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Dense gas clumps skew galaxy metallicity measurements

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-07 19:00 UTC pith:2EINOOZS

load-bearing objection Systematic comparison of O+ and O2+ abundance prescriptions on combined SITELLE+MUSE data; the T_e,[N II]=T_e,[O II] assumption is circular but the downstream sensitivity appears modest. the 1 major comments →

arxiv 2607.05295 v1 pith:2EINOOZS submitted 2026-07-06 astro-ph.GA

Toward Unbiased Abundance Measurements in Inhomogeneous H\,II Regions

Eric Habjan , Christopher Faesi , Kathryn Kreckel , J. Eduardo M\'endez-Delgado , Francesco Belfiore , Ryan J. Vaught , Brent Groves , Fabian Scheuermann
show 5 more authors
Thomas G. Williams Ralf S. Klessen Amirnezam Amiri Kathryn Grasha Simon Glover
This is my paper
classification astro-ph.GA
keywords HII regionschemical abundanceselectron densityelectron temperatureabundance discrepancy factordensity inhomogeneitiesoxygen abundancecollisionally excited lines
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper argues that the standard method for measuring electron densities in HII regions — the [S II] nebular doublet ratio — systematically underestimates the true density because it is insensitive to dense clumps of gas within these regions. Those same clumps bias the [O II] temperature diagnostic hot, which in turn causes singly ionized oxygen abundances to be underestimated by roughly 0.6 dex. The authors show that by instead using the [O II] auroral-to-nebular line ratio as a density diagnostic (which remains sensitive to gas up to ~10^5 cm^-3) combined with the [N II] electron temperature (which is insensitive to density inhomogeneities), one recovers O+ abundances consistent with recombination-line expectations. For doubly ionized oxygen, they show that the [S III] temperature serves as a reliable proxy for T_e,[O III], and that the empirical temperature T_0(O2+) from recombination-line calibrations produces O2+ abundances that minimize temperature-inhomogeneity biases. Together, these prescriptions reduce the abundance discrepancy between collisionally excited and recombination lines to a few tenths of a dex, rather than the order-of-magnitude offsets that have historically plagued direct abundance measurements. The authors also find a tentative correlation between a proxy for the abundance discrepancy factor and metallicity, suggesting that chemically evolved HII regions may host larger abundance discrepancies.

Core claim

The central mechanism is that density inhomogeneities — dense clumps within HII regions — suppress nebular emission lines like [S II] and [O II]λ3727 (which have critical densities of ~10^3 cm^-3) while leaving auroral lines like [O II]λ7325 (sensitive up to ~10^5 cm^-3) relatively unaffected. This differential suppression causes the [S II] density diagnostic to underestimate the true average density by ~10^3 cm^-3, and when that underestimated density is fed into the [O II] temperature calculation, the temperature is overestimated by ~0.3×10^4 K, cascading into a ~0.6 dex underestimate of O+ abundance. The paper demonstrates that swapping in the [O II] auroral-to-nebular ratio as the密度diagn

What carries the argument

The [O II] auroral-to-nebular line ratio (λ7325/λ3727) serves as both a density diagnostic (when combined with T_e,[N II]) and a temperature diagnostic, but its behavior differs dramatically depending on which density is assumed: paired with n_e,[S II] it yields biased temperatures, while paired with T_e,[N II] it yields densities insensitive to inhomogeneities. The [S III] temperature and the empirical T_0(O2+) calibration serve as the corresponding inhomogeneity-insensitive temperature diagnostics for the O2+ zone.

Load-bearing premise

The derivation of the [O II] auroral-to-nebular density assumes that the [N II] and [O II] zones share the same electron temperature (T_e,[N II] = T_e,[O II]). The paper's own data show these temperatures are only weakly correlated (ρ = 0.38) and differ systematically by ~0.3×10^4 K. If this equality fails for individual HII regions with density inhomogeneities, the derived densities — and hence the reference O+ abundances — would be biased.

