Çakmak et al

*Hikmet Çakmak

# Astrophysical Parameters and Dynamical Evolution of Open Clusters: NGC 2587,
Col 268, Mel 72, Pismis 7

Hikmet Çakmak  Orhan Güneş  Yüksel Karataş  Charles Bonatto  Department of
Astronomy and Space Sciences, Istanbul University, Science Faculty, 34119,
Üniversite-Istanbul, Turkey  Istanbul Medeniyet University, Faculty of Arts
and Humanities, Department of History of Science, Kadikoy, Istanbul, Turkey
Universidade Federal do Rio Grande do Sul, Departamento de Astronomia, CP
15051, RS, Porto Alegre 91501-970, Brazil<EMAIL_ADDRESS>Çakmak H
Güneş O  Karataş Y  Charles Bonatto

(XXX; XXX; XXX)

###### Abstract

We determined astrophysical and dynamical parameters of the open clusters
(OCs) NGC 2587, Collinder 268 (Col 268), Melotte 72 (Mel 72), and Pismis 7
from Gaia DR2 photometric/astrometric data and a new technique, fitCMD.

fitCMD provides (Z, Age(Gyr)) as (0.025, 0.45) for NGC 2587, (0.0025, 0.5) for
Col. 268, (0.011, 1.25) for Mel 72, and (0.008, 1.00) for Pismis 7,
respectively. As compared to Gaia DR2 distances , the obtained photometric
distances from fitCMD provide somewhat close distances. For NGC 2587 and Mel.
72, both distances are in good concordance.

Except for NGC 2587, the ages of the remaining OCs are higher than their
relaxation times, which suggests that they are dynamically relaxed. NGC 2587
did not undergo dynamical evolution. Mel 72 and Pismis 7 with relatively flat
MF slopes indicate signs of a somewhat advanced dynamical evolution, in the
sense that they appear to have lost a significant fraction of their low-mass
stars to the field. Pismis 7’s negative/flat MFs indicates that its high mass
stars slightly outnumber its low mass ones. Given its mild dynamical
evolution, the high mass stars move towards the central region, while low-mass
stars are continually being lost to the field. Col 268 presents small
dimensions which suggest a primordial origin. The outer parts of Mel 72 and
Pismis 7 - with large cluster radii expand with time, while Mel 72’s core
contracts because of dynamical relaxation (Figs. 10e–f).

###### keywords:

open clusters and associations: individual NGC 2587, open clusters and
associations: individual Col 268, open clusters and associations: individual
Mel 72, open clusters and associations: individual Pismis 7

††articletype: Article Type

, , , and (XXXX), Revealing Dynamical Properties of Open Clusters: NGC 2587,
Col 268, Mel 72, Pismis 7, AN, XXX;XX:X–X.

##  1 Introduction

Astrophysical parameters (reddening, distance, age), structural parameters
(core and cluster radii, $R_{core}$ and $R_{RDP}$), and overall masses, mass
function slopes ($\chi$, MF slopes), relaxation times $(t_{rlx})$ and
evolutionary parameters $(\tau)$ of the open clusters (OCs) are needed for the
interpretation of the dynamical evolution. The stars inside the OCs undergo
internal and external perturbations such as stellar evolution, mass
segregation, and encounters with the disk and Giant Molecular Clouds (GMCs)
(Lamers & Gieles, 2006), (Gieles et al., 2007). All these process produce a
varying degree of mass loss that may lead to the cluster dissolution into the
field. Mass segregation preferentially drives the low mass stars to the outer
parts of the clusters.

Our sample OCs, NGC 2587, Col. 268, Mel. 72, and Pismis 7 were studied in
2MASS $JHK_{s}$ by Bukowiecki et al. (2011, hereafter, B11) and Kharchenko et
al. (2013, hereafter, K13). Tadross (2011) published the astrophysical
parameters of NGC 2587 in 2MASS. The astrophysical parameters of Mel 72 were
derived in $UBVI$ photometry by Hasegawa et al. (2008) and Piatti et al.
(2009). B11 published their astrophysical/structural parameters, and their
masses/MF slopes, except for Pismis 7. K13 also gave their astrophysical
parameters together with their core radii (Tables 6–7).

The likely members of four OCs have been determined from Gaia DR2
astrometric/photometric data (Brown et al., 2018; Lindegren et al., 2018). To
study the issues above, the well-determined cluster parameters (heavy element
abundance, colour excess, distance modulus/distance, age and mass) are needed.
For this, we employ an approach fitCMD, designed by Bonatto (2019). fitCMD
uses the isochrones of Bressan et al. (2012) (hereafter B12) and $G$,
$G_{BP}$, $G_{RP}$ filters.

This paper is organized as follows. The cluster membership technique is
presented in Section 2. The derivation of reddenings, distance
moduli/distances, ages of four OCs from fitCMD algorithm is explained in
Section 3. The obtained cluster dimensions, masses/mass function slopes,
relaxation times/ evolutionary parameters of four OCs are given in Sections
4–5. A Discussion/Conclusion is presented in Section 6 together with the sub-
sections; a comparison with the literature and the dynamical evolution.

Table 1: Equatorial and Galactic coordinates of four OCs. Cluster | $\alpha(2000)$ | $\delta(2000)$ | $\ell$ | $b$
---|---|---|---|---
| (h m s) | $(^{\circ}\,{}^{\prime}\,{}^{\prime\prime})$ | $(^{\circ})$ | $(^{\circ})$
NGC 2587 | 08 23 22.9 | -29 31 02.1 | 249.46 | 4.46
Col 268 | 13 18 11.4 | -67 05 00.0 | 305.54 | -4.35
Mel 72 | 07 38 31.3 | -10 41 51.4 | 227.84 | 5.38
Pismis 7 | 08 41 08.0 | -38 42 08.6 | 259.05 | 1.99
Table 2: The median proper motion components, proper motion radii and parallaxes/distances of the likely members of four OCs for this paper (top rows) and Cantat-Gaudin et al. (2020) (bottom rows). Cluster |  $\mu_{\alpha}$ (mas/yr) |  $\mu_{\delta}$ (mas/yr) |  $\Delta r$ (mas/yr) |  $\varpi$ (mas) | $d(kpc)$
---|---|---|---|---|---
NGC 2587 |  -4.27$\pm$0.08 |  3.57$\pm$0.08 | 0.21 |  0.31$\pm$0.03 |  3.25$\pm$0.08
Col 268 |  -5.48$\pm$0.08 |  -0.40$\pm$0.07 | 0.19 |  0.37$\pm$0.06 |  2.67$\pm$0.13
Mel 72 |  -4.18$\pm$0.08 |  3.66$\pm$0.07 | 0.21 |  0.38$\pm$0.05 |  2.61$\pm$0.08
Pismis 7 |  -3.31$\pm$0.09 |  2.77$\pm$0.10 | 0.15 |  0.19$\pm$0.04 |  4.82$\pm$0.35
NGC 2587 |  -4.26$\pm$0.09 |  3.59$\pm$0.11 |  |  0.30$\pm$0.05 |  3.39$\pm$0.59
Col 268 |  -5.48$\pm$0.09 |  -0.40$\pm$0.08 |  |  0.36$\pm$0.04 |  2.80$\pm$0.30
Mel 72 |  -4.16$\pm$0.11 |  3.68$\pm$0.09 |  |  0.37$\pm$0.06 |  2.71$\pm$0.40
Pismis 7 |  -3.31$\pm$0.09 |  2.79$\pm$0.13 |  |  0.15$\pm$0.06 |  6.67$\pm$2.44

