# Investigating stellar variability in the open cluster region NGC 381

Maurya, Jayanand Aryabhatta Research Institute of observational sciencES
(ARIES). Nainital, Uttrakhand, India  School of Studies in Physics and
Astrophysics, Pt. Ravishankar Shukla University, Raipur, Chattisgarh 492 010,
India  Astronomy & Astrophysics Division, Physical Research Laboratory,
Ahmedabad, 380009, State of Gujarat, India  Joshi, Yogesh C  Aryabhatta
Research Institute of observational sciencES (ARIES). Nainital, Uttrakhand,
India  Panchal, A  Aryabhatta Research Institute of observational sciencES
(ARIES). Nainital, Uttrakhand, India  Gour, A. S  School of Studies in Physics
and Astrophysics, Pt. Ravishankar Shukla University, Raipur, Chattisgarh 492
010, India

###### Abstract

We study variable stars in the field of the open cluster NGC 381 using
photometric data observed over 27 nights and identify a total of 57 variable
stars out of which five are member stars. The variable stars are classified
based on their periods, amplitudes, light curve shapes, and locations in the
H-R diagram. We found a rich variety of variable stars in the cluster. We
identified a total of 10 eclipsing binaries out of which 2 are Algol type (EA)
while 8 are W UMa type (EW) binaries. The estimated ages of these EW binaries
are greater than 0.6 Gyr which is in agreement with the formation time
constraint of $\geqslant$ 0.6 Gyr on short-period eclipsing binaries. The
estimation of the physical parameters of the three EW type binaries is done
using PHOEBE model-fitting software. The pulsating variable stars include one
each from $\delta$ Scuti and $\gamma$ Dor variability class. We determined the
pulsation modes of pulsating variables with the help of the FAMIAS package. We
obtained 15 rotational variables stars comprising four dwarf stars identified
on the basis log(g) versus log(Teff) diagram. These dwarf stars are found to
have generally larger periods than the remaining rotational variables.

stars: variable-open cluster:NGC 381-binaries:eclipsing -techniques:
photometric

††software: IRAF (Tody, 1986, 1993), DAOPHOT II (Stetson et al., 1990), PHOEBE
(Prša & Zwitter, 2005), FAMIAS (Zima, 2008).

##  1 Introduction

Open clusters are known to host a rich variety of the variable stars (Xue et
al., 2019; Sandquist et al., 2020; Joshi et al., 2020a). These stars play a
very crucial role in understanding the stars’ intrinsic properties like
pulsation, rotation, and emission. In addition to intrinsic properties,
extrinsic properties of stars like geometrical effects in the case of
eclipsing binaries can also be understood through variability study. Period-
Luminosity relations of pulsating stars like Cepheids and RR Lyrae have been
distinctly useful in the distance measurement and the Galactic structure study
(Minniti et al., 2020; Skowron et al., 2019; Iorio & Belokurov, 2019). The
intermediate-age clusters host a large number of $\delta$ Scuti stars as their
Main Sequence (MS) turn-offs lie close to the $\delta$ Scuti instability
strip. $\delta$ Scuti-type variables are A$3$-F$0$ spectral class main-
sequence or subgiant stars belonging to the lower classical instability strip.
Intermediate age open clusters are the best observing targets for the
asteroseismological study of $\delta$ Scuti-type stars (Arentoft et al.,
2005). $\gamma$ Dor stars are relatively a new class of main-sequence or sub-
giant pulsating stars having spectral class A7-F5 and effective temperature
Teff between 6700 to 7400 K. These stars are cooler than $\delta$ Scuti stars
and lie near the red edge of the $\delta$ Scuti instability strip.

Eclipsing binary stars are quite common in open clusters as well. They are
located above the single-star main sequence in the color-magnitude diagram of
the clusters and can be used as a possible distance indicator (Pietrzyński et
al., 2019) and in measurements of stellar masses and diameters (Panchal &
Joshi, 2021). Eclipsing binaries have also been useful in the calibration of
theoretical stellar models (Torres et al., 2010). The evolution from eclipsing
binaries of Algol-type (EA) to W UMa type binaries (EW) provides an
opportunity to understand angular momentum loss, tidally locked system, and
mass transfer within binary systems (Jiang et al., 2014; Jiang, 2020; Chen et
al., 2020). The binary systems are also known to have pulsating companions
like $\delta$ Scuti (Liakos & Niarchos, 2017; Miszuda et al., 2021). The
evolution of $\delta$ Scuti star in a binary system is affected by mass
transfer and tidal interaction. The $\delta$ Scuti star in a binary system is
very useful in theoretical modeling as the physical parameters of binary
components can be accurately determined. Some eclipsing binaries exhibit
asymmetry in the maxima of the light curves which is known as O’ Connell
effect due to stellar spots on the surface of the components (Joshi et al.,
2016).

The photometric variability is also caused by the modulation of the brightness
due to rotation. Rotational variables belonging to the same cluster population
are found to show a period-magnitude relation (Soares-Furtado et al., 2020).
These rotational variables have an uneven distribution of the cooler spots on
the surface which produces variation in the magnitude due to the inclination
angle of observation. The cooler spots and chromospheric activities generally
happen in the late-type stars (Long et al., 2021). Drake (2006) found that
rotation periods of chromospheric active stars decrease as the distance from
the Galactic plane increases. This relation between period and distance from
the Galactic plane was attributed to the fact that stars at low altitudes
would be more evolved than stars at higher altitudes (Drake, 2006). Thus, the
study of variable stars provides us a tool to investigate the interior of
stellar evolution and the external properties of stars.

In the present study, we carried out a variability search and analysis of the
open cluster NGC 381 for which the photometric study has earlier been carried
out in Maurya & Joshi (2020). This is an intermediate-age sparse open cluster
with an estimated age of 447 $\pm$ 52 Myr (Maurya & Joshi, 2020). It is
located at a distance of 1131 $\pm$ 48 pc (Maurya & Joshi, 2020). The present
variability study has been carried out in an $\sim$ 18 $\times$ 18 arcmin2
region around the cluster center (RA = 01:08:19.57; DEC = +61:35:18.24). This
is the very first variability study of the cluster NGC 381 which is part of
our ongoing project of the variability investigations of the poorly studied
open clusters (Joshi et al., 2020a). In our previous study, we found a total
of 132 member stars of the cluster after applying a 50$\%$ probability cut-off
in this region. However, Cantat-Gaudin et al. (2018) found a total of 148
member stars belonging to the cluster NGC 381 distributed over a much larger
region than the observed region in the present study. The present paper is
structured as follows. We discuss observations, data reduction, and the
transformation of the instrumental magnitude to the standard magnitude in
Section 2. The methods and techniques applied to detect variable stars in the
present data set are described in Section 3. We discussed the locations of the
variable stars on the H-R diagram and their implications in Section 4. The
classification and detailed analysis of the variable stars are discussed in
Section 5. We discuss and summarize our analysis in Section 6.

Table 1: The log for observed time series data of NGC 381 region. The ’L’ and ’S’ denote long and short exposures respectively. Date | V filter | I filter
---|---|---
| Number of frames | Number of frames
01/10/2017 | 9L, 9S | 3L, 3S
02/10/2017 | 36L, 37S | 11L, 12S
12/10/2017 | 28L, 28S | 6L, 6S
14/10/2017 | 4L, 4S | 4L, 4S
21/10/2017 | 3L, 3S | 3L, 2S
22/10/2017 | 32L, 32S | -
23/10/2017 | 37L, 42S | 45L, 44S
29/10/2017 | 23L, 24S | 25L, 25S
30/10/2017 | 50L, 42S | -
02/11/2017 | 13L, 15S | 8L, 11S
03/11/2017 | 9L, 10S | 10L, 10S
09/11/2017 | 10L | -
12/11/2017 | 6L, 6S | 17L, 17S
14/11/2017 | 6L, 6S | 7L, 7S
16/11/2017 | 24L, 21S | 26L, 26S
17/11/2017 | 4L, 4S | -
24/11/2017 | 8L | -
05/01/2018 | 19L, 11S | 13L, 13S
06/01/2018 | 17L, 16S | 10L, 10S
13/01/2018 | 3L | 3L
22/01/2018 | 3L | 3L, 3S
07/10/2018 | 10L | 3L
20/10/2018 | 7L | -
01/12/2018 | 18L | 19L
04/12/2018 | 21L, 12S | 12L, 12S
07/12/2018 | 8L | -
14/01/2019 | 4L, 4S | 5L, 5S

##  2 Observations and Data analysis

The observations for the current study were taken using 1.3-m Devasthal Fast
Optical Telescope (DFOT). The DFOT telescope was mounted with a 2k$\times$2k
CCD having wide coverage of $\sim 18^{{}^{\prime}}$ $\times$
$18^{{}^{\prime}}$ over the sky. The data for the variability analysis was
collected during 27 nights spanning more than a year starting from 01 Oct.
2017. The observations were accumulated in 738 and 437 frames of the Johnson-
Cousins V and I bands, respectively. We searched both long and short period
variables using the collected data. The exposure times were varying between
180-220 and 80-120 sec for long exposures in V and I bands, respectively
whereas short exposures for both the bands were in the range of 10-30 sec. The
observation log for the observations of NGC 381 is given in Table 1.

