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A binary planetesimal is a gravitationally bound pair of planetesimals. Its internal two-body problem and its centre of mass orbit respond differently to external forces, including gas drag.

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 dust birth ring is a belt of parent planetesimals whose collisions supply dust. Radiation pressure gives the fragments different orbits even when their release velocities match those of their parents.

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.

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.

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.

Assume negligible fragment release speed relative to a circular parent planetesimal. Its initial speed is $v_b=\sqrt{G{M}_{\star }/r_b}$, but radiation pressure reduces the grain's gravitational parameter to $(1-\beta )G{M}_{\star }$. Its specific orbital energy and specific angular momentum areFor $0\le \beta <1/2$, the resulting dust orbit released from a circular parent ring hasRelease is at periapsis: the inherited tangential speed exceeds the new circular speed. Low-$\beta$ grains remain near the dust birth ring; higher bound $\beta$ gives increasingly eccentric orbits extending well outside it. Uniform release longitude produces an approximately axisymmetric dust halo, with long residence times near apoapsis. At $\beta =1/2$ the conservative release orbit is parabolic; above this radiation-pressure blowout threshold it is initially unbound. For $\beta \ge 1$, zero or repulsive effective gravity also prevents a bound ellipse.

The left panel shows one release longitude; rotating it around the ring fills the halo. At longer times Poynting–Robertson drag also carries bound grains inward, so an initially outward-spanning release distribution does not imply an empty inner region in a continuously supplied, collision-free population.

Let $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$ and $r=|\mathbf{r}|$. Newton's second law for the two planetesimals givesSubtracting yields the forced relative two-body problemThe internal gravitational forces cancel from the centre of mass equation but add in the relative equation; external forces affect the internal orbit through their difference in acceleration.

Let $v_K=\sqrt{\mu /a}$ be the speed of the circular Kepler orbit at radius $a$. The release velocity relative to the planetesimal isThe particle starts at the same position as its parent, so the change in specific orbital energy isUsing the velocity components from part (c),Since $C=-\mu /(2a)$ and $C^{\prime }=-\mu /(2a^{\prime })$, rearrangement gives