Figure 1: The $\mu_{\alpha}$ versus $\mu_{\delta}$, parallax histogram plus
Gaia CMD for NGC 2587 and Col 268. The red circles indicate the cluster
region. The parallax histograms of the potential members inside the red
circles are shown in the bottom left. The cluster stars inside
$\varpi\pm\sigma_{\varpi}$ mas (the vertical/horizontal lines on the parallax
histograms) are our likely members. A single stellar cluster sequence of the
probable members (filled dots of bottom right panels) is clearly seen on the
$(G,G_{BP}-G_{RP})$ CMDs.

Figure 2: The $\mu_{\alpha}$ versus $\mu_{\delta}$, parallax histogram plus
Gaia CMD for Mel 72 and Pismis 7. The symbols and their meanings are the same
as Fig.1.

##  2 Separation of the Cluster Members

In order to separate the cluster members of NGC 2587, Col 268, Mel 72 and
Pismis 7, we have obtained their Gaia DR2 astrometric/photometric data (Brown
et al., 2018) for a large-area (15 arcmin) from
VizieR111http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/246.. Their
cluster stars are shown on the ($\mu_{\alpha}$, $\mu_{\delta}$) diagrams
(hereafter, VPD) (Figs. 1–2). The potential cluster members show a more
concentrated structure, whereas field stars have a more scattered
distribution. The eye-fitted proper motion radii (red circles) represent the
cluster region. They have been constructed via the mathematical equations,
$x=x_{0}+r\cos(\theta)$ and $y=y_{0}+r\sin(\theta)$. Here, (x0, y0) are the
median values of ($\mu_{\alpha}$, $\mu_{\delta}$) (mas yr-1), the radius
$r=\sqrt{\sigma_{\mu\alpha}^{2}+\sigma_{\mu\delta}^{2}}$ mas yr-1, and
$\theta$ =$0^{\circ}$ to $360^{\circ}$. These propor motion radii are listed
in Col. 4 of Table 2, and they are a good compromise for the likely cluster
members. The median values of the proper motion histograms on top/right of the
VPDs (vertical/horizontal blue lines) make it easier to determine the proper
motion value of the cluster’s center. The median values
($\varpi\pm\sigma_{\varpi}~{}mas$) of the parallax histograms (bottom-left
panels) for the cluster stars inside the red circles classify the probable
members. By applying Gaussian Mixture Model (GMM) Pedregosa et al. (2011) to
the probable members inside $\varpi\pm\sigma_{\varpi}$ mas, their membership
probabilities, P($\%$) have also been determined in order to check. They have
greater than 80$\%$ (vertical dashed lines of Fig. 3). The GMM 222$P$ is
defined $\Phi_{c}$ /$\Phi$. Here $\Phi=\Phi_{c}+\Phi_{f}$ is the total
probability distribution. c and f subscripts for cluster and field parameters,
respectively. The used parameters for estimation of $\Phi_{c}$ and $\Phi_{f}$
are $\mu_{\alpha}$, $\mu_{\delta}$, $\varpi$, $\sigma_{\mu\alpha}$,
$\sigma_{\mu\delta}$, $\sigma_{\varpi}$. model considers that the distribution
of proper motions of the stars in a cluster’s region can be represented by two
elliptical bivariate Gaussians. The expressions used can be found in the
papers of Balaguer-Nunez et al. (1998), Wu et al. (2002), Sariya et al. (2012)
and Dias et al. (2018). These probable members provide good single stellar
sequence on the Gaia DR2 $(G,G_{BP}-G_{RP})$ colour–magnitude diagrams (CMDs)
(bottom-right panels of Figs. 1–2), and thus they are considered for
determining the astrophysical parameters. Note that the scatter of the members
on the $(G,G_{BP}-G_{RP})$ of Pismis 7 is similiar to Fig. 2 of Ahumada
(2005). The estimated median equatorial coordinates from the likely members of
our OCs are listed in Table 1. These coordinates are similar to those of WEBDA
(Mermilliod, 1992).

Instead of the inverse of Gaia-DR2 parallaxes, the distances of four OCs are
obtained from the posterior probability density functions
(PDFs)333https://www2.mpia-hd.mpg.de/ calj/gedr3-distances/main.html given by
Bailer-Jones et al. (2018) and Bailer-Jones et al. (2021). For this, we use
the global zero point of $-$0.017 mas (Lindegren et al., 2021). The median
proper motion components/the proper motion radii and the median
parallaxes/distances of four OCs are listed in Table 2. Within the errors,
their median values are in compatible with those of Cantat-Gaudin et al.
(2018) and Cantat-Gaudin et al. (2020) (bottom panel of Table 2). However,
note that both median parallaxes of Pismis 7 are somewhat different.

##  3 Derivation of Reddening, Distance, Age and Mass

In order to obtain the astrophysical parameters from Gaia DR2
$(G,G_{BP}-G_{RP})$ for the probable members of four OCs, an approach
(fitCMD), improved by Bonatto (2019) is utilised. fitCMD is to transpose
theoretical initial mass function $(IMF)$ properties for the isochrones of
given age and metallicity to their observational CMDs. Based on initial mass
function (IMF) properties of the B12 PARSEC
isochrones444http://stev.oapd.inaf.it/cgi-bin/cmd., fitCMD searches for the
values of cluster stellar mass $(m_{cl})$, $Age$, global metallicity ($Z$),
foreground reddening $E(G_{BP}-G_{RP})$ , distance modulus $(m-M)_{G}$, and
magnitude-dependent photometric completeness that produce the artificial and
observational CMDs. The adopted parameter ranges are presented in Table 3. The
photometric completeness limits which are estimated from Fermi function are
given in row 9 of Table 3.

