# A comprehensive kinematic model of the LMC disk from star clusters and field
stars using
Gaia DR3: Tracing the disk characteristics, rotation, bar, and the outliers

S. R. Dhanush Indian Institute of Astrophysics, Bangalore, 560034, India
Pondicherry University, R.V. Nagar, Kalapet, 605014, Puducherry, India  A.
Subramaniam Indian Institute of Astrophysics, Bangalore, 560034, India  S.
Subramanian Indian Institute of Astrophysics, Bangalore, 560034, India

###### Abstract

The internal kinematics of the Large Magellanic Cloud (LMC) disk have been
modeled by several studies using different tracers with varying coverage,
resulting in a range of parameters. Here, we modeled the LMC disk using 1705
star clusters and field stars, based on a robust Markov Chain Monte Carlo
(MCMC) method, using the Gaia DR3 data. The dependency of model parameters on
the age, coverage, and strength of the clusters are also presented. This is
the first comprehensive 2D kinematic study using star clusters. Red clump (RC)
stars and young main-sequence stars are also modeled for comparison. The
clusters and field stars are found to have distinctly different kinematic
centers, disk inclination, position angle of the line of nodes, and scale
radius. We also note a significant radial variation of the disk parameters.
Clusters and young stars are found to have a large residual proper motion and
a relatively large velocity dispersion when compared to the RC field
population, which could be due to perturbation from the bar and spiral arms.
We traced the presence of large residual proper motion and non-circular motion
among clusters likely to be due to the bar and detected a decrease in the
scale radius as a result of the possible evolution of the bar. The
kinematically deviant clusters point to a spatio-temporal disturbance in the
LMC disk, matching with the expected impact factor and time of the recent
collision between the LMC and the Small Magellanic Cloud.

(galaxies:) Magellanic Clouds — galaxies: star clusters: general — galaxies:
kinematics and dynamics — galaxies: interactions — galaxies: evolution

##  1 Introduction

The Magellanic Clouds (MCs) consist of two irregular dwarf galaxies observable
from the Southern Hemisphere. The Large Magellanic Cloud (LMC), the larger of
the pair, is situated at 49.59$\pm$0.09 kpc away (Pietrzyński et al. 2019,
hereafter, P20), while its smaller companion, the Small Magellanic Cloud
(SMC), lies at a distance of 62.44$\pm$0.47 kpc (Graczyk et al., 2020). The
close proximity of the MCs to the Milky Way (MW) offers a unique opportunity
to study them and gain valuable insights into the mechanisms that drive galaxy
mergers and their consequences for galaxy evolution.

The LMC-SMC-MW is an interacting system (Putman et al., 1998; Diaz & Bekki,
2012; Hammer et al., 2015). Traditionally, the kinematics of the LMC was
thought to be modified by its interactions with the MW. In the last couple of
decades, it was believed that the LMC was experiencing its first encounter
with the MW, as indicated in the studies by Kallivayalil et al. (2006a, b),
and Besla et al. (2007). However, the recent study by Vasiliev (2023)
indicates the potential scenario where the MCs are in their second passage
around the MW. Either way, the current morphology of the LMC is predominantly
influenced by the interactions with the neighboring SMC rather than the MW
(Besla et al. 2012, hereafter B12; Yozin & Bekki 2014; Choi et al. 2018a, b).

There have been significant efforts to understand the internal kinematics of
the LMC. They were previously studied primarily using the H I gas and stars in
the LMC (Kim et al., 1998; van der Marel et al., 2002; Olsen & Massey, 2007;
Indu & Subramaniam, 2015). Recently, Wan et al. (2020, hereafter W20) used the
all-sky data release of SkyMapper (DR1) to investigate the kinematics of the
LMC across various stellar populations, including carbon stars, RGB stars, and
young stars. Also, the kinematics of central regions of the LMC is studied by
Niederhofer et al. (2022, hereafter N22) using the Visible and Infrared Survey
Telescope for Astronomy (VISTA) survey of the Magellanic Clouds system (VMC;
Cioni et al. 2011). Recent studies with Gaia Early Data Release 3 (EDR3) and
Data Release 3 (DR3), revealed the structure and velocity maps of the LMC
(Gaia Collaboration et al. 2021, hereafter G21; Jiménez-Arranz et al. 2023,
hereafter J23). Also, Choi et al. (2022, hereafter C22) performed the 3D
kinematic model of the LMC, with the red clump population using Gaia EDR3.
Their findings suggest a direct LMC-SMC collision in the last 250 Myr.

Star clusters are excellent tracers of star formation history. So far, star
clusters are used to trace the cluster formation (CF) peaks (Glatt et al.
2010; Nayak et al. 2016; Dhanush et al. 2024, hereafter D24), chemical
enrichment of the disk (Hill et al., 2000; Johnson et al., 2006; Palma et al.,
2015; Chilingarian & Asa’d, 2018; Narloch et al., 2022) and in a few cases,
kinematics of the LMC disk using radial velocity estimations (Schommer et al.,
1992; Bennet et al., 2022). The availability of the proper motion (PM) data
from the Gaia DR3 catalog opens up a new method to explore the LMC disk
kinematics using star clusters. Recently D24 parameterised 1710 star clusters
in the LMC using the Gaia DR3 data. The PM of the cluster estimated using the
median PM value of the members therefore presents a golden data set to explore
the LMC disk kinematics using tangential velocity. As the ages of these
clusters are already estimated, the data set will allow us to explore the
dependency of kinematic properties on the age of the cluster. D24 also
identified a nearby field region for all the 1710 clusters, and this
population can be used as a control/ comparison population. This study
therefore explores the LMC disk kinematics using the unique data set of D24.

The aim of this study is to model the LMC disk and estimate the kinematic
parameters using a robust Markov Chain Monte Carlo (MCMC) method. The data set
allows us to explore their dependency on various factors such as age, spatial
coverage of the population as well as the richness of clusters. The kinematic
parameters of the LMC disk are derived for the first time using star