The variable stars in the field of the bulge cluster NGC 6558

We made a survey of the variable stars in a $13.2 \times 13.2$ arcmin$^2$ centered on the field of the Galactic bulge cluster NGC 6558. A total of 78 variables was found in the field of the cluster. Many of these variables are included in the Catalogue of Variable Stars in Galactic Globular Clusters (Clement et al. 2001), OGLE or Gaia-DR3 data releases. A membership analysis based on the proper motions of Gaia-DR3 revealed that many of these variables do not belong to the cluster. We employed the data from the aforementioned surveys and our own data in the VI photometric system to estimate the periods, which along with the light curves morphology and position in a deferentially dereddened colour-magnitude diagram(CMD), help classifying the variable types. Two new member variables were found; an eclipsing binary (V18) and a semi-regular SR/L (V19). In the end we conclude that only 9 variables are likely cluster members. Member variables were used to discuss the mean metallicity and distance of the parental cluster and find the average values.


INTRODUCTION
The globular cluster NGC 6558 resides in the bulge of the Galaxy, very near to the Galactic center (Bica et al. 2016).From high resolution spectroscopic analysis of four red giants in NGC 6558 Barbuy et al. (2018) concluded that the cluster shows abundance pattern typical of the oldest inner bulge clusters and could be among the oldest objects in the Galaxy.Dynamically, the orbit of NGC 6558 is confined to less that 1.5 kpc (see the orbital integration in Fundamental parameters of Galactic globular cluster by Baumgardt et al. (2023) 1 from the Galactic center, confirming the cluster is a local resident with all rights.
Being the bulge regions very rich in stars and dust, the images of bulge clusters are highly contaminated with field stars and are generally subject to heavy interstellar differential reddening.Hence, to properly study the colour magnitude Diagram (CMD) of a bulge cluster, its stellar populations or families of variable stars, or to use specific groups of stars as indicators of physical properties, requires a thorough membership analysis and the calculation of a reddening map across the field of the system (e.g.Alonso-García et al. (2012); Yepez et al. (2023)).Once this is achieved, then some of the variable stars can be used as indicators of cluster metallicity and distance, and their positions in the CMD can be compared with theoretical calculations to inferred evolutionary stages and internal stellar structure, in particular for the stars in the horizontal branch (HB).Previous studies carried out by our team towards the estimation of the main physical parameters; reddening, distance and metallicity employing their variable star populations are described in detail in the works by Arellano Ferro (2022) and Arellano Ferro (2024).The results for the bulge clusters NGC 6333, NGC 6401 and NGC 6522 are reported in the papers by Arellano Ferro et al. (2013), Tsapras et al. (2017) and Arellano Ferro et al. (2023) respectively.

Bosque Alegre and Las Campanas Data
The data for this work were obtained with the 1.54 m telescope of the Bosque Alegre Astronomical Station (EABA), of the National Observatory of Cordoba, Argentina, during August-September 2018 and between June and August 2019, for a total of eleven nights.Two detectors were employed; in 2018 a CCD KAF-16803 with 4096 × 4096 pixeles, while in 2019 a CCD KAF-6303E with 3072 × 2048 pixeles.The corresponding fields were 16.9 × 16.9 arcmin2 and 12.6 × 8.4 arcmin 2 respectively.Also observations were performed with the 1m telescope SWOPE from Las Campanas Observatory, Chile, during three nights in June 2018.The dectector was a CCD E2V 231-81 with 4096 x 4112 pixels and a field of 14,8 × 14,9 arcmin 2 .See Table 1 for the log of the observations.In the following we shall refer to EABA seasons as BA18 and BA19, and to Las Campanas seasons as SWOPE.

Data Reduction
To extract high-precision time-series photometry for all point sources in the field of our images, we employed the Difference Image Analisis (DIA) and the pipeline DanDIA (Bramich 2008); (Bramich et al. 2013); (Bramich et al. 2015)..The reference  and  images are obtained by stacking up images of the best quality in the collection.Then, individual images are subtracted from the reference to create individual differential images where the flux of all detected point sources is measured.The total flux in ADU/s is calculated as: where  ref is the reference flux (ADU/s),  diff () the differential flux (ADU/s) and () is the the photometric scale factor.To convert fluxes to instrumental magnitudes we used: where   () is the instrumental magnitude of the star at time .

OGLE and 𝐺𝑎𝑖𝑎-DR3 photometric data
We have also extensively used data of the variable stars in the field of the cluster identified in the OGLE and -DR3 data bases, and shall be discussed in the following sections.

Transformation to the VI standard system
The instrumental light curves can be converted into the standard system employing local standard stars in the field of the cluster.We found 9, 10 and 12 standard stars for the images in the seasons SWOPE, EABA2018 and EABA2019 respectivelly, in the catalogue of (Stetson 2000) 2 , which have been set into the Johnson-Kron-Cousins standard system using the equatorial standards from Landolt (1992).The transformation equations are of the form  - = A( -) + B, and  - = C( -) + D, and the season constants are reported in Table 2.
In Table 3 we include a small portion of the time-series VI photometry obtained in this work.The full table shall be available in electronic form in the Centre de Donnés astronomiques de Strasbourg database (CDS).

MEMBERSHIP ANAYSIS
Being in the Galactic bulge, the colour-magnitude diagram (CMD) of NGC 6558 is heavily contaminated by field stars all subject to a remarkable differential reddening.To produce a clean and useful CMD a membership analysis and a local reddening map are in order.
The membership analysis was performed using the positions and proper motions available in the -DR3 and employing the method of Bustos Fierro & Calderón (2019).The method 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 ( t , t ) and proper motions (  * ,  ) 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.
There are 45,506.point sources within an 8 arcmin field centered in the cluster.However, proper motions are available for only 28,334.Most of the stars were found field stars and only 495 were identified as likely cluster members.

DIFFERENTIAL REDDENING AND THE CMD
To properly deredden the colour magnitude diagram of this bulge cluster, it is necessary to consider the effects of differential reddening.Fortunately, NGC 6558 has been included in the thorough reddening study of the inner Galaxy by Alonso-García et al. (2012).We used their reddening map to differentially deredden the VI CMD obtained from our photometry.
In Fig. 2 we display on the left panel the CMD dereddened with a constant value  (−) = 0.40.On the right panel we have applied the differential corrections guided by the reddening map, i.e.  ( − ) +    , where    corresponds to the differential corrections for the corresponding position of the star.The improvement on the dispersion at the RGB and the HB are obvious

THE VARIABLE STARS IN NGC 6558
The field of NGC 6558, like in most of the bulge clusters, is very rich in variable stars.The advent of missions like  and OGLE have detected a large number of them.However, many of these variables are not linked to the cluster but are merely projected against it.Our aim in the following sections is to critically evaluate the membership of them.The challenge is to achieve good photometric values that entitle us to position the variables in our observed and properly dereddened VI CMD, which along with the proper motion analysis, their pulsational type and, in the case of RR Lyrae, the estimation of their distance via the Fourier decomposition of their light curves, should allow us to pronounce about their membership status.

The catalogued variables (Clement et al. 2001)
The catalogue of variable stars in globular clusters (CVSGC) (Clement et al. 2001) in its April 2016 update, lists 17 variables and labels two of them (V2 and V7) as probably non-variable.The rest are 8 RRab, 3 RRc, and 4 long period variables labeled "L?"

Other variables in the field of NGC 6558 according to OGLE, and 𝐺𝑎𝑖𝑎
In the  mission (Gaia Collaboration et al. 2016) and the Optical Gravitational Lensing Experiment (OGLE) (Udalski et al. 1992) many variables were detected in the field of NGC 6558.We have cross-matched those variables with point sources measured in our photometry and have combined the photometric data to build the light curves.In the case of OGLE III (Soszyński et al. 2013) and IV (Soszyński et al. 2014), VI magnitudes are available.For the case of -DR3 data, a transformation into the VI system was necessary.This was achieved using the transformation equations of (Riello et al. 2021).
We identified 56 variables in the OGLE data base within the field of NGC 6558 and for the present purpose of the paper we shall identify them with the prefix 'O'.Similarly we noticed 13 stars announced as variable in -DR3, these stars we call them with the prefix 'G'.Then we aim to confirm their variability and type, and Variable stars in the field of the cluster, members and no members, are shown by colour symbols according to the following code: RRab-blue; RRc-green; RGB-red;  Sct (G10) and member RGB (V19) -yellow; binaries -turquoise; member binary V18 (O20)-magenta.Solid symbols are used for members and empty ones for field stars (triangles) or stars of unknown status due to lack of proper motion (squares).The isochrone is from VandenBerg et al. ( 2014) for [Fe/H]=−1.35and an age of 12.0 Gyrs.Red ZAHB is from the models built from the Eggleton code (Pols et al. 1997(Pols et al. , 1998;;Schroder et al. 1997), and calculated by Yepez et al. (2022).The vertical black lines at the ZAHB mark the empirical red edge of the first overtone instability strip (Arellano Ferro et al. 2015, 2016).to check their membership.The table cross matching the variables in the CVSGC, OGLE and , indicating their membership status, and their light curves are reported in Appendix A at the end of the paper.
In the process we discovered two variables not reported neither in  nor in OGLE databases and which we identified as N1 and N2.In our membership analysis these stars were found to be field stars.Their light curves are shown in Fig. 3. Judging by their periods and light curve morphology they resemble RRc variables, however we note that for N1 the amplitude in the −band is a bit larger than in the −band (Amp  = 0.110 ± 0.002 mag and Amp  = 0.170 ± 0.003 mag), which is unusual.On the other hand, N2 exhibits expected amplitude difference with amplitude in the -band being larger (Amp  = 0.193 ± 0.005 mag and Amp  = 0.151 ± 0.003 mag).
In the identification chart of Fig. 4 we include all stars listed in the CVSGC (2016 edition) plus the two new member variables discovered in this work (V18 and V19).Note that V2 and V7 in fact do not show signs of variability, and that 10 of them are not cluster members, as indicated in Table A1.

