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The independent dimensionless variable is $S=s/s_{\infty }$. Since $g{Z}_i=s_{\infty }G{Z}_i$ and $dS/ds=1/s_{\infty }$, the chain rule gives $d(g{Z}_i)/ds=d(G{Z}_i)/dS$. The derivative in the normalized equation must consequently be with respect to $S$. Substituting the gas history in the retained-ejecta chemical evolution equation yieldsThis step allows a varying retention fraction $f(S)$; no derivative of $f$ is needed because $f$ enters the instantaneous metal source and loss terms.

Set $H=G{Z}_i$ and now take $f=S$. The coefficient in this first-order ordinary differential equation decomposes asHence an integrating factor is $\mu (S)=S^{1/(1-\beta )}/(1-S)$. With $q=1+1/(1-\beta )=(2-\beta )/(1-\beta )$, multiplication by this integrating factor givesThe general solution has an additional term $C/S^q$. Finite initial gas-phase metallicity rules it out. Consequently, for $0<S<1$,The upper limit is the dimensionless $S$, not the unnormalized stellar mass. On each compact subinterval of $0\le S<1$, the geometric series $1/(1-u)={\sum }_{k\ge 0}{u}^k$ converges uniformly, permitting integration term by term. ThusIn particular $Z_i\sim y_{i}S/(q+1)$ at small $S$. For positive stellar yield, the series is strictly increasing, so the inverse needed for the metallicity distribution function exists. Differentiating either representation gives $Z_i^{\prime }=y_i/(1-S)-q{Z}_i/S>0$. It diverges logarithmically as gas exhaustion approaches; the trace-metal approximation eventually becomes inadequate.

The quadratic gas-history enrichment model is not a closed gas reservoir: its gas mass initially increases. This can be made consistent with the metal equation by pristine galactic gas inflow. Indeed gas accounting gives $dg/ds=-(1-\beta f)/(1-\beta )+d{M}_{\mathrm{in}}/ds$. Since $dg/ds=1-2S$ and $f=S$, the required inflow is $d{M}_{\mathrm{in}}/ds=q(1-S)\ge 0$. It adds no metals and therefore leaves the preceding derivation intact.

Use the instantaneous recycling approximation and treat the interstellar medium as well mixed. If star formation initially consumes mass $dM$, returning a fraction $\beta$ leaves locked stellar mass $ds=(1-\beta )dM$. The original metal mass removed from the interstellar medium is $Z_{i}dM$. Of this original material, $\beta f{Z}_{i}dM$ returns and remains in the galaxy. In addition, the stellar yield produces fresh metal mass $y_{i}ds$, of which the retained part is $f{y}_{i}ds$. Therefore metal accounting givesThis is the retained-ejecta chemical evolution equation. It assumes that the same retention fraction applies to original and newly synthesized metals. Pristine galactic gas inflow can change $g$ without contributing to $d(g{Z}_i)$; enriched galactic gas inflow would require an additional metal source. Here $0\le \beta <1$, and $0\le f\le 1$.

A star inherits the gas-phase metallicity at its birth. Suppose enrichment is monotone and a fixed initial mass function, with an appropriate surviving tracer selection, produces $\kappa$ observable stars per unit locked mass. For two birth abundances $Z_a<Z_b$, the relative count isThus $s(Z_i)$ is proportional to the cumulative metallicity distribution function; its derivative gives the differential metallicity distribution function. The normalized distribution after final locked mass $s_{\infty }$ is $s_{\infty }^{-1}ds/d{Z}_i$. For logarithmic abundances, the logarithmic metallicity distribution instead has $dN/d{\log }_{10}{Z}_i=(\ln 10)Z_{i}dN/d{Z}_i$.

The fixed initial mass function and survival selection matter: a mass distribution is not automatically the number distribution of every presently observable star. If enrichment is nonmonotone, add the contributions $\kappa |ds/d{Z}_i|$ from all birth-time branches attaining that abundance, rather than using one inverse.

