# A time-series _VI_ study of the variable stars of the
globular cluster NGC 6397111Based on observations at Estación Astrofísica de
Bosque Alegre, Argentina, and at Las Campanas Observatory, Chile

J. A. Ahumada  Observatorio Astronómico, Universidad Nacional de Córdoba,
Laprida 854, X5000BGR, Córdoba, Argentina<EMAIL_ADDRESS>A.
Arellano Ferro  Universidad Nacional Autónoma de México, Instituto de
Astronomía, AP 70-264, CDMX 04510, México<EMAIL_ADDRESS>I. H. Bustos
Fierro  Observatorio Astronómico, Universidad Nacional de Córdoba, Laprida
854, X5000BGR, Córdoba, Argentina<EMAIL_ADDRESS>C. Lázaro
Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna,
Tenerife, Spain, _and_
Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain
<EMAIL_ADDRESS>M. A. Yepez  Universidad Nacional Autónoma de México, Instituto de
Astronomía, AP 70-264, CDMX 04510, México<EMAIL_ADDRESS>K. P. Schröder
Universidad de Guanajuato, Departamento de Astronomía, Guanajuato 36000,
México<EMAIL_ADDRESS>J. H. Calderón  Observatorio Astronómico,
Universidad Nacional de Córdoba, Laprida 854, X5000BGR, Córdoba, Argentina,
_and_
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Buenos
Aires, Argentina<EMAIL_ADDRESS>
###### Abstract

We present a new time-series _VI_ CCD photometry of the globular cluster NGC
6397, from which we obtained and analysed the light curves of 35 variables
carefully identified in the cluster field. We assessed the membership of the
variables with an astrometric analysis based on _Gaia_ DR2 data. The cluster
colour-magnitude diagram was differentially dereddened and cleaned of non
members, which allowed us to fit isochrones for [Fe/H]$=-2.0$ dex in the range
13.0–13.5 Gyr, for a mean reddening $E(B-V)=0.19$, and a distance of 2.5 kpc.
This distance was confirmed using the period-luminosity relation for the
cluster’s five SX Phoenicis variables (V10, V11, V15, V21, and V23) present
among its blue stragglers, yielding $2.24\pm 0.13$ kpc. We also modelled the
light curves of four eclipsing binaries (V4, V5, V7, and V8), and gave the
parameters of the systems; the contact binaries V7 and V8 have distances
consistent with that of the cluster. NGC 6397 appears to harbour no RR Lyrae
stars, being its horizontal branch remarkably blue, much like that of its
analogous cluster, M10. To match the blue tail of the horizontal branch
population, models of 0.64–0.66 $M_{\odot}$ with mass loss at the RGB are
required, indicating rather thin shell masses for the HB stars.

###### keywords:

globular clusters: individual (NGC 6397) , binaries: eclipsing , Stars:
variables: RR Lyrae , blue stragglers , horizontal-branch

††journal: New Astronomy

##  1 Introduction

Galactic globular clusters, one of the most luminous remnants of the formation
epoch of our Galaxy, play a central role in several areas of astronomy. Being
its stellar components at the same distance and of similar age, the stellar
mass distribution allows the comparison of stars in different evolutionary
stages, they are tracers of the structure of the galactic halo, they are good
laboratories for the study of stellar dynamical processes, they give key
information on the early evolution of our Galaxy and other galaxies, and their
ages and compositions provide severe constraints on cosmological models. Much
investigation has been focused on the understanding of the variable stars that
globular clusters typically harbour, in particular—but not only—the RR Lyrae
stars, from the observational and theoretical points of view. These stars can
be used to derive independent estimates of several properties for individual
stars and for the cluster as a whole.

In a series of works, mostly summarized in Arellano Ferro et al. [2017],
hereafter AF17, our group has been employing CCD time series photometry of
globular clusters to update their variable star census and to obtain
independent and homogeneous estimates of their metallicity and distance.

In this paper we analyse time-series observations of NGC 6397 (C1736$-$536 in
the IAU nomenclature), located at
$\alpha=17^{\mathrm{h}}~{}40^{\mathrm{m}}~{}42.^{\\!\\!{\mathrm{s}}}09$,
$\delta=-53^{\circ}~{}40^{{}^{\prime}}~{}27.^{\\!\\!{{}^{\prime\prime}}}6$
(J2000), $l=338.^{\\!\\!\circ}17$, $b=-11.^{\\!\\!\circ}96$. With [Fe/H]
$\approx-2.0$ [Carretta et al., 2009], it is one of the most metal-poor
globular clusters. Its high King central concentration parameter ($c=2.50$,
Trager et al. 1995) indicates that it is one of the Milky Way clusters that
would have suffered core collapse [Djorgovski & King, 1986]. Its foreground
reddening $E(B-V)$ is 0.16–0.19 mag according to the galactic dust
calibrations of Schlafly & Finkbeiner [2011] and Schlegel et al. [1998], and
it has an estimated age of about 13.4 Gyr [Gratton et al., 2003], being thus
one of the oldest Galactic globular clusters. However, a recent determination
by Correnti et al. [2018], that uses IR WFC3@HST data and isochrone fittings,
brings the cluster to $12.6\pm 0.7$ Gyr, an age akin to that of most similar
systems. At $\sim 2.5$ kpc [Gratton et al., 2003], this cluster is the second
nearest to the Sun, after NGC 6121 (M4).

The _Catalogue of Variable Stars in Globular Clusters_
222http://www.astro.utoronto.ca/c̃clement/read.html (CVSGC, Clement et al.
2001) lists 36 variables in the field of the cluster, of which only one or
perhaps two are of RR Lyrae type. Variables V1 (V639 Ara) and V2 (V825 Ara)
were discovered by Bailey [1902]. Star V3, an RRab variable, was mentioned by
H. H. Swope in unpublished correspondence with H. B. Sawyer, who assigned the
number three in her 2nd catalogue [Sawyer, 1955]; V3 is considered to be a
field star and is designated as V826 Ara in the _Moscow General Catalogue of
Variable Stars_ 333http://www.sai.msu.su/gcvs/gcvs/vartype.htm (GCVS, Samus et
al. 2017). Stars V4 to 11 were announced by Kaluzny [1997], although V10 had
already been identified as the blue straggler BS#11 by Lauzeral et al. [1992].
Stars V10 and V11 are pulsating variables of the SX Phoenicis type. Variables
12–24 were identified by Kaluzny & Thompson [2003]. Stars V15, V21, and V23
are SX Phe variables; stars V22 and V23 are also listed by Lauzeral et al.
[1992] as blue stragglers BS#8 and BS#16. Star V22, however, is listed in the
CVSGC as a field RRc? variable. The ellipsoidal variable V16 is the optical
component of the millisecond pulsar J1740$-$534; based on radial velocity
determinations, Kaluzny & Thompson [2003] and Kaluzny et al. [2008] conclude
that it is a cluster member. Finally, Kaluzny et al. [2006] announced stars
V25 to 36, among them one irregular variable (V26), two eclipsing binaries
(V30 and V32), two optical counterparts of cataclismic variables (V33 and
V34), and the rest variable stars of possibly assorted although unconfirmed
types.

