1. INTRODUCTION

The globular cluster M56 (NGC 6779) is among the most metal-poor clusters in the Galaxy, and is relatively nearby and hence a bright system lying well within the Galactic disc ([Fe/H]≈ -1.98 (^{Searle & Zinn 1978}); d=8.4 kpc (^{Hatzidimitriou et al. 2004}); (l = 62.66 ^◦ , b = 8.34 ^◦ ). Therefore, it is subject to a considerable reddening (E(B − V ) = 0.26) (^{Kron & Guetter 1976}). Typical of very metalpoor clusters, its horizontal branch (HB) shows a very long and populated blue tail and a scarcely populated instability strip (IS). As a consequence, very few RR Lyrae stars are expected to be found in the system. In fact, presently only two RRab and two RRc are listed in the Catalogue of variable stars in globular clusters (CVSGC) of ^{Clement et al. (2001)}. Other variables included in the CVSGC are one W Vir (CW), six semi-regular or slow variables (SR or L), one RV Tauri (RV) and one SX Phoenicis star, for a total of 14 variables. The variability of V2 has been doubted by several authors and we shall report our conclusion in the present paper. The most recent light curve analysis of the known variables is that by ^{Pietrukowicz et al. (2008)}, which shall be a substantial basis for our present analysis. It has been argued, on kinematical grounds, that M56 might be of extragalactic origin due to the occurrence of a massive accretion event by the Milky Way, around 10 Gyrs ago (^{Deason et al. 2013}). A study conducted by ^{Piatti & Carballo-Bello (2019)} concluded that M56 shows evidence of extratidal features like tails and extended haloes, as a consequence of the merger event. Also, ^{Massari et al. (2019)} showed that the age-metallicity relation for Galactic globular clusters is bifurcated, meaning that metal-rich clusters with disc-like kinematics are more likely to have been formed in situ, while metal-poor clusters with halo-like kinematics are more likely to have been accreted, M56 being one of the latter.

In the present study we propose the determination of the cluster mean metallicity and distance, as indicated by the light curve morphology of its RR Lyrae stars via the Fourier decomposition and ad-hoc calibrations. We also aim to employ the Gaia-eDR3 proper motions to distinguish cluster members from field stars in our images of the cluster and to produce a cleaner colour-magnitude diagram (CMD) that should enable some observational and theoretical considerations regarding the structure and stellar distribution of the horizontal branch (HB). We will also discuss the position of M56 relative to the so-called Oosterhoff Gap and its relation with its probable extragalactic origin. Alternative distance determination methods involving the variable stars will be discussed and a theoretical approach of the representation of the HB blue tail as a consequence of mass loss in the RGB will close our work.

2. OBSERVATIONS AND DATA REDUCTIONS

The data used for the present paper were obtained with the 0.84-m Ritchey-Chrétien telescope of the Observatorio Astronómico Nacional San Pedro Mártir on two epochs performed approximately one and a half months apart, from August 4-5 2019 for the first, and from September 27-30 2019 for the second. The first set of images consisted of 255 in the V filter and 659 in the I filter, while the second set consisted of 258 images in both V and I filters. The estimated average seeing of the first set of images was 1.4" while for the second set was 1.8". The exposure times ranged between 7s-60s for the first set in both filters, while for the second set it ranged between 40s-60s also in both filters. Alongside with the Mexman filter wheel, the detector used was a 2048 × 2048 pixel ESOPO CCD (e2v CCD42-90) with 1.7 e ⁻ /ADU gain, a readout noise of 3.8 e ⁻ , a scale of 0.444 arcsec/pix corresponding to a field of view (FoV) of ≈ 7.4 arcmin².

