Isochrone ﬁtting of Galactic globular clusters – V. NGC 6397 and NGC 6809 (M55)

We ﬁt variouscolour–magnitudediagrams(CMDs) of the Galactic globularclusters NGC 6397 and NGC 6809 (M55) by isochrones from the Dartmouth Stellar Evolution Database (DSED) and Bag of Stellar Tracks and Isochrones (BaSTI) for 𝛼 –enhanced [ 𝛼 /Fe] = + 0 . 4. For the CMDs, we use data sets from HST , Gaia , VISTA, and other sources utilizing 32 and 23 photometric ﬁlters for NGC 6397 and NGC 6809, respectively, from the ultraviolet to mid-infrared. We obtain the following characteristics for NGC 6397 and NGC 6809, respectively: metallicities [Fe/H] = − 1 . 84 ± 0 . 02 ± 0 . 1 and − 1 . 78 ± 0 . 02 ± 0 . 1 (statistic and systematic uncertainties); distances 2 . 45 ± 0 . 02 ± 0 . 06 and 5 . 24 ± 0 . 02 ± 0 . 18 kpc; ages 12 . 9 ± 0 . 1 ± 0 . 8 and 13 . 0 ± 0 . 1 ± 0 . 8 Gyr; reddenings 𝐸 ( 𝐵 − 𝑉 ) = 0 . 178 ± 0 . 006 ± 0 . 01 and 0 . 118 ± 0 . 004 ± 0 . 01 mag; extinctions 𝐴 V = 0 . 59 ± 0 . 01 ± 0 . 02 and 0 . 37 ± 0 . 01 ± 0 . 04 mag; extinction-to-reddening ratio 𝑅 V = 3 . 32 + 0 . 32 − 0 . 28 and 3 . 16 + 0 . 66 − 0 . 56 . Our estimates agree with most estimates fromthe literature. BaSTI gives systematically higher [Fe/H] and lower reddenings than DSED. Despite nearly the same metallicity, age, and helium enrichment, these clusters show a considerable horizontal branch (HB) morphology diﬀerence, which must therefore be described by another parameter. This parameter must predominantly explain why the least massive HB stars (0.58–0.63 solar masses) are only found within NGC 6809. Probably they have been lost by the core-collapse cluster NGC 6397 during its dynamical evolution and mass segregation. In contrast, NGC 6809 has a very low central concentration and, hence, did not undergo this process.


INTRODUCTION
In Gontcharov, Mosenkov & Khovritchev (2019, hereafter Paper I), Gontcharov, Khovritchev & Mosenkov (2020, hereafter Paper II), Gontcharov et al. (2021, hereafter Paper III), and Gontcharov et al. (2023, hereafter Paper IV) we used theoretical stellar evolution models and their corresponding isochrones to fit colour-magnitude diagrams (CMDs) for the Galactic globular clusters (GCs) NGC 288, ★ E-mail: georgegontcharov@yahoo.com NGC 362, NGC 5904 (M5), NGC 6205 (M13), NGC 6218 (M12), NGC 6362,and NGC 6723.This series of papers is inspired by the recent appearances and improvements for models/isochrones and photometric data sets of individual cluster members in ultraviolet (UV), optical, and infrared (IR) bands.In particular, we use photometric data from the Hubble Space Telescope (HST; Piotto et al. 2015, Nardiello et al. 2018, hereafter NLP18, Simioni et al. 2018, hereafter SBA18) Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) as the unWISE catalogue (Schlafly et al. 2019), various groundbased telescopes by Stetson et al. (2019, hereafter SPZ19), and other sources.Moreover, the precise parallaxes and proper motions (PMs) from HST and Gaia EDR3 1 allow us an accurate selection of GC members.To fit CMDs we use theoretical models of stellar evolution, namely the Dartmouth Stellar Evolution Database (DSED, Dotter et al. 2007) 2 and a Bag of Stellar Tracks and Isochrones (BaSTI, Pietrinferni et al. 2021) 3 .These models are presented by user-friendly online tools in order to calculate isochrones for low metallicity, various levels of helium abundance, and -enhancement, which are typical in GCs (Monelli et al. 2013;Milone et al. 2017).These isochrones reproduce different stages of stellar evolution, namely the main sequence (MS), turn-off (TO), subgiant branch (SGB), red giant branch (RGB), horizontal branch (HB), and asymptotic giant branch (AGB).Best-fitting isochrones provide us with age, distance, reddening, and metallicity [Fe/H] for a cluster dominant population or a mix of populations.We cross-identify data sets to estimate systematic differences between them, convert the derived reddenings into extinction for each filter we consider, and draw an empirical extinction law (i.e. a dependence of extinction on wavelength) for each combination of cluster, data set, and model.
In this paper, we fit the pair of GCs NGC 6397 and NGC 6809 (also known as Messier 55 or M55).These clusters are considerably contaminated by foreground and background stars (in particular, those of Sagittarius dwarf galaxy, see Siegel et al. 2011).Hence, their data sets should be cleaned with PMs and parallaxes.Accordingly, we can fit isochrones directly to a bulk of certain cluster members in very clean CMDs, without calculation of any fiducial sequence.Since these clusters are similar in metallicity, age, helium enrichment, and reddening, it is fruitful to consider their relative estimates.Moreover, this similarity makes NGC 6397 and NGC 6809 interesting to find an explanation for their significant HB morphology difference besides metallicity, age, and helium enrichment.
Previous isochrone fittings of the clusters since Dotter et al. 1 The photometry and astrometry of GCs are exactly the same in Gaia EDR3 and DR3. 2 http://stellar.dartmouth.edu/models/ 3 http://basti-iac.oa-abruzzo.inaf.it/index.html(2010) and corresponding reddening and age values are presented in Table 1.Their results can be compared with ours (see Sect. 5).Unfortunately, earlier fittings by Alcaino et al. (1997); Anthony-Twarog & Twarog (2000); Gratton et al. (2003); Richer et al. (2008) for NGC 6397 and by Piotto & Zoccali (1999) for NGC 6809 seem to be obsolete due to incomplete or too simple models.NGC 6397 and NGC 6809 are among metal-poor GCs, with [Fe/H]<-1.7.They are valuable candidates to verify modern models/isochrones in such a low-metallicity regime, especially in application to the data sets never fitted before.
Most of the studies in Table 1 fit Victoria-Regina (VR, VandenBerg & Denissenkov 2018) or DSED isochrones to the HST/ACS data sets. 4Our study stands out, since it engages the BaSTI isochrones (together with the DSED ones) and many more data sets, which have appeared or improved recently.The number of data sets and photometric measurements, fitted in our study, is an order of magnitude higher than in any study before.
The outline of this paper is as follows.We present some properties of NGC 6397 and NGC 6809 in Sect.2, theoretical models and isochrones used -in Sect.3, and data sets used -in Sect. 4. The results of our isochrone fitting are introduced and discussed in Sect. 5.In Sect.6, we summarize our main findings and conclusions.