Write the gross star formation rate as $\psi =k{M}_g$, with $k>0$ constant. Use the instantaneous recycling approximation: a fraction $F$ returns promptly, while a fraction $1-F$ stays in long-lived stars and remnants. Assume $0\le F<1$, instantaneous mixing, and no galactic gas inflow or galactic outflow. If $M_s$ denotes the permanently locked stellar mass, mass conservation givesSet $\lambda =(1-F)k$. Integration, with $M_g(0)=M_0$, gives the exponential closed-box gas consumption lawThe locked mass is $M_s=M_0(1-e^{-\lambda t})$, so $M_g+M_s=M_0$ at every time. The decay time is $[(1-F)k{]}^{-1}$, longer than the gross gas-processing time $k^{-1}$ because returned gas can form subsequent generations. The formal $F=1$ limit retains all gas and locks no mass; a finite stellar yield defined per locked mass then needs separate treatment.

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.

Let $G$ be the remaining gas mass, $S$ the formed stellar mass, and $G_0$ the initial gas mass; initially $S=0$ and the gas-phase metallicity is zero. Take the stellar yield $y$ and the mass-loading factor $\nu$ to be constant, with homogeneous mixing and no galactic gas inflow. With the specified zero bulk return, mass conservation givesNew metals add $ydS$, while star formation and the galactic outflow remove metals at the existing gas-phase metallicity $Z$. ThusThe leaky-box model of galactic chemical evolution therefore givesFor the usual gas fraction of a galaxy, $\mu =G/(G+S)$, the denominator is the baryonic mass still in the system, which decreases under the galactic outflow. Since $G_0=G+(1+\nu )S$, we have $G_0/G=1+(1+\nu )(1-\mu )/\mu$. In the remaining-mass convention,If instead $\mu$ denotes $G/G_0$ relative to the fixed initial reservoir, the same calculation gives $\boxed{Z=y\ln (1/\mu )/(1+\nu )}$. Stating the denominator matters: these two gas fraction of a galaxy conventions coincide for the closed-box model of galactic chemical evolution, recovered at $\nu =0$, but differ when gas escapes. Neglecting bulk return while retaining a nonzero stellar yield is the intended trace-metal approximation.

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 has $M=M_0{e}^{-Z/y}$ and $S=M_0-M$, giving $dN/dZ=(\kappa M_0/y)e^{-Z/y}$. To compare shapes without confusing different total stellar counts, the figure normalizes both metallicity distribution functions to unit integral after complete gas exhaustion and uses $x=Z/y$:The right panel also shows the corresponding cumulative fractions, $1-e^{-x}$ and $\min (x/2,1)$.

Pristine galactic gas inflow replenishes the fuel and dilutes freshly produced metals, so enrichment per unit newly formed stellar mass is slower and asymptotes to $2y$. More stars are formed at later abundances; the proportional-infall model of galactic chemical evolution makes twice as many stars as the closed box for the same initial gas mass in this limit. Consequently the normalized low-abundance fraction is smaller: for $Z\ll y$, it is $Z/(2y)$ rather than approximately $Z/y$. Their unnormalized $dN/dZ$ actually agree at $Z=0$ for the same $M_0$, which is why stating the normalization matters. At a finite stopping abundance $Z_f$, use $p_Z=1/Z_f$ for infall and $p_Z=e^{-Z/y}/[y(1-e^{-Z_f/y})]$ for the closed box, each truncated at its own chosen stopping point.