3. STAR MEMBERSHIP USING GAIA-EDR3

Before studying the CMD of M56, it is convenient to be able to separate field stars from the likely cluster members. We have approached this goal by employing the high-quality astrometric data and proper motions in the Gaia-eDR3 data base (^{Gaia Collaboration 2021}) and the method of ^{Bustos Fierro & Calderón (2019)}, which is based on the Balanced Iterative Reducing and Clustering using Hierarchies (BIRCH) algorithm developed by ^{Zhang et al. (1996)}. This algorithm detects groups of stars in a 4D space of physical parameters - projection of celestial coordinates (X _t , Y _t ) and proper motions (µ _α , µ _δ ). Figure 1 illustrates the resulting vector-point diagram (VPD) showing the motion of the cluster relative to the field stars. Also the resulting Gaia-CMD of member stars is shown, which is consistent with that of a globular cluster, giving support to the cluster members selection. Unlike more traditional methods, which are based on the fitting of a probability density function and a cut-off at a certain minimum probability to consider stars as reliable cluster members, our method is based on a clustering algorithm as a first stage and a detailed analysis of the residual overdensity as a second stage; member stars extracted in the first stage are labelled M1, and those extracted in the second stage are labelled M2. The analysis was carried out for a 25 arcmin radius field centered in the cluster. We considered 76 235 stars from which 6552 were found to be likely members, out of these only 4695 were in the FoV of our images. We were able to measure and produce light curves for 3632 member stars.

Fig. 1 The left side panel displays the vector-point diagram (VPD) of the proper motions of the stars measured by Gaia-eDR3 in M56. Stars within a 25 arcmin radius were used to perform a membership analysis. Blue dots correspond to stars with a high probability of being cluster members while the grey dots correspond to field stars. The panel on the right shows the corresponding CMD where the blue symbols for member stars highlight the characteristic shape of the globular cluster. For details, see § 3. The colour figure can be viewed online. 

From the distribution of field stars in phase space we estimated the number that is expected to be located in the same region of the sky and of the VPD as the extracted members; therefore, they could have been erroneously labelled as members. Within the M1 stars the resulting expected contamination is 77 (1.6%) and within the M2 stars it is 167 (9.0%); therefore, for a given extracted star its probability of being a cluster member is 98% if it is labelled M1, or 91% if it is labelled M2.

To evaluate the completeness of our DIA photometry we compared the number of stars we detected and measured in the FoV of our images, with the ones measured by Gaia-eDR3. Of the 76 235 stars considered for the membership analysis, only 12 433 are in our FoV, and we were able to measure 4845. In Figure 2 we confront these two samples as a function of magnitude. We can see that our DIA photometry is complete down to G=16.5 mag, which comfortably includes the horizontal portion of the HB, and it is fairly complete down to G=18.5 mag, which encompasses the vertical portion of the HB in the CMD.

Fig. 2 Red histogram represents the number of stars in our FoV per G magnitude interval as measured by Gaia- eDR3. Black histogram is the magnitude distribution, converted from V to G, of the stars measured in this work through our DIA approach. See the discussion on completeness in § 3. The colour figure can be viewed online. 

4. THE VARIABLE STARS IN M56

4.1. Light Curves of Known Variables

Of the 14 known variables in M56, listed in Table 1, only V1-V6 and V12-V14 are within the FoV of our images. Variables V7, V8, V9, V10 and V11 are outside the FoV of our images and have been identified as being field variables in the CVSGC and have also been confirmed as such by the membership analysis performed in § 3. For the sake of completeness, we tried to recover their light curves from the Gaia- DR2 photometry (^{Gaia Collaboration et al. 2018}) but we were only able to do it for V8, V10 and V11. The variables V7 and V9 do not possess the variability flag associated by Gaia and therefore were not measured. For V8, V10 and V11 we proceeded to transform their Gaia G, G _BP , and G _RP bands into the VI Johnson standard system to make them consistent with our photometry. For this purpose, we used the relations provided by J.M. Carrasco and which can be found in the Gaia-DR2 documentation¹ (2018: Gaia team). The light curves of the known variables and the ones recovered with Gaia are displayed in Figure 3.