What would settle it

If simultaneous recombination-line and collisionally-excited-line measurements of O+ in the same HII regions show that the reference prescription (T_e,[N II], n_e,[OII]λ7325/λ3727, [O II]λ3727) still systematically underestimates O+ by more than ~0.2 dex, the claim that this prescription is insensitive to inhomogeneities would fail.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Existing PHANGS-MUSE oxygen abundances derived with MUSE-only lines ([O II]λ7325, n_e,[S II], T_e,[O III]) may appear to agree with recombination-line calibrations by coincidence — compensating biases in O+ and O2+ cancel — rather than through physical correctness.
  • The common assumption that log(N/O) = log(N+/O+) may be off by ~0.08 dex, with N+/O+ ≈ 0.84 × N/O in this sample, affecting nitrogen-to-oxygen ratio studies across nearby galaxies.
  • If the tentative correlation between the proxy ADF and metallicity holds, the abundance discrepancy problem would be most severe in metal-rich HII regions — precisely where many strong-line calibrations are anchored.
  • Galaxy-to-galaxy variation in n_e,[OII] and T_e,[N II] suggests that HII regions within a single galaxy share similar physical conditions, which could systematize abundance biases on galactic scales.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If dense clumps are the root cause, the proxy ADF should correlate with the fraction of [O II]λ3727 flux suppressed by collisional de-excitation — testable by comparing auroral-to-nebular ratios across HII regions with independently measured density structure (e.g., from radio recombination lines or high-resolution dust maps).
  • The assumption T_e,[N II] = T_e,[O II] used to derive n_e,[OII]λ7325/λ3727 is circular in a weak sense: the density diagnostic depends on a temperature equality that the paper's own data show is violated (ρ = 0.38, systematic offset ~0.3×10^4 K). A self-consistent iterative scheme — solving for n_e and T_e simultaneously without imposing the equality — could test whether the recovered abundances a
  • The galaxy-to-galaxy segregation in density and temperature suggests that strong-line calibrations trained on one galaxy sample may carry environment-dependent biases when applied to others, especially at high metallicity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 8 minor

Summary. This manuscript investigates how density and temperature inhomogeneities in H II regions bias direct-method oxygen abundance determinations. Combining [O II] measurements from SITELLE with a full optical suite from MUSE for five nearby galaxies, the authors derive electron temperatures and densities using auroral lines ([N II]λ5755, [S III]λ6312, [O II]λλ7320,7330). They propose a 'reference' O+ abundance prescription using T_e,[N II], n_e,[OII] (from the [O II] auroral-to-nebular ratio), and [O II]λ3727, arguing it is less sensitive to inhomogeneities than standard prescriptions using T_e,[O II] and n_e,[S II]. They also validate T_e,[S III] as a proxy for T_e,[O III] and explore a proxy ADF–metallicity correlation. The analysis is methodologically careful, with Monte Carlo error propagation and quality checks against literature calibrations.

Significance. The paper addresses a timely and important problem: the abundance discrepancy factor (ADF) and its connection to ISM inhomogeneities. The combination of SITELLE [O II] data with PHANGS-MUSE is a genuine observational advance. The quality checks ([O II]λ7320/7330 ratio comparison, [S III] temperature validation against Rogers+2021) are well-executed and lend credibility to the measurements. The identification of a potential proxy ADF–metallicity correlation, if confirmed, would be a meaningful result. However, the central claim—that the reference O+ prescription is 'insensitive to density inhomogeneities'—rests on an assumption (T_e,[N II] = T_e,[O II]) that the paper's own data contradict, and the sensitivity of the downstream abundance to this assumption is not quantified. This gap limits the strength of the conclusions at present.

