1. Introduction

The southern globular cluster NGC 1851 is a remarkably bright system located in the constellation Columba, and in the outer region of the Milky Way, at a distance of about 12.0 kpc from the Sun (α = 5^h14^m06.76^s, δ = −40^o02^′ 47.6^′′, J2000; 1 = 244.51^o, b = −35.03^o). It is a highly concentrated system (^{Kuzma et al. 2018}) characterized by a diffuse halo extending to more than 10 times the tidal radius, although according to ^{Marino et al. (2014)}, stars dynamically linked to the cluster are present to at least 2.5 tidal radii. The Galactic orbit of the cluster is very eccentric, e = 0.7, and the lack of a tidal tail has triggered the suggestion that the cluster may be a stripped dwarf galaxy nucleus accreted by the Milky Way (^{Kuzma et al. 2018}). However, a tail that seems to emerge from NGC 1851 was detected by ^{Carballo-Bello et al. (2018)}, which may be interpreted as a tidal remnant of a tentative progenitor dwarf galaxy host of NGC 1851. Its rather young age, 9.2 Gyr according to ^{Koleva et al. (2008)} or 11.0 Gyr according to ^{VandenBerg et al. (2013)}, seems to support the hypothesis of an extra Galactic origin. From a chemo-dynamical analysis, ^{Callingham et al. (2022)} associated NGC 1851 to the ancient major merger event of the Milky Way Gaia-Enceladus-Sausage (^{Belokurov et al. 2018}; ^{Helmi et al. 2018}).

The horizontal branch (HB) of NGC 1851 possesses a moderate population of hot blue tail stars and a dense red clump nearly twice as populated; hence its HB structural parameter L = −0.36 (^{Arellano Ferro 2024}) is consistent with its Oosterhoff type Oo I and metallicity [Fe/H]=-1.3, following the trend defined by other Galactic clusters of the same Oo-type and similar metallicity.

NGC 1851 harbours a large number of RR Lyrae stars, 48 according the the 2020 edition of the Catalogue of Variable Stars in Globular Clusters (CVSGC, ^{Clement et al. 2001}). It may also contain 4-5 long period variables near the tip of the red giant branch (RGB). One of our goals in this paper is to employ the variable stars as indicators of the mean metallicity and distance of the parental cluster; hence it is of relevance to ask whether all the variables reported in the CVSGC are cluster members, since the large majority of them were discovered before a detailed membership analysis was a feasible possibility. In the present paper we propose a membership analysis based on the Gaia-DR3 proper motions before we produce a cleaner colour-magnitude diagram (CMD) and then to study the distribution of the variables and their physical properties.

2. Observations and image reductions

2.2. Locking Our Photometry to the Standard System

For the BA18 season we were able to produce light curves for 4623 point sources in the V -band and 1578 in the I-band. For the BA19 season we measured 2305 light curves in the V -band and 1949 in the I-band. These instrumental light curves were transformed to the standard Johnson-Kron-Cousins system defined by ^{Landolt (1992)}, by employing the local standard stars in the field of NGC 1851 provided in the catalogue of ^{Stetson (2000) 8}. We identified 179 and 101 standards in BA18 and BA19 respectively for which we have VI photometry. Figure 1 shows the dependence of the standard minus instrumental magnitudes with the instrumental colour (v − i), from which the transformation equations, in-scribed in the figure legend, were calculated. These equations were employed to convert all instrumental light curves into the standard system.

Fig. 1 Transformation relationship between VI instrumental and standard photometric systems, calculated for the BA18 and BA19 seasons with 179 and 101 standards respectively, taken from the collection of ^{Stetson (2000)}. 

Table 2 displays a small portion of the time-series VI photometry obtained in this work. The full table will be made available in electronic form in the Centre de Donnés Astronomiques de Strasbourg database (CDS).

TABLE 2 TIME-SERIES VI PHOTOMETRY FOR THE VARIABLE STARS OBSERVED^*  

Variable Star ID	Filter	HJD (d)	M std (mag)	m ins (mag)	σ m (mag)
V1	V	2458467.53771	16.000	18.889	0.014
V1	V	2458467.55788	16.097	18.985	0.009
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
V1	I	2458467.53148	15.433	18.884	0.020
V1	I	2458467.53416	15.525	18.976	0.012
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
V3	V	2458467.53771	16.264	19.155	0.019
V3	V	2458467.55788	16.206	19.097	0.012
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
V3	I	2458467.53148	15.719	19.179	0.026
V3	I	2458467.53416	15.775	19.235	0.018
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.

