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Investigating the Colour-Magnitude Relations for White Dwarf Stars in SDSS Photometry

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Three empirical colour-magnitude relations give DA white dwarf distances to 0.26-0.37 mag scatter.

desk verdict A useful set of SDSS-only colour-magnitude relations for DA white dwarfs, honestly fit but with the key generalization claim unvalidated. read the letter →

arxiv 2504.21075 v1 pith:G4YA6OXS submitted 2025-04-29 astro-ph.SR astro-ph.GA

O. Caliskan , S Gokmen This is my paper · ORCID
classification astro-ph.SRastro-ph.GA
keywords whitedwarfscolour-magnituderelationsSDSSphotometryGaiaDR3photometricdistancesDAinterstellarextinctiondistancedetermination
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

The paper claims that three empirical colour-magnitude relations (CMRs) fitted to 5,516 DA-type white dwarfs in SDSS photometry, with absolute magnitudes anchored by Gaia DR3 parallaxes, are accurate enough to be used for distance determination. Each relation expresses an absolute magnitude as a quadratic polynomial in a pair of dereddened SDSS colour indices; the fits reach $R^{2}$ values between 0.86 and 0.95 with scatter of 0.26 to 0.37 mag. If the claim holds, these relations provide photometric distances for faint white dwarfs in SDSS fields where Gaia parallax errors are too large to be useful. The paper further reports that distances computed from the new relations agree with Gaia and Bayesian distance catalogues to within about 30 pc, while the earlier catalogue that supplied the input sample systematically overestimates distances beyond roughly 400 pc.

What carries the argument

The central object is a bivariate quadratic colour-magnitude relation of the form $M = a x^2 + b y^2 + c x y + d x + e y + f$, where $x$ and $y$ are pairs of dereddened SDSS colour indices. The calibration uses 5,516 stars selected to have $g_0 < 21$ mag, relative parallax error $\sigma_\varpi/\varpi \le 0.10$, and positions inside theoretical 0.3, 0.6, and 0.9 $M_\odot$ DA white-dwarf cooling tracks, a cut intended to remove unresolved binaries and contaminants. Apparent magnitudes are dereddened using a Milky Way dust map with the $V$-band absorption scaled by distance through an exponential dust-height law. The machinery converts two measured colours plus an apparent magnitude into an absolute magnitude and therefore a distance modulus, bypassing the need for a trigonometric parallax.

What would settle it

Apply Equations (5)-(7) to a blind sample of DA white dwarfs with $g_0 > 21$ mag or distances outside the 17-793 pc calibration range, and compare the photometric distances with independent parallaxes of better than 1% from future Gaia releases or space astrometry. A systematic offset in absolute magnitude larger than the quoted 0.26-0.37 mag scatter, or scatter that grows toward fainter magnitudes or redder colours, would falsify the claim that the relations are reliable for distance determination.

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Extended reading notes

Core claim

For hydrogen-rich (DA) white dwarfs, the SDSS absolute magnitudes $M_g$, $M_r$, and $M_i$ are predictable from two colour indices each. The paper derives three relations: $M_g$ from $(u-g)_0$ and $(g-r)_0$; $M_r$ from $(g-r)_0$ and $(r-i)_0$; and $M_i$ from $(r-i)_0$ and $(i-z)_0$. The $M_g$ relation is the most precise, with $R^2=0.951$ and a standard deviation of 0.263 mag, and it spans the widest absolute-magnitude range; the redder-index relations are less accurate because white dwarfs emit little light at long wavelengths. Distances implied by the relations are within a median of $-1$ pc and $-2.4$ pc of Gaia DR3 and Bayesian distances, with standard deviations near 30 pc, whereas the catalogue from which the sample was drawn overestimates distances beyond about 400 pc. The paper's central claim is that these CMRs can be reliably used to determine distances of DA white dwarfs observed in SDSS photometry.

Load-bearing premise

The fitted coefficients are unbiased only if restricting the calibration sample to stars inside the 0.3, 0.6, and 0.9 solar-mass theoretical DA tracks removes binaries and contaminants without excluding a genuine part of the white-dwarf sequence; if that cut is wrong, the coefficients inherit the bias.

