Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 72 / 1 / ii

Let $Z$ be the well-mixed gas-phase metallicity, measured as a metal mass fraction. Removing gas into stars takes metals at rate $Z\psi$; prompt return restores existing metals at rate $ZF\psi$, together with newly synthesized metals at rate $Y(1-F)\psi$. Thus the closed-box model of galactic chemical evolution has metal budgetExpand the left side and use ${\dot{M}}_g=-(1-F)\psi$. The terms proportional to existing gas-phase metallicity cancel, leavingSince $\psi =k{M}_g$, initially pristine gas obeysThe stellar yield here is per permanently locked mass, so inserting another factor $1-F$ into $Y$ would double-count recycling. The approximation treats metals as a trace constituent and ignores finite stellar-return delays. Apply it while $Y\lambda t\ll 1$; the formal unlimited linear growth is not a physical prediction for almost entirely metallic gas.