1D D U Gorakhpur University, Gorakhpur, Uttar Pradesh,India
2Aryabhatta Research Institute of Observational Sciences, Nainital,
Uttarakhand

# Photometric and Kinematic study of the open cluster NGC 1027

Apara Tripathi1 Neelam Panwar2 Saurabh Sharma2 Brijesh Kumar2 Shantanu
Rastogi1

###### Abstract

We present photometric and kinematic analysis of an intermediate age open
cluster NGC 1027 using $UBV(RI)_{c}$ and Gaia Early Data Release 3 (EDR3)
data. Structural and fundamental parameters such as cluster center, cluster
extent, reddening, age and distance are estimated in this study. Cluster
center is found about 2 arcmin away from the center reported earlier. Radius
has been estimated to be about 8.00 arcmin(2.65pc). Using proper motion Gaia
EDR3 data, membership probabilities has been derived for the stars in the
region of cluster radius. We find mean proper motion of the cluster to be
$\sim$(-0.84, 2.04) mas yr-1 in (RA, DEC). We find 217 most probable
(P${}_{\mu}>$ 70%) cluster members with mean parallax 0.892 $\pm$ 0.088 mas.
Out of these, 160 members have counterparts in our optical observations. Few
stars having P${}_{\mu}>$ 70% are found out of the cluster radius showing
imprints of dynamical evolution. The color-color and color-magnitude diagrams
for the cluster members found within 8.00 arcmin have been constructed using
$UBV(RI)_{c}$ photometry and Gaia EDR3 data. This yields a reddening
E($B$-$V$) $\sim$ 0.36 mag, age $\sim$ 130 Myr and distance $\sim$ 1.14 kpc.
The mass function slope in the cluster region is $\Gamma$ $\sim$ -1.46 $\pm$
0.15, which is similar to other Galactic open clusters. The dynamical study
shows lack of faint stars in its inner region leading to mass segregation
effect. A comparison of dynamical age with cluster age indicates that NGC 1027
is a dynamically relaxed cluster suggesting that mass segregation may be
imprint of its dynamical relaxation.

###### keywords:

Star: Open star cluster: Color-Magnitude diagrams: individual: NGC 1027, Mass
function-Mass segregation

<EMAIL_ADDRESS>2022

#### 000 2022 1– 1

##  1 Introduction

Stars in the Galaxy often form in star clusters. They contain stars of diverse
mass range that are gravitationally bound and presumably formed from the same
molecular material, at same time and located at the same distance and move
together (Harris & Pudritz, 1994; Friel, 1995; Bate $et~{}al$., 2003; Carraro
& Costa, 2007; Cantat-Gaudin & Anders, 2020; Pang $et~{}al$., 2020).
Observations suggest that the age of Galactic star clusters range from a few
Myr to few Gyr, indicating that they have been forming all along the life of
our Galaxy. External forces of interaction with the galactic tidal field and
two body relaxation effect changes the structure of open cluster probing them
to be good targets to investigate astrophysical processes involved in
formation and evolution of stars and Galaxies (Miholics $et~{}al$., 2014;
Tripathi $et~{}al$., 2013, 2014).

Intermediate-age open clusters (OCs) are located in the disc of the Galaxy.
Study of these type of OCs play a crucial role in advancing our understanding
of the physical processes involved such as change in its structure, mass
function, mass segregation and develop the mechanism that govern their
evolution by time. They probe the dynamical and chemical evolution of the
Galactic disc in age and metallicity. Their study also improvise important
constraints for the evolutionary astrophysical models. (Sagar $et~{}al$.,
1986; Phelps & Janes, 1994; Lada & Lada, 2003; Chen $et~{}al$., 2003; Joshi
$et~{}al$., 2016; Castro-Ginard $et~{}al$., 2018; Angelo $et~{}al$., 2019;
Cantat-Gaudin & Anders, 2020)

NGC 1027 ($\alpha_{J2000}$ \- 02:42:40, $\delta_{J2000}$ +61:35:42; l $\sim$
135.758 and b $\sim$ \+ 1.525 ), also known as Melotte 16, Collinder 30, OCL
357, MWSC 225 open star cluster, is located in the eastern part of Cassiopeia
constellation near the northern celestial pole, adjacent to two prominent
nebulae, the Heart (IC 1805) and Soul Nebulae (IC 1848). However, NGC 1027 is
not physically associated with these two nebulae as they are more distant (d
$\sim$ 2.1 kpc; Chauhan $et~{}al$., 2011; Panwar $et~{}al$., 2017) compared to
the NGC 1027 (d$\sim$ 1 kpc; Maciejewski & Niedzielski, 2007). Some basic
parameters of the cluster as reported in WEBDA, A site Devoted to Stellar
Clusters in the Galaxy and the Magellanic Clouds are listed in Table
1.Ruprecht (1966) considered this cluster as class iii 2 p, i.e., a less
populated open cluster with noticeable concentration and a medium range in
magnitude of the stars Trumpler (1930).

