Galaxy evolution

Galaxy evolution is the change of a galaxy's stellar population, gas supply, structure and mass over cosmic time. Star formation consumes cold gas; evolving stars return chemically enriched material, while stellar feedback and active-galactic-nucleus feedback can heat or expel it. Galaxy mergers rearrange stellar orbits and may concentrate gas, changing both structure and subsequent star formation. Observers combine stellar ages, optical spectra and star formation rate diagnostics, then compare populations at different redshifts. Such comparisons require care: a distant population is a statistical sample of earlier galaxies, rather than a time sequence of the same objects.^[ref]

galactic feedback couples star formation or black-hole accretion to the surrounding gas through energy, momentum and mass exchange. It includes stellar feedback and active-galactic-nucleus feedback. A feedback loop may heat, stir, expel or compress gas, so its sign depends on how the gas supply and cooling respond; energy injection alone does not establish a lasting reduction of star formation.

A galaxy with negligible present star formation can still gain stars through mergers. Stellar mass assembly and in-situ star formation therefore describe different histories; quenching the latter does not freeze the former.

A gas-budget model in which inflow competes with star formation, gas return and outflows. A zero instantaneous gas derivative defines a balance nullcline; it need not define a fixed point of the evolving stellar and black-hole masses.

With constant inflow $\Phi$, no mass return or outflow, and $\Psi =ϵM_g$, mass conservation gives $M_g=\Phi /ϵ+(M_0-\Phi /ϵ)e^{-ϵt}$. Positive $ϵ$ makes the gas equilibrium stable, with perturbations decaying on timescale ${ϵ}^{-1}$. The star formation rate tends to the inflow, not necessarily to zero.

For ${\dot{M}}_g+ϵM_g=\Phi (t)$, multiplication by $e^{ϵt}$ and integration gives $M_g=M_0{e}^{-ϵt}+{\int }_0^t{e}^{-ϵ(t-s)}\Phi (s)ds$. Exponential memory filters the past inflow; the constant-inflow equilibrium is one specialization.

For initially no stars, $M_{\star }=\Phi t+(M_0-\Phi /ϵ)(1-e^{-ϵt})$. Adding the gas solution verifies $M_g+M_{\star }=M_0+\Phi t$. Thus equilibrium of the gas reservoir coexists with continuing linear growth of stellar mass.

In a system where gas and stellar masses both evolve, the locus ${\dot{M}}_g=0$ gives the instantaneous gas-balance nullcline. Motion along the other state variables generally changes the equilibrium reservoir, so the nullcline is not itself an exact constant-gas trajectory.

For an Eddington-luminous hole with $M_{\mathrm{BH}}=\beta M_s$ and wind mass flux $\lambda L$, define $A=\lambda \beta (4\pi G{m}_{p}c/{\sigma }_T)$. The gas budget ${\dot{M}}_g=\Phi -ϵM_g-A{M}_s$ has positive instantaneous balance only for $M_s<\Phi /A$. A positive constant-gas trajectory is incompatible with ongoing ${\dot{M}}_s=ϵM_g$ unless the secular black-hole coupling is neglected.

Galaxy morphological evolution changes the relative prevalence and internal structure of discs, spheroids, bars, and irregular systems over time. Comparing epochs requires consistent wavelength, spatial resolution, and selection. Galaxy mergers, gas accretion and internal instabilities can all contribute.

The observed size distribution of galaxies changes with epoch at a given galaxy stellar mass. Growth of an individual object and changes in which objects enter a selected sample both contribute. Minor galaxy mergers can deposit stars at large radii, increasing size while adding relatively little mass; stellar mass loss and dissipative evolution provide other mechanisms.

A galaxy red sequence is the relatively narrow red locus in a galaxy colour-magnitude diagram, populated largely by systems with little recent star formation. Its colour, abundance, and galaxy stellar mass evolve. Dust and selection must be distinguished from old stellar ages; growth can involve quenching and later assembly of existing stars.

A starburst galaxy undergoes an episode of rapid star formation. An instantaneous-burst idealization places all newly formed stars at one stellar age, so their present evolutionary-state fractions follow from the initial mass function at that fixed age rather than from an average over formation times.

A galaxy merger combines two or more galaxies into a remnant. Their orbital angular momentum and energy are redistributed into internal stellar and halo motions, escaping matter, and, when present, dissipative gas. A bound pair produced by tidal capture of galaxies need not coalesce immediately.

An interacting pair of spiral galaxies undergoing a galaxy merger. Their long tidal tails contain stars displaced by the encounter. They are also a nearby example of a starburst galaxy, with vigorous star formation accompanying the interaction.^[ref]

The Hubble Space Telescope image below combines observations from its Wide Field Camera 3 and Advanced Camera for Surveys. Bright young stars, emitting regions of the interstellar medium and obscuring interstellar dust reveal different aspects of the encounter. The close view emphasizes the interacting central regions; wider views show the long outer tidal tails.^[ref]

A gas-rich galaxy merger involves a substantial reservoir of gas capable of radiative cooling. Dissipation and gravitational torques permit gas inflow and enhanced star formation, unlike the nearly collisionless stellar evolution in a dry galaxy merger. A surviving or replenished gas reservoir can subsequently form a galactic disk.

Gas can shock and radiate during a galaxy encounter, allowing it to change orbital energy in ways unavailable to an ideal collisionless stellar component. Gravitational torques can transport its angular momentum and fuel central star formation or accretion. The outcome depends on orbit, gas content and internal structure.

A major galaxy merger joins galaxies with comparable masses. Rapid changes in the gravitational potential can redistribute stellar orbital energies through violent relaxation, substantially disturbing preexisting galactic disks. The outcome depends on gas content, orbit and subsequent accretion; a major merger does not universally prohibit disk rebuilding.

In a dilute, homogeneous population with number density $n$, an effective encounter cross section $\Sigma =\pi R_{\mathrm{enc}}^2$ gives the rate per target galaxyHere $R_{\mathrm{enc}}$ is the maximum encounter impact parameter, not automatically one galaxy's visible radius. If the cross section depends on relative speed, average the product instead: ${\Gamma }_{\mathrm{enc}}=n⟨\Sigma (v_{\mathrm{rel}})v_{\mathrm{rel}}⟩$. Gravitational focusing can create such speed dependence.

With independent encounters at a constant rate, a Poisson process givesThe approximation ${\Gamma }_{\mathrm{enc}}T$ is valid only for rare encounters. This is an encounter probability; it is not automatically a galaxy merger probability. Passing galaxies can remain unbound, and a bound pair can take additional time to coalesce.

As a simple additional model, suppose each encounter independently produces a merger within the relevant window with a fixed conditional probability $p$. The probability generating function of the Poisson count givesso the corresponding merger probability is $1-e^{-p{\Gamma }_{\mathrm{enc}}T}$. A realistic $p$ depends on encounter geometry, relative energy, angular momentum, gas and the allowed delay. It cannot be inferred from the geometric encounter rate alone.

For an entire volume $V$ of identical potential targets, the total distinct encounter-pair rate is $nV{\Gamma }_{\mathrm{enc}}/2$, because summing the per-target rate counts each unordered pair twice. This pair correction is separate from the conditional chance of merging.

Tidal capture converts sufficient relative orbital energy into internal tidal heating to leave two initially unbound galaxies on a bound orbit. For identical spherical galaxies of mass $m_g$, a distant rapid flyby of impact parameter $p$ and relative speed $v$ gives the approximate capture condition $pv<[(32/3)G^2{m}_g^2⟨r^2⟩{]}^{1/4}$. This requires the impulse approximation and weak gravitational focusing; it is not a general criterion for subsequent coalescence.

Two equal galaxies each receive the tidal impulse heating of a spherical galaxy, while their relative orbital energy at infinity is $m_g{v}^2/4$. Transfer of more internal energy than that orbital energy gives $pv<[32G^2{m}_g^2⟨r^2⟩/3{]}^{1/4}$. Slow encounters require adiabatic shielding of tidal encounters, so the impulsive inequality does not apply there.

