---
title: Abundance Discrepancy Factor (ADF)
url: https://www.emergentmind.com/topics/abundance-discrepancy-factor-adf
type: topic
---

# Abundance Discrepancy Factor (ADF)

The **Abundance Discrepancy Factor** (ADF) is the observational measure of the mismatch between ionic abundances derived from heavy-element **recombination lines** (RLs, or ORLs in much planetary-nebula work) and those derived from **collisionally excited lines** (CELs) for the same ion. In ionized nebulae, RL-based abundances are systematically higher than CEL-based abundances, making the ADF a central quantity in nebular abundance analysis and in the calibration of gas-phase metallicity scales. In H II regions the discrepancy is usually modest, whereas in planetary nebulae it can become extreme, reaching factors of tens to hundreds in a small subset of objects [1612.03633; 1510.07516; 2201.05627].

## 1. Definition, notation, and scope

In H II-region studies, the ADF is commonly written as a logarithmic abundance offset,
\[
{\rm ADF}({\rm X}^{i}) = {\rm log}({\rm X}^{i}/{\rm H}^+)_{\rm RLs} - {\rm log}({\rm X}^{i}/{\rm H}^+)_{\rm CELs},
\]
or equivalently
\[
\mathrm{ADF(X^{i+}) \equiv log(X^{i+}/H^+)_{RLs} - log(X^{i+}/H^+)_{CELs}}.
\]
A positive value means that the RL abundance exceeds the CEL abundance, which is the usual observational outcome [1612.03633; 1610.08497]. In much planetary-nebula literature, the same comparison is written as a linear ratio,
\[
\mathrm{ADF}(X^{i+}) = \frac{(X^{i+}/H^+)_{\mathrm{ORLs}}{(X^{i+}/H^+)_{\mathrm{CELs}}},
\]
with the largest body of work focused on \(\mathrm{O}^{2+}\) because both optical RLs and bright CELs are available for that ion [1309.1663; 2201.00902].

The best-studied reference ion is therefore \(\mathrm{O}^{2+}\). Other ions with observational ADF determinations include \(\mathrm{O}^{+}\), \(\mathrm{C}^{2+}\), and \(\mathrm{Ne}^{2+}\) in H II regions, and \(\mathrm{O}^{+}\), \(\mathrm{Ne}^{2+}\), and other heavy ions in planetary nebulae, although those measurements are generally sparser or more sensitive to aperture, atomic-data, and recombination-correction systematics [1612.03633; 1309.1663].

The discrepancy is astrophysically important because the adopted oxygen abundance scale underpins chemical evolution models, stellar-yield inferences, luminosity-metallicity and mass-metallicity relations, strong-line calibrations, and even primordial helium work [1612.03633]. The ADF is therefore not a minor spectroscopic detail but a systematic uncertainty in nebular abundance analysis.

## 2. Observational determination

The ADF is constructed by deriving the same ionic abundance from two line classes with very different emissivity dependences. CELs are bright and observationally efficient, but their emissivities depend strongly on electron temperature. RLs are intrinsically faint but much less temperature-sensitive. In H II regions, heavy-element optical RLs typically have intensities of order \(0.0001\) to \(0.001\) times that of H\(\beta\), which is why high-S/N spectra from large telescopes or deep IFU observations are required [1612.03633].

For oxygen and carbon, the classic optical RL diagnostics are **O II multiplet 1 around \(4650\) Å** and **C II \(\lambda 4267\)**. The canonical CEL comparison for \(\mathrm{O}^{2+}\) uses the bright [O III] nebular lines and the [O III] auroral line to define the standard optical \(T_e\) method [1610.08497; 1702.01120; 1003.2567]. For carbon, historical CEL abundances often came from ultraviolet transitions, which introduced additional aperture and reddening systematics in early ADF(C\(^{2+}\)) work [1610.08497].

Recent work has expanded the diagnostic basis. Far-infrared [O III] \(52\,\mu{\rm m}\) and \(88\,\mu{\rm m}\) lines provide a CEL-based oxygen abundance scale that is only weakly temperature-sensitive, and ultraviolet CEL ratios such as O III] \(\lambda\lambda1661,1666\) and C III] \(\lambda\lambda1907,1909\) allow internally consistent CEL determinations of \(\mathrm{C}^{2+}/\mathrm{O}^{2+}\) [2304.09898; 2508.03954]. Spatially resolved IFU and long-slit data have further turned the ADF from a single integrated number into a mapped quantity, enabling direct comparison with \(T_e\), \(n_e\), morphology, and kinematics [1205.0509; 2112.00480; 2407.06385; 2603.22382].

