Saturating closed-box enrichment law ($Z_g=1-{\mu }^Y$)

A finite-metal-fraction variant of the closed-box model of galactic chemical evolution uses effective stellar yield $Y(1-Z_g)$, rather than constant yield. With pristine initial gas and $Y>0$, metal accounting gives $d{Z}_g/d\log (1/\mu )=Y(1-Z_g)$ and hence $Z_g=1-{\mu }^Y$. The mass-weighted mean stellar metallicity is $[Y-(Y+1)\mu +{\mu }^{Y+1}]/[(Y+1)(1-\mu )]$. At final gas fraction ${\mu }_f$, the metallicity distribution function is $(1-z{)}^{1/Y-1}/[Y(1-{\mu }_f)]$ on $0<z<1-{\mu }_f^Y$. For trace abundances, $Z_g\simeq -Y\log \mu$ and the usual closed-box metallicity distribution is recovered to leading order.