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For a pristine closed-box model of galactic chemical evolution, $M_{g}dZ=Yd{M}_s$ and $d{M}_s=-d{M}_g$ imply $M_g=M_0{e}^{-Z/Y}$ independently of the star formation rate. The resulting metallicity distribution function is exponential on $0\le Z\le Z_f$. Its normalized density is $e^{-Z/Y}/[Y(1-e^{-Z_f/Y})]$. A fixed initial mass function and survival selection are needed to turn locked stellar mass into a stellar number distribution. Logarithmic abundance intervals have the additional factor $(\ln 10)Z$.

Changing from linear stellar metallicity to its logarithm multiplies the metallicity distribution function by its Jacobian determinant. A flat distribution in linear metallicity rises exponentially in logarithmic metallicity. The commonly used iron abundance coordinate is equivalent to total-metallicity scaling only if relative element abundances are fixed.

For initially pristine gas, zero bulk mass return, a constant stellar yield $y$ and a well-mixed galactic outflow of mass-loading factor $\nu$, gas conservation gives $d{M}_g=-(1+\nu )d{M}_s$ and metal accounting gives $M_{g}dZ=yd{M}_s$. Consequently $M_g=M_{g,0}{e}^{-Z/y_e}$ with $y_e=y/(1+\nu )$. Since $d{M}_s/dZ=M_g/y$, complete depletion gives a normalized mass-weighted metallicity distribution function $p(Z)=e^{-Z/y_e}/y_e$ and mean stellar metallicity $y_e$. A time-independent initial mass function makes the distribution of surviving tracers proportional to this distribution; a sample selected by light need not have the same weights.

For $\nu =1/2$ and initially pristine gas, $M=M_0(1-Z/(2y))$ and the formed stellar mass is $S=2(M_0-M)=M_{0}Z/y$. If a fixed initial mass function supplies $\kappa$ surviving tracers per unit formed mass, then $N(<Z)=\kappa M_{0}Z/y$ and $dN/dZ=\kappa M_0/y$ until the current gas abundance. Complete exhaustion gives a uniform normalized metallicity distribution function on $0<Z<2y$. This is uniform in linear metallicity, not in its logarithm.

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.

There is a genuine inconsistency in this part. For any nonnegative metallicity distribution function, its maximum is at least its mean. A population of positive mean solar metallicity cannot have a maximum equal to half that mean. For example, a uniform abundance distribution on $[0,2Z_{\odot }]$ has mean $Z_{\odot }$ and maximum $2Z_{\odot }$. Thus the printed maximum-to-mean ratio cannot be proved. The following calculation explains the supplied function, gives the consistent distribution, and also treats the literal birth-metallicity interpretation separately.

Let $f(\mu )$ denote the supplied abundance function. Algebraically it isWith $\delta =1-\mu$, a Taylor series of $(1-\delta {)}^{Y+1}$ givesConsequentlyThis is an early-consumption expansion: its fractional correction is $(1-Y)\delta /3+O({\delta }^2)$. Small yield alone does not make the expression linear throughout most of the gas-consumption history. At fixed $0<\mu <1$, the small-$Y$ expansion instead is

In a well-mixed closed-box model of galactic chemical evolution, the gas fraction of a galaxy also labels the permanently locked stellar mass: $d{M}_s=-M_{g,\mathrm{init}}d\mu$. Newly formed stars inherit the current gas-phase metallicity $Z_g(\mu )$. Therefore the mass-weighted mean stellar metallicity satisfies the cumulative and birth stellar metallicities relationApplying the derivative to $f$ givesThus the supplied function has the natural interpretation of a cumulative mean, not an individual star's birth abundance. Direct integration of $1-u^Y$ verifies that it gives exactly $f$. This is the saturating closed-box enrichment law: it corresponds to effective stellar yield $Y(1-Z_g)$, with $Z_g$ measured as a fraction. The usual constant-yield trace-metal model instead gives $Z_g=-Y\log \mu$. Both agree to first order when $Y|\log \mu |\ll 1$; their exact formulae should not be identified outside that regime.

Let the final gas fraction be ${\mu }_f>0$ and ${\delta }_f=1-{\mu }_f$. In the consistent mean-metallicity interpretation, ${\overline{Z}}_{\ast }({\mu }_f)=Z_{\odot }$ while the most recently formed stars have the maximum $z_f=1-{\mu }_f^Y$. At early consumption,so the corrected result isThe assumption here includes ${\delta }_f\ll 1$, equivalently $2Z_{\odot }/Y\ll 1$ in this approximation. Exactly, $z_f=1-{\mu }_f^Y$ and $Z_{\odot }=f({\mu }_f)$ determine the maximum implicitly; it remains larger than the mean, but its ratio to the mean need not equal two after substantial consumption.

For the mass-weighted metallicity distribution function, stars formed while the gas fraction falls by $-d\mu$ have probability $dP=-d\mu /{\delta }_f$. Writing their individual abundance as $z=1-{\mu }^Y$ gives $\mu =(1-z{)}^{1/Y}$ and the change of variablesThe density is zero outside this interval. Its cumulative probability is $[1-(1-z{)}^{1/Y}]/(1-{\mu }_f)$, so it integrates to one; integrating $zF(z)$ reproduces $f({\mu }_f)$. A star-count distribution has the same form if the initial mass function and survival selection give a fixed number of observed stars per unit locked mass. Luminosity-weighted samples require different weights.

