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The paper claims that the Fourier transform method applied to He I absorption lines in medium-resolution survey spectra can reliably measure v sin i for Be-type stars, yielding a 479-star catalog and a deconvolved equatorial rotation distri

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2026-08-01 23:00 UTC pith:YVL6TMFP

load-bearing objection Useful new vsini catalog for 479 Be stars, but the deconvolved velocity and gamma_e results are built on a Gaussian guess and shouldn't be cited as robust. the 4 major comments →

arxiv 2607.15540 v1 pith:YVL6TMFP submitted 2026-07-17 astro-ph.SR

Investigation of projected rotational velocities of Be-type stars in LAMOST DR7

Zhicun Liu , Jiao Li , Jiaming Liu , Xiao-Long Wang , Guozhen Hu , Wenyuan Cui This is my paper
classification astro-ph.SR
keywords Be-type starsprojected rotational velocityv sin iFourier transform methodhelium absorption linesHα emission morphologystellar rotationcritical rotation
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.

The paper tries to establish that projected rotational velocities (v sin i) of Be-type stars can be extracted from medium-resolution survey spectra by locating the first zero of the Fourier transform of four He I absorption lines (4922, 5015, 5047, and 6678 Å). Applying this method to 479 Be-type stars, it derives a catalog of v sin i values and finds that the deconvolved true equatorial rotation distribution is single-peaked near v ≈ 260 km/s, not bimodal. For 105 stars with atmospheric parameters, the mean equatorial rotation is about 0.74 of the critical velocity, with evidence for two subpopulations. The paper also reports that Be stars with double-peaked Hα emission spin faster than those with single-peaked emission, and that v sin i distributions do not differ significantly across field, cluster, and OB-association environments. A sympathetic reader would care because a large homogeneous Milky Way Be-star rotation catalog, plus a single-peaked rotation distribution, bears directly on whether Be stars form by single-star spin-up, binary mass transfer, or both.

Core claim

The central claim is that the first zero (or minimum) of the Fourier transform of He I λ4922, 5015, 5047, and 6678 Å line profiles is set by rotational broadening alone, making the FT method a reliable v sin i estimator for Be-type stars even at the medium resolution of large surveys. The paper validates this against literature rotational velocities for 56 stars and then presents v sin i for 479 Be-type stars. After deconvolving the projected distribution under the assumption of randomly oriented rotation axes, the true equatorial velocity distribution shows a single peak at v ≈ 260 km/s. For the 105 stars with measured effective temperatures and surface gravities, the mean ratio of equatori

What carries the argument

The named central object is the Fourier transform method: for a rotationally broadened spectral line, the transform has a first zero whose position is inversely related to v sin i, and the paper assumes this zero can be identified in medium-resolution spectra even when other broadening mechanisms are present. The four He I lines used lie in the blue and red arms of the survey's medium-resolution coverage. Downstream, an iterative deconvolution procedure (assuming a Gaussian model and random inclinations) converts the observed v sin i distribution into an estimate of the true equatorial velocity distribution, and a Monte Carlo comparison against critical-velocity fractions quantifies the rota

Load-bearing premise

The load-bearing premise is that the He I lines used are photospheric and free of circumstellar disk or shell contamination, so the first Fourier zero is governed by rotation alone; if those lines are filled in or blended with disk features, all v sin i values and every derived distribution are biased.

What would settle it

Compare FT-derived v sin i against high-resolution profile fits for a sample of Be stars with strong Hα emission and known shell signatures; if the FT values deviate systematically for shell stars or vary with disk phase for the same star, the contamination-free assumption is falsified. A simpler version: check whether the first Fourier zero shifts when the same star's He I line is observed at epochs of different disk strength.

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

If this is right

  • A homogeneous catalog of 479 Galactic Be-type stars with v sin i and uncertainties becomes available for formation and evolution studies.
  • Be stars with double-peaked Hα emission rotate faster on average than single-peaked ones, implying that emission morphology encodes rotation state.
  • The deconvolved equatorial velocity distribution peaking near 260 km/s, rather than a bimodal distribution, constrains the relative importance of spin-up channels.
  • The mean equatorial rotation of about 0.74 of critical, with a bimodal γ distribution, suggests at least two subpopulations (slow recent accretors vs fast evolved Be stars).
  • The v sin i–Hα peak separation correlation is stronger for shell stars, supporting a geometric (inclination) interpretation of shell spectra.