For the members, a Gaia DR2 CMD with the IMF is built for each isochrone (at
$DM=0$ and no reddening). By CMD cell- containing stars with mass in the range
$(m1,m2)$, the following parameters are obtained, the number of stars per
cluster mass $n_{H}=N_{1,2}/m_{cl}$ and the average mass ($<m>$), which
basically involves integrating the mass function ($\phi(m)=dN/dm$) between
$(m1,m2)$: $N_{1,2}=\int_{m1}^{m2}\phi(m)~{}dm$. Each CMD cell contains the
respective relative density (number per cluster mass) of occurrence of stars,
which is equivalent to the classical Hess diagram,
$H_{M}=H_{M}(m_{cl},Age,Z)$. The parameter search consists on finding the
absolute minimum of the residual hyper-surface $R_{H}$ defined as
$R_{H}=(H_{obs}-H_{sim})^{2}$, where $H_{obs}$ is the observed Hess diagram
and $H_{sim}=H_{sim}(m_{cl},Age,Z,CE,DM,k_{F},m_{TO})$ is the simulated one.
Here, $CE$, $DM$, $k_{F}$, $m_{TO}$ mean the colour excess, distance modulus,
the steepness of the descent, and turn-off magnitude, respectively. The
adopted thresholds/Hess cell numbers/Hess widths for Gaia colours/magnitudes
are listed in rows 7–8 of Table 3. The values for seven parameters at the
absolute minimum are assumed to represent those of the star cluster. In
practical terms, locating the minima of $R_{H}$ is equivalent to finding the
parameters that minimize the quantity $S_{R}$. Here,
$S_{R}=\sum_{mag,col}W(mag)\times\ R_{H}(mag,col)$ on the colour/magnitude
plane. The sum runs over all Hess cells and $W(mag)$ is the statistical weight
of each cell. $W(mag)$ corresponds to the inverse of the observed Hess density
computed at the respective magnitude of each cell, i.e.
$W(mag)=1/\sum_{col}H_{obs}(mag,col)$. Residual minimization between observed
and synthetic CMDs – by means of the global optimization algorithm $SA$ – then
leads to the best-fitting parameters. $SA$ (Simulated Annealing) is an
iterative and statistical technique. For any given step $k$, it randomly
selects a new set of parameters ($M^{k}_{cl}$, $Age^{k}$, $Z^{k}$, $CE^{k}$,
$DM^{k}$, $k^{k}_{F}$, $m^{k}_{TO}$) from the respective ranges. thus, $SA$
concentrates on smaller parameter ranges, centred around the most promising
values, and a new step $(k+1)$ is taken. Iterations stop when $SA$ meets the
convergence criterion: five consecutive repetitions of the same value of
$S_{R}$. However, for the statistical nature of $SA$, fitCMD should be
repeated a few times to minimize the probability of getting stuck into a deep,
but secondary minimum.

Table 3: The parameters which are used for fitCMD. Cluster | NGC 2587 | Col 268 | Mel 72 | Pismis 7
---|---|---|---|---
Cluster (stellar) mass (m⊙) | 100-1000 | 10-10000 | 100-1000 | 10-10000
$(m-M)_{G}$ (mag) | 10.0-15.0 | 10.8-15.1 | 10.0-15.0 | 12.0-17.0
$E(G_{BP}-G_{RP})$ (mag) | 0.00-1.00 | 0.0074-1.76 | 0.00-1.00 | 0.00-2.00
Age (Myr) | 100 - 1000 | 100-1000 | 100-10000 | 120-13500
$Z$ $(Z_{\odot}=+0.0152)$ | 0.0001 - 0.03 | 0.0001-0.03 | 0.0001-0.03 | 0.0001-0.03
$A_{V}$ (mag) | 3.1 | 3.1 | 3.1 | 3.1
$(G_{BP}-G_{RP})$ threshold/Hess Cell/Cell width (mag) | (0.08-1.41)/67/0.020 | (0.37-2.20)/92/0.020 | (0.19-1.64)/73/0.020 | (0.63-2.58)/98/0.020
$G$ threshold/Hess Cell/Cell width (mag) | (11.75-18.06)/64/0.100 | (11.15-17.58)/65/0.100 | (12.31-19.40)/71/0.101 | (12.00-18.83)/69/0.101
Photometric completeness (mag) | $G<17.17$ | $G<16.46$ | $G<17.02$ | $G<18.08$
Usable members | 118 | 128 | 273 | 204

Figure 3: The membership probability histogram P($\%$) of four OCs. The
vertical blue dotted lines

show the likely members which are greater than 80$\%$ inside
$\varpi\pm\sigma_{\varpi}$ mas.

The efficiency of fitCMD for input parameters is tested with simulated CMDs,
built with pre-defined values of seven parameters. Individual stellar masses
are attributed according to Kroupa’s IMF (Kroupa, 2001) \- for masses larger
than $0.1~{}m_{\odot}$ \- until the sum matches $m_{cl}$; the respective
magnitudes are taken from the PARSEC isochrone corresponding to $Age$ and $Z$.
Typical photometric uncertainties are added for each Gaia DR2 filters and
their uncertainties $(\sigma_{k})$. If the age, metallicity, distance modulus
and colour excess are reasonably well determined, the ratio
observed/artificial Hess cells content corresponds to $m_{cl}$.

The obtained best-fitting astrophysical parameters $E(G_{BP}-G_{RP})$,
$E(B-V)$, $Z$, $(V-M_{V})_{0}$, d (pc) and Age (Gyr) are given in Table 6. The
observed $(G,G_{BP}-G_{RP})$ CMDs (Hess diagrams) are shown in Figs. 4–5. The
B12 isochrones (solid red lines) fit well the main sequence (MS), turn$-$off
(TO) and Red Giant/Red Clump (RG/RC) regions.

Figure 4: $(G,G_{BP}-G_{RP})$ CMDs (Hess diagram) for NGC 2587 and Col 268.
The blue dots represent the probable members (the bottom rows of Table 3).
Solid black lines denote the B12 isochrones. The coloured shaded regions show
the relative stellar densities (number per cluster mass) within CMD cells.

Figure 5: $(G,G_{BP}-G_{RP})$ CMDs (Hess diagram) for Mel 72 and Pismis 7. The
symbols and their meanings are the same as Fig. 4.