PROPERTIES OF THE CLUSTERS
Table 2 presents some properties of NGC 6397 and NGC 6809.
The populations of each cluster differ in helium abundance .Milone et al. (2017) estimated the fraction of the first (primordial) population of stars as 0.345 ± 0.036 in NGC 6397 and 0.311 ± 0.029  Schlafly & Finkbeiner (2011) 0.16 0.12 ( − ) (mag) from Meisner & Finkbeiner (2015) 0.31 0.15 ( − ) (mag) from Gontcharov et al. (2022) 0.17 0.12 in NGC 6809.VandenBerg & Denissenkov (2018) found the fractions of three populations in NGC 6809 as 51%, 41%, and 8% with = 0.25, 0.265, and 0.28, respectively.Mucciarelli et al. (2014) derived a nearly primordial average = 0.24 ± 0.02 for NGC 6397 using a large data set of helium abundances obtained with a high-resolution spectrograph.Milone et al. (2018) found a small average helium difference between the populations, as well as a small maximum internal helium variation: Δ = 0.006 ± 0.009 and Δ = 0.008 ± 0.011 for NGC 6397 and Δ = 0.014 ± 0.008 and Δ = 0.026 ± 0.015 for NGC 6809.Lagioia et al. (2021) found no evidence of intrinsic broadening of the AGB due to helium abundance variation for both NGC 6397 and NGC 6809.However, such broadening of the RGB is seen in some of our CMDs and may suggest that NGC 6809 has some stars with rather high , as discussed in Sect.4.4.Kaluzny et al. (2014) derived ≈ 0.25 for NGC 6809 from their consistent mass-radius, mass-luminosity, and CMD fitting of the cluster's eclipsing binary V54.We verify in Sect. 5 that ≈ 0.26 also does not contradict to the properties of V54.In Sect.3, we take this information into account to select appropriate for our isochrone-to-data fitting.
Table 2 demonstrates that both NGC 6397 and NGC 6809 have rather accurate metallicity estimates from spectroscopy by Carretta et al. (2009).However, later [Fe/H] estimates demonstrate some issues.For example, for NGC 6809, Rain et al. (2019) found very low [Fe/H]= −2.01 ± 0.02 from their analysis of UVES spectra of 11 stars, while Wang et al. (2017) found [Fe/H]= −1.86 ± 0.06 using the same technique applied to UVES and GIRAFFE spectra.Moreover, Mészáros et al. (2020) derived average [Fe/H]= −1.89 ± 0.09 and −1.76 ± 0.07 for NGC 6397 and NGC 6809, respectively, from high-resolution spectra of a hundred RGB stars of each cluster.Mészáros et al. (2020) found that their [Fe/H] are about 0.15 dex systematically higher than those from Carretta et al. (2009) and from the compilation of Harris (1996), not only for NGC 6397 and NGC 6809, but for the bulk of GCs in their sam-ple.They noted that most of the [Fe/H] difference can be explained through the choice of the reference solar abundance mixture, 6 while the remaining difference may be due to wrong calibrations because of wrong reddening estimates or due to a systematic difference in the temperature scales used, or due to some effects which are not modeled yet.Besides, [Fe/H] estimates from photometry are not always consistent with each other and do not always agree with the estimates of Carretta et al. (2009): e.g.Correnti et al. (2018) found [Fe/H]= −1.88 ± 0.04 for NGC 6397 from isochrone fitting of IR photometry of the faint MS.Moreover, Lovisi et al. (2012) used high-resolution spectra of NGC 6397 stars at various stages and found [Fe/H]= −2.12 ± 0.01 for the TO, while −1.20 ± 0.22 for blue stragglers and −1.94 ± 0.14 for the HB.To explain such discrepancies, Jain et al. (2020) analyze a variation of metallicity between the TO and RGB through the use of hundreds stellar spectra and conclude that at the low metallicity regime of [Fe/H]≈ −2 both synthetic and empirical stellar spectra need to be improved against a considerable systematics.Also these discrepancies may be due to diffusive processes combined with convection affecting in different ways the stars in distinct evolutionary stages (Cassisi & Salaris 2020).Thus, we see that the current absolute accuracy of the iron scale seems to be no higher than ±0.1 dex and, hence, needs improvement and verification by different methods.
Photometry can provide [Fe/H] estimates for such a verifica- 6 Mészáros et al. (2020) and Carretta et al. (2009) are based on the solar metallicity estimates from Grevesse, Asplund & Sauval (2007) and Gratton et al. (2003), respectively.Gontcharov & Mosenkov (2018) show that models of the Galaxy combined with stellar evolution models make solar metallicity tightly related to estimates of reddening/extinction across the whole Galactic dust layer above or below the Sun.Namely, the higher the reddening (more dusty Galaxy) the lower the solar metallicity.In combination with the discussion of Mészáros et al. (2020), this shows a relation between [Fe/H] of GCs and the amount of dust in our Galaxy.
tion.The slopes of the RGB and faint MS (> 3 mag fainter than TO), are sensitive to [Fe/H]. 7We get [Fe/H] as an isochrone fitting parameter (along with reddening, age, and distance) in CMDs with well-populated bright RGB or faint MS.The average values of the derived [Fe/H] estimates are used for fitting the remaining CMDs.We consider it separately for each model.However, both the bright RGB and faint MS are affected by helium enrichment (Savino et al. 2018), crowding or poor astrometry at the cluster field centres, saturation and completeness effects, and systematic errors of photometry.These effects may result in a systematic uncertainty of about 0.2 dex in our [Fe/H] estimate obtained from the pair of a CMD and a model.
Table 2 indicates a relatively small foreground and differential reddening for NGC 6397 and NGC 6809.However, the reddening estimates in Table 2 are not fully consistent when taking into account their stated precision as a few hundredths of a magnitude, which is confirmed by Gontcharov & Mosenkov (2018).Namely, the outliers are a very high reddening estimate by Meisner & Finkbeiner (2015) for NGC 6397 and a very low estimate by Harris (1996) for NGC 6809.Interestingly, in Paper IV, the reddening estimate by Meisner & Finkbeiner (2015) is the higher outlier for NGC 6362, which is located in the same fourth Galactic quadrant as NGC 6397, while the reddening estimate by Harris (1996) is the lower outlier for NGC 6723, which is located in the same first Galactic quadrant as NGC 6809.This may be a result of spatial variations of dust medium properties.
Among the reddening estimates in Table 2, a pair is taken from the new version of our 3D analytical model of dust spatial distribution (Gontcharov et al. 2022), whose predictions agree with isochrone-to-CMD reddening estimates for most middle-and highlatitude GCs.
A mild differential reddening across the field of these clusters is shown by Alonso-García et al. (2012); Milone et al. (2012) and Bonatto, Campos & Kepler (2013, hereafter BCK13).We correct data sets with a large number of cluster members for differential reddening in Sect.4.5.

THEORETICAL MODELS AND ISOCHRONES
We use the following theoretical stellar evolution models and corresponding -enhanced isochrones to fit the CMDs of NGC 6397 and NGC 6809: (i) BaSTI (Hidalgo et al. 2018;Pietrinferni et al. 2021) with various [Fe/H] and helium abundance, [ /Fe]= +0.4,initial solar ⊙ = 0.0172 and ⊙ = 0.2695, diffusion, overshooting, mass loss efficiency = 0.3, where is the free parameter in Reimers law (Reimers 1975).As in Paper III and in Paper IV, we also apply the BaSTI extended set of zero-age horizontal branch (ZAHB) models with the same mass for the helium core and the same envelope chemical stratification but different values for the total mass.This set seems to be a realistic description of stochastic mass loss between the MS and HB.
(ii) DSED (Dotter et al. 2008) with various [Fe/H] and helium abundance, [ /Fe]= +0.4,solar ⊙ = 0.0189 and no mass loss.DSED gives no realistic ZAHB with a stochastic mass loss taken into account.However, DSED provides the HB and AGB isochrones for some filters.We do not use these isochrones for our fitting, following a recommendation by the DSED team (Dotter, private communication).Yet, we present these isochrones in some our CMD figures in order to show that an acceptable description of the HB and AGB is possible with DSED too.
In order to fix an appropriate for isochrone fitting of each cluster, we fit isochrones with different to all CMDs under consideration and conclude that BaSTI isochrones with > 0.25 better fit the blue AGB domain, but only for NGC 6809 and only for data sets covering the whole cluster field, while both BaSTI and DSED isochrones with > 0.25 better fit the faint RGB domain in all CMDs.As noted in Paper IV, this widening of the faint RGB may be due to a segregation into two populations, with higher and lower helium abundance.The remaining CMD domains are fitted by the = 0.267 and 0.25 isochrones equally well, since these isochrones almost coincide.Thus, for our isochrone-to-CMD fitting, we adopt = 0.25 for an unresolved mix of the populations in both the clusters, except the blue AGB and faint RGB, which are fitted with = 0.267 (see Sect. 4.4).Only the isochrones with = 0.25 are shown in our CMD figures for clarity.
We fit the isochrones for a grid within −2.4 <[Fe/H]< −1.5 with a step of 0.1 dex, distances within ±0.8 kpc from the Baumgardt & Vasiliev (2021) estimates with a step of 0.01 kpc, reddenings between zero and twice the highest reddening estimate from Table 2 with a step of 0.001 mag, and ages within 8 − 18 and 8 − 15 Gyr for BaSTI and DSED, respectively, with a step of 0.5 Gyr.