Based on the wide-field $BV$ CCD photometric survey, Maciejewski & Niedzielski
(2007) reported basic parameters of 42 open clusters (OCs) including NGC 1027.
They estimated reddening E($B$-$V$) $\sim$ 0.41$\pm$0.12 mag, log age $\sim$
8.4 and distance $\sim$ 1.03 kpc for NGC 1027. Liu & Pang (2019) cataloged
2443 cluster candidates using Star cluster Hunting Pipeline (SHiP) and
classified NGC 1027 as an old class iii star cluster with a prominent MS turn-
off in the color-magnitude diagram (CMD). He suggested for re investigation of
Class iii star cluster candidates.

Photometric studies of OCs are found to be fundamental tools for determining
their structural parameters. These parameters play an important role towards
understanding galactic structure and evolution. However, most of the OCs are
affected by field star contamination. Therefore finding out member stars of
the cluster becomes necessary to look into the cluster properties (Carraro
$et~{}al$., 2008; Kharchenko $et~{}al$., 2013; Cantat-Gaudin, T. $et~{}al$.,
2018). Several methods by different researches has been suggested to determine
membership of cluster stars (Medhi & Tamura, 2013; Gao, 2014). Proper motion
diagram is a reliable tool to identify cluster members on the basis of
astrometric precision and membership determination on kinematic method (e.g.,
Cantat-Gaudin, T. $et~{}al$., 2018; Castro-Ginard $et~{}al$., 2018; Cantat-
Gaudin & Anders, 2019; Bisht $et~{}al$., 2019; Castro-Ginard $et~{}al$., 2020;
Joshi $et~{}al$., 2020). This precise learning of cluster membership gives
information about distribution of stellar mass in the OCs and throw light on
dynamic evolution of star clusters. Numerous research studies have obtained
present-day MF for a number of OCs (Dib $et~{}al$., 2017; Bostancı $et~{}al$.,
2018; Joshi $et~{}al$., 2020), yet, it is still not clear weather initial mass
function (IMF) is universal in time and space or depends upon different star
forming conditions (Elmegreen, 2000; Kroupa, 2002; Bastian $et~{}al$., 2010).
Study of the distribution of low and high mass stars towards the cluster
region provides a clue to the mass segregation in OCs.

In the present work $UBVR_{c}I_{c}$ photometry for NGC 1027 has been done for
the first time. We investigate the spatial structure, basic parameters,
extent, reddening and age to understand the dynamical evolution of the cluster
using the kinematic Gaia EDR3 data. Our high quality CCD photometry data go
down to $V$ $\sim$ 22 mag. We estimate membership probabilities of stars in
the cluster NGC 1027 down to $G\sim$ 20 mag. The high probability members are
utilized to examine mass function and mass segregation in the cluster NGC
$\sim$ 1027.

Table 1: Parameters listed in WEBDA for cluster NGC 1027. Cluster Parameter | Values
---|---
Trumpler class | IIIp
RA(J2000) | 05:36:18
DEC(J2000) | +34:08:24
Longitude(l/deg) | 164.535
Latitude(b/deg) | 1.072
Distance(d/pc) | 1318
Reddening(E(B-V)/mag) | 0.36
Age(Log(t/yr)) | 7.487

The organization of the manuscript is as follows. Section 2 provides a brief
description of the observations, data reduction procedures and astrometric
calibration. Results and analysis including radial density profile, kinematic
members, reddening, CMDs, age and distance to the cluster is discussed in
Section 3. We discuss the results including a discussion on the presence of
kinematic members beyond the cluster radius in section 4 and Section 5
summarises the results.

##  2 Observations & Data Reduction

###  2.1 UBVRcIc photometric observations

The CCD UBVRcIc data used in this study is taken using 104-cm Sampurnanand
Telescope (ST) of the Aryabhatta Research Institute of Observational Sciences
(ARIES), Nainital. The telescope covers about 13 $\times$ 13 arcmin2 on the
sky. In order to improve signal-to-noise ratio, binned mode of 2 $\times$ 2
pixel is used during the observations. The observing log is given in Table 2.
Several number of bias and twilight flat-field frames were taken in all the
filters along with multiple short and long exposure frames of the target and
standard fields. To calibrate the magnitude of stars in cluster field standard
field SA 104 of Landolt (1992) is observed. Basic steps are performed using
the standard tasks within the IRAF data reduction package for pre-processing
of data frames which includes bias subtraction, flat fielding and cosmic ray
removal.

