 Cumulative and birth stellar metallicities

In a well-mixed closed-box model of galactic chemical evolution, gas consumed into locked stellar mass supplies a natural cumulative mass coordinate $\delta$. The mean stellar metallicity is ${\overline{Z}}_{\ast }(\delta )={\delta }^{-1}{\int }_0^{\delta }{Z}_{\mathrm{birth}}(s)ds$, giving the displayed derivative relation. If the birth abundance increases linearly, $Z_{\mathrm{birth}}=b\delta$, its cumulative mean is half the current maximum. More generally the maximum of any nonnegative abundance distribution cannot lie below its mean. A fixed initial mass function and the same survival selection are required to identify this mass-weighted distribution with a star-count distribution.