Initial data sets
The following data sets (hereafter twin data sets, see Table 3) are used for both the clusters: (i) the HST Wide Field Camera 3 (WFC3) UV Legacy Survey of Galactic Globular Clusters (the 275 , 336 , and 438 filters) and the Wide Field Channel of the Advanced Camera for Surveys (ACS; the 606 and 814 filters) survey of Galactic globular clusters (Piotto et al. 2015), (NLP18),8 with additional photometry of the same RGB, SGB, and MS stars of NGC 6397 in the WFC3 467 , ACS 435 , and ACS 625 filters (Libralato et al. 2022), 9(ii) Parallel-Field Catalogues (the ACS 475 and 814 filters) of the HST UV Legacy Survey of Galactic Globular Clusters (SBA18), 10(iii) photometry from various ground-based telescopes processed by SPZ19,11 with the NGC 6397 and NGC 6809 data sets processed within the same pipeline and presented recently, 12  (iv) Gaia DR3 photometry in the , BP and RP filters (Riello et al. 2021), 13  (v) SkyMapper Southern Sky Survey DR3 (SMSS, SMSS DR3) photometry in the SMSS , SMSS , SMSS , SMSS , and SMSS filters (Onken et al. 2019), 14  (vi) VISTA and VISTA photometry of the VISTA Hemisphere Survey with the VIRCAM instrument on the Visible and Infrared Survey Telescope for Astronomy (VISTA,VHS DR5; (McMahon et al. 2013)), 15  (vii) Wide-field Infrared Survey Explorer (WISE) photometry in the 1 filter from the unWISE catalogue (Schlafly et al. 2019). 16  The following data sets are used for one of the clusters (see Table 3): (i) HST/ACS photometry of NGC 6397 in the 606 and 814 filters (Richer et al. 2008), 17  (ii) photometry of NGC 6397 in the 439 and 555 filters from the HST Wide Field and Planetary Camera 2 (WFPC2) (Piotto et al. 2002), 18  (iii) Strömgren photometry of NGC 6397 with the 1.54-m Danish Telescope, European Southern Observatory (ESO), La Silla (Grundahl et al. 1999, hereafter GCL99), (iv) CCD Strömgren and photometry of NGC 6397 with the 0.9-m telescope at Cerro-Tololo Inter-American Observatory (CTIO) (Anthony-Twarog & Twarog 2000, hereafter AT2000), 19  (v) photometry of NGC 6397 with the FOcal Reducer and low dispersion Spectrograph 2 (FORS2) mounted at the Very Large Telescope (VLT) UT1 of the European Southern Observatory (ESO) (Nardiello et al. 2015), 20  (vi) photometry of NGC 6397 with the 0.9-m CTIO telescope (Kaluzny 1997), 21  (vii) photometry of NGC 6397 with the 1.54-metre telescope of the Bosque Alegre Astrophysical Station of the Córdoba Observatory, National University of Córdoba, Argentina and 1-metre Swope Telescope of Las Campanas Observatory, Chile (Ahumada et al.

2021),
(viii) the Two Micron All Sky Survey (2MASS) photometry of NGC 6397 in the 2MASS filter (Skrutskie et al. 2006), (ix) photometry of NGC 6809 with the 2.5-m du Pont telescope at Las Campanas Observatory (Kaluzny et al. 2010), 22  (x) photometry of NGC 6809 with the 1-m Swope telescope of Las Campanas Observatory (Narloch et al. 2017), 23  (xi) the fiducial sequences for NGC 6809 in the filters derived by Mandushev et al. (1996) and Mandushev (1998, hereafter MFR) from the photometry with the 2.5-m Du Pont telescope of the Las Campanas Observatory.
Three data sets with the HST/ACS photometry, i.e. presented by (i) Richer et al. (2008), (ii) SBA18, and (iii) NLP18 together with the photometry from Libralato et al. (2022) for some of the same stars, have little, if any, common stars.The Strömgren photometry data sets of GCL99 and AT2000 are independent.All of the data sets from SPZ19, Nardiello et al. (2015), Kaluzny (1997), Kaluzny et al. (2010), Narloch et al. (2017), and MFR are independent.The SPZ19 data sets contain photometry from various initial data sets, but not from the others under consideration.
Each star has photometry in some but not all filters.In total, 32 and 23 filters are used for NGC 6397 and NGC 6809, respectively, spanning a wavelength range between the UV and middle IR.Table 3 presents the effective wavelength eff in nm, number of stars and the median photometric precision (after the cleaning of the data sets, which is described below) for each filter.We calculate the median precision from the precision statements by the authors of the data sets.Then we apply it to evaluate the uncertainties of our results (see appendix A of Paper II and Sect.5).
To clean the data sets, we generally follow the recommendations of their authors to select single star-like objects with reliable photometry.Typically, stars with a photometric uncertainty < 0.12 mag are selected, while for some data sets we apply a higher or lower cut level between 0.08 and 0.2 mag.For the HST WFC3 and ACS photometry, we use stars with |sharp| < 0.15, membership probability > 0.9 or −1, and quality fit > 0.9.For the SPZ19 data sets, we use stars with DAOPHOT parameters < 3 and |sharp| < 0.3.For the SMSS DR3 data sets, we select star-like objects (i.e. with ClassStar> 0.5) and with flags < 8.For the data set of GCL99, we select stars with < 3 and |sharp| < 0.3.For the data set of Nardiello et al. (2015), we select stars with the quality of point spread function fit parameter < 0.5.For the data set of Kaluzny (1997), we select stars with all quality flags 0. For the Narloch et al. (2017) data set, we use stars with a cluster member probability higher than 0.5.

