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$R_{\rm e}$. II. Understanding the IC 3475 galaxy type, including ultra-diffuse galaxy, structural scaling relations

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Ultra-diffuse galaxies are the faint tail of the early-type galaxy sequence, not a separate class.

desk verdict Graham makes a real contribution by quantifying the Sersic-based case that UDGs are faint ETGs, but the main 'expectation' curve is partly calibrated on the UDGs it claims to explain. read the letter →

arxiv 2504.16593 v2 pith:UZVFTLRR submitted 2025-04-23 astro-ph.GA

Alister W. Graham This is my paper · ORCID
classification astro-ph.GA
keywords ultra-diffusegalaxiesearly-typeSersicprofilessize-luminosityrelationsurfacebrightnessdwarfgalaxyscalingrelationsIC3475
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that relatively gas-poor ultra-diffuse galaxies are not anomalously large but simply occupy the faint end of the curved size–luminosity relation of early-type galaxies. Their large effective radii follow automatically from the Sérsic model once one knows how central surface brightness and Sérsic index vary with absolute magnitude across the whole sequence. The paper also explains the growing scatter toward faint magnitudes in the magnitude–size plane and the smaller scatter in magnitude–isophotal-radius diagrams as consequences of the same relations. The stakes are classification and formation: if the claim holds, UDGs do not need a separate formation channel such as failed massive haloes or tidally puffed dwarfs, and apparent two-class splits among UDGs are selection artefacts rather than evidence for distinct origins. That matters for interpreting the many low-surface-brightness galaxies that upcoming deep surveys will deliver.

What carries the argument

The central machinery is the Sérsic $R^{1/n}$ light-profile model, used as a bridge between three pieces: the empirical magnitude–central surface brightness relation, the empirical magnitude–Sérsic-index relation, and the analytic identity $\langle\mu\rangle_e = \mu_0 + 2.5b/\ln 10 - 2.5\log f(n)$, with $f(n)=ne^b\Gamma(2n)/b^{2n}$ and $b(n)$ fixed by enclosing half the light. Feeding the empirical relations into the magnitude–size equation yields a derived curve, Equation (10), that is not a fit: $\log R_{e,\rm eq} = 0.139M_B - 0.5\log[f(n)] + 0.217b + 1.812$, with $n = 10^{-(14.0+M_B)/10}$ from the magnitude–index relation. The same equations convert offsets in $n$ and $\mu_e$ at fixed magnitude into offsets in $\mu_0$ and hence into changes in $R_e$, which is the mechanism that explains the scatter and the apparent UDG subclasses.

What would settle it

Take a sample of faint dwarf galaxies selected purely by luminosity, with no cut on surface brightness or size, and measure their Sérsic parameters. If the $M_B$\u2013$\log n$ or $M_B$\u2013$\mu_0$ relations turn over or steepen fainter than about $M_B = -16$ mag, then the derived $R_e(M_B)$ curve is not the correct baseline and the claim that UDGs have expected sizes loses its footing. Alternatively, a single UDG whose measured $n$ and $\mu_0$ cannot reproduce its observed $R_e$ through the derived relation would falsify the reduction.

Watch

Extended reading notes

Core claim

The paper claims that relatively gas-poor ultra-diffuse galaxies are not a distinct class with unexpectedly large sizes. They are the faint extension of the same early-type-galaxy sequence: given the empirical relations between absolute magnitude and central surface brightness, and between absolute magnitude and Sérsic index, the Sérsic $R^{1/n}$ model predicts the observed large effective radii without any additional formation mechanism. The paper further claims that the scatter about the magnitude–size relation is the mapped consequence of the scatter in $\mu_0$ at fixed magnitude, driven by offsets in both $n$ and effective surface brightness $\mu_e$, and that the proposed split of gas-poor UDGs into two classes reflects this scatter rather than two genuinely different origins. It also attributes the reduced scatter in magnitude–isophotal-radius diagrams to the insensitivity of faint isophotal radii to exactly those perturbations in $n$ and $\mu_e$ that move $R_e$.

Load-bearing premise

The load-bearing premise is that the linear magnitude–surface-brightness and magnitude–Sérsic-index relations, anchored on brighter early-type galaxies, continue unchanged into the ultra-diffuse regime; because the index relation was refit after adding the UDG data, the predicted size curve is not fully independent of the galaxies it is used to validate.

