= Solution
Assume the <stellar yield> is common to the compared <stellar populations> and compare their mass-weighted <stellar metallicities>. Inverting the <metallicity distribution in a leaky box> result gives
$$
\boxed{\nu=\frac{y}{\langle Z\rangle_*}-1.}
$$
For example, with $y=Z_\odot$, a solar-metallicity massive <elliptical galaxy> requires $\nu\simeq0$, whereas $\langle Z\rangle_*<0.01Z_\odot$ in a <dwarf galaxy> requires $\nu>99$. A supersolar <stellar metallicity> requires a correspondingly supersolar <stellar yield>: outflow alone cannot raise $\langle Z\rangle_*$ above $y$. If $y=2Z_\odot$, a massive <elliptical galaxy> with $\langle Z\rangle_*$ between $Z_\odot$ and $2Z_\odot$ has $0\le\nu\le1$, while the same low-metallicity <dwarf galaxy> needs $\nu>199$.
\b[For a yield of order solar, the required mass loading runs from order zero or unity in massive spheroids to order one hundred or more in faint dwarfs.] Without a specified <stellar yield>, the robust comparison is
$$
\frac{1+\nu_{\rm dwarf}}{1+\nu_{\rm giant}}
=\frac{\langle Z\rangle_{*,\rm giant}}{\langle Z\rangle_{*,\rm dwarf}}
\gtrsim100.
$$
This explains the sense of the <galaxy mass--metallicity relation>; <galactic gas inflow>, incomplete depletion, and changes in the <initial mass function> can also change observed <stellar metallicity>.
Back to article page