## 3. H II regions

In the H II-region literature, \(\mathrm{ADF}(\mathrm{O}^{2+})\) is typically between **0.10 and 0.35 dex** [1612.03633]. Representative values collected in the review literature include Orion, M8, 30 Dor, N66C, and NGC 5253, and the discrepancy is positive for multiple ions, not only \(\mathrm{O}^{2+}\) [1612.03633].

The Magellanic Clouds provide one of the clearest homogeneous extragalactic datasets. In the Large Magellanic Cloud, \(\mathrm{ADF}(\mathrm{O}^{2+})\) ranges from **\(0.14 \pm 0.05\)** in 30 Doradus to **\(0.31 \pm 0.03\)** in N44C, while in the Small Magellanic Cloud it ranges from **\(0.29 \pm 0.10\)** to **\(0.35 \pm 0.13\)**. The LMC values cluster near \(\sim 0.2\) dex, whereas the SMC values cluster near \(\sim 0.3\) dex, with low internal dispersion in each galaxy [1610.08497]. In the same dataset, ADF(C\(^{2+}\)) is more uncertain and more dispersed because the carbon CEL abundances came from ultraviolet data, but the SMC values are again larger than the LMC ones [1610.08497].

A broader comparison against oxygen abundance suggests that \(\mathrm{ADF}(\mathrm{O}^{2+})\) is not monotonic with metallicity. The distribution shows a minimum around
\[
12 + \log({\rm O/H}) \sim 8.5,
\]
with larger ADFs at both lower and higher metallicity, producing the “seagull shape” emphasized in the review literature. The low-metallicity rise appears clearer than the high-metallicity branch, which is strongly affected by dispersion and by the confounding role of low-ionization objects [1612.03633]. This metallicity dependence is one reason the Magellanic-Cloud measurements are important: the SMC falls precisely in the regime where elevated ADFs are observed [1610.08497].

Spatially resolved work shows that the H II-region ADF can be either fairly smooth or strongly environment-dependent. In long-slit data for M8 and M17, \(\mathrm{ADF}(\mathrm{O}^{2+})\) remains fairly constant along the observed slit positions, with mean values around **0.37–0.40 dex**; by contrast, the integrated spectrum of NGC 7635 yields a remarkably high value of about **0.59 dex** [1003.2567]. The first resolved ADF(\(\mathrm{O}^{2+}\)) map of an H II region, for M8 from SDSS-V/LVM, gives a global mean of **\(\sim 0.47 \pm 0.02\) dex** and radial variations between **\(\sim 0.35\)** and **\(\sim 0.50\)** in the Her 36-dominated central region [2603.22382]. At still smaller scales, proplyd spectroscopy in Orion shows that dense ionized structures can strongly bias CEL diagnostics: once the high density of proplyd 177-341 is treated properly and the background is subtracted, the intrinsic \(\mathrm{ADF}(\mathrm{O}^{2+})\) is about zero, whereas the surrounding nebular background retains the usual positive ADF [1205.0509].

The comparison with young stars is mixed and environment-dependent. In Orion, RL-based C/H and O/H are more consistent with solar and stellar abundances than CEL-based values [1612.03633]. In the Magellanic Clouds, by contrast, B-type stellar abundances agree better with nebular CEL-based oxygen abundances than with RL-based ones, and RL-based carbon can be too high relative to the stellar benchmark if a dust correction is included [1610.08497]. The review literature summarizes this as a possible metallicity-dependent reversal: RL-based nebular metallicities appear to agree better with stellar metallicities in the high-abundance regime, whereas CEL-based metallicities agree better at low abundance [1612.03633].

Far-infrared tests complicate any universal interpretation. In Mrk 71, the far-IR [O III] abundance agrees with the optical CEL abundance and not with the RL abundance, while the directly inferred \(t^2\) is consistent with zero. For that object, the long-standing temperature-fluctuation explanation of the ADF is ruled out [2304.09898].

## 4. Planetary nebulae

Planetary nebulae exhibit the full dynamic range of ADF phenomenology. Ordinary planetary nebulae typically have ADFs of order **2–3**, but a minority reach values many times larger, extending to **2–3 orders of magnitude** in extreme cases [2201.05627]. The contrast with H II regions is therefore fundamental: planetary nebulae include an extreme-ADF population not seen in normal H II-region samples [1612.03633].