major comments (1)
  1. §3.1.1, paragraph beginning 'We derive n_e,[OII]λ7325/λ3727...': The density n_e,[OII] is derived from the [O II]λ7325/λ3727 ratio by fixing T_e = T_e,[N II] under the assumption T_e,[N II] = T_e,[O II]. The paper's own data contradict this equality: Table 2 gives T_e,[N II] = 0.15(±0.05)×T_e,[O II] + 0.63(±0.15) with ρ = 0.38±0.09, and §3.1.3 reports that T_e,[O II] is on average ~0.3×10^4 K higher than T_e,[N II]. The paper argues this offset is caused by density inhomogeneities biasing T_e,[O II] high, which is a reasonable physical argument, but it creates a circular structure: T_e,[N II] is used both to derive n_e,[OII] and to compute the final O+ abundance. Critically, the paper never tests how sensitive n_e,[OII]—or the downstream reference O+ abundance—is to the assumed temperature. The ~0.06 dex shift observed when swapping n_e,[OII] for n_e,[S II] (§4.1.2) bounds the density-sS
minor comments (8)
  1. §3.1.1: The diagnostic is referred to as both '[O II]λ7325/λ3727' and '[O II]λ7235/λ3727' (with transposed digits) in the same paragraph. Fix the typo.
  2. Table 2 caption: The T_e,[S III]–T_e,[N II] fit gives a slope of 0.92±0.07, but §3.1.3 states 'T_e,[S III] ~ 0.95×T_e,[N II]'. These are consistent but the slight inconsistency in the quoted coefficient should be reconciled.
  3. §3.2.2: 'The average gas temperature T_0(O2+) is calculated using all 202 T_e,[N II] detections' — please clarify whether the MC uncertainties on T_0(O2+) account for the uncertainty in the MD23a calibration coefficients (Eq. 2), or only for the propagated T_e,[N II] errors.
  4. §4.1.2: The average reference O+/H+ is reported as '8.2 [12 + log(O+/H+)] with a standard deviation of 0.9'. A standard deviation of 0.9 dex is very large; please comment on whether this reflects genuine astrophysical scatter or is dominated by measurement uncertainty.
  5. §5.1: 'the median value for NGC 3627 is ~7×10^4 K' appears to be a typo for ~7×10^3 K, given the context of T_e,[N II] values.
  6. Figure 2: Only three galaxies (NGC 628, NGC 2835, NGC 4535) appear in the T_e–T_e plots. The caption should explicitly state that NGC 3351 and NGC 3627 have no detections, for consistency with Figures 3–5.
  7. §4.3.2, third paragraph: The statement that MUSE-only abundances are '~0.01 dex higher' than reference, followed by 'This close agreement is driven by compensating differences,' is later described as 'coincidental rather than physically motivated.' The text would benefit from a clearer statement of why this cancellation occurs and whether it is expected to hold in other samples.
  8. References: Several arXiv preprints are cited with future dates (e.g., Rosales-Ortega et al. 2026, Habjan et al. 2026). Please ensure these are updated to final published references where available.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for a careful and constructive report. The central concern—quantifying the sensitivity of n_e,[OII] and the downstream reference O+ abundance to the assumed T_e,[N II] = T_e,[O II] equality—is well-taken, and we agree that the manuscript must address it explicitly. Below we respond point by point.

read point-by-point responses

  1. Referee: §3.1.1: The density n_e,[OII] is derived by fixing T_e = T_e,[N II] under the assumption T_e,[N II] = T_e,[O II], but the paper's own data contradict this equality (Table 2; §3.1.3 reports T_e,[O II] ~0.3×10^4 K higher than T_e,[N II] on average). The paper argues this offset is caused by density inhomogeneities biasing T_e,[O II] high, creating a circular structure: T_e,[N II] is used both to derive n_e,[OII] and to compute the final O+ abundance. Critically, the paper never tests how sensitive n_e,[OII]—or the downstream reference O+ abundance—is to the assumed temperature. The ~0.06 dex shift observed when swapping n_e,[OII] for n_e,[S II] (§4.1.2) bounds the density-sensitivity, but no analogous bound is placed on the temperature-sensitivity of n_e,[OII] itself.