Editorial extensions

If this is right

  • Photometric distances from these CMRs can be assigned to SDSS DA white dwarfs too faint for precise Gaia parallaxes, extending distance measurements to the survey's limiting magnitude.
  • When both blue colours are available, the $M_g$ relation from $(u-g)_0$ and $(g-r)_0$ is the preferred estimator because it is the most precise and covers the widest magnitude range.
  • Distances from the new relations agree with Gaia DR3 and Bayesian distance catalogues to within about 30 pc in the calibration region, so the relations can serve as a cross-check on future astrometric distance catalogues.
  • The comparison implies that the distances in the input catalogue are systematically too large beyond about 400 pc, which affects any science built on those catalogue distances.
  • Galactic white-dwarf population studies can adopt the relations as a spectral-fitting-free distance estimator for SDSS-selected stars.

Reading between the lines

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

  • The calibration sample is limited to stars with better than 10% Gaia parallaxes, so it is biased toward the nearest, brightest white dwarfs; using the relations beyond $g_0\approx 21$ or beyond roughly 800 pc assumes the colour-absolute-magnitude relation is unchanged outside the calibrated range, a point the paper does not directly test.
  • A scatter of 0.26 to 0.37 mag in absolute magnitude translates to roughly 12-17% uncertainty in distance, which is the floor for any Galactic structure or luminosity-function work that adopts these relations.
  • The same two-colour polynomial calibration could be rebuilt from Gaia DR3 parallaxes for white dwarfs observed in other photometric surveys, since the method depends only on the availability of parallax-anchored training stars and matched photometry.
  • The reported offset in the input catalogue's distances beyond 400 pc, if confirmed, suggests that earlier conclusions drawn from those distances in that regime may need revisiting.
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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

4 major / 5 minor

Summary. The paper derives empirical colour-magnitude relations (CMRs) for DA white dwarfs in the SDSS ugriz system, using a sample of 5,516 stars selected from the Anguiano et al. (2017) catalogue after cross-matching with Gaia DR3 and applying quality cuts. Three quadratic relations with one cross term are fitted for M_g, M_r, and M_i as functions of two colour indices each (Eqs. 5–7, Table 1). Reported in-sample R^2 values are 0.86–0.95 with residual scatters 0.26–0.37 mag. The paper claims these relations can be reliably used for distance determination of faint SDSS white dwarfs, and validates them by comparing distances with Gaia DR3, Bailer-Jones et al. (2021), and Anguiano et al. (2017).

Significance. If the relations generalize, they would provide a practical tool for photometric distances to faint SDSS DA white dwarfs where Gaia parallaxes are imprecise, and the paper provides a full coefficient table plus machine-readable fits in the text. The authors are also appropriately honest in the Summary that the Gaia/BJ21 comparisons are not fully independent, and they flag the A17 offset. However, the central applicability claim — that in-sample R^2 and sigma imply reliable distances for the faint target population — is only partially supported, because no out-of-sample or faint-star test is performed and the only external benchmark shows a distance-dependent offset that the authors attribute entirely to the external catalogue.