Where Pith is reading between the lines

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

  • If the assumed random-inclination model is wrong (e.g., Be disks bias observed inclinations), the deconvolved peak at 260 km/s could be an artifact; a testable extension is to compare against independent inclination measurements from interferometry for a subset.
  • The method's reliance on uncontaminated He I lines implies the catalog may be least reliable exactly for shell stars, whose disks cover the photosphere; a targeted comparison with high-resolution line fits for shell stars would settle this.
  • Because the sample is magnitude- and S/N-limited and only includes stars with clear He I absorption, selection against weak-lined or heavily veiled Be stars could bias the derived distribution; the paper's future low-resolution OB catalog may offer a selection-corrected check.
  • The reported lack of v sin i differences across environments is consistent with Be formation being dominated by processes independent of local stellar density; if true, binary-driven spin-up should be common in all environments.

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

4 major / 5 minor

Summary. Liu et al. present projected rotational velocities for 479 Be-type stars from LAMOST DR7 medium-resolution spectra, using the Fourier-transform zero method on He I lines at 4922, 5015, 5047, and 6678 Å. They validate the method on 56 stars with literature vsini values, using high-resolution spectra degraded to R~7500, and provide the measured vsini table. They then compare vsini distributions across Hα morphologies and stellar environments, deconvolve an apparent true equatorial velocity distribution with a four-iteration Lucy method under a Gaussian initial estimate, derive a gamma_e = v/v_crit distribution for 105 stars with stellar parameters, and examine the vsini–Hα peak-separation correlation.

Significance. If the vsini measurements are sound, the machine-readable catalog of 479 Be-type stars is a useful homogeneous addition to the literature, complementing smaller high-resolution samples. The explicit validation comparison and the caution about low vsini values are positive features. However, the distributional conclusions — the single peak near 260 km/s and the bimodal gamma_e distribution with mean 0.74 — are not yet supported by the analysis as written because they depend on a barely documented deconvolution and on an internally inconsistent Gaussian Monte Carlo setup. The catalog is likely publishable after those claims are either made robust or removed.

major comments (4)
  1. [Section 4.3 and Figure 8] The central claim of a single-peaked true rotation distribution at v≈260 km/s is not established by the presented analysis. The text states that the Lucy (1974) deconvolution was run 'assuming a Gaussian distribution and using four iterations,' but the initial Gaussian parameters are not given and no convergence test is shown. With four iterations the solution remains close to the initial guess; if that guess is centered near 260 km/s, the peak is inherited rather than measured. Please specify the initial guess (mean and sigma), show results for a range of iteration counts (e.g., 10, 50, 100), and test at least one non-Gaussian or uninformative initial guess. This is required to support conclusion (4) and the corresponding abstract statement.
  2. [Section 4.4 and Appendix C] The gamma_e analysis is internally inconsistent and the 'deconvolved' distribution is not defined. Equation (C4) as printed is not a Gaussian: the exponent lacks the minus sign. The text immediately after Eq. (C4) sets sigma=0.002, while Table C1 uses sigma=0.02. More importantly, the Monte Carlo described in Appendix C fits gamma_cent by comparing N_crit counts with observed/corrected vsini under an assumed Gaussian gamma_e distribution; this procedure cannot by itself yield the bimodal deconvolved gamma_e distribution shown in Figure C1, and no deconvolution algorithm is described for that figure. Please specify the gamma_e deconvolution, correct Eq. (C4), and show sensitivity to the Gaussian assumption and to the sigma value. The abstract's 0.74 mean and conclusion (5) depend on this.
  3. [Section 3.2 and Figure 4] The validation does not test the pipeline on the actual LAMOST MRS spectra. The 56-star comparison was made by taking high-resolution literature spectra and degrading them to R~7500; this does not capture LAMOST-specific effects such as continuum normalization, pixel sampling, line blending, or circumstellar contamination. Because Be stars can have disk emission or shell features that fill in photospheric He I lines, the assumption that the He I lines are purely rotationally broadened needs a direct check in the 479-star sample. Please provide quantitative residuals/outlier statistics for the 56-star comparison and either cross-match a subset of LAMOST targets with independent vsini measurements or perform a line-profile contamination test (e.g., line symmetry, comparison between He I lines, or exclusion of shell stars).
  4. [Sections 3.2 and 4.1, Figures 5 and 8] The distributional analysis includes stars with vsini<50 km/s, for which the paper itself advises caution: only three validation stars are below 50 km/s, and the text says users should be cautious for such stars. The low-velocity tail contributes to the K-S tests, to the deconvolution, and to the mean values. Please either restrict distributional claims to the validated range or add a conservative treatment of the vsini<50 km/s subsample and show that the conclusions are unchanged.
minor comments (5)
  1. [Section 4.1] 'The overall distribution appears broadly unbimodal' should read 'broadly unimodal' or 'does not appear bimodal.'
  2. [Figure 9 caption] The legend description ('black and blue squares', circled numbers ①,②, ...) does not correspond clearly to the visible figure; please simplify and match the figure symbols.
  3. [Figure 8] The axis label 'vsini / v' obscures the distinction between observed vsini and deconvolved v; label the axes as 'vsini (observed)' and 'v (deconvolved)'.
  4. [Equation (C4)] The exponent in Eq. (C4) should include a minus sign: exp[-(gamma_e - gamma_cent)^2 / (2 sigma^2)]. As printed, the expression is not a normalized Gaussian.
  5. [Section 3.2] 'user should be cautious' should be 'users should be cautious.'

