2013 Vol. X No. XX, 000–000

11institutetext: National Astronomical Observatories/Yunnan Observatory,
Chinese Academy of Sciences, P. O. Box 110, 650011 Kunming, China; e-mail:
<EMAIL_ADDRESS>
22institutetext: Key Laboratory for the Structure and Evolution of Celestial
Objects, Chinese Academy of Sciences 33institutetext: University of the
Chinese Academy of Sciences, Beijing 100049, China

# Two particular EA-type binaries in the globular cluster $\omega$ Centauri

K. Li 112233 S.-B. Qian 112233

###### Abstract

We analyzed the $B$ and $V$ light curves of two EA-type binaries V211 and
NV358 using the WD code for the first time. Our analysis shows that V211 is a
typical Algol-type binary and NV358 is a well detached binary system. As the
two binaries are definite proper motion members of $\omega$ Centauri, we
estimated their physical parameters, obtaining $M_{1}=1.13\pm 0.03M_{\odot},$
$R_{1}=0.98\pm 0.01R_{\odot}$ and $M_{2}=0.33\pm 0.01M_{\odot},$
$R_{2}=0.92\pm 0.01R_{\odot}$ for V211; $M_{1}=1.30\pm 0.05M_{\odot},$
$R_{1}=1.03\pm 0.01R_{\odot}$ and $M_{2}=0.58\pm 0.02M_{\odot},$
$R_{2}=0.78\pm 0.01R_{\odot}$ for NV358. On the color-magnitude diagram of
$\omega$ Centauri, V211 is located in the faint blue straggler region and its
primary component is more massive than a star at the main-sequence turnoff.
Therefore, V211 is a blue straggler and should be formed by mass transfer from
the secondary component to the primary. The age of NV358 is less than 1.93
Gyr, indicating that it is much younger than first-generation stars in
$\omega$ Centauri. As NV364 in $\omega$ Centauri, NV358 should be a second-
generation binary.

###### keywords:

galaxies: globular clusters: individual($\omega$ Centauri) — stars: binaries:
close — stars: binaries: eclipsing — stars: blue stragglers — stars:
individual(V211, NV358)

##  1 Introduction

Eclipsing binaries are very rare in globular clusters, but they play a
significant role in the dynamical evolution of globular clusters and stellar
populations studies. They provide the source of energy that can against and
avoid core collapse in globular clusters (Goodman & Hut 1989). They are also
particularly interesting. Some Algol type eclipsing binaries such as NJL5,
V239 in $\omega$ Centauri and V228 in 47 Tuc (Helt et al. 1993; Li & Qian
2012; Kaluzny et al. 2007) have been identified to be blue stragglers (BSs).
They are good tools to comprehend the BS formation theories. Detached double
line spectroscopic eclipsing binaries in globular clusters provide the
opportunity to calculate the age of the cluster based on their well determined
physical properties of mass and radius (see Thompson et al. 2001, 2010).

BS stars appear to be anomalously younger than other stars of their
population. In the color-magnitude diagrams (CMDs) of clusters, these stars
are bluer and brighter than the main-sequence turnoff. They were first noticed
in the globular cluster M3 by Sandage (1953). Now, not only in globular and
open clusters, dwarf galaxies (Piotto et al. 2004; De Marchi et al. 2006;
Momany et al. 2007), but also in the field (Carney et al. 2005), investigators
have discovered BSs. The formation of BSs is still controversial. Several
mechanisms have been proposed to explain BS formation. The two most popular BS
formation mechanisms are mass transfer in close binary systems (McCrea 1964;
Carney et al. 2001) and direct collision between two stars (Hills & Day 1976).
Perets & Fabrycky (2009) discussed another possibility of BS formation in
primordial (or dynamically) hierarchical triple star systems. By studying this
type of objects, one can reveal the dynamical history of a cluster and the
role of dynamics on the stellar evolution. BSs statistics can also provide
some constrains for initial binary properties. The bimodal BS radial
distribution in many globular clusters (e.g. Ferraro et al. 2004; Mappelli et
al. 2006; Lanzoni et al. 2007; Beccari et al. 2011) has been observed. A
scenario that BSs in the dense core were formed in collisions, whereas BSs in
the low density cluster outskirts were formed by mass transfer in close
binaries has been suggested to explain this.