TABLE 1 DATA OF VARIABLE STARS IN M56 IN THE FOV OF OUR IMAGES 

Star ID	Type	< V > (mag)	< I > (mag)	AV (mag)	AI (mag)	P (days)	HJDmax + 2450000	α (J2000.0)	δ (J2000.0)	Membership status
V1	CW	15.47	14.60	0.89	0.54	1.510315	8756.7828	19:16:39.33	+30:12:16.6	M1
V2	CST?	-	-	-	-	-	-	19:16:37.34	+30:11:35.2	M1
V3	SR	13.144	-	-	-	-	-	19:16:37.82	+30:12:33.9	M1
V4	RRc	16.06	15.38	0.28	0.18	0.423552	8700.8311	19:16:27.45	+30:08:21.1	M1
V5	SR	13.154	-	-		-	-	19:16:36.59	+30:08:47.3	M1
V6	RVB	12.754	11.644	0.68	0.62	-	-	19:16:35.77	+30:11:38.9	M1
V7 1,2,6	Lb	-	-	-	-	-	-	19:16:58.74	+30:07:31.0	FS
V8 1,2	Lb	15.155	11.035	0.66	0.28	-	-	19:16:28.70	+30:05:24.2	FS
V9 1,2,6	Lb	-	-	-	-	-	-	19:16:50.11	+30:19:58.8	FS
V10 1,2	RRab	17.265	16.535	1.08	0.73	0.598900	7088.5956	19:16:02.72	+30:12:25.6	FS
V11 1,2	SX Phe	15.775	15.205	0.53	0.66	0.075625	7281.6951	19:16:03.66	+30:15:40.5	FS
V12	RRab	16.04	15.22	0.42	0.30	0.906231	8753.6200	19:16:17.25	+30:09:23.7	M1
V13	SR	14.584	13.104	-	-	-	-	19:16:38.74	+30:10:59.0	M1
V14	RRc	16.09	15.40	0.28	0.17	0.378104	8699.8962	19:16:29.84	+30:12:27.4	M1
V153	SX Phe	19.11	18.45	0.25	0.23	0.045524	8756.7720	19:16:41.55	+30:12:08.7	M1
V16 1,3	EB	18.50	17.45	0.25	0.20	0.336064	8756.7021	19:16:34.09	+30:09:09.5	FS
V173	EB	18.47	17.77	0.43	0.39	0.319831	8756.6109	19:16:34.11	+30:10:25.4	M2
V183	EB	15.35	14.03	-	-	0.367116	8756.7379	19:16:38.70	+30:11:09.9	M1
V19 1,3	RRc	20.13	19.02	0.47	0.19	0.273647	8756.7900	19:16:30.19	+30:12:34.5	FS

¹Field Star. ²Out of our FoV. ³New variable. ⁴Magnitude weighted mean. ⁵Transformed into V I Johnson standard system from Gaia-DR2 photometry. ⁶Not measured by Gaia-DR2.

Fig. 3 Light curves of all known variables in M56 previous to the present work. The colour code is: lilac for data from Gaia-DR2 photometry transformed into V, black and red correspond to two nights from August 2019, and blue to data from September 2019. Irregular or probably non-periodic giants are plotted as function of HJD. The periods are expressed in fraction of a day. The colour figure can be viewed online. 

We remark that the red giants V3, V5 and V6 were saturated in our I images and as such we have no I-band photometry.

4.2. The Search for New Variables

The small number of reported variables present in M56 prompted a rigorous search in order to provide a more complete sample for our analysis. We approached this task by using the string-length method (^{Burke et al. 1970}, ^{Dworetsky 1983}). We phased each light curve in our data with periods between 0.02 d and 1.7 d in steps of 10 ⁻⁶ d. In each case, the length of the line joining consecutive points, called the string-length and represented by the parameter S _Q , was calculated. The best phasing occurs when S _Q is minimum, and corresponds to the best period our data can provide. This method led us to the discovery of five new bona fide variables: one SX Phe, three EB and one RRc, and were named V15-V19. According to the method described in § 3, V16 (EB) and V19 (RRc) are field variables. Their light curves are displayed in Figure 4. The ones deemed to have a high probability of being cluster members, (coded M1 and M2 in Table 1), are in the expected region of the CMD (Figure 5), in good agreement with their variability type. All variables listed in Table 1 that are contained in our FoV are identified in the cluster chart of Figure 6.

Fig. 4 Light curves of the variables newly found in this work: one SX Phe (V15), three EB (V16-V18) and one RRc (V19). The colour code is as in Figure 3. The periods are expressed in fraction of a day. The colour figure can be viewed online. 