The image cleaning procedures like bias correction, flat fielding, and cosmic
rays removal were completed using suitable IRAF packages. We used DAOPHOT II
packages and processes described by Stetson (1987) to obtain instrumental
magnitude through the point-spread function (PSF). The PSF technique is
preferred for crowded fields like star clusters. The stars on the edges of
some frames can contaminate the magnitudes of the adjacent stars and give a
false sense of variability. We, therefore, removed all the data points within
20 pixels from the edges of each frame. We used DAOMATCH and DAOMASTER
packages of DAOPHOT II to cross-match the same stars in different frames. The
cluster NGC 381 was observed on 10 Oct. 2017 with Landolt’s standard field SA
98 for the photometric calibration. The standardized data of NGC 381 was used
in its photometric study (Maurya & Joshi, 2020). We converted the instrumental
magnitude of the stars detected in frames of variability data to the standard
magnitude by comparison with the standard data used in Maurya & Joshi (2020).
The following conversion formula is used for obtaining the standard magnitudes
of stars:

V = a1 $\times$ v + a2

a1 and a2 used in the above relation are conversion coefficients. V is the
standard magnitude while v is the instrumental magnitude of the same star. The
conversion coefficients a1 and a2 are calculated through least-square fits for
each frame of variability data. We did not use color terms in the conversion
formula as the effect of the color terms was found to be very insignificant in
the magnitude calibration. The I band data were reduced using the same methods
as discussed for the reduction of V band data. However, the cluster was
observed mostly in V-band. The number of I-band frames collected at any
specific night were less than 20 for most of the nights. Due to more frames
and coverage, V-band data was used for most of the analysis except for FAMIAS
package. We used both V and I band data for the mode identification through
the FAMIAS package which requires multi-band data. Only V-band observations
were used for variability search. We produced light curves for a total of 5183
stars. All the light curves were inspected for the presence of variability.

##  3 Identification

We used the Lomb-Scargle algorithm (Lomb, 1976; Scargle, 1982) to search the
periodic signal in the time series data. The Lomb-Scargle method is very
efficient for periodicity search in unevenly spaced data. The period
corresponding to the frequency of the maximum power in the power spectrum is
generally used for phase calculation and phase-folding of the light curves. We
checked all the periodic variables by phase folding the light curves for their
periods. We chose only those variables which show good periodic brightness
variation throughout the entire phase.

We found a total of 57 periodic variables with periods ranging 41 minutes to
10.74 days in NGC 381 region. We verified the periods and the light curves of
these periodic variables using NASA Exoplanet Archive Periodogram service
(Akeson et al., 2013) and found that the two periods obtained were generally
in agreement. Once the period of a star is known, we determined the phase of
each epoch. We generated phase-folded light curves after binning in intervals
of generally 0.01 phase. We calculated the mean magnitude for each phase bin.
The binned light curves are useful in better identification of variables due
to smoothing out of scattered points. We inspected the light curves of all the
identified periodic variables for twice the estimated period to find primary
and secondary minima.

We investigated the light curves of all the stars detected in the region.
These variables were carefully examined for any blending by the neighboring
stars and stars contaminated due to light from the neighboring star were
discarded. The light curves having periods of 1/n day were excluded from the
variability analysis to avoid false variability arising solely due to the
aliasing effect. Here, n in the 1/n stands for a positive integer. The light
curves were plotted after folding the data with phase. We examined the light
curves of all the periodic variables for the presence of the primary and
secondary minima as the signature of binary systems by plotting phase folded
light curves corresponding to the twice of the obtained periods. The amplitude
of the variation in periodic variables was calculated by taking half of the
difference in the mean magnitudes of the three faintest and the three
brightest points in the curves (Joshi et al., 2020a). We have provided the
basic parameters like period, amplitude, and magnitude in Table 2.

Table 2: The basic parameters of the identified variable stars. The columns denote star id, RA, DEC, V magnitude, period, amplitude, Teff, log (Teff), bolometric magnitude, luminosity, membership, and types of variability. ’M’ indicates cluster member while ’F’ marks field star. ID | RA (J2000) | DEC (J2000) | V | Period |  $\Delta$V | Teff | log(Teff) | Mbol | log(L/L⊙) | Membership | Type
---|---|---|---|---|---|---|---|---|---|---|---
| (hh:mm:ss) | (dd:mm:ss) | (mag) | (day) | (mag) | (K) | (K) |  |  |  |
199 | 17.38816 | 61.47656 | 19.659 | 0.64170 | 0.117 | 6058 | 3.78234 | 5.138103 | -0.163241 | F | Misc
298 | 17.37635 | 61.51619 | 14.970 | 2.88585 | 0.274 | 7156 | 3.74758 | 1.067196 | 1.465122 | F | EA
528 | 17.34962 | 61.53601 | 19.578 | 0.63944 | 0.240 | 5866 | 3.76832 | 6.143452 | -0.565381 | F | Misc
535 | 17.34960 | 61.46318 | 18.186 | 0.88731 | 0.103 | 6708 | 3.82659 | 3.906007 | 0.329597 | F | Misc
600 | 17.34102 | 61.63476 | 17.804 | 0.30660 | 0.071 | 4873 | 3.68779 | 0.307142 | 1.769143 | F | EW
608 | 17.34195 | 61.47071 | 19.597 | 0.20628 | 0.104 | 5649 | 3.75199 | 5.342264 | -0.244906 | F | Rot
610 | 17.34198 | 61.44597 | 18.093 | 0.88574 | 0.095 | 6352 | 3.80293 | 3.072912 | 0.662835 | F | Misc
732 | 17.32448 | 61.71775 | 19.529 | 0.30934 | 0.259 | 6642 | 3.82227 | 4.805819 | -0.030327 | F | EW
737 | 17.32681 | 61.43044 | 16.723 | 5.90783 | 0.068 | 7300 | 3.86335 | 1.245203 | 1.393919 | F | Misc
749 | 17.32584 | 61.44954 | 16.834 | 0.85538 | 0.061 | 4331 | 3.62835 | 7.357904 | -1.051162 | F | Rot
1155 | 17.27404 | 61.67363 | 18.785 | 0.30390 | 0.056 | 5382 | 3.73098 | 5.404430 | -0.269772 | F | Rot
1173 | 17.27376 | 61.50248 | 16.161 | 7.97406 | 0.042 | 5058 | 3.70359 | 0.672518 | 1.622993 | F | Rot
1217 | 17.26697 | 61.65434 | 16.034 | 1.70109 | 0.037 | 6271 | 3.71063 | 3.371889 | 0.543244 | F | Rot
1397 | 17.25031 | 61.47081 | 9.858 | 2.48932 | 0.265 | 8059 | 4.19605 | -4.264955 | 3.597982 | F | EA
1399 | 17.25022 | 61.43972 | 14.344 | 15.60373 | 0.065 | 6029 | 3.76993 | 4.701243 | 0.011503 | F | Rot
1441 | 17.24511 | 61.51020 | 16.333 | 2.28997 | 0.064 | 5007 | 3.69958 | 0.278268 | 1.780693 | F | Misc
1596 | 17.22836 | 61.60860 | 14.118 | 0.43500 | 0.022 | 7158 | 3.82861 | 3.137557 | 0.636977 | M |  $\gamma$ Dor
1602 | 17.22763 | 61.58761 | 15.413 | 1.13352 | 0.030 | 6922 | 3.80548 | 3.324183 | 0.562327 | F | Misc
1689 | 17.21949 | 61.44122 | 19.499 | 0.85503 | 0.120 | 5739 | 3.75883 | 4.550551 | 0.071779 | F | Misc
1785 | 17.20845 | 61.56744 | 17.962 | 0.47661 | 0.207 | 5841 | 3.76648 | 3.387451 | 0.537020 | F | EW
1812 | 17.20693 | 61.43172 | 17.971 | 0.46948 | 0.080 | 5075 | 3.70547 | 3.594283 | 0.454287 | F | Rot
1935 | 17.19160 | 61.43677 | 14.743 | 18.86972 | 0.059 | 4966 | 3.71715 | 5.490398 | -0.304159 | F | Misc
2018 | 17.17960 | 61.61100 | 20.054 | 0.38967 | 0.153 | 4119 | 3.61478 | 7.402637 | -1.069055 | F | Misc
2058 | 17.17713 | 61.43163 | 18.790 | 0.46970 | 0.107 | 6259 | 3.79651 | 3.889816 | 0.336073 | F | Misc
2092 | 17.17362 | 61.44512 | 16.624 | 0.46971 | 0.065 | 6489 | 3.80542 | 3.705286 | 0.409886 | F | Misc
2245 | 17.15525 | 61.64929 | 15.623 | 0.96225 | 0.023 | 7869 | 3.83518 | 2.063115 | 1.066754 | F | Misc
2302 | 17.14846 | 61.71435 | 13.743 | 1.46727 | 0.042 | 5976 | 3.77641 | 4.751433 | -0.008573 | F | Rot
2458 | 17.13336 | 61.67006 | 19.243 | 0.27405 | 0.106 | 5012 | 3.69791 | 6.883659 | -0.861464 | F | EW
2472 | 17.13248 | 61.60869 | 13.045 | 1.27248 | 0.026 | 7574 | 4.02109 | 0.742912 | 1.594835 | M | Misc
2483 | 17.13197 | 61.57956 | 15.163 | 0.07504 | 0.037 | 6731 | 3.81758 | 1.983478 | 1.098609 | F |  $\delta$-Scuti
2597 | 17.11615 | 61.71983 | 19.354 | 0.76046 | 0.110 | 5804 | 3.76370 | 4.313651 | 0.166540 | F | Misc
2618 | 17.11641 | 61.58124 | 17.374 | 0.24802 | 0.020 | 4825 | 3.68353 | 0.905611 | 1.529756 | F | Rot
2673 | 17.10656 | 61.67836 | 11.645 | 12.19928 | 0.042 | 7956 | 3.88981 | 0.749634 | 1.592147 | M | Misc
2699 | 17.10342 | 61.68298 | 19.325 | 0.30427 | 0.081 | 5658 | 3.75265 | 5.028546 | -0.119419 | F | Rot
2999 | 17.07245 | 61.51315 | 15.389 | 2.93362 | 0.035 | 6516 | 3.79020 | 4.144697 | 0.234121 | M | Rot
3005 | 17.07080 | 61.60435 | 19.634 | 0.31405 | 0.150 | 4802 | 3.68145 | 5.670018 | -0.376007 | F | EW
3472 | 17.02022 | 61.49097 | 19.207 | 0.38882 | 0.126 | 5653 | 3.75225 | 5.138588 | -0.163435 | F | EW
3535 | 17.01155 | 61.57847 | 17.657 | 0.32339 | 0.112 | 5784 | 3.70447 | 5.917628 | -0.475051 | F | EW
3680 | 16.99572 | 61.62203 | 19.166 | 0.84519 | 0.216 | 7238 | 3.85964 | 3.430055 | 0.519978 | F | Misc
3717 | 16.99284 | 61.54328 | 18.603 | 0.58746 | 0.089 | 4873 | 3.68783 | 5.478819 | -0.299528 | F | Rot
3792 | 16.98570 | 61.44255 | 15.835 | 0.68694 | 0.049 | 4795 | 3.65312 | 6.642276 | -0.764910 | F | Rot
4049 | 16.95194 | 61.57502 | 13.343 | 2.94373 | 0.024 | 7988 | 3.90244 | -0.484138 | 2.085655 | F | Misc
4070 | 16.95224 | 61.44262 | 17.795 | 0.31974 | 0.046 | 5273 | 3.72202 | 1.591900 | 1.255240 | F | Misc
4243 | 16.92789 | 61.53868 | 17.463 | 2.13002 | 0.084 | 5615 | 3.77448 | 4.823012 | -0.037205 | F | Misc
4281 | 16.91943 | 61.67900 | 18.635 | 0.24194 | 0.054 | 7211 | 3.85798 | 3.439219 | 0.516313 | F | Misc
4283 | 16.92179 | 61.55971 | 19.408 | 0.84517 | 0.131 | 6280 | 3.79796 | 4.464289 | 0.106285 | F | Misc
4361 | 16.90875 | 61.60868 | 15.017 | 0.40748 | 0.024 | 6392 | 3.81621 | 3.751129 | 0.391548 | M | Misc
4405 | 16.90401 | 61.55529 | 16.117 | 0.76060 | 0.032 | 5239 | 3.83875 | 2.403936 | 0.834332 | F | Rot
4525 | 16.88284 | 61.72299 | 20.046 | 0.26657 | 0.158 | 4546 | 3.65762 | 5.943748 | -0.485499 | F | Misc
4705 | 16.86359 | 61.57545 | 18.034 | 0.55897 | 0.076 | 5209 | 3.70465 | 3.444240 | 0.514304 | F | Misc
4829 | 16.84602 | 61.71936 | 19.438 | 0.34551 | 0.114 | 4409 | 3.64436 | 3.735335 | 0.397866 | F | EW
5091 | 16.81309 | 61.56333 | 14.800 | 0.78344 | 0.062 | 6335 | 3.80175 | 3.469250 | 0.504300 | M | Misc
5166 | 16.80137 | 61.66182 | 16.948 | 0.89365 | 0.065 | 6170 | 3.80247 | 3.381778 | 0.539289 | F | Misc
5199 | 16.79518 | 61.70208 | 18.722 | 1.13805 | 0.126 | 6601 | 3.81958 | 3.488482 | 0.496607 | F | Misc
5370 | 16.77350 | 61.65828 | 19.460 | 1.14294 | 0.142 | 5029 | 3.70151 | 6.353474 | -0.649389 | F | Misc
5423 | 16.77012 | 61.50882 | 15.765 | 0.46019 | 0.032 | 4423 | 3.64572 | -0.725947 | 2.182379 | F | Rot
5451 | 16.76296 | 61.61349 | 16.935 | 1.13334 | 0.054 | 10619 | 4.02610 | 0.287079 | 1.777169 | F | Misc
Figure 1: The light curve RMS as a function of the magnitudes for the stars in
the NGC 381 region. The y-axis in the figure is on the logarithmic scale. The
red larger points denotes the variable stars whereas blue points in the
background are for the non-variable stars.

We found a total of 57 periodic variables after all the mentioned
examinations. We have determined the light curve root mean square (RMS) to
investigate the spread in data points for variable stars. The plot for the
light curves RMS against the mean magnitudes of the stars is shown in Figure
1. It is evident in the figure that the variable stars are generally
associated with larger standard deviation in the magnitudes compared to non-
variable stars. The finding chart for all the 57 variable stars is given in
Figure 2.

Figure 2: The finding chart for 57 periodic variables detected in NGC 381
region. The variable stars are marked by the circle and labeled with star IDs.
The right ascension and declination are given in degrees.

The obtained periodic variables include 39 short period (p$<$1 day) and 18
long period (p$>$1 day) variables. The mean magnitudes of these variable stars
vary from 9.858 mag to 20.054 mag in the V band. The estimated periods of
these variables are in the range of $\sim$3 hours to $\sim$19 days while
amplitude variation is between 0.022 to 0.256 mag. We found 10 out of 57
periodic variables as eclipsing binary stars. The light curves of the
variables other than eclipsing binaries and RS CVn stars are shown in Figure
3. Six stars out of these 57 variables are member stars as mentioned in Table
2. The 51 remaining variable stars belong to the field population found in the
cluster NGC 381 region.

##  4 Variable stars and H-R diagrams

The H-R diagram basically represents the empirical relation between spectral
type and luminosity. The other forms of the H-R diagram can be temperature
versus luminosity or color index versus absolute magnitude. The location of a
star on the H-R diagram is affected by the factors like stellar wind, magnetic
field, rotation, and chemical abundance besides initial mass and age. The
positions of the variable stars on the H-R diagrams provide us with
information about the evolutionary stage of the stars. The H-R diagrams have
also been immensely useful in classifying the variable stars as they offer
constrained regions for different classes of variable stars.