The traditional opinion about globular clusters is that all stars within a
globular cluster are thought to share the same age and initially homogeneous
chemical composition and they are simple stellar populations. This has one
noticeable exception: $\omega$ Cen, the most massive cluster in the Milky Way,
whose stars show a large spread in metallicity. At present, the situation has
been much more complex, and it is now identifiable that almost all the
globular clusters so far examined in detail have at least two stellar
generations. Clear evidence for multiple main sequences (Bedin et al. 2004)
and giant branches (Nataf et al. 2011), and unusual horizontal branch
(D’Antona et al. 2005) and subgiant branch (Milone et al. 2008; Moretti et al.
2009) morphologies all can be explained in straightforward ways by the
presence of multiple generations of stars. The first second-generation binary
named NV364 in $\omega$ Centauri has been identified by Li et al. (2012).

$\omega$ Centauri is one of the most metal poor globular clusters in the Milky
Way, with [Fe/H]=-1.53, and has an interstellar reddening $E(B-V)=0.12$ and a
distance modulus $(m-M)_{v}=13.94$ (Harris 1996(2010 edition)). V211, with a
period of 0.576235 $d$, was first identified in the outskirts of the cluster
by Kaluzny et al. (1996) during a search for variable stars in the central
part of the globular cluster $\omega$ Centauri. NV358, with a period of
0.59964 $d$, was first discovered in the outer region of the cluster by
Kaluzny et al. (2004) during a photometric survey for variable stars in the
field of this cluster. On the CDM of the cluster, V211 is located in the faint
BS region, while NV358 occupies a position of the bright BS domain. Light
curves of several eclipsing binaries in $\omega$ Centauri have been analyzed,
but except for the two. In this paper, we show the investigation of $B$ and
$V$ light curves of the two binaries taken from Kaluzny et al. (2004) using
the Wilson-Devinney code.

##  2 Light curve analysis of the two binaries

$\omega$ Centauri was observed using the 1.0-m Swope telescope at Las Campanas
Observatory by Kaluzny et al. (2004) under The Cluster AgeS Experiment (CASE)
project during the interval from 1999 February 6/7 to 2000 August 9/10. $B$
and $V$ light curves of 301 variables were obtained and the photometric data
are available on the VizieR Result page. As no photometric analysis has been
carried out for the two binaries, V211 and NV358, which have good enough
photometric data and are located in the BS region on the CMD of $\omega$
Centauri, we chose them to do further analysis. Using the fourth version of
the W-D program (Wilson & Devinney 1971; Wilson 1990, 1994; Wilson & Van Hamme
2003) which is a good tool for the modeling of eclipsing binaries based on
real photometric and spectroscopic (radial velocity) data, we analyzed the $B$
and $V$ light curves of the two binaries taken from Kaluzny et al. (2004) for
the first time. Some of the photometric data that have serious derivation from
the phased light curves (most in $B$ band) were deleted. The ephemerides used
to calculate the phases of the two binaries were

$Min.I=2451283.7791+0.576235E,$ (1) $Min.I=2451284.1098+0.599640E.$ (2)

During our solutions, the effective temperature of the primary component,
$T_{1}$, was determined based on the values of dereddened $B-V$ at the
secondary minimum of the two binaries described below. The gravity-darkening
coefficients of the components were taken to be 1.0 for radiative atmosphere
($T\geq 7200$) from von Zeipel (1924) and 0.32 for convective atmosphere
($T<7200$) from Lucy (1967). The bolometric albedo coefficients of the
components were fixed at 1.0 and 0.5 for radiative and convective atmospheres
following Ruciński (1969). The bolometric and bandpass limb-darkening
coefficients of the components were also fixed (van Hamme 1993). Starting with
the solutions by mode 2, we found that the solutions of V211 usually converged
when the secondary component fills its Roche lobe and that of NV358 quickly
converged. So, the final iterations of V211 were made in mode 5, which
corresponds to semi-detached configuration, while that of NV358 was made in
mode 2. The quantities varied in the solutions of the two stars were the mass
ratio $q$, the effective temperature of the secondary component $T_{2}$, the
monochromatic luminosity of primary component in $B$ and $V$ bands $L_{1}$,
the orbital inclination $i$ and the dimensionless potential of the primary
component $\Omega_{1}$. The dimensionless potential of the secondary component
$\Omega_{2}$ is also a variable for NV358.