Editorial extensions

If this is right

  • If UDGs are the faint end of the ETG sequence, then the UDG designation thresholds (for example, $R_e \gtrsim 1.5$ kpc and $\mu_0 \gtrsim 24$ mag arcsec$^{-2}$) mark a portion of one continuous distribution rather than a physically new class.
  • The size–magnitude trend seen within surface-brightness-selected UDG samples is produced by the selection slice and can run roughly orthogonal to the true ETG relation, so it should not be read as evidence for a distinct scaling law.
  • The larger scatter in dynamical mass indicators such as $\sigma^2 R_e / G$ at faint magnitudes, and correlations with globular-cluster richness, trace the same $\mu_0$ scatter; arbitrary sample divisions can create apparent mass-dependent trends.
  • Isophotal radii measured at a sufficiently faint surface brightness level scatter less than effective radii at fixed magnitude, making them a stabler size indicator for faint ETGs and UDGs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable consequence of the continuity claim is that a large census of dwarf galaxies selected by luminosity alone should reproduce the same linear $M_B$\u2013$\log n$ and $M_B$\u2013$\mu_0$ relations at $M_B \gtrsim -15$ mag, with no break or second sequence.
  • If the explanation of UDG size via low $n$ and faint $\mu_0$ is correct, gas-rich UDGs with ongoing star formation may follow the same structural relations but shifted in $\mu_0$ because young stellar populations or disc structure alter the central light profile; the paper only gestures at this through its H I-bearing UDG comparison.
  • The scatter-decomposition method could be applied to the mass–size plane at fixed stellar mass rather than luminosity, which would separate the role of mass-to-light ratio variations from structural scatter.
  • Forward-modelling the $\mu_0$\u2013$R_e$ selection cuts could turn the paper's qualitative selection-bias argument into a quantitative completeness correction for deep surveys.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper argues that gas-poor ultra-diffuse galaxies (UDGs) are not a distinct class with unexpectedly large sizes, but rather the low-luminosity continuation of the early-type galaxy (ETG) sequence. Using the Sersic R^1/n model together with empirical M_B-mu0,B and M_B-log n relations, the author derives a curved M_B-R_e relation (Equations 10 and 13) and claims that UDGs and NUDGs lie on this curve. The paper further explains the scatter about the M_B-R_e relation through scatter in n and mu_e, explains the smaller scatter in M_B-R_iso diagrams, and argues that the two UDG classes defined by Buzzo et al. (2025) are an artifact of slicing the intrinsic scatter about the M_B-mu0 relation. The work is presented as a follow-up to the author's Paper I framework.

Significance. If the claims hold, the paper provides a useful unification: UDGs would not require separate formation channels, and their large effective radii would follow from Sersic profiles plus the empirical M_B-mu0 and M_B-log n relations. The algebraic derivation from Equations 3 to 10 is internally consistent and is a genuine strength, and the discussion of sample-selection effects in the M_B-R_e diagram (Figure 2c versus 1c) is a valuable cautionary contribution. The central weakness is that the derived 'expected' curve is partly calibrated on the UDG/NUDG sample that it is then used to validate; the paper's abstract describes the baseline as defined by brighter early-type galaxies, which is not strictly what the analysis does.