A post-core-collapse, dynamically evolved globular cluster, NGC 6397 boasts of
harbouring about two dozen blue stragglers near the cluster centre, revealed
by ground-based CCD photometry [Auriere et al., 1990, Lauzeral et al., 1992],
as well as at least 79 X-ray sources detected by NASA’s Chandra X-ray
telescope within its half-mass radius [Grindlay et al., 2001]. Based on HST
data, Cohn et al. [2010] published optical identifications for the 79 X-ray
sources, among them stars V7, V12, V13, V14, V16, V17, V20, V24, V26, V30,
V31, V33, V34, V35, and V36. In particular, stars V12 and V13 are the optical
counterparts to cataclismic variables CV1 and CV6 of Grindlay et al. [2001],
and stars V33 and 34 are the optical counterparts to CV3 and CV2, in which
Shara et al. [2005] found dwarf nova-like eruptions.

For the sake of completeness, it is also worth mentioning these more recent
developments. First, the search for planetary transits of Nascimbeni et al.
[2012], who identified 12 new variables among low-main-sequence stars, all of
them considered to belong to the field, based on the proper motions and
positions in the colour-magnitude diagram. Second, the photometric survey by
Martinazzi et al. [2017] who, also among the stars of the lower main sequence
of NGC 6397, found 412 suspected variables. And third, the report by Marcano
et al. [2021] of optical modulations in the companion to the X-ray source U18,
which gives support to the interpretation that it is the second “redback”
millisecond pulsar known in the cluster.

In this paper we present a new time-series photometry of NGC 6397. By
analysing the light curves of the cluster variables, we want to obtain
information on the variable stars and their parent system. This is the layout
of the paper. In § 2 we present our observations and describe the DIA
photometric process; § 3 deals with the colour-magnitude diagram, including
the description of the _Gaia_ DR2 method employed to select the cluster
members, and the matching of isochrones; in § 4 we give the accurate
identification and membership of all the cluster variables; § 5 is devoted to
the modelling of four eclipsing binaries in the field of the cluster; in § 6
we derive the distance to the cluster using several methods; in § 7 we discuss
the structure of the cluster horizontal branch, and present its modelling
using the Eggleton codes with mass loss during the red giant stage; finally,
in § 8 we give an account of the results of the work.

##  2 Data and reductions

###  2.1 Observations

The observations were performed in two sites. First, the Bosque Alegre
Astrophysical Station of the Córdoba Observatory, National University of
Córdoba, Argentina, whose 1.54-metre telescope was used on 19 nights between
June 3, 2017 and August 5, 2018. We used a camera Alta F16M with a detector
KAF-16803 of $4096\times 4096$ square 9-micron pixels, binned $2\times 2$,
with a scale of 0.496 arcsec/pix after binning. Due to coma, the binned Bosque
Alegre images were trimmed to $1160\times 1160$ pixels, for a useful field of
view (FoV) of $9.7^{\prime}\times 9.7^{\prime}$. Second, Las Campanas
Observatory, Chile, whose 1.0-metre Swope Telescope was used on the night of
June 28, 2018, with the CCD E2V 231-84 of $4096\times 4112$ pixels also binned
$2\times 2$; the scale is 0.435 arcsec/pixel after binning, and the FoV is
$15.0^{\prime}\times 14.9^{\prime}$. A total of 748 ($V$) and 737 ($I$) images
were acquired along the 20 nights. The log of observations is in Table 1,
where the dates, number of frames, exposure times, and average nightly seeing
are listed.

Table 1: Distribution of observations of NGC 6397

Date | $N_{V}$ | $t_{V}$ | $N_{I}$ | $t_{I}$ | Seeing
---|---|---|---|---|---
|  | [s] |  | [s] | [′′]
20170603 | 9 | 400 | 11 | 200 | 1.5
20170616 | 6 | 400 | 7 | 200 | 2.6
20170630 | 8 | 400 | 9 | 200 | 1.3
20170701 | 12 | 400 | 12 | 200 | 1.8
20170728 | 12 | 400 | 12 | 200 | 1.8
20170818 | 9 | 400 | 11 | 200 | 2.4
20170819 | 18 | 400 | 18 | 200 | 1.8
20170820 | 24 | 90,160,300,400 | 23 | 50,60,120,200 | 1.4
20170826 | 5 | 120 | 6 | 60 | 1.4
20170901 | 22 | 120 | 25 | 60 | 1.9
20170915 | 27 | 120 | 26 | 60 | 1.8
20170922 | 37 | 120 | 37 | 60 | 1.3
20171006 | 17 | 120 | 6 | 60 | 1.7
20180518 | 63 | 40,120 | 58 | 20,60 | 1.6
20180519 | 75 | 40,200 | 77 | 20,100 | 1.5
20180623 | 56 | 60 | 55 | 30 | 2.2
20180628∗ | 193 | 5,10 | 193 | 1,3 | 1.3
20180803 | 60 | 60 | 60 | 30 | 1.3
20180804 | 55 | 60 | 52 | 30 | 1.2
20180805 | 40 | 60 | 39 | 30 | 1.2

Notes. Columns $N_{V}$ and $N_{I}$ give the number of images taken with the
$V$ and $I$ filters. Columns ${t}_{V}$ and ${t}_{I}$ provide the exposure
time, or range of exposure times. The average seeing is listed in the last
column. Observations are from Bosque Alegre Astrophysical Station, except the
asterisked night.

###  2.2 Difference image analysis

To carry out high-precision photometry for all the point sources in our
collection of images of NGC 6397, we applied the recognised technique of
difference image analysis (DIA). As in previous papers, we employed the
DanDIA444DanDIA is built from the DanIDL library of IDL routines available at
http://www.danidl.co.uk. pipeline for the data reduction process [Bramich,
2008, Bramich et al., 2013]. The Bosque Alegre and Las Campanas images were
reduced independently. Reference images for the $V$ and $I$ filters, and for
each observatory, were built by stacking the best-quality images in our
collection. Sequences of difference images in each filter were then built by
subtracting the respective convolved reference image from the rest of the
series. Differential fluxes for each star detected in the reference image were
measured on each difference image. Finally, light curves for each star were
constructed from the series of differencial images, by calculating the total
flux $f_{\mbox{\scriptsize tot}}(t)$ at each epoch $t$ from:

$f_{\mbox{\scriptsize tot}}(t)=f_{\mbox{\scriptsize
ref}}+\frac{f_{\mbox{\scriptsize diff}}(t)}{p(t)},$ (1)

where $f_{\mbox{\scriptsize ref}}$ is the reference flux,
$f_{\mbox{\scriptsize diff}}(t)$ is the differential flux, and $p(t)$ is a
photometric scale factor. Conversion to instrumental magnitudes is achieved
using:

$m_{\mbox{\scriptsize ins}}(t)=25.0-2.5\log\left[f_{\mbox{\scriptsize
tot}}(t)\right],$ (2)

where $m_{\mbox{\scriptsize ins}}(t)$ is the instrumental magnitude of the
star at time $t$. All fluxes are in ADU s-1, and uncertainties are propagated
as usual. The procedure is described, with details and caveats, by Bramich et
al. [2011].

###  2.3 Photometric calibrations

To correct for possible systematic errors, we applied the methodology
developed by Bramich & Freudling [2012] to solve for the magnitude offsets to
be subtracted from each photometric measurement made on a given image. This is
a first-order correction to the systematic uncertainty introduced into the
photometry of an image, due to an error in the fitted value of the scale
factor $p$ (Equation 1). In the present case, the systematic error corrections
were found to be negligible, always smaller than 1 mmag, except in the case of
the $I$ photometry from Las Campanas, where a correction of 6 mmag was
calculated and applied for stars brighter than the 15th magnitude.