###  2.1 V211

The temperature of the primary component, $T_{1}$, was determined from the
dereddened color index $(B-V)_{1}$ using the program provided by Worthey & Lee
(2011). Bellini et al. (2009) determined the membership probability of V211 as
90%. We adopted an interstellar reddening of $E(B-V)=0.12$ and a metallicity
value of [Fe/H] = -1.53 (Harris 1996(2010 edition)) for $\omega$ Centauri. The
color index at the secondary minimum is a good approximation to the color
index of the primary star. Based on the phased data, the color index at the
secondary minimum is measured to be $(B-V)_{1}=0.393$, leading to
$T_{1}=7035K$. The bolometric albedo and the gravity-darkening coefficients of
the components were set as $A_{1}=A_{2}=0.5$ and $g_{1}=g_{2}=0.32$ for
convective atmosphere. Bolometric and bandpass square-root limb-darkening
parameters of the components taken from van Hamme (1993) were listed in Table
1. A $q$-search method was used to determine the mass ratio of V211. Solutions
were carried out for a series of values of the mass ratio (0.2, 0.25, 0.3,
0.4, 0.5, 0.6, 0.7). The relation between the resulting sum $\Sigma$ of
weighted square deviations and $q$ is plotted in Fig. 1. The minimum value was
obtained at $q=0.30$. Therefore, we fixed the initial value of mass ratio $q$
at 0.30 and made it an adjustable parameter. Then, we executed a differential
correction until it converged and final solutions were derived. The final
photometric solutions are listed in Table 1. The comparison between observed
and the theoretical light curves is shown in Fig. 1.

###  2.2 NV358

The color index of NV358 at the secondary minimum is 0.142. Bellini et al.
(2009) determined the membership probability of NV358 is 99%. The
$(B-V)_{1,0}$ of the primary was fixed at 0.022. Using the same method of
V211, we fixed the effective temperature of the primary component of NV358 at
$T_{1}=8918K$. The bolometric albedo and the gravity-darkening coefficients of
the components were set to $A_{1}=A_{2}=1.0$ and $g_{1}=g_{2}=1.0$ for
radiative atmosphere. Bolometric and bandpass square-root limb-darkening
parameters of the components taken from van Hamme (1993) were listed in Table
2. A $q$-search method was also used to determine the mass ratio of NV358. The
final photometric solutions are listed in Table 2. Fig. 2 shows the comparison
between observed and the theoretical light curves.

Because Johnson et al. (2009) found large metallicity spread for stars in the
cluster $\omega$ Centauri, a metallicity value of [Fe/H] = -1.0 was also used
to determine the effective temperature of the primary component, and
$T_{1}=7104K$ for V211 and $T_{1}=8923K$ for NV358 were obtained. The
solutions based on the metallicity of [Fe/H] = -1.0 are listed in Tables 1 and
2. We find that the additional solution results of this metallicity value are
in accordance to previous solutions. Therefore, the results using [Fe/H] =
-1.53 are adopted to be the final solution.

Figure 1: Left panel shows $q$-search for V211. Right panel displays observed
(open symbols) and theoretical (solid lines) light curves of V211 in $BV$
passbands.

