= Two-size fragmentation cascade
{title2=$\dot F=\alpha SF-\beta F^2$}
A simplified <collisional cascade> tracks large objects $S$ and small fragments $F$. A fragment can disrupt a large object and multiply, while fragment–fragment collisions remove material. With $\dot S=-\kappa SF$ and $\dot F=\alpha SF-\beta F^2$, elimination of time gives a <linear differential equation>, $dF/dS-(\beta/\kappa)F/S=-\alpha/\kappa$. It has solution $F=C S^k-(\alpha/\kappa)S/(1-k)$ for $k=\beta/\kappa\ne1$. Its maximum occurs at $F=(\alpha/\beta)S$; early exponential growth does not imply permanent exponential growth after the large-object reservoir is depleted.
Back to article page