Table 2: Description of the optical observation for NGC 1027 and standard field, with dates and exposure times for each passband Cluster/ | Date | Filter | Exp(sec)
---|---|---|---
Standard field |  |  | x no.of Frames
NGC 1027 | 24/25 Jan 06 | V | 120x3
|  | B | 180x3
|  | I | 60x2
|  | R | 60x2,300x3
|  | U | 300x2
| 24 Dec 05 | V | 900x2
|  | B | 1200x3
|  | I | 240x3
|  | R | 300x3
|  | U | 1800x2
SA104 | 24/25 Jan 06 | V | 130x5, 180x1
|  | B | 200x5,240x1
|  | I | 60x7
|  | R | 100x5,120x2
|  | U | 300x4, 240x1

Airmass covers range of 1.2 to 1.3 for Landolt (1992) standards. 9 standard
stars of Landolt (1992) is taken from the observed field of SA 104. The mean
FWHM varies about 2 arcsec over the entire obsation period.

###  2.2 Photometric and Astrometric Calibration

Photometry of bias-subtracted and flat field corrected CCD frames is carried
out using DAOPHOT-ii software(Stetson, 1987, 2000). Quantitative values for
the brightness of stars are obtained by aperture photometry and profile
fitting photometry.

Measurements of bright stars which are saturated in deep exposure frames have
been taken from short-exposure frames. For translating the observed aperture
magnitudes to the standard magnitudes, least-square linear regressions
outlined by Stetson (2000), are fitted. The instrumental magnitude calibration
equations are as follows:
$v=V+4.22\pm 0.01-(0.04\pm 0.01)(B-V)+(0.21\pm 0.01)X\\\ $ $b=B+4.65\pm
0.01-(0.06\pm 0.01)(B-V)+(0.33\pm 0.01)X\\\ $ $i=I_{c}+4.67\pm 0.01-(0.02\pm
0.01)(V-I_{c})+(0.09\pm 0.01)X\\\ $ $r=R_{c}+4.13\pm 0.01-(0.04\pm
0.01)(V-R_{c})+(0.16\pm 0.01)X\\\ $ $u=U+6.86\pm 0.01-(0.03\pm
0.01)(U-B)+(0.61\pm 0.01)X\\\ $ here V, B, $R_{c},I_{c}$ and U are standard
magnitudes and v, b, i, r, u represent instrumental aperture magnitudes
normalized for 1 sec of exposure time and X is the airmass. Second order color
correction terms are ignored as they are small in comparision to the
photometric data reduction error. Second order color correction terms are
small in comparison to the other photometric data reduction error. Hence, they
are ignored. Zero-points for local standards are evaluated accounting the
aperture growth curve, difference in exposure times and atmospheric
extinction. Errors in zero-points and color coefficients are $\sim$ 0.01 mag.
A final catalogue of stellar objects identified in atleast two filters were
obtained with shape-defining parameters 0.2 $\leq$ sharpness $<$ 1.0 and
goodness of fit estimator $\chi$ $<$ 5\. The accuracy of the morphology and
position of the cluster features in CMDs depends on the variation of
photometric errors with magnitudes. Photometric errors are approximated to be
related with the signal-to-noise ratio of the source.

The internal errors as derived from $DAOPHOT$ in $U$, $B$, $V$, $R_{c}$ and
$I_{c}$ are plotted against $V$ mag in Figure 1. This shows that the
photometric error is less than 0.01 mag in $B$ up to $V$ $\sim$17 mag; less
than 0.01 mag in $V$, $R$ and $I$ up to $V$ $\sim$18 mag; and for $U$-band it
is less than 0.01 mag up to $V$ $\sim$16 mag.

Figure 1: Left panel shows photometric errors corresponding to the brightness
measurement at $U$, $B$, $V$, $R_{c}$, and $I_{c}$ against $V$-band
brightness. Errors on the y-axis represent the internal error as estimated by
$DAOPHOT$ routine. Right panel shows the plot of proper motions in both $RA$
and $Declination$ directions and their errors versus $G$ magnitude. Blue
symbol represent stars having membership probability $>$ 70$\%$.

Astrometric solutions are used to get celestial coordinates of all the stars
in J2000.0. We use the $CCMAP$ and $CCTRAN$ tasks of $IRAF$ to obtain the
celestial coordinates of the detected stars. The sources detected in different
wavebands are band merged using a matching radius of $\sim$ 2 arcsec. In
total, 5913 stars are detected in all five bands. To determine the
completeness of our $UBVR_{C}I_{C}$ data, we used the magnitude histograms
(Panwar $et~{}al$., 2017). We found that completeness of the data in $V$, $B$,
$I_{c}$, $R_{C}$ and $U$ band is limited to about 20 mag, 20.5 mag, 18 mag, 19
mag and 20 mag, respectively.

###  2.3 Kinematic data from Gaia EDR3

The Early Gaia Data Release 3 (Gaia Collaboration, 2020) came to public in the
year 2020. This data consists of five parametric astrometric solutions which
include positions on the sky ($\alpha$, $\delta$) parallaxes and proper motion
(PM) ($\mu_{\alpha}$cos $\delta$, $\mu\delta$) with a limiting magnitude of G
$\sim$21 mag. The Gaia EDR3 data are comparatively more accurate than the
Second Gaia Data Release (Gaia Collaboration $et~{}al$., 2018).

For analysis we have cross matched optical CCD data with Gaia EDR3 (Gaia
Collaboration, 2020). Parallax uncertainties are in the range up to 0.02-0.03
milliarcsec (mas) for sources at G $\leq$ 15 mag and 0.07 mas for sources at G
$\sim$ 17 mag. The uncertainties in the respective proper motion components
are upto 0.01 - 0.02 mas $yr^{-1}$ for G $\leq$ 15 mag, 0.05 mas $yr^{-1}$ for
$\sim$ 20 mag. Proper motion and their corresponding errors are plotted
against Gmag in right panel of Figure 1.

##  3 Data Analysis & Results

###  3.1 Cluster Center and Radial Density Profile

To obtain the cluster parameters, first we estimate the cluster center using
the $Python$ Kernel density estimation (KDE). For this, we use the Gaia EDR3
data for the sources with G-band magnitude uncertainty $\leq$0.1 mag within 40
arcmin radius around the cluster center as mentioned in the $WEBDA$. Our
analysis shows the cluster center at $\alpha$, $\delta$ $\sim$ 02:42:43.9,
+61:39:34 (about 2 arcmin away from the center mentioned in $WEBDA$). Cluster
radius ($r_{cl}$) is taken as the distance from cluster center to where the
average cluster contribution becomes negligible with respect to the background
stellar field.

Cluster radius as well as extent of field-star contamination is estimated
using spatial surface density profile of stars. For quantitative estimation of
the cluster size we have taken the concentric circular regions around the
estimated cluster center. Star counts within each annular ring of width about
36 arcsec is divided by the respective areas of the annular rings to obtain
the number density. To estimate the structural parameters, King (1962) surface
density profile is fitted to the radial distribution of stars. The fit is done
using a non-linear least squares fit routine which uses the errors as weight.
The radial density profile can be represented by :

$f(r)=\frac{f_{0}}{1+(\frac{r}{r_{c}})^{2}}+f_{b}$ (1)

where f0 is the central density region, rc is the core radius of the cluster
and fb is the background density.

The best fit solution along with uncertainties for density distribution is
shown in Figure 2. The radial density profile of the cluster NGC 1027
decreases and flattens around 8 arcmin. Therefore, for further analyses we
have considered the cluster radius as 8 arcmin, which is lower than 10.3
arcmin, the value reported before by Maciejewski & Niedzielski (2007). For NGC
1027, f0 = 2.2 stars/arcmin-2, rc = 2.4 arcmin and fb = 2.3 stars/arcmin-2.

###  3.2 Kinematic members of the cluster NGC 1027 with Gaia EDR3

We use Gaia EDR3 data for the sources within the cluster region to select the
cluster members and to determine mean proper motion of cluster NGC 1027
(radius $\leq$ 8 arcmin). We use the proper motion in RA
($\mu_{\alpha}^{\star}$) and proper motion in declination ($\mu$ δ) for the
stars within the cluster radius to generate the vector point diagram (VPD),
where $\mu_{\alpha}^{\star}$ $\equiv$ ${\mu}_{\alpha}\cos(\delta)$. We
consider only those sources with proper motion uncertainty less than 0.5 mas
yr-1 and G-band magnitude uncertainty less than 0.1 mag. The VPD for those
sources is shown in Figure 3. The over-density of sources can be easily
noticed in Figure 3. The proper motion of the stars in the cluster region
peaks at $(\mu_{\alpha}^{\star}$, $\mu_{\delta}$) $\sim$(-0.84, 2.04) mas
yr-1.

Figure 2: Left panel: The color maps are two-dimensional kernel density
estimations (KDEs) of the spatial distribution of the bright GAIA EDR3 sources
(G $\leq$ 17 mag). Right panel: The radial density profile for the bright
stars in the region. Dashed curve represents the least-square fit of the King
(1962) profile to the observed data points. The error bars represent $\pm$
$\sqrt{N}$ errors. The black line indicates the density of field stars.