Figure 2: Left panel shows $q$-search for NV358. Right panel displays observed (open symbols) and theoretical (solid lines) light curves of NV358 in $BV$ passbands. Table 1: Photometric solutions for V211 in the globular cluster $\omega$ Centauri Parameters | [Fe/H] = -1.53 | Errors | [Fe/H] = -1.0 | Errors
---|---|---|---|---
$g_{1}=g_{2}$ | 0.32 | Assumed | 0.32 | Assumed
$A_{1}=A_{2}$ | 0.5 | Assumed | 0.5 | Assumed
$x_{1bol}$ | 0.093 | Assumed | 0.093 | Assumed
$x_{2bol}$ | 0.265 | Assumed | 0.265 | Assumed
$y_{1bol}$ | 0.632 | Assumed | 0.632 | Assumed
$y_{2bol}$ | 0.441 | Assumed | 0.441 | Assumed
$x_{1B}$ | 0.173 | Assumed | 0.173 | Assumed
$x_{2B}$ | 0.757 | Assumed | 0.757 | Assumed
$y_{1B}$ | 0.706 | Assumed | 0.706 | Assumed
$y_{2B}$ | 0.109 | Assumed | 0.109 | Assumed
$x_{1V}$ | 0.059 | Assumed | 0.059 | Assumed
$x_{2V}$ | 0.445 | Assumed | 0.445 | Assumed
$y_{1V}$ | 0.726 | Assumed | 0.726 | Assumed
$y_{2V}$ | 0.398 | Assumed | 0.398 | Assumed
$T_{1}(K)$ | 7035 | Assumed | 7104 | Assumed
$q(M_{2}/M_{1})$ | 0.2941 | $\pm 0.0094$ | 0.2942 | $\pm 0.0093$
$T_{2}(K)$ | 5219 | $\pm 24$ | 5257 | $\pm 24$
$i$ | 85.166 | $\pm 0.416$ | 85.172 | $\pm 0.415$
$L_{1}/(L_{1}+L_{2})(B)$ | 0.8739 | $\pm 0.0008$ | 0.8737 | $\pm 0.0008$
$L_{1}/(L_{1}+L_{2})(V)$ | 0.8160 | $\pm 0.0013$ | 0.8160 | $\pm 0.0013$
$\Omega_{1}$ | 3.7041 | $\pm 0.0562$ | 3.7055 | $\pm 0.0566$
$\Omega_{2}$ | 2.4532 | Assumed | 2.4534 | Assumed
$r_{1}(pole)$ | 0.2922 | $\pm 0.0048$ | 0.2921 | $\pm 0.0049$
$r_{1}(side)$ | 0.2972 | $\pm 0.0052$ | 0.2971 | $\pm 0.0052$
$r_{1}(back)$ | 0.3002 | $\pm 0.0054$ | 0.3001 | $\pm 0.0054$
$r_{2}(pole)$ | 0.2597 | $\pm 0.0023$ | 0.2596 | $\pm 0.0023$
$r_{2}(side)$ | 0.2704 | $\pm 0.0024$ | 0.2704 | $\pm 0.0024$
$r_{2}(back)$ | 0.3031 | $\pm 0.0024$ | 0.3031 | $\pm 0.0024$
Table 2: Photometric solutions for NV358 in the globular cluster $\omega$ Centauri Parameters | [Fe/H] = -1.53 | Errors | [Fe/H] = -1.0 | Errors
---|---|---|---|---
$g_{1}=g_{2}$ | 1.0 | Assumed | 1.0 | Assumed
$A_{1}=A_{2}$ | 1.0 | Assumed | 1.0 | Assumed
$x_{1bol}$ | 0.435 | Assumed | 0.435 | Assumed
$x_{2bol}$ | 0.082 | Assumed | 0.082 | Assumed
$y_{1bol}$ | 0.254 | Assumed | 0.254 | Assumed
$y_{2bol}$ | 0.646 | Assumed | 0.646 | Assumed
$x_{1B}$ | 0.061 | Assumed | 0.061 | Assumed
$x_{2B}$ | 0.125 | Assumed | 0.125 | Assumed
$y_{1B}$ | 0.792 | Assumed | 0.792 | Assumed
$y_{2B}$ | 0.753 | Assumed | 0.753 | Assumed
$x_{1V}$ | 0.039 | Assumed | 0.039 | Assumed
$x_{2V}$ | 0.036 | Assumed | 0.036 | Assumed
$y_{1V}$ | 0.701 | Assumed | 0.701 | Assumed
$y_{2V}$ | 0.743 | Assumed | 0.743 | Assumed
$T_{1}(K)$ | 8918 | Assumed | 8923 | Assumed
$q(M_{2}/M_{1})$ | 0.4442 | $\pm 0.0125$ | 0.4443 | $\pm 0.0125$
$T_{2}(K)$ | 7249 | $\pm 47$ | 7255 | $\pm 48$
$i$ | 76.033 | $\pm 0.272$ | 76.037 | $\pm 0.272$
$L_{1}/(L_{1}+L_{2})(B)$ | 0.8247 | $\pm 0.0016$ | 0.8248 | $\pm 0.0016$
$L_{1}/(L_{1}+L_{2})(V)$ | 0.7873 | $\pm 0.0019$ | 0.7875 | $\pm 0.0020$
$\Omega_{crit}$ | 2.7669 | Assumed | 2.7672 | Assumed
$\Omega_{1}$ | 4.0500 | $\pm 0.0447$ | 4.0463 | $\pm 0.0448$
$\Omega_{2}$ | 3.4006 | $\pm 0.0607$ | 3.4027 | $\pm 0.0607$
$r_{1}(pole)$ | 0.2761 | $\pm 0.0035$ | 0.2764 | $\pm 0.0035$
$r_{1}(point)$ | 0.2861 | $\pm 0.0041$ | 0.2864 | $\pm 0.0041$
$r_{1}(side)$ | 0.2805 | $\pm 0.0037$ | 0.2808 | $\pm 0.0038$
$r_{1}(back)$ | 0.2841 | $\pm 0.0039$ | 0.2845 | $\pm 0.0040$
$r_{2}(pole)$ | 0.2072 | $\pm 0.0063$ | 0.2071 | $\pm 0.0063$
$r_{2}(point)$ | 0.2192 | $\pm 0.0082$ | 0.2190 | $\pm 0.0082$
$r_{2}(side)$ | 0.2103 | $\pm 0.0067$ | 0.2101 | $\pm 0.0067$
$r_{2}(back)$ | 0.2166 | $\pm 0.0077$ | 0.2165 | $\pm 0.0077$