Gaia DR3 cluster members
Similar to Paper IV, accurate Gaia DR3 parallaxes and PMs are used to select cluster members and derive systemic parallaxes and PMs.The distribution of the Gaia DR3 data sets, selected within the truncation radii of the clusters, over the PM components is presented in Fig. 1.The cluster members are those inside the red circles.It is seen that members can be separated from the fore-and background stars.
We now briefly describe the selection of the members.As seen in Table 2, Moreno, Pichardo & Velázquez (2014) and Bica et al. (2019) provide different estimates for the tidal radii of these GCs.Therefore, first, we consider initial Gaia DR3 samples within initial radii which exceed any previous estimate.The cluster centre coordinates are taken from Goldsbury et al. (2010).
Second, we find empirical truncation radii of 41 and 18 arcmin for NGC 6397 and NGC 6809, respectively, as the radii where the  star count surface density drops to the Galactic background.All the data sets, except for the MFR data set with fiducial sequences only, are truncated at these radii to reduce contamination from nonmembers.
Third, we leave only stars with PMs; duplicated_source= 0 (Dup=0), i.e. sources without multiple source identifiers; astrometric_excess_noise< 1 ( < 1); a renormalized unit weight error not exceeding 1.4 (RUWE< 1.4); available data in all three Gaia filters with a precision < 0.12 mag; and a corrected excess factor phot_bp_rp_excess_factor (i.e.E(BP/RP)Corr) between −0.14 and 0.14 (Riello et al. 2021).Note that this cleaning removes almost all stars of the Gaia DR3 data sets within a central arcminute of both the cluster fields.
Fourth, foreground and background stars are rejected as those with an inappropriate parallax (see Paper IV).
Fifth, we begin with the initial systemic PM components cos( ) and from Vasiliev & Baumgardt (2021, hereafter VB21), calculate the standard deviations cos( ) and of the PM components for the cluster members, cut off the sample at 3 , and recalculate the weighted mean systemic PM components.We repeat this procedure iteratively until it stops losing stars in the 3 cut.Faint cluster members with less certain PMs make a negligible contribution to the weighted mean systemic PMs.
The final empirical standard deviations cos( ) = 0.38 and = 0.33 mas yr −1 for NGC 6397 and NGC 6809, respectively (averaged for the PM components) are reasonable, but significantly higher than the mean stated PM uncertainties (0.15 and 0.21 mas yr −1 , respectively), which may mean an underestimation of the latters.
The CMDs of the Gaia DR3 stars from Fig. 1 are shown in Fig. 2. It is seen that Sagittarius dwarf galaxy is a major background contaminant for both the clusters.For NGC 6809 it is clearly seen in Fig. 1.Fig. 2 shows that the galaxy's members also dominate among bright non-members ( < 16 mag) of NGC 6397: the RGB of the galaxy is seen as a bulk of stars several magnitudes fainter than the RGB of NGC 6397.
Fig. 3 presents the final BP − RP versus RP CMDs for the Gaia DR3 clusters members after correction for differential reddening described in Sect.4.5.
Our final weighted mean systemic PMs are presented in Table 5 in comparison to those from VB21 and Vitral (2021).Being obtained from Gaia DR3 by different approaches, these estimates are, nevertheless, consistent within ±0.01 mas yr −1 , i.e. well beneath the Gaia DR3 PM systematic errors (about 0.02 mas yr −1 ), which are estimated by VB21.Since only the statistic uncertainties are evaluated for ours and Vitral (2021)'s estimates, we adopt the dominating systematic uncertainties as the final ones of our PMs.
Similarly, we adopt the total uncertainty of Gaia DR3 par-Table 4. The results of our isochrone fitting for two models and some key CMDs for both the clusters.The colour is the abscissa and the magnitude in the redder filter is the ordinate in all the CMDs, except the AT2000 CMDs where − , − or − are the abscissas and is the ordinate.Each derived reddening is followed by its empirical systematic uncertainty and corresponding ( − ), given in parentheses and calculated using extinction coefficients from Casagrande & VandenBerg (2014, 2018a,b) or CCM89 with V = 3.     6 for  comparison with other estimates in Sect.5.4.Table 7 contains the final lists of the Gaia DR3 cluster members.

Cluster members in other data sets
Almost all authors of the data sets under consideration made an effort to select cluster members.This cleaning is acceptable for the MFR and Piotto et al. (2002) data sets.The original star-by-star data for the MFR data set are not available (we use its fiducial sequences).Anyway, we cannot cross-identify both the data sets with Gaia to improve cluster member selection.This may introduce some additional systematic errors into corresponding results.The level of these errors is estimated from the comparison of our results for various data sets in Sect. 5.The CMDs for these data sets are presented in Figs 4 and 5, respectively.Two observational runs of Mandushev et al. (1996) and Mandushev (1998) 4).
The Gaia DR3 cluster members are among only bright stars of the data sets of SBA18, while faint stars of these data sets draw rather clear CMDs.Therefore, we decide to derive cluster parameters from combined CMDs: we use only Gaia DR3 cluster members among  The remaining data sets are cross-identified with those of Gaia DR3 to reveal cluster members.The improvement is seen from a typical Fig. 10, where contaminated CMDs of NGC 6397 and NGC 6809 for the SPZ19 data sets are compared with the same CMDs for only Gaia DR3 members of these data sets.However, this improvement comes at the expense of a few faint magnitudes lost.
The Gaia membership identification is especially important for NGC 6397 and NGC 6809 in order to overcome a bias due to a non-uniform distribution of the Sagittarius dwarf galaxy stars over their CMDs, as seen in Fig. 10.

Isochrone-to-data fitting
Owing to the accurate selection of the cluster members, our CMDs are more well defined than typical CMDs in the pre-HST and pre-Gaia era.Therefore, we can fit isochrones directly to a bulk of cluster members, without needing to calculate a fiducial sequence.In this case, the best solution corresponds to a minimal sum of the residuals between isochrone's and data set points.We select the best isochrone among those calculated for the parameter grid mentioned in Sect.3.
We have to exclude three CMD domains from the direct fitting: the blue HB, RR Lyrae variables, and blue stragglers, marked I, III, and IV in Fig. 11, respectively.The blue HB, i.e. the area bluer than the turn of the observed HB downward, is excluded, since even its best prediction deviates from the observations in typical CMD when its other domains are fitted well, as seen in Fig. 11.We fit the HB stars between the areas I and II by the BaSTI HB models with = 0.25, while the stars in the area II (blue AGB) are better fitted by the AGB isochrones with higher = 0.267, as noted in Sect.3. Another CMD domain fitted with higher = 0.267 is the faint RGB, marked V in Fig. 11 and mentioned in Sect.3.All the remaining stars are fitted by isochrones with = 0.25.
To balance the contributions of different CMD domains, we assign a weight to each data point.The weight is inversely proportional to the number of stars of a given magnitude for a given data set, i.e. it reflects the luminosity function of a given data set.
Since we fit a zigzag pattern of an isochrone to a zigzag pattern of the bulk of stars, different parts of them are more sensitive to different parameters.Namely, reddening and distance correlate with the overall shift of the pattern along the abscissa (i.e.colour) and ordinate (i.e.magnitude), respectively.Therefore, nearly vertical and nearly horizontal parts of the pattern are more sensitive to the determination of reddening and distance, respectively.Similarly, [Fe/H] is more sensitive to the slopes of the RGB and faint MS.Finally, age correlates with the length of the SGB, as well as with the HB-SGB and SGB-MS magnitude differences, although different definitions of each of these quantities are possible (e.g. the SGB-MS magnitude difference can be defined as the one between the middle of the SGB and the MS of the same colour).
We have checked that the results of the isochrone-to-data fitting obtained by two methods, with and without fiducial sequences, almost coincide.Namely, the derived [Fe/H], ages, distances, and reddenings [converted into ( − )] agree within 0.1 dex, 0.5 Gyr, 80 pc, and 0.01 mag, respectively.The results obtained without fiducial sequences are presented in Table 4.

