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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 givesFor example, with $y=Z_{\odot }$, a solar-metallicity massive elliptical galaxy requires $\nu \simeq 0$, whereas $⟨Z{⟩}_{\ast }<0.01Z_{\odot }$ in a dwarf galaxy requires $\nu >99$. A supersolar stellar metallicity requires a correspondingly supersolar stellar yield: outflow alone cannot raise $⟨Z{⟩}_{\ast }$ above $y$. If $y=2Z_{\odot }$, a massive elliptical galaxy with $⟨Z{⟩}_{\ast }$ between $Z_{\odot }$ and $2Z_{\odot }$ has $0\le \nu \le 1$, while the same low-metallicity dwarf galaxy needs $\nu >199$.

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 isThis 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.

The closed-box model of galactic chemical evolution assumes a fixed total baryonic mass, no gas inflow or outflow, instantaneous and homogeneous mixing, a constant stellar yield, a fixed initial mass function, and usually the instantaneous recycling approximation. It predicts $Z=y\log (1/\mu )$ for gas fraction $\mu$, but real galaxies violate these assumptions.

Evidence for galactic outflows includes blueshifted absorption, broad or split emission lines, extraplanar ionized and molecular gas, X-ray bubbles, and metal-enriched circumgalactic material. The low baryon fractions and low effective yields of dwarf galaxies, together with the galaxy mass--metallicity relation, provide indirect evidence for preferential gas and metal loss. Sustained star formation for longer than a closed reservoir's depletion time, low-metallicity high-velocity clouds, metallicity dilution during starbursts, the G-dwarf problem, and circumgalactic or intergalactic accretion signatures imply continuing galactic gas inflow.

Let $f(t)=A{e}^{-t/\tau }$ be the inflow rate, let the outflow rate be $\lambda \psi$, and let $R$ be the promptly returned fraction. The total baryonic, gas, and metal masses obeyThe first metal term is newly synthesized material; the second locks existing metals into long-lived stars and removes them in an ambient-composition wind. Expanding the final derivative and substituting ${\dot{M}}_g$ cancels those common terms, leaving

The observed Kennicutt–Schmidt law relates star-formation and gas surface densities approximately by ${\Sigma }_{\mathrm{SFR}}\propto {\Sigma }_g^{1.4}$, with a nearly linear relation to molecular gas over many resolved regimes. For the simple integrated model requested here, take a constant depletion coefficient $S$ so thatThen, with $\alpha =S(1-R+\lambda )$,The integrating factor $e^{\alpha t}$ gives, for $\alpha \ne 1/\tau$,At resonance, $\alpha =1/\tau$, the continuous limit isThe first term is depletion of the initial reservoir; the second is gas supplied by the exponentially declining inflow and subsequently consumed or expelled.

The galaxy mass--metallicity relation is the observed tendency for more massive galaxies to have larger gas-phase metallicity and stellar metallicity. Gas metallicity is commonly inferred from nebular emission-line ratios in star-forming H II regions, often quoted as $12+{\log }_{10}(\mathrm{O}/\mathrm{H})$ and measured within a finite spectroscopic aperture. Stellar metallicity comes from stellar absorption features or population-synthesis fits and is luminosity weighted unless the analysis explicitly reconstructs a mass-weighted distribution. Galaxy stellar mass is inferred from photometry or a spectral-energy-distribution fit and depends on the adopted initial mass function. Radial metallicity gradients, dust, line calibration, and aperture selection must consequently be matched before samples are compared.

The usual physical explanation is that a shallow potential well lets a low-mass galaxy lose a larger fraction of newly synthesized metals in galactic outflows. The closed-box model of galactic chemical evolution is therefore replaced by a leaky-box model of galactic chemical evolution withwhere $\lambda$ is the mass-loading factor. Under the instantaneous recycling approximation, let $y_z$ be the stellar yield, absorb the returned mass fraction into the definitions, and suppose the escaping gas has the current gas metallicity $Z_g$. Then mass conservation and metal conservation areSubstitution of the first equation into the second cancels the terms that merely transfer pre-existing metals and leavesBecause the mass of metals locked into stars obeys $d{M}_{z,\ast }=Z_{g}d{M}_{\ast }$, integration gives $M_{z,\ast }=M_{\ast }{Z}_{\ast }=\int Z_{g}d{M}_{\ast }$. The total newly made metal mass is partitioned between present gas, stars, and the outflow:Writing the gas-to-stellar mass ratio as $r_g=M_g/M_{\ast }$ therefore producesThus simultaneous gas and stellar metallicities, together with the gas fraction and an assumed nucleosynthetic yield, estimate the integrated mass loading. The corresponding effective yield is $y_{\mathrm{eff}}=y_z/(1+\lambda )$, and the leaky box has

The G-dwarf problem is that the local Milky Way disk contains far fewer low-metallicity long-lived G dwarfs than the constant-yield closed-box metallicity distribution predicts. A yield that rises with metallicity may initially sound promising because enrichment would accelerate after the first generations. In fact it worsens the problem. In a closed box, $d{M}_{\ast }=-d{M}_g$ andFor $y_z=kZ$ and a nonzero initial metallicity $Z_0$ at gas mass $M_{g,0}$,Hence the cumulative mass of stars born below metallicity $Z$ isIf the system begins at $Z_0=0$, the assumed yield also vanishes and enrichment never starts. For $Z_0>0$, the metallicity distribution function haswhich puts still more stellar mass near the low-metallicity floor. Metal-poor gas inflow, pre-enrichment, and selective outflow are therefore more plausible ingredients in resolving the G-dwarf problem.