###  2.4 Transformation to the standard system

Standard stars in the field of NGC 6397 are included in the online collection
of Stetson (2000),555http://www3.cadc-ccda.hia-iha.nrc-
cnrc.gc.ca/community/STETSON/standards which were used to transform our
instrumental _vi_ magnitudes into the Johnson-Kron-Cousins _VI_ system. The
mild colour dependences of the standard _minus_ instrumental magnitudes are
shown in Fig. 1 for the observations from Bosque Alegre and Las Campanas. The
transformation equations are given in the figure itself. However, given the
non-significant colour dependence for the Bosque Alegre data, we opted to
transform them to the standard system using the linear relations: $V=(0.980\pm
0.004)\times v-(2.012\pm 0.082)$ and $I=(0.985\pm 0.004)\times i-(2.926\pm
0.068)$, since they provide a good agreement with Las Campanas observations,
of better spatial resolution.

Figure 1: Colour dependence of the transformations for the $V$ and $I$ filters
between the instrumental and the standard photometric systems. The _left_ and
_right_ panels correspond to the transformations for the observations in
Bosque Alegre and Las Campanas, respectively. We employed a set of standard
stars of Stetson [2000] present in the field of NGC 6397.

##  3 The colour-magnitude diagram

###  3.1 Cluster stellar membership

We applied the method of Bustos Fierro & Calderón [2019] to identify probable
members in the field of the cluster. It uses the high quality astrometric data
available in _Gaia_ DR2, implementing the Balanced Iterative Reducing and
Clustering using Hierarchies (BIRCH) algorithm [Zhang et al., 1996] in a four-
dimensional space of physical parameters—positions and proper motions—that
detects groups of stars in the 4D-space. In this way we extracted 31,047
_Gaia_ stellar sources that are likely cluster members, from a much larger
sample of 472,304 stars within 60 arcmin around the cluster. A total of 17,482
stars are in the field of our Las Campanas images, of which 12,971 are
considered members. Figure 2 shows the Vector-Point Diagram (VPD) of cluster
and field stars, and the corresponding _Gaia_ colour-magnitude diagram.

Figure 2: Membership of stars in the NGC 6397 field, based on _Gaia_ DR2 data
(see § 3.1). The _left_ panel shows the Vector-Point Diagram (VPD) for 472,304
stars within 60 arcmin around the cluster. Red dots represent stars considered
likely members. The cluster clearly stands out against the vast number of
field stars. The _right_ panel displays the colour-magnitude diagram in the
_Gaia_ photometric system.

###  3.2 Cluster reddening

NGC 6397 belongs to a group of globular clusters located toward the centre of
the Milky Way; hence, it is affected by interstellar reddening of a patchy
nature. To differentially deredden our photometry, we employed the
differential reddening grid calculated by Alonso-García et al. [2012], that
covers nearly the whole field of our images from the Swope telescope. The
resolution of the grid is 5.4′′ ($0.^{\\!\\!{\mathrm{s}}}36$) in RA and 3.6′′
in DEC. The procedure to deredden each star starts by first averaging the four
neighbouring differential reddening values given in the grid, and then adding
a mean overall reddening. The mean value of $E(B-V)$ suggested by Alonso-
García et al. [2012] for NGC 6397 is $0.18$. The Galactic dust distribution
and the calibrations of Schlafly & Finkbeiner [2011] and Schlegel et al.
[1998] suggest values of 0.166 and $0.188\pm 0.003$, respectively. In Fig. 3
we show the CMD before (left panel) and after (right panel) the reddening
correction and membership cleaning. In the latter we adopted $E(B-V)=0.19$ and
a distance of 2.5 kpc, to place the two isochrones for 13.0 and 13.5 Gyr. To
deredden the colour $(V-I)$ we employed the ratio $E(V-I)/E(B-V)=1.259$.
Although differential reddening is mild across the NGC 6397 area, a sharper
distribution of stars is noticeable after this correction.

###  3.3 Isochrones and ZAHB fitting

The colour-magnitude diagram (CMD) of NGC 6397, built with data from the Swope
telescope (the better quality data) is shown in Fig. 3. In the left panel the
member and non-member stars are represented by black and light blue points. In
the right panel we present the CMD after the non-members have been removed,
and the member stars have been subjected to the differential reddening
correction ($\S$ 3.2). The CMD diagram appears very clean, which enabled a
proper matching of isochrones and ZAHB models. These best fitting models were
taken from the grid of the Victoria-Regina models of VandenBerg et al. [2014]
for [Fe/H] $=-2.0$, $Y=0.25$, and [$\alpha$/Fe] $=+0.4$, and ages of 13.0 and
13.5 Gyr. The isochrones match very well the stellar unreddened distribution
for a distance of 2.5 kpc and $E(B-V)=0.19$. It is pertinent to note here that
Brown et al. [2018] recently derived, from the $V$ luminosity at the main-
sequence turnoff, an absolute age of $13.4\pm 1.2$ Gyr for the cluster.

In the CMD it is evident a very blue horizontal branch void of any RR Lyrae
stars. The only two RR Lyrae known in the field of the cluster (V3 and V22)
are clearly non-members and background objects, as they appear in the blue
straggler area, near the turnoff. The structure of the HB will be discussed in
$\S$ 7.

Figure 3: Colour-magnitude diagram of NGC 6397, built from Las Campanas data.
The _left_ panel shows the observed, reddened stellar distribution. Black and
blue dots represent cluster members and field stars respectively, according to
the membership analysis of $\S$ 3.1. In the _right_ panel only cluster members
are plotted, after their colours were individually corrected by differential
reddening ($\S$ 3.2), and globally shifted an average colour excess
$E(B-V)=0.19$. The star sequences appear well matched by four isochrones
calculated using the models of VandenBerg et al. [2014], for ages (left to
right): 12.0, 12.5, 13.0, and 13.5 Gyr, and [Fe/H] $=-2.0$, $Y=0.25$, and
[$\alpha$/Fe] $=+0.4$. The isochrones were located at a distance of 2.5 kpc.
At the HB, three continuous black loci, also positioned at 2.5 kpc, show the
evolutionary tracks for total masses of 0.64, 0.65, and 0.66 $M_{\odot}$,
calculated as explained in § 7. The resulting ZAHB is shown in red, and for
comparison we include, as a dashed line, the ZAHB computed from VandenBerg et
al. [2014] models, for the same parameters. The known variables measured in
the present study are labelled with different symbols and colours: RRab
(_blue_); RRc (_green_); SX Phe (_lilac_); $\gamma$ Dor-type? and cataclysmic
(_red_); contact binaries and ellipsoidal (_orange_).

##  4 Variable stars in the field of NGC 6397

The 36 stars catalogued in the CVSGC [Clement et al., 2001], their coordinates
and _Gaia_ DR2 identifications (when available), and their membership as
estimated in this paper, are listed in Table 2. Among them, only V1 and V3
appear flagged as variables in _Gaia_ DR2. Finding charts for all variables,
except V1, are in Fig. 4.

