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A collision cross-section is the effective area that converts a target number density and incident particle speed into a collision rate. It can depend on particle species, relative speed, and long-range forces, and need not equal a purely geometric collision cross-section.

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.

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.

Suppose the inclined population samples its arguments of periapsis and nodal orientations, or that sufficiently slow precession supplies this sampling. Near the planet the vertical amplitude is $H\simeq a_{p}I$. For uniformly distributed vertical phase, $z=H\sin \psi$ has probability density function $[\pi \sqrt{H^2-z^2}{]}^{-1}$, so the midplane volume probability is ${\Sigma }_c/(\pi H)$.

In the genuinely three-dimensional regime $R_p\ll H\ll a_p$, the geometric collision cross-section is $\pi R_p^2$. The relative velocity differs from the coplanar value only at order $I^2$ for the specified high-speed encounters. Therefore the inclined comet collision time isAt fixed ${\rho }_p,M_{\star },I$ the leading scaling is $Q^{3/2}{M}_p^{-2/3}{a}_p^2$, with no leading $q$ dependence when $q\ll a_p$. The displayed factor retains the first dependence on $q/a_p$. Also $t_{\mathrm{col},\mathrm{I}}/t_{\mathrm{col}}\simeq 2a_{p}I/R_p$ in this averaging convention.

Inclination alone does not specify a collision time for one fixed orbit. If its orbital nodes miss the planet's orbit, a tilted Kepler orbit can have no collisions at all. The finite result is an orientation-averaged population result, or requires precession or scattering to sample those orientations. When $a_{p}I\lesssim R_p$, the three-dimensional estimate crosses over to the coplanar result instead of tending to zero.

A coplanar target sweeps a strip of width $2R_p$ through the comet's local position probability. Hence the relevant geometric collision cross-section is a length, and the rare collision rate per comet is ${\Gamma }_{\mathrm{col}}=2R_p{\Sigma }_c(a_p)v_{\mathrm{rel}}$.

Using the corrected phase-mixed comet surface density and the notation $s=q/a_p$, $\gamma =\sqrt{3-2\sqrt{2s}}$, the coplanar comet collision time isFor $q\ll a_p$, this becomes ${\pi }^2{Q}^{3/2}{a}_p/(2\sqrt{3}{R}_p\sqrt{G{M}_{\star }})$, so $t_{\mathrm{col}}\propto Q^{3/2}{M}_p^{-1/3}{a}_p$ at fixed stellar mass and planetary mass density. This is the mean waiting time for one orbiting comet; the mean interval between impacts from $N_c$ such independent comets is $t_{\mathrm{col}}/N_c$. It assumes rare encounters and orbital phase mixing, not a Poisson process derived from a single deterministic trajectory.

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.

Work in the rest frame of the hole and neglect stellar velocities compared with $v$. In addition to $v\gg {\sigma }_{\ast }$, assume the galactic encounter speed is nonrelativistic, $v\ll c$. Treat a star as a test particle with specific angular momentum $\ell =bv$. At its pericentre $r_{\min }$, its radial velocity vanishes. Conservation of energy and conservation of angular momentum giveThe second term is gravitational focusing. For the rough Newtonian estimate in the question, assume capture when the pericentre reaches the Schwarzschild radius, $r_S=2G{M}_{\mathrm{BH}}/c^2$. It follows thatThe approximation keeps the focusing term because $v\ll c$.

Let ${\rho }_{\ast }=m{n}_{\ast }$ be the stellar mass density. Multiplying the geometric collision cross-section by the incoming mass flux gives the collisionless black-hole capture rateFor a supersonic gas with negligible sound speed, Bondi--Hoyle--Lyttleton accretion has capture radius $r_a=2G{M}_{\mathrm{BH}}/v^2$ and rateConsequentlyFor comparable densities and $v=200\mathrm{km}{\mathrm{s}}^{-1}$, the suppression is approximately $4\times 10^{-7}$. Gas can collide, dissipate orbital energy, and be captured from a region of size $GM/v^2$; collisionless particles generally leave on an unbound orbit unless their angular momentum is small enough for direct capture. Chandrasekhar dynamical friction can slow the hole by transferring energy to a stellar wake, but does not mean that the stars producing that wake are swallowed.

There is an important accuracy limitation in the requested numerical coefficient. Newtonian orbital mechanics evaluated at a relativistic event horizon cannot give the exact capture boundary. In Schwarzschild spacetime, a slowly incoming massive particle has dimensionless energy $\mathcal{E}\simeq 1$ and timelike geodesic effective potentialThe double turning point satisfying $V=1$ and $dV/dr=0$ occurs at $r=4r_g$ and $\ell =4G{M}_{\mathrm{BH}}/c$. The exact low-speed boundary is therefore $b_{\mathrm{cap}}\simeq 4G{M}_{\mathrm{BH}}/(cv)$, twice the rough value; it makes the rate ratio $4({\rho }_{\ast }/{\rho }_g)(v/c{)}^2$. The key suppression scaling is unchanged.

Direct capture of ordinary stars or dark matter is therefore an inefficient route to rapid growth of a supermassive black hole under these conditions. Tidal disruption events can capture stellar debris without swallowing an intact point-like star, and special dense environments can modify the supply. These qualifications do not make collisionless capture equivalent to dissipative gas accretion.

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 $Q_D^{\ast }$.

The impactors' number density per diameter is $K{D}_{\mathrm{im}}^{-\alpha }/V$. With gravitational focusing neglected, the geometric collision cross-section is $\pi (D+D_{\mathrm{im}}{)}^2/4$. The planetesimal collision rate is consequentlywhere