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The gas fraction of a galaxy is commonly $\mu =M_g/(M_g+M_s)$, using the gas and stellar masses remaining in the system. In a leaky-box model of galactic chemical evolution, it differs from the fraction $g=M_g/M_{g,0}$ of the initial gas reservoir. For an initially star-free box with outflow rate $\alpha {\dot{M}}_s$, $\mu =(1+\alpha )g/(1+\alpha g)$. These different fractions give different-looking metallicity formulae and effective yields.

The qualitative prediction of galactic chemical evolution is useful: later stellar populations tend to inherit more enriched gas, and old stellar populations can preserve low stellar metallicity. The closed-box metallicity distribution, however, predicts many more metal-poor long-lived dwarfs in the solar neighbourhood than are observed. This is the G-dwarf problem, also apparent with other long-lived tracers. A single well-mixed box cannot describe the different age and abundance distributions of the galactic disk, galactic bulge, and stellar halo of the Milky Way. Nor need its exponentially declining star formation rate match the history of ongoing disk star formation.

Prolonged relatively metal-poor galactic gas inflow supplies fuel while diluting the enrichment, allowing many stars to form after the earliest enrichment stage; pre-enrichment of the initial reservoir also removes a pristine tail. Galactic outflows lower the effective yield, but a constant well-mixed outflow alone still gives an exponential metallicity distribution function, and is not automatically a cure for the G-dwarf problem. Spatially varying star formation, radial transport and migration, and selection of local survivors complicate a comparison with one global box. A varying initial mass function or stellar yield changes the enrichment law. Finally, finite stellar lifetimes and Type Ia supernovae make iron enrichment delayed, so an observed iron-abundance distribution is not interchangeable with the instantaneous total-metal distribution. Gas exchange, spatially distinct populations, and delayed enrichment are essential qualifications to the closed-box prediction.

Use the instantaneous recycling approximation with a well-mixed gas reservoir, constant stellar yield $p$, no gas inflow, and an irreversible galactic outflow. Let $M_s$ be the mass locked into long-lived stars and remnants after prompt recycling, and put $\psi (t)={\dot{M}}_s(t)\ge 0$. The yield $p$ is the newly synthesized mass of heavy elements returned to the gas per unit increase of $M_s$. Thus $M_h$ in $Z=M_h/M_g$ denotes the heavy-element mass in the gas, excluding metals already locked into stars.

Assume $M_s(0)=0$, $M_g(0)=M_0$, and $Z(0)=Z_0$; initially pristine gas means $Z_0=0$. Take the outflow rate to be $\alpha \psi$ with constant $\alpha \ge 0$, and assume its metallicity equals the current gas-phase metallicity $Z$. This is the well-mixed leaky-box model of galactic chemical evolution. The parameter $\alpha$ is a mass-loading factor relative to the net rate of locking mass into stars. If a gross star formation rate $\Psi$ and a prompt returned fraction $\mathcal{R}$ are used instead, $\psi =(1-\mathcal{R})\Psi$ and a wind rate $\eta \Psi$ corresponds to $\alpha =\eta /(1-\mathcal{R})$.

Conservation of mass givesand the baryonic mass remaining in the box is $M_g+M_s=M_0-\alpha M_s$. Newly made metals enter the gas at rate $p\psi$, while pre-existing metals are locked into stars at rate $Z\psi$ and leave in the wind at rate $\alpha Z\psi$. HenceUsing $M_h=Z{M}_g$ and the product rule,The cancellation expresses the fact that a well-mixed wind removes gas and metals in the same proportion, while fresh stellar production raises the abundance in the remaining gas.

Divide this equation by ${\dot{M}}_g=-(1+\alpha )\psi$ along the evolving reservoir, or integrate in time through intervals with no star formation. The resulting gas-phase metallicity isEquivalently, its fully explicit dependence on a prescribed net star formation rate isvalid while $M_g(t)>0$. With pre-existing stars, replace $M_s(t)$ in the gas-mass relation by $M_s(t)-M_s(0)$. Without specifying $\psi (t)$ or a gas-consumption law, the model fixes $Z$ as a function of gas mass but does not determine a unique function of time.

For the closed-box model of galactic chemical evolution, set $\alpha =0$. Then $M_g=M_0-M_s$ andAt the same remaining fraction $g=M_g/M_0$ of the initial gas reservoir, the leaky box has the logarithmic coefficient $p/(1+\alpha )$, lower than the true stellar yield by the wind factor. This coefficient is often called an effective yield, but the definition of the gas fraction must be specified.

In particular, the observable gas fraction of a galaxy is often $\mu =M_g/(M_g+M_s)$, using the mass still present, rather than $g$. In our initially star-free model,ThereforeFor $0<\mu <1$ and $\alpha >0$, the leaky-box enrichment is smaller at the same $\mu$: $\ln [1+\alpha (1-\mu )]\le \alpha (1-\mu )<\alpha \ln (1/\mu )$. If effective yield is defined observationally as $(Z-Z_0)/\ln (1/\mu )$, it is consequently not simply the constant $p/(1+\alpha )$ at all $\mu$.

As an explicit time example, choose a linear gas-consumption law $\psi =ϵM_g$, with constant $ϵ>0$. The mass and metallicity equations giveThe closed box with the same $ϵ$ has the same metallicity at a given time, while the leaky box exhausts its gas faster. Thus a comparison at fixed time depends on the chosen star formation rate, even though the comparison at fixed gas fraction above is unambiguous.

Finally, the assumption about wind metallicity is essential. If the expelled gas has abundance $Z_w$ instead, its metal-loss rate is $\alpha Z_w\psi$, and the same bookkeeping yieldsA wind enriched relative to the ambient gas lowers the enrichment rate; the logarithmic solution derived above applies to $Z_w=Z$.

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.