Circularity Check

1 steps flagged

vsini catalog is externally benchmarked, but the headline deconvolved single-peak v distribution is partly supplied by a Gaussian input to a four-iteration Lucy deconvolution rather than inferred from the data.

specific steps
  1. fitted input called prediction [Section 4.3, Figure 8]
    "To estimate the distribution of v, we adopted the iterative procedure of Lucy (1974) assuming a Gaussian distribution and using four iterations. ... The deconvolved rotational velocity distribution of our entire sample of Be-type stars does not exhibit a bimodal distribution, but rather a peak at v≈260 km·s−1."

    The 'deconvolved' v distribution is the output of Lucy deconvolution whose only stated input is a Gaussian distribution, stopped after four iterations. At that early stopping point the solution is still largely the initial guess, and the paper gives no Gaussian mean/sigma or convergence test. Since a Gaussian is unimodal by definition, the reported single peak at ~260 km/s and the conclusion 'no bimodal distribution' are inherited from the assumed input (with an unstated fitted location), not independently recovered from the data. The per-star vsini measurements themselves are validated externally, so the circularity affects this distributional claim rather than the catalog.

full rationale

The core vsini values are not circular: the FT method is applied to He I lines and validated against 56 Be stars with independent high-resolution literature vsini (Section 3.2, Figure 4), so the catalog is externally benchmarked. Sample construction uses earlier catalogs including some from the same group (Wang et al. 2022, Liu et al. 2024, 2025), but these are source lists used for selection, not the fitted rotation parameters, and no uniqueness theorem or ansatz is imported from self-citations. K-S and correlation results are internal statistical descriptions of the measured vsini. The only load-bearing circularity-like step is Section 4.3/Figure 8: a Gaussian distribution is assumed as the Lucy-deconvolution input and after four iterations the paper reports a single-peaked v distribution peaking near 260 km/s. This peak is substantially an attribute of the input prior rather than an unforced inference. The gamma_e analysis in Appendix C uses a similar Gaussian Monte Carlo fit and contains a sigma inconsistency (sigma=0.002 in Eq. C4 vs sigma=0.02 in Table C1), but the bimodal gamma_e deconvolution there shows the algorithm can produce non-Gaussian structure, so the issue is partial rather than complete. Overall, a moderate circularity score is appropriate: the catalog is independent, while the headline distributional claim is partly self-fed by a Gaussian initial guess.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central vsini measurements rest mainly on the FT method assumptions and sample selection. The gamma_e result additionally depends on several fitted or imported parameters: the Gaussian gamma_cent grid, the sigma inconsistency, the Dufton et al. gravity-darkening polynomial, and the Lucy deconvolution iteration/Gaussian prior. No new physical entities are introduced.

