1. Introduction

NGC 2419 (C0734+390 in the IAU nomenclature) (α = 07^h38^m08.47^s, δ = +38^◦52^′56.8^′′, J2000; = 180.37^◦, b = +25.24^◦) is a large and very luminous globular cluster at about 80 kpc from the Galactic center and hence it is among the most distant clusters in the outer halo of our Galaxy. It is noted for being a very loose or extended system given its brightness M_V and half-light radius, which led ^{van den Bergh & Mackey (2004)} to suggest its extra galactic origin, and the likelihood of it being a stripped core of a former spheroidal dwarf galaxy. These suggestions however have not been supported by ^{Ripepi et al. (2007)} on the ground of the following arguments: the cluster is of the Oo II type while extra galactic clusters and dwarf galaxies reside within the Oosterhoff gap (^{Catelan 2009}); the lack of sub-structures in the main sequence bellow the TO (i.e. lack of multiple populations); the thinness of the red giant branch (ruling out multiple chemical compositions among cluster stars); and the apparent absence of extratidal structures. In Ripepi et al’s opinion NGC 2419 is a normal metal-poor Galactic globular cluster.

The variable star population of NGC 2419 is quite rich. There are 101 variables registered in the Catalogue of Variable Stars in Globular Clusters (CVSGC) (^{Clement et al. 2001}), 75 of which are RR Lyrae, 1 Pop II or CW star, 12 SX Phe, 5 long-period red giants, 3 eclipsing binaries, 2 δ Scuti stars and 3 non-classified. A thorough analysis of these variable stars, based on high-quality images from the HST, and from two large ground telescopes; the Galileo (TNG) 3.5 and the Subaru 8.2m, was carried out by ^{Di Criscienzo et al. (2011)}.

Our team has obtained VI CCD images of NGC 2419 between 2005 and 2022 for a total of 405 and 327 in V and I respectively, using the 2m telescope of Indian Astronomical Observatory (IAO). Naturally the depth and accuracy of our photometry is not comparable to that used by ^{Di Criscienzo et al. (2011)}, and due to crowding we were not able to recover the light curves for all known variables. It may seem pretentious to draw conclusions from Lyrae stars at V ∼ 20.5, which nears the faint limit of our data. Nevertheless, we can use our photometry to provide a new discussion of the cluster membership of the stars in the field of our images, particularly those of variable nature, and provide independent estimates for the mean cluster reddening, average metallicity and distance.

Our approach to the determination of mean M_V and [Fe/H] of RRL stars, is the Fourier light curve decomposition, of both the fundamental mode and first overtone pulsators RRab and RRc respectively. This, and the employment of well established semi-empirical calibrations and their zero points between the Fourier parameters and the physical quantities, provide individual stellar estimations of the distance and [Fe/H]; hence, the average of these values for tested cluster member variables leads to proper average values for the parent cluster. The present paper is a report of our results.

2. Observations and image reductions

2.2. Transformation to the Standard System

The instrumental photometry was transformed to the standard Johnson-Kron-Cousins photometric system (^{Landolt 1992}) VI, using local standard stars in the fields of the target clusters. These standard stars have been taken from the extensive collection of ^{Stetson (2000) 6}. We found 554 standard stars with instrumental light curves in the field of our images. The standard minus instrumental magnitudes and the light dependence with the (v − i) colour is displayed in Figure 1. The corresponding transformation equations are also given in the figure.

Fig. 1 Transformation relationship between instrumental and standard photometric systems. These equations were calculated using 554 local standard stars from the collection of ^{Stetson (2000)}. The colour figure can be viewed online. 

3. Stellar membership analysis

Distinguishing the true cluster members from the field stars projected on the field of view of a cluster is relevant since one is interested in a clean CMD that represents the structure of the system. Presently, this challenge is on reach given the high quality proper motions available in the Gaia mission (^{Gaia Collaboration et al. 2023}).

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 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.

This method was applied to the stars within a radius of 15 arc minutes from the center of NGC 2419. This field contains 3965 Gaia sources but only 3129 have a measurement of the proper motion, 1584 of which were found to be likely cluster members. In Figure 2 the corresponding Vector Point Diagram (VPD) and colour-magnitude diagram (CMD) show-ing the cluster member and field stars are shown. The farthest member from the center is about 6 arc minutes away which, at a distance of 84.0 kpc (see our distance determination in § 7) corresponds to a distance of about 140 pc. In spite of its half-light radius at about 17.9 pc, comparable to other clusters in the halo (^{van den Bergh & Mackey 2004}), these star members at such large distances are the indication of the extended halo that led ^{van den Bergh Mackey (2004)} to suggest its extra galactic origin and presently being the remains of a former dwarf galaxy tidally stripped by the Milky Way. In Figure 3, the radial distance distribution of member stars in NGC 2419 is displayed and shows its extended nature. For comparison we included the similar distribution of large halo cluster NGC 1851. The membership analysis reveals the bound nature of a subtle cluster halo as large as 140 pc from the cluster center.