A circular area of radius 0.7 mas yr-1 around the peak of the over-density in
the VPD is used to select the probable cluster members, and the remaining
sources in the VPD are considered as field stars. To determine the membership
probability of the stars in the cluster region, we use the approach discussed
in Pandey $et~{}al$. (2020). Assuming a distance of $\sim$ 1.14 kpc (see Sec.
3.4) and a radial velocity dispersion of 1 km s-1 for open clusters (Girard
$et~{}al$., 1989), a dispersion ($\sigma_{c}$) of $\sim$ 0.19 mas yr-1 in the
PMs of the cluster can be obtained. We obtain $\mu$xf = -0.08 mas yr-1,
$\mu$yf = -0.33 mas yr-1, $\sigma$xf = 1.05 mas yr-1 and $\sigma$yf = 2.02 mas
yr-1 for the probable field members. These values are further used to
construct the frequency distributions of the cluster stars
(${\phi}_{c}^{\nu}$) and field stars (${\phi}_{f}^{\nu}$) by using the
equation given in Yadav $et~{}al$. (2013) and then the value of membership
probability for the ith star is calculated using the equation given below:

$P_{\mu}(i)=\frac{{n_{c}\times\phi^{\nu}_{c}(i)}}{n_{c}\times\phi^{\nu}_{c}(i)+n_{f}\times\phi^{\nu}_{f}(i)}$
(2)

where nc and nf are the normalized number of probable cluster members and
field members, respectively. In Figure 4 (upper panel), we have plotted the
estimated membership probability for all the Gaia sources within the cluster
radius as a function of G-band magnitude as blue dots. Gaia sources with high
membership probability (P${}_{\mu}>$ 70%) are shown with red circles. 217
stars has been found with (P${}_{\mu}>$ 70%). There is a clear separation
between the cluster members and field stars toward the brighter part,
supporting the effectiveness of this technique. A high membership probability
extends down to G $\sim$ 19 mag, whereas toward the fainter limits, the
probability gradually decreases. A majority of the stars with high membership
probability follow a tight distribution in the VPD.

Figure 3: Proper motion vector-point diagram for the Gaia sources located
within the cluster radius. Blue color bar shows the distribution of the proper
motions for the stars in the control field of the similar area. Figure 4: Top
Panel: Membership probability for the stars located within the cluster area
(radius $\sim$ 8′) plotted as a function of G-band magnitude. Bottom panel:
The blue dots show the distribution of parallax values for the Gaia sources
within the cluster as a function of G magnitude. The error bars show
respective uncertainties in the parallax values. The red circles represent the
probable cluster members (P${}_{\mu}>$ 70%). The green squares represent the
probable cluster members with good parallax estimates ($\varpi$/$\sigma$ ϖ $>$
5).

We have also plotted the parallax for all the Gaia sources as a function of
G-band magnitude (dots in Figure 4, bottom panel). The respective
uncertainties in the parallax values are shown with the error bars. The red
circles represent sources having membership probability P${}_{\mu}>$ 70%. We
estimate the cluster distance using the parallax values of the cluster members
having high membership probability (P${}_{\mu}>$ 70%) and good parallax
accuracy ($\varpi$/$\sigma_{\varpi}$ $>$5). These sources are shown with the
red circles in Figure 4 (bottom panel). The median parallax value of these
sources is 0.892 $\pm$ 0.088 mas. We estimate the cluster distance after
correcting the median parallax value for the known parallax offset of $\sim$
-0.015 (Stassun & Torres, 2021). The distance estimate for the cluster using
Gaia data comes out to be $\sim$ 1.14 $\pm$ 0.11 kpc, which is in agreement
with that reported by (Cantat-Gaudin & Anders, 2020)). Using a matching radius
of $\sim$ 1 arcsec, we get $\sim$ 1610 optical $UBVR_{c}I_{c}$ counterparts of
Gaia sources with membership probability information within the cluster
radius. Out of these, about 160 stars are cluster members.

###  3.3 Reddening toward the NGC 1027: (U-B)/(B-V) Color-Color Diagram

Figure 5: Two color diagram for NGC 1027. Red color circle denotes member
stars present within 8 arcmin radius

Knowledge of reddening is essential for estimation of distance and age of the
cluster. The reddening, E($B$-$V$) in the cluster region is determined using
the two color diagram (TCD) ($U$-$B$) versus ($B$-$V$) for the member stars
within 8 arcmin (Figure 5). To avoid large uncertainties we have skipped stars
having photometric error larger than 0.1 mag in $UBV$ band. We adopt the slope
of the reddening vector E($U$-$B$)/E($B$-$V$) = 0.72 and fit the intrinsic
zero-age-main sequence (ZAMS) with the solar metallicity (Schmidt-Kaler, 1982)
for main-sequence stars by shifting the ZAMS for different color excesses. The
best fit corresponding to E($B$-$V$)=0.36 mag is shown with the thick curve in
Figure 5. The dashed curve shows the ZAMS shifted for a reddening E($B$-$V$)
of 0.7 mag. Our estimated value of color excess E($B-V$) = 0.36 $\pm$ 0.04 mag
is comparable to E($B-V$) = 0.41$\pm$ 0.12 mag estimated by Maciejewski &
Niedzielski (2007). Line-of-sight extinction, AG, and the reddening,
E($G_{BP}-G_{RP}$), are estimated using the relations given in (Wang & Chen,
2019). We use $A_{G}$ = 1.21 and E($G_{BP}-G_{RP}$) = 0.35 for the further
analyses of Gaia EDR3 data.