Figure 3: The phase folded light curves of periodic variable stars identified
in the NGC 381 region. The plots are given for twice of the phase value for a
good presentation. The star IDs, periods (p), and variability types are
denoted at the top of each light curve. The original light curves and Lomb-
Scargle periodograms are shown in the appendix of this paper. The light curves
for the eclipsing binaries are given in Figure 6. The light curves for stars
1602 and 4049 are shown in Figure 12. These two stars are reported to be RS
Canum Venaticorum-type (RS CVn) variables in the literature (Watson et al.,
2006).

###  4.1 Color-magnitude diagram

The absolute G versus (GBP \- GRP) color-magnitude diagram (CMD) has been
useful in the classification of variable stars. The CMD is particularly
important as it provides constrained regions for variable types like eclipsing
binaries, rotational, eruptive, and cataclysmic variables besides pulsating
variables (Gaia Collaboration et al., 2019; Sanghi et al., 2021). The
pulsating variables like $\delta$-Scuti and $\gamma$ Dor have very localized
clumps on CMD. The separate locations of variables on the CMD help in removing
ambiguity due to the similarity in light curves of two different classes of
variables (Sanghi et al., 2021). We used absolute G magnitudes, GBP, and GRP
band data for 35 variables provided by Anders et al. (2019). In the study, the
absolute G magnitudes were calculated using precise Gaia DR2 parallaxes. These
magnitudes for the variable stars are given in Table 3. We have shown the CMD
constructed from these 35 variable stars in Figure 4.

Table 3: G, GBP, and GRP magnitudes are taken from Gaia eDR3 mag. The absolute G magnitudes denoted by G0 are used from Anders et al. (2019). The first column gives star IDs. ID | G | GBP | GRP | (GBP-GRP) | G0
---|---|---|---|---|---
| (mag) | (mag) | (mag) | (mag) |
298 | 14.3877 | 14.7998 | 13.7793 | 1.020500 | 0.666100
600 | 17.5191 | 18.0799 | 16.8012 | 1.278700 | 0.488836
610 | 17.7972 | 18.4208 | 17.0336 | 1.387200 | 3.048655
737 | 16.5466 | 17.0495 | 15.8812 | 1.168300 | 1.267372
749 | 16.1014 | 17.1122 | 15.1064 | 2.005800 | 7.622625
1173 | 15.6201 | 16.4421 | 14.7247 | 1.717400 | 0.709279
1217 | 15.7076 | 16.2976 | 14.9249 | 1.372700 | 3.387695
1397 | 9.8681 | 10.002 | 9.6066 | 0.395400 | -2.927836
1399 | 14.1543 | 14.5896 | 13.5438 | 1.045800 | 4.637926
1441 | 15.8335 | 16.6254 | 14.955 | 1.670400 | 0.321563
1596 | 13.9532 | 14.3141 | 13.4158 | 0.898300 | 3.036791
1602 | 15.2371 | 15.6453 | 14.6385 | 1.006800 | 3.246058
1785 | 17.6741 | 18.3018 | 16.8906 | 1.411200 | 3.367079
1812 | 17.5073 | 18.2624 | 16.6488 | 1.613600 | 3.587901
1935 | 14.4212 | 15.0251 | 13.6831 | 1.342000 | 5.468198
2092 | 16.4239 | 16.8871 | 15.7813 | 1.105800 | 3.641529
2245 | 15.5027 | 15.7892 | 15.0404 | 0.748800 | 1.96228
2302 | 13.9527 | 13.9487 | 12.8531 | 1.095600 | 4.802494
2472 | 12.9203 | 13.2031 | 12.4731 | 0.730000 | 0.992314
2483 | 14.9365 | 15.416 | 14.2758 | 1.140200 | 1.914954
2618 | 16.7424 | 17.5992 | 15.8243 | 1.774900 | 0.91183
2673 | 11.5709 | 11.7849 | 11.2164 | 0.568500 | 0.711613
2999 | 15.1195 | 15.5919 | 14.4665 | 1.125400 | 4.026561
3535 | 17.0551 | 17.7956 | 16.2115 | 1.584100 | 5.800122
3792 | 15.3578 | 16.0753 | 14.5375 | 1.537800 | 6.807869
4049 | 13.2593 | 13.4677 | 12.8803 | 0.587400 | -0.627663
4070 | 17.3223 | 18.103 | 16.4568 | 1.646200 | 1.619608
4243 | 16.9251 | 17.6806 | 16.0811 | 1.599500 | 4.692011
4361 | 14.7457 | 15.2224 | 14.0984 | 1.124000 | 3.604831
4405 | 15.5044 | 16.3757 | 14.5575 | 1.818200 | 2.561388
4705 | 17.3425 | 18.225 | 16.4173 | 1.807700 | 3.404837
5091 | 14.7421 | 15.1451 | 13.9861 | 1.159000 | 3.56836
5166 | 16.5781 | 17.1719 | 15.8359 | 1.336000 | 3.255291
5423 | 14.9632 | 16.0395 | 13.9387 | 2.100800 | -0.506022
5451 | 16.661 | 17.1892 | 15.9542 | 1.235000 | 0.634745

The CMD for variable stars in the present study indicates the absence of stars
like the long-period pulsating variable, $\alpha$ Cygni, R Coronae Borealis,
V361 Hydrae, and ZZ Ceti stars as constrained regions for these types of stars
lie outside the current CMD (Gaia Collaboration et al., 2019). We have
considered the position of variable stars on the CMD during their
classification.

Figure 4: Plot of absolute-G versus (GBP \- GRP) color-magnitude diagram for
the 35 variable stars identified in the NGC 381 region.

###  4.2 H-R diagram

Figure 5: H-R diagram of variable stars found in NGC 381 region. The
instability strips corresponding to slowly pulsating B type, $\delta$ Scuti,
and $\gamma$ Dor stars are shown by red dotted curve, continuous line, and
dashed line, respectively. The blue filled points denote the stars which were
available in Anders et al. (2019) and Bai et al. (2019) catalogs. The open
circles represent stars whose effective temperatures were calculated using
Torres (2010) relations.

We examined log(L/L⊙) versus log(Teff) H-R diagram in the present study to
know the evolutionary stage of the identified variable stars. The effective
temperature, Teff, of 24 stars were available in the catalog provided by Bai
et al. (2019). These temperatures were calculated using a large training
sample for regression of about four million stars which makes their estimated
effective temperatures more precise than the effective temperatures of Gaia
DR2. We used effective temperatures from Anders et al. (2019) catalog for
additional 11 stars. The log(Teff) for the remaining 22 variable stars were
calculated from (B-V)0 color through Torres (2010) relations using distances
from Bailer-Jones et al. (2018) and reddening from 3D reddening map given by
Green et al. (2019) which provides reddening E(g-r) with very good resolution
for given right ascension, declination, and distance. We calculated E(B-V)
from E(g-r) using extinction ratio relations given by Wang & Chen (2019). We,
then, calculated (B-V)0 values from (B-V) color of the star using E(B-V).
These values of log(Teff) were used in calculation of logarithmic luminosity,
log(L/L⊙), using the method described by Torres (2010). The logarithmic
luminosity values were calculated using the following equations:

Mbol = Mv+BCv

log(L/L⊙) = -0.4(Mbol-M${}_{bol_{\odot}}$)

The bolometric magnitude Mv is calculated from magnitude in V band, distance,
and Av extinction. The values of distance and extinction were taken from
Anders et al. (2019) which were estimated with Gaia DR2 data. The bolometric
correction (BCv) values were calculated using empirical relations given by
Torres (2010). The M${}_{bol_{\odot}}$ in the above equations is the
bolometric magnitude of the Sun which is taken as 4.73 mag (Torres, 2010).
Once we had calculated log(Teff), the values of log(L/L⊙) were calculated
using the above described relations through Torres (2010) method of BCv
calculation. We plotted the H-R diagram using the calculated log(Teff) and
log(L/L⊙) as shown in Figure 5. The instability strips corresponding to
$\delta$-Scuti, $\gamma$ Dor, SPB, and Cephei stars are also marked in the H-R
diagram.

The instability strips belonging to high mass stars like Slowly Pulsating B
type (SPB) stars do not host any star in the H-R diagram of NGC 381. As most
of the variable stars in the NGC 381 region belong to the field population,
the lack of high mass variable stars is along with the expectations. We found
three variable stars in the gap between SPB and $\delta$-Scuti strips which is
theoretically inhibited for pulsation. The variable stars in this gap have
also been found in previous studies (Balona et al., 2011; Joshi et al., 2020b)
and we have discussed this phenomenon in Section 5. There are few variable
stars in the instability strips for low mass variables like $\delta$-Scuti and
$\gamma$ Dor. The locations of variable stars in the H-R diagram as well as
CMD have been used for the classification of variables. In the following
section, we individually describe the characteristics of the different classes
of variables found in our study.

##  5 Classification

The classification of variable stars is necessary for their comparative study.
The variable stars have been classified based on period, amplitude, effective
temperature, mass, the shape of the light curves, and location on the H-R
diagram. Pulsating variable stars are known to be located in various
instability strips related to the pulsating mechanism on the H-R diagram.
Additionally, the locations of the variable stars on the CMD diagram were also
considered for the classification of the variability class. Despite of our
best efforts, we acknowledge that there can be some ambiguity in the
classifications of the variable stars for instance the rotational variable can
also show primary and secondary minima. Similarly, there can be ambiguity in
the classification of the $\delta$-Scuti and $\gamma$ Dor stars as they belong
to closely placed instability strips on the H-R diagram. The gaps in the data
can also affect the shape of the light curves especially for the low amplitude
variables.

###  5.1 Variability caused by Eclipses and Rotation

####  5.1.1 Eclipsing Binaries

Eclipsing binaries have been greatly useful in the precise determination of
masses, luminosities, radii, and distances (Pietrzyński et al., 2019; Stassun
& Torres, 2021). We identified a number of eclipsing binaries from the
inspection of the light curves. The types of photometric eclipsing binaries
are identified through specific features like the size of the primary and
secondary minima in the light curves. These specific features can be
theoretically explained by Roche lobes geometry. In eclipsing binary of Algol
type (EA), one of the component stars has filled or overflowed the Roche lobe.
The light curves of these variables are identified by the clear beginning and
end of the maxima and minima. The light curves of eclipsing binaries of
$\beta$-Lyrae type (EB) do not show any definite start or end of eclipses as
light continuously varies. The EBs are semi-detached systems having
gravitationally distorted components and hence have continuously varying light
curves. The W Ursae Majoris-type (EW) eclipsing binaries or W UMa stars are
marked by almost equal primary and secondary minima in the light curves. EW
binaries are contact binaries consisting of ellipsoidal components with
periods generally less than 1 day and no clear onset or end of the eclipses.
It has been found that EA binaries evolve to EW through angular momentum loss
and mass transfer (Qian et al., 2018).

We found two EA and eight EW binaries in NGC 381 region. The high number of EW
binaries in the NGC 381 region compared to EA binaries is consistent with the
fact that a study by Chen et al. (2020) detected EW binaries many times more
than EA binaries. The light curves for obtained eclipsing binaries are given
in Figure 6. The light curves of binary stars 298 and 1397 are marked with
clear beginning and end of primary secondary minima. Though all three types of
eclipsing binaries have wide-spread locations on CMD, however, EA-type
binaries have been found to be located at the most extreme than the other
types. The binary star ID 1397 is located on the CMD in the region mostly
occupied by EA-type binaries (e.g., Gaia Collaboration et al., 2019).
Therefore we classified star 1397 as an EA-type eclipsing binary. The light
curve properties of binary star 298 suggest it to be an EA-type eclipsing
binary which was also supported by its location on the CMD. The light curves
of the remaining 8 eclipsing binaries having IDs 600, 732, 1785, 2458, 3005,
3472, 3535, and 4829 show characteristics of EW-type binaries. The stars
having ID 600, 1785, and 3535 are also shown on the CMD and their locations on
the CMD are consistent with their classification as EW stars. The positions of
the EA binary stars 298 and 1397 on the CMD are significantly higher than the
locations of EW binary systems indicating primary stars of earlier spectral
type in the case of EA binaries compared to primary stars of EW systems. We
found periods of EW-type binaries to be significantly smaller than the EA-type
binaries as expected by theories of EA to EW evolution (Chen et al., 2020).
The periods of all these binaries are found to be greater than the observed
period cut-off of 0.23 days for eclipsing binaries (Rucinski, 2007; Norton et
al., 2011). This sharp period cut-off is considered to be related to the
nuclear evolution time-scale of the primary component (Chen et al., 2020).

Figure 6: The original light curves (left subplot), Lomb-Scargle periodograms
(middle subplot), and phase folded light curves (right subplots) of each
eclipsing binary star found in the NGC 381 region. The periods (p) and
corresponding frequencies (f) are also shown in the subplots. Star IDs are
given at the top of each plot.

####  5.1.2 Physical parameters of Eclipsing Binaries

#### Through empirical relations

It was found that absolute parameters of the eclipsing binaries depend on the
period and the mass ratios of binary systems (Latković et al., 2021). We
calculated the physical parameters of EW stars using Latković et al. (2021)
empirical relations given as follows:

M1 = (2.94 $\pm$ 0.21)P + (0.16 $\pm$ 0.08)

M2 = (0.15 $\pm$ 0.17)P + (0.32 $\pm$ 0.06)

R1 = (3.62 $\pm$ 0.13)P + (0.04 $\pm$ 0.05)

R2 = (1.56 $\pm$ 0.13)P + (0.16 $\pm$ 0.05)

L1 = (13.98 $\pm$ 0.75)P + (3.04 $\pm$ 0.27)

L2 = (3.66 $\pm$ 0.26)P + (0.69 $\pm$ 0.09)

We also calculated the ages of these EW stars using the method described in
Yildiz & Doğan (2013) and used by Latković et al. (2021). The secondary
components of EW-type binaries are currently less massive than the primary
star. However, the secondary stars are found to be more luminous than the
luminosity expected for the current mass. This extra luminosity is interpreted
as a hint of its mass before mass transfer. The ML is defined as the mass of
an isolated star with the luminosity equivalent to the luminosity of the
secondary star. The quantity $\delta$M which is equal to ML \- M2 is used to
find the initial mass of the secondary star, M2i. The values of ML and M2i are
calculated as follows:

ML = $\Bigg{(}\frac{L_{2}}{1.49}\Bigg{)}^{0.237}$

M2i = M2 \+ 2.5($\delta$M - 0.07)0.64

The initial mass ratio can be calculated in the form as follows:

$\frac{1}{q_{i}}$ = $\frac{M_{1}-(M_{2i}-M_{2})(1-\gamma)}{M_{2i}}$

where $\gamma$ is the ratio of the mass loss from the system to the (M2i \-
M2) value which we considered as 0.664 for these calculations (Latković et
al., 2021). The value of mass ratio was found to vary with mass difference
$\delta$M for EW type eclipsing binary while there was no correlation for EA
type eclipsing binaries (Yildiz & Doğan, 2013). The correction was applied for
mass difference $\delta$M ¡ 0.35 (refer to Figure 7 of Yildiz & Doğan (2013))
for EW stars. We calculated $\delta$M and found that all the EW stars found in
our study have $\delta$M ¿ 0.35 so we did not use the corrections. We employed
the following relation for the calculation of the ages of binary components:

tMS = $\frac{10}{M^{4.05}}$(0.0056(M + 3.993)3.16 \+ 0.042)

where M is the mass of the component of which age is to be determined. The age
of the binary system itself for EW type star is calculated as age = tMS(M2i)
i.e. the age of the binary system is calculated by putting M = M2i in the
above equation. However, if the age of the binary system calculated through
this method comes to be greater than the age of the primary star then the age
of the primary star should be taken as the age of the binary star system
(Latković et al., 2021).

The primary temperature of EW systems has been found to be correlated with the
orbital period (Jayasinghe et al., 2020a; Latković et al., 2021). We
calculated the primary temperatures of EW stars having periods below 0.5 days
using the following relation given by Latković et al. (2021):

T1 = -(2752$\pm$320)$\times$P + (6745$\pm$100)

where T1 is the temperature of primary and P is the orbital period of the EW
binary. The values of the physical parameters calculated using the empirical
relations are given in Table 4. We caution about the reliability of the
obtained parameters corresponding to the star with ID 600 as parameters of EW
stars with periods greater than 0.5 days have been hardly reliable (Latković
et al., 2021). The ages of remaining stars are found to be greater than 0.6
Gyr which is consistent with the formation time constraint of $\geq$ 0.6 Gyr
for short-period binaries (Hwang & Zakamska, 2020). The obtained temperature
of the primary component corresponds to the spectral class G. These
temperature values are consistent with the classification of these stars as EW
binaries because the spectral classes of both the components of an EW system
are generally found to be later than the F class.

Table 4: The calculated parameters of eclipsing binaries using empirical relations. The star id is given by ID. The labels M, R, L, and T denotes mass, radius, luminosity, and temperature. The suffix 1 denote primary star and 2 marks secondary star. The age is given in second from last column. ID | M1 | M2 | R1 | R2 | L1 | L2 | Age | T1
---|---|---|---|---|---|---|---|---
| (M⊙) | (M⊙) | (R⊙) | ((R⊙) | (L⊙) | (L⊙) | (Gyr) |
600 | 1.963 | 0.412 | 2.260 | 1.117 | 11.613 | 2.934 | 0.589 | -
732 | 1.069 | 0.366 | 1.160 | 0.643 | 7.365 | 1.822 | 0.755 | 5894
1785 | 1.561 | 0.391 | 1.765 | 0.904 | 9.703 | 2.434 | 0.647 | 5433
2458 | 0.966 | 0.361 | 1.032 | 0.588 | 6.871 | 1.693 | 0.786 | 5991
3005 | 1.083 | 0.367 | 1.177 | 0.650 | 7.430 | 1.839 | 0.751 | 5881
3472 | 1.303 | 0.378 | 1.448 | 0.767 | 8.476 | 2.113 | 0.697 | 5675
3535 | 1.111 | 0.369 | 1.211 | 0.664 | 7.561 | 1.874 | 0.744 | 5855
4829 | 1.176 | 0.372 | 1.291 | 0.699 | 7.870 | 1.955 | 0.727 | 5794

#### Through model fitting

Eclipsing binaries exhibit variation in brightness with time due to their
orbital geometry and alignment from the observer’s point of view. We can
determine many physical parameters and properties like period, mass, distance,
and radii accurately utilizing photometric and spectroscopic methods. The
unique features of the evolution and formation of eclipsing binaries can be
studied via continuous monitoring. In our work, we identified 10 eclipsing
binaries comprising 8 EW-type of eclipsing variables. Out of these eight EW
binaries, the V-band photometric light curves of the two EWs are analyzed with
the help of the PHOEBE software package. The rest of the identified eclipsing
binaries require more photometric observations as some parts of light curves
are missing in them.

PHOEBE is one of the most popular tools used for light curves and radial
velocity (RV) curves modeling of eclipsing binaries. It is based on
Wilson–Devinney code which is written in FORTRAN. We used PHOEBE 1.0 which
provides a GUI and scripter tool for modeling eclipsing binaries. As the two
binaries 0732 and 1785 are reported as EWs, we selected an over-contact binary
not in thermal contact as the fitting model in PHOEBE. The temperature of the
systems is determined using the color-temperature relation provided by (Torres
et al., 2010). The estimated $T_{eff}$ is used as the temperature of the
primary component. The calculation of mass-ratio (q=$m_{2}/m_{1}$) is the
first step in modeling the photometric light curve. The accurate determination
of the q-parameter requires RV observation but in absence of it, photometric
light curves can also be used to constrain the q-parameter around real value.
It has been observed that the photometric mass ratio is reliable for total
eclipsing binaries only (Hambálek & Pribulla, 2013). So, as the inclination
angle decreases the uncertainty in the photometric mass-ratio increases. But
due to the absence of RV data, we are dependent only on the q-search method.
In this method, some known parameters like the time of minima, period,
$T_{eff}$ of primary, surface albedo, and gravity brightening are fixed. The
other quantities like inclination (i), $T_{eff}$ of secondary, surface
potential ($\Omega$), luminosity ($l_{1}$, and $l_{2}$) are set as free
parameters. The q is varied in small steps and corresponding to each q best-
fit model is obtained with the number of iterations. The cost function
corresponding to the best fit model for each q is noted. The q which
corresponds to the minimum cost function is assumed as the actual mass ratio.
The process of q-search is repeated for both of the binary systems and the
estimated q for 0732 and 1785 are 0.42 and 0.26, respectively. The PHOEBE-
scripter was run multiple times with these q-parameters to get the final
solution. The fitted light curves are plotted with the observed data in Figure
7.

Both systems 0732 and 1785 show a small difference in the levels of primary
and secondary minima, so, some temperature difference between both the
components can be expected. The final model shows a temperature difference of
almost 500 and 150 for the binaries 0732 and 1785, respectively. The fill-out
factor is found to be 0.31 and 0.53 for 0732 and 1785 respectively. The mass
and radii of the primary component are determined using the relations by
(Gazeas, 2009). The parameters determined from our observations using PHOEBE
and these relations are given in Table 5. The system 1785 is also observed by
Zwicky Transient Facility (ZTF, Bellm et al. (2019) survey between 18 May 2018
to 31 May 2022 in zi, zr, and zg band. Most of the photometric observations
are in zr-band (1173 data points). To compare the modeling results from our
observed data for system 1785, we analyze 1785 ZTF zr-band data with PHOEBE.
The _catflags_ flag in the ZTF catalog informs user about quality of data
111https://irsa.ipac.caltech.edu/onlinehelp/ztf/help.pdf. The photometric
observations corresponding to _catflags_ =0 are used for the modeling. For the
ZTF data of the system 1785, the q-search method estimated the q value to be
0.24($\pm$0.03). The secondary temperature and inclination for the system 1785
are determined as 4965($\pm$135) K and 71${}^{\circ}(\pm$2), respectively,
using the ZTF data. Thus, the modeling results from ZTF data are similar to
the results obtained from our observed data.

Table 5: The physical parameters estimated by PHOEBE model fitting on the light curves of the EW binary-systems ID 0732 and 1785. Parameters | ID 0732 | ID 1785
---|---|---
Period (days) | 0.30935 | 0.47661
q | 0.42$\pm$0.03 | 0.26$\pm$0.05
i | 71$\pm$1 | 72$\pm$5
$T_{1}$ (K) | 6642 | 5179
$T_{2}$ (K) | 6116$\pm$154 | 5003$\pm$55
$\Omega_{1}$ | 2.64$\pm$0.03 | 2.29$\pm$0.02
$\Omega_{2}$ | $\Omega_{1}$ | $\Omega_{1}$
$L_{1}/(L_{1}+L_{2})$ | 0.752 | 0.794
$M_{1}~{}(M_{\odot})$ | 1.06 | 1.51
$M_{2}~{}(M_{\odot})$ | 0.44 | 0.39
$R_{1}~{}(R_{\odot})$ | 1.03 | 1.65
$R_{2}~{}(R_{\odot})$ | 0.71 | 0.92

All the results are determined using the photometric mass ratio and V-band
photometric data. For the modeling, binned light curves are used and there is
also some scattering in data points. These results can be considered
preliminary results as more accurate results will require multi-band
photometry of these systems. The use of long-term multi-band photometric data
can also help in the analysis of other long-term effects like period
variation, surface activity, etc.

Figure 7: The light curves of the two EW binary systems whose physical
parameters are estimated using PHOEBE code model fittings. The blue continuous
curves show the best model fits of the PHOEBE model fitting code. The IDs for
each binary system are given at the top of each subplot.

###  5.2 Rotational Variables

Stars showing variability caused by axial rotation of surface having non-
uniform brightness are known as rotational variables. The variation in the
surface brightness may be due to the presence of spots and other magnetic
activities. The rotational rate of the surface of a star decreases with
increasing age and this property is used as an age indicator for individual
stars (Barnes, 2007). The late-type stars (G or K type) with true color (B-V)0
values larger than 0.5 mag (Mamajek & Hillenbrand, 2008) are classified as
rotational variables. In our study, We found total 15 rotational variables
having IDs 608, 749, 1155, 1173, 1217, 1399, 1812, 2302, 2618, 2699, 2999,
3717, 3792, 4405, and 5423. The locations of these 15 stars on the CMD were
compatible with their classification as rotational variables.

The rotational variable star 2999 is also a member star which gives us
opportunity to utilize the period-age relations known to apply for the
rotational variables. We used empirical period-age relations given by Angus et
al. (2019) to calculate the period for the star 2999. The GBP-GRP value for
the star was found to be 1.1254 mag as given in Table 3. We took age of the
cluster to be 447 $\pm$ 52 as previously determined by us (Maurya & Joshi,
2020). Using the period-age relation, the period of the rotational variable
2999 was predicted to be 8.8450 days which is much larger than the period
value of 2.93362 days determined in the present study.

Stars 2618 and 5423 locations on the CMD were in the region populated by only
rotational ellipsoidal variables. This indicates that these two stars can be
classified as rotational ellipsoidal type variables which are close binaries
exhibiting variability without eclipses due to distortion of orbital motion by
the companion star. These variables are known to have generally smaller
amplitude compared to contact binaries (Drake et al., 2017) which is also the
case with stars 2618 and 5423. The stars 1173 and 1812 were located in the
region occupied by rotational ellipsoid variables and RS CVn variables on CMD.
These two stars did not show distortion of the eclipses in light curves
generated from current data to ascertain their classification as RS CVn stars.

We have shown the distributions of the rotational variables in log (g) versus
log (Teff) diagram in Figure 8. We noticed that four stars having IDs 1217,
1399, 2302, and 2999 are separated from others in the log (g) versus log
(Teff) plot. These four stars satisfy the criteria for dwarf stars given by
Ciardi et al. (2011). The magnetic braking causes a decline in the rotation
rates of stars with increasing ages (Skumanich, 1972). The dwarf stars having
spectral class later than F have been found to be long-period variables
generally with the period of the order of weeks compared to the smaller period
of early-type dwarf (Ciardi et al., 2011). The dwarf stars 1217, 1399, 2302,
and 2999 also have comparatively larger periods than the periods of other
rotating variables as the effective temperatures given in Table 2 indicate
that these are spectral class G stars. The star 2618 has a mass of 1.36
$M_{\odot}$ (Anders et al., 2019) and the smallest period of 0.248 days in our
list of identified rotational variables. An important transition occurs around
1.3 $M_{\odot}$ stars above which stars are generally fast-rotating. In these
stars, convective envelopes become so thin that it becomes unable to produce
strong magnetic braking through magnetic winds (van Saders & Pinsonneault,
2013).

Figure 8: Plot of log(g) versus log(Teff) diagram for rotational variables
whose log(g) and Teff values are available in Anders et al. (2019) and Bai et
al. (2019) catalogs.

###  5.3 Variability caused by Pulsations

####  5.3.1 $\delta$-Scuti stars

Figure 9: Plot of power spectrum for star 2483. The f1 denotes frequency
binary system while frequency f2 corresponds to $\delta$-Scuti type
variability.

$\delta$-Scuti stars are short period pulsating variables found in the
theoretical instability strip of $\delta$-Scuti stars on the H-R diagram.
These variables have periods less than 0.3 days and amplitude of the order of
a few 0.1 mag. $\delta$-Scuti stars are A0-F5 spectral type stars lying on the
main sequence or post-main sequence stages of the H-R diagram (Breger, 2000)
driven by $\kappa$ mechanism of pulsation. $\delta$-Scuti stars are
intermediate mass (1.5-2.5 M⊙) stars (Bedding et al., 2020) with effective
temperature in range of 6300-8600 K (Uytterhoeven et al., 2011). Star 2483 is
located in the $\delta$-Scuti instability strip of the H-R diagram for NGC
381. This star exhibits properties of $\delta$-Scuti for a period of 0.07504
days while it also shows the light curve of the binary system for a period of
0.25379 days. The power spectrum for 2483 is shown in Figure 9. The amplitude
corresponding to the period of 0.07504 days was found to be 0.037 mag as also
listed in Table 2. The effective temperature and mass of this star are
reported to be 6731 K and 1.696 M⊙, respectively (Bai et al., 2019; Anders et
al., 2019). On the basis of the period, amplitude, and the location on the H-R
diagram star 2483 can be classified as a $\delta$-Scuti variable star. This
star is located within the region assigned for $\delta$-Scuti stars on the CMD
(Sanghi et al., 2021) while it is very close to the constrained region for
$\delta$-Scuti stars on the CMD given by Gaia Collaboration et al. (2019).
Therefore, we classified star 2483 as a $\delta$-Scuti variable. This star is
an interesting target for further follow-up spectroscopic studies as it seems
to be binary star with a $\delta$-Scuti component.

Figure 10: Plot illustrating Eclipsing nature of the $\delta$-Scuti star 2483.
The light curves is constructed for the period of 0.25379 day.

####  5.3.2 $\gamma$ Dor variable stars

Figure 11: The plots of amplitude ratios for mode identification through
FAMIAS package using multi-band photometric data. The left and right panel
show plots of the amplitude ratios for stars 2483 and 1596, respectively. The
mode l=0, 1, 2, and 3 are shown by red, green, blue, and cyan respectively.

$\gamma$ Dor stars are g-mode pulsating stars with spectral class A7-F5 lying
in the $\gamma$ Doradus instability strip in the H-R diagram. The location of
the $\gamma$ Dor variables on the H-R diagram is important for understanding
the interior of stars as it is the transition zone where convective cores and
radiative envelopes energy transfer changes to radiative cores and convective
envelopes energy transfer. This property associated with $\gamma$ Dor stars
helps in constraining the theoretical models of heat transfer in stars. The
pulsation in $\gamma$ Dor stars is non-radial g-modes and pulsation periods of
these stars are between 0.3-3.0 days. The typical amplitude of variation for
$\gamma$ Dor stars are $\sim$0.1 mag. These stars are less massive than
$\delta$ Scuti stars having a mass range from 1.5 M⊙ to 1.8 M⊙. The variable
star with ID 1596 in our list of variables exhibits properties of a $\gamma$
Dor star as suggested by its amplitude, period, and location in the $\gamma$
Doradus instability strip. The location of this star on the CMD was in the
region mostly occupied by $\gamma$ Dor stars. The period was found as 0.43500
day for the star 1596. The effective temperature for star 1596 was 6739 K
reported by Anders et al. (2019) which is in the range of temperature
6700-7400 K expected for the $\gamma$ Dor stars. Therefore, we classified star
1596 to be $\gamma$ Dor star. The pulsation mode identification of this
$\gamma$ Dor star was done using multi-band photometric data through FAMIAS
software (Zima, 2008) as discussed in the following section.

####  5.3.3 Mode identification of pulsation

We used time-series data obtained in V and I bands for the identification of
pulsation modes of $\delta$-Scuti and $\gamma$ Dor stars through FAMIAS
software packages (Zima, 2008). The results are compared with prediction of
models for l = 0,1,2,3,4. The effective temperatures, surface gravity, and
metallicity values required for model fitting in FAMIAS were taken from Anders
et al. (2019). The multi-band photometric mode determination using FAMIAS is
based on the relation between amplitude ratios and phase differences obtained
for various photometric bands (Watson, 1988). We identified 2483 as a
$\delta$-Scuti star while 1596 was characterized as a $\gamma$ Dor star. The
plots of amplitude ratios versus phase differences are given in Figure 11 for
both the pulsating stars. The $\delta$-Scuti star 2483 was found to be
pulsating in l = 0 which corresponds to radial pulsation (Ziaali et al., 2019;
Jayasinghe et al., 2020b). We could not find a good fitting for star 1596,
however, it seems to be pulsating in l = 0 mode. The low harmonic degree
pulsation of l=0 is typical for $\delta$-Scuti and $\gamma$ Dor stars (Sánchez
Arias et al., 2017).

###  5.4 Misc Variable

The variable stars which could not be classified in any one type of
variability class were called miscellaneous type variables. A total of 30
variable stars were classified as Miscellaneous in the present study. Stars
2472, 2673, and 5451 lie between SPB and $\delta$-Scuti instability strips.
This region in the H-R diagram is prohibited for pulsation according to
standard stellar models. Mowlavi et al. (2016) classified these stars as fast-
rotating pulsating B (FaRPB) variable stars based on a spectroscopic study
which revealed that the majority of these stars were fast-rotating. FaRPB
stars are known to be the short period (p¡0.55 d) variables (Mowlavi et al.,
2016; Joshi et al., 2020a). However, these variables were classified as
miscellaneous variables since they have periods significantly larger than 0.55
days. The light curves of stars 1689 and 1935 show a hint of eclipsing binary
but we could not confirm their exact nature and assigned the Misc category to
them. Star 2245 was located near the blue edge of the $\delta$-Scuti
instability strip. However, its period of 0.96225 days was significantly
higher than the period of less than 0.3 days expected for $\delta$-Scuti
stars. Star 737 lie close to the blue edge of the $\delta$-Scuti instability
strip, however, it has a period of 5.908 days which is very high for any
$\delta$-Scuti star.