Fig. 2 Gaia-DR3 VPD (left panel) and CMD (right panel) of the cluster NGC 2419. Red and gray points correspond to likely cluster members and field stars respectively, determined as described in § 3. A total of 3865 Gaia point sources within 15 arc minutes are displayed, while 1584 were found to be cluster members. The colour figure can be viewed online. 

Fig. 3 Star members distribution of the extended cluster NGC 2419 compared with the large halo cluster NGC 1851. The horizontal axis represents the radial distance projected on the plane of the sky. The colour figure can be viewed online. 

We have been able to obtain VI photometry for 1107 point sources in the field of our images. These light curves include 74 of the known variables in the cluster (^{Clement et al. 2001}), and the general data for these variables is included in Table 2. A CMD with these variables identified has been produced and will be discussed in the following sections.

4. Variable stars measured in the present study

Given the size of our telescope and sky conditions at the time of the observations which limited the deepness and resolution of our imaging data, some of the well known variables remain blended and it was not possible to measure them well. It should be noted that the HB level of this distant cluster is fainter than 20th magnitude, which is close to the faint limit of our capabilities. Variables in Table 2 are mostly RR Lyrae (RRab and RRc), but also a sample of red giant branch variables (RGBs) is available, plus one W Vir (V18) and one double mode or RRd star (V39). The identifications of these variables are given in the charts of Figure 4. Their light curves are displayed in the Appendix, where we have distinguished the data from the several observing runs.

Fig. 4 Identification chart of variable stars in the field of NGC 2419. The approximate size of each images is 6.9 × 6.9, 2.5 × 2.5 and 1.6 × 1.6 square arc minutes. 

5. The Oosterhoff type of NGC 2419

Given its low metallicity the cluster likely belongs to the Oo II type. This is in fact confirmed by the period average of its RRab and RRc stars of 0.665 d and 0.343 d respectively. The log P - Amplitude diagram (Bailey's diagram) is shown in Figure 5. It is clear that the distribution of stars, particularly that of the RRab, falls towards the sequences of evolved star (^{Cacciari et al. 2005}) which is the characteristic of Oo II type clusters.

Fig. 5 Period-Amplitude diagram for the RR Lyrae in NGC 2419. Blue and green symbols represent RRab and RRc stars respectively. The star V40 is not a cluster member. In the top panel, the curves to the right are the locy for RRab stars (unevolved continuous and evolved segmented) in M3 according to ^{Cacciari et al. (2005)}. The black parabola for the RRc stars was calculated by ^{Kunder et al. (2013b)} from 14 OoII clusters and ^{Arellano Ferro et al. (2015)} calculated the red parabolas from a sample of RRc stars in five OoI clusters. In the bottom panel, the continuous and segmented blue lines were constructed by ^{Kunder et al. (2013a)}. The black and green parabolas were calculated by ^{Yepez et al. (2020)} and ^{Deras et al. (2019)} respectively, using 35 RRc stars from eight OoII clusters. The colour figure can be viewed on-line. 

6. On the cluster reddening

Given its Galactic location, NGC 2419 is not much affected by interstellar reddening. To estimate the colour excess of individual RRab stars, we have taken advantage of the fact that the V - I colour curve of RRab stars is constant near its minimum, between the phases 0.5 and 0.8 (^{Sturch 1966}). By employing the calibration of (V - I) _{0, min} = 0.58 ± 0.02 (^{Guldenstschuh et al. 2005}), we calculated the average colour excess for 12 cluster member RRab and found E(B - V) = 0.07 ± 0.07. This average is comparable with the predicted values from the calibrations of ^{Schlafly & Finkbeiner (2011)} and ^{Schlegel et al. (1998)}, 0.052 ± 0.001 and 0.061 ± 0.001 respectively. In this work we adopted E(B - V) = 0.06 or E(V - I) = 1.259 × E(B - V), which leads to E(V - I) = 0.08.

7. Physical parameters of the RR Lyrae stars from the light curves Fourier decomposition

Our approach to the calculation of relevant stellar physical parameters for the RR Lyrae stars is via the Fourier decomposition of their V-band light curves. Of particular interest are the mean [Fe/H] and M_V of the RR Lyrae sample, as they are good representations of the mean cluster metallicity and distance.