Fig. 5 The colour-magnitude diagram of M56. The left panel shows all the stars measured in the FoV of our images. Non-member field stars identified by the method described in § 3 are shown in purple colour, while the likely members are represented by black dots. The colour code for the variable stars is as follows: in teal the CW, in green the RRc, in blue the RRab, in orange the newly discovered variables, in red the SR, and in yellow a suspected but still unconfirmed variable. The right panel is the intrinsic CMD for E(B − V ) = 0.26, where isochrones for ages 12.0 (cyan), 12.4 (blue), 13.0 (green) and 13.5 (red) Gyrs, [Fe/H]=-2.0, Y=0.25 and [α/Fe]=+0.4, built from the models of ^{VandenBerg et al. (2014}), have been added by shifting them to a distance of 10 kpc to reproduce the observations. The red ZAHB was calculated from the Eggleton code (^{Pols et al. 1997}; Pols et al. 1998; ^{Schröder et al. 1997}). The blue ZAHB is from the models of VandenBerg et al. (2014). The thin black lines are evolutionary tracks for a helium core mass of 0.50M _⊙ and total masses of 0.64, 0.65 and 0.66M _⊙ also from the Eggleton code. The empirical border between first overtone and fundamental mode, i.e., the red edge of the first overtone instability strip (FORE), is shown as two vertical black lines. (See § 7 for a full discussion). The colour figure can be viewed online. 

Fig. 6 Identification chart of variables in our FoV of M56. The field is 5.92×5.92 arcmin². Expansion of the digital version is recommended for clearness. 

5. THE FOURIER DECOMPOSITION APPROACH TO THE PHYSICAL PARAMETERS OF RR LYRAE STARS

The use of Fourier light curve decomposition has been a well tested approach towards the determination of key physical parameters of RR Lyrae stars. The amplitudes and displacements, A _k and φ _k , and the period P of a series of harmonics of the form:

are intimately correlated with stellar values of [Fe/H[, distance, and other relevant physical parameters. In equation 5, m(t) is the magnitude of the star at time t, P is the period of pulsation, and E ₀ is the epoch. To calculate A _k and φ _k of each harmonic, we made use of a least-squares fit approach. The Fourier parameters are defined as φ _ij = jφ _i − iφ _j , and R _ij = A _i /A _j . For the RR Lyrae stars, the Fourier coefficients are listed in Table 2.

These parameters, inserted in ad-hoc calibrations for RRab and RRc stars, can lead to individual values of the physical parameters reported in Table 3. The specific calibrations and their zero points are given and discussed in great detail in several of our previous papers (e.g. ^{Arellano Ferro et al. (2017)} and ^{Deras et al. (2019)}) and shall not be repeated here.

M56 is a particularly weak case for Fourier de- composition since it contains a very limited sample of member RR Lyrae: one RRab (V12) and two RRc (V4 and V14). The value of [Fe/H]_ZW, i.e., the metallicity in the scale of ^{Zinn & West (1984)}, is obtained from the calibration of ^{Morgan et al. (2007)} for RRc stars and ^{Jurcsik & Kovács (1996)} for the RRab star. A value in the high resolution spectroscopic scale [Fe/H]_spec can be obtained from the calibrations of ^{Nemec et al. (2013)}. In all cases, the key Fourier parameters are φ ₃₁ and P . For V4, the value φ ₃₁=5.730 (Table 2) is far outside the range of the RRc stars that define the calibration, or in fact it is anomalously large among RRc stars in general (e.g. see Figure 2 of ^{Clement et al. (1992)}). This fact is most likely due to its peculiar light curve shape, with maxima more acute than in typical RRc starscompare with the light curve of V14). For the RRab star V12 we also face some inconvenience: the value of its compatibility parameter, as defined by ^{Jurcsik & Kovács (1996)} and ^{Kovács & Kanbur (1998)} is D _m =4.3. These authors recommend values smaller than 3.0 to warrant the consistency of a given light curve with the morphology of those of the calibrators. Hence V12 seems also a bit off the limits.

As a result we have a metallicity estimation via the Fourier approach for one RRc star (V14); [Fe/H]_Spec = −2.03 ± 0.14. This value is, however, consistent with the standard value reported by ^{Harris (1996)} of −1.98.

