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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.

The geometric collision cross-section in a coplanar calculation is a length $2R_p$, rather than an area. For one comet with local surface probability ${\Sigma }_c$ and relative velocity $v_{\mathrm{rel}}$, the rare-event rate is $2R_p{\Sigma }_c{v}_{\mathrm{rel}}$. Without gravitational focusing, the resulting mean collision time scales as $Q^{3/2}{M}_p^{-1/3}{a}_p$ for deep planet-crossing orbits at fixed stellar mass and planetary mass density.

At infinity, the meteorite's incoming asymptote has speed $V$ and perpendicular distance $b$ from the Earth’s centre. The magnitude of its specific angular momentum is the magnitude of $\mathbf{r}\times \mathbf{v}$. On the asymptote, this is $bV$, so conservation of angular momentum gives $\boxed{h=bV}$. Its mechanical energy at infinity is $E=m{V}^2/2$. Comparing with part (ii) givesFor $b>0$ this Kepler orbit is a hyperbola. The smallest $r=1/u$ occurs at periapsis, where $\cos (\theta -{\theta }_0)=1$. ThusThe middle equality follows by substituting $e^2-1=b^2{V}^4/(G^2{M}^2)$. A trajectory entering the Earth satisfies $r_{\min }<R$, equivalentlyTherefore the sufficient collision condition isThis is gravitational focusing: Newtonian gravity increases the collision radius measured at infinity. Equality is the ideal grazing trajectory, which touches the surface; a strict inequality penetrates it. For $b=0$ the incoming motion is radial and collides, with $r_{\min }=0$ as the limiting value of the above expression.

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.

Synchronous radius and its limits. Kepler's third law for a circumplanetary orbit givesHere $G$ is the gravitational constant. A planet-synchronous orbit repeats after one spin period. A planet-stationary orbit additionally requires a circular orbit, zero orbital inclination to the equator and prograde motion. Then an antenna fixed on the ground points continuously at the same satellite; merely matching the orbital period does not give this property.

Write the planetary radius as $R_p=(3M/(4\pi \rho ){)}^{1/3}$. Requiring the semi-major axis to exceed $R_p$ gives $P^2>3\pi /(G\rho )$. The stellar tidal force restricts the circumplanetary orbit to the Hill sphere, of radius $r_H=a(M/(3M_{\star }){)}^{1/3}$. ThusThese are the surface and Hill-radius constraints in the idealized spherical, small-$M/M_{\star }$ model. Long-term prograde stability generally requires a radius appreciably inside the Hill sphere; for nonzero orbital eccentricity, the surface constraint applies to the periapsis, not just to the semi-major axis.

Collision time and the launch population. A phase-mixed isotropic swarm occupies a shell of radial scale $e{a}_s$. Its number density scales as $N_s/(e{a}_s^3)$, its relative speed as $a_s/P$, and its geometric collision cross-section as $r^2$. Consequently the total collision rate scales as $N_s^2{r}^2/(Pe{a}_s^2)$. To give the numerical normalization used here, adopt an effective shell volume $V=4\pi e{a}_s^3$. At a fixed position the velocity directions are uniformly distributed in the tangent plane, so their mean relative speed is $4v_K/\pi =8a_s/P$. For an unordered pair, the collision cross-section is $4\pi r^2$, givingThus the mean interval between collisions is $A/N_s^2$ in this large-population kinetic model. The effective shell width fixes an order-one coefficient: small orbital eccentricity and random planes alone do not specify a unique radial probability density. Phase mixing, negligible gravitational focusing, $r\ll e{a}_s$, and uncorrelated encounters are implicit in this estimate. Exactly identical orbital periods with perfectly fixed phases do not themselves produce a memoryless collision process.

For nearly planet-stationary orbits with $e\ll I\ll 1$, the swarm volume is smaller by a factor of order $I$, while the relative speed is smaller by the same factor, since vertical motion of scale $I{v}_K$ dominates the eccentric motion. The two changes cancel in the rate $n\sigma v$: there is no parametric factor $I$ in the collision time within the same phase-mixed kinetic approximation. Numerical factors and phase correlations can differ. This is not an argument that bringing all satellites into one nearly circular plane makes their phases random.

With $N_s=Rt$, an Inhomogeneous Poisson process has cumulative hazard functionThe survival function of the first collision is $\exp [-H(t)]$. Setting the expected number of collisions to one givesThis is a characteristic first-event population, with probability $1-e^{-1}$ of an earlier event. It is not a median: the median has an additional factor $(\log 2{)}^{1/3}$.