free parameters (6)
  • gamma_cent grid in Monte Carlo = ~0.80 (corrected), ~0.85 (uncorrected)
    Seven gamma_cent values from 0.6 to 0.9 are tested, and the value whose simulated Ncrit counts best match the observed counts is adopted. This is a fit to the same vsini data (Appendix C, Table C1).
  • sigma of Gaussian gamma_e distribution = 0.002 (text/Eq. C4) or 0.02 (Table C1 caption)
    The width of the assumed Gaussian gamma_e model controls the simulated Ncrit counts; the paper states two different values in different places.
  • Lucy deconvolution iteration count and Gaussian prior = 4 iterations; Gaussian distribution
    The true rotation velocity distribution is obtained by Lucy (1974) deconvolution assuming a Gaussian and stopping at four iterations; no sensitivity test is shown (Section 4.3).
  • S/N threshold for sample selection = S/N >= 40
    Chosen from visual inspection of synthetic spectra at S/N 30-90 (Appendix A); this selection affects which stars enter the sample.
  • Empirical vsini uncertainty prescription = 10 km/s for vsini<=100; 10% above
    Adopted from validation scatter and stated as empirical; used for Table 1 values (Section 3.2).
  • Gravity-darkening correction polynomial = Coefficients not tabulated; imported from Dufton et al. 2022
    A cubic polynomial from Dufton et al. (2022) is applied to correct vsini for gravity darkening before computing gamma_e; the coefficients are not given in this paper (Section 4.4).
axioms (6)
  • domain assumption Rotation axes of Be-type stars are randomly oriented in space.
    Required for Lucy deconvolution of vsini into equatorial v (Section 4.3); field/cluster samples may deviate from isotropy.
  • domain assumption The first zero of the Fourier transform of each He I line is determined only by rotational broadening.
    Foundational assumption of the FT method, adopted from Simon-Diaz & Herrero (2007); the paper does not re-test it for Be disk contamination (Section 3.1).
  • domain assumption The He I 4922, 5015, 5047, and 6678 absorption lines are photospheric and uncontaminated by circumstellar emission or shell features.
    Be stars have disks that can fill or contaminate lines; shell stars in the sample are especially at risk. Not explicitly tested (Sections 3.1, 4.1).
  • ad hoc to paper The equatorial rotational velocity distribution is Gaussian with fixed sigma for Monte Carlo gamma_e inference.
    Equation C4 imposes p(gamma_e) Gaussian; this is a modeling choice, not derived from stellar physics.
  • domain assumption Stellar masses from solar-metallicity PARSEC isochrones, using Payne Teff/logg from Xiang et al. (2022), are accurate enough for critical velocities.
    Critical velocity and gamma_e depend on mass and gravity; evolved binaries or shell stars may violate single-star isochrone assumptions (Section 4.4).
  • domain assumption The critical-velocity formula v_c = sqrt(2/3)(GMg)^(1/4) with the polar radius assumed unaffected by rotation.
    The paper follows Rivinius et al. (2013) and uses v_c as a lower-limit critical velocity; oblateness and gravity darkening are simplified (Appendix C, Eqs. C1-C3).

pith-pipeline@v1.3.0-alltime-deepseek · 21049 in / 11978 out tokens · 133787 ms · 2026-08-01T23:00:10.773066+00:00 · methodology

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read the original abstract

Stellar rotation plays a key role in the transfer of angular momentum, and a large sample of Be-type stars with reliable projected rotational velocities is crucial for understanding their formation and evolution. In this work, we derive the projected rotational velocities ($v$\,sin\,$i$) of 479 Be-type stars using the Fourier transform method, based on their LAMOST Medium-resolution Survey (MRS) spectra. Our results suggest that the Fourier transform method can provide reliable $v$\,sin\,$i$ values for Be-type stars by analyzing the \ion{He}{1}\,lines at 4922, 5015, 5047, and 6678 \,\AA in their LAMOST MRS spectra. A K-S test indicates that Be-type stars with different H$\alpha$ emission line morphologies exhibit different $v$\,sin\,$i$ distributions, and Be-type stars with double-peaked emission have a higher fraction of rapid rotators than those with single-peak emission. The $v$\,sin\,$i$ distributions of our Be-type stars in the field, OB associations, and clusters show no significant differences. The deconvolved $v$\,sin\,$i$ distribution of our entire Be-type star sample does not exhibit a bimodal distribution but rather a single peak at $v\approx260$\,km$\cdot$s$^{-1}$. Based on the analysis of 105 stars in our sample, we find that the mean equatorial rotational velocity is 0.74 times the critical velocity. Furthermore, we investigate the relationship between $v$\,sin\,$i$ and the H$\alpha$ peak separation velocity for Be-type stars exhibiting double-peak H$\alpha$ emission lines, using Pearson and Spearman rank correlation coefficients.