###  3.4 Age and distance of the cluster

The CMDs are important tools to estimate the cluster’s age and distance. We
have constructed the optical CMDs using cluster members from our optical
catalogue and Gaia data. $V$ vs ($B-V$) and $V$ vs ($V-I$) color-magnitude
diagrams for the sources in the cluster region (black dots) along with the
member stars (red circles) shown in the Figure 6. $G_{BP}-G_{RP}$ 7 is plotted
using the cluster members with membership probabilities Pμ $>$ 70%. The CMDs
of target cluster show a well-defined narrow main sequence (MS) that extends
from $V\sim$10.5 mag to $V\sim$20 mag. It is seen that the field star
contamination becomes more evident for stars $V$ $\geq$ 17 mag.

Distance and age estimation of the cluster have been derived by using MS
isochrone for solar metallicity taken from Padova PARSEC version 1.2S database
of stellar evolutionary tracks & isochrones of (Pastorelli $et~{}al$., 2020).
This is corrected for the derived reddening values from the TCD. We tried to
fit the MS isochrones of different ages over the member stars (shown with the
red circles in Figure 6 and 7). The visual best fit of the MS to the blue part
of the intrinsic CMDs gives distance modulus ($V$-MV). The best fit isochrone
yield a distance modulus ($V$-MV) = 10.28 mag indicating a distance of $\sim$
1.14 Kpc for the cluster and an log age of 8.11 ($\sim$ 130 Myr) corrected for
the adopted distance and reddening. Our estimated distance for the target
field is comparable to the value of 1.03 Kpc whereas the age estimation is
lower than the value 252 Myr derived by (Maciejewski & Niedzielski, 2007).

Figure 6: $V$/$(B-V)$ (left panel) and $V$/($V$-$I$) (right panel) color
magnitude diagrams for the stars in the cluster NGC 1027 (radius $\leq$ 8
arcmin). The red circles represent high membership probability members from
Gaia EDR3. Figure 7: $G$/($G_{BP}$-$G_{RP}$) color-magnitude diagram for the
stars in the cluster region. The red circles represent Gaia sources with Pμ
$>$ 70\. The thick curve represent the main sequence isochrone (Padova-Parsec)
for log age of 8.11 corrected for the adopted distance and reddening.

###  3.5 Luminosity Function and Mass function

Stellar mass cannot be measured directly. Stellar models are required to
transform magnitude of a star in a given wavelength into masses. With the help
of CMDs, we derive luminosity function (LF) of probable main sequence cluster
members and MF using mass-luminosity relation from theoretical evolutionary
models. We have used Gaia EDR3 high membership probability members to
determine LFs for the cluster. We have used $G$ versus ($G_{B_{P}}-G_{R_{P}}$)
CMD to construct the LF. Field star contamination and membership uncertainty
affects luminosity and mass function of open star clusters (Scalo, 1998),
therefore, to avoid the contribution of the field stars in our sample of
cluster members, we have used only kinematic and photometric confirmed members
of the cluster for the LF and MF estimation. To determine the completeness of
the data, we constructed the magnitude histogram method (Panwar $et~{}al$.,
2017). Using the magnitude histogram, we found that completeness of the data
limits to $G$ $\sim$ 18.5 mag. For estimation of LF, magnitude of the MS stars
were converted to absolute magnitudes using estimated distance modulus and
reddening. Conversion of LF into present day MF is done using the latest
version of the theoretical isochrones of Padova’s stellar evolutionary tracks
and isochrones (Pastorelli $et~{}al$., 2020). To construct MF, we have
converted absolute mag bins to mass bins. Consequently, we transform the LF to
MF for all the main-sequence stars ($\sim$ 180 stars) having mass 0.55 $\leq$
M/M⊙ $\leq$ 2.1. MF slope is given by using a power-law

$\Gamma=d\log N(\log m)/d\log m$ (3)

where $N(\log m)$ is the number of stars per unit logarithmic mass interval.
dN is the probable cluster members in a mass bin dM with central mass M.
Observational results suggest that the MF slopes is similar to that given by
Salpeter (1955), that is, $\Gamma$ = -1.35. The resulting mass function for
our target cluster is shown in Figure 8. For the cluster NGC 1027, we obtain
that the mass distribution has a best-fitting slope $\Gamma$ = -1.46 $\pm$
0.15, similar to the Salpeter value.

