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ChronoFlow: A Data-Driven Model for Gyrochronology

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A neural density model trained on ~7,600 open-cluster rotation periods dates star clusters to ~15% (0.06 dex) and individual stars to ~0.7 dex, learning the full rotation distribution instead of fitting a spin-down law.

desk verdict A genuinely useful catalog and a flexible gyrochronology model; just don't treat the 0.06 dex as an absolute age accuracy — it's precision on the fiducial age scale. read the letter →

arxiv 2412.12244 v2 pith:W4BQRIPN submitted 2024-12-16 astro-ph.SR astro-ph.GAastro-ph.IMcs.LG

Phil R. Van-Lane , Joshua S. Speagle , Gwendolyn M. Eadie , Stephanie T. Douglas , Phillip A. Cargile , Catherine Zucker , Yuxi (Lucy) Lu , Ruth Angus This is my paper · ORCID
classification astro-ph.SRastro-ph.GAastro-ph.IMcs.LG PACS 97.10.Kc98.20.Di
keywords gyrochronologystellarrotationopenclustersagesnormalizingflowsmachinelearningmagneticbrakingGaiaDR3
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 claims that stellar ages can be read from rotation periods far more flexibly than analytic spin-down laws allow, by training a neural density estimator on the largest standardized catalog of rotating stars in open clusters yet assembled: about 7,600 stars across 30 clusters and associations spanning 1.5 Myr to 4 Gyr. The model, ChronoFlow, learns the full probability distribution of rotation period at each color and age, including the dispersion that real coeval stars show, rather than fitting a single spin-down track. Leave-one-cluster-out tests recover the literature ages of clusters to a statistical scatter of 0.06 dex (~15%), and individual stellar ages to about 0.7 dex. The paper argues this makes gyrochronology competitive with isochrone dating for coeval populations and viable for individual low-mass stars, and it folds the systematics from dust maps, membership, and calibration ages into a total error budget of 0.08 dex.

What carries the argument

The load-bearing object is the conditional normalizing flow: a neural spline flow with three transform layers and eight bins trained by minimizing the negative log-likelihood of $P(P_{\mathrm{rot}} \mid C_0, \sigma_{C_0}, \tau)$, with the flow density blended against a uniform background component weighted by each star's cluster membership probability and a fixed 5% outlier probability. Conditioning the density on color and photometric uncertainty is what insulates the model from selection effects in color and age, because at every age the conditional density integrates to one along the color axis rather than reflecting where stars happen to be observed. For inference, the same learned density is plugged into a factorized Bayes equation, and the cluster posterior is built as the product of individual stellar likelihoods times a uniform age prior over 1 Myr to 13.8 Gyr. The training catalog is equally load-bearing: 16 literature rotation catalogs harmonized onto Gaia DR3 photometry, de-reddened with three-dimensional dust maps, with HDBScan membership probabilities where available.

What would settle it

Hold out from training a cluster whose age is known independently of isochrone fitting, for example from the lithium-depletion boundary or from eclipsing binaries, and test whether ChronoFlow recovers that age within its claimed 0.06 dex scatter as it does for the isochronal labels in leave-one-cluster-out tests. If the scatter does not reproduce against independent ages, the claimed precision is an artifact of training-label agreement.

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Extended reading notes

Core claim

ChronoFlow models the conditional distribution $P(P_{\mathrm{rot}} \mid C_0, \sigma_{C_0}, \tau)$ — rotation period given de-reddened Gaia color, photometric uncertainty, and age — as a conditional normalizing flow, so it captures the observed width and shape of the rotation sequence at fixed color and age instead of collapsing it to a mean track. Inserted into the Bayesian identity $P(\tau \mid P_{\mathrm{rot}}, C_0, \sigma_{C_0}) \propto P(P_{\mathrm{rot}} \mid C_0, \sigma_{C_0}, \tau)\,P(C_0 \mid \tau, \sigma_{C_0})\,P(\tau)$, the learned density yields posterior age distributions for single stars, and the product of stellar likelihoods across a cluster's members yields its age posterior. In leave-one-cluster-out tests the inferred cluster ages scatter around the fiducial literature ages with an intrinsic scatter of 0.06 dex and no systematic offset, while individual stellar posteriors carry a median $1\sigma$ width of about 0.7 dex, with 79% of the fiducial stellar ages falling inside the $1\sigma$ interval. The authors present this as evidence that a fully data-driven, dispersion-aware model can forward-model rotational evolution and date both coeval populations and individual stars across a wider parameter space than existing empirical models.