Figure 6: Stellar RDPs (filled dots) of four OCs. Solid line shows the best-
fit King profile. Horizontal bar: stellar background level measured in the
comparison field. Shaded region: $1\sigma$ King fit uncertainty. Figure 7:
$\phi(m)$ versus $m~{}(m_{\odot})$ for the overall regions of four OCs. Here
$\phi(m)=dN/dm$ (stars $m_{\odot}^{-1}$).

##  4 Dimension and Mass Function

By utilising the stellar radial density profiles (RDPs), we have derived
structural parameters of four OCs. The fitted RDPs of four OCs to the relation
of King (1966) have been displayed in Fig. 6. For this, the three-parameter
function, $\sigma(R)=\sigma_{bg}+\sigma_{0}/(1+(R/R_{core})^{2}$, given by
King (1966) is considered. For this equation, $\sigma_{bg}$ is the residual
background density. $\sigma_{0}$ and $R_{core}$ are the central density of
stars and the core radius, respectively. The meanings of solid line,
horizontal bar and the shaded area are mentioned in caption of Fig. 6. We
obtained the cluster radii $(R_{RDP})$ by comparing the RDP level with
background and by measuring the distance from the centre. The $R_{core}$,
$\sigma_{bg}$, and $\sigma_{0}$ of four OCs have been derived from the fitted
King profile to the observational RDPs (Fig. 6). The large uncertainties
within $R<1^{\prime}$ in the RDPs are due to their low star contents in their
central parts. These structural parameters are listed in Cols. 3-10 of Table
7.

From the obtained masses from fitCMD (Sect. 3), the mass function relation of
$\phi(m)(stars~{}m_{\odot}^{-1})$ versus m($m_{\odot}$) is displayed in Fig. 7
for the overall parts of four OCs. Table 4 provides the mass information. The
observed number of stars and the corresponding total mass (obviously, the mass
is computed by the simulation) are listed in Cols. 4-5. The mass range (Col 3)
refers to the mass which is actually present in the observed CMDs, while
$m_{tot}(m_{\odot})$ and star number (Cols. 6-7) refer to fitCMD’s full
simulation (including stars with $>0.08~{}m_{\odot}$). Ideally, the simulated
number of stars should be exactly the same as the observed, but sometimes
there are residual field stars. The full simulation corresponds to the whole
mass range assumed to be present in the cluster. For instance, in the case of
NGC 2587, the observed CMD contains 118 stars with $m_{tot}=180~{}m_{\odot}$
in the mass range: 0.96–3.09 $m_{\odot}$; if we had access to the full range,
the stars would have masses between 0.08 $m_{\odot}$ to 3.09 $m_{\odot}$.
Therefore, fitCMD computes the values for this full range as 1621 stars
storing 681 $m_{\odot}$.

The MF slopes shown in the plots are the observed ones, i.e., those computed
directly from the observed CMDs. The theoretical IMF is used only for the
purpose of estimating the completeness-corrected mass (by estimating the
difference in the number of stars actually detected at a given magnitude with
respect to the expected one). Therefore, all the other parameters are
unaffected by this procedure.

The MFs of NGC 2587 (Fig. 7) presents a break followed by slope flattening for
masses in the range $0.96\leq m_{\odot}\leq 3.09$, which is noticeable
particularly in the overall region. Up to $m\sim 1.5m_{\odot}$, its overall
MFs is very steep negative $(\chi=-2.60\pm 0.46)$. After $m>1.5m_{\odot}$,
this MFs becomes very steep positive $(\chi=+1.98\pm 0.63)$. Since the break
occurs at $m\sim 1.5m_{\odot}$, it is not the same as that implied by Kroupa’s
mass function $(m\sim 0.5m_{\odot})$. This break is due to completeness
affecting masses lower than $m\sim 1.5m_{\odot}$. We use the MFs for
$m>1.5m_{\odot}$ as representative of the overall cluster. This mass range
does not seem to be critically affected by completeness. The overall MF slopes
of Col 268 and Mel 72 are quite positive/flat $(\chi=+0.61\pm 0.23)$ and
$(\chi=+0.54\pm 0.23)$, respectively. Pismis 7’s overall MFs is negative/flat
$(\chi=-0.76\pm 0.25)$.

Table 4: Mass information for the overall regions of four OCs. Their meanings are explained in Sect.4. Cluster | r (pc) |  mass range-$m(m_{\odot})$ | N | $m_{tot}(m_{\odot})$ | N | $m_{tot}(m_{\odot})$
---|---|---|---|---|---|---
NGC 2587 | 0.0–13.7 | 0.96-3.09 | 118 | 180 | 1621 | 681
Col. 268 | 0.0–5.0 | 0.84-2.52 | 128 | 170 | 1901 | 748
Mel. 72 | 0.0–14.9 | 0.66-2.06 | 273 | 310 | 2546 | 950
Pismis. 7 | 0.0–13.7 | 1.06-2.18 | 204 | 230 | 2575 | 973

##  5 Relaxation time and Evolutionary parameter

The relaxation times, $t_{rlx}$ (Myr) of four OCs are obtained from a
relation, $t_{rlx}\approx
0.04\left(\frac{N}{lnN}\right)\left(\frac{R}{1pc}\right)$. Here N is the
number of stars located inside the cluster radius, $R_{RDP}$. As an indicator
of dynamical evolution, the evolutionary parameters are estimated from the
relation of $\tau=Age/t_{rlx}$. The relations of these time scales can be
found in the works of Bonatto et al. (2005), Bonatto & Bica (2006) and Bonatto
& Bica (2007). By adopting $\sigma_{v}\approx 1\,kms^{-1}$ as an upper value
explicitly (Bonatto & Bica, 2011), instead of $\sigma_{v}\approx 3\,kms^{-1}$
of Binney et al. (1998), these time scales are estimated and listed in Table
5.

$t_{rlx}$ gives the time required for the stars in the core/halo to travel
from one end of these regions to the other (Stars move at $\sigma_{v}\approx
1\,kms^{-1}$). Mass segregation (known as migrating from the cores to halo) is
directly related to $t_{rlx}$ and $\tau$. $\tau$ also depends on both age and
$t_{rlx}$. There is a negative relationship between $t_{rlx}$ and $\tau$.
Large $\tau$ and small $t_{rlx}$ correspond to advanced mass segregation,
accordingly, small $\tau$ and large $t_{rlx}$ mean small scale mass
segregation.

From the propagating the errors in Age (Col. 5 of Table 6), radii and N (Col.6
of Table 4) into $t_{rlx}$ and $\tau$, the uncertainties of the evolutionary
parameters $(\tau)$ are estimated. The errors in ages and radii are at a
similar ($\sim 3\%-13\%$) level. If the uncertainties in the number of stars
(N) become to be larger, this is responsible for a large uncertainty in
$t_{rlx}$ (Table 5) and, consequently, a large uncertainty in the evolutionary
parameter. Here, $t_{rlx}$ and $\tau$ are considered simply as an order of
magnitude estimate.