6. COMMENTS ON THE DISTANCE TO M56

The distance to M56 has been estimated by several independent approaches, namely, the absolute magnitude estimation of the RR Lyrae stars via the Fourier decomposition of their light curves with the results V12 (9.6 kpc), V4 (8.9 kpc), V14(9.0 kpc) (see also Table 3); the positioning of the isochrones and ZAHB on the intrinsic CMD (10 kpc); the P -L in the I filter for the RR Lyrae stars (9.0 kpc) (^{Catelan et al. 2004}) and the application of three independent and well known P -L relations of SX Phe stars (^{Poretti et al. 2008}, ^{Arellano Ferro et al. 2011}, and ^{Cohen & Sarajedini 2012}). The calculation for the SX Phe stars V11 and V15 (new variable) deserves a further discussion. Assuming that these two stars are pulsating in the fundamental mode, these calibrations lead to the distances of 3.46 ± 0.14 kpc and 11.5 ± 0.4 kpc, respectively. First or second overtone pulsation assumptions lead to larger distances. It is clear, then, that V11 is definitively not a cluster member, in agreement with the determination from the Gaia-eDR3 proper motions described in § 3. V15 seems to lie about 1-2 kpc behind the cluster and may or may not belong to the cluster; hence, we did not include it in the calculation of the average distance shown in Table 4.

7. THE CMD OF M56 AND ITS HB

From the PSF-fitting photometric analysis of our images in § 2, we were able to recover 4845 stars, 3630 (≈ 75%) of which have a high probability to be cluster members. This allowed us to create a clean CMD by removing the field stars (Figure 5). We used the models from ^{VandenBerg et al. (2014)} to generate a ZAHB and four isochrones of ages ranging from 12.0 - 13.5 Gyrs with a metallicity of [Fe/H]_ZW = -2.0, Y=0.25 and [α/Fe]=0.4. Also, using the Eggleton code (^{Pols et al. 1997}; Pols et al. 1998; ^{Schröder et al. 1997}) we generated another ZAHB for a core mass of 0.50M ₀ , and three evolutionary tracks corresponding to total masses of 0.64, 0.65, and 0.66M ₀ . The isochrones and the ZAHBs were shifted to a distance of 10 kpc using a reddening E(B−V ) = 0.26. The small discrepancy between the ZAHB from the models of VandenBerg and Eggleton is due to the assumed core mass of the progenitor star. Although the sample of RR Lyrae stars is small, the fact that the RRc and RRab stars are on the first overtone and fundamental sides of the first overtone blue edge (FOBE), is consistent with the mode distribution found in other OoII type clusters, e.g. ^{Yepez et al. (2020)}. A useful quantity to help describe the morphology of the HB is the Lee parameter (^{Lee 1990}) defined as L = (B−R)/(B + V + R), where B is the number of stars on the blue side of the instability strip (IS) or the first overtone blue edge (FOBE), R is the number of stars on the red side of the IS or fundamental mode red edge (FMRE) and V is the number of variable stars within the IS. In the case of M56, this yields an L = 0.82. Figure 7 shows a plot of L vs [Fe/H]_ZW where globular clusters previously studied by our group following the same approach as the one used in this work, are plotted. The triangle in Figure 7 is defined by ^{Catelan (2009)} as a region that is preferentially populated by clusters thought to be of extragalactic origin. M56 falls well above this triangular region. The HB of M56 is predominately blue, which is usually seen in most metal-poor Oosterhoff II (OoII) type clusters. While it is not possible to discuss the Oo-type of M56 given the scarce number of RR Lyrae stars, the extremely large period of the only RRab star V12, makes the case most peculiar even among Oo-II type clusters. Its position on the L vs [Fe/H]_ZW plane, however, makes it difficult to support its possible extragalactic origin on these grounds.

Fig. 7 The HB structure parameter L as a function of [Fe/H]_ZW. The two lines define theoretically the socalled Oosterhoff Gap (^{Bono et al. 1994}), a region in the L - [Fe/H]_ZW plane that is devoid of bona fide Galactic globular clusters. The triangle represents a region found by ^{Catelan (2009)} that is populated by globulars identified in systems outside of the Galaxy. M56 is coloured in red and its position was determined by the estimated values of the metallicity and L in this work (0.82, -1.96). Circles and squares are used for inner and outer halo clusters, respectively. Symbols with a black rim represent globular clusters where the fundamental and first modes of pulsations are well segregated on the HB, while in those without a rim the modes of pulsation are mixed in the either/or region. Open symbols correspond to clusters yet to be studied by our group, and their current positions are subject to change. For a detailed discussion, see § 7. The colour figure can be viewed online. 