    Authors: The referee correctly identifies that the assumption T_e,[N II] = T_e,[O II] is contradicted by our own data, and that the resulting structure—where T_e,[N II] enters both the density derivation and the final abundance—needs explicit justification and sensitivity testing. We agree and will revise the manuscript accordingly. Specifically, we will add a quantitative sensitivity test: we will re-derive n_e,[OII] and the reference O+ abundance using T_e,[O II] (the directly measured, inhomogeneity-biased temperature) instead of T_e,[N II], and report the resulting shift in both n_e,[OII] and O+/H+. This will place a direct bound on how much the downstream abundance depends on the temperature assumption. We expect this shift to be modest relative to the ~0.61 dex effect from using T_e,[O II] directly in the abundance calculation (§4.1.2, rightmost panel of Figure 4), because the density dependence of the [O II]λ7325/λ3727 ratio is logarithmic while the abundance dependence on T_e is exponential—but this must be shown quantitatively rather than asserted. We will also add a new figure or table showing n_e,[OII] derived at T_e,[N II] ± 0.3×10^4 K (the observed offset) to bracket the plausible range. Regarding circularity: we agree the logic is not fully independent and will restructure the discussion to be transparent about this. The physical argument is that T_e,[N II] is the least density-biased temperature available (the [N II] nebular lines have critical densities ~10^5 cm^-3, well above the ~10^3 cm^-3 inhomogeneity scale), so using it as the temperature for the density derivation is the least-biased choice, even if the resulting density and abundance are not fully independent. We will state this explicitly rather than leaving it implicit. Finally, we will soften the claim revision: yes

Circularity Check

0 steps flagged

No significant circularity found; one minor self-citation chain (MD23a calibration) is acknowledged by the authors as partly driving correlation but is not load-bearing for the central claim.

full rationale

The paper's central claim is that combining T_e,[N II] with n_e,[OII]λ7325/λ3727 and [O II]λ3727 yields O+ abundances 'insensitive to density inhomogeneities,' and that T_e,[S III] reliably estimates T_e,[O III]. Walking the derivation chain: (1) n_e,[OII] is derived from the [O II]λ7325/λ3727 ratio using the assumption T_e,[N II] = T_e,[O II] (Section 3.1.1). This is a physical assumption, not a circular definition—n_e is derived from a different line ratio than the one used to compute T_e,[N II], and the assumption is externally testable (the paper itself tests it via Table 2, finding ρ=0.38). (2) The reference O+ abundance uses T_e,[N II], n_e,[OII], and [O II]λ3727—three independently measured quantities. The O+ abundance is not defined in terms of itself. (3) The comparison with the MD23a calibration (Eq. 2, T_0(O2+) = 1.17×T_e,[N II] − 3340) does create a shared dependence on T_e,[N II], which the authors explicitly acknowledge: 'likely due to both methods having a strong dependence on T_e,[N II].' However, the MD23a calibration is derived from RL-based metallicities in independent H II region spectra (MD23a), not from the present paper's data, so it is not circular by construction—it is an external empirical relation. The ρ=0.91 correlation is partly expected but not forced by construction, since the CEL-derived metallicity also depends on n_e,[OII], [O II]λ3727 fluxes, and the ICF, none of which enter the MD23a calibration. (4) T_e,[O III] is estimated from T_e,[S III] via Eq. 1 (from B24, an independent calibration using CHAOS data), and T_0(O2+) from T_e,[N II] via Eq. 2 (from MD23a). These are independent empirical calibrations, not self-definitions. The paper does not fit any parameter to its own data and then call the fit a prediction. The self-citations to MD23a/b and RV24 (Méndez-Delgado and Rickards Vaught are co-authors) provide methodological context but are not load-bearing for the derivation—the PyNeb calculations stand on their own given the stated assumptions. The T_e,[N II] = T_e,[O II] assumption is a correctness risk (contradicted by the paper's own data), not a circularity: it is an assumption that could be wrong, not a definition that is true by construction. Overall, the derivation is self-contained against external benchmarks, with one minor self-citation that is acknowledged and not load-bearing.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, or forces. It works entirely within the standard framework of CEL-based nebular abundance analysis, using atomic data from PyNeb and standard ICF prescriptions. The free parameters are conventional choices (low-density floor, fitting bounds) rather than physically motivated new parameters.