Not all PNe with unusual central stars show extreme discrepancies. In deep studies of [WC] and wels nebulae, \(\mathrm{ADF}(\mathrm{O}^{++})\) is moderate, in the range **1.2 to 4**, and no correlation is found with surface brightness, diameter, excitation class, density, metallicity, N/O, or [WC] spectral type [1309.1663]. A preliminary MIKE survey of a dozen [WC]/wels PNe likewise found oxygen ADFs always below 4 and no direct evidence for the cold, dense H-poor clumps proposed in some earlier interpretations [1109.1957]. Young high-density nebulae such as Vy 2-2, Hu 2-1, Vy 1-2, and IC 4997 occupy an intermediate regime, with \(\mathrm{ADF}(\mathrm{O}^{+2})\) between **\(1.85 \pm 1.05\)** and **\(5.34^{+1.27}_{-1.08}\)** [2201.00902].

The extreme tail is represented by objects such as NGC 6153, NGC 6778, Hf 2-2, M 1-42, A30, and Sp 3. NGC 6153 has a characteristic **ADF \(\sim 10\)** and was used as a benchmark for explicit multiphase abundance analysis [2407.06385]. NGC 6778 shows an integrated \(\mathrm{adf}(\mathrm{O}^{2+})\) of **17.9** and rises to about **40** close to the central star [1510.07516]. Sp 3 has an extreme oxygen ADF of **\(24.6^{+4.1}_{-3.4}\)** despite a **4.81 d** central-binary period, which is inconsistent with the purported trend that longer-period post-common-envelope PNe should exhibit normal ADFs [1908.08724]. Hf 2-2 and M 1-42 have long been treated as canonical high-ADF nebulae with values of about **70** and **20**, respectively, and resolved studies find local \(\mathrm{ADF}(\mathrm{O}^{2+})\) values as high as \(\sim 300\) in Hf 2-2 and \(>100\) in M 1-42 near the central sightlines [2509.16201]. In the born-again planetary nebula A30, the polar knots reach an ADF of about **700**, whereas the equatorial knot J4 yields only \(\mathrm{ADF}(\mathrm{O}^{2+}) \sim 22\) and a conservative upper limit of **35**, demonstrating strong internal segregation of the discrepancy [2201.05627].

Spatially resolved IFU and echelle spectroscopy show that these extreme ADFs are usually centrally peaked and associated with distinct ORL-emitting plasma. MUSE maps of NGC 6778, M 1-42, and Hf 2-2 show that both \(\mathrm{ADF}(\mathrm{O}^{+})\) and \(\mathrm{ADF}(\mathrm{O}^{2+})\) peak toward the center, where recombination-based temperatures are lowest and heavy-element ORLs are strongest [2112.00480]. UVES PV analyses of Hf 2-2 and M 1-42 confirm two plasma components: a normal nebular plasma that emits both forbidden and permitted lines, and an additional, denser, cooler, centrally concentrated component that emits the permitted lines of O I, C II, N II, O II, and Ne II [2509.16201]. A complementary UVES reanalysis of Hf 2-2, M 1-42, and NGC 6153 finds CEL temperatures near \(10^4\) K, Balmer and Paschen jump temperatures below that, and O II and N II ORL temperatures \(<2000\) K, with ORL-emitting regions close to the nebular center [2512.04533].

## 5. Physical interpretations and discriminants

The classical explanation is **spatial temperature fluctuations**, parameterized by \(t^2\) in the Peimbert formalism. In this picture, RL abundances are closer to the true abundances because their emissivities are almost identical to that of H\(\beta\) over the usual H II-region temperature range, whereas CEL emissivities are strongly temperature-sensitive and therefore biased by unresolved thermal structure [1612.03633]. This framework remains central in both H II-region and PN work.

Other interpretations remain active. The review literature discusses the \(\kappa\)-distribution hypothesis, but also notes that this idea has been seriously questioned [1612.03633]. A chemically inhomogeneous scenario due to semi-ionized, dense, cool, metal-rich clumps interpreted as unmixed supernova ejecta has also been proposed for H II regions, with the distinct implication that RL and CEL abundances bracket the true value rather than tracing the same plasma [1612.03633]. A related idea invokes high-density clumps without abundance contrast [1612.03633]. In H II regions, very dense ionized structures such as proplyds show directly that collisional de-excitation can depress CEL abundances so strongly that the intrinsic ADF tends to zero once the correct density structure is used [1205.0509].