Which population collides next? Immediately after the first disruption let $S=N_{s0}-2$ and $F=2x^{-3}$. The latter follows from mass conservation for equal-density spherical pieces. Using the same unordered-pair counting and geometric collision cross-sections as above, define $b=(1+x{)}^2/2$; thenHere $S^2,F^2$ stand for the usual large-population approximations to $S(S-1),F(F-1)$. The factor two distinguishing identical and different species is essential. The probability that the next collision is fragment–fragment isIt is the most likely type if $S<4/[x(1+x{)}^2]$ and $S<2/x^2$. It has probability greater than one half ifFor $x\ll 1$, a strongly fragment-dominated next event therefore requires $Sx\ll 4$; the fragment–satellite comparison is more restrictive than the satellite–satellite comparison. If $S=0$, the next event must be fragment–fragment, provided fragments remain.

Population equations and the normalization discrepancy. Every satellite–satellite event produces $2x^{-3}$ fragments and destroys two satellites. Every satellite–fragment event produces a net $x^{-3}-1$ fragments and destroys one satellite; every fragment–fragment event destroys two fragments. Consistent collision counting therefore givesFor $x\ll 1$, $b\simeq 1/2$ and $x^{-3}-1\simeq x^{-3}$. The last two terms in $A{\dot{N}}_f$ are then twice those in the printed equation. This is a genuine factor-of-two inconsistency: equal-size fragment pairs have a collision cross-section smaller by $x^2$, so their event rate must be $x^2{N}_f^2/A$ if the satellite event rate is $N_s^2/A$; destroying both fragments necessarily gives the sink $-2x^2{N}_f^2/A$.

If the printed equation is taken as a prescribed approximate rate model instead, its implicit event rates are $N_s^2/A$, $N_f{N}_s/(4A)$ and $x^2{N}_f^2/(2A)$. Under precisely that mixed normalization, its corresponding satellite equation isThe printed source term also neglects the consumed fragment in a satellite–fragment event, a legitimate relative $O(x^3)$ approximation. That approximation does not repair the pair-counting discrepancy.

The ensuing cascade. This is a two-size fragmentation cascade. First use the printed approximate model, dropping satellite–satellite events as requested. Put $S=N_s$, $F=N_f$, $k=4x^2$ and take $x<1/2$. ThenIntegrating this linear differential equation, with $F_0=2x^{-3}$ and $S_0=N_{s0}-2>0$, givesInitially the collisional cascade grows if $S_0>8x^2$, with approximate early exponential growth time $4A{x}^3/S_0$ when the satellite population is nearly fixed. The fragment population reaches its maximum atAfterwards satellites are depleted and fragment–fragment losses dominate. For $S_0>0$ the continuum solution has $S\to 0$, $F\to 0$, withWhen no satellites remain initially, $F(t)=F_0/[1+x^2{F}_{0}t/A]$ directly. Small integer populations eventually invalidate these deterministic differential equations.

The consistently counted model has, to leading order in $x$, exactly the same curve $F(S)$ and peak, but both retained time derivatives are twice as large. Its growth time is $2A{x}^3/S_0$, and $F(t)\sim A/(2x^{2}t)$. Keeping $(1+x{)}^2$ and the consumed fragment replaces $k$ by $4x^2/(1+x{)}^2$ and $x^{-3}$ in the curve by $x^{-3}-1$. This explicitly separates the physical collision bookkeeping from the printed normalization while giving the evolution under both conventions.

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.

Work in the planet's rest frame, neglecting its recoil and tidal forces during the close encounter. A hyperbolic Kepler orbit has positive specific orbital energy $\epsilon =v_{\infty }^2/2$ and specific angular momentum $h=b{v}_{\infty }$, where $b$ is the impact parameter. Its orbital eccentricity isAt the pericentre distance, $r_{\min }=h^2/[\mu (1+e_h)]$. Substituting $h^2={\mu }^2(e_h^2-1)/v_{\infty }^2$ yieldsThe two asymptotes have true anomalies $\pm f_{\infty }$, where $\cos f_{\infty }=-1/e_h$. Their directions of motion give the gravitational scattering angle $\theta =2f_{\infty }-\pi$, so $\sin (\theta /2)=-\cos f_{\infty }=1/e_h$. ThereforeAs a useful independent form, $b^2=r_{\min }^2+2\mu r_{\min }/v_{\infty }^2$, displaying gravitational focusing. Here $\mu =G{M}_p$ to the stated small-comet accuracy; the exact relative-coordinate formula replaces it by $G(M_p+M)$.

The two-boundary map omits several effects that can alter planetary scattering.

The initial semi-major axis, orbital eccentricity, orbital inclination and orbital phase determine whether encounters occur and their relative velocities. A strongly bound comet needs more energy to escape; a nearly parabolic one needs less. The planetary radius and mass density, the finite comet radius, and gravitational focusing determine collision probabilities. Tidal disruption, atmospheric gas drag, sublimation and physical fragmentation can destroy a body before a nominal point-particle scattering sequence is completed.