Table 2: Variable stars in the field of NGC 6397

Variable Id. | Variable Typea |  $V$a | RAb | Decb |  _Gaia_ DR2 Source Id. | Memb.c
---|---|---|---|---|---|---
|  | [mag] | (J2000.0) | (J2000.0) |  |
V1 | M | 13.36 | 17:41:04.75 |  $-53$:32:58.6 | 5921753573179938304 | m?
V2 | RV? | 12.88 | 17:40:10.86 |  $-53$:47:33.9 | 5921745812182394496 | f
V3 | RRab | 16.07 | 17:40:16.98 |  $-53$:41:03.6 | 5921748045565559040 | f
V4 | EW | 16.35 | 17:41:08.83 |  $-53$:42:34.3 | 5921750072784246400 | f
V5 | E | 18.70 | 17:41:05.55 |  $-53$:33:36.0 | 5921753573174589440 | f
V6 | EW | 17.16 | 17:40:53.35 |  $-53$:43:39.6 | 5921744884469398656 | f
V7 | EW | 17.09 | 17:40:43.93 |  $-53$:40:35.5 | 5921745296766522752 | m
V8 | EW | 16.24 | 17:40:39.27 |  $-53$:38:47.2 | 5921754161593008384 | m
V9 | EW | 16.25 | 17:40:02.25 |  $-53$:35:45.3 | 5921755295470448256 | f
V10 | SX Phe | 16.00 | 17:40:37.55 |  $-53$:40:36.5 | 5921748217344441856 | m
V11 | SX Phe | 15.43 | 17:40:44.14 |  $-53$:40:40.8 | 5921745296766500224 | m
V12d | CV? | 17.52 | 17:40:41.62 |  $-53$:40:20.0 | $\dots$ | $\dots$
V13 | CV? | 19.43 | 17:40:48.94 |  $-$53:39:49.5 | 5921751137936013056 | $\dots$
V14 | E | 19.25 | 17:40:46.50 |  $-$53:41:15.7 | $\dots$ | $\dots$
V15 | SX Phe | 15.43 | 17:40:45.42 |  $-53$:40:25.3 | 5921751172297381120 | m
V16 | ELL | 16.65 | 17:40:44.63 |  $-53$:40:41.9 | 5921745296766500864 | m
V17 | ELL? | 16.23 | 17:40:43.82 |  $-53$:41:16.6 | 5921745296766429824 | m
V18 | E | 15.75 | 17:40:43.64 |  $-53$:40:28.0 | 5921751172301644032 | $\dots$
V19 | E | 17.12 | 17:40:42.85 |  $-53$:40:23.8 | 5921751172301639808 | $\dots$
V20 |  $\gamma$ Dor? | 15.83 | 17:40:41.70 |  $-53$:40:33.6 | 5921745296786376320 | m
V21 | SX Phe | 15.47 | 17:40:41.59 |  $-53$:40:23.9 | 5921745296786384128 | $\dots$
V22 | RRdc | 16.15 | 17:40:41.15 |  $-53$:40:42.3 | 5921745296766606336 | f
V23 | SX Phe | 15.52 | 17:40:39.34 |  $-53$:40:47.0 | 5921748251704126464 | $\dots$
V24e |  $\gamma$ Dor? | 16.17 | 17:40:39.10 |  $-53$:40:23.4 | $\dots$ | $\dots$
V25 | ? | 17.98 | 17:41:10.18 |  $-53$:39:30.9 | 5921750652596773888 | f
V26 | ? | 16.24 | 17:40:43.05 |  $-53$:38:31.7 | 5921751305431697920 | f
V27 | ELL? | 18.25 | 17:41:13.81 |  $-53$:41:14.5 | 5921750244578290816 | f
V28 | ? | 15.13 | 17:41:02.73 |  $-53$:39:47.5 | 5921751000496772224 | f
V29 | ELL? | 19.72 | 17:40:59.67 |  $-53$:40:39.0 | 5921751034856719488 | $\dots$
V30 | E | 17.64 | 17:40:54.53 |  $-53$:40:44.7 | 5921750931784538112 | f
V31 | ELL? | 15.99 | 17:40:42.62 |  $-53$:40:27.5 | 5921745296766546176 | $\dots$
V32 | E | 16.13 | 17:40:40.33 |  $-53$:41:25.6 | 5921745262406760960 | m
V33 | CV? | 18.3 | 17:40:42.71 |  $-53$:40:18.7 | 5921751172296037632 | $\dots$
V34 | CV | 16.2 | 17:40:42.40 |  $-53$:40:28.7 | 5921745296766570368 | $\dots$
V35f | ELL | 18.79 | 17:40:43.35 |  $-53$:41:55.6 | $\dots$ | $\dots$
V36 | ? | 16.53 | 17:40:44.14 |  $-53$:42:11.7 | 5921745155038332032 | m

Notes.
aCVSGC [Clement et al., 2001].
bCoordinates are mostly from _Gaia_ DR2, and the rest are from the HST archive
image u5dr0401r.
cThis work.
dBright _Gaia_ source 5921745296786384896 is at $1^{\prime\prime}$ towards the
South.
e _Gaia_ source 5921748251704160512 is at $0.7^{\prime\prime}$ towards the NE.
fBright _Gaia_ source 5921745262406612608 is at $1.3^{\prime\prime}$ towards
the SE.

Several of these stars, such as V12, 13, 14, 33–36, are in the observed field,
but are too faint or too near a bright companion to be properly detected or
measured in our data. Besides, stars V2, 5, 6, and 9 are outside the trimmed
Bosque Alegre field, i.e., they were observed during only one night at Las
Campanas (cf. Table 1); in that night, V2 and V6 did not show any variations.
We shall therefore limit our light-curve analysis to those stars—listed in
Table 3—of variability clearly detected with our photometry. In all cases we
attempted a new determination of the period based on our data, using the
string-length method [Dworetsky, 1983]. However, a trial and error approach
showed that, in some cases, the light curve was better phased with the period
given in the CVSGC. In Table 4 we report the $V$ and $I$ time-series
photometry for all variables in our field of view; the full table, which also
includes the parameters of Equation 1 computed by DanDIA, is published only in
electronic format. Individual variables are discussed next, and the light
curves are shown in Figs. 5 and 6.

Figure 4: Identification charts for the variables in NGC 6397. _Left_ , the
periphery of the cluster ($15^{\prime}\times 15^{\prime}$). _Right_ , the
cluster core ($1.7^{\prime}\times 1.7^{\prime}$). Only star V1 is outside the
larger field. Table 3: Data on the variable stars of NGC 6397 reported in this
work

Variable Id. | Variable Type | $<V>^{a}$ | $<I>^{a}$ | $A_{V}$ | $A_{I}$ | $P$ | HJDmax
---|---|---|---|---|---|---|---
|  | [mag] | [mag] | [mag] | [mag] | [d] |  $+$245 0000.
V3 | RRab | 16.176 | 15.526 | 1.222 | 0.895 | 0.494748 | 8257.9189
V4 | EW | 16.537 | 15.390 | 0.667 | 0.585 | 0.422742 | 8258.8602
V5 | E | 18.853 | 16.167 | 1.057 | 0.681 | 0.27c | 8297.7744
V7 | EW | 17.657 | 16.694 | 0.905 | 0.836 | 0.2699c | 8257.8506
V8 | EW | 16.414 | 15.713 | 0.421 | 0.431 | 0.271240 | 7963.7078
V10 | SX Phe | 15.907b | 15.375b | 0.070 | $\dots$ | 0.030030 | 7908.8962
V11 | SX Phe | 15.340 | 14.794 | 0.053 | 0.035 | 0.038262 | 8033.5452
V15 | SX Phe | 15.525b | 14.990b | 0.070 | $\dots$ | 0.025240 | 8258.8130
V16 | ELL | 16.767 | 15.741 | 0.123 | 0.149 | 1.489390 | 8257.8884
V18 | E | 15.701b | 15.042b | 0.218 | 0.167 | 0.7867c | 8297.6893
V20 |  $\gamma$ Dor? | 15.863b | 15.275b | $\dots$ | $\dots$ | 0.8612c | 8033.5452
V21 | SX Phe | 15.634 | 14.828 | 0.363 | 0.114 | 0.038961 | 7984.6207
V22 | RRd | 16.059 | 15.472 | 0.184 | 0.123 | 0.46497 | 8258.7534
|  |  |  |  |  | 0.30669 |
V23 | SX Phe | 15.523 | 14.961b | 0.081 | $\dots$ | 0.038005 | 8033.5091
V24 |  $\gamma$ Dor? | 16.174b | 15.316b | 0.123 | $\dots$ | 0.4572c | 8033.5398
V25 | ? | 17.987b | 16.566b | 0.190 | $\dots$ | 1.2306c | 7998.5528

Notes.
aIntensity-weighted mean.
bMagnitude-weighted mean ($\overline{V}$, $\overline{I}$).
cCVSGC [Clement et al., 2001].