Figure 8: Mass functions of stars using probable member main-sequence stars.
The error bars represent 1/$\sqrt{N}$

Figure 9: Left panel: Normalized cumulative radial distribution of the member
stars in the cluster region. Right panel: DSS2-R image of the open cluster NGC
1027. The red and blue contours show the density distribution of low-mass ($<$
1.2 M⊙) and relatively higher-mass ($\geq$ 1.2 M⊙) members, respectively. The
green circle shows the cluster radius and magenta dashed circle represents the
area (radius $\sim$ 40 arcmin), used for the search of kinematic members
outside cluster radius.

##  4 Discussion

Observational investigation of the OCs reveal that in general, massive stars
are concentrated toward the nucleus in comparison to low-mass stars which are
mostly distributed in the corona region of the cluster (Soderblom $et~{}al$.,
1998; Sharma $et~{}al$., 2000).

This segregation of massive stars towards cluster’s center can happen due to
the equipartition of energy (dynamical evolution) or due to the star formation
processes itself.

We have analysed the radial distribution of probable member stars (with
membership probability $>$ 70 %) in two mass bins, i.e., stars with masses
$<$1.2 M⊙ and $\geq$1.2 M⊙, to look for mass segregation effect in NGC 1027
(cf. Figure 9 (left panel)). The distribution clearly demonstrates that stars
in the mass bin with massive stars ( $\geq$1.2 M⊙) are concentrated towards
the cluster center compared to low mass stars (mass bin $<$ 1.2 M⊙),
suggesting mass segregation effect in the NGC 1027 cluster. To investigate
whether the mass segregation in this cluster is due to star formation
processes or due to dynamical relaxation, we estimated the dynamical
relaxation time ($T_{E}$), i.e., time in which the individual star exchange
sufficient energy to achieve the velocity distribution of a Maxwellian
equilibrium. We have used the method given by Binney & Tremaine (1987) to
estimate the dynamical relaxation time ($T_{E}$), i.e.,

$T_{E}=\frac{N}{8logN}\times T_{cross}$ (4)

where, $T_{cross}$ = D/$\sigma_{V}$ is the crossing time, N is the total
number of stars in the cluster region of diameter $D$, and $\sigma_{V}$ is the
velocity dispersion, with a typical value of 3 km s-1 (Sharma $et~{}al$.,
2020; Bisht $et~{}al$., 2017). Using the above equation, we estimated the
value of $T_{E}$ as $\sim$10 Myr for the NGC 1027 cluster. Comparing the
cluster age ($\sim$130 Myr) with its dynamical relaxation time ($\sim$10 Myr),
we found that the cluster is dynamically relaxed and the observed mass
segregation is due to the result of dynamical evolution.

Due to the mass segregation process, low-mass members may become more
vulnerable for evaporation and hence ejected out of the cluster more easily
[e.g., see(Mathieu, 1984)]. This stellar evaporation results in a continuous
decrease of the total mass of the cluster, and hence the gravitational binding
energy of the cluster. However, external disturbances, e.g., the tidal force
from nearby giant molecular clouds or star clusters, passages through Galactic
spiral arms or disks, or a shear force by Galactic differential rotation,
increase the stellar evaporation of the cluster. For a cluster, the tidal
radius is the defined as a distance at which two gravitational forces; one
towards the Galactic center and another towards the cluster center; balances.
Thus, tidal radius is estimated to separate gravitationally bound and unbound
stars to a cluster. The tidal radius of a star cluster in the solar
neighborhood is given by the equation:

$r_{t}=\left[{\frac{GM_{C}}{2(A-B)^{2}}}\right]^{1/3}$ (5)

where, $G$ is the gravitational constant, MC is the total mass of the cluster,
and A, B are the Oort constants where A = 15.3 $\pm$ 0.4 km s-1 kpc-1, B =
-11.9 $\pm$ 0.4 km s-1 kpc-1 (Bovy, 2017). For the total cloud mass of
$\sim$164 M⊙ estimated from the MF distribution, the corresponding tidal
radius comes out to be $\sim$7.8 pc. If we assume that $\sim$ 50% of the
cluster mass is missed due to data incompleteness, the tidal radius will be
$\sim$9 pc. Therefore, the tidal radius of the cluster is much larger than the
present estimate of the cluster radius (Rcl $\sim$2.65 pc), or in other words,
this cluster is already started to loose its low-mass members due to dynamics.

Thus, there is a possibility of finding member stars beyond the observed
cluster boundary. For this, we searched for the stars sharing the similar
proper motion and distance as cluster members, outside the observed cluster
boundary.