The stars 1602 and 4049 have been classified as RS CVn variables in the AAVSO
International Variable Star Index (VSX) catalog (Watson et al., 2006). The RS
CVn variables are eclipsing binary systems with the primary star of the
spectral class of F-G type. These binary systems are known to have asymmetric
light curves caused by chromospheric activities and rotation. The distortion
in the light curves is thought to be caused by the uneven distribution of the
surface cool spots (Eaton & Henry, 2007; Eaton & Hall, 1979). RS CVn variables
show spectral emission lines of Ca II H and K (Zhang & Gu, 2008). The light
curves of these stars are the results of spot modulation of the eclipses. The
LCs for these two variable stars are shown in Figure 12. As it is clear from
the figure that the light curves of these stars do not exhibit clear eclipses
so these stars could be rotational or non-eclipsing binaries with large spots
on the one or both of the components. However, the effective temperatures of
the stars 1602 and 4049 were reported to be 6922 and 7988 K by Bai et al.
(2019) indicating that these systems are earlier than K spectral type which is
a potential spectral class for RS CVn stars (Chen et al., 2020). The locations
of stars 1602 and 4049 on the CMD are compatible with their classification as
RS CVn variables. The RS CVn stars with larger (GBP \- GRP) color values
generally have greater absolute G magnitudes (Gaia Collaboration et al.,
2019). This tendency is also found to be true in the case of RS CVn stars
identified in the present study as can be seen in Table 3. Considering all
these properties of the stars 1602 and 4049 we classify them as Miscellaneous
variables.

Figure 12: Possible RS CVn variable stars. The light curves in upper panel are
for the obtained period while light curves in lower panel are plotted for
twice the obtained period. The star IDs and the periods are given on the top
of each plots.

###  5.5 Comparison

We searched variable stars identified by us in the previously available
variable stars catalogs. We found 11 out of 57 variable stars identified by us
to be reported in VSX catalog. There were also eight variable stars common
with the catalog of periodic variable stars provided by Chen et al. (2020)
using the ZTF data. The periods and classifications of these stars are given
in Table 6 for comparison. The periods determined in zg and zr bands of ZTF
were found to be differ significantly in Chen et al. (2020) catalog for few
stars. The observed data points in zr band were more than twice of the data
points in zg band for these eight stars so we mentioned only periods in zr
band in Table 6. We found that the classifications of the variable stars given
in these catalogs are generally in agreement with the our classifications. We
could not find evidence for the variables 1602 and 4049 to be RS CVn stars as
they are reported in these catalogs. The periods given in the catalogs are
mostly in agreement with the periods determined by us. The remaining 46
variables are not listed in any variable stars catalogs so they can be
considered as new variable stars identified by us.

Table 6: The comparison of periods and variability type of variable stars reported in the VSX catalog and the ZTF catalog of periodic variable stars provided by Chen et al. (2020) . The periods and variability types from the VSX and the ZTF catalogs are denoted by prefixes VSX and ZTF. The periods and variability types determined by us are given in the second last and the last columns, respectively. ID | PVSX | TypeVSX | PZTF | TypeZTF | P | Type
---|---|---|---|---|---|---
| (day) |  | (day) |  |  |
298 | - | EA | 8.54994 | EA | 2.88585 | EA
732 | 0.30936 | EW | - | - | 0.30934 | EW
1397 | 2.48934 | EA/DM | - | - | 2.48932 | EA
1602 | 0.53806 | RS CVn | 0.53806 | RS CVn | 1.13352 | Misc
1785 | 0.47660 | EW | 0.47660 | EW | 0.47661 | EW
2458 | 0.27406 | EW | 0.27405 | EW | 0.27405 | EW
2483 | 0.07504 |  $\delta$-Scuti | - | - | 0.07504 |  $\delta$-Scuti
3005 | 0.31404 | EW | 0.31404 | EW | 0.31405 | EW
3472 | 0.38883 | EW | 0.38883 | EW | 0.38882 | EW
3535 | 0.31240 | EW | 0.32340 | EW | 0.32339 | EW
4049 | 9.26598 | RS CVn | 9.26598 | RS CVn | 2.94373 | Misc

We also check Transiting Exoplanet Survey Satellite (TESS) archive to compare
our observations. We found six of our detected variable stars namely 1397,
1596, 2302, 2472, 2673, and 4049 in TESS archives having Simple Aperture
Photometry (SAP) fluxes estimated using QLP pipelines. The cadence for these
stars is 1800 seconds in the archived data. We could extract very nice light
curve for the star 1397 which is a bright star with TESS magnitude (Tmag)
$\sim$ 9.7 mag (Paegert et al., 2022) showing periodicity of amplitude
$\sim$0.3 mag (see Figure 13). The TESS data for stars 1596 and 2673 also
produced good light curves, however, the period for 2673 was different than
ours. The time-series PSF fluxes of the stars 2999, 4361, and 5091 extracted
using a PSF-based Approach to TESS High quality data Of Stellar cluster
(PATHOS) pipeline were also available in TESS archive. The light curve
generated using TESS data for star 2999 was consistent with period obtained
from the our observed data. We could not extract any signal of variability for
remaining five stars probably due to presence of systematic artifacts in SAP
fluxes compared to Pre-search Data Conditioning SAP (PDCSAP) fluxes (Lin et
al., 2019).

Figure 13: The light curves for variable stars found in TESS archives. The
star IDs and periods are denoted at the top of each plot. The TESS mag (Tmag)
for the star ID 2999 is calculated from the PSF flux provided in the TESS
archives using the relations given by (Handberg et al., 2021).

##  6 Summary and Conclusion

We extensively observed the open cluster region NGC 381 in V and I bands for
the variability study. The region was observed using 1.3-m DFOT telescope
equipped with 2k$\times$2k CCD having a large field of view of $\sim
18^{{}^{\prime}}$ $\times$ $18^{{}^{\prime}}$ suitable for variability search.
The data were collected on 27 nights over a span of more than one year from 01
October 2017 to 14 January 2019. The data was reduced using PSF technique and
standard magnitudes were obtained. In the present study, we identified a total
of 57 periodic variable stars in the cluster region NGC 381. These variable
stars were classified on the basis of the shape of the light curves, period,
amplitude, and location on HR diagrams. We found one $\delta$-Scuti star and
one $\gamma$ Dor star. The pulsation modes of these variable stars were
examined through FAMIAS package using Johnson V and Cousin I band data and the
$\delta$-Scuti star was found to be pulsating in possibly the first overtone
mode. To detect eclipsing binaries, we checked the light curves of all the
periodic variables for twice the period obtained. The stars were qualified as
eclipsing binaries only after distinct identification of primary and secondary
minima. This way we found a total of 10 eclipsing binaries comprising 2
eclipsing binaries of Algol type and 8 W UMa type eclipsing binaries. The
physical parameters of the EW type binaries were estimated using empirical
relations and PHOEBE model fitting packages. We classified late-type variables
as rotational variables if the color (B-V)0 was found to be larger than 0.5
mag. We identified total 15 rotational variables in the present study. The
rotational variability is thought to be present due to the uneven distribution
of magnetic cool spots on the surface of the stars. There were also some
variables that could not be classified in any particular type of the variables
and we classified them as miscellaneous type variables. A total of 30
variables were classified in this class. However, more time-series data
complemented with the spectroscopic study is necessary to characterize these
variables.

## Acknowledgements

This work presents results from the European Space Agency (ESA) space mission
Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis
Consortium (DPAC). Funding for the DPAC is provided by national institutions,
in particular the institutions participating in the Gaia MultiLateral
Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia.
The Gaia archive website is https://archives.esac.esa.int/gaia. This paper
includes data collected by the TESS mission, which are publicly available from
the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope
Science Institute (STScI). Funding for the TESS mission is provided by the
NASA’s Science Mission Directorate. Based on observations obtained with the
Samuel Oschin 48-inch and the 60-inch Telescope at the Palomar Observatory as
part of the Zwicky Transient Facility project. ZTF is supported by the
National Science Foundation under Grant No. AST1440341 and AST-2034437 and a
collaboration including Caltech, IPAC, the Weizmann Institute for Science, the
Oskar Klein Center at Stockholm University, the University of Maryland, the
University of Washington, Deutsches Elektronen-Synchrotron and Humboldt
University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan,
the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence
Livermore National Laboratories, Lawrence Berkeley National Laboratories, and
IN2P3, France. Operations are conducted by COO, IPAC, and UW.

Figure 14: The mag versus JD and Lomb-Scargle periodogram plots for the 45
variable stars whose phase folded light curves are shown in Figure 3. The star
IDs and phase folding frequencies are given at the top of the subplots.

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