7.1. Post-Evolutionary ZAHB Models

In order to offer some input on the stellar mass distribution on the HB, as well as on the mass structure (core and envelope masses) for the stars of the HB and their ulterior evolution towards higher luminosities, we have calculated some evolutionary tracks using the Eggleton code (^{Pols et al. 1997}; Pols et al. 1998; ^{Schröder et al. 1997}) with a modified Reimers law for the mass loss with η = 0.8 × 10 ⁻¹³ (^{Schröder & Cuntz 2005}) as a consequence of the He-flash episodes during the RGB stages. A detailed description of this procedure has been given by ^{Arellano Ferro et al. (2020)}. To compare our theoretical predictions with the observed stellar distributions, we converted the CMD into an HRD, i.e., we converted the plane (V − I)-V into log(T _eff )-log(L/L _⊙). To this end, we adopted the (V − I) _o -log(T _eff ) and BC-log(T _eff ) calibrations of ^{VandenBerg & Clem (2003)}. The resulting HRD of the HB region is shown in Figure 8. It should be noted that the scatter in the HB is larger on its blue side, i.e., the stars corresponding to the vertical blue tail in the HB in the CMD. Since they are considerably fainter, some may or may not be truly members of the HB. In comparison, the horizontal portion where the RR Lyrae stars reside is neat and well represented by the theoretical loci. While in fact we have modelled cases for core masses of 0.49, 0.50 and 0.51M _⊙, we display only the case for 0.50M _⊙ as we believe it produced the best representation of the data.

Fig. 8 HRD of the horizontal branch of M56. These models were created using a helium core mass of 0.50M _⊙. Four tracks for total masses of 0.56 (black), 0.60 (orange), 0.64 (cyan) and 0.68M _⊙ (lilac) are shown. The blue ZAHB is from the models of ^{VandenBerg et al. (2014)}. Theoretical border lines of the IS are from ^{Bono et al. (1994)} for the fundamental mode (blue) and the first overtone (green). The colour code of the variable stars is as in Figure 4. The fiducial marks at A (≈ 2.5 Myrs), B (leaving the HB) and C (at the base of the AGB) along the tracks help to visualise the evolutionary times involved; on the 0.56M _⊙ track, the time from A to C is about 100.9 Myrs, thus the post-ZAHB age of V1 is about 100.8 Myrs. The crossing time from B to C takes roughly 2 Myrs. The colour figure can be viewed online. 

In Figure 8 there are also shown the positions of the RRab V12 (blue dot), RRc V4 and V14 stars (green dots) and the only CW V1 star (teal dot), along with the theoretical borders of the instability strip for the fundamental and first overtone modes (^{Bono et al. 1994}). The red line is the ZAHB for the stars with a He-core of 0.50M _⊙. The blue line corresponds to the ZAHB calculated with the models of ^{VandenBerg et al. (2014)} for [Fe/H]=-2.0, Y=0.25 and [α/Fe]=+0.4; this ZAHB starts with a core mass of 0.49M _⊙ that decreases towards the blue and for this reason it lies a bit below of the red line. Black, orange, cyan and lilac loci are tracks for total masses of 0.56, 0.60, 0.64, 0.68 M _⊙, respectively, i.e. for envelopes with 0.06, 0.10, 0.14 and 0.18 M _⊙. The RR Lyrae stars lie between tracks with envelopes of 0.14-0.18 M _⊙. The CW star, V1, seems well represented by a track with a very thin envelope (0.06M ₀ ) which has originated at the bluest part of HB. Considering an age range of 12.0 - 13.5 Gyrs, the probable progenitor star of the stars on the HB had an initial mass of ≈ 0.81 - 0.83 M _⊙ at the ZAMS. Such a star lost between 0.15 and 0.26 M _⊙ during the He flashes in its evolution from the RGB towards the ZAHB.

8. SUMMARY

In this work we have performed V I CCD photometry of 4845 sources present in the globular cluster M56 and we summarise our results as follows: