Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 1 / i

Let $G$ be the remaining gas mass, $S$ the formed stellar mass, and $G_0$ the initial gas mass; initially $S=0$ and the gas-phase metallicity is zero. Take the stellar yield $y$ and the mass-loading factor $\nu$ to be constant, with homogeneous mixing and no galactic gas inflow. With the specified zero bulk return, mass conservation givesNew metals add $ydS$, while star formation and the galactic outflow remove metals at the existing gas-phase metallicity $Z$. ThusThe leaky-box model of galactic chemical evolution therefore givesFor the usual gas fraction of a galaxy, $\mu =G/(G+S)$, the denominator is the baryonic mass still in the system, which decreases under the galactic outflow. Since $G_0=G+(1+\nu )S$, we have $G_0/G=1+(1+\nu )(1-\mu )/\mu$. In the remaining-mass convention,If instead $\mu$ denotes $G/G_0$ relative to the fixed initial reservoir, the same calculation gives $\boxed{Z=y\ln (1/\mu )/(1+\nu )}$. Stating the denominator matters: these two gas fraction of a galaxy conventions coincide for the closed-box model of galactic chemical evolution, recovered at $\nu =0$, but differ when gas escapes. Neglecting bulk return while retaining a nonzero stellar yield is the intended trace-metal approximation.