Table 4: Time-series $V$ and $I$ photometry for the variables in our field of
view

Variable Id. | Filter | HJD | $m_{\mathrm{std}}$ | $m_{\mathrm{ins}}$ | $\sigma_{m}$
---|---|---|---|---|---
|  | [d] | [mag] | [mag] | [mag]
V2 | $V$ | 2458297.56732 | 13.249 | 14.943 | 0.004
V2 | $V$ | 2458297.57256 | 13.242 | 14.937 | 0.005
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
V2 | $I$ | 2458297.58010 | 9.556 | 10.820 | 0.002
V2 | $I$ | 2458297.58886 | 9.552 | 10.816 | 0.002
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
V3 | $V$ | 2457908.85026 | 16.365 | 18.750 | 0.006
V3 | $V$ | 2457908.85498 | 16.377 | 18.762 | 0.012
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
V3 | $I$ | 2457908.84190 | 15.620 | 18.825 | 0.012
V3 | $I$ | 2457908.84429 | 15.626 | 18.831 | 0.012
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮

Notes. The columns $m_{\mathrm{std}}$ and $m_{\mathrm{ins}}$ are the standard
and instrumental magnitudes, and $\sigma_{m}$ is the error (the same) for
both. This is an excerpt from the full table, which is available only with the
electronic version of the article.

###  4.1 RR Lyrae stars

V3. This is the first RRab star known in the NGC 6397 area; however, there is
firm evidence that it does not belong to the cluster. This is confirmed by its
position in the CMD, near the cluster turnoff.

V22. The second RR Lyrae variable present in the cluster area, it also appears
in the blue straggler zone of the CMD and was originally identified as one of
them by Lauzeral et al. [1992]. In the CVSGC the star is listed as a probable
double-mode RR Lyrae, but no periods are suggested. While Kaluzny & Thompson
[2003] discussed the possibility of V22 being a pulsating multiperiodic
variable related to $\gamma$ Doradus stars, Kaluzny et al. [2006] classified
it as an RRd, with $P_{0}=0.52$ d and $P_{1}=0.344$ d ($P_{1}/P_{0}=0.66$). In
our analysis we found that a period $P_{0}=0.46497$ d produces the light curve
shown in Fig. 5; there is, however, evidence of a second period $P_{1}$ of
0.30669 d, with which we also derive a ratio $P_{1}/P_{0}=0.66$. Both periods
are listed in Table 3.

###  4.2 SX Phoenicis stars

V10. With V11, this star was originally identified as an SX Phe variable by
Kaluzny [1997]. A low amplitude modulation with a periodicity of about 0.287 d
is evident in the data from Las Campanas. On top of this, clear short-period
variations with a period of 0.030 d are detected. This period agrees with that
reported in the CVSGC (0.0308 d). An amplitude modulation of the short period
variations is explained by the presence of a secondary frequency of period
0.03378 d, which is likely a non-radial mode. A model built with the said
three periods is shown in Fig. 6.

V11. Clear monoperiodic variations are observed in the light curves (Figs. 5
and 6), phased with a period of 0.038262 d [Kaluzny et al., 2006].

V15. Identified as an SX Phe variable by Kaluzny & Thompson [2003], with V21
and 23; all of them are in the blue straggler region of the CMD. The small
variations in brightness are displayed in Fig. 6. The two periods 0.02524 d
and 0.02200 d represent rather well the data periodicity and the amplitude
modulations. This is a very short period for an SX Phe variable. Kaluzny &
Thompson [2003] found for V15 a period of 0.0215 d and discussed the
possibility that it is a pulsating hot dwarf rather than an SX Phe.

V21. Mild variations in the $V$-band are seen in this SX Phe star which are
properly folded with a period $P=0.038961$ d (see Fig. 5). This period
represents well the periodic behaviour of the variations, but some mean
magnitude modulations are also seen in Fig. 6, which may be due to a secondary
periodicity not detected in our data.

V23. Small-amplitude variations are clear in this SX Phe star, showing a
periodicity of 0.03805 d (Fig. 6).

###  4.3 $\gamma$ Doradus stars?

In the CMD (Fig. 3), V20 and V24 are found among the blue straggler stars and
in the main sequence near the turnoff point, respectively. Light variations of
these two stars were reported by Kaluzny et al. [2006], giving periods of
0.8612 and 0.4572 d. These rather long periods for a typical SX Phe star led
these authors to speculate that V20 and 24 belong to a new type of Population
II variables, counterpart of the Population I $\gamma$ Doradus class, i.e.,
nonradially oscillating stars locate near the intersection of the red edge of
the classical instability strip and the main sequence.

V20. Due to seeing conditions and faintness of the star, it could not be
measured in Bosque Alegre images, although it was observed at Las
Campanas—albeit only five hours. The Swope light curve is in Fig. 5. A mild
variation consistent with the period 0.8612 d is clearly seen, consistent with
the variability and period of Kaluzny et al. [2006].

V24. Like V23, it presents small-amplitude variations of the SX Phe type. Our
data are not properly phased with the period 0.4572 d reported by Kaluzny et
al. [2006], but the combination of this value and the SX Phe-like, short-
period of 0.02754 d produce the rather good representation shown in Figs. 5
and 6. In the cluster CMD (Fig. 3), the star is towards the red edge of the
main sequence which, together with its proper motion, make clear that V24 does
not belong to the cluster. There is a star of similar brightness at
$0.7^{\prime\prime}$ towards the NE, _Gaia_ source 5921748251704160512 (cf.
Fig. 1 of Kaluzny & Thompson 2003). Star V24, on the other hand, is not a
_Gaia_ DR2 source.

###  4.4 Eclipsing Binaries and Ellipsoidal Variables

V4, V7, V8. Neat eclipses are observed for these three contact binaries or EW-
type stars; the periods are 0.42274 d, 0.2699 d, and 0.27124 d, respectively.
Star V7, discovered and identified as an W UMa binary by Kaluzny [1997], is
not detected in our Bosque Alegre observations. Our (Las Campanas) period for
V7 coincides with Kaluzny & Thompson’s [2003]. In § 5 we present and discuss a
modelling of these light curves.

V5. Its Swope light curve in $I$ shows clear eclipses; the one in $V$,
however, is much noisier (Fig. 5). They were phased with the period given in
the CVSGC, $P=0.27$ d. The star is too red to appear in our CMD. An analysis
of the light curves of this binary is presented in § 5.

V6, V9, V14. No variations were detected in our one-night data from Las
Campanas for V6 and V14. On the other hand, the Swope light curve of the
eclipsing binary V9 is fairly incomplete (Fig. 5) and therefore was not
analysed. Nevertheless, the three stars are included in the CMD.