Differential reddening
Differential reddening (DR) across the fields of both clusters is taken into account following the method of BCK13.Briefly, the cluster field is divided into a cell grid, with the angular resolution being higher in regions containing more stars.Then, the stellar-density Hess diagram (including photometric errors) of each cell is matched to the average (whole field) diagram by applying shifts along the reddening vector that are subsequently converted into DR in the cell.By design, the method assumes that differences in the CMDs can be accounted for entirely by DR.However, other variations of CMD over cluster field, such as photometry zero-point variations, point-spread function variations, telescope focus change, distortion, telescope breathing, stellar population variations, and other reasons, discussed by Anderson et al. (2008), are difficult to separate from DR.
It appears that only data sets with at least 3000 stars provide sufficient coverage of the cluster fields and, hence, draw rather precise DR maps, i.e. the data sets of NLP18, GCL99, Gaia DR3, SPZ19, SMSS, Libralato et al. ( 2022 • the DR maps have little to do with each other as for different data sets, as for different CMDs/colours of the same data set; • DR variations are mostly small: within Δ ( − ) = ±0.04mag after conversion by use of any reliable extinction law; • some CMDs/colours show strong gradients over the fields (e.g. for GCL99 − in Fig. 12) or sharp extremes of DR in a small area (e.g.red peaks for SPZ19 − in Fig. 12).
The peaks in the SPZ19 DR maps can be explained as a manifestation of initial observational data sets (SPZ19 data set combines them) covering small parts of the field and having significant systematics over the field.Given that DR is not large across the fields of NGC 6397 and NGC 6809, these findings show that the effects mentioned above are more important than DR itself in the CMDs of both the clusters.Anyway, our correction of the data sets for DR reduces the scatter of their stars around their ridge lines or best-fitting isochrones in CMDs, e.g. in Fig. 13.Note that the mean DR correction for all the CMDs is exactly zero.This leads to a negligible shift of bulk of stars in the CMDs and does not change an average reddening over the field, which is presented in Table 4.
Since each data set and CMD/colour draw its own DR map, it is not surprising that our DR maps differ from those of Alonso-García et al. (2012).

RESULTS
As in our previous papers, we fit isochrones to a hundred CMDs with different colours.The results for adjacent CMDs appear consistent.We present some interesting CMDs with isochrone fits in Figs 3-10 and Figs 14-16.Other CMDs are presented online or can be provided on request.
We present the obtained [Fe/H], distances, reddenings, and ages for the most important CMDs in Table 4.For comparison, we convert the obtained reddenings into ( − ), given in parentheses, by use of extinction coefficients from Casagrande & VandenBerg (2014, 2018a,b) or Cardelli, Clayton & Mathis (1989, hereafter CCM89) with V = 3. 1. 25  Table 4 provides the empirical systematic uncertainty of obtained reddening after its value.This systematic uncertainty is defined as the maximal deviation of the best-fitting isochrone from the bulk of the stars along the reddening vector (i.e.nearly along the colour).The systematic uncertainty never drops below 0.03 mag and it is usually larger than the predicted statistic uncertainty.The latter is described in the balance of uncertainties (see appendix A of Paper II).The largest values in such pairs of the systematic and statistic uncertainties are shown by the extinction error bars in Figs 17 and 18 with resulting empirical extinction laws.

Issues
The VISTA photometry for the brightest stars is biased, as discussed in Paper IV.This is seen in Fig. 14.However, accurate parameters of the clusters can be obtained by use of the remaining VISTA stars.
In Paper IV, we discussed drawbacks of UV, UV-optical, optical-IR, and IR-IR CMDs with respect to (w.r.t.) a typical optical CMD (with filters within 430 < eff < 1000 nm).Especially, the UV CMDs are highly affected by the multiple population chemical patterns (see Sbordone et al. 2011 andCassisi et al. 2013 for the first results, as well as a recent discussion by VandenBerg, Casagrande & Edvardsson 2022).However, some of these CMDs can give reliable [Fe/H], age, reddening, and distance estimates (see Table 4).
Fig. 15 shows an example UV CMD with the HST/WFC3 336 − 438 colour, where almost all domains are fitted by both the BaSTI and DSED isochrones with reasonable residuals.The maximal colour offset of the best-fitting isochrones from these data is 0.05 mag and it is the same for both the models and both the clusters.Moreover, the maximal colour offsets are < 0.05 mag for the UV CMDs with the − Strömgren colour from the GCL99 and AT2000 data sets, as well as for the CMDs with the − colour from the SPZ19 data sets.For comparison, such offsets were 0.08-0.10mag for the UV CMD with HST/WFC3 336 − 438 colour for NGC 6362 and NGC 6723 in Paper IV. Thus, it seems that the BaSTI and DSED UV isochrones better fit UV CMDs for low metallicity GCs (NGC 6397 and NGC 6809) than for those with a higher metallicity (NGC 6362 and NGC 6723).However, Table 4 shows that some distance and age estimates derived from the UV CMDs are unreliable.Thus, the results from UV, UVoptical, optical-IR, and IR-IR pairs (including those in Table 4) are not used for our final estimates.
We do not use the SPZ19 photometry in the filter for our final estimates due to its lower precision than in the filters.Similar to Paper IV, our cross-identification of data sets, which use the same or similar filters, reveals some systematic differences up to 0.04 mag in magnitudes and colours for some data sets.They are common and expected (SPZ19).Our DR corrections reduce these differences and, hence, confirm that they are mostly due to some systematic errors of the data sets (see Sect. 4.5).We do not take 25 Extinction-to-reddening ratio V ≡ V / ( − ) = 3.1 is defined for early type MS stars, while the observed ratio V / ( − ) depends on intrinsic spectral energy distribution of stars under consideration (Casagrande & VandenBerg 2014).For rather cool and metal-poor stars of the GCs under consideration the observed reddening is calculated as ( − ) = V /3.48, while the extinction coefficients are calculated for the median effective temperature 6400 K of the cluster members.into account the residual systematics after the DR correction, since we find little, if any, influence of these systematics to the derived parameters, as seen in Table 4 and in Figs 17 and 18.In particular, similar to Paper IV and in contrast to Paper III, we do not adjust the data sets.The adjustment would slightly decrease the scatter of the derived parameters.However, without the adjustment, we better see the real influence of the data set systematics.Yet, we eliminate the   Accordingly, following the discussion of Mészáros et al. (2020), our estimates support the reference solar abundance mixture from Grevesse, Asplund & Sauval (2007), but not from Gratton et al. (2003).