^* The standard and instrumental magnitudes are listed in Columns 4 and 5, respectively, corresponding to the variable stars in Column 1. Filter and epoch of mid-exposure are listed in Columns 2 and 3, respectively. The uncertainty in m _ins, which also corresponds to the uncertainty in M _std, is listed in Column 6. A full version of this table is available at the CDS database.

3. Variable stars in this study

We were able to measure VI magnitudes for 1434 stars identified as cluster members in the field of our images. These cluster members will be used to produce the CMD diagram. We could measure 47 of the 55 variables in the cluster listed by ^{Clement et al. (2001)} in the CVSGC. As can be seen in Table 1, the observations were carried out under limited seeing conditions, and as a result our DIA analysis was unable to retrieve useful data for stars that are very faint, stars near the cluster center, or stars blended with a brighter neighbour. We could not measure the RRab stars V30, V36, V39, V40, V41, V43 and V44.

From an independent exploration we detected clear variability of three faint stars that we temporarily call F1, F2 and F3.

The Gaia-DR3 lists 22 variable stars in the field of the cluster that do not match any of the V55 already known in the CVSGC. Gaia photometry is available for 21 of these stars. For the sake of clarity we list these 21 stars in Table 3 along with their Gaia source identification and equatorial coordinates. We arbitrarily number these variables with a prefix ‘G’. We explore their light curves from our own photometry and from Gaia data, and can confirm with confidence the variability and nature for only five of them.

All the reported variables in field of NGC 1851 are listed in Table 4, along with their variable types, mean magnitudes, amplitudes and ephemerides whenever possible. Included in the table are also the stars that we were unable to measure and the non-confirmed Gaia variables, since we are providing their equatorial coordinates and a proper field identification.

The cluster membership status of all these variables is discussed in the following section.

We offer a finding chart of all the variables in Figure 2. Their light curves are displayed in the Appendix, where we shall distinguish the data from different seasons. In the Appendix we also discuss individual peculiar or outstanding variables.

Fig. 2 Identification chart of variable stars in the field of NGC 1851. The approximate size of the images is 9.7 × 9.7 and 1.6 × 1.6 square arc minutes. 

4. Stellar membership analysis

In current times, the membership analysis of large numbers of stars in the field of a given globular cluster is possible thanks to the high quality of proper motions available in the Gaia mission (^{Gaia Collaboration et al. 2023}). Sieving the likely cluster members and the field stars, enables the production of cleaner CMDs, and hence a better perspective of the stellar distributions and evolutionary properties. This is of particular interest for specific groups of variable stars, e.g. RR Lyrae stars in the horizontal branch (HB).

The method developed by ^{Bustos Fierro & Calderón (2019)} to determine the stellar membership is based on a two step approach: (1) it finds groups of stars with similar characteristics in the four-dimensional space of the gnomonic coordinates (X _t,Y _t) and proper motions (µ _α∗ ,µ _δ ) employing the Balanced Iterative Reducing and Clustering using Hierarchies (BIRCH) clustering algorithm (^{Zhang et al. 1996}); and (2) in order to extract likely members that were missed in the first stage, the analysis of the projected distribution of stars with different proper motions around the mean proper motion of the cluster is performed.

Figure 3 shows the corresponding vector point diagram (VPD) and CMD, distinguishing the likely cluster members from the field stars. We considered a 30 arc minute radius field from the cluster center, which contains 25238 Gaia point sources out of which 11220 were found to be likely cluster members.

Fig. 3 Gaia-DR3 VPD (left panel) and CMD (right panel) of the cluster NGC 1851. Red and gray points correspond to likely cluster members and field stars, respectively, determined as described in § 4. A total of 25238 Gaia point sources within 30 arc minutes are displayed, while 11220 were found to be cluster members. The colour figure can be viewed online. 