V16. This ellipsoidal variable is the optical component of the millisecond
pulsar J1740$-$5340—see discussion in Kaluzny & Thompson [2003] and Kaluzny et
al. [2008]. In Fig. 5 our light curve has been folded with the period listed
by Kaluzny et al. [2006], i.e., 1.35406 d.

V17. This star is listed as a probable elliptical variable in the CVSGC. In
our data, however, we did not detect any variations. It appears near the
cluster turnoff.

V18. This eclipsing variable is a cluster blue straggler. Kaluzny & Thompson
[2003] reported an unusual light curve (see their Fig. 10); they propose that
it is a detached or semidetached system composed of stars with very similar
surface brightness, rather than a contact binary. Our light curves have been
phased with $P=0.78669$ [Kaluzny et al., 2006].

V19. No variations are noticed in our data. We therefore cannot confirm its
reported variable nature, which appears to be of rather small amplitude (cf.
Fig. 8 of Kaluzny & Thompson 2003).

###  4.5 Cataclismic Variables

V12, V13, V33, V34. These stars were identified by Kaluzny et al. [2006] with
the cataclismic variables CV1, CV6, CV2, and CV3 of Grindlay et al. [2001],
respectively. Although they are rather faint for the precision of our
photometry, or are blended with a brighter neighbour (V12), we were
nevertheless able to detect and measure stars V13, V33, and V34, and include
them in the CMD. However, we could not find any variations. The mild, long
term or sporadic variations given by Kaluzny et al. [2006]—their Fig. 3—rather
explain the lack of variations in our data.

###  4.6 Others

V2. According to the CVSGC, this is a long-period (39.5977 d) variable of
possibly RV Tau type. We covered it during only one night at Las Campanas; no
variations were observed in its light curves, from which we estimated
$<V>=13.25$ and $<I>=9.55$. Its colour is too red to be represented in our
CMD.

V25. The light curve (Fig. 5) was phased with a period of 1.2306 d [Kaluzny et
al., 2006]. The slight variations are not conclusive as to how classify this
star.

V28. We could not phase properly the light curve of this rather bright
($V\approx 15.2$ mag) star with the period 25.997 d suggested by Kaluzny et
al. [2006]. A plot of $V$ vs. HJD (Fig. 5) shows clear variations, and one
full cycle suggests a much longer period of $P=97.78$ days. This new period is
not rigid but is rather a characteristic time of variation, as stochastic
variations are hinted by our data. The star is very red and is outside the CMD
limits of Fig. 3. It was classified by Kaluzny et al. [2006] as a pulsating
star, but could in fact be a spotted variable.

###  4.7 Variables not detected

V26, 27, 29–32, 35–36. We were not able to detect variations using published
periods or after searching for them with the string-length method. They are
nevertheless represented in our CMD, with the exception of V35, which is too
faint and too near a bright star.

Figure 5: Light curves in _VI_ of some variables in the field of NGC 6397.
Black and red symbols stand for Bosque Alegre and Las Campanas observations,
respectively. Note that V20 and V28 are not phased but plotted vs. HJD; the
solid curve in V20 is the portion of a $P=0.8612$ d model. Figure 6: Light
curves in $V$ of six SX Phoenicis stars in the field of NGC 6397. The long
boxes contain $\sim 7.5$ hours of Swope data.

##  5 Modelling contact binaries in NGC 6397

Figure 5 shows the $V$, $I$ light curves of the eclipsing binaries V4, V5, V7,
and V8, among other stars. They evidently have morphologies typical of
semidetached and contact binaries. To model the systems, we used the code
BinaRoche [Lázaro et al., 2009], which simultaneously fits both the $V$ and
$I$ light curves.

The BinaRoche model takes into account the deformation of the stars, in the
Roche model geometry, and the inhomogeneous temperature distribution over the
stellar surfaces, considering gravity darkening and heating effects. The
gravity darkening exponents can be declared free parameters, but the standard
values—$\beta=0.25$ for a star with a radiative envelope, and $\beta=0.08$ if
the star has a convective envelope—usually produce satisfactory fits. The
codes adopts $T\propto g^{\beta}$, with $T$ and $g$ being the local effective
temperature and surface gravity, respectively. The filling factor $S_{i}$ of
each star, which is also a free parameter of the model, is defined as the
ratio between its polar radius ($r_{p}$) and the polar radius of its critical
Roche lobe ($R_{p}$): $S_{i}=(r_{p}/R_{p})_{i}$, $i=1,2$. The mass of the
primary component—in solar mass units—and the mass ratio $q=M_{2}/M_{1}$ are
parameters of the model, since the stellar surface fluxes depend on
$T_{\mathrm{eff}}$, $\log g$, and [Fe/H]. The code adopts stellar atmospheres
fluxes from the BaSel library [Lejeune et al., 1998]. Expressing the masses of
the binary components in solar masses allows the derivation of the distance to
the system in parsecs.

The analysis starts assuming the binaries are members of NGC 6397, i.e, they
have the cluster’s metallicity and intestellar extinction. The adoption of a
$E(B-V)$ value, and the relative contribution of the primary component to the
total flux in $V$ and $I$, a result of the light curves fits, allows us to
obtain the intrinsic colour $(V-I)_{0}$ of the primary star from the observed
colour of the binary at the maximum. Then, the effective temperature
$T_{\mathrm{eff},1}$ of the primary is estimated from the calibrations
$(V-I)_{0}$ vs. $T_{\mathrm{eff}}$ of Huang et al. [2015] if
$-0.8\leq\mathrm{[Fe/H]}\leq+0.3$, or Casagrande et al. [2010] when [Fe/H]=
-2.0.

To estimate the mass of the primary component, we adopted the relation of
Torres et al. [2010] for stars in detached, non-interacting binary systems
with accurately measured masses and radii. They give polinomial fits for $\log
M(M_{\odot})$ and $\log R(R_{\odot})$, in terms of $T_{\mathrm{eff}}$, $\log
g$, and [Fe/H]. Unfortunately, these relations do not include systems with
metallicity as low as $-2.0$ dex, i.e., that adopted for NGC 6397, which
introduces an additional uncertainty. The mass of the secondary star is
derived from the free parameter $q=M_{2}/M_{1}$. Since the relative flux of
the primary and $\log g_{1}$ are results of the model, a few iterations are
necessary to reach stable values of $M_{1}$ and $T_{\mathrm{eff,1}}$.

Next we discuss the individual systems.

V4. Assuming the star to be a cluster member, with $E(B-V)=0.19$ and
$\mathrm{[Fe/H]}=-2.0$, we obtained $T_{\mathrm{eff,1}}\approx 5200$ K and
$M_{1}\approx 0.59M_{\odot}$. But with these parameters, the calculated
distance to the system turns out to be $d\approx 1290$ pc, much lower than the
accepted distance of NGC 6397 (see § 6). We therefore considered another
solution, supposing that the binary is a field star of solar metallicity,
i.e., with $\mathrm{[Fe/H]}=0.0$. Adopting a reasonable guess of
$E(B-V)=0.10$, we recomputed the binary parameters, which are listed in Table
5, while the observed and model light curves are shown in Fig. 7. The filling
factors $S_{1}$ and $S_{2}$ indicate that V4 is a semidetached system, with a
secondary oversized component filling its Roche lobe. Even considering the
uncertainty in $E(B-V)$, from the derived distance it seems very likely that
V4 is not a cluster member.