major comments (4)
  1. [Sec. 4, Fig. 8] The comparison with Gaia DR3 and Bailer-Jones et al. (2021) is not an independent validation because the same Gaia parallaxes were used to compute the absolute magnitudes in the calibration (Eq. 4). The paper acknowledges this in Sec. 5, but the conclusion that the CMRs are 'reliably and accurately utilised for distance determination' still rests on that comparison. The only truly independent benchmark, A17, shows a median offset of -30 pc and a growing systematic bias beyond 400 pc. The authors attribute the entire bias to A17, but the calibration sample is concentrated at d < 392 pc (90%), so a distance-dependent extrapolation error in the CMR is an equally plausible reading. A quantitative test is needed: e.g., restrict the distance comparison to an out-of-sample set of fainter DA white dwarfs with lower-quality Gaia parallaxes, or at least split the calibration sample by distance and show that the CMR fitted on the inner 68% predicts the outer 32% without bias.
  2. [Sec. 3.2, Fig. 4] The sample-cleaning step in Sec. 3.2 excludes stars outside the 0.3, 0.6, and 0.9 Msun Holberg & Bergeron (2006) tracks, removing 1,773 of 7,289 stars. The manuscript does not define the quantitative boundary of the exclusion (e.g., how far in colour or magnitude a star must lie from the tracks to be rejected), nor does it test how sensitive the fitted coefficients are to the chosen tracks or to the exclusion width. Since the tracks themselves are theoretical models and the excluded population plausibly includes unresolved binaries, thick-disc or halo white dwarfs, and high-mass remnants, the fitted CMR coefficients inherit any track or boundary bias. A robustness test (e.g., refit with 0.2/0.8 Msun tracks, or with a different exclusion width) is needed to support the claim that the calibration sample is unbiased.
  3. [Sec. 3.3, Eqs. (5)–(7)] The central claim in Sec. 3.3 that R^2 = 0.86–0.95 and sigma = 0.26–0.37 mag imply the CMRs 'can be reliably and accurately utilised for distance determination' is an in-sample statement. R^2 measures scatter of the calibration data about the fitted surface; it does not account for extrapolation to the fainter, more distant population (g0 near 21, distances beyond ~400 pc) that the paper targets in Sec. 5. The authors should either provide an out-of-sample validation (e.g., a hold-out set, or a test against a fainter Gaia-selected WD sample with less precise parallaxes) or soften the claim to 'in-sample precision' and discuss the extrapolation risk explicitly.
  4. [Sec. 2 and Sec. 3.1] The paper uses the Gaia DR3 parallax without applying the global zero-point correction (e.g., Lindegren et al. 2021) and uses d = 1000/varpi without a Bayesian or Lutz-Kelker treatment. For a sample with sigma_varpi/varpi < 0.1 this introduces a distance-dependent bias at the few-percent level. Since the calibration absolute magnitudes are computed from these distances (Eq. 4), the CMR zero-points f1, f2, f3 absorb any parallax zero-point offset. The comparison with BJ21 distances in Fig. 8 would be a more meaningful test if the authors also showed the effect of applying the zero-point correction to their calibration; as written, the agreement with BJ21 partly reflects that both use the same uncorrected Gaia parallaxes. Please quantify the impact of the zero-point correction on the fitted coefficients.
minor comments (5)
  1. [Sec. 2, Fig. 1] The caption of Figure 1 refers to red dotted lines showing distance thresholds, but the text says 68%, 90%, and 95% of the sample lie within 287, 392, and 450 pc; the thresholds should be labeled in the figure itself or in the caption with the exact percentile values.
  2. [Sec. 3.1, Fig. 2] The text says 'the uncertainties of the selected white dwarf stars in three colour indices as a function of the g-apparent magnitude ... are plotted in Figure 1', but the figure is Figure 2 and shows five panels, not three. Please correct the cross-reference and the count.
  3. [Sec. 3.3, Eq. (6)] The last term in Eq. (6) is written as e2(i-r)0, but the relation is for M_r as a function of (g-r)0 and (r-i)0; this should be e2(r-i)0. The Table 1 coefficients are labelled e2, so the equation is probably a typo, but it should be fixed.
  4. [Abstract] The abstract ends with 'in the bf range of 0.86 to 0.95'; 'bf' appears to be a LaTeX markup artifact and should be removed.
  5. [Sec. 5, Summary] The text says 'the relation constructed from the (u-g)0 and (g-r)0 colour indices spans a broader absolute magnitude range than those based on redder filters, such as (M_r, M_i)', but M_r and M_i are not colour indices; they are absolute magnitudes. Please rephrase to 'the relations for M_r and M_i'.

Circularity Check

1 steps flagged · score 4.0 of 10

Gaia/BJ21 distance agreement is an in-sample validation loop; the only external benchmark (A17) shows an offset that the paper attributes away.

  1. fitted input called prediction [Section 4 (Discussion), Fig. 8; cf. Eq. (4) and Section 3.3]
    "While the distances of stars were determined from the trigonometric parallaxes provided in the Gaia DR3 catalogue using the relation d(pc) = 1000/ϖ (mas), the distances from BJ21 were obtained by extracting the corresponding values from the BJ21 catalogue based on the equatorial coordinates of the stars. ... The comparison of the three distance datasets obtained from the literature with those calculated using the CMRs determined in this study is presented in Figure 8."

    The CMRs in Eqs. (5)-(7) were fitted to absolute magnitudes built from the same Gaia DR3 parallaxes via Eq. (4) (Mg = G0 - 5 log(1000/ϖ) + 5). For the same 5,516 calibration stars, distance from the CMR is d = 10^{0.2(g0 - M_pred + 5)}, so agreement with d = 1000/ϖ is guaranteed to within the fit residuals by construction. The reported median differences of -1 pc (Gaia) and -2.4 pc (BJ21) therefore re-express the in-sample scatter of Fig. 7 in distance units; they are not an independent confirmation. The paper's Summary concedes that Gaia and BJ21 are 'not fully independent', but the agreement is still cited as evidence of reliability, while the only external benchmark (A17) shows a substantial offset that the authors attribute to A17.

full rationale

The calibration itself is a standard empirical multiple regression of Gaia-parallax-based absolute magnitudes on SDSS colours; this is not circular and no load-bearing self-citation or uniqueness theorem is used. The circular validation loop is in Section 4: CMR distances are computed for the same 5,516 stars whose Gaia parallaxes defined the fitted absolute magnitudes. Agreement with Gaia DR3 and BJ21 is therefore an algebraic consequence of the fit, not independent evidence; the paper explicitly acknowledges this non-independence. The only external benchmark, Anguiano et al. (2017), shows a systematic offset beyond 400 pc, and the paper dismisses A17 rather than treating it as a failed extrapolation. Consequently the reliability claim leans on in-sample R² values (0.86-0.95) and a closed-loop distance comparison, while the stated application to faint stars is an untested extrapolation. This is a mild-to-moderate circular validation rather than a definitional derivation, so the score is 4 rather than higher.

Assumptions & free parameters 18 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard photometric calibration assumptions: parallax-based distances without zero-point correction, adopted extinction maps and coefficients, theoretical DA mass tracks used as a purity filter, and a fixed polynomial functional form. No new physical entities are introduced.

free parameters (18)
  • a1 = -2.3287
    Regression coefficient for (u-g)_0^2 in Equation (5), fit to 5,516 calibration stars.
  • b1 = 1.0313
    Regression coefficient for (g-r)_0^2 in Equation (5), fit to 5,516 calibration stars.
  • c1 = 0.6356
    Regression coefficient for the cross term (u-g)_0 (g-r)_0 in Equation (5), fit to 5,516 calibration stars.
  • d1 = 1.5255
    Regression coefficient for (u-g)_0 in Equation (5), fit to 5,516 calibration stars.
  • e1 = 5.5578
    Regression coefficient for (g-r)_0 in Equation (5), fit to 5,516 calibration stars.
  • f1 = 12.4927
    Intercept in Equation (5), fit to 5,516 calibration stars.
  • a2 = -4.6110
    Regression coefficient for (g-r)_0^2 in Equation (6), fit to 5,516 calibration stars.
  • b2 = 0.25683
    Regression coefficient for (r-i)_0^2 in Equation (6), fit to 5,516 calibration stars.
  • c2 = 3.0200
    Regression coefficient for the cross term (g-r)_0 (r-i)_0 in Equation (6), fit to 5,516 calibration stars.
  • d2 = 3.36816
    Regression coefficient for (g-r)_0 in Equation (6), fit to 5,516 calibration stars.
  • e2 = 2.1026
    Regression coefficient for the final colour term in Equation (6), fit to 5,516 calibration stars.
  • f2 = 13.0225
    Intercept in Equation (6), fit to 5,516 calibration stars.
  • a3 = -2.4688
    Regression coefficient for (r-i)_0^2 in Equation (7), fit to 5,516 calibration stars.
  • b3 = 0.1963
    Regression coefficient for (i-z)_0^2 in Equation (7), fit to 5,516 calibration stars.
  • c3 = -5.3970
    Regression coefficient for the cross term (r-i)_0 (i-z)_0 in Equation (7), fit to 5,516 calibration stars.
  • d3 = 4.2925
    Regression coefficient for (r-i)_0 in Equation (7), fit to 5,516 calibration stars.
  • e3 = 0.3183
    Regression coefficient for (i-z)_0 in Equation (7), fit to 5,516 calibration stars.
  • f3 = 13.2530
    Intercept in Equation (7), fit to 5,516 calibration stars.
assumptions (5)
  • domain assumption Parallax-to-distance conversion d(pc)=1000/varpi is unbiased for the selected sample.
    Used in Section 3.1 and Equation (4) with no Lutz-Kelker or Gaia zero-point correction.
  • domain assumption Schlafly and Finkbeiner (2011) dust maps with Bahcall and Soneira scaling and Cardelli et al. coefficients describe the true extinction for every sightline.
    Equations (1) through (3) apply this extinction model to all SDSS and Gaia bands.
  • domain assumption Holberg and Bergeron (2006) DA tracks for 0.3, 0.6, and 0.9 Msun correctly separate single white dwarfs from binaries and contaminants.
    Section 3.2 and Figure 4 exclude 1,773 stars using these tracks; the exact exclusion boundary is not quantified.
  • ad hoc to paper The adopted quadratic polynomial form with one cross term is adequate to represent the white dwarf colour-magnitude relation.
    Equations (5) through (7) assume this functional form with no model comparison or residuals versus colour diagnostics.
  • domain assumption The remaining 5,516 stars are representative of the DA white dwarf population for which the CMR will be used.
    Selection thresholds and track cuts may bias the sample toward a specific mass and temperature range.