An independent membership analysis for a large number of globular clusters, based on the proper motions of Gaia-DR3, was performed by ^{Vasiliev & Baumgardt (2021)}. These authors provided membership probabilities for each star in the cluster field. In Columns 9 and 10 of Table 4 we list the membership status according the method from ^{Bustos Fierro & Calderón (2019)} (B&C) and the probabilities from ^{Vasiliev & Baumgardt (2021)} (V&B). With a few exceptions, the match is good. The exceptions that call for attention are V8, V25, V37, V42, V44, V45 and V48, and their membership status deserves a few comments. We should note that the coordinates given in the CVSGC are the starting point for a match with Gaia and that for stars V37, V38, V42, V45, V46 and V48 only X,Y coordinates are listed; hence the matching is sometimes more dubious. In the present work we provide equatorial coordinates for these six stars

We should consider the fact that at the reported coordinates in Table 4, we recover in our photometry the light curve of a variable star of the expected type, period and light curve morphology, and that their position in the CMD also becomes a sound membership indicator tool. Therefore, we conclude that V8 is a cluster member, whereas V25, based on its accurate parallax is a field star, much closer than the cluster. For RR Lyrae stars V37, V42, V45, V47, V48, V50 and V53 there are good matches with Gaia sources with good proper motions, but in all cases the corresponding stars are much brighter than the HB, as can be appreciated in the CMD; hence they are all foreground field RR Lyrae stars. Given that at least one of the two membership identification methods indicate that they are field stars, we opted for labeling them as such. Similarly off the HB there are the RR Lyrae V14, V33, and V51. However in these cases both B&C and V&B approaches identify them as very likely cluster members. Since these are located near the central region of the cluster, we cannot rule out contamination of our photometry by close unresolved neighbours and we opted for considering them likely cluster members. We were unable to reliably measure the star V44 in the central region of the cluster, and hence its membership status remains unknown (UN).

Regarding the variable stars reported by Gaia-DR3, listed in Table 3, all but G17, G18 and G21 were found to be cluster members by both B&C and V&B. All stars marked with ‘UN’ by B&C lack proper motion in the Gaia database; hence its membership cannot be assessed from that information. In Column 11 of Table 4 we list our final membership assessment.

5. The Oosterhoff Type of NGC 1851

The period averages for the RRab and RRc stars in NGC 1851 are <P ab> = 0.57 ± 0.06 days and <P c> = 0.31 ± 0.04 days. These numbers point to an Oosterhoff type Oo I for this cluster. In the log P - amplitude plane, or Bailey’s diagram, of Figure 4 we plot amplitudes and periods for all RR Lyrae stars measured in the this work (Table 4). The distribution of star on this plane clearly favours the un-evolved sequences and the Oo I type of this cluster, in agreement with the average periods and the metallicity of the cluster of about [Fe/H]_UV = −1.25, (see § 7).

Fig. 4 Period-amplitude diagram for RR Lyrae stars in NGC 1851. Blue and green circles represent RRab and RRc stars, respectively. The colour figure can be viewed online. 

6. On the Cluster Reddening

Given its Galactic location, the reddening of NGC 1851 is relatively low compared to other globular clusters closer to the Galactic plane, where the density of dust and gas is higher. Independent estimates of the cluster reddening consistently report low values of E(B − V); for example, ^{Harris (1996)} and ^{Walker (1998)} give a value of E(B − V) = 0.02. The latter stresses that there are no compelling evidences for values too different from this estimation. The calibrations of ^{Schlegel et al. (1998)}, and ^{Schlafly & Finkbeiner (2011)} give values of 0.037 and 0.032, respectively. We have not attempted a reddening determination from the colour (V − I) being constant between phases 0.5−0.8 for RRab stars (^{Sturch 1966}) and the subsequent calibration of ^{Guldenschuh et al. (2005)}, since our I light curves are scanty and phase gaps are present. Hence we adopted the value of E(B −V) = 0.032 ± 0.002 from the calibration by ^{Schlafly & Finkbeiner (2011)}.

7. Physical Parameters of the RR Lyrae Stars from the Light Curves Fourier Decomposition

The Fourier decomposition of the light curves of RR Lyrae stars, both RRab and RRc, is a well established approach towards the determination of some physical parameters, mainly the metallicity [Fe/H], the luminosity (or absolute magnitude M_V and hence the distance), as well as the mass and mean stellar radius. The Fourier decomposition technique, as well as the semi-empirical calibrations and their zero points leading to the physical parameters, have been presented and discussed in detail in the papers by ^{Arellano Ferro et al. (2010)}. A summary of the results for 40 clusters calculated homogeneously for over more than a decade can be found in ^{Arellano Ferro (2024)}. The interested reader is referred to those works for the the involved details.