Table 5: Relaxation times and evolutionary parameters of overall regions of four OCs. Cluster | $t_{relax}\,(Myr)$ | $\tau$
---|---|---
NGC 2587 | 1120.7$\pm$814.0 | 0.4$\pm$0.3
Col 268 | 250.0$\pm$142.5 | 2.0$\pm$1.2
Mel 72 | 592.7$\pm$341.0 | 2.1$\pm$1.2
Pismis 7 | 148.8 $\pm$105.0 | 6.7$\pm$4.8
Table 6: Comparison of the astrophysical parameters to the literature. Cols.
1-4 represent the cluster name, reddenings, the metal/heavy element
abundances, true distance moduli $(V-M_{V})_{0}$, and their corresponding
heliocentric distances, respectively. Col. 5 gives the ages
$(\log(A)/A~{}(Gyr))$. The isochrones, photometry and the references are
listed in Cols. 6-8, respectively.

Cluster |  $E(B-V)$ & $E(G_{BP}-G_{RP})$ |  [M/H]$~{}~{}\&~{}~{}$ Z |  $(V-M_{V})_{0}$ & $d$ (pc) | $\log(A)\&A~{}(Gyr)$ | Isochrone . | Phot. | Reference
---|---|---|---|---|---|---|---
NGC 2587 |  $0.11$${}\,/\,{}$ | $0.13$ |  $0.22$${}\,/\,{}$ | $0.025$ |  $12.48$${}\,/\,{}$ | $3128$ |  $8.65$${}\,/\,{}$ | $0.45$ | Bressan et al.(2012) | Gaia DR2 | This paper
|  $0.23$${}\,/\,{}$ | $-$ | - |  $11.20$${}\,/\,{}$ | $1740$ |  $8.00$${}\,/\,{}$ | $0.10$ | Bonatto et al.(2004) | 2MASS | 5
|  $0.10$${}\,/\,{}$ | $-$ |  $0.02$${}\,/\,{}$ | $0.02$ |  $12.18$${}\,/\,{}$ | $2700$ |  $8.70$${}\,/\,{}$ | $0.50$ | Lejuene&Schaerer (2001) | UBVI | 6
|  $0.09$${}\,/\,{}$ | $-$ | solar |  $12.55$${}\,/\,{}$ | $3250$ |  $8.60$${}\,/\,{}$ | $0.40$ | Girardi et al.(2002) | 2MASS | 3
|  $0.10$${}\,/\,{}$ | $-$ | solar |  $12.19$${}\,/\,{}$ | $2700$ |  $8.70$${}\,/\,{}$ | $0.50$ | Marigo et al.(2008) | 2MASS | 4
|  $0.15~{}(A_{V}=0.45)$${}\,/\,{}$ | $-$ | solar |  $12.53$${}\,/\,{}$ | $3210$ |  $8.50$${}\,/\,{}$ | $0.32$ | Bressan et al.(2012) | Gaia DR2 | 10
Col 268 |  $0.53$${}\,/\,{}$ | $0.66$ |  $-0.78$${}\,/\,{}$ | $0.0025$ |  $10.97$${}\,/\,{}$ | $1564$ |  $8.70$${}\,/\,{}$ | $0.50$ | Bressan et al.(2012) | Gaia DR2 | This paper
|  $0.36$${}\,/\,{}$ | $-$ | - |  $11.23$${}\,/\,{}$ | $1740$ | - | - | $ubvyH_{\beta}$ | 8
|  $0.23$${}\,/\,{}$ | $-$ | solar |  $-$${}\,/\,{}$ | $1900$ |  $8.95$${}\,/\,{}$ | $0.60$ | Girardi et al.(2002) | 2MASS | 9
|  $0.24$${}\,/\,{}$ | $-$ | solar |  $11.10$${}\,/\,{}$ | $1680$ |  $8.85$${}\,/\,{}$ | $0.71$ | Girardi et al.(2002) | 2MASS | 3
|  $0.34$${}\,/\,{}$ | $-$ | solar |  $11.40$${}\,/\,{}$ | $1810$ |  $8.79$${}\,/\,{}$ | $0.62$ | Marigo et al.(2008) | 2MASS | 4
|  $0.37~{}(A_{V}=1.16)$${}\,/\,{}$ | $-$ | solar |  $12.10$${}\,/\,{}$ | $2630$ |  $8.38$${}\,/\,{}$ | $0.24$ | Bressan et al.(2012) | Gaia DR2 | 10
Mel 72 |  $0.16$${}\,/\,{}$ | $0.19$ |  $-0.14$${}\,/\,{}$ | $0.011$ |  $11.79$${}\,/\,{}$ | $2277$ |  $9.10$${}\,/\,{}$ | $1.25$ | Bressan et al.(2012) | Gaia DR2 | This paper
|  $0.20$${}\,/\,{}$ | $-$ | solar |  $12.38$${}\,/\,{}$ | $3000$ |  $8.80$${}\,/\,{}$ | $0.60$ | MAI | $UBVI_{KC}$ | 1
|  $0.08$${}\,/\,{}$ | $-$ |  $-0.37$${}\,/\,{}$ | $0.008$ |  $12.51$${}\,/\,{}$ | $3180$ |  $9.20$${}\,/\,{}$ | $1.60$ | Bertelli et al.(2004) | BVI | 2
|  $0.07$${}\,/\,{}$ | $-$ | solar |  $12.03$${}\,/\,{}$ | $2550$ |  $8.95$${}\,/\,{}$ | $0.89$ | Girardi et al.(2002) | 2MASS | 3
|  $0.23$${}\,/\,{}$ | $-$ | solar |  $13.06$${}\,/\,{}$ | $3960$ |  $8.86$${}\,/\,{}$ | $0.72$ | Marigo et al.(2008) | 2MASS | 4
|  $0.14$${}\,/\,{}$ | $0.18$ | solar |  $12.22$${}\,/\,{}$ | $2340$ |  $9.00$${}\,/\,{}$ | $1.00$ | Marigo et al.(2017) | Gaia DR2 | 11
|  $0.13~{}(A_{V}=0.41)$${}\,/\,{}$ | $-$ | solar |  $12.14$${}\,/\,{}$ | $2680$ |  $8.99$${}\,/\,{}$ | $0.98$ | Bressan et al.(2012) | Gaia DR2 | 10
Pismis 7 |  $0.70$${}\,/\,{}$ | $0.86$ |  $-0.28$${}\,/\,{}$ | $0.008$ |  $12.79$${}\,/\,{}$ | $3614$ |  $9.00$${}\,/\,{}$ | $1.00$ | Bressan et al.(2012) | Gaia DR2 | This paper
|  $0.69$${}\,/\,{}$ | $-$ | solar |  $13.46$${}\,/\,{}$ | $4920$ |  $8.70$${}\,/\,{}$ | $0.50$ | Girardi et al.(2000) | BVRI | 7
|  $0.36$${}\,/\,{}$ | $-$ | solar |  $14.07$${}\,/\,{}$ | $6690$ |  $8.90$${}\,/\,{}$ | $0.79$ | Girardi et al.(2002) | 2MASS | 3
|  $0.94$${}\,/\,{}$ | $-$ | solar |  $13.70$${}\,/\,{}$ | $4780$ |  $8.70$${}\,/\,{}$ | $0.50$ | Marigo et al.(2008) | 2MASS | 4
|  $0.52~{}(A_{V}=1.60)$${}\,/\,{}$ | $-$ | solar |  $13.32$${}\,/\,{}$ | $4610$ |  $8.89$${}\,/\,{}$ | $0.78$ | Bressan et al.(2012) | Gaia DR2 | 10