free parameters (2)
  • Low-density floor for n_e = 100 cm^-3
    When n_e < 10^2 cm^-3 (the low-density limit for both [S II] and [O II] diagnostics), the density is set to 100±100 cm^-3 (Section 3.1.1). This is a standard convention but is a choice that affects ionic abundance calculations.
  • Gaussian width bounds for auroral line fitting = σ_λ,SL < σ_λ < σ_λ,SL + 0.5 Å
    The allowed range for the Gaussian velocity dispersion in the auroral line fitting (Section 2.3) is chosen to exclude unphysically broad lines. The upper bound of +0.5 Å is an ad hoc parameter.
axioms (5)
  • domain assumption T_e,[N II] = T_e,[O II]
    Used in Section 3.1.1 to derive n_e,[OII]λ7325/λ3727 from the [O II] and [N II] auroral-to-nebular ratios. Stated as a prediction of simple photoionization models (Morisset+2015, Vale Asari+2016, Ferland+2017, MD23b, RV24, Caldalti+2025).
  • domain assumption T_e,[S III] reliably estimates T_e,[O III] via Eq. 1
    The empirical relation T_e,[O III] = 0.80×T_e,[S III] + 0.20 from B24 (Eq. 1) is adopted without independent validation in this dataset. Used throughout Section 4.2 for O2+ abundance derivations.
  • domain assumption T_0(O2+) from T_e,[N II] via Eq. 2 represents the temperature at which RLs and CELs agree
    The MD23a calibration (Eq. 2) is adopted as a proxy for RL-derived abundances. The paper's own T_e,[N II] range (0.6-1.3×10^4 K) extends below the MD23a calibration range (0.8-1.3×10^4 K), so the relation is extrapolated for some regions (Section 4.3.2).
  • standard math ICF(O+/O2+) from Izotov+2006 correctly corrects for unseen oxygen ionization stages
    Used in Section 4.3.1 to compute total oxygen abundances. Standard ICF assuming no significant O3+ contribution.
  • domain assumption The [O II]λ7320/7330 ratio is insensitive to density up to n_e ≈ 10^5.5 cm^-3
    Used in Section 2.3 as a quality check; the measured ratio of 1.27±0.03 is compared to the theoretical prediction of 1.24 from Kisielius+2009.

pith-pipeline@v1.1.0-glm · 33656 in / 3396 out tokens · 216336 ms · 2026-07-07T19:00:07.489801+00:00 · methodology

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read the original abstract

Probing the chemical content of the interstellar medium (ISM) in nearby galaxies provides key insight into their chemical evolution and informs our interpretation of galaxies at higher redshift. However, nonlinear structure in the ISM, including density and temperature inhomogeneities, can bias chemical abundance measurements and systematically affect empirical calibrations derived from them. In this work, we investigate biases in $T_e$-derived oxygen abundance determinations and explore the physical properties that correlate with them. We combine $\mathrm{[O\,II]}\lambda\lambda3726, 3729$ measurements from SITELLE with a full suite of optical emission lines obtained with MUSE. From auroral emission lines ($\mathrm{[N\,II]}\lambda5755$, $\mathrm{[S\,III]}\lambda6312$, and $\mathrm{[O\,II]}\lambda\lambda7320, 7330$) and nebular emission lines (including $\mathrm{[N\,II]}\lambda6584$ and $\mathrm{[S\,III]}\lambda9069$), we derive electron densities, temperatures, and chemical abundances for a sample of $\mathrm{H\, II}$ regions in five galaxies. We find that densities derived from the $\mathrm{[O\,II]}$ auroral-to-nebular ratio are $\sim10^3$ cm$^{-3}$, which is higher than the standard $\mathrm{[S\,II]}$ densities derived from nebular doublet ratios. We demonstrate that combining the $\mathrm{[N,II]}$ electron temperature with the density inferred from the $\mathrm{[O\,II]}$ auroral-to-nebular line ratio yields singly ionized oxygen abundances consistent with literature expectations for a prescription insensitive to density inhomogeneities. We also find that the $\mathrm{[S\,III]}$ temperature provides a reliable estimate of $T_{e,\mathrm{[O\,III]}}$, enabling robust measurements of doubly ionized oxygen abundances. Overall, these results indicate that the abundance discrepancy factor could be higher in more chemically evolved $\mathrm{H\, II}$ regions.