major comments (3)
  1. [Section 3, Equations (9) and (10); Figure 2c] The central 'expected' size-luminosity curve is not independent of the UDG/NUDG sample it is used to validate. Equation 9 is introduced as a slightly revised version of Equation 1 that 'better matches the fuller ensemble' after the UDG and NUDG data are added, and Equation 10 is then derived from Equation 9. The abstract's phrasing that UDG sizes are 'in line with expectations from ... brighter early-type galaxies' is therefore only partially supported: the brighter-ETG baseline has been recalibrated with the very objects whose consistency is the conclusion. At a central surface brightness typical of UDGs (mu0,B around 26 mag arcsec^-2), Equation 9 gives M_B about 0.4 mag fainter than Equation 1, moving log R_e by roughly 0.06 dex through the 0.139 M_B term in Equation 10; the shift is small but is set by the same sample. Please recompute the curve using only the ETG sample (e.g., Equation 1 or a refit with UDG/NUDG points excluded), extrapolate to the faint regime, and report the residuals of the UDG/NUDG points about that curve. If the revised relation is retained, quantify how much of the apparent agreement is due to the refit.
  2. [Section 3, Equations (1), (2), (9)-(10)] The derivation assumes that the M_B-mu0,B and M_B-log n relations are linear and universal across the full magnitude range, including the faint UDG regime. This assumption is load-bearing and is not tested. Because the UDG selection cuts (R_e,maj >= 1.5 kpc and mu0,g >= 24 mag arcsec^-2) slice the parameter space at faint magnitudes, a curvature or a change of slope in either relation at the faint end could be absorbed into Equation 9 and propagated into the derived curve. Please test for such bending by fitting the relations to the brighter ETG sample alone and extrapolating, and by reporting binned residuals in the UDG regime. If a bend is present, the continuity conclusion would need to be revised.
  3. [Figure 2c and Section 4.1] The claim that there is no distinct separation between IC 3475-type galaxies, UDGs, and dwarf/ordinary ETGs is currently supported by visual inspection and by the placement of the derived curve. A quantitative statement is needed: report the RMS scatter of the UDG/NUDG points about the derived M_B-R_e curve, and also about the ETG-only extrapolation, ideally compared with the scatter of the brighter ETGs at the same magnitudes. The rebuttal of the two UDG classes would also be strengthened by showing that the residuals of Class A and Class B objects about Equations (9) and (2) are statistically indistinguishable.
minor comments (4)
  1. [Section 2, paragraph introducing the UDG/NUDG designation] The sentence beginning 'Galaxies from from Buzzo et al. (2024, 2025)' contains a duplicated 'from' and an unmatched parenthesis; please correct this.
  2. [Throughout, especially Section 4] There are several typographical errors, including 'propogate', 'dynamical massses', and 'Triangal i'; these should be fixed during proofreading.
  3. [Figures 2 and 3, panel labels] The label 'derived (not a fit)' is technically true of the Sersic algebra, but because the input M_B-mu0 relation is refit to the full sample in Equation 9, the label may mislead readers. Consider using 'derived from fits to other relations' or explicitly listing Equations 9 and 2 as the inputs.
  4. [Equations (3) and (4)] The (1+z)^4 surface-brightness correction is mentioned but then disappears from the subsequent equations; please state explicitly that the cosmological correction is negligible for the nearby sample or carry the term through the derivation for clarity.

Circularity Check

1 steps flagged · score 5.0 of 10

Expected Re curve is not independent of the UDGs: Eq. 9 is refit after adding UDG/NUDG data, then used to derive Eq. 10.

  1. fitted input called prediction [Section 3, paragraph introducing Equations 9 and 10]
    "The following slightly revised version of Equation 1 better matches the fuller ensemble of data shown in Figure 2a: MB = 0.59µ0,B,obs− 28.1,σ = 0.8mag. (9) By feeding this empirical relation into Equations 4 and 6, one derives the following relation shown by the black curve in Figure 2c. log Re,eq(kpc) = 0.139MB− 0.5 log[f (n)] + 0.217b + 1.812, (10) with n = 10−(14.0+MB)/10.0 coming from Equation 2."

    Equation 10, the curve used to show that UDG sizes are 'in line with expectations', is obtained by substituting Equation 9 into the Sersic equations. Equation 9 is not the bright-ETG relation (Equation 1) but a 'slightly revised version' that 'better matches the fuller ensemble', i.e., it is refit after the UDG/NUDG data are added. Thus the 'expected' Re(M) is a transform of a fit that already includes the target galaxies; any systematic offset of UDGs in the M-mu0 plane is absorbed into the baseline and then announced as agreement. The bright-only Equation 1 would shift MB by about -0.36 mag at mu0 ~ 24 (log Re by ~0.07 dex), but this independent test is not shown. The prediction is therefore partially forced by construction.

full rationale

The paper's central quantitative step is the derivation of the size-luminosity curve from the Sersic model plus empirical M-mu0 and M-log n relations. Equation 10 is not a fit to Re, which is good; however, Equation 9 is explicitly revised to 'better match the fuller ensemble' of Figure 2a, which includes the UDG/NUDG sample. Feeding that revised relation into the Sersic equations means the 'expected' Re curve is partly calibrated with the same galaxies whose consistency with the curve is the conclusion. The unrevised Equation 1 differs from Equation 9 by about 0.36 mag in MB at mu0 ~ 24, moving log Re by about 0.07 dex toward the UDGs, so the effect is modest and the qualitative continuity claim would likely survive; but the paper never presents the bright-only extrapolation test. There is also no evidence that Equation 2 (M-log n) was refit after adding UDGs, so the n-relation and Sersic geometry provide independent content. Therefore the circularity is partial: the quantitative prediction is partly forced, not fully tautological. No other load-bearing self-citations or uniqueness imports were found; the self-citations to Paper I and Graham et al. (2006) are used for context and previous derivation, not to forbid alternatives. Score 5 reflects partial construction-based prediction, not a fully circular derivation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim depends on empirical fits (Equations 2 and 9) and the Sersic model, plus assumed photometric transformations. No new physical entities are introduced. The key structural weakness is that one of the defining fits was updated using the very UDG data the paper explains.

free parameters (3)
  • Slope and intercept of the MB-mu0,B,obs relation = 0.59 and -28.1 (Equation 9); earlier 0.63 and -28.7 (Equation 1)
    Fitted to the ETG and UDG data in Figure 2a; the central size-luminosity curve is derived from this fit, so the match between UDGs and the curve is partly set by this fit.
  • Slope and intercept of the MB-log(n) relation = -10.0 and -14.0 (Equation 2)
    Empirical fit from Graham and Guzman 2003; used with Equation 9 to derive Equation 10.
  • Scatter sigma about the MB-mu0 relation = 0.8 mag
    Used to generate the dashed lines and to interpret the spread of UDGs as scatter rather than distinct classes.
assumptions (4)
  • domain assumption The Sersic R1/n model (Equation 5) describes the light profiles of all ETGs and UDGs, with Equations 6 to 8 connecting mu0, mu_e, n, and Re.
    The derivation of the size-luminosity curve (Equation 10) relies entirely on the Sersic model; deviations such as truncated or anti-truncated profiles and tidal debris are acknowledged in Section 3.3.
  • domain assumption The empirical MB-mu0 and MB-log(n) relations (Equations 2 and 9) are linear and universal across the full magnitude range, including the faint UDG regime.
    Equation 9 is explicitly refit with UDGs included, so its extension to faint magnitudes is an assumption rather than an independent prediction; the paper does not test for a bend or break at MB fainter than -15.
  • domain assumption Photometric conversions from g-band AB to B-band Vega (B_AB = g_AB + 0.35 and B_Vega = B_AB + 0.12) apply to all UDGs and NUDGes.
    Section 2; derived from assumed (g-z) colors near 0.9 to 1 mag. The paper quotes about 0.04 mag scatter, but a systematic error would shift the UDG points relative to the ETG-based curves.
  • domain assumption The 'triangular cutout' in Figure 2a is a selection artifact rather than a real population gap in the MB-mu0 plane.
    The paper invokes this cutout to explain the artificial UDG size-luminosity relation; if the cutout reflects a real bimodality, the two-class interpretation could be revived.

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Pith. "Pith review of $R_{\rm e}$. II. Understanding the IC 3475 galaxy type, including ultra-diffuse galaxy, structural scaling relations." pith.science (2026). https://pith.science/paper/UZVFTLRR

@misc{pith2026250416593,
  author       = {Pith},
  title        = {Pith review of: $R_\rm e$. II. Understanding the IC 3475 galaxy type, including ultra-diffuse galaxy, structural scaling relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UZVFTLRR}},
  note         = {Machine review of arXiv:2504.16593}
}
abstract

It is explained why relatively gas-poor ultra-diffuse galaxies (UDGs), a subset of IC 3475 galaxy types, do not have unexpectedly large sizes but large sizes that are in line with expectations from the curved size-luminosity relation defined by brighter early-type galaxies (ETGs). These UDGs extend the faint end of the (absolute magnitude, $\mathfrak{M}$)-log(S\'ersic index, $n$) and $\mathfrak{M}$-(central surface brightness, $\mu_{\rm 0}$) relations defined by all ETGs, leading to the large effective half-light radii, $R_{\rm e}$, in these UDGs. It is detailed how the scatter in $\mu_{\rm 0}$, at a given $\mathfrak{M}$, relates to variations in the galaxies' values of $n$ and effective surface brightness, $\mu_{\rm e}$. These variations map into changes in $R_{\rm e}$ and produce the scatter about the $\mathfrak{M}$-$R_{\rm e}$ relation at fixed $\mathfrak{M}$. Similarly, the scatter in $\mathfrak{M}$, at fixed $\mu_{\rm 0}$ and $n$, can be mapped into changes in $R_{\rm e}$. The suggestion that there may be two types of relatively gas-poor UDGs appears ill-founded, arising from the scatter about the $\mathfrak{M}$-$\mu_{\rm 0}$ relation. The increased scatter about the faint end of the $\mathfrak{M}$-$R_{\rm e}$ relation and the smaller scatter about $\mathfrak{M}$-(isophotal radii, $R_{\rm iso}$) relations are explained. Artificial and potentially misleading size-luminosity relations for UDGs are also addressed. Finally, expected trends with dynamical mass, and evolutionary pathways towards relatively gas-rich galaxies, are briefly discussed. Hopefully, the understanding presented here will prove helpful for interpreting the many low surface brightness galaxies that the Large Synoptic Survey Telescope will detect.

Figures

Figures reproduced from arXiv: 2504.16593 by the authors.

Figure 1
Figure 1. Absolute magnitude versus central surface brightness (Equation 1, panel a), Sersic index (Equation ´ 2, panel b), and (equivalent axis) effective half-light radius (Equation 4, panel c). The dashed lines in panel a) ensnare roughly the ±2σ scatter about the ETGs, and they have been mapped into panel c). The arrows reveal how a change in absolute magnitude at fixed µ0,B and n results in a corresponding change in Re, … view at source ↗
Figure 2
Figure 2. Extension of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Variant of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: An example of how variations/offsets in µe and n for a surface brightness profile will result in the offset to µ0. For a given M, with an associated µ0 and n from the M-µ0 and M-log(n) relations, the horizontal offsets δµ0 seen in Figures 1a–3a are attributable to the …
Figure 6
Figure 6. Figure 6: Breaking down the scatter in the MB-µ0,B diagram for ETGs. Symbols have the same meaning as in [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: Variant of Figure 2a, in which the symbol size is now proportional to the ellipticity (= 1 − b/a) such that galaxies that appear round have a small symbol size. In the inset panel, the grey triangles represent 28 UDGs predominantly in clusters, while the pink squares r…
Figure 8
Figure 8. Figure 8: Absolute magnitude versus the ‘equivalent axis’ isophotal radius at µB = 26 mag arcsec−2 obtained from each galaxy’s Sersic profile. The central black curve ´ is derived using Equations 2 and 9, while the outer grey curves denote a ±0.125 dex offset in isophotal radius…
Figure 9
Figure 9. Figure 9: Five representative B-band light profiles of ETGs with different Sersic indices ´ are shown (thick solid curves). For each value of n, an associated absolute magnitude and central surface brightness are assigned from Equations 2 and 9, from which the effective surface …
Figure 10
Figure 10. Figure 10: Absolute magnitude versus the ‘equivalent axis’ isophotal radius at µB = 31 mag obtained from each galaxy’s Sersic profile. The central black curve is derived ´ using Equations 2 and 9, while the outer grey curves denote a ±0.25 dex offset in isophotal radius at fixed…
Figure 11
Figure 11. Figure 11: Given MB ∝ σ 2 for dETGs (Davies et al. 1983; Graham 2013, and refer￾ences therein), one can appreciate how the scatter about the MB-Re relation, shown here, results in different dynamical masses (σ 2Re/G), and in turn, mass-dependent trends, that are subject to arbit…
Figure 12
Figure 12. Figure 12: B-band (Vega) central surface brightness versus effective half light radius for the UDGs, NUDGes and ETGs presented herein (Section 2), along with the disc￾component of the S0 and spiral galaxies compiled by Graham & de Blok (2001) and the exponential models fit to th…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.