Figures

Figures reproduced from arXiv: 2607.15540 by Guozhen Hu, Jiaming Liu, Jiao Li, Wenyuan Cui, Xiao-long Wang, Zhicun Liu.

Figure 1
Figure 1. Figure 1: The LAMOST continuum-normalized blue- (left panel) and red-arm (right panel) medium-resolution spectra of two Be-type stars (J044634.59+303656.0 and J013131.58+594607.5) in our sample. Some important line features, such as He I 4922, 5015, 5047, 6678 ˚A, and Hα are also marked. 0.5 0.3 0.1 0.1 0.3 0.5 0.7 0.9 J-H 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 H-K 474 Be-type stars [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The J-H versus H-K color-color diagram for 474 of 479 Be-type stars in this work. The Be-type stars in the blue area correspond to the area − 0.2< (H–K) < 0.5 and − 0.3 < (J–H) < 0.5 defined by Chen et al. (2016). To derive the v sin i values of Be-type stars using the Fourier transform method, we selected the Be-type stars whose spectra have a sufficient signal-to-noise ra￾tio (S/N≥40), Hα emission, and c… view at source ↗
Figure 3
Figure 3. Figure 3: Left: observed spectral line profiles (solid black line) and theoretical rotational profiles (solid red line) of the He I 6678 ˚A line for J064043.22+094601.6 (top; S/NR=382.33) and J203906.45+414059.7 (bottom; S/NR=62.68). Right: Fourier transform of the observed profiles (solid black line) and the theoretical rotational profiles (dashed red lines) for J064043.22+094601.6 with v sin i = 197.6 km·s −1 (top… view at source ↗
Figure 5
Figure 5. Figure 5: The v sin i distribution of 479 Be-type stars. The yellow, black and red dashed lines represent the 15th per￾centile (80 km·s −1 ), the mean value (201 km·s −1 ) and the 85th percentile (315 km·s −1 ) of v sin i, respectively. km ·s −1 and v sin i= 106.5 km ·s −1 . The central verti￾cal dashed blue line shows the position of the first zero. 3.2. Validation To validate the reliability of v sin i for Be-type… view at source ↗
Figure 6
Figure 6. Figure 6: Left: histogram (left y-axis) and cumulative fractions (right y-axis) of v sin i for 479 Be-type stars with different Hα emission line morphologies. The number of Be-type stars in each category is also marked. Right: from top to bottom, the LAMOST MRS spectra show Hα emission profiles of four Be-type stars, each with a different Hα emission line morphology. The colors used for the histogram and cumulative … view at source ↗
Figure 7
Figure 7. Figure 7: The histogram (left Y-axis) and cumulative fractions (right Y-axis) of v sin i for Be-type stars in the field (black), clusters (green), OB associations (red), and the combined sample of clusters and OB associations (blue). The gray dashed line indicates where the cumulative fractions equal one [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The distribution of rotation velocities of 479 Be￾type stars. The black histogram represents the observed v sini distribution binned to 40 km·s −1 . The red solid line marks the deconvolved v sini (v), which was obtained using the iterative procedure of Lucy (1974), assuming a Gaussian distribution and using four iterations. random rotation axes. This method has been used to optimize the fit to observation… view at source ↗
Figure 9
Figure 9. Figure 9: From left to right, the mean values of v sini for field and cluster Be-type stars in the MW, LMC, and SMC are compared. The black and blue squares represent the mean v sin i of the field and cluster Be-type stars in the different literature, respectively. The numbers (➀, ➁, · · · ) in each metallicity region correspond to specific labels. For example, ➀ refers to Dachs et al. (1992), where the number in pa… view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of v sin i and ∆Vα for 108 Be￾type stars (orange circles) and 97 Be-type stars with shell (blue circles). The blue and orange dashed lines represent the fitting results of the correlation between v sin i and ∆Vα. The different color histograms in the top and right panels correspond to the marks of the main panel. tended circumstellar envelope of ionized gas, increasing the separation between … view at source ↗

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