Other planets can hand a comet from one scatterer to another, eject it, or lift its periapsis clear of the original scatterer's orbit. Mean-motion resonances and secular perturbations can protect objects from encounters or correlate kicks, contradicting the independent random walk assumption. Planetary migration changes the encounter geometry over time.

Stellar flybys and a galactic tide can change distant comet periapses, allowing new encounters or detaching an object from the planetary region. Stellar mass evolution changes both binding and planetary orbits. Finally, the age and the supply rate of new comets determine whether the observed population is a residual one or continuously replenished. Thus the mass-radius map is a useful conditional classification, not a complete survival law.

Put ${\mu }_{\star }=G{M}_{\star }$. Conservation of specific orbital energy and specific angular momentum givesThe last comet expression is the vis-viva equation at the planet's orbit. The planet has only tangential velocity, while the comet's tangential component is $v_t=h/a_p$. Consequently its prograde relative velocity is exactly, before planetary deflection,In the near-parabolic encounter limit $Q\gg a_p$, this reduces toThe displayed approximation also requires $Q\gg a_p$. The assumptions $Q\gg q$ and $q\ll a_p$ alone do not guarantee it: an orbit with $Q=a_p$ encounters the planet near apoapsis and has $v_{\mathrm{rel}}\simeq v_p$, not $\sqrt{3}{v}_p$.

If all coplanar near-parabolic planet-crossing comets are admitted, so that $0\le q\le a_p$, the tangential component ranges from $-\sqrt{2}{v}_p$ to $+\sqrt{2}{v}_p$ when retrograde orbits are included. ThusFor prograde orbits alone the upper limit is $\sqrt{3}{v}_p$; for the stated deep-crossing limit $q\ll a_p$, both prograde and retrograde encounters approach $\sqrt{3}{v}_p$. These are the speeds outside the planet's gravitational well. The surface impact speed is $v_{\mathrm{imp}}=\sqrt{v_{\mathrm{rel}}^2+v_{\mathrm{esc},\mathrm{p}}^2}$, where $v_{\mathrm{esc},\mathrm{p}}$ is the planetary escape velocity; under negligible gravitational focusing the two speeds agree.

With negligible comet radius and mass, the planet's radius and escape velocity areGravitational focusing changes the limiting collision impact parameter from $R_p$ to $R_p\sqrt{1+v_{\mathrm{esc},\mathrm{p}}^2/v_{\mathrm{rel}}^2}$. Thus it is negligible when $v_{\mathrm{esc},\mathrm{p}}^2\ll v_{\mathrm{rel}}^2$.

Put $s=q/a_p$ and $\gamma =\sqrt{3-2\sqrt{2s}}$ in the $Q\gg a_p$ prograde approximation. Substituting $v_{\mathrm{rel}}=\gamma \sqrt{G{M}_{\star }/a_p}$ givesEquivalently $2(M_p/M_{\star })(a_p/R_p)\ll {\gamma }^2$. For $q\ll a_p$, use ${\gamma }^2\simeq 3$. If negligible gravitational focusing is required for every coplanar near-parabolic orbit, including nearly tangential prograde encounters, use the smaller value ${\gamma }^2=(\sqrt{2}-1{)}^2$.

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 $D_{\mathrm{cc}}\ll D$.

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.

Ignoring gravitational focusing, a galaxy sweeps a cylinder of volume $\pi R^2{v}_{\mathrm{rel}}T$. Multiplying this by the number density $N$ of target galaxies and averaging the relative speed gives the mean encounter countFor independent encounters modeled by a Poisson process, the probability of at least one galaxy merger is $1-e^{-\mu }$. The printed expression $P=\mu$ is its rare-encounter approximation $\mu \ll 1$; it is not an exact probability for arbitrary $T$.

For an illustrative present-day field population take an effective merger impact parameter $R=20\mathrm{kpc}=0.02\mathrm{Mpc}$, $N=10^{-2}{\mathrm{Mpc}}^{-3}$, and $⟨v_{\mathrm{rel}}⟩=200\mathrm{km}{\mathrm{s}}^{-1}$. These are assumed order-of-magnitude inputs, not precise observational measurements. Taking a Hubble time of $T\simeq 1.4\times 10^{10}\mathrm{yr}\simeq 4.4\times 10^{17}\mathrm{s}$, the given distance conversion yieldsand henceThus the geometric field estimate is of order $10^{-5}$ to $10^{-4}$ per Hubble time. It scales as $R^{2}N⟨v_{\mathrm{rel}}⟩T$ and is very sensitive to environment and the adopted effective merger radius. Enhanced density in groups, gravitational focusing, and the evolution of the galaxy population are all omitted; this number is not a prediction of the full cosmological merger fraction.