We estimated the membership probability and found $\sim$460 stars with
membership probability $>$ 70% upto a radius of 40 arcmin (cf. Figure 9, right
panel).

To study the cluster dynamics and mass segregation, we divided these stars
into two mass bins, stars having mass less than 1.2M⊙ and stars having mass
greater than 1.2M⊙, and created the respective stellar density distribution
maps. The stellar density distributions for above two mass bins are over-
plotted on a DSS2-R band image shown in the Figure 9 (right panel).

There is a clear evidence of mass segregation as the stars having mass greater
than 1.2M⊙ are concentrated toward the center compared to the relatively low-
mass stars.

The spatial distribution of the low mass members in the cluster NGC 1027 show
slightly tailed structure (cf. Figure 9, right panel). This might have caused
due to the tidal striping of member stars from the cluster. Tidal striping of
stars can happen in a cluster due to the tidal forces from a nearby massive
object, disk crossing, or differential rotation of the Galaxy. Thus, the tidal
stripping depends on the mass and age of a cluster. The ratio of the number of
cluster inside the tidal radius to those outside, can tell us about the amount
of tidal stripping already done in a cluster. For example, for Blanco 1 (age
$\sim$ 100 Myr) cluster, it is found to be $\sim$3 (Zhang $et~{}al$., 2020),
whereas, for Coma Berenicesa cluster (age $\sim$ 700 Myr) this ratio is
estimated as 0.6 (Tang $et~{}al$., 2019). However, in case of an old massive
star cluster Hyades (age $\sim$ 800 Myr), this ratio is estimated as $\sim$1.1
(Röser $et~{}al$., 2019). NGC 1027 is relatively young and appears to be more
dynamically bound. The observed radius of the cluster suggest that it is a
dynamically relaxed cluster that show mass-segregation and has already lost
some of its member stars and currently it is in the process of dissolution due
to the influence of external tidal interactions.

##  5 Summary and conclusion

We present the photmetric and kinematic study of the intermediate age open
cluster NGC 1027 using $UBVRI_{c}$ data taken with 1.04m Sampunanand telescope
and Gaia EDR3 data. The results of the study are summarize as follows:

  1. 1.

Using two-dimensional KDEs analysis of the Gaia EDR3 data, we have obtained
the cluster center that is about 2 arcmin away from the center mentioned in
$WEBDA$. The cluster extent is estimated to be about 8.0 arcmin.

  2. 2.

The distribution of proper motion of the stars in the cluster region shows a
clear separation between the cluster members and field stars. The mean proper
motion for the cluster has been estimated to be ($\mu_{\alpha}^{\star}$ ,
$\mu_{\delta}$ )= (- 0.84, 2.04) mas yr-1.

  3. 3.

Based on the membership analysis, 217 member stars are identified
(P${}_{\mu}>$ 70%) in the cluster region. Their mean parallax value is 0.892
$\pm$ 0.088 mas corresponding to a distance of $\sim$ 1.14 kpc.

  4. 4.

The ($U$-$B$)/($B$-$V$) color-color diagram for the cluster members give a
reddening E($B$-$V$) $\sim$ 0.36 $\pm$ 0.04 mag towards the cluster.

  5. 5.

The CMDs for the cluster members, constructed using UBVRI photometric data and
Gaia EDR3 data, show that NGC 1027 is an intermediate age OC with an age
$\sim$ 130 Myr and located at a distance of $\sim$ 1.14 kpc.

  6. 6.

The LFs and MFs are determined by considering the members selected from Gaia
proper motion database and photometry. The mass function slope for the
cluster, $\Gamma$ =-1.46 $\pm$ 0.15, is in good agreement with that given by
Salpeter (1955) for the Galactic OCs.

  7. 7.

We have studied the radial distribution of stellar masses within the cluster
region and found that relatively massive stars are dominantly distributed in
the inner regions of the cluster suggesting mass segregation.

  8. 8.

We report presence of astrometric members of the target field outside the
clusters radius. The observed small size of the cluster as compare to its
tidal radius, indicates mass-segregation, slightly elongated morphology and a
comparison of dynamical relaxation time with the age of cluster. We conclude
that the cluster is under the influence of external tidal interactions.

## Acknowledgement

We are thankful to anonymous referee for providing the comments that
significantly improve the manuscript. NP acknowledges the financial support
received through the SERB CRG/2021/005876 grant. This research has made use of
the WEBDA database, operated at the Department of Theoretical Physics and
Astrophysics of the Masaryk University. This work has made use of data from
the European Space Agency (ESA) mission Gaia https://www.cosmos.esa.int/gaia),
processed by the Gaia. Data Processing and Analysis Consortium. AT also like
to acknowledge Prof. U S Pandey for his scientific suggestions and
corrections.

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