  * (1):Piatti et al. (2010), (2):Hasegawa et al. (2008); (3): Bukowiecki et al.(2011); (4): Kharchenko et al. (2013); (5): Tadross (2011); (6):Piatti et al. (2009); (7): Ahumada (2005);(8): Moffat et al. (1975), (9) :Bica et al. (2008), (10): Cantat-Gaudin et al. (2020). (11): Hendy and Tadross (2021). MAI (Col. 6) means the morphologic age index.

Table 7: Structural parameters and literature comparison of four OCs. Col. 2:
arcmin to parsec scale. The symbol $*\,\prime^{-2}$ in Cols. 7-8 mean
$stars~{}arcmin^{-2}$. Comparison field ring and the correlation coefficient
(CC) are listed in Cols. 9-10. Total mass, mass function slope (MFs), total
star number, references, respectively are given in Cols. 11-14.

Cluster | $(1^{\prime})~{}(pc)$ | $R_{core}$ | $R_{RDP}$ | $R_{core}$ | $R_{RDP}$ | $\sigma_{0K}$ | $\sigma_{bg}$ | $\Delta R$ | CC | $m_{tot}(m_{\odot})$ | MFs | N | Reference
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | $pc$ | $pc$ | $\prime$ | $\prime$ | $*\,\prime^{-2}$ | $*\,\prime^{-2}$ | $\prime$ |  |  |  |  |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$
NGC 2587 | 0.91 |  1.40$\pm$0.60 |  13.70$\pm$0.50 |  1.5$\pm$0.7 |  15.10$\pm$0.60 |  3.9$\pm$2.1 |  6.63$\pm$0.05 | 16-25 | 0.69 | 681 |  1.98$\pm$0.63 | 1621 | This paper
|  |  0.69$\pm$0.10 |  5.53$\pm$0.79 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 174 | -1.13 |  105$\pm$135 | 1
|  |  1.71$\pm$0.49 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 2
Col 268 | 0.45 |  0.90$\pm$0.2 |  5.0$\pm$0.30 |  2.0$\pm$0.5 |  11.0$\pm$0.6 |  6.0$\pm$1.6 |  17.84$\pm$0.09 | 12-31 | 0.95 | 748 |  0.61$\pm$0.23 | 1901 | This paper
|  |  0.97$\pm$0.14 |  5.62$\pm$0.72 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 478 | 2.05 |  1369$\pm$1230 | 1
|  |  0.75$\pm$0.27 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 2
|  |  0.60$\pm$0.20 |  3.70$\pm$0.30 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 3
Mel 72 | 0.66 |  1.10$\pm$0.20 |  15.00$\pm$0.90 |  1.7$\pm$0.3 |  22.6$\pm$1.4 |  9.5$\pm$1.3 |  5.63$\pm$0.05 | 23-31 | 0.99 | 950 |  0.54$\pm$0.23 | 2546 | This paper
|  |  0.80$\pm$0.10 |  8.2$\pm$1.00 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 647 | 0.70 |  7279$\pm$616 | 1
|  |  2.21$\pm$0.38 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 2
|  |  0.31$\pm$0.06 |  3.41$\pm$0.10 |  0.45$\pm$0.09 |  5.0$\pm$0.15 | $-$ | $-$ | $-$ | $-$ | 223 |  -2.20$\pm$0.10 | 168 | 4
Pismis 7 | 1.05 |  0.90$\pm$0.20 |  13.70$\pm$0.60 |  0.9$\pm$0.2 |  13.0$\pm$0.6 |  16.7$\pm$4.6 |  8.27$\pm$0.06 | 14-21 | 0.99 | 973 |  -0.76$\pm$0.25 | 2575 | This paper
|  |  1.50$\pm$0.18 |  14.05$\pm$2.05 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1
|  |  2.10$\pm$0.95 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 2

  * (1): Bukowiecki et al.(2011); (2): Kharchenko et al. (2013); (3):Bica et al. (2008); (4): Hendey and Tadross (2021).

Figure 8: The differences between this paper and literature for $E(B-V)$,
$d~{}(kpc)$, and $Age$ (Gyr) in Table 6. Figure 9: Spatial distribution of our
sample OCs (filled circles). The schematic projection of the Galaxy is seen
from the North pole. $(X_{GC},~{}Y_{GC})$ kpc show the Galactocentric
cartesian coordinates. Figure 10: For four OCs, $R_{RDP}$ versus $R_{core}$
(panel a), $\tau$ versus $t_{rlx}$ (panel b), $m_{tot}(m_{\odot})$ versus
$t_{rlx}$ (panel c), $\chi$ versus $t_{rlx}$ (panel d),
$(R_{core},~{}R_{RDP})$ versus Age (panels e–f). The meanings of the symbols
in panels (a), (e), (f) are explained in the text.

##  6 Discussion and Conclusion

###  6.1 Comparison with the literature

The astrophysical parameters of four OCs have been determined from fitCMD
algorithm. A comparison of $E(B-V)$, $E(G_{BP}-G_{RP})$, $Z$, $(V-M_{V})_{0}$,
$d~{}(pc)$ and $Age$ (Gyr) values of four OCs with the literature is presented
in In Table 6/Fig. 8. The reddening comparison has been given in $E(B-V)$,
which is converted from Gaia/2MASS. The small/large differences between our
values and the literature can be seen from Table 6/Fig. 8. We leave the detail
comparison to the reader.