Let $\psi (t)$ be the star formation rate, $M$ the remaining gas mass, and $S$ the mass put into stars. With zero returned bulk gas, pristine galactic gas inflow, and no galactic outflow, mass conservation givesA well-mixed gas reservoir loses existing metals at rate $Z\psi$ and gains newly synthesized metals at rate $y\psi$; the incoming gas brings none. ThusFor $0<\nu <1$, eliminate time and integrate:The proportional-infall model of galactic chemical evolution therefore gives, for initially pristine gas,The displayed form requires $Z_0=0$; more generally $Z=y/\nu +(Z_0-y/\nu )(M/M_0{)}^{\nu /(1-\nu )}$. Also $S=(M_0-M)/(1-\nu )$ for an initially star-free reservoir, so $M/M_0$ is not the gas fraction of a galaxy $M/(M+S)$. As a check, the $\nu \to 0$ limit is the closed-box model of galactic chemical evolution relation $Z=y\log (M_0/M)$; at $\nu =1$ the gas mass is constant and one must instead solve $\dot{Z}=(y-Z)\psi /M_0$. Retaining a nonzero stellar yield while setting bulk mass return to zero is the prescribed trace-metal approximation, not a literal claim that stellar ejecta have no mass.

The G-dwarf problem is the observed shortage of metal-poor, long-lived disc stars relative to a simple closed-box model of galactic chemical evolution. In the fully evolved comparison, the half-rate proportional-infall model of galactic chemical evolution lowers the low-$Z$ fraction from $1-e^{-Z/y}$ to $Z/(2y)$, and prevents the closed-box tail from extending to arbitrarily high stellar metallicity. It therefore reduces the relative weight of the earliest metal-poor generation and the very metal-rich tail.

This particular constant-proportional-infall model is an improvement, not a complete fit. It still forms a metal-poor population immediately from its initially pristine reservoir, with nonzero constant $dN/dZ$ all the way down to zero, and predicts a broad linear distribution followed by an abrupt upper cutoff instead of a generally smooth, concentrated observed distribution. Comparison with observed abundance histograms must first use the same coordinate: its logarithmic metallicity distribution rises in proportion to $Z$ up to that cutoff, rather than remaining flat in logarithmic bins. Local G-dwarf measurements exhibit a prominent logarithmic-abundance peak and a paucity of low-abundance stars; translating iron abundances into total $Z$ also requires an abundance-ratio model. More realistic histories can vary the galactic gas inflow and star formation rate, allow recycling and delayed enrichment, and account for migration and selection of different stellar populations; simply tuning a constant stellar yield does not remove all these shape deficiencies.

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.

With well-mixed gas-phase metallicity $Z$, zero mass return, star formation rate $\psi$, pristine galactic gas inflow $\nu \psi$, and constant stellar yield $y$, mass conservation and metal accounting give $\dot{M}=-(1-\nu )\psi$ and $M\dot{Z}=(y-\nu Z)\psi$. Thus, for $0<\nu <1$,The initial gas-phase metallicity $Z_0=0$ gives the usual pristine-infall expression. Neglecting returned bulk mass while retaining a nonzero metal yield is a trace-metal idealization.

Let $S=s/s_{\infty }$, let the interstellar medium have mass $g=s_{\infty }S(1-S)$, and retain fraction $f=S$ of returned material. For $0\le \beta <1$ put $q=(2-\beta )/(1-\beta )$. The integrating factor for the metal mass is $S^{1/(1-\beta )}/(1-S)$. A finite initial gas-phase metallicity removes the homogeneous solution and gives the displayed expression, equivalently $Z=y{\sum }_{n\ge 1}{S}^n/(q+n)$ for $0\le S<1$. The increasing initial gas mass requires pristine galactic gas inflow: gas accounting gives $d{M}_{\mathrm{in}}/ds=q(1-S)$. A fixed initial mass function makes the birth metallicity distribution function proportional to $ds/dZ$ when enrichment is monotone.

Under the instantaneous recycling approximation, forming mass $dM$ locks $ds=(1-\beta )dM$ into stars. If a fraction $f$ of returned material remains in the interstellar medium, the retained original metal mass is $\beta fZdM$, and retained newly made metal mass is $fyds$, where $y$ is the stellar yield per locked mass. Subtracting the metal mass $ZdM$ initially consumed gives the displayed equation. Pristine galactic gas inflow adds gas but no metals; enriched inflow adds its metal mass separately. The retained fraction applies to both old and newly made metals in this idealization.