In the present paper we have limited the calculation of physical parameters to those stars that have proven to be likely cluster members, according to the discussion offered in §4 and summarized in Table 4.

8. The Colour Magnitude Diagram

The observed CMD of NGC 1851 built from our VI photometry with only likely cluster member stars was dereddened assuming E(B − V) = 0.03 mag. The resulting intrinsic CMD is displayed in Figure 5. All variable stars are plotted with the colours and symbols code in the caption. This diagram helps to confirm the non-membership of many stars, as discussed in previous sections, since their positions are at odds with their variable type in many cases. We remind the reader that our final membership assessment is given Column 11 of Table 4.

Fig. 5 Color-Magnitude Diagram (CMD) of NGC 1851. Variables stars in the field of the cluster are plotted with colour symbols according to the following code:solid blue and green circles represent RRab and RRc star respectively; red circles are for SR/L variables near the tip of the RGB. The star V25, classified as an eclipsing binary, is shown with a yellow circle. Turquoise colour is use for three newly identified variables, and purple open triangles for variables reported in Gaia that were not identified in our photometry or were not confirmed as variables. The Red ZAHB was constructed by ^{Yepez et al. (2022)} using the models built from with the Eggleton (^{Pols et al. 1997}, 1998; ^{Schröder et al. 1997}). The green and blue vertical nearly vertical lines on the HB are the theoretical first overtone and fundamental mode instability strips respectively (^{Bono et al. 1994}).en The isochrone is from VandenBerg et al. (2014) for [Fe/H]=-1.35 and an age of 12.0 Gyrs. The vertical black lines at the ZAHB mark the empirical red edge of the first overtone instability strip (^{Arellano Ferro et al. 2015}, ²⁰¹⁶). The colour figure can be viewed online. 

We call attention to the distribution of RRab and RRc stars on the HB. Considering exclusively the stars that are clear cluster members, we see that some RRab are located in the bimodal region of the instability strip, i.e. in the intersection of the fundamental and first overtone instability strips. The theoretical bounds of these strips are indicated by the green and blue borders calculated by ^{Bono et al. (1994)}. The empirical border of the first overtone red edge (FORE) is indicated by the two vertical black lines in the HB (^{Arellano Ferro et al. 2015}, ²⁰¹⁶) and matches well with the theoretical FORE. RRab stars crossing to the blue of the FORE are a characteristic of some Oo I type clusters, like NGC 1851, but this does not happen in Oo II clusters, where the RRab remain to the red of the FORE, i.e. off the bimodal region (see ^{Yepez et al. 2022} and references there in for a discussion). This characteristic of Oo II clusters is probably a consequence of the more advanced stage of their evolution to the red, towards the AGB.

Of the 21 stars in the field of NGC 1851 marked in Gaia-DR3 as variables, we found a counterpart measured in our photometry for 15 of them. The others, were either too faint or unresolved, given the spatial resolution of our images. For a proper comparison we transformed the Gaia photometric data into VI using the transformation equations of ^{Riello et al. (2021)}. We could confirm the variability and variable type of 5 of them; V57-V61 in Table 4. The remaining 10 are plotted in the DCM with purple open triangles and are distributed all across the diagram. The Gaia cadence is not designed for the identification of some variables; therefore, the authenticity of these variables will have to be confirmed with proper time-series observations on images of resolution higher than ours.

The theoretical ZAHB shown in the figure as a red continuous locus was calculated by ^{Yepez et al. (2022)} using the models built with the Eggleton code (^{Pols et al. 1997}, ¹⁹⁹⁸; ^{Schröder et al. 1997}) for a metallicity of z=0.001, a core mass of 0.5 M/M _⊙, and a range of total masses of 0.59-0.68 M/M _⊙. The isochrone is from VandenBerg et al. (2014) for [Fe/H]=−1.35 and an age of 12.0 Gyr.

All the above loci have been drifted to a distance of 11.95 kpc and represent well the observed distribution of the cluster member stars.