The Fourier decomposition is performed by fitting the observed light curve with a Fourier series model of the form:

where m(t) is the magnitude at time t, P is the period, and E is the epoch. A linear minimization routine is used to derive the best-fit values of the amplitudes A_k and phases φ_k of the sinusoidal components. From the amplitudes and phases of the harmonics in eq. 1, the Fourier parameters, defined as φ _ij = jφ _i − iφ _j , and R _ij = A _i /A _j , are computed. Subsequently, the low-order Fourier parameters can be used in combination with semi-empirical calibrations to calculate [Fe/H] and M _V . The employed calibrations for the mass, T _eff, and radii are summarized in the papers by ^{Arellano Ferro et al. (2010)} and ^{Arellano Ferro et al. (2011)}, while the calibrations for [Fe/H] and M _V and their zero points are most recently discussed by ^{Arellano Ferro (2022)} and ^{Arellano Ferro (2024)}.

8. Discussion: The Color Magnitude Diagram

In Figure 6 we display two versions of the observed CMD. The panel on the left shows the stellar distribution of every point source that we were able to measure in our collection of VI images. This includes cluster member stars as well as field stars. Through the membership analysis described in § 3, we could distinguish the most likely cluster members (red dots in the figure) from the field stars (small light blue dots). In the right panel, we have plotted only the cluster members and have dereddened the diagram by assuming E(V − I) = 0.08. The variable stars contained in Table 4 are plotted with colours according to their pulsational type, as coded in the figure caption. When a variable is found to be a cluster member, the colour symbol has a smaller red dot in the center, otherwise the star had no proper motion reported in Gaia-DR3, and hence its membership status is unknown, or it is likely a field star.

Fig. 6 Observed CMD of NGC 2419. The left panel shows the field stars (small blue dots) and likely cluster member stars (red dots), according to our membership analysis. The right panel shows only the member stars in the dereddened plane. We use E(B − V) = 0.08. Blue and green circles represent RRab and RRc stars, respectively. Black circles are used for SR stars. The only CW star is the yellow circle. Black X corresponds to V80, whose variability we could not confirm. The vertical black lines at the ZAHB mark the empirical red edge of the first overtone instability strip (^{Arellano Ferro et al. 2015}, ²⁰¹⁶). Also shown are the theoretical fundamental (blue lines) and first overtone (green lines) instability strip borders from ^{Bono et al. (1994)}. Isochrones are from ^{VandenBerg et al. (2014)} for an age of 12.5 Gyr (black) and 13.0 Gyr (turquoise) with [Fe/H]= -1.8 and -2.0, respectively. Red ZAHB is from the models built from the Eggleton code (^{Pols et al. 1997}, ¹⁹⁹⁸; ^{Schröder et al. 1997}), and was calculated by ^{Yepez et al. (2022)}. The black evolutionary track corresponds to a model of total mass 0.54 M _⊙ and a core of 0.50 M _⊙. These very thin envelope models explain well type II Cepheids such as V18 (^{Yepez et al. 2022}). The colour figure can be viewed online. 

We also include in this dereddened diagram the two isochrones from ^{VandenBerg et al. (2014)} for ages of 12.5 Gyr (black) and 13.0 Gyr (turquoise) with [Fe/H]= -1.8 and -2.0, respectively. Red ZAHB is from the models built from the Eggleton code (^{Pols et al. 1997}, ¹⁹⁹⁸; ^{Schröder et al. 1997}), and calculated by ^{Yepez et al. (2022)}. The black evolutionary track corresponds to a model with a central core mass of 0.50 M_⊙ and a thin envelope of 0.04 M_⊙ and it was selected as it best represents the position of the W Vir star V18 confirming the conclusions reached by ^{Yepez et al. (2022)}, that type II Cepheids can be interpreted as products of post-HB evolution driven by burning of very low mass hydrogen and helium shells. According to ^{Bono et al. (2020)}, W Vir stars may be post-early asymptotic giant branch stars (see also the case of the CWB star V81 in NGC 7006 (^{Arellano Ferro et al. 2023})), although the case of V18 seems more like that of a post-HB which according to the model has taken 114 million years from the HB to reach its present position.

A qualitative comparison of our limited CMD with the deep and detailed CMD, Figure 2 of ^{Di Criscienzo et al. (2011)} (hereinafter diC11), leads to the following instructive observations. First, NGC 2419 has a well developed HB blue tail from where V18 has most likely evolved.