Reddenings and the photometric distances which are found from fitCMD for NGC
2587 and Mel 72 are in good agreement with the ones of Cantat-Gaudin et al.
(2020). Differences of the reddenings and distances for Col 268 and Pismis 7
are up to 0.16–0.18 mag and $\sim 1.0$ kpc, respectively. This stems from the
usage of $Z$ values of fitCMD, instead of solar abundance. For Mel 72, our
reddenings/distances/ages are in good agreement with Hendey & Tadross (2021).
The obtained photometric distances from fitCMD provide somewhat close
distances, as compared to Gaia DR2 distances (Col.6 of Table 2). For NGC 2587
and Mel 72, both distances are in good concordance. The differences of both
distances are at a level of $\sim 1.0$ kpc for Col 268 and Pismis 7.

The results for Pismis 7 are neccessarily rather tentative. It is the most
distant and the most heavily reddened cluster and clearly shows that its stars
are differentially reddened.

As emphasized by Paunzen & Netopil (2006) and Moitinho (2010), the small/large
discrepancies in ages and distance moduli/distances (Table 6) result from the
adopted isochrones and the reddening values, which are found from CMDs.
Although the B12 Padova isochrones are used for this paper and the papers of
B11 and K13, large/small differences in ages can be explained by the derived
reddenings.

The dimensions, total masses, MF slopes and star numbers of four OCs are
presented in Table 7 together with the literature values. The inconsistency of
these parameters with the literature may be partly explained by the detected
star numbers (Col. 13 of Table 7).

###  6.2 Dynamical Evolution of Four OCs

The relations between dynamical evolution parameters are presented in Figs.
10(a)–(f). The relations, the horizontal lines, the labels $R1$-$R4$, "C" and
filled grey points of panels (a), (e), (f) are from Güneş et al. (2017) (their
figs. 11, 14, and 17). From panel (a), NGC 2587, Mel 72 and Pismis 7 do not
follow the relation of Güneş et al. (2017). These OCs can be partly attributed
to clusters with large radii retaining their masses. Whereas Col 268 with the
small dimensions is quite close to the relation (dashed diagonal).

The relaxation times/evolutionary parameters of four OCs exhibit similarity
because they are all within one sigma of one another (Table 5 and panel b).
Except NGC 2587, the ages of the remaining OCs are higher than their
relaxation times. So, they are dynamically relaxed. From panel c, they seem to
be less massive OCs than $m_{tot}=1000m_{\odot}$. Mass function gives the
distribution of the stars in the cluster. In this context, NGC 2587’s steep
MFs $(\chi=+1.98)$ means that its low-mass stars outnumber its massive ones
(panel d). This OC did not undergo dynamical evolution due to its small
evolutionary time. This explains the reason that this OC lies in $R1$/$R3$
with large cluster/small core radii (panels e and f). The relatively flat (as
compared to $\chi=+1.35$ of Salpeter (1955) or $\chi=1.30$ of Kroupa (2001))
MF slopes of Col 268 and especially Mel 72 (panel d) suggest that both OCs
present signs of a somewhat advanced dynamical evolution, in the sense that
they appear to have lost a significant fraction of their low-mass stars to the
field.

Col 268 with the small dimensions $(R_{core},~{}R_{RDP})=(0.9,~{}5.0)$ pc is
intrinsically small. Instead of shrinking in size and mass with time, it may
have a primordial origin which may be related to high molecular gas density in
Galactic directions (Camargo et al., 2010; van den Berg et al., 1991). Note
that Col 268 (0.5 Gyr) falls in the range "C" (panels e–f).

Pismis 7’s negative/flat MFs (panel d) implies that its high-mass stars
slightly outnumber its low-mass stars. Due to its $\tau=6.7$ value, Pismis 7
shows a sign of mild dynamical evolution. In this context, its high mass stars
move towards the central region, while its low mass stars are continually
being lost to the field. Regarding Pismis 7, it is the most distant OC, so
completeness may affect the detection of its low-mass content. However, since
its age is about 1 Gyr, mass segregation also may played some role in
depleting its low-mass stars. The combination of these two factors may explain
its MFs.

In panels e–f, the cluster dimension versus cluster age is given. As discussed
by Camargo et al. (2010), this kind of relationship provides some information
for cluster survival/dissociation rates. From this perspective, some clusters
expand with time, while some seem to shrink. According to Mackey & Gilmore
(2003), the dynamical evolution of the core/cluster radii of the clusters
started at 500-600 Myr (the label "C", panels e–f), and continued to 1 Gyr. A
bifurcation occurs at an age $\sim 1$ Gyr. Pismis 7 locates at the
bifurcation. Mel 72 locates in $R2$ region. Therefore, the outer parts of Mel
72 and Pismis 7 expand with time. However, Mel 72’ core contracts because of
dynamical relaxation, dependig on its $R4$ location. However, Pismis 7’s core
may expand with the time because its massive stars move towards its central
parts, given its mild dynamical evolution.

Considering the locations of four OCs in the Galaxy (Table 1/Fig. 9), NGC
2587, Mel 72 and Pismis 7 locate at third quadrant (outside solar circle),
which is a region with low density of GMCs. So, except for Pismis 7, they do
not seem to expose to the external dynamical effects much, such as tidal
stripping due to disk and bulge crossings plus encounters with GMCs. Col 268
did not seem to lost its star content much because of the presence of massive
GMCs, and tidal effects from disk and Bulge crossings as external process,
taking care its direction $(\ell=305^{\circ}.54,~{}R_{GC}=7.6~{}kpc)$555We
adopt the value $R_{\odot}=8.2\pm 0.1$ kpc of Bland-Hawthorn & Gerhard (2016)
for their $R_{GC}$ distances.

Star clusters usually present crowding, especially towards the central region.
This means that, near the center, some fraction of the faintest stars will not
be detected because of crowding. At the outer parts where crowding is less
important, one can get more faint stars observed. However, two factors should
be considered on our results. A large uncertainty in $t_{rlx}$ and $\tau$
reduces the reliability of the interpretation of the dynamic evolution of
these OCs. In this respect, for the interpretations we consider their mean
values as a simple approach. Binaries widen the main sequence of the OCs by
0.75 mag, so the theoretical isochrones are fitted to the mid-points of CMDs
of the OCs, rather than the faint or blue sides (Carney et al., 2001). Due to
a consequence of the dynamical evolution in OCs, multiple systems tend to
concentrate in central regions (Takahasi & Portegies Zwart, 2000). Because of
a significant fraction of binaries in the central parts of OCs, the number of
low-mass stars is underestimated with respect to the high mass stars (Bonatto
et al., 2005).