Figure 7: Light curves and fits of four eclipsing binaries in NGC 6397. The
parameters of the models are in Table 5.

V5. Our light curve in $V$ is rather faint and noisy, but the colour $(V-I)$
out of eclipse indicates that it is a rather red object. Again, starting with
the assumption that it is a member of NGC 6397, adopting $E(B-V)=0.19$ and
$\mathrm{[Fe/H]}=-2.0$, we derived parameters corresponding to a very low mass
system, with a primary of $M_{1}\approx 0.30~{}M_{\odot}$ and
$T_{\mathrm{eff,1}}\approx 3700$ K, and a distance $d\approx 550$ pc; this
discards it as a member of the cluster. We repeated the process considering
the binary as a field star with solar metallicity and $E(B-V)=0.05$. The new
results are in Table 5, and the observed and model light curves are shown in
Fig. 7. The filling factors of the model suggest a detached binary. The right
side of the secondary eclipse in $I$ looks depressed relative to the model,
but due to the low quality of the light curves we did not attempt to reproduce
it. Like V4, it seems quite sure that V5 is also a field binary.

V7. The light curves show the rounded morphology typical of contact systems,
but the depths of the eclipses are not well defined due to dispersion in the
data. As before, we adopted $E(B-V)=0.19$ and $\mathrm{[Fe/H]}=-2.0$. The
observed and model light curves are shown in Fig. 7, and the parameters from
the light curves fits are in Table 5. The filling factors of both components
indicate that V7 is a contact system, and the resulting distance of $\approx
2.6$ kpc is consistent with the assumed membership to NGC 6397.

V8. The light curves (Fig. 7) present a rounded morphology with flat eclipses.
The parameters derived from the modelling are in Table 5, assuming that the
binary belongs to the cluster. The filling factors of both components indicate
that V8 is a contact system with a low mass ratio. The distance ($\approx 2.3$
kpc) makes V8 a likely cluster member.

The binary parameters of Table 5 are presented without error bars, as the
unknown systematic uncertainties involved—photometry, adopted reddening and
metallicity, model atmosphere fluxes and limb darkening, adopted relations for
$T_{\mathrm{eff,1}}$ and $M_{1}$—make pointless any statistical estimation of
errors. We can nevertheless consider, for example, the light curve of V8. By
decreasing the observed $(V-I)$ at maxima in 0.02 mag we obtain $M_{1}\approx
0.96M_{\odot}$, $T_{\mathrm{eff,1}}\approx 7600$ K, and $d\approx 2480$ pc. We
can therefore expect that uncertainties of the order of $\delta M_{1}\approx
0.03M_{\odot}$, $\delta T_{\mathrm{eff,1}}\approx 200$ K, and $\delta d\approx
150$ pc would be rather conservative.

Additionally, we can compare the distances in Table 5 with those derived by
other independent methods involving binaries. For that purpose we adopted the
period-luminosity (P-L) relations of Rucinski & Duerbeck [1997] and Rucinski
[2000] for W UMa binaries with Hipparcos parallaxes and for W UMa systems in
globular clusters, respectively. It must be pointed out that, as V4 and V5 are
a semidetached and a detached system in our solutions, using W UMa P-L
relations is not entirely appropriate; even so, we carried out the comparison
with them as well. The metallicities used in the P-L relation of Rucinski &
Duerbeck [1997] are those in Table 5. The distances obtained from the P-L
relations in the bands $V,I$ are listed in Table 6. We see that for the two W
UMa systems, V7 and V8, the distances expected from the mean P-L relations are
in fairly good agreement with our estimations in Table 5 and in § 6, and
therefore with the standard distance of NGC 6397. On the other hand, even if
they are not contact systems, V4 and V5 seem to be much nearer than the
cluster.

Table 5: Parameters derived from modelling with BinaRoche the $V$, $I$ light
curves of close binaries in NGC 6397

Variable | [Fe/H]a | $E(B-V)^{a}$ | $M_{1}$ | $R_{1}$ | $T_{\mathrm{eff,1}}$ | $S_{1}$ | $M_{2}/M_{1}$ | $R_{2}$ | $T_{\mathrm{eff,2}}$ | $S_{2}$ | $i$ | $d^{b}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | [$M_{\odot}$] | [$R_{\odot}$] | [K] |  |  | [$R_{\odot}$] | [K] |  | [deg] | [pc]
V4 | $0.00$ | 0.10 | 0.82 | 0.76 | 5000 | 0.74 | 1.00 | 1.05 | 4875 | 1.00 | 80.5 | 1525
V5 | $0.00$ | 0.05 | 0.42 | 0.48 | 3475 | 0.75 | 0.46 | 0.41 | 3135 | 0.90 | 88.0 | 490
V7 | $-2.00$ | 0.19 | 0.68 | 0.75 | 5950 | 1.00 | 0.75 | 0.66 | 5615 | 1.00 | 87.5 | 2495
V8 | $-2.00$ | 0.19 | 0.94 | 0.90 | 7420 | 1.00 | 0.32 | 0.54 | 5886 | 1.00 | 87.3 | 2350

Notes.
aAdopted values for the model.
bMean value of distance from $V$ and $I$.

Table 6: Distances in parsecs derived from P-L relations for W UMa binaries in
NGC 6397

Variable |  $d_{V}$ (RD97) |  $d_{I}$ (RD97) |  $d_{V}$ (R00)
---|---|---|---
V4 | 1630 | 1755 | 1627
V5 | 231 | 312 | 230
V7 | 2354 | 2383 | 2622
V8 | 2070 | 2020 | 2305

Notes. The values $d_{V}$ and $d_{I}$ are distances derived from light curves
in $V$ and $I$. RD97: Rucinski & Duerbeck [1997]; R00: Rucinski [2000].

##  6 Distance to the cluster

###  6.1 From the SX Phoenicis P-L relation

The distance to the cluster can be estimated using the Period-Luminosity (P-L)
relation for its SX Phe stars. The variables V10, V11, V15, V21, and V23 are,
given their proper motions in _Gaia_ DR2 and positions in the CMD, very
convincing SX Phe members of the system. For the calculations we employed
three independent calibrations of the P-L relation, namely Poretti et al.
[2008] (_long-dash_), Arellano Ferro et al. [2011] (_solid_), and Cohen &
Sarajedini [2012] (_short-dash_), as shown in Fig. 8, for the fundamental
(_black_), first overtone (_blue_), and second overtone (_lilac_) modes,
respectively. The positioning of the first and second overtone loci was done
assuming the ratios between the periods $P_{1}/P_{0}=0.783$ and
$P_{2}/P_{0}=0.571$ [Jeon et al., 2003, 2004]. It is rather clear from their
distribution on the P-L plane that V10, V11, and V21 are fundamental
pulsators, while the main periods in V23 and V15 correspond to the first and
second overtone, respectively. Assuming these modes, and using the three
calibrations for each mode, we ended up with three distance determinations for
each variable—all very similar—that yield a grand average of $2.24\pm 0.13$
kpc. This result compares very well with the distance of 2.3 kpc in Harris
[1996], and with the recent measurement by Brown et al. [2018], based on
WFC3@HST geometrical parallaxes, of $2.39\pm 0.10$ kpc.

###  6.2 From HB and isochrones fitting

In § 3.3 we showed that a proper matching of the Victoria isochrones to the
star distribution in the CMD (Fig. 3) results by adopting a reddening
$E(B-V)=0.19$ and a distance of 2.5 kpc, in good agreement with the value
obtained using the SX Phoenicis variables in § 6.1.

