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

A star inherits the gas-phase metallicity at its birth. Set $y_e=y/(1+\nu )$; the preceding calculation gives $G(Z)=G_0{e}^{-Z/y_e}$ and $dS/dZ=G(Z)/y$. If the initial mass function is fixed, let $\kappa$ be the number of the counted stars per unit formed stellar mass; this includes any restriction to surviving long-lived tracers. The metallicity distribution in a leaky box is thereforeThe total eventual formed stellar mass is $G_0/(1+\nu )$, so $N_{\infty }=\kappa G_0/(1+\nu )$ and the equivalent normalized metallicity distribution function isIf $N(Z)$ means a cumulative count rather than a count per stellar metallicity interval, integration yieldsAt a finite observation time these expressions end at the current gas-phase metallicity $Z_f$ and the normalized distribution divides by $1-e^{-Z_f/y_e}$. A distribution per logarithmic stellar metallicity instead multiplies $dN/dZ$ by $(\ln 10)Z$; it is not the same shape as the distribution per linear stellar metallicity.