##  7 Acknowledgments

We thank our referee for his/her valuable suggetsions/comments. This paper has
made use of results from the European Space Agency (ESA) space mission Gaia,
the data from which were processed by the Gaia Data Processing and Analysis
Consortium (DPAC). Funding for the DPAC has been provided by national
institutions, in particular the institutions participating in the Gaia
Multilateral Agreement. The Gaia mission website is http:
//www.cosmos.esa.int/gaia. This paper has also made use of the WEBDA database,
operated at the Institute for Astronomy of the University of Vienna. This
publication also makes use of SIMBAD database-VizieR
(http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/246.).

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\bm{C}_{{d(s2)}(22)(1)}&=\begin{bmatrix}\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&-\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\\\
-\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\end{bmatrix}.\end{split}$

Thus using Remark S.1 and following the proof of Lemma 4 in Martin and
Eccleston,  (1998), since $\bm{H}_{t}\bm{{1}}_{t}=\bm{{0}}_{t\times 1}$, we
get that if $d$ is uniform on periods,

$\displaystyle\begin{split}\bm{C}_{{d(s2)}(11)(1)}&=\begin{bmatrix}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{1}\right)\bm{T}_{d}&-\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{2}\right)\bm{T}_{d}\\\
-\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{2}\right)\bm{T}_{d}&\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{4}\right)\bm{T}_{d}\end{bmatrix},\\\
\bm{C}_{{d(s2)}(12)(1)}&=\bm{C}^{{}^{\prime}}_{{d(s2)}(21)(1)}=\begin{bmatrix}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&-\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\\\
-\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\end{bmatrix},\text{
and}\\\
\bm{C}_{{d(s2)}(22)(1)}&=\begin{bmatrix}\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&-\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\\\
-\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\end{bmatrix}.\end{split}$
(S.49)

So, if $d\in\varOmega_{t,n,p}$ is a design uniform on periods, then from
(S.48) and (S.49), we get that for $f_{1},f_{2}=1,2$,
$\bm{{C}}_{d(s2)(f_{1}f_{2})(1)}=\bm{\tilde{C}}_{d(s2)(f_{1}f_{2})}$. Thus
using the expression of the matrices $\bm{{C}}_{d(s2)}$ and
$\bm{\tilde{C}}_{d(s2)}$ as given in Lemma S.3 and Lemma S.5, respectively, we
get that $\bm{{C}}_{d(s2)}=\bm{\tilde{C}}_{d(s2)}$. ∎

######  Lemma S.8.

Let
$\bm{V}_{1}^{*}=\bm{V}_{1}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{1}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{1}^{-1}\bm{J}_{p\times
p}\bm{V}_{1}^{-1}$ and
$\bm{V}_{R}^{*}=\bm{V}_{R}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{R}^{-1}\bm{{1}}_{p}\right)^{-1}\times$
$\bm{V}_{R}^{-1}\bm{J}_{p\times p}\bm{V}_{R}^{-1}$. Then
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)>0$
and
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)>0$.
Here, $\bm{J}_{p\times p}=\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$ and
$\bm{H}_{p}=\bm{I}_{p}-\frac{1}{p}\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$.

###### Proof.

Here, $\bm{V}_{1}$ is a symmetric and positive definite matrix. Since
$pr^{\perp}\left(.\right)$ is a symmetric and idempotent matrix,
$\bm{V}_{1}^{*}=\bm{V}_{1}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{1}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{1}^{-1}\bm{J}_{p\times
p}\bm{V}_{1}^{-1}$ can be written as

$\displaystyle\bm{V}_{1}^{*}$
$\displaystyle=\bm{W}_{1}^{{}^{\prime}}\bm{W}_{1},$ (S.50)

where $\bm{W}_{1}=pr^{\perp}\left(\bm{V}_{1}^{-1/2}\bm{J}_{p\times
p}\right)\bm{V}_{1}^{-1/2}$ and $\bm{J}_{p\times
p}=\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$. Since
$\bm{H}_{p}=\bm{I}_{p}-\frac{1}{p}\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$ is a
symmetric and idempotent matrix,
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)=tr\left(\bm{W}_{2}^{{}^{\prime}}\bm{W}_{2}\right)\geq
0$, where $\bm{W}_{2}=\bm{W}_{1}\bm{\psi}\bm{H}_{p}$. It can be easily seen
that $tr\left(\bm{W}_{2}^{{}^{\prime}}\bm{W}_{2}\right)=\bm{{0}}_{p\times p}$
if and only if $\bm{W}_{2}=\bm{{0}}_{p\times p}$. Now, suppose
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)=0$.
So, $\bm{W}_{1}\bm{\psi}\bm{H}_{p}=\bm{{0}}_{p\times p}$, and thus from
(S.50), we get

$\displaystyle\bm{V}_{1}^{*}\bm{\psi}\bm{H}_{p}=\bm{{0}}_{p\times p}.$ (S.51)

From the expression of $\bm{V}_{1}^{*}$, we get that the row sums of
$\bm{V}_{1}^{*}$ is $0$. Thus using (S.51), by calculation we get

$\displaystyle\begin{split}\left(\bm{V}_{1}^{*}\right)_{1,2}&=\cdots=\left(\bm{V}_{1}^{*}\right)_{1,p}=-\frac{1}{p}\left(\bm{V}_{1}^{*}\right)_{1,1},\\\
\vdots\\\
\left(\bm{V}_{1}^{*}\right)_{p,2}&=\cdots=\left(\bm{V}_{1}^{*}\right)_{p,p}=-\frac{1}{p}\left(\bm{V}_{1}^{*}\right)_{p,1},\end{split}$
(S.52)

where $\left(\bm{V}_{1}^{*}\right)_{l_{1},l_{2}}$ denotes the
$(l_{1},l_{2})^{th}$ element of $\bm{V}_{1}^{*}$. Using the property that the
row sums of $\bm{V}_{1}^{*}$ is $0$ in (S.52), we get that
$\bm{V}_{1}^{*}=\bm{{0}}_{p\times p}$. Thus by calculation we get

$\displaystyle\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{1}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{J}_{p\times
p}\bm{V}_{1}^{-1}$ $\displaystyle=\bm{I}_{p}.$ (S.53)

The above equation (S.53) leads to a contradiction. Hence, our assumption that
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)=0$ is
wrong. Thus we get,
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)>0$.

Since $\bm{V}_{R}$ is a positive definite matrix, similarly we can prove that
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)>0$. ∎