Metallicity Distribution Function Explained

• The MDF is defined as the frequency distribution of stellar metallicities, encoding the fossil record of chemical enrichment and star formation history.
• It is constructed from spectroscopic and photometric observations using techniques like kernel-density estimation and binning to correct for biases.
• Analyzing MDF shapes—including mean, dispersion, and tails—provides critical constraints on galaxy evolution models and feedback processes.

The metallicity distribution function (MDF) quantitatively describes the frequency of stars as a function of metallicity within a stellar system. It encodes the fossil record of chemical enrichment and star-formation history for galaxies, star clusters, and resolved fields. The MDF is formally defined as the number of stars per unit logarithmic metallicity, typically expressed as $dN/d[\mathrm{M/H}]$ or $dN/d[\mathrm{Fe/H}]$, where $[\mathrm{M/H}] \equiv \log_{10}(Z/Z_\odot)$ is the logarithmic total metallicity relative to Solar, and $[\mathrm{Fe/H}]$ refers specifically to iron. The shape, mean, dispersion, and higher moments of the MDF are primary constraints on theories of galaxy evolution, hierarchical assembly, and feedback processes.

1. Definitions and Mathematical Formalism

The MDF for a resolved stellar population is defined as the probability distribution or normalized histogram of stellar metallicity. For a sample of $N$ stars with metallicities $\{z_i\}$, one typically estimates the MDF as

$$\mathrm{MDF}(z) = \frac{1}{N} \sum_{i=1}^N \delta(z-z_i),$$

where $\delta$ is the Dirac delta function, or through binned or kernel-density estimates for discrete observations. The independent variable $z$ is frequently $[\mathrm{Fe/H}]$ or $[\mathrm{M/H}]$, depending on the available element abundances. In integrated light or from spectral decomposition, the MDF can be recovered as the probability density $P(Z|r)$ at position $r$ for metallicity $Z$: $P(Z|r) = \frac{dN}{dZ}(Z;r),$ with $\int P(Z|r)\,dZ=1$ by construction (Mejía-Narváez et al., 2020).

Analytic models for the MDF include the closed-box (Simple) model, leaky-box (including outflows), and infall/accretion models (e.g., Lynden-Bell 1975), which predict forms such as: $$\frac{dN}{d[\mathrm{Fe/H}]} \propto 10^{[\mathrm{Fe/H}]} \exp\left(-10^{[\mathrm{Fe/H}]}/p\right),$$ with $p$ the effective yield (Kirby et al., 2010, Ross et al., 2015). These models may be extended to include pre-enrichment, time-dependent accretion, and more complex gas flows.

2. Measurement Techniques: Resolved Stars and Integrated Populations

MDFs are constructed using either spectroscopic or photometric metallicity determinations. For resolved galaxies, individual red-giant-branch (RGB), red-clump (RC), or main-sequence turn-off (MSTO) stars provide the input metallicities. Calibration methodologies include:

• Spectroscopic abundance analyses, via equivalent-widths or synthetic spectral fitting, yielding $\alpha$/Fe], [M/H]" title="" rel="nofollow" data-turbo="false" class="assistant-link">Fe/H.
• Photometric metallicity inference by mapping colors (e.g., $(V-I)$, $(g-i)$, CaHK indices) onto grids of isochrones with fixed age and $\alpha$-enhancement (Mould et al., 2010, Fu et al., 2021, Youakim et al., 2020).
• Integrated-light measurements, or, in IFU surveys such as CALIFA, kernel-density reconstructions of the composite stellar population at each spatial resolution element (Mejía-Narváez et al., 2020).

For synthetic populations or simulation particles, each stellar population is treated as a mono-metallicity, mono-age simple stellar population (SSP), assigned weights according to their contribution to observable samples (e.g., upper RGB bias) (Choi et al., 2022).

Corrections for completeness, selection biases, and photometric errors are essential; completeness functions $C(z)$ are used to recover the intrinsic MDF: $$\mathrm{MDF}_{\rm corr}(z) = \mathrm{MDF}_{\rm raw}(z)/C(z),$$ as in ultra-low-metallicity halo samples (Yong et al., 2012).

3. Empirical Behavior of the MDF Across Galaxy Types

Massive Galaxies (Ellipticals and Bulges)

The MDF in the outer halos of giant ellipticals, exemplified by NGC 5128, is broad (dispersion $\sim$0.77 dex), with a median $[\mathrm{M/H}] \approx -0.4$ and a metal-poor tail to $[\mathrm{M/H}] \sim -2.0$ (Choi et al., 2022). Simulations reproduce these properties when including both accreted populations (dominant in the halo) and a minor, narrow in situ, metal-rich component formed rapidly at early times.