For planetary nebulae with high ADFs, the weight of the resolved evidence favors explicit **multiphase gas**. In NGC 6153, a tailored two-phase analysis separates a warm CEL-emitting phase from a cold RL-emitting phase and introduces the **abundance contrast factor** (ACF), defined as the abundance ratio between the two plasma components once the H\(\beta\) partition is accounted for; the ACF is, on average, **0.9 dex higher than the ADF** [2407.06385]. MUSE and UVES studies of NGC 6153, Hf 2-2, M 1-42, and NGC 6778 consistently find centrally concentrated ORL emission, lower temperatures from recombination diagnostics, and kinematic differences between CEL- and ORL-emitting zones, supporting a cold, metal-rich component embedded within the normal nebular plasma [2112.00480; 2407.06385; 2512.04533]. In Hf 2-2 and M 1-42, the additional plasma component contains masses of \(\mathrm{N}^{2+}\) and \(\mathrm{O}^{2+}\) ions at least as large as those in the normal nebular plasma, which the authors identify as the main reason those nebulae have larger ADFs than NGC 6153 [2509.16201].

Not all fluctuation-based models reproduce the full phenomenology. Simulations with realistic continuous density and temperature fluctuations can reproduce filling factors in the observed range and may be consistent with H II-region ADFs, but they do not satisfactorily reproduce the ADFs of planetary nebulae, especially the extreme tail. In those models, negative density-temperature correlation can increase the optical ADF, but it also tends to drive the inferred filling factor above unity, which is not generally observed [1912.05542].

The empirical relation to central-star evolution is also significant. Extreme ADFs are increasingly associated with close binaries and with born-again systems. NGC 6778 strengthens the link between high ADFs and central-star binarity [1510.07516], while Sp 3 shows that extreme ADFs are not confined to the shortest post-common-envelope periods and that selection effects likely affect any simple period-ADF trend [1908.08724]. The review literature therefore treats low-ionization, dynamically disturbed, or binary-shaped nebulae as potentially special environments in which the ADF is amplified [1612.03633].

## 6. Abundance scales, ratios, and current status

The ADF has immediate consequences for the abundance scale itself. If RL and CEL methods differ systematically by **0.1–0.35 dex** in H II regions, and by much more in extreme PNe, then absolute O/H and C/H abundances are method-dependent and require an explicit physical choice [1612.03633]. In some low-metallicity H II regions, stellar benchmarks and far-IR tests support the CEL scale over the RL scale [1610.08497; 2304.09898]. In Orion and some higher-metallicity environments, RLs are favored by the comparison with local stellar and solar abundances [1612.03633]. The review conclusion is therefore deliberately non-universal: the best abundance indicator may depend on metallicity regime and nebular environment [1612.03633].

At the same time, some abundance ratios are much more stable than absolute abundances. New ultraviolet-plus-optical work finds a clear ADF for \(\mathrm{C}^{2+}\), with
\[
\langle ADF(C^{2+})\rangle = 0.40\pm0.02~\mathrm{dex},
\qquad
\langle ADF(O^{2+})\rangle = 0.35\pm0.02~\mathrm{dex},
\]
and shows that UV CEL-based and optical RL-based \(\mathrm{C}^{2+}/\mathrm{O}^{2+}\) agree to within about **\(0.05\pm0.03\) dex** and generally within **\(<0.1\) dex** object by object [2508.03954]. Absolute C/H and O/H remain uncertain at the ADF level, but C/O is comparatively robust if carbon and oxygen suffer similar discrepancies [2508.03954].

The present observational picture is therefore stratified by object class. In H II regions, the ADF is systematic but usually modest, often smooth on resolved scales, and plausibly linked to metallicity, temperature structure, density inhomogeneities, or combinations thereof [1612.03633; 2603.22382]. In planetary nebulae, especially the high-ADF subset, the discrepancy is frequently spatially and kinematically segregated and is most naturally described as a multiphase problem involving a cold, metal-rich component [2112.00480; 2407.06385; 2512.04533]. The review literature explicitly warns against a single universal mechanism and argues that the “story may be more complex than a single universal mechanism” [1612.03633].

Taken together, the ADF remains a major unsolved problem in ionized-nebula astrophysics, but it is no longer merely an integrated ORL/CEL mismatch. It is an empirical tracer of the thermal, chemical, and dynamical structure of nebulae, and it defines the conditions under which absolute nebular abundances can—or cannot—be placed on a common scale across local H II regions, planetary nebulae, and high-redshift galaxy spectroscopy [1612.03633; 2304.09898; 2508.03954].

Source: https://www.emergentmind.com/topics/abundance-discrepancy-factor-adf