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With gross star formation rate $\psi =k{M}_g$, constant returned mass fraction $F$, and the instantaneous recycling approximation, the net locked mass rate is $(1-F)\psi$. Mass conservation gives ${\dot{M}}_g=-(1-F)k{M}_g$. For constant stellar yield $Y$ per unit locked mass and initially pristine gas, metal conservation gives $M_g\dot{Z}=Y(1-F)\psi$, so gas-phase metallicity grows as $Z=Y(1-F)kt$. Gross star formation and permanent locking have different normalizations.

The galaxy mass--metallicity relation is the empirical tendency for galaxies with greater galaxy stellar mass to have higher gas-phase metallicity and stellar metallicity. Mass-dependent metal loss, gas inflow, star-formation efficiency, and recycling all contribute.

A linear gas-consumption law assumes that the net rate of locking mass into long-lived stars is proportional to the gas mass, $\psi =ϵM_g$, with constant $ϵ$. In a leaky-box model of galactic chemical evolution with ambient-metallicity outflow $\alpha \psi$, gas mass decays as $M_g=M_{g,0}{e}^{-(1+\alpha )ϵt}$, while gas-phase metallicity grows as $Z=Z_0+pϵt$. The wind changes the time to consume gas but, under these assumptions, not the enrichment rate at a given time.

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 $Z=0$.

Let $Z$ be the well-mixed gas-phase metallicity, measured as a metal mass fraction. Removing gas into stars takes metals at rate $Z\psi$; prompt return restores existing metals at rate $ZF\psi$, together with newly synthesized metals at rate $Y(1-F)\psi$. Thus the closed-box model of galactic chemical evolution has metal budgetExpand the left side and use ${\dot{M}}_g=-(1-F)\psi$. The terms proportional to existing gas-phase metallicity cancel, leavingSince $\psi =k{M}_g$, initially pristine gas obeysThe stellar yield here is per permanently locked mass, so inserting another factor $1-F$ into $Y$ would double-count recycling. The approximation treats metals as a trace constituent and ignores finite stellar-return delays. Apply it while $Y\lambda t\ll 1$; the formal unlimited linear growth is not a physical prediction for almost entirely metallic gas.

Divide metal enrichment by gas depletion in the closed-box model of galactic chemical evolution:The logarithm appears on integrating from $M_0$ and $Z=0$:With the gas fraction of a galaxy $\mu =M_g/M_0$, this isIt gives $Z=0$ at $\mu =1$ and increasing gas-phase metallicity as gas is consumed, independently of the time dependence of the star formation rate. For this closed initially star-free system, $M_0=M_g+M_s$, so this initial-reservoir fraction equals the present gas-to-total baryonic fraction. That equality fails in an open system.

A primordial light-element abundance determination seeks gas or stars with little subsequent chemical processing. The three abundances require different observations and corrections.

For helium, measure helium and hydrogen recombination emission in metal-poor extragalactic ionized regions. Correct line ratios for electron temperature, density, extinction, collisional excitation, underlying stellar absorption and unobserved ionization stages. Convert the number ratio $y=n_{\mathrm{He}}/n_{\mathrm{H}}$ to helium mass fraction $Y\simeq 4y/(1+4y)$, allowing for heavy elements when necessary. Since stars produce helium and metals, extrapolate a sample toward zero gas-phase metallicity to infer $Y_p$. The sensitivity to atomic data and these corrections is illustrated by Peimbert and collaborators' helium determination.

For deuterium, fit isotope-shifted hydrogen and deuterium absorption lines in relatively simple, metal-poor quasar absorption systems. The deuterium feature is displaced by about $-82\mathrm{km}{\mathrm{s}}^{-1}$ from the corresponding hydrogen line. Multiple transitions and velocity components help distinguish deuterium from unrelated hydrogen contamination. The fitted neutral-column ratio estimates D/H in predominantly neutral gas. Since stellar processing destroys deuterium, low-metallicity, minimally processed absorbers are favored; the technique is exemplified by Pettini and collaborators' deuterium measurement.

For lithium, measure the ${}^7\mathrm{Li}$ absorption feature near $670.8\mathrm{nm}$ in warm, old, metal-poor halo stars. Stellar-atmosphere models convert line strength to Li/H. The Spite plateau supplies a nearly constant low-metallicity level, but stellar metallicity trends, atmospheric temperature scales, departures from local thermal equilibrium and stellar lithium depletion must be considered. Thus the photospheric level is not automatically the initial abundance. Ryan and collaborators' halo lithium measurements illustrate the measured abundance and chemical-evolution corrections.

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.

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.

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.

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 $Z_w=Z$.