Burst transport under inverse-size radial drag ($r^2(D,t)=r_0^2-2\gamma t/D$)

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.

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.

Let the initial dust mass be $m_d$, let its differential number distribution be $K{D}^{-\alpha }$ with $\alpha <4$ between $D_{\min }$ and $D_{\max }$, and put $\eta =D_{\min }/D_{\max }$. Under burst transport under inverse-size radial drag, complete absorption at a fixed radius occurs at $t=TD/D_{\max }$, with $T=t_r(D_{\max })$. The arriving mass is proportional to $D^{3-\alpha }dD$. Substitute $D=D_{\max }t/T$ and $dD/dt=D_{\max }/T$ and use mass normalization of a power-law size distribution to prove the displayed rate for $\eta T\le t\le T$, with zero rate elsewhere. Its integral is exactly $m_d$. Resonant delays, scattering losses, a finite release duration or grain destruction change this ideal pulse.