9. Conclusions

The presence of variable field stars projected against a Galactic globular cluster is very common, and while such contamination by field stars in the Galactic bulge globular clusters can be remarkably high (e.g. see the case of NGC 6558 ^{Arellano Ferro et al. 2024}) due to the richness of the bulge of field variable stars, particularly of RR Lyrae, it can also be noticeable in more isolated globular clusters in the outer regions of the Milky Way. Such is the case of NGC 1851, as we have demonstrated in the present work. An ad hoc membership analysis based on the proper motions and parallaxes available in Gaia-DR3, complemented with mean magnitudes and colours in the V − (V − I) CMD, has shown that of the 55 variables originally listed in the CVSGC, 8 have been found to be clearly field stars, and for 6 more the membership cannot be solidly assessed due to the lack of proper motion data or to blending with bright neighbours, particularly in the central regions of the cluster.

Three variables not detected before were identified, but only one turned out to be a cluster member. We named it V56 and classified it tentatively as an SX Phe star. Among the 21 variables reported by Gaia not included in the CVSGC, we confirmed the variability of two RRab and three long term L variables. Since they turned out to be cluster members we assign to them variable names V57-V61.

Identifying variable cluster members is rewarding, since they can be used with confidence, as indicators of average physical quantities representative of the parental cluster. Here we estimated the mean metallicity and distance of NGC 1851 via the Fourier decomposition of RR Lyrae light curves, to find [Fe/H]_ZW = −1.35±0.22 dex and d = 11.9±0.6 kpc. A few comments on the position of NGC 1851 relative to the Oosterhoff gap are in order, since the cluster has been associated with an CMa dwarf galaxy (^{Martin et al. 2004}). We noted before that the average period of the member RRab stars is <P ab> = 0.57 ± 0.06 d, which with the metallicity [Fe/H]_ZW = −1.35 places the cluster among the Oo I clusters and slightly off the Oosterhoff gap marked by ^{Catelan (2009}, see his Figure 5). On the other hand, let us consider the structural, or horizontal branch type parameter, defined as HBt = (B − R)/(B + V + R), ^{Lee et al. (1994)}, where B and R are the number of stars to the blue and to the red of the instability strip respectively, and V represents the number of RR Lyrae in the instability strip (Lee et al. 1994; ^{Demarque et al. 2000}). In the [Fe/H]-HBt plane Catelan (2009, his Figure 7) identified a region devoid of Galactic globular clusters, but populated otherwise by clusters associated with neighbouring galaxies, and termed this region as “forbidden” or as the “Oosterhoff gap”. We should recall here that the Oo I clusters NGC 1851 and NGC 2808, as well as the Oo II clusters NGC 2298 and NGC 1904, have been suggested by Martin et al. (2004) to be associated to the Canis Major dwarf galaxy accreted by the Milky Way. More recently, ^{Callingham et al. (2022)} have associated the first three to the Gaia-Enceladus-Sausage merger event and to the Helmi merger (^{Helmi et al. 2018}) for the case of NGC 1904.

NGC 1851 and NGC 2808 have well developed HB blue tails but prominent red clumps; hence their H Bt values are very red, i.e. negative, whereas NGC 1904 and NGC 2298 have massive blue tails but lack a red clump. Therefore, their H Bt values are very blue, hence large and positive.

Considering the updated version of the [Fe/H]-H Bt plane (^{Yepez et al. 2022}, see their Figure 11), and plotting these four clusters with the coordinates ([Fe/H],H Bt); NGC 1851 (−1.35,−0.36, this work), NGC 2808 (−1.15, −0.49, ^{Catelan 2009}), NGC 1904 (−1.68,+0.74, ^{Arellano Ferro 2024}) and NGC 2298 (−1.96,+0.96, ^{Torelli et al. 2019}), it is evident that none of these four clusters occupy the Oosterhoff gap.

We are faced with two possible conclusions; these clusters are not associated to external galaxy mergers of the MW beyond the spatial coincidence, or else the globular clusters of extragalactic origin can occupy regions in the [Fe/H]-H Bt or [Fe/H]-< P _ab > planes other than the Oosterhoff gap defined by ^{Catelan (2009)}, as in fact some are seen in his Figures 5 and 7. This reinforces the view that the Oosterhoff gap retains its meaning only in Galactic terms. Hence, we do not find compelling evidence, from these arguments, for an association of NGC 1851 (and perhaps neither of NGC 2808, NGC 1904 and NGC 2298) to the large accretion events that seem to have sculpted the Galactic halo.