= Quadratic gas-history enrichment model
{title2=$Z(S)=yS^{-q}\int_0^S u^q(1-u)^{-1}\,du$}
Let $S=s/s_\infty$, let the <interstellar medium> have mass $g=s_\infty S(1-S)$, and retain fraction $f=S$ of returned material. For $0\leq\beta<1$ put $q=(2-\beta)/(1-\beta)$. The <integrating factor> for the metal mass is $S^{1/(1-\beta)}/(1-S)$. A finite initial <gas-phase metallicity> removes the homogeneous solution and gives the displayed expression, equivalently $Z=y\sum_{n\geq1}S^n/(q+n)$ for $0\leq S<1$. The increasing initial gas mass requires pristine <galactic gas inflow>: gas accounting gives $dM_{\rm in}/ds=q(1-S)$. A fixed <initial mass function> makes the birth <metallicity distribution function> proportional to $ds/dZ$ when enrichment is monotone.
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