THE COLOUR-MAGNITUDE DIAGRAM
In the CMD of Fig. 2 we included all the variables in the field of NGC 6558 coded as described in the caption.All stars whose variability has been confirmed are plotted using their intensity-weighted means <  > 0 and corresponding colour <  > 0 − <  > 0 .Filled and empty symbols were used for members and either field stars or of unknown status due to the lack of proper motions in the -DR3 database.
All member variables occupy the expected regions according to their variable type.We note the distribution of RRab stars relative to the empirical first overtone red edge (FORE) indicated in the DCM by the vertical black lines in the HB.See Fig. 5 for an expanded version of the HB region.There is evidence that at least two RRab stars V3 and V4, are sitting to the left of the FORE, i.e. in the bimodal or "either-or" region, shared by fundamental and first overtone mode pulsators.Both stars show signs of amplitude modulations.The period analysis of V4 (RRLYR-14866) using Period04 (Lenz &  Breger 2004) shows a periodic modulation of the light curve equal to  BL = 26.73±0.01day (known as the Blazhko effect, Blažko 1907).V1, fundamental mode variable, seems a border case between blue and red parts of the instability strip.(Bono et al. 1994) .See Section 9 for a discussion.
from the Fourier decomposition of the best observed members (V1, V3, V4 and V17).The above average period and metallicity, when plotted on the plane [Fe/H] vs <P  > of figure 5 of Catelan (2009), places the cluster in the Oosterhoff gap, suggesting that NGC 6558 is of the intermediate Oosterhoff type or Oo-Int.We note that in the work by Catelan (2009), NGC 6558 was classified as Oo I due to the lower average pulsation period for associated fundamental mode pulsators.
A revision of the membership probabilities assigned by Vasiliev & Baumgardt (2021) for stars in the field of NGC 6558, confirm the membership status of V3, V16 and V17, but assigns very low probabilities to V1 and V4.If these two stars are not considered members the average <P  > becomes 0.677 d, which would only highlight NGC 6558 as a peculiar cluster in the [Fe/H] vs <P  > plane, i.e. too high a metallicity for its average period.Two clusters with a similar property are the moderate metal-rich Oo III clusters NGC 6388 and NGC 6441 which, in spite their extended HB blue tails (Piotto et al. 1997;Rich et al. 1997), they display a prominent red-clump which is absent in NGC 6558.
In our opinion and based on our membership analysis, the five RRab members lead to a value of <P  >=0.60±0.04,which puts the cluster in the Oosterhoff gap in the [Fe/H] vs <P  > plane and NGC 6558 should be considered of an Oo-int nature.

