= Solution
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> with
$$
\dot M_{\rm out}=\lambda\dot M_*,
$$
where $\lambda$ 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 are
$$
dM_g=-(1+\lambda)dM_*,
$$
$$
d(M_gZ_g)=y_z\,dM_*-(1+\lambda)Z_g\,dM_*.
$$
Substitution of the first equation into the second cancels the terms that merely transfer pre-existing metals and leaves
$$
M_g\,dZ_g=y_z\,dM_*.
$$
Because the mass of metals locked into stars obeys $dM_{z,*}=Z_g\,dM_*$, integration gives $M_{z,*}=M_*Z_*=\int Z_g\,dM_*$. The total newly made metal mass is partitioned between present gas, stars, and the outflow:
$$
y_zM_*=Z_gM_g+M_*Z_*+\lambda\int Z_g\,dM_*.
$$
Writing the <gas-to-stellar mass ratio> as $r_g=M_g/M_*$ therefore produces
$$
\boxed{y_z=r_gZ_g+(1+\lambda)Z_*},
\qquad
\boxed{\lambda=\frac{y_z-r_gZ_g}{Z_*}-1}.
$$
Thus 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 $y_{\rm eff}=y_z/(1+\lambda)$, and the leaky box has
$$
Z_g=y_{\rm eff}\log\frac{M_{g,0}}{M_g}.
$$
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, $dM_*=-dM_g$ and
$$
M_g\,dZ=y_z(Z)\,dM_*.
$$
For $y_z=kZ$ and a nonzero initial metallicity $Z_0$ at gas mass $M_{g,0}$,
$$
\frac{dZ}{Z}=-k\frac{dM_g}{M_g},
\qquad
\boxed{Z=Z_0\left(\frac{M_{g,0}}{M_g}\right)^k}.
$$
Hence the cumulative mass of stars born below metallicity $Z$ is
$$
\boxed{M_*(\lt Z)=M_{g,0}\left[1-\left(\frac{Z_0}{Z}\right)^{1/k}\right]}.
$$
If the system begins at $Z_0=0$, the assumed yield also vanishes and enrichment never starts. For $Z_0>0$, the <metallicity distribution function> has
$$
\frac{dM_*}{dZ}=\frac{M_{g,0}}{k}Z_0^{1/k}Z^{-1-1/k},
$$
which 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.
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