Reddening and extinction
We verify the agreement of reddening estimates from all CMDs with each other and with an extinction law by combining all derived reddening estimates into empirical extinction laws.These laws are presented in Figs 17 and 18.
Similar to our previous papers, in order to draw these laws, we cross-identify all possible data sets with the 2MASS, VISTA and unWISE data sets and calculate extinctions in all filters from the derived reddenings and IR extinctions.For example, where ( − 1) is obtained from a CMD, while very low extinction W1 in the 1 filter is slightly upgraded iteratively with upgrade of extinction law.
Note that some data set pairs cannot be cross-identified.The main reasons are a very small common field or common magnitude range of such data sets.Namely, the NLP18 data set for NGC 6397 contains only rather faint stars in a small field and, hence, has only a few common stars with 2MASS.Other such  The common HST/ACS 814 filter for the NLP18, Richer et al. (2008), SBA18, andLibralato et al. (2022) data sets allows us to process their reddenings together by adopting the same extinction F814W .Extinctions derived for all HST filters, detectors, and data sets are shown together in Figs 17 and 18 by the red diamonds.These extinction estimates agree with each other following the same smooth extinction laws without outliers.This is a robust confirmation of the systematic accuracy of our reddening and extinction estimates at the level of a few hundredths of a magnitude.Thus, we use all the HST results for its optical filters ( eff > 430 nm) for our final estimates of [Fe/H], age, reddening, and distance.
2MASS versus unWISE and VISTA versus unWISE are crossidentified via common Gaia DR3 cluster members.The VISTA-unWISE CMDs represent a very short wavelength baseline and, hence, provide uncertain age, distance, and [Fe/H].Therefore, fixing these parameters for the VISTA-unWISE pair, we derive only the reddening ( VISTA − 1) as the average difference w.r.t.RP and from SPZ19: The extinctions in Figs 17 and 18 show a low scatter of a few hundredths of a magnitude around the CCM89 extinction law with best-fitting V = 3.3 and 2.9 for NGC 6397 and NGC 6809, respectively.
We derive our final reddening and extinction estimates for the clusters through the use of all 19 and 14 independent optical CMDs for NGC 6397 and NGC 6809, respectively.Table 8 presents the final reddening estimates.The DSED estimates are systematically We calculate our final V estimates as the averages of all its direct measurements (6 for NGC 6397 and 4 for NGC 6809) using equation ( 1) and its counterparts for other IR filters and the Strömgren filter, which is very close to the one.We obtain V = 0.59 ± 0.01 ± 0.02 and 0.37 ± 0.01 ± 0.04 mag (statistical and model-to-model uncertainties) for NGC 6397 and NGC 6809, respectively.Accordingly, the ratio of these V and ( − ) estimates, V = 3.32 +0.32  −0.28 and 3.16 +0.66 −0.56 (total uncertainty) for NGC 6397 and NGC 6809, respectively.
The systematic uncertainty 0.1 dex for [Fe/H] (see Sect. 5.2) is the dominant contribution to systematic uncertainty of all our reddening and extinction results, which is equivalent to ( − ) = 0.01 and V = 0.03.Our ( − ) estimates agree with those in Table 2 for both the clusters, except very high estimate by Meisner & Finkbeiner (2015) for NGC 6397 and very low estimate by Harris (1996) for NGC 6809.We find no reason for these outliers.Also, our ( − ) estimates agree with those from isochrone fitting in Table 1, except very low ( − ) by Martinazzi et al. (2014) for NGC 6397 and by Valcin et al. (2020) for NGC 6809.
Despite the rather sparse HB populations in NGC 6397 and NGC 6809, the statistic uncertainty of their derived distances is lower than the systematic uncertainty.The latter can be estimated from the scatter of the previous estimates of the distance moduli or distances presented in the compilation of all GC distance determinations by Baumgardt & Vasiliev (2021). 26All recent (since Dotter et al. 2010) estimates of the distance modulus for NGC 6397 by isochrone-to-CMD fitting, except outlying 12.12 from Valcin et al. (2020), are within 11.95-12.05(including our own 11.95).Similarly, those for NGC 6809, except outlying 13.70 from Valcin et al. (2020), are within 13.53-13.67(including our own 13.60).Assuming this scatter is due to some systematics, we accept a systematic uncertainty of distance moduli as ±0.05 and ±0.07 for NGC 6397 and NGC 6809, respectively, which converts to ±62 and ±178 pc distance systematic uncertainty.Such large systematics may be due to contamination of the HB and SGB of both the clusters by the MS of Sagittarius dwarf galaxy, as seen in Fig. 10.Note that our distance estimates agree with the most probable distance estimates of Baumgardt & Vasiliev (2021) presented in Table 2: within 31 and 108 pc, i.e. 0.8 and 1.5 of their stated statistical uncertainties for NGC 6397 and NGC 6809, respectively, and well inside the systematic uncertainties.Both ours and Baumgardt & Vasiliev (2021)'s estimates for both the clusters differ considerably from those of Harris (1996) (see Table 2) and, hence, can update them.
We convert our distance estimates into parallaxes to compare them in Table 6 with our parallax estimates from the Gaia DR3 astrometry (see Sect. 4.2) and with those from VB21.A good agreement between the parallaxes within the stated uncertainties is seen.Note that for such nearby GCs, the parallax estimates from the Gaia DR3 astrometry have nearly the same precision as those from our isochrone fitting, unlike more distant GCs in our previous studies, whose isochrone fitting parallaxes are more precise.
The systematic uncertainty of age was discussed and estimated in Section 3.1 of Paper IV.This should take into account the scatter of the previous age estimates in Table 1 and others [e.g.NGC 6397's age of 12.8 ± 0.50 ± 0.75 Gyr (statistic and systematic uncertainties) from a population synthesis study of the white dwarf population by Torres et al. (2015) and 13.4 ± 0.7 ± 1.2 from the luminosity at the TO by Brown et al. (2018)].Taking into account this scatter and the discussion of age uncertainty by VandenBerg & Denissenkov (2018) and Valcin et al. (2020), we assign 0.8 Gyr as the systematic uncertainty of our derived ages.

Eclipsing binary
NGC 6809 contains the detached eclipsing binary V54, whose component masses and radii were measured accurately by Kaluzny et al. (2014).Hence, we can verify the cluster parameters by fitting isochrones to the precise V54 mass-radius relation.Fig. 19 presents our fitting of V54 with the DSED and BaSTI isochrones for various [Fe/H], , and age.A better fit is seen for higher age or lower [Fe/H] or higher .The latter, about = 0.26, is most fruitful to obtain the best fit.

Relative estimates
Similar to Paper IV, we consider the relative estimates for the cluster parameters separately derived for each model.Systematic errors of the models must be canceled out in such relative estimates.Therefore, the relative estimates may be more accurate than the absolute ones.
We use 8 independent CMDs of 5 twin data sets with accurate photometry in optical filters in order to derive relative estimates for the cluster parameters: Table 10 presents the relative estimates.The models are consistent in them, i.e. the distribution of the combined sample of the DSED and BaSTI relative estimates for each parameter is nearly Gaussian and each uncertainty of the combined sample agrees with those of the models.This confirms that, indeed, systematic errors of the models are canceled out in the relative estimates.
The final uncertainties of the relative estimates are the standard deviations from Table 10 divided by the square root of the number of the CMDs and models used (8 CMDs by 2 models).The relative estimates show that NGC 6809 is 2.75 ± 0.02 kpc further, Δ ( − ) = 0.057 ± 0.004 less reddened, 0.06 ± 0.12 Gyr older (i.e. of nearly the same age), and with 0.06 ± 0.03 dex higher [Fe/H] (i.e. of nearly the same metallicity) than NGC 6397.For comparison, the right column of Table 10 presents the absolute differences of the parameters in the sense 'NGC 6809 minus NGC 6397' derived from optical CMDs, as described in Sect.5.2-5.4.A good agreement between the relative estimates and absolute differences is evident.

HB morphology difference
NGC 6397 and NGC 6809 show a considerable HB morphology difference (e.g.see Figs 3,7,10,11,14,and 15) despite nearly the same metallicity and age, which are usually considered as the first and second parameters to explain such a difference (see Paper IV and references therein).Moreover, these clusters have similar low helium enrichments.Therefore, we should describe this HB difference by another parameter, other than metallicity, age, or helium enrichment.
The HB morphology can be presented as the HB types (see Lee, Demarque & Zinn 1994 (2019). 27All these characteristics represent the HBs of NGC 6397 and NGC 6809 as rather blue and similar.Hence, these indices seem to be an incomplete description of the observed significant HB morphology difference of these clusters.
Given −2.0 <[Fe/H]< −1.7, age about 12.5-13.5Gyr, and ≈ 0.25 for both NGC 6397 and NGC 6809, and taking into account the stochastic nature of the mass loss before the HB, e.g.described by the BaSTI ZAHB predictions (see Sect. 3), one obtains a realistic possible scatter of the HB stars within a wide range of masses, at least 0.50−0.78⊙ for our GCs.This possible scatter should be compared to the observed scatter.The latter is estimated from our fitting of the observed colour distribution of the HB and AGB stars by the BaSTI isochrones: 0.63−0.67 and 0.58−0.68⊙ for NGC 6397 and NGC 6809, respectively. 28These estimates are obtained consistently, within ±0.02 ⊙ , for all CMDs with many HB stars.These estimates show a good agreement with those modeled by Gratton et al. (2010) Thus, the main HB morphology difference between these clusters seems to be a narrower HB mass range of NGC 6397 w.r.t.NGC 6809.Note that the wider mass range of the HB stars in NGC 6809 means a larger amount of stars evolved from them, e.g. of RR Lyrae variables and red AGB stars.The latter is seen in our CMDs: NGC 6397 has no RR Lyrae, while there are several in NGC 6809; NGC 6397 contains twice fewer number of the AGB stars on the red side of the RR Lyrae gap.Yet, the difference in the highest mass is rather small (0.67 versus 0.68 ⊙ ).Hence, first of all, the desired HB parameter must explain the existence of the bluest HB stars of 0.58 − 0.63 ⊙ in NGC 6809, but not in NGC 6397.Note that these stars make up the HB blue tail (i.e. the bluest part of the HB on the blue side of the HB knee), which is observed for NGC 6809, but not for NGC 6397.Hence, a natural explanation for the HB morphology difference between these clusters is that either 27 is calculated from cumulative number distributions along the HB in the magnitude and − color.This index varies between 0 and 14 for the most red and blue HB. 28All stars in the middle part of the NGC 6809's HB far from the AGB appear RR Lyrae variables after our star-by-star inspection.
NGC 6397 has lost or NGC 6809 has acquired the bluest HB stars.The desired parameter may be related to a peculiar evolution of these stars, i.e. with an extreme mass-loss, segregation of stellar masses due to cluster evolution, or other peculiar processes.
This loss or acquisition of low mass HB stars may relate to the fact that NGC 6397 is a core-collapse cluster, i.e. the one with a highly compact, bright core, with a surface brightness constantly increasing towards the cluster centre.In contrast, NGC 6809, without core collapse, has a very low central concentration of stars and a roughly flat surface brightness of the cluster core.The core collapse relates to the mass segregation during cluster evolution, when massive stars tend to clump in the cluster centre, while less massive ones populate the outskirts, sometimes escaping the cluster (Martinazzi et al. 2014), as well as to the increase of dynamical interactions among stars in the dense core of post-core-collapse cluster (Meylan & Heggie 1997).The current mass of NGC 6397 is just 10 per cent of its initial mass (Dieball et al. 2017).The HB stars are the least centrally concentrated population and absent in the central area of the core of NGC 6397 (Dieball et al. 2017).Hence, most low-mass HB stars may be lost in the dynamical evolution and mass segregation of NGC 6397.Yet, the additional parameter (after metallicity, age and helium enrichment) is still an issue.Our study may provide sufficient input data to solve it.

CONCLUSIONS
This study continues the series of Paper I, Paper II, Paper III, and Paper IV in the estimation of key parameters of Galactic globular clusters via fitting theoretical isochrones to observed multiband photometry.We have analyzed the low metallicity pair NGC 6397 and NGC 6809 (Messier 55) with similar metallicity, age, helium enrichment, and extinction.The cluster members have been carefully selected through HST and Gaia DR3 proper motions and parallaxes.Accordingly, we provided the lists of reliable members of the clusters, their median parallax (0.416 ± 0.010 and 0.203 ± 0.010 mas for NGC 6397 and NGC 6809, respectively), and systemic proper motions with their total (systematic plus statistic) uncertainties in mas yr We employed the photometry in 32 and 23 filters for NGC 6397 and NGC 6809, respectively, from the HST, Gaia DR3, SMSS DR3, 2MASS, VISTA VHS DR5, unWISE, and other data sets.These filters span a wide wavelength range from the UV to mid-IR, namely from about 230 nm to 4060 nm.As in our previous studies, we crossidentified some data sets with each other.As a result, we could (i) estimate systematic differences between the data sets and (ii) use the 2MASS, VISTA and unWISE photometry with a very low extinction for determination of the extinction in all other filters and drawing of empirical extinction laws.
We fitted the data by the DSED and BaSTI theoretical models of stellar evolution for [ /Fe]= 0.4 with nearly primordial helium abundance ≈ 0.25.As a result, we obtained [Fe/H], reddening, age, and distance as the parameters.BaSTI provides metallicity Δ[Fe/H]≈ 0.06 dex systematically higher than DSED and reddening Δ ( − ) ≈ 0.01 mag systematically lower than DSED.
An important result of this study is the agreed parameters of NGC 6397 and NGC 6809 derived from successful fitting of two recent isochrone sets to all recent photometric data sets, most of which have never been fitted before.To derive reddening, age, and distance, we use 19 and 14 independent CMDs, while 16 and 10 ones to derive [Fe/H] for NGC 6397 and NGC 6809, respectively.
There are pairs of similar data sets for the clusters, which are obtained with the same telescope and/or processed within the same pipeline.We used these data sets to derive very precise relative estimates for the parameters.NGC 6809 appears 2.75 ± 0.02 kpc further, Δ ( − ) = 0.057 ± 0.004 less reddened, 0.06 ± 0.12 Gyr older (i.e. of the same age), and with 0.06 ± 0.03 dex higher [Fe/H] (i.e. of the same metallicity) than NGC 6397.
Despite nearly the same metallicity, age, and helium enrichment, these clusters show a considerable HB morphology difference, which must therefore be described by another parameter.Primarily, this parameter must explain the existence of the least massive HB stars of the blue tail (0.58-0.63 solar mass) only in NGC 6809.Probably such stars have been lost by the core-collapse cluster NGC 6397 in its dynamical evolution and mass segregation, unlike NGC 6809, which has a very low central concentration.

Figure 1 .
Figure 1.The distribution of the Gaia DR3 data sets, selected within the truncation radii of the clusters, over the PM components (mas yr −1 ), after the remaining cleaning of the sample.To make the figure clearer, we only show stars with precise PM components (< 1.5 mas yr −1 ) and photometry in all filters (< 0.07 and < 0.1 mag for NGC 6397 and NGC 6809, respectively).The weighted mean PM and the selection area for the clusters are shown by the white crosses and red circles, respectively.The blue circle shows a concentration of the Sagittarius dwarf galaxy members behind NGC 6809.

Figure 2 .
Figure 2. BP − RP versus CMDs for the stars included (red symbols) and excluded (black symbols) by their parallaxes and PMs after the remaining cleaning of the Gaia DR3 samples.For clearer figure, only stars with precise PM components (< 1.5 mas yr −1 ) and photometry in all filters (< 0.07 and < 0.1 mag for NGC 6397 and NGC 6809, respectively) are shown.A significant background of the clusters is the MS stars (at BP − RP ≈ 1 and < 16 mag) and giants (at BP − RP ≈ 1.4 and < 16 mag) of Sagittarius dwarf galaxy.

Figure 3 .
Figure 3. BP − RP versus RP CMDs for the Gaia DR3 clusters members after correction for differential reddening.The isochrones from BaSTI (red) and DSED (green) for = 0.25 are calculated with the best-fitting parameters from Table 4.

Figure 4 .
Figure 4. − versus CMD with the fiducial sequences from the Mandushev et al. (1996) (black circles) and Mandushev (1998) (purple circles) observational run for NGC 6809.The isochrones from BaSTI (red) and DSED (green diamonds) for = 0.25 are calculated with the best-fitting parameters from Table 4. Appropriate DSED predictions for the HB and AGB are shown by the green squares for illustration purposes.
, shown by different colours, indicate an insignificant systematic difference of 0.019 mag between their TO colours.We cannot fit the Piotto et al. (2002) faintest MS stars by any reliable BaSTI or DSED isochrone and, hence, ignore these stars.Richer et al. (2008), NLP18, and Libralato et al. (2022) have cleaned their HST data sets from non-members by use of dedicated HST PMs: their CMDs are presented in Figs 6, 7, and 8, respectively.Although imperfect, their membership selection cannot be

Figure 5 .
Figure 5. HST/WFPC2 439 − 555 versus 555 CMD from the Piotto et al. (2002) data set for NGC 6397.The isochrones from BaSTI (red) and DSED (green diamonds) for = 0.25 are calculated with the bestfitting parameters from Table 4. Appropriate DSED predictions for the HB and AGB are shown by the green squares for illustration purposes.The black horizontal line shows the cut of the faintest MS stars without a reliable isochrone fitting.

Figure 6 .
Figure 6.The same as Fig. 5 but for the HST/ACS 606 − 814 colour from the Richer et al. (2008) data set for NGC 6397.The black horizontal line shows the cut of the faintest MS stars without a reliable isochrone fitting.

Figure 7 .
Figure 7. HST/ACS 606 − 814 versus 814 CMD from the (NLP18) data set.The isochrones from BaSTI (red) and DSED (green diamonds) for = 0.25 are calculated with the best-fitting parameters from Table 4. Appropriate DSED predictions for the HB and AGB are shown by the green squares for illustration purposes.

Figure 9 .
Figure 9.The same as Fig. 3 but for the HST/ACS 475 − 814 versus 814 CMDs with the Gaia DR3 cluster members from the SBA18 data sets (black) and remaining faint stars from the SBA18 data sets (grey).

Figure 10 .
Figure 10.− versus CMDs with the initial SPZ19 data sets (left) and the Gaia DR3 cluster members from the SPZ19 data sets (right).The isochrones from BaSTI (red) and DSED (green diamonds) for = 0.25 are calculated with the best-fitting parameters from Table 4. Appropriate DSED predictions for the HB and AGB are shown by the green squares for illustration purposes.
),Narloch et al. (2017),Kaluzny (1997), Kaluzny et al. (2010), PS1, 2MASS, VISTA, and all their cross-identifications, with the exception of the NGC 6397 data sets of Piotto et al. (2002) and Ahumada et al. (2021) with insufficient information about stellar coordinates.Four examples of DR maps for NGC 6397 are shown in Fig. 12.All DR maps of both the clusters show that:

Figure 11 .
Figure 11.A central part of the − versus CMDs with the Gaia DR3 cluster members from the SPZ19 data sets (i.e. the data from Fig. 10).The isochrones from BaSTI (red) and DSED (green) for = 0.25, the BaSTI HB for = 0.25 (magenta), and isochrones from BaSTI (orange) and DSED (blue) for = 0.267 are calculated with the best-fitting parameters from Table 4.The grey areas are the CMD domains of the (I) blue HB, (II) blue AGB, (III) RR Lyrae, (IV) blue stragglers, (V) and faint RGB, which are discussed in the text.

Figure 12 .
Figure 12.The differential reddening maps for the same NGC 6397 field obtained by use of four CMDs.

Figure 13 .
Figure 13.A central part of the Gaia DR3 BP − RP versus RP CMD for NGC 6397 from Fig. 3 (a) before and (b) after differential reddening correction.

Figure 14 .
Figure 14.The same as Fig. 3 but for the Gaia -VISTA − VISTA colour.The black horizontal line shows the cut of bright stars with systematically erroneous photometry.

Figure 16 .
Figure 16.The same as Fig. 5 but for the − colour from the (a) Kaluzny et al. (2010) and (b) Narloch et al. (2017) data sets for NGC 6809.The black horizontal lines show the cut of bright stars with their biased colour.

Figure 17 .
Figure 17.The empirical extinction laws for NGC 6397 from the isochrone fitting drawn by the different models.The data sets are: HST WFC3 and ACS by Richer et al. (2008); Nardiello et al. (2018); Simioni et al. (2018); Libralato et al. (2022) -red diamonds; Gaia -yellow snowflakes; SPZ19 -blue squares; SMSS -blue inclined crosses; HST WFPC2 by Piotto et al. (2002) -black inclined crosses; Nardiello et al. (2015) -open brown squares; Kaluzny (1997)open green diamonds; GCL99 -green circles; AT2000 -open red circles; Ahumada et al. (2021) -blue upright crosses; IR data sets from 2MASS, VISTA, and unWISE -purple upright crosses.The vertical lines denote the effective wavelengths of the and filters.The black curve shows the extinction law of CCM89 with V = 3.3 tied to the obtained V , which is shown by the horizontal line.

Figure 19 .
Figure 19.The mass-radius relation (in solar mass and radius) for the eclipsing binary V54 of NGC 6809 (black symbols with error bars) fitted by the BaSTI (red) and DSED (green) isochrones for various [Fe/H], , and age (the left and right isochrone in each model pair is for 13.5 and 13 Gyr, respectively).
for definition) of these clusters (0.98 for NGC 6397 and 0.87 for NGC 6809 from Mackey & van den Bergh 2005 with an uncertainty about ±0.1 from Torelli et al. 2019), or as their median colour difference between : 0.625 − 0.661 and 0.645 − 0.682 from HST and ground-based observations of NGC 6397, respectively, while 0.61 − 0.70 from ground-based observations of NGC 6809.Moreover, our mass estimates for the HB stars of NGC 6397 agree with the estimate 0.64−0.66⊙ obtained by Ahumada et al. (2021) from their modeling of the HB blue tail with mass loss at the RGB.

Table 2 .
Some properties of the clusters under consideration.

Table 3 .
The effective wavelength eff (nm), number of stars, and the median precision of the photometry (mag) for the data sets and filters under consideration.For data sets cross-identified with Gaia DR3, only Gaia cluster members are counted.
1. [Fe/H] is given only for CMDs, which allow its calculation as a fitting parameter.The complete table is available online.

Table 5 .
The cluster systemic PMs (mas yr −1 ).The statistic uncertainties are given for the PMs from this study and from Vitral (2021), while the total (statistic plus systematic) uncertainty is given for the PMs from VB21.The latter is adopted by us as the most realistic estimate of the PM uncertainties.

Table 6 .
Various parallax estimates (mas) for NGC 6397 and NGC 6809 with their total (statistic and systematic) uncertainties.

Table 7 .
The list of the Gaia DR3 members of NGC 6397 and NGC 6809.The complete table is available online.

Table 9 .
Our NGC 6397 and NGC 6809 age (Gyr)and distance (kpc) estimates from optical CMDs.All the uncertainties are standard deviations of one measurement.

Table 10 .
The relative estimates for the obtained [Fe/H] (dex), age (Gyr), distance (kpc), and ( − ) (mag) in the sense 'NGC 6809 minus NGC 6397'.All the values are shown with uncertainty of one measurement.The right column presents the differences of the parameters in the sense 'NGC 6809 minus NGC 6397' derived from all optical CMDs.± 0.21 for NGC 6809, which is introduced by Torelli et al.