Load-bearing premise

The load-bearing premise is that the literature cluster ages used as training labels are accurate and that the rotation-age relation is universal across clusters at fixed color, so if the isochronal fiducial ages are systematically biased, or if metallicity or formation conditions shift rotation at fixed age and color, the measured 0.06 dex scatter reflects agreement with those labels rather than absolute age accuracy.

Editorial extensions

If this is right

  • Cluster ages from rotation alone reach about 15% statistical precision, which is competitive with isochrone fitting for low-mass main-sequence populations where isochrones are weakest.
  • Individual stellar ages to roughly 0.7 dex become available for main-sequence FGKM stars, enough to place the Sun at $5.1^{+1.7}_{-1.4}$ Gyr, consistent with its true age.
  • The learned densities act as evolutionary tracks that reproduce known features such as stalled spin-down near $C_0 \approx 1$–$2$ and delayed convergence for the reddest stars, giving physical spin-down models a direct target to explain.
  • Previously uncalibrated systems can be dated immediately: the paper reports $245^{+40}_{-34}$ Myr for M34, $132^{+20}_{-18}$ Myr for NGC 2516, $121^{+48}_{-25}$ Myr for NGC 6709, and $142^{+26}_{-21}$ Myr for the Theia 456 stream.
  • The standardized catalog of 7,615 rotators, with membership probabilities and propagated photometric errors, is released as a public benchmark for future gyrochronology work.

Reading between the lines

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

  • If the claimed precision survives contact with ages measured independently of isochrone fitting — lithium-depletion boundaries, eclipsing binaries, asteroseismology — gyrochronology could become the default cheap age estimator for low-mass stars across the disk; the 0.7 dex single-star scatter and the bias against intermediate-period fast rotators would still limit precision uses such as exoplanet
  • The five outlier clusters are worth attention as a self-test: ChronoFlow's ages for M34 and NGC 2516 side with a younger subset of the literature, which indicates the model can disagree with its own training labels in particular cases rather than merely memorizing them.
  • A concrete next test that the paper's own caveats point to is metallicity: because all its clusters are near-solar, ChronoFlow cannot yet say whether rotation-age relations differ with composition, so rotation data for metal-poor or metal-rich clusters would either confirm universality or force metallicity into the model.
  • The same conditioning architecture could be retargeted at other age-sensitive observables, such as activity indices, which the paper flags as a future direction.
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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 presents ChronoFlow, a conditional normalizing-flow model for the joint distribution of rotation period and color at fixed age, trained on a newly compiled catalog of approximately 7,600 stars in 30 open clusters and associations (1.5 Myr to 4 Gyr). The authors standardize Gaia DR3 photometry, de-redden using 3D dust maps, assign cluster membership probabilities, and adopt fiducial ages from four PARSEC-based literature catalogs. They validate the model with leave-one-cluster-out cross-validation for cluster ages and batch LOOCV for individual stellar ages, report systematic uncertainties due to dust maps, membership, and calibration ages, and apply the model to M34, NGC 2516, NGC 6709, and Theia 456. The central quantitative claims are a cluster-age statistical uncertainty of 0.06 dex (about 15%), an individual stellar age uncertainty of 0.7 dex, and a total cluster-age error budget of 0.08 dex after adding 0.06 dex of systematics.

Significance. The paper makes a strong empirical contribution: it assembles the largest standardized open-cluster rotation catalog to date, makes the code publicly available, and introduces a flexible probabilistic model that captures the observed rotational dispersion without imposing a parametric spindown law. The LOOCV scheme is a meaningful test of generalization to unseen clusters, and the systematic tests in Section 6 and Appendix G are careful and well documented. The comparisons to GPgyro and gyro-interp help establish the practical regime of the model. If the headline error budget is properly qualified as precision on the adopted literature age scale, the work provides a useful empirical baseline and a forward-modeling tool for gyrochronology.

major comments (3)
  1. [§5.1, Table 3, §G.4]
  2. [§5.1, Figure 11]
  3. [§5.2, Figures 13–16]
minor comments (4)
  1. [Figure 5 caption]
  2. [Figure 12 caption]
  3. [§6.1, §6.2]
  4. [§5.1.1]

Circularity Check

0 steps flagged · score 1.0 of 10

No construction-level circularity: the age predictions are a genuinely held-out supervised regression, but the error budget is scale-relative to the adopted literature ages.

full rationale

ChronoFlow's derivation is a supervised density-estimation pipeline: rotation periods, colors, and photometric uncertainties are inputs; literature cluster ages in Section 2.4 are training labels; the conditional normalizing flow is trained with the loss in Eq. (10); and cluster/stellar age posteriors follow from Bayes' rule in Eq. (5). The headline 0.06 dex cluster-age uncertainty comes from leave-one-cluster-out cross-validation (Section 5.1), where the held-out cluster's own age label was not used to fit the model, so the recovery is not forced by construction. The systematic budget in Table 3 is likewise measured by retraining under alternative dustmaps, membership cuts, and age scales (Appendices G.1-G.4), not by re-fitting the predicted ages. The main caveat, which the paper itself acknowledges in Section 6 and Appendix G.4 ('Since cluster ages are inherently model-dependent, the choice of model calibration ages affects age inference'), is that a common multiplicative offset in all training-age labels would leave the LOOCV residuals essentially unchanged, so the 0.06 dex statistical and 0.08 dex total figures bound precision on the adopted age scale rather than absolute age accuracy against an external zero-point. The external Sun check (Section 5.2) is a single anchor with roughly 1.5 Gyr uncertainty. This is a calibration identifiability limitation, not a case where a prediction reduces by construction to its inputs; the self-citation to Van-Lane et al. (2023) is only a proof-of-concept reference and is not load-bearing here.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central constraints are the chosen age labels, the universal-relation assumption, and a handful of hand-set thresholds. No new physical entities are introduced.

free parameters (6)
  • Outlier probability pout = 0.05
    Universal outlier probability in the mixture loss (Eq. 11), set by hand to account for binaries and contaminants; not fit to data.
  • Color cut for age inference = C0 = 0.5 (text says C0 > 0.5; figure captions say exclude (BP-RP)0 < 0.5)
    Stars blueward of the Kraft break are excluded from age posteriors because they have radiative envelopes; the exact cut is chosen ad hoc based on Beyer and White (2024).
  • Duplicate-period quality cut = df < 0.2
    Stars with two rotation periods differing by more than 20% are dropped (Section 2.1); threshold chosen to remove the discrepant tail.
  • Default cluster membership probability = 0.9 (and 0.5, 0.7, 0.84, 0.85 in different clusters)
    Adopted per-cluster membership priors for catalogs without quantitative probabilities (Appendix E).
  • Dust conversion constant = 0.829 (Eqn. 1)
    Ratio derived partly from a private communication (Xiangyu Zhang 2024), so not fully public.
  • Normalizing flow hyperparameters = 3 transform layers, 8 bins, 5000 epochs, 1e-3 to 5e-7 learning rate
    Chosen via validation loss experiments (Section 4); hand-picked architecture.
assumptions (6)
  • standard math Bayes' theorem and probability calculus
    Used to build the age inference framework in Eqns. (3)-(5).
  • domain assumption Uniform age prior on [1, 13800] Myr
    Eqn. (6); assumes no prior information on stellar age beyond the range.
  • domain assumption Color distribution P(C0|tau) is uniform in [-0.05, 3.8] and independent of age
    Eqns. (7)-(8); the paper acknowledges this is not strictly true because the bluest stars turn off the main sequence.
  • domain assumption The rotation-age relation is universal across clusters at fixed de-reddened color
    Joint training on all clusters in Section 4 assumes a single underlying distribution.
  • domain assumption Fiducial literature cluster ages from G+18, B+19, CG+20, L+23 are accurate
    Used as training labels and as the test targets in LOOCV, Sections 2.4 and 5.1.
  • domain assumption De-reddening conversions in Eqns. (1)-(2) are correct
    Underpins all colors; partly based on private communication and literature chains.

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Cite this review

Pith. "Pith review of ChronoFlow: A Data-Driven Model for Gyrochronology." pith.science (2026). https://pith.science/paper/W4BQRIPN

@misc{pith2026241212244,
  author       = {Pith},
  title        = {Pith review of: ChronoFlow: A Data-Driven Model for Gyrochronology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4BQRIPN}},
  note         = {Machine review of arXiv:2412.12244}
}
abstract

Gyrochronology is a technique for constraining stellar ages using rotation periods, which change over a star's main sequence lifetime due to magnetic braking. This technique shows promise for main sequence FGKM stars, where other methods are imprecise. However, the observed dispersion in rotation rates for similar coeval stars has historically been difficult to characterize. To properly understand this complexity, we have assembled the largest standardized data catalog of rotators in open clusters to date, consisting of $\approx$8,000 stars across 30 open clusters/associations spanning ages of 1.5 Myr to 4 Gyr. We have also developed ChronoFlow: a flexible data-driven model which accurately captures observed rotational dispersion. We show that ChronoFlow can be used to accurately forward model rotational evolution, and to infer both cluster and individual stellar ages. We recover cluster ages with a statistical uncertainty of 0.06 dex ($\approx$15%), and individual stellar ages with a statistical uncertainty of 0.7 dex. Additionally, we conducted robust systematic tests to analyze the impact of extinction models, cluster membership, and calibration ages. These contribute an additional 0.06 dex of uncertainty in cluster age estimates, resulting in a total error budget of 0.08 dex. We apply ChronoFlow to estimate ages for M34, NGC 2516, NGC 6709, and the Theia 456 stellar stream. Our results show that ChronoFlow can precisely estimate the ages of coeval stellar populations, and constrain ages for individual stars. Furthermore, its predictions may be used to inform physical spin down models. ChronoFlow is publicly available at https://github.com/philvanlane/chronoflow.

Figures

Figures reproduced from arXiv: 2412.12244 by the authors.

Figure 1
Figure 1. Distribution of the fractional difference (df ) in Prot for stars having two measurements. The value we use as a quality cut (0.2) is illustrated by the dashed grey line. The value of 0.2 was chosen to exclude the tail of stars with very discrepant Prot measurements, while keeping the large majority of these stars. In this work we present our updated model: ChronoFlow, which models rotational evolution flexibly with… view at source ↗
Figure 2
Figure 2. De-reddened color-rotation space for all stars in our final catalog, including the young subgroups of Taurus and Sco-Cen. Stars are color-coded according to the age of their cluster. Older stars appear to have generally converged onto slowly rotating sequences more than younger stars. pius, or ρ Ophiucus, but were not part of the Ratzenb¨ock et al. (2023) or Krolikowski et al. (2021) catalogs (which we use as our so… view at source ↗
Figure 3
Figure 3. Distribution of (BP −RP)0 uncertainties, includ￾ing propagation of parallax, dustmap, and conversion errors. There is a minimum value of 0.01415 indicated by the red dashed line; this is the DR3 systematic floor corresponding to errors of 0.01 in the BP and RP bands. This plot is trun￾cated at 0.2 mag, but there is a thin tail that extends to 1.24 mag. a cut to remove pre-MS (PMS) stars. Although the rotational beha… view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: Literature age estimates for our clusters, demonstrating the coverage in our catalog. The values from the four main catalogs are highlighted explicitly as red, green, pink, and orange points, and the average value is shown as a black diamond (with an error bar showing …
Figure 5
Figure 5. Figure 5: A zoomed in view of M67 from [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: A comparison of the distribution described in Eqn. (8) for different values of σC0 . Larger σC0 results in a wider, smoother prior around the expected C0 limits. probabilistic model that excels at modeling complex density distributions. We trained ChronoFlow to learn P…
Figure 7
Figure 7. Figure 7: P(Prot | C0, σC0 , τ ) (background shading) from ChronoFlow at each cluster’s fiducial literature age. Observations from our catalog are plotted as circles (sized by cluster membership probability). We calculate this using representative values for σC0 and pcl of 0.028…
Figure 8
Figure 8. Figure 8: Analagous to [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Age probability distributions for a sample star from the NGC 2281 cluster (log τ = 9.1, Prot = 10.6 d, C = 0.88). The posterior is clearly likelihood-driven but also smoothed out by the prior, and peaks at an age slightly older than the fiducial cluster age (vertical g…
Figure 11
Figure 11. Figure 11: Inferred cluster ages from LOOCV compared to the fiducial values from literature. Subgroups of the Taurus and Sco-Cen regions are plotted in grey, and the five anomalous clusters are plotted in red. The best fit line is τinf = (1.00 ± 0.02) · τfid − (0.00 ± 0.06), wit…
Figure 12
Figure 12. Figure 12: A comparison of the five anomalous clusters from [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: Individual stellar age residuals from posterior estimates. Grey stars are members of the Taurus and Sco-Cen associations, and all other clusters are plotted in teal. In the top panel, stars are uniformly distributed along the age axis in order of increasing age, inste…
Figure 16
Figure 16. Figure 16: Average standard deviation of individual stellar posteriors as a function of color and inferred age. Data are averaged over bins of width 0.2 in both dimensions. Grey shaded regions are bins with no data. the cluster ages as the true ages of these stars. However, this…
Figure 15
Figure 15. Figure 15: Average residuals as a function of color and inferred age. Data are averaged over bins of width 0.2 in both dimensions. Grey shaded regions are bins with no data. precision. Precision notably suffers for stars having (BP − RP)0 ≳ 3.0 or (BP − RP)0 ≲ 0.75. Therefore, w…
Figure 17
Figure 17. Figure 17: Cluster age inference results from each model when applied to the subset of our data catalog in the models’ effective regimes. The outlier clusters described in §5.1.1 are included. Only stars with a Gaia DR3 teff gspphot measurement were included in this exercise, he…
Figure 18
Figure 18. Figure 18: A comparison of the stellar residuals between ChronoFlow, gyro-interp, and GPgyro in τinf − (BP − RP)0 space, averaged across bins of 0.2 in both dimensions. The residual is the inferred age from each model minus the fiducial age according to our catalog. The left pan…
Figure 19
Figure 19. Figure 19: HDBScan Cluster membership probabilities as a function of color and distance. As expected, there is weaker confidence in stars that are redder and farther away, since their astrometry is less precise. try, they are fundamentally biased lower towards fainter (i.e. redd…
Figure 20
Figure 20. Figure 20 [PITH_FULL_IMAGE:figures/full_fig_p021_20.png]
Figure 21
Figure 21. Figure 21: “Evolutionary” tracks calculated from our model for eight different (BP − RP)0 values. The background shading represents the conditional probability density P(Prot | C0, τ ) for a representative σC0 value of 0.028. Contours represent the 10th, 33rd, 50th, 67th, and 90…
Figure 22
Figure 22. Figure 22: Probability densities P(Prot | C0, τ ) extrapolated to the age of the universe for different characteristic C0 values. The vertical line represents are oldest training age (M67). We have plotted the Sun in the third panel for reference, showing that it fits our extrap…
Figure 23
Figure 23. Figure 23: Cluster CMDs in increasing order of cluster age (excluding the Taurus and Sco-Cen regions). Both the G0 magnitudes and (BP − RP)0 colours are de-reddened using the extinctions from Edenhofer et al. (2023). Isochrones are generated using the fiducial cluster ages from …
Figure 24
Figure 24. Figure 24: Proper motion distributions for each cluster. Cluster members are plotted against a 2000-star sample drawn randomly from between 4σ and 10σ in distance from the cluster center. Marginal distributions for µRA and µDEC are plotted along the axes for both cluster members…
Figure 25
Figure 25. Figure 25: Distribution of stellar log probabilities for each cluster. Clusters shaded grey are sub-groups of the Taurus and Sco-Cen regions. Clusters are ordered from best fit and the top to worst fit at the bottom; some exhibit a clear bimodality [PITH_FULL_IMAGE:figures/full…
Figure 26
Figure 26. Figure 26: Comparison of the stellar log probability distributions for each cluster, calculated using observed data (teal triangles) and 200 samples (per observed star) simulated by ChronoFlow (gold squares). Clusters are ordered by difference between observed and sampled, where…
Figure 27
Figure 27. Figure 27: The effect of varying σC0 on model predictions. Probability density slices in (BP − RP)0 - Prot space are shown as a function of age and σC0 . The pink bars indicate the relative number of stars in each bin (scaled exponentially) [PITH_FULL_IMAGE:figures/full_fig_p04…
Figure 28
Figure 28. Figure 28: Comparison of the extinction on our catalog stars calculated using Edenhofer et al. (2023) and Green et al. (2019). The median uncertainty is shown in both directions in the upper left for reference (note that the Edenhofer uncertainty is much smaller than the Bayesta…
Figure 29
Figure 29. Figure 29: Cluster age posterior probabilities for the Pleiades for each of our four test cases comparing the E23 and B19 dustmaps [PITH_FULL_IMAGE:figures/full_fig_p042_29.png]
Figure 31
Figure 31. Figure 31: A comparison of the LOOCV results for three clusters when the test data were selected exclusively from individual membership catalogs: Curtis et al. 2020 (C+20), Godoy-Rivera et al. 2021 (GR+21), Long et al. 2023 (L+23), and Rampalli et al. 2021 (R+21). The vertical g…
Figure 32
Figure 32. Figure 32: A comparison of the mean and variation (1σ shown) in residuals from five different LOOCV models. The pink lines indicate the standard deviation in literature values used for each cluster across the models. The young sub-groups of Taurus and Scorpius are not shown here…

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Forward citations

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · cited by 4 Pith papers

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    This corresponds to stars with a Gaia RUWE value of <1.4, which is a cut we apply in our final selection anyways

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    We also exclude four Pleiades stars that L23 sourced directly from the Hartman et al. (2010) catalog, but for which the original rotation period was measured in other works (Prosser et al. 1993; a reference labelled P95 which we assume to be Prosser et al. 1995; and Messina 2001), and for which Hartman et al. (2010) were not able to measure rotation perio...

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    If there are multiple matches, we consider the angular and magnitude differences, and select the best match based on both criteria

    If a star had a DR2 ID in the source catalog, we found the nearest neighbors in DR3 using the gaiaedr3.dr2 neighbourhood table. If there are multiple matches, we consider the angular and magnitude differences, and select the best match based on both criteria. We find a match for 2,495 rotators this way

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    and Decl

    If the star had a DR2 ID but no neighbor in the above step, or did not have a Gaia DR2 ID in the source catalog, we used topcat (Taylor 2005) and/or the Gaia archive search to find the closest match within 1” using the R.A. and Decl. values provided with the catalog. In some cases this radius had to be extended to 2” to find a match

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    There were 89 stars in the Godoy-Rivera et al. (2021) catalog for which no R.A. or Decl. was provided in that paper, but for which we found measurements in the rotation source catalogs for those rotators. 17 of these had Gaia DR3 matches within 1”, however four of those 17 were ambiguous so we exclude those from our sample. 36 V an-Lane et al. The other 7...

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    stability values

    The crossmatches were all double checked using apparent magnitudes in the G and V bands where available (we converted G to V and vice versa where necessary using the G − V and B − V relationships presented in table 5.8 of van Leeuwen et al. 2018). In some cases, better matches...

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    We instead used Lithium depletion ages for the clusters that have such estimates available (α Persei, IC 2391, IC 2602, NGC 2451A, NGC 2547, and Pleiades). These estimates come from Stauffer et al. (1998), Stauffer et al. (1999), Oliveira et al. (2003), Jeffries & Oliveira (20...

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    A model where we prioritize the B+19 and G+18 ages where available, then L+23, then CG+20. We use the average of B+19 and G+18 for 11 clusters (where both have estimates), just B+19 for eight clusters, G+18 for four clusters, L+23 for two clusters, and CG+20 for the last three...

    2023

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