Our photometry does not reach these faint regions of the CMD. If we consider the variable star distribution near the HB in diC11 CMD, we may conclude that all RR Lyrae in their study are cluster members. However, diC11 did not perform a membership analysis. In our CMD of Figure 6, several RR Lyrae appear noticeably brighter than the HB (e.g. V37, V38, V40, V41, V48, V51, V56 and V69). It is not difficult to corroborate from the identification chart of Figure 4, that these stars are tenants of the cluster central regions and hence are most likely blended, making our photometry further limited and hence producing spurious positions in our CMD. These conditions may have also limited the possibilities of the Gaia mission to measure them, as none of them have proper motions reported in Gaia-DR3; thus, we have assigned them the unknown (UN) membership status in Table 2. The membership of these variables in NGC 2419 should be corroborated in the future. The rest of the variables in Table 2, all likely members, are distributed closer to the HB with some scatter. This scatter is comparable to that observed in Figure 2 of diC11 if plotted at the same scale; therefore, the RR Lyrae population shows signs of evolution off the ZAHB.

The mass distribution along the ZAHB is determined by the amount of mass lost during the Heflash events at RGB, as it is clearly demonstrated by the models of ^{Silva Aguirre et al. (2008)}. The more mass is lost, the less massive the star is when settling towards the bluer regions of the ZAHB. This is consistent, for instance, with the fact that bluer (hotter) RRc stars are less massive than redder (cooler) RRab stars. The two modes are in principle separated by the first overtone red edge (FORE) of the instability strip. Depending upon the exact total mass at exhaustion of core-Helium, RRc and RRab stars can either be neatly separated by the FORE or can share the inter-order or bimodal instability strip (i.e. the intersection of the first overtone and fundamental mode instability strips). These concepts are graphically illustrated in Figure 3 of ^{Caputo et al. (1978)}.

The empirical position of the FORE (^{Arellano Ferro et al. 2015}, ²⁰¹⁶) is shown in Figure 6 by two black vertical lines. Also shown are the theoretical borders of the instability strip for the first overtone and fundamental modes from ^{Bono et al. (1994)}. Note that the theoretical and empirical FORE match very well. It was found by ^{Arellano Ferro et al. (2019)} that in all OoII type clusters studied by them, RR Lyrae pulsating modes are well segregated by the FORE, whereas such clear segregation happens only in some OoI clusters, while in others the two modes share the inter-order or “either-or” region. This was interpreted by these authors as indicating that in OoII clusters the RR Lyrae stars always start their ZAHB evolution from less massive bluer stars, while in OoI clusters they exhibit a wider mass distribution, and hence segregation may or may not occur. It remains unclear what physics constrain these two options but it is likely connected with the mass-loss processes during the RGB.

In the case of NGC 2419, the RRc and RRab stars are all mixed across the HB. This can be seen in the CMD of Figure 6, but also in the CMD in Figure 2 of diC11. For a Oo II cluster, the location of RRab stars in the inter-order region is contrary to the argument given in the earlier paragraph. Hence, NGC 2419 is an unconventional Oo II cluster exhibiting mixing of the modes in the instability strip.

9. Conclusions

Using the Gaia-DR3 proper motions of stars in the field of NGC 2419, we have been able to confirm the membership of most of the variables known in the cluster. Several of the known variables are in crowded environments and, due to blending with neighbours, we were not able to resolve them; thus, they may appear in odd positions in the CMD. These stars also lack Gaia-DR3 proper motion data; thus, we cannot confirm their membership status (e.g. V38, V41, V48, V51, V56, V69). V89, with a proper motion measurement, was found to be a likely field star.

The radial distribution of member stars, clearly demonstrates the extended reach of the cluster to distances of about 140 pc, making of NGC 2419 one of the largest clusters in the Galaxy.

From the Fourier light curve decomposition of clearly resolved RR Lyrae stars, we determined the average metallicity and distance [Fe/H]_UV = −1.90 ± 0.27 and D = 86.3 ± 5.0 kpc from the RRab light curves and [Fe/H]_UV = −1.88 ± 0.30 and D = 83.1 ± 8.1 kpc from the RRc stars. These determinations are in excellent agreement with the well established results [Fe/H] _Spec = −2.2 (^{Carretta et al. 2009}), and D = 88.47 ± 2.40 kpc (^{Baumgardt & Vasiliev 2021}).

Our post He-flash models show that the W Virginis star V18 has evolved from a ZAHB blue tail progenitor with a very thin shell; this progenitor has a total mass of 0.54 M _⊙ but a shell of only 0.04 M _⊙. These results confirm the conclusions from ^{Deras et al. (2022)} for the Pop II cepheids of M56 or from ^{Yepez et al. (2022)} for M14, that thin shells are a required condition for the generation of Pop II cepheid pulsations.