##  3 Results and Discussions

Based on the $B$ and $V$ light curves, photometric solutions for the two EA-
type binaries, V211 and NV358, have been derived. It is shown that V211 is a
typical Algol-type binary and NV358 is a well detached binary system, the
primary and secondary components of NV358 fill 68.3% and 81.4% of their
critical Roche lobes, respectively. Using the fractional luminosities of the
components from the $B$ and $V$ light-curve solutions, quoted above, and
adopting the observed $B$ and $V$ magnitudes of the two binaries at maximum
light, we found the following visual magnitudes of the components of the two
binaries: $V_{1}=18.331\pm 0.002$, $B_{1}=18.686\pm 0.001$, $V_{2}=19.948\pm
0.008$ and $B_{2}=20.788\pm 0.007$ for V211, $V_{1}=17.330\pm 0.003$,
$B_{1}=17.449\pm 0.002$, $V_{2}=18.751\pm 0.010$ and $B_{2}=19.131\pm 0.010$
for NV358. The mean source of the errors is the respective uncertainties in
the solutions. Fig. 3 shows the positions of the individual eclipsing
components on the CMD (Noble et al. 1991) of $\omega$ Centauri. V211 is
located in the faint BS region, while NV358 occupies a position in the bright
BS domain.

###  3.1 Physical parameters of the two binaries

According to<EMAIL_ADDRESS>proper-motion catalog of the globular cluster $\omega$
Centauri conducted by Bellini et al. (2009), V211 (designation 298164) and
NV358 (designation 129801) are definite proper-motion members of the cluster
with respective probabilities of 90% and 99%. Using the same method as Liu et
al. (2011), we estimated the physical parameters of the two binaries by the
light-curve program of the WD code. First, we calculated the absolute
bolometric magnitudes of the two binaries based on the $V$-band absolute
magnitudes of them. Second, using the light-curve program of the W-D code, we
could also obtain the absolute bolometric magnitudes of the two binaries,
which can be compared to the results in the previous step. When the results of
the two steps are consistent with each other, the physical parameters of the
two binaries are obtained. The physical parameters of the two binaries are
listed in Table 3, where the errors represent the uncertainty of $q$. The mean
physical parameters of the two binaries are as follows: for V211,
$M_{1}=1.13\pm 0.03M_{\odot},$ $R_{1}=0.98\pm 0.01R_{\odot}$ and
$M_{2}=0.33\pm 0.01M_{\odot},$ $R_{2}=0.92\pm 0.01R_{\odot}$; for NV358,
$M_{1}=1.30\pm 0.05M_{\odot},$ $R_{1}=1.03\pm 0.01R_{\odot}$ and
$M_{2}=0.58\pm 0.02M_{\odot},$ $R_{2}=0.78\pm 0.01R_{\odot}$.