FOURIER LIGHT CURVE DECOMPOSITION OF MEMBER RR LYRAE
We identified five RRab member stars of the cluster.In this section we perform a Fourier decomposition of their light curves and use their Fourier parameters and ad-hoc semi-empirical calibrations to estimate their metallicity and distance.For briefness we will not repeat here the description of this approach since it has been thoroughly described by Arellano Ferro et al. (2010) and summarized by Arellano Ferro (2024).
In Table 4 the Fourier coefficients of the five member RRab are listed as it is their consistency parameter   defined by (Jurcsik & Kovács 1996).According to these authors, the [Fe/H] calibration for RRab stars is applicable for values of   < 3.0.We have relaxed this criterion a bit to   < 5.0 to increase the possibilities of our sample.
We recall that the iron abundance value obtained photometrically from this calibration is given into the scale of Jurcskic and Kovács, [Fe/H] JK , which can be converted into the scale of Zinn & West (1984) via [Fe/H] JK = 1.431[Fe/H]ZW + 0.88 (Jurcsik & Kovács 1996) and in turn it can be transformed into the spectroscopic scale of Carretta et al. (2009)  ZW .In Table 5 we report the mean physical parameter obtained for the 5 RRab stars.For comparison, we have included in column 4 of Tab. 5 the iron values estimated by Dékány et al. (2021) [Fe/H] DK , for the same RR Lyrae in NGC 6558.These authors used the photometric Fourier decomposition approach considering newly calculated calibrations of the -band light curve parameters.The average [Fe/H] DK matches well our results for [Fe/H] UVES .

ON THE DISTANCE TO NGC 6558
The P-L () calibration for RR Lyrae stars can be used to produce independent estimates of the distance.We have employed two calibrations available in the literature.Catelan et al. (2004) et al. 1993).For the calculation, we have adopted the value of the metallicity [Fe/H] UV = -1.20,obtained from the Fourier approach (see Table 5, and the average  ( −)= 0.40.We applied the above calibration to the five RRab stars in Table 5 and the RRc star V6.The period of the latter was fundamentalized using the ratio  1 / 0 = 0.75.The resulting mean distance was 8.10 ± 0.22 kpc, which is in good agreement with the result from the Fourier approach of 8.47 ± 0.46 kpc.
On the other hand, a recent P-L () calibration calculated by Prudil et al. (2024) using data published by the Gaia-DR3 is of the form   = 0.197 − 1.292 log  + 0.196 [Fe/H].When this equation is applied to the six cluster members RRab and RRc, a mean distance of 7.72 ± 0.20 kpc was found.We note that for the treatment of reddening and differential reddening, we used reddening maps and laws from Schlegel et al. (1998) and Alonso-García et al. (2012, see Section 4).
With the P-L relations in the aforementioned paper, Prudil et al. (2024) derived reddening maps and reddening laws for the following four colours: ( −  s ), ( −  s ), ( − ) and ( BP − ).Using the newly reddening maps and reddening laws, solely based on individual RR Lyrae variables toward the Galactic bulge, we have obtained their distances, using a similar approach as used by Prudil et al. (2019).The results are shown in Table 7.In the last column, we have averaged the results for colours ( − s )and ( − ) which in turn lead to a grand average of 8.73±0.29.Comparing these results with those in Table 6 we point to the agreement, within the respective uncertainties, with the distance obtained from the Fourier light curve decomposition approach.
It should be noted that in their compilation of globular cluster distances, Bumgardt et al. 20233 the distances determined for NGC 6558 prior to 1998, tend to be smaller than 8 kpc and the authors report an average of 7.79 ± 0.18 kpc.In their paper Baumgardt & Vasiliev (2021) the reported mean distance is 7.47 ± 0.29 kpc.In either case our calculations via the Fourier approach and the P-L() relationship render a distance a bit larger than 8 kpc, as stated  (Catelan et al. 2004) 7.72±0.20P-l () (Prudil et al. 2024) 7.79±0.18Baumgardt et al. (2023) compilation 7.47±0.29 (Baumgardt & Vasiliev 2021) above.For a clearer reference, in Table 6 we summarize all the above distances and methods