###  6.3 From the modelling of eclipsing binaries

As described in § 5, eclipsing binaries V7 and V8 are likely cluster members;
their distances, derived from the modelling of the light curves and employing
P-L relations for W UMa stars, are between 2.35 and 2.6 kpc, again consistent
with the results in §§ 6.1–6.2.

Figure 8: Period-Luminosity relations for SX Phe stars: Poretti et al. [2008]
(_long-dash_), Arellano Ferro et al. [2011] (_solid_), and Cohen & Sarajedini
[2012] (_short-dash_), for the fundamental (_black_), first overtone (_blue_),
and second overtone (_lilac_) modes. Figure 9: Enlargement of the HB region in
the CMD (Fig. 3), showing in detail the matching to the observations by the
Eggleton evolutionary tracks and ZAHB, and by the Victoria ZAHB. Discussion is
in § 7.

##  7 The structure of the horizontal branch

We now want to draw attention to the similarity between the CMD of NGC 6397
(Fig. 3) and that of NGC 6254 (M10, see Fig. 6 of Arellano Ferro et al. 2020).
In particular, the structure of the HB is worth examining. Both clusters
display a well-developed blue tail and a virtually non-existent horizontal
region, with the consequent absence of RR Lyrae stars, with the sole exception
of the RRc V22 in M10. The likeness extends to the age ($\approx$ 13 Gyr, as
suggested by the turnoff isochrone fitting).

As we did for M10, we modelled the HB of NGC 6397 with the evolution code,
parametrization, and prescription of the RGB mass-loss (a modified Reimers law
with $\eta$ = $0.8\times 10^{-13}$) of Schröder & Cuntz [2005]. The code was
originally developed by Peter Eggleton (1971, 1972, 1973), and further
improved and tested by Pols et al. [1997, 1998] and Schröder et al. [1997].

The very blue end of the HB of NGC 6397 suggests rather thin hydrogen shell
masses, consistent with the best-match HB models plotted in the CMD (Fig. 3,
or the magnification in Fig. 9), of total mass between 0.64 and 0.66
$M_{\odot}$. This values are similar to those we found for M10 (see Fig. 8 of
Arellano Ferro et al. 2020). In Fig. 9 we added the theoretical ZAHB line
derived from our Eggleton (Cambridge) code, and compared it to a standard ZAHB
of the well-established Victoria models, for [Fe/H] $=-2.0$, [$\alpha$/Fe]
$=+0.4$, and $Y=0.25$ [VandenBerg et al., 2014]. Note that HB stars evolve
upwards of their ZAHB starting points, and such a line should, therefore,
ideally demarcate a lower boundary to the distribution of observed HB stars in
the CMD. While our ZAHB line does so, the Victoria ZAHB appears to leave a
gap, indicating that the respective models are less luminous than the observed
HB stars of NGC 6397 by almost 0.2 mag. This seems to be at odds with the fact
that both codes (Eggleton and Victoria) are mutually consistent in their
results. They share the Eggleton approach of an adaptive, non-lagrangian mesh,
which places more mesh points in regions with large temperature and pressure
gradients. However, the standard Victoria ZAHB uses He-core masses of
approximately 0.49 $M_{\odot}$, as D. VandenBerg pointed out in a private
communication. The models we computed for M10 and this cluster, by contrast,
reach a slightly higher He-core mass of 0.51 $M_{\odot}$. This is a small
difference in mass (about 4%) but, by virtue of the mass-luminosity
relation—much like in main-sequence stars—there is a change in brightness
which is fully responsible for the aforementioned difference of 0.2 mag.

It is pertinent to note that we systematically stop our tip-RGB evolution
models just before the He-flash, when the already slowly burning helium
exceeds a luminosity of 10 $L_{\odot}$ [Arceo-Díaz et al., 2015]. Beyond this
point, the exponential growth of the He-burning leads to the He-flash and to a
non-equilibrium transition towards the ZAHB, which most evolution codes do not
cover. There is, in other words, a sort of final “He-simmering” time at the
tip of the RGB, especially when the hydrogen shell around the core has been
much reduced by mass loss; this should be expected, e.g., in old globular
clusters, hence causing the very blue HB stars in M10 and NGC 6397. This final
“tip-RGB time” makes the helium cores of our models a little more massive than
those of the conventional Victoria ZAHB models.

##  8 Summary and conclusions

We have presented the photometric analysis of a new time-series of _VI_ CCD
images of the post-core-collapse globular cluster NGC 6397. The data reduction
and the photometry were carried out with the Differential Image Analysis (DIA)
procedure. In an already extensive series of papers by our group (see, e.g.,
AF17), we have shown that DIA is a powerful tool to generate massive numbers
of good-quality light curves for all the objects present in a specific sky
field, with particular interest in the light curves of variable stars.
Unfortunately, NGC 6397 does not harbour RR Lyrae stars, which are proper for
a Fourier light curve decomposition analysis; however, we had at our disposal
another types of variables and diverse tools to take advantage toward the
determination of the cluster distance and metallicity.

First, we cleaned the CMD making use of the powerful _Gaia_ DR2 astrometric
solution and the BIRCH algorithm, selecting the most likely members (almost
13,000 stars in Las Campanas data, § 3.1). After a differential dereddening of
the CMD (§ 3.2), isochrones for 13.0–13.5 Gyr and [Fe/H] $=-2.0$ from
VandenBerg et al. [2014] were then matched to the main sequence, turnoff, and
RGB (Fig. 3, _right_). The matching is good for a distance of 2.5 kpc and a
mean reddening $E(B-V)=0.19$. An independent method for deriving the cluster
distance involves the Period-Luminosity relation for the SX Phoenicis
variables present among its blue straggler population (§ 6.1). Using three
different P-L relations for five SX Phe stars (V10, V11, V15, V21, and V23),
we obtained a mean value of $2.24\pm 0.13$ kpc. Yet another distance
determinations come from the cluster’s eclipsing binaries V4, V5, V7, and V8.
A first method uses the modelling of their light curves with the BinaRoche
code (§ 5); it turns out that, while V4 and V5 are a semidetached and a
detached system fairly nearer than the cluster, V7 and V8 are contact binaries
at 2.6 and 2.3 kpc, thus being likely cluster members (Table 5). These results
are furthermore confirmed by using a P-L relation for W UMa binaries (Table
6).

The blue horizontal branch of NGC 6397 is similar to that of M10 which,
however, harbours just one RR Lyrae [Arellano Ferro et al., 2020], while NGC
6397 has none. The models of 0.64–0.66 $M_{\odot}$ with mass loss at the RGB
[Schröder & Cuntz, 2005], located at the distance and reddening suggested by
the isochrones fitting to the TO and RGB, provide a good matching of the HB
stars. Comparing with M10, this means that a minor range of mass loss at the
RGB—i.e., a minor range of mass of the remaining envelopes above de He-
core—represents better the observations. By contrast, in M10 that range is
wider, 0.56–0.62 $M_{\odot}$; it must be pointed out, however, that the HB of
M10 is richer and extends longer towards faint magnitudes than that of NGC
6397. The difference of mass loss during the RGB stage might be abscribed to
magnetic fields in the stars’ chromospheres that somehow hinder the normal
losses [Arellano Ferro et al., 2020].

## Acknowledgements

AAF acknowledges the support from DGAPA-UNAM grant through project IG100620.
It is a pleasure to thank Dr. Don A. VandenBerg for the use of his model
interpolation software and for an instructive discussion on horizontal branch
models.

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