Table 3: Physical parameters of V211 and NV358 Parameters | Values | Errors | Values | Errors
---|---|---|---|---
| V211 |  | NV358 |
$M_{1}$ ($M_{\odot}$) | 1.13 | $\pm 0.03$ | 1.30 | $\pm 0.05$
$M_{2}$ ($M_{\odot}$) | 0.33 | $\pm 0.01$ | 0.58 | $\pm 0.02$
$R_{1}$ ($R_{\odot}$) | 0.98 | $\pm 0.01$ | 1.03 | $\pm 0.01$
$R_{2}$ ($R_{\odot}$) | 0.92 | $\pm 0.01$ | 0.78 | $\pm 0.01$
$A$ ($R_{\odot}$) | 3.30 | $\pm 0.02$ | 3.68 | $\pm 0.04$
$L_{bol1}$ ($L_{\odot}$) | 2.03 | $\pm 0.05$ | 5.86 | $\pm 0.13$
$L_{bol2}$ ($L_{\odot}$) | 0.54 | $\pm 0.01$ | 1.46 | $\pm 0.13$
$\log{g_{1}}$ ($cgs$) | 4.51 | $\pm 0.01$ | 4.52 | $\pm 0.01$
$\log{g_{2}}$ ($cgs$) | 4.03 | $\pm 0.01$ | 4.42 | $\pm 0.02$
$M_{bol1}$ ($mag$) | 3.98 | $\pm 0.01$ | 2.83 | $\pm 0.01$
$M_{bol2}$ ($mag$) | 5.41 | $\pm 0.01$ | 4.34 | $\pm 0.04$
$M_{bol}$ ($mag$) | 3.722 | $\pm 0.010$ | 2.589 | $\pm 0.016$
$m_{v_{max}}$ ($mag$) | 18.11 | Assumed | 17.07 | Assumed

###  3.2 The particularity of the two binaries

Both of the two binaries are very interesting targets. V211 is located in the
faint BS region on the CMD of $\omega$ Centauri and is a definite member star
of this cluster. Therefore, V211 is an eclipsing BS. The mass of the primary
component of V211 is $1.13M_{\odot}$, which is larger than that of a star at
the main sequence turnoff ($M_{TO}=0.8M_{\odot}$). Liking other eclipsing
Algol BS such as NJL5, V239 in $\omega$ Centauri and V228 in 47 Tuc (Helt et
al. 1993; Li & Qian 2012; Kaluzny et al. 2007), V211 should be formed by mass
transfer from the secondary component to the primary inducing a reversal of
the original mass ratio so that the current primary was originally the less
massive component. NV358 occupies a position in the bright BS domain of the
CMD of $\omega$ Centauri, and the two components of NV358 also stay in the BS
region. Both are bluer than the main-sequence stars in $\omega$ Centauri,
indicating that they are younger and metal richer. We have derived that the
surface gravity of the primary component is $\log{g}=4.52$ $cgs$, suggesting
that it is a main-sequence star. Then, we can estimate its age using the
equation

$t_{MS}=\frac{3.37\times 10^{9}}{(M/M_{\odot})^{2.122}},$ (3)

derived from Yıldız (2011). An age of $T_{a}\leq 1.93$ Gyr is obtained for the
primary component. It is believed that short period binary systems are formed
by a fragmentation process and are unlikely formed from a capture process.
Therefore, the components of NV358 should be of the same age. We adopt a value
of 1.93 Gyr for the age of NV358. The age of $\omega$ Centauri is $16\pm 3$
Gyr (Noble et al. 1991), and almost at the same time the first-generation
stars in $\omega$ Centauri formed. Therefore, NV358 is much younger than the
first-generation stars in $\omega$ Centauri. As NV364 in $\omega$ Centauri (Li
et al. 2012), we deduce that NV358 is also a second-generation binary.

In summary, V211 is an eclipsing BS, making it an important object to
comprehend the hypothesis that BSs are formed by mass transfer in close
binaries. At the same time, V211 was discovered in the out region of $\omega$
Centauri, so it can improve the scenario that BSs in the low density cluster
outskirts were formed by mass transfer in close binaries. NV358 is a second-
generation binary, making it an evidence of multiple populations of the
cluster $\omega$ Centauri. NV358 was discovered in the outer region but not in
the center of $\omega$ Centauri, improving the result that $\omega$ Centauri
was previously the nucleus of a nucleated dwarf galaxy and later it has been
completely destroyed by a gravitational interaction between the Milky Way and
the nucleated dwarf galaxy so that only its nucleus is now observed (Li et al.
2012). In the future, we hope to obtain the spectroscopic data, making it
possible to evaluate more accurate physical parameters of the two binaries.
This will provide evidence needed to improve our results.

Figure 3: Positions of the two binaries in the CMD for $\omega$ Centauri. The
solid square gives the position of V211 and the solid triangle shows the
position of NV358, open symbols represent the positions of the respective
components.

###### Acknowledgements.

This work is partly supported by Chinese Natural Science Foundation (Nos.
11203066, 11133007, 11103074, 10973037 and 10903026) and by the West Light
Foundation of the Chinese Academy of Sciences. Thanks for the photometric data
published in Kaluzny et al. (2004).

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