MODELLING THE HORIZONTAL BRANCH
There is a clear correlation between the HB extension to the blue and the metallicity among globular clusters (Sandage & Wallerstein 1960); the lower the metallicity, the bluer the HB.The temperature of a HB star depends both on the metallicity (lower metallicity produces lesser opacities and then more compact, hotter, and bluer shells), but also on the mass of a H-rich shell.It was noted by Schröder & Cuntz (2005) that by comparing models of a given metallicity but different shell masses it can be shown that lower shell masses also produce bluer HB stars.
The role of mass loss during the He-flashes events at the RGB upon the colour of a He-core burning star settling on the HB, has been illustrated by Silva Aguirre et al. (2008).The more mass is lost in the RGB the bluer the star will be on the ZAHB when the helium core gets ignited.
In Fig. 5 the five cluster member RR Lyrae are shown on an expanded version of the HB region.The stars are well represented by models with a core mass of 0.50  ⊙ and a range of shell mass between 0.10 and 0.18  ⊙ .The main-sequence progenitor of these stars is a 0.82-0.86 ⊙ star that reached the RGB after approximately 13 and 11 Gyrs respectively, where it lost some 20 to 30% of its mass before evolving to the ZAHB.
In older globular clusters, HB stars have developed from slightly lower mass stars on the main sequence.Since the degenerate Helium-core needs in all cases ∼ 0.5  ⊙ to start the central He-burning, then the resulting H-rich shell mass are smaller and bluer.We have argued (Arellano Ferro et al. 2020) that the range of colour on the HB population, larger than expected from a suitable model and their moves on the HRD (Vandenberg & Durrell 1990), can be explained by slight variations of the mass loss on the RGB, which produces certain shell-mass range of HB stars in the same globular cluster.A moderate star-to-star variation of mass-loss would make a simple explanation for the extended colour distributions on the HB.We have also speculated that mass-loss in the RGB may be modulated by the presence of magnetic fields in red giants (Konstantinova-Antova et al. 2013).(Cacciari et al. 2005) and (Kunder et al. 2013a): likewise are the red and black parabolas for RRc stars (Kunder et al. 2013b) and (Yepez et al. 2020).The distribution of stars do not favour neither Oo I nor Oo II types.The RRab variables V1 and V4 were found to be cluster members in our analysis, but were given a low membership probability by the analysis of Vasiliev & Baumgardt (2021) .See discussion in section 6.

SUMMARY AND CONCLUSIONS
Since the stellar field in the Galactic bulge is densely populated, the number of variable stars in the field of a Galactic bulge cluster can seem very large, however, it is very likely that many of those variables do not pertain to the cluster.That such is the case of NGC 6558 has been demonstrated in this work.In the Catalogue of Variable Stars in Galactic Globular Clusters, OGLE and -DR3 data, we identified 78 stars in the field of NGC 6558.However a proper motion analysis and a thorough exploration of this variables on the CMD of the cluster shows that only 9 of them seem to be truly cluster members.
We have then Fourier decomposed the light curves of the five RRab members to estimate the mean distance and metallicity of the cluster as 8.47 ± 0.46 kpc and [Fe/H] ZW = −1.33 ± 0.11, or in the spectroscopic scale of Carretta et al. (2009) [Fe/H] UVES = −1.20 ± 0.13.
A cross identification of all variables in the field of the cluster between their OGLE and -DR3 numbers is provided and VI light curves are displayed and made available in the Centre de Donnés astronomiques de Strasbourg database (CDS).Their ephemerides, AR and DEC, and mean VI magnitudes and amplitudes are also tabulated. of DGAPA-UNAM through projects IG100620 and IN103024.We have made an extensive use of the SIMBAD and ADS services, for which we are thankful.Data Availability: The data underlying this article shall be available in electronic form in the Centre de Donnés astronomiques de Strasbourg database (CDS), and can also be shared on request to the corresponding author.membership status is indicated by an asterisk in column 3.In Table A2 there are reported the intensity weighted mean magnitudes, amplitudes and epochs for all variables in the field, regardless they are members or not.Although long-period variability cannot be ruled out when plotted vs HJD in stars G3, G15, G19 and G26, further data would be required to establish them as true variables.Clear variations are found in G8 which phases nicely with a period of 70.985 days, and G10 for which a period of 0.178146 d was found.However, from their proper motion analysis they seem to be field stars.

Figure 1 .
Figure 1.Results from the membership analysis (section 3).Left and right panels show the vector point (VPD) and CMD diagrams respectively.Red dots represent the likely cluster members while the gray dots represent the field stars.
Fig. 1 displays the results in the Vector-Point diagram (VPD) and the resulting Colour-Magnitude diagram (CMD).The identified likely members clearly trace the major features in the CMD, such as the horizontal branch (HB) and the red giant branch (RGB).

Figure 2 .
Figure2.The CMDs built from cluster members measured by our VI photometry.The panel on the left shows the CMD dereddened with an average  ( −  ) = 0.40 while the one to the right has been differentially dereddened as described in section 4. The improvement after the corrections is clear.Variable stars in the field of the cluster, members and no members, are shown by colour symbols according to the following code: RRab-blue; RRc-green; RGB-red;  Sct (G10) and member RGB (V19) -yellow; binaries -turquoise; member binary V18 (O20)-magenta.Solid symbols are used for members and empty ones for field stars (triangles) or stars of unknown status due to lack of proper motion (squares).The isochrone is from VandenBerg et al. (2014) for [Fe/H]=−1.35and an age of 12.0 Gyrs.Red ZAHB is from the models built from the Eggleton code(Pols et al. 1997(Pols et al. , 1998;;Schroder et al. 1997), and calculated byYepez et al. (2022).The vertical black lines at the ZAHB mark the empirical red edge of the first overtone instability strip(Arellano Ferro et al. 2015, 2016).