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Pith. "Pith review of Investigating the Colour-Magnitude Relations for White Dwarf Stars in SDSS Photometry." pith.science (2026). https://pith.science/paper/G4YA6OXS

@misc{pith2026250421075,
  author       = {Pith},
  title        = {Pith review of: Investigating the Colour-Magnitude Relations for White Dwarf Stars in SDSS Photometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G4YA6OXS}},
  note         = {Machine review of arXiv:2504.21075}
}
abstract

In this study, colour-magnitude relations (CMRs) for DA-type white dwarfs in the Sloan Digital Sky Survey (SDSS) photometric system were investigated. For this purpose, the SDSS data for 20,247 white dwarf stars, as provided in the study by Anguiano et al. (2017), were matched with the Gaia third data release (Gaia DR3) catalogue to obtain trigonometric parallax ($\varpi$) data. The SDSS $ugriz$ magnitudes of the stars were corrected for interstellar extinction using dust maps provided for the Milky Way, and distances from the Sun to the stars were calculated. The SDSS magnitudes were thus corrected for the effects of interstellar extinction. For the calibration of the stars, 5,516 white dwarf stars were selected, with apparent magnitudes brighter than $g_0=21$ mag and relative parallax errors measured to better than $\sigma_\varpi/\varpi=0.1$. Subsequently, three separate CMRs were derived for the absolute magnitudes $M_{\rm g}$, $M_{\rm r}$, and $M_{\rm i}$, each calibrated to two-colour indices. The coefficient of determination ($R^2$) of the obtained CMRs are highly reliable in the bf range of 0.86 to 0.95. Moreover, the standard deviations of the differences between the absolute magnitudes obtained from the relations and the original ones of the calibration stars range from 0.26 to 0.37 mag.

Figures

Figures reproduced from arXiv: 2504.21075 by the authors.

Figure 1
Figure 1. Relative parallax errors (𝜎𝜛/𝜛) as a function of distance for 7,289 white dwarf stars, based on trigonometric parallaxes from Gaia DR3. The red dotted lines indicate the distance thresholds within which 68%, 90%, and 95% of the sample stars are located. calibrations for absolute magnitudes, each based on different colour indices. These new calibrations offer improved precision over previous methods and enable the in… view at source ↗
Figure 2
Figure 2. The variation of the colour index errors of selected white dwarf stars with 𝑔-apparent magnitudes. (a) (𝑢 − 𝑔)err × 𝑔, (b) (𝑔 − 𝑟)err × 𝑔, (c) (𝑟 − 𝑖)err × 𝑔, (d) (𝑖 − 𝑧)err × 𝑔, and (e) cumulative distribution of the star sample. 3. ANALYSIS 3.1. Photometric Absorptions Determination In this study, the uncertainties of the selected white dwarf stars in three colour indices as a function of the 𝑔-apparent magnitude … view at source ↗
Figure 3
Figure 3. Histograms of the original (𝐴∞(𝑉)) (a) and reduced absorption (𝐴d (𝑉)) values (b) of selected 5,516 white dwarf stars. 3.2. HR Diagram of the White Dwarfs In this study, the positions of selected white dwarfs on the colour-magnitude diagram (CMD) constructed from Gaia photometric and astrometric data were examined. The absolute magnitudes 𝑀𝐺 of the stars were estimated using the following equation 𝑀𝐺 = 𝐺0 − 5 × log … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The positions of the 5,516 white dwarf stars on the 𝑀G × (𝐺BP − 𝐺RP) CMD. The red curves show three different mass tracks from Holberg & Bergeron (2006). Dark and light blue dots represent selected and scattered white dwarf stars [PITH_FULL_IMAGE:figures/full_fig_p007…
Figure 5
Figure 5. Figure 5: The locations of 5,516 white dwarf stars in equatorial (top panel) and Galactic (bottom panel) coordinates. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: The positions of the 5,516 white dwarf stars on the two-colour diagrams as a function of their absolute magnitudes. (a) (𝑢 − 𝑔)0 × (𝑔 − 𝑟)0, (b) (𝑔 − 𝑟)0 × (𝑟 − 𝑖)0, and (c) (𝑟 − 𝑖)0 × (𝑖 − 𝑧)0. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Comparison of the calculated and the original SDSS absolute magnitudes (upper panels) and distribution of the absolute magnitude residuals (Δ𝑀) concerning the original absolute magnitudes (lower panels) for 5,516 white dwarf stars. The solid black line represents one-t…
Figure 8
Figure 8. Figure 8: Comparison of the distances to the 5,516 white dwarfs calculated in this study with those of Gaia (a), BJ21 (b), and A17 (c), respectively (top panel) and illustration of the distance differences (bottom panel). The solid black line represents one-to-one lines, and the…

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Pith tools

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