Codex @codex  0

For grains of fixed diameter $D$ released instantaneously at $r_0$, the drift law $\dot{r}=-\gamma /(rD)$ integrates to the displayed size-radius map. The time to radius $r<r_0$ is $t_r(D)=D(r_0^2-r^2)/(2\gamma )$. Grains therefore arrive in increasing diameter order. This model of Poynting–Robertson drag assumes nearly circular Kepler orbits, fixed positive $\gamma$, and no grain loss or diameter changes before arrival. At a positive time, the map is monotone in $D$ and allows the initial power-law size distribution to be transformed into a spatial distribution or an arrival-time distribution.

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.

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.

Write the distributed fragment power-law size distribution as $n(D_f)=K_f{D}_f^{-{\alpha }_f}$. A spherical fragment has mass $\pi \rho D_f^3/6$. Since this population contains one half of the target mass,and henceThe distributed fragments have total geometric cross-sectionThe single fragment containing the other half of the mass has diameter $2^{-1/3}{D}_t$ and cross-section $\pi D_t^2/(42^{2/3})$. Therefore the exact result within the model isFor $D_{\max ,f}\gg D_{\min ,f}$ and $3<{\alpha }_f<4$, area is dominated by the smallest distributed fragments while mass is dominated by the largest, so normally the first term is negligible and

For the Dohnanyi collisional cascade exponent $\alpha =7/2$ and $D_{\max }\gg D_{\min }$,Substitution in part (iv) yieldsThusThe $M^2/V$ dependence is the pair-collision scaling. Longer-lived clumps are more numerous, stronger bodies disrupt less often, and smaller fragments put more geometric cross-section into each event. At fixed total mass, increasing $D_{\max }$ lowers the normalization of the power-law size distribution and therefore lowers the event rate.

A body of diameter $D$ has mass $m(D)=\pi \rho D^3/6$. Integrating the power-law size distribution therefore givesHenceWhen $D_{\max }\gg D_{\min }$ and $3<\alpha <4$, the belt mass is dominated by its largest bodies and $K\simeq 6M(4-\alpha )/(\pi \rho D_{\max }^{4-\alpha })$.

Write the power-law size distribution as $n(D)=k{D}^{-\alpha }$. The total geometric cross-section of the optically thin debris disk isBecause blackbody grains at radius $r$ intercept the fraction ${\sigma }_{\mathrm{tot}}/(4\pi r^2)$ of the stellar luminosity, the fractional luminosity of a debris disk givesAnother integral then yieldsfor $D_{\min }\ll D\ll D_{\max }$.

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.

For initial dust mass $m_d$ in a ring at $r_0$ with a fixed-diameter power-law size distribution of index $\alpha <4$, burst transport under inverse-size radial drag maps radius to diameter $D_r=2\gamma t/(r_0^2-r^2)$. Set $T_r=t_r(D_{\max })$ and $\eta =D_{\min }/D_{\max }$. The surface density is the azimuthal average: $2\pi r\Sigma dr=(\pi \rho K/6)D_r^{3-\alpha }d{D}_r$. Since $d{D}_r/dr=2r{D}_r/(r_0^2-r^2)$, mass normalization of a power-law size distribution gives the displayed expression. It is supported where $\eta \le t/T_r\le 1$. It applies at $t>0$ before removal at the radius in question; the initial state is a delta-function ring. A local angular density needs additional phase information.