 Dust-burst accretion pulse

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.