For trace abundances, $(1-z{)}^{1/Y-1}\simeq e^{-z/Y}$ to leading order at fixed $z/Y$. The corresponding ordinary closed-box metallicity distribution isIn the early-consumption regime $z_f\ll Y$, this becomes approximately uniform, $F\simeq 1/(Y{\delta }_f)$ on $[0,Y{\delta }_f]$. Its mean is half its upper endpoint, again producing the corrected maximum $2Z_{\odot }$.

If the supplied $f(\mu )$ is instead retained literally as the individual birth abundance, set $q=f(\mu )$ rather than $z=1-{\mu }^Y$. The observation-time population mean is then ${\delta }_f^{-1}{\int }_{{\mu }_f}^{1}f(u)du$, not $f({\mu }_f)$. Its exact distribution can be specified parametrically usingPositivity follows because $f$ is the average of the increasing function $1-(1-\delta {)}^Y$. Its support is $0<q<f({\mu }_f)$, and normalization follows directly from $dP=d\delta /{\delta }_f$. At early consumption it is uniform with $F_{\mathrm{literal}}\simeq 2/(Y{\delta }_f)$, maximum $Y{\delta }_f/2$ and mean $Y{\delta }_f/4$. Even this literal alternative gives maximum twice the population mean, never half. The distinction between birth abundance and cumulative mean resolves the calculation; the printed inverse maximum-to-mean ratio is false under either interpretation.

A newly formed stellar population inherits the current gas-phase metallicity. Eliminate time between locked-mass growth and enrichment:For $Y>0$, integration from pristine gas gives $M_g=M_0{e}^{-Z/Y}$ and hence the closed-box metallicity distributionTo obtain numbers rather than mass, specify a fixed initial mass function and count a long-lived tracer population. Let $\eta$ be its number of surviving stars per unit locked stellar mass. At the final gas-phase metallicity $Z_f$,with zero density outside that interval. The cumulative count is $N(<Z)=\eta M_0(1-e^{-Z/Y})$ within the interval, and the normalized metallicity distribution function isThese equations do not use $\psi (t)$: any positive star formation rate gives the same distribution under the same chemical assumptions. If the tracer number is instead normalized per gross formed mass, replace $\eta$ by that normalization divided by $1-F$. Short-lived stars require an age-dependent survival factor and generally do not follow this number distribution. In a logarithmic metallicity distribution, $dN/d{\log }_{10}Z=(\ln 10)ZdN/dZ$; its interior peak is at $Z=Y$, whereas the distribution per linear interval decreases from $Z=0$.

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.

A star inherits the gas-phase metallicity at its birth. Set $y_e=y/(1+\nu )$; the preceding calculation gives $G(Z)=G_0{e}^{-Z/y_e}$ and $dS/dZ=G(Z)/y$. If the initial mass function is fixed, let $\kappa$ be the number of the counted stars per unit formed stellar mass; this includes any restriction to surviving long-lived tracers. The metallicity distribution in a leaky box is thereforeThe total eventual formed stellar mass is $G_0/(1+\nu )$, so $N_{\infty }=\kappa G_0/(1+\nu )$ and the equivalent normalized metallicity distribution function isIf $N(Z)$ means a cumulative count rather than a count per stellar metallicity interval, integration yieldsAt a finite observation time these expressions end at the current gas-phase metallicity $Z_f$ and the normalized distribution divides by $1-e^{-Z_f/y_e}$. A distribution per logarithmic stellar metallicity instead multiplies $dN/dZ$ by $(\ln 10)Z$; it is not the same shape as the distribution per linear 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.

For $\nu =1/2$, the gas-phase metallicity relation reduces to $Z=2y(1-M/M_0)$. Elimination of $M$ from mass conservation givesWith a fixed initial mass function, let $\kappa$ be the number of surviving low-mass tracer stars per unit formed stellar mass. Successive stars inherit the monotonically increasing gas-phase metallicity, so the cumulative and differential counts are respectivelywhere $Z_f=2y(1-M_f/M_0)$ is the abundance at the epoch when star formation stops. Above $Z_f$ the cumulative count is constant, and the differential count vanishes. In the complete gas-exhaustion limit, $Z_f\to 2y$ and $N_{\mathrm{total}}=2\kappa M_0$, so the normalized metallicity distribution function is $p_Z=1/(2y)$ on $0<Z<2y$; at a finite stopping epoch it is $1/Z_f$. The distribution is flat in linear metallicity. A logarithmic metallicity distribution instead satisfies $dN/d{\log }_{10}Z=(\ln 10)ZdN/dZ$ and is not flat.

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.

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.

A finite-metal-fraction variant of the closed-box model of galactic chemical evolution uses effective stellar yield $Y(1-Z_g)$, rather than constant yield. With pristine initial gas and $Y>0$, metal accounting gives $d{Z}_g/d\log (1/\mu )=Y(1-Z_g)$ and hence $Z_g=1-{\mu }^Y$. The mass-weighted mean stellar metallicity is $[Y-(Y+1)\mu +{\mu }^{Y+1}]/[(Y+1)(1-\mu )]$. At final gas fraction ${\mu }_f$, the metallicity distribution function is $(1-z{)}^{1/Y-1}/[Y(1-{\mu }_f)]$ on $0<z<1-{\mu }_f^Y$. For trace abundances, $Z_g\simeq -Y\log \mu$ and the usual closed-box metallicity distribution is recovered to leading order.