= Solution
Assume the physical coefficients $Q_a,Q_b$ are positive. Differentiate the <catastrophic disruption threshold>:
$$
\frac{dQ_D^*}{dD}=-aQ_aD^{-a-1}+bQ_bD^{b-1}
=D^{-a-1}[-aQ_a+bQ_bD^{a+b}].
$$
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 $Q_D^*$ tends to infinity at both endpoints $D\to0$ and $D\to\infty$. The <strength-gravity disruption transition> is therefore
$$
\boxed{D_w=\left(\frac{aQ_a}{bQ_b}\right)^{1/(a+b)}.}
$$
At this diameter, $bQ_bD_w^b=aQ_aD_w^{-a}$, so $Q_w^*=[(a+b)/b]Q_aD_w^{-a}$. 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 $\tfrac12(D_{\rm im}/D_w)^3v_{\rm rel}^2$. Setting this to $Q_w^*$ gives the <catastrophic projectile diameter>
$$
\boxed{D_{\rm wc}=D_w\left(\frac{2Q_w^*}{v_{\rm rel}^2}\right)^{1/3}
=\left[\frac{2(a+b)Q_a}{b v_{\rm rel}^2}\right]^{1/3}
\left(\frac{aQ_a}{bQ_b}\right)^{(3-a)/[3(a+b)]}.}
$$
Here $v_{\rm rel}$ 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 $(\rho_{\rm target}/\rho_{\rm projectile})^{1/3}$. For $a=1/2$, $b=3/2$,
$$
D_w=\left(\frac{Q_a}{3Q_b}\right)^{1/2},\qquad
Q_w^*=\frac43Q_a\left(\frac{Q_a}{3Q_b}\right)^{-1/4},
$$
so collecting the powers gives
$$
\boxed{D_{\rm wc}=2\,3^{-3/4}Q_a^{3/4}Q_b^{-5/12}v_{\rm rel}^{-2/3}.}
$$
The 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> $M/2$ gives
$$
\frac M2=\int_{D_{\min}}^{D_{\max}}\frac{\pi\rho D^3}{6}KD^{-\alpha}\,dD
=\frac{\pi\rho K}{6(4-\alpha)}(D_{\max}^{4-\alpha}-D_{\min}^{4-\alpha}),
$$
$$
\boxed{K=\frac{3M(4-\alpha)}{\pi\rho(D_{\max}^{4-\alpha}-D_{\min}^{4-\alpha})}
\simeq\frac{3M(4-\alpha)}{\pi\rho D_{\max}^{4-\alpha}}.}
$$
The approximation uses $(D_{\min}/D_{\max})^{4-\alpha}\ll1$. 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:
$$
\frac d{dt}r^2=-\frac{2\gamma}{D},\qquad
r^2(D,t)=r_0^2-\frac{2\gamma t}{D},\qquad
\boxed{t_p(D)=\frac{D(r_0^2-r_p^2)}{2\gamma}.}
$$
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,
$$
\sigma_{\rm acc}=\pi R_{\rm pl}^2\left(1+\frac{v_{\rm esc}^2}{v_\infty^2}\right),\qquad
v_{\rm esc}^2=\frac{2GM_{\rm pl}}{R_{\rm pl}}.
$$
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 $dD_t/dt=D_{\max}/T$. Hence the <dust-burst accretion pulse> follows from the arriving <mass> per diameter:
$$
\dot M_p=\frac{\pi\rho K}{6}D_t^{3-\alpha}\frac{D_{\max}}{T}
=\frac{(4-\alpha)M}{2T[1-\eta^{4-\alpha}]}\left(\frac tT\right)^{3-\alpha},\qquad \eta T\le t\le T.
$$
Dropping the small lower-cutoff normalization correction gives the requested expression
$$
\boxed{\dot M_p=\left(2-\frac\alpha2\right)\frac M{t_p(D_{\max})}
\left[\frac t{t_p(D_{\max})}\right]^{3-\alpha},\qquad
t_p(D_{\min})\le t\le t_p(D_{\max}).}
$$
The 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 write
$$
H_r=r_0^2-r^2,\quad T_r=t_r(D_{\max})=\frac{D_{\max}H_r}{2\gamma},\quad
y=\frac t{T_r},\quad D_r=D_{\max}y=\frac{2\gamma t}{H_r}.
$$
At fixed $t>0$, the size-radius map is monotone and $dD_r/dr=2rD_r/H_r$. 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> gives
$$
2\pi r\Sigma(r,t)=\frac{\pi\rho K}{6}D_r^{3-\alpha}\frac{dD_r}{dr}
=\frac{\pi\rho K}{3}\frac{rD_r^{4-\alpha}}{H_r}.
$$
It follows that
$$
\boxed{\Sigma(r,t)=\frac{(4-\alpha)M}{2\pi(r_0^2-r^2)[1-\eta^{4-\alpha}]}
\left[\frac t{t_r(D_{\max})}\right]^{4-\alpha},\qquad
\eta\le\frac t{t_r(D_{\max})}\le1.}
$$
The 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.
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