= Critical initial mass function slope for impulsive disruption
{title2=$\alpha_{\rm crit}\simeq1.791$}
For an <initial mass function> $dN\propto m^{-\alpha}dm$ on $0.1<m/M_\odot<100$, complete loss of material initially above $10M_\odot$ removes fraction
$$
F(\alpha)=\frac{100^{2-\alpha}-10^{2-\alpha}}{100^{2-\alpha}-0.1^{2-\alpha}},\qquad F(2)=\frac13.
$$
The <impulsive disruption by stellar mass loss> threshold is $F=1/2$, with unique finite root $\alpha\simeq1.79101236$. Decreasing $\alpha$ shifts mass to larger stars and increases $F$, so the idealized system is globally unbound for $\alpha<\alpha_{\rm crit}$. The derivation requires rapid, well-mixed loss and absence of a confining external potential.
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