Figure 3 .
Figure3.Two newly detected variables, N1 and N2, which do not pertain to NGC 6558.Their light curves are phased with periods 0.303797 d and 0.342412 d respectively.Blue points represent observations, Fourier fits to observed data are marked in red, and green lines (bottom panel) stand for the difference between Fourier fits to illustrate colour variation.

Figure 4 .
Figure 4. Identification chart of the cluster member variable stars NGC 6558.The field of the left panel is about 13.2×13.2arcmin 2 whereas the central region in the right panel is approximately 3.3×3.3arcmin 2 According toArellano Ferro et al. (2019)(their figure8) orDeras et al. (2022) (their figure7), the presence of fundamental mode RRab stars in the bimodal region is an exclusive characteristic of some Oo I clusters, and does not occur in Oo II type clusters.This comment is of relevance considering that the Oosterhoff type of NGC 6558 cannot be assessed clearly: in the Bailey's diagram of Fig. 6 member stars are few and their distribution do not decant in favour of either Oo I or OoII sequences; the average period of the 5 member RRab stars is 0.60 ± 0.04 day which is the border between Oostherhoff types.The empirical border [Fe/H] ZW ∼ −1.5 between the metal poor Oo II and the metal richer Oo I (Arellano Ferro 2024), and the value of [Fe/H] ZW = −1.33 ± 0.11 obtained

Figure 5 .
Figure5.Horizontal branch region of NGC 6558.Blue and green symbols represent RRab and RRc cluster member stars respectively.Red continuous line and vertical black lines are the ZAHB and first overtone red edge described in the caption of Fig.2.Segmented lines are evolutionary tracks with total and core masses given in the figure legend.Blue and green vertical loci represent the instability strip borders for the fundamental and first overtone respectively(Bono et al. 1994) calculated the equation   = 0.471 − 1.132 log  + 0.205 log , with log  = [/] − 1.765; [/] = [Fe/H] − log(0.638f + 0.362) and log f = [/Fe], from where we adopted [/Fe]=+0.4 (Salaris

Figure 6 .
Figure 6.Period-Amplitude diagram.Continuous and segmented loci to the right are indications of Oo I and Oo II types respectively for the RRab stars distribution(Cacciari et al. 2005) and(Kunder et al. 2013a): likewise are the red and black parabolas for RRc stars(Kunder et al. 2013b) and(Yepez et al. 2020).The distribution of stars do not favour neither Oo I nor Oo II types.The RRab variables V1 and V4 were found to be cluster members in our analysis, but were given a low membership probability by the analysis ofVasiliev & Baumgardt (2021)

Figure B1 .
Figure B1.Light curves of variables reported the CVSGC(Clement et al. 2001) (2016 edition), plus the two new cluster member variables noticed in this work, V18 and V19.Except for V4, where a Blazhko modulation of 26.73 days was found (see section 6), we have not detected secondary frequencies among the scattered light curves.In all cases the scatter is intrinsic to the photometric uncertainties.

Figure B2 .
Figure B2.Light curves of the 56 variables identified in the OGLE database.A period modulation cannot be ruled out in O50 but our data are insufficient to quantify it.

Figure
Figure B2.Continued

Table 1 .
Log of the observation of NGC 6558 * .Columns   and   record the number of images acquired while   y   indicate the typical exposure times.The average nightly seeing is given in the last column.

Table 3 .
Time-series VI photometry for the variables stars observed in this work *

Table 4 .
Fourier coefficients for RRab stars.The numbers in parentheses indicate the uncertainty on the last decimal place.The deviation parameter  m is given in th elast column.

Table 5 .
Physical parameters for the RRab stars.The numbers in parentheses indicate the uncertainty on the last decimal place.Iron values from Dékány et al. (2021).The extreme value of V3 is likely due to blending and was not averaged.*Not included in the average of [Fe/H] due to uncertain  31 .

Table 6 .
Summary of distance estimates of NGC 6558.

Table A1 .
Crossmatch between OGLE and -DR3 identifications of variable stars in the field of NGC 6558.* m-cluster member, f-field star, u-unknown due to lack of proper motion