A dry galaxy merger joins galaxies with little cold gas, so dissipation and merger-triggered star formation are weak. Conservation of energy followed by re-virialization can make repeated dry minor mergers increase a galaxy's radius faster than its mass.

In a dry galaxy merger, zero orbital energy at infinite separation and retention of all mass and energy give $E_F=E_I+E_A$. If the progenitors and remnant satisfy the virial theorem, their gravitational radii obey $R_F/R_I=(1+\eta {)}^2/(1+ϵ\eta )$, where $\eta =M_A/M_I$ and $ϵ=⟨v_A^2⟩/⟨v_I^2⟩$. The latter average is weighted by satellite mass. Nonzero orbital energy, escaping stars, or dissipative losses change the energy accounting.

A minor galaxy merger has a small satellite-to-host mass ratio. Its stars are commonly deposited at large host radii after tidal stripping and Chandrasekhar dynamical friction.

For repeated minor galaxy mergers with negligible internal satellite binding energy and zero orbital energy at infinity, parabolic dry-merger energy accounting leaves total energy nearly constant. The virial theorem then gives gravitational radius $R_g\propto M^2$ and mean-square speed proportional to $M^{-1}$. Small satellite mass does not by itself imply negligible internal binding energy.

Stellar feedback is the transfer of energy, momentum, mass, and newly formed elements from stars to surrounding matter through radiation, winds, supernovae, and cosmic rays.

Depositing supernova energy into dense unresolved gas can radiate it away before a numerically resolved expansion develops. Equal injected energy therefore need not imply equal outflow momentum or heating at different resolution. A feedback model should represent the coupling and cooling phase it intends to approximate, and comparisons should include gas recycling and the resulting stellar and circumgalactic properties.

Before local expulsion, a linear star-formation law $\psi =\nu {\rho }_g$ without return gives ${\rho }_g={\rho }_0{e}^{-\nu t}$. If $q=\gamma E_{\mathrm{SN}}$ is the energy supplied per unit stellar mass, accumulated energy is $q{\rho }_0(1-e^{-\nu t})$. A fixed binding energy per unit remaining gas mass $B(r)$ gives the threshold $q(e^{\nu t}-1)=B(r)$. This assumes a specified retained gravitational profile and neglects exchange between volumes.

The local shell-energy prescription sets $B(r)=GM(<r)/r$ rather than the full escape work. It predicts simultaneous quenching for $M(<r)\propto r$, but a centre-to-edge front for $M(<r)\propto r^2$. Exterior-shell contributions explain why this ordering can differ from a true escape-potential calculation.

With test-parcel escape work $B(r)=-\Phi (r)$, outer gas in a spherical isolated attractive potential is less tightly bound and quenches first under identical local specific heating. A truncated inverse-square density gives $r_q=r_0\exp [1-q(e^{\nu t}-1)/(2k)]$ after edge onset. The result is a fixed-potential closure, not a self-consistent potential update after mass loss.

For flux $F$ and effective dust opacity $\kappa$, radiative acceleration is $\kappa F/c$. Comparing it with the gravitational acceleration gives an Eddington luminosity condition appropriate to dusty gas. In an optically thick starburst galaxy, feedback can limit the star formation rate per area and drive a galactic outflow, while the gas supply limits the duration. A surface-density limit is not a universal ceiling on total star formation rate: the star-forming area matters. Radiation-supported starburst disk calculations give characteristic optically thick infrared fluxes near $10^{13}{L}_{\odot }{\mathrm{kpc}}^{-2}$.

If a virialized, well-mixed self-gravitating system instantaneously retains mass fraction $ϵ>1/2$, homologous revirialization gives $R^{\prime }/R=ϵ/(2ϵ-1)$. At or below one half, the idealized remnant is unbound.

Take an initially isolated virialized self-gravitating stellar population with kinetic energy $T=|W|/2$ and Newtonian gravitational potential energy $W<0$. Instantaneous spatially uniform removal of mass, without kicks to surviving material, leaves fraction $f$ with $T^{\prime }=fT$ and $W^{\prime }=f^{2}W$. Hence $E^{\prime }=|W|f(1/2-f)$: the homologous whole-system energy is negative only for $f>1/2$. Positive whole-system energy does not forbid a bound core after selective escape, and slow mass loss follows the adiabatic mass-loss expansion law instead.

For an initial mass function $dN\propto m^{-\alpha }dm$ on $0.1<m/M_{\odot }<100$, complete loss of material initially above $10M_{\odot }$ removes fractionThe impulsive disruption by stellar mass loss threshold is $F=1/2$, with unique finite root $\alpha \simeq 1.79101236$. Decreasing $\alpha$ shifts mass to larger stars and increases $F$, so the idealized system is globally unbound for $\alpha <{\alpha }_{\mathrm{crit}}$. The derivation requires rapid, well-mixed loss and absence of a confining external potential.

If a self-gravitating system loses mass slowly compared with its orbital times, adiabatic invariance gives $MR=\text{constant}$ in the homologous approximation, so retaining fraction $ϵ$ expands the system by $R^{\prime }/R=1/ϵ$.

A galactic outflow carries gas, metals, energy, and momentum away from a galaxy under stellar or active-galactic-nucleus feedback. Whether the material escapes or later returns depends on its speed and the galactic potential.

A galactic wind is a large-scale flow carrying gas, energy and momentum away from a galaxy. Stellar winds and supernovae can maintain a hot, pressurized component that expands out of a star-forming region; an active galactic nucleus can supply another driving mechanism. Cooler entrained gas can coexist with the hot component.

Observations must distinguish the wind from unrelated point sources and foreground emission. In M82, ROSAT measurements separated extended soft X-ray emission from compact sources and traced emission away from the galactic plane. The emitting gas need not be the freely expanding wind itself: shocked clouds can dominate the detected radiation.^[ref]

Gas lifted out of a galactic disk can remain bound and return rather than escape into the intergalactic medium. This recycling carries metals and angular momentum between the disk and its surroundings and delays the availability of gas for subsequent star formation.

Self-gravitating energy is nonlocal. Shell assembly assigns $-GM(<r)dM/r$ and counts pairs once; the density $\rho \Phi /2$ is another total-energy allocation. Removing a test parcel from a fixed potential instead requires $-\Phi dM$. These prescriptions need not give the same radial feedback threshold, even though their units agree.

If stellar feedback supplies energy $e_{\ast }$ per unit formed stellar mass, an energy-driven outflow requires $\nu v_{\mathrm{esc}}^2/2\lesssim e_{\ast }$. Fixed energy coupling therefore suggests mass-loading factor $\nu \propto v_{\mathrm{esc}}^{-2}$. A fixed momentum supply $p_{\ast }$ instead gives $\nu v_{\mathrm{esc}}\lesssim p_{\ast }$ and momentum-driven outflow scaling $\nu \propto v_{\mathrm{esc}}^{-1}$. These are launch and escape estimates, not universal wind laws; subsequent entrainment, cooling and fallback change the escaping mass.

The mass-loading factor is the ratio of outflow rate to star-formation rate, $\lambda ={\dot{M}}_{\mathrm{out}}/{\dot{M}}_{\ast }$. Its value can depend on galaxy mass, radius, gas phase, and the surface through which the flow is measured.

Galactic gas inflow supplies a galaxy from its circumgalactic medium, cooling halo, or intergalactic filaments. It can sustain star formation and dilute the metallicity of the existing interstellar gas.

Galactic chemical evolution follows the production, transport, dilution, locking, and loss of chemical elements as stars form and die and gas enters or leaves a galaxy.

An alpha element is an element such as oxygen, neon or magnesium associated with nucleosynthesis involving helium nuclei and massive-star burning. Core-collapse supernovae promptly return substantial abundances of these elements. Their abundance relative to iron can distinguish a rapid enrichment history from one with substantial delayed Type Ia supernova enrichment.

Massive stars produce prompt enrichment through core-collapse supernovae, whereas a Type Ia supernova follows binary stellar evolution and contributes substantial iron after a distribution of delays. A stellar population formed rapidly before much delayed enrichment can have a larger alpha-element to iron ratio than a stellar population formed over a prolonged interval. Relative abundance also depends on the initial mass function, stellar yields, and gas exchange; this is a chemical clock with model assumptions.

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.

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 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.

Galactic metallicity measures the abundance of elements heavier than helium in a galaxy. Gas and stars retain different time-weighted records of enrichment.

Gas-phase metallicity is the heavy-element abundance of interstellar gas. In star-forming galaxies it is commonly inferred from nebular emission lines in H II regions, often through oxygen abundance $12+{\log }_{10}(\mathrm{O}/\mathrm{H})$.

Stellar metallicity is the heavy-element abundance locked in stars. Integrated-light measurements are luminosity weighted unless a stellar-population reconstruction converts them to a mass-weighted quantity.

In a well-mixed closed-box model of galactic chemical evolution, gas consumed into locked stellar mass supplies a natural cumulative mass coordinate $\delta$. The mean stellar metallicity is ${\overline{Z}}_{\ast }(\delta )={\delta }^{-1}{\int }_0^{\delta }{Z}_{\mathrm{birth}}(s)ds$, giving the displayed derivative relation. If the birth abundance increases linearly, $Z_{\mathrm{birth}}=b\delta$, its cumulative mean is half the current maximum. More generally the maximum of any nonnegative abundance distribution cannot lie below its mean. A fixed initial mass function and the same survival selection are required to identify this mass-weighted distribution with a star-count distribution.

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.

The instantaneous recycling approximation treats short-lived stars as returning their ejecta immediately after formation while long-lived stars retain their mass indefinitely. It turns delayed chemical enrichment into local differential equations.

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.

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.

The stellar yield is the newly synthesized metal mass returned to gas per unit mass locked into long-lived stars, under a specified recycling convention and initial mass function.

Let a fraction $R$ of gross stellar birth mass return immediately, and let the new metal source be $bR\Psi$. The yield per permanently locked mass is then $y=bR/(1-R)$. Metal conservation in an initially pristine closed box gives $Z=y\ln (M_0/M_g)$. A coefficient defined by the metal source must not be confused with a yield normalized per locked stellar mass; the source convention must be stated, especially when taking $R\to 0$ or $R\to 1$ limits.

The closed-box model evolves a fixed total baryonic mass with no inflow or outflow. Under instantaneous recycling and constant yield it predicts $Z=y\log (1/\mu )$ for gas fraction $\mu$.

This fraction compares gas mass with gas plus permanently locked stellar mass in a stated galactic reservoir. Under instantaneous recycling with a constant locking fraction and star formation proportional to gas mass, it decays as $e^{-\alpha ϵt}$. Gross cumulative star formation is not the denominator's surviving stellar mass.

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.

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 $(\ln 10)Z$.

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.

A fixed-efficiency closed box with permanent locking fraction $\alpha$ predicts exponential decay of gas and gross star formation. Backward extrapolation therefore grows exponentially with lookback time. It requires a stated present age and assumes the same gas reservoir and efficiency over the entire interval. Cosmological inflow, outflow, mergers and changing gas conditions generally invalidate such an extrapolation over many depletion times.

The leaky-box model adds a galactic outflow to the closed box. If the outflow has the ambient gas metallicity and constant mass-loading factor $\lambda$, its effective yield is reduced to $y/(1+\lambda )$ when returned fractions are absorbed into the definitions.

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.

The effective yield is the yield that a simple closed-box relation would need to reproduce an observed gas fraction and metallicity. Outflow and pristine inflow generally lower it below the nucleosynthetic yield.

The G-dwarf problem is the deficit of low-metallicity long-lived G dwarfs near the Sun relative to the prediction of a constant-yield closed-box model. Prolonged accretion of metal-poor gas is a standard ingredient in its resolution.

A metallicity distribution function gives the number or mass of stars per metallicity interval. Its interpretation depends on sample selection, stellar lifetimes, spatial migration, and whether metallicity means total heavy-element abundance or a particular abundance ratio.

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.