ isFor an infinitesimally neighboring orbit,This derivative is evaluated within the first-order orbital-shape model; derivatives of the neglected quadratic terms give relative $O(Z_f)$ corrections near crossing. The least separation over longitude is consequently $\delta a[1-|z+a{\partial }_{a}z|]$. The differential secular orbit-crossing criterion is $|z+a{\partial }_{a}z|=1$. Since $Z_f\propto a$ and $A\propto a^{3/2}$,The first term stays bounded by $4Z_f\ll 1$, whereas the second grows in proportion to time. To leading order in $Z_f$, crossing begins when $(3/2)Z_{f}At=1$. ThusMore precisely, in this first-order orbital-shape approximation the exact first crossing lies within a fractional $O(Z_f)$ window of this value, since the magnitude differs from $(3/2)Z_{f}At$ by at most $4Z_f$. Higher-order Laplace coefficients give additional corrections. The local derivation takes the neighboring-orbit limit before the long-time limit; for a finite separation it requires $\delta a/a_1\ll Z_f$ near crossing. A broad continuous disk contains such neighboring orbits. The crossings first appear in directions selected by the phase of $z+a{\partial }_{a}z$, not simultaneously at every longitude.

For the collision energy estimate adopt the phase-averaged mean orbital eccentricity $\overline{e}=4Z_f/\pi \simeq 5\alpha e_{\mathrm{pl}}/\pi$. The specified velocity estimate givesUsing the projectile kinetic energy per target mass convention for specific impact energy, equal-size and equal-density bodies have equal masses and $Q=\frac{1}{2}{v}_{\mathrm{col}}^2$. A catastrophic planetesimal collision requires $Q\gtrsim Q_D^{\ast }$, soThe numerical coefficient is only illustrative because the collision speed was specified only to order of magnitude. If the disruption threshold is instead defined using the centre of mass kinetic energy per combined mass, equal masses give $Q_R=v_{\mathrm{col}}^2/8$ and the coefficient becomes $8{\pi }^2/25$ for a threshold expressed in that convention. The robust result is a lower bound of order $Q_D^{\ast }{a}_{\mathrm{pl}}/(G{M}_{\star }{e}_{\mathrm{pl}}^2)$.

Within the inner-disk approximation, catastrophic collisions require sufficiently large $a_1/a_{\mathrm{pl}}$, not arbitrarily small radii. Although the Keplerian speed rises inward, the secularly induced orbital eccentricity falls faster: $v_{\mathrm{col}}^2\propto \alpha$. The lower bound must be much smaller than one to leave a domain compatible with $\alpha \ll 1$. A small planet orbital eccentricity, a large disruption threshold, or a distant planet can eliminate that domain. The planet mass controls the time to crossing but cancels from the eventual forced eccentricity amplitude and this collision-energy criterion. The threshold alone does not ensure that crossing has occurred within the disk's available lifetime.

In the gravitational focusing regime, take $d\ll D$ and $v_{\mathrm{rel}}\ll v_{\mathrm{esc}}$ at large separation. Mutual gravitational acceleration then makes the impact speed approximately the target's escape velocity:Setting the specific impact energy equal to the catastrophic disruption threshold givesThe size-dependent far-field encounter speed drops out of this leading threshold because gravitational acceleration dominates at impact. Gravitational focusing also increases the collision cross-section by approximately $1+v_{\mathrm{esc}}^2/v_{\mathrm{rel}}^2$, but this concerns the rate rather than the energy threshold. These formulas assume the disruption-energy convention remains per unit target mass and that gravitationally accelerated impact energy couples to disruption as specified by $Q_D^{\ast }$.

Let $d$ be an impactor diameter and assume it is much smaller than target diameter $D$, with equal material mass density. Neglect gravitational focusing and take the encounter speed to be the stated target-size law $v_{\mathrm{rel}}=v_p{D}^p$. The specific impact energy isA catastrophic planetesimal collision requires $Q\ge Q_D^{\ast }=Q_b{D}^b$. Therefore the catastrophic projectile diameter isUse $\max (D_{\mathrm{cc}},D_{\min })$ as the actual lower size cutoff when integrating a rate. If this exceeds $D_{\max }$, the specified population contains no destructive impactors. The small-impactor approximation itself requires $D_{\mathrm{cc}}\ll D$.

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