= Proportional-infall model of galactic chemical evolution
{title2=$\dot M_{\rm in}=\nu\psi$}
With well-mixed <gas-phase metallicity> $Z$, zero mass return, <star formation rate> $\psi$, pristine <galactic gas inflow> $\nu\psi$, and constant <stellar yield> $y$, <mass conservation> and metal accounting give $\dot M=-(1-\nu)\psi$ and $M\dot Z=(y-\nu Z)\psi$. Thus, for $0<\nu<1$,
$$
Z=\frac y\nu+\left(Z_0-\frac y\nu\right)(M/M_0)^{\nu/(1-\nu)}.
$$
The initial <gas-phase metallicity> $Z_0=0$ gives the usual pristine-infall expression. Neglecting returned bulk mass while retaining a nonzero metal yield is a trace-metal idealization.
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