Early-type spirals with massive spheroidal components (e.g., the Sombrero galaxy, NGC 4594) display MDFs nearly indistinguishable from those of giant ellipticals at fixed mass: peaked at $[\mathrm{Fe/H}] \approx -0.5$, $\sigma \approx 0.5$ dex, with extended metal-poor tails (Mould et al., 2010). Both closed-box models with outflow and accretion models can reproduce the broad observed distributions, provided early, rapid gas accretion or significant gas loss is invoked.

The Milky Way: Disk, Bulge, and Halo

In the Milky Way disk, the MDF shape varies systematically with galactocentric radius: negative skewness (metal-poor tail) in the inner disk ($R<7$ kpc), nearly symmetric (Gaussian) near the solar radius, and positive skewness (metal-rich tail) in the outer disk ($R>11$ kpc). The midplane MDF peaks range from $[\mathrm{Fe/H}] \approx +0.23$ in the inner disk to $[\mathrm{Fe/H}] \approx -0.43$ at $R=13-15$ kpc, with $\sigma \sim 0.18-0.24$ dex (Hayden et al., 2015, Loebman et al., 2015, Martinez-Medina et al., 2016).

The MDF of the bulge is complex and highly skewed, covering $-3.0 < [Fe/H] < +1.0$. It is best described by multi-component Gaussian mixtures, with dominant metal-rich “boxy/peanut” subcomponents (mean near +0.15 and –0.25), an inner thick disk, and minor contributions from metal-weak thick disk and inner halo (Ness et al., 2015). The mean metallicity decreases with vertical distance from the plane due to changing population mix, and the vertical gradient is $-0.45$ dex kpc$^{-1}$.

In the halo, MDFs are peaked at $[\mathrm{Fe/H}] \approx -1.6$ (inner halo) with a smooth, exponential metal-poor tail: $$\log_{10} N \propto S[\mathrm{Fe/H}], \text{ with } S \simeq 1.0-1.5,$$ in both photometric and spectroscopic surveys (Youakim et al., 2020, Chiti et al., 2021). The spatial invariance of the MDF at the lowest metallicities supports a scenario of well-mixed early accretion for extremely metal-poor populations.

Dwarf Galaxies

Dwarf spheroidals exhibit diverse MDF shapes, depending on their star formation and dynamical histories. More luminous systems (e.g., Fornax, Leo I) show narrower, peaked MDFs requiring accretion/infall or pre-enrichment to fit, while less luminous dwarfs (e.g., Sextans, Draco) have lower yields and broader distributions, indicating strong gas outflow (Kirby et al., 2010, Ross et al., 2015). Sharp cutoffs or secondary peaks in UFDs or systems with interrupted star formation (e.g., Reticulum II) provide evidence for extended chemical evolution and the impact of global reionization or Type Ia SN enrichment (Luna et al., 19 Jun 2025).

Ultra-faint dwarfs, when sampled adequately, may display bimodal MDFs as in Reticulum II, with peaks separated by up to $\sim$1 dex, interpreted as signatures of multi-burst star-formation separated by periods of supernova-driven chemical enrichment and feedback (Luna et al., 19 Jun 2025).

4. Physical Interpretation: Enrichment, Feedback, Migration, and Assembly

The MDF encapsulates integrated processes:

• Enrichment and Feedback: The width and shape constrain the timescale and efficiency of star formation, SNe yields, outflow efficiency, and the potential for infall of pristine or metal-poor gas (Ross et al., 2015, Kirby et al., 2010). Closed-box models only fit the metal-poor tail; successful fits require prompt early enrichment, infall/accretion, or gas loss.
• Two-phase Assembly: In massive galaxies, the MDF is decomposable into a metal-rich, in situ population from early rapid formation, and a metal-poor, accreted population from disrupted satellites (Choi et al., 2022). Minor merger-driven heating or AGN feedback affects radial mixing and the formation of secondary peaks.
• Radial Migration: Disk MDFs reflect secular and non-secular processes: the changing shape and skewness with radius show the imprint of radial migration, whereby inner, metal-rich stars relocate to the outer disk. Pure orbital blurring is insufficient to explain MDF inversion; “churning” (i.e., changes in guiding radii due to bar/spiral resonance overlap) is required (Hayden et al., 2015, Martinez-Medina et al., 2016, Loebman et al., 2015).
• Hierarchical Assembly and Halo Structure: MDF modeling can reconstruct the assembly history and mass spectrum of destroyed progenitors by exploiting the mass–metallicity relation and the MDF as a mixture of Gaussian or more complex components (Deason et al., 2023). Substructure discovery in the halo utilizes distinct MDF peaks identified as debris from individual accretion events, though mapping between peak metallicity and progenitor mass/luminosity is complicated by redshift evolution and internal gradients (Kim et al., 22 Aug 2025).