Figures

Figures reproduced from arXiv: 2607.05295 by Amirnezam Amiri, Brent Groves, Christopher Faesi, Eric Habjan, Fabian Scheuermann, Francesco Belfiore, J. Eduardo M\'endez-Delgado, Kathryn Grasha, Kathryn Kreckel, Ralf S. Klessen, Ryan J. Vaught, Simon Glover, Thomas G. Williams.

Figure 1
Figure 1. Figure 1: — Example fits of each auroral line from Region 144 in NGC 2835; [N II]λ5755 in blue (S/N ∼ 18), [S III]λ6312 in green (S/N ∼ 12), [O II]λ7320 in orange (S/N ∼ 14), and [O II]λ7330 in red (S/N ∼ 13). The MUSE DAP nebular spectrum is shown as the black line and each panel shows the fits for each of the four auroral lines in our spectral range. Each of these fits were done using a Gaussian with an added line… view at source ↗
Figure 2
Figure 2. Figure 2: — Each of our directly measured electron temperatures plotted against each other. NGC 628 temperatures are shown in red, NGC 2835 in black and NGC4535 in green. There are no detections for NGC 3351 and NGC 3627. The dashed gold lines are the WLS fits to the Te – Te relationships, which are shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: — ne,[S II] plotted against ne,[OII]λ7325/λ3727. NGC 628 is shown in red, NGC 2835 in black, and NGC 4535 in green. There are no detections for NGC 3351 and NGC 3627. The WLS fit is in gold and the shaded region represents the 1 σ uncertainty. The Spearman rank correlation coefficient ρ is shown in the top left corner and a 1-1 line is plotted in black. MD23b. Another potential explanation for this factor … view at source ↗
Figure 4
Figure 4. Figure 4: — The reference O+ abundance derived using Te,[N II], [O II]λ3727 and ne,[OII]λ7325/λ3727 is shown on the y-axis plotted against our three other O+ abundances described in Section 4.1. The electron temperature, density, and emission lines used for each O+ on the x-axis are printed in the bottom right corner of each panel. NGC 628 O+ abundances are shown in red, NGC 2835 in black, and NGC4535 in green. Ther… view at source ↗
Figure 5
Figure 5. Figure 5: — The reference O2+ abundance derived using T0(O2+) and ne,[OII]λ7325/λ3727 is shown on the y-axis plotted against our three other O2+ abundances described in Section 4.2. The electron temperature, density, and emission lines used for each O2+ on the x-axis are printed in the bottom right corner of each panel. NGC 628 O2+ abundances are shown in red, NGC 2835 in black, and NGC4535 in green. There are no de… view at source ↗
Figure 6
Figure 6. Figure 6: — The MD23a metallicity calibration is on the y-axis is plotted against CEL direct metallicities. The electron temperature, density, and emission lines used for each metallicity on the x-axis is printed in the bottom right corner of each panel. The MD23a calibration plotted against our CEL direct abundances gives a proxy for the discrepancy between RL and CEL derived abundances. NGC 628 metallicites are sh… view at source ↗
Figure 7
Figure 7. Figure 7: — On the y-axis N+/O+ (ne,[OII]λ7325/λ3727) is plotted against N+/O+ (ne,[S II]) in the left panel and the elemental nitrogen abundance N/O in the right panel. NGC 628 H ii region data points are shown in red, NGC 2835 in black, and NGC4535 in green; no NGC 3351 or NGC 3627 H ii regions had detections. A linear polynomial is fit to each relation, shown by the gold dotted line, with the 1 σ uncertainty in t… view at source ↗
Figure 8
Figure 8. Figure 8: — In the left panel, N+/O+ (ne,[S II]) is plotted against the second direct metallicity in [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: — On the y-axis we plot the proxy ADF, defined as the ratio of O2+ (T0(O2+), ne,[OII]λ7325/λ3727) and O2+ (Te,[O III], ne,[OII]λ7325/λ3727). The proxy ADF is plotted against ne,[S II] and colored by the MD23a metallicity calibration. The Spearman rank correlation coefficient in shown in the upper left corner of each plot. 5.3. Temperature Inhomogeneities The [O III]λ4363 auroral line is outside the spectra… view at source ↗

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