5. Applications: Galaxy Archeology, Chemodynamical Modeling, and Substructure Discovery

Galaxy Formation and Evolution

The MDF provides stringent constraints on chemical-evolution models, requiring the inclusion of mass-dependent star-formation timescales, delayed feedback prescriptions, and spatially resolved mixing and migration (Mejía-Narváez et al., 2020). Broad or multi-modal MDFs can only be reproduced in models with temporally and spatially varying accretion and feedback efficiencies.

Substructure Identification

Chemo-dynamical mapping, where MDFs are combined with orbital parameters, enables identification of discrete substructures in the Galactic halo, corresponding to distinct accretion events or partially disrupted satellites (Kim et al., 22 Aug 2025). Gaussian mixture modeling of the MDF in orbital phase space yields robust peaks associated with specific progenitors. However, the relation between metallicity peak and progenitor parameters is degenerate due to internal gradients and the cosmological evolution of the mass–metallicity relation.

Progenitor Demographics from Accreted MDFs

For the Milky Way stellar halo, statistical modeling of the MDF as a mixture of components with parameters set by the mass–metallicity relation allows estimation of the total number and mass spectrum of destroyed satellites. Large samples of $\sim 10^4$–$10^5$ halo stars are required to achieve sensitivity to progenitors down to the ultra-faint dwarf mass scale (Deason et al., 2023).

Resolved Galaxy Surveys

Integral field spectroscopy and wide-field photometric surveys permit spatially resolved MDF measurements across thousands of galaxies (e.g., CALIFA). The MDF extends beyond simple radial gradients, revealing multi-modal distributions associated with composite star-formation histories, bulge-disk superpositions, and radial migration in both early- and late-type galaxies (Mejía-Narváez et al., 2020).

6. Model Limitations, Systematics, and Future Prospects

Model Limitations and Caveats

No single analytic form can fully capture the complexity of observed MDFs in galaxies with hierarchical assembly or multiple star-formation episodes (Choi et al., 2022, Kirby et al., 2010, Luna et al., 19 Jun 2025). Internal abundance gradients, metallicity-dependent selection effects, and incompleteness at low [Fe/H] impact the empirical MDF, requiring careful forward modeling and corrections (Yong et al., 2012).

Simulations vs. Observations

Simulated MDFs are sensitive to assumptions about star-formation thresholds, yields, feedback, and the resolution of enrichment processes. Discrepancies in median metallicity, width, and low-metallicity tail often reflect these modeling choices (Calura et al., 2012).

Future Prospects

The MDF remains a central diagnostic in the era of precision Galactic archeology. Upcoming all-sky surveys (e.g., DESI, 4MOST, PFS, Gaia) will deliver samples of $10^5$–$10^6$ stars with well-determined metallicities, enabling MDF-based reconstruction of accretion events, star-formation bursts, and chemical evolution to unprecedented depth (Deason et al., 2023, Kim et al., 22 Aug 2025). Next-generation simulations with improved chemical enrichment physics, merger histories, and feedback prescriptions will further refine theoretical predictions for the MDF in both resolved and integrated systems (Choi et al., 2022).

Table: Characteristic MDF Properties in Selected Environments

Environment	Median [M/H]/[Fe/H]	Dispersion (dex)	MDF Shape
NGC 5128 outer halo (Choi et al., 2022)	–0.38	~0.77	Broad, pronounced tail
Milky Way disk, R=5–7kpc (Hayden et al., 2015)	+0.11 ([Fe/H])	~0.22	Negative skew
Milky Way halo (Youakim et al., 2020)	–1.6 ([Fe/H])	n/a	Exponential metal-poor tail
Dwarf spheroidal, Leo I (Ross et al., 2015)	–1.34 ([M/H])	0.21	Narrow, sharp cutoff
UFD Reticulum II (Luna et al., 19 Jun 2025)	–3.0/–2.1 (bimodal)	0.3/≤0.25	Bimodal with large gap

Broadly, the MDF is an observational bridge between the fossil record of chemical enrichment and the dynamical assembly of galaxies. Its detailed shape provides key leverage on the timing, efficiency, and mechanisms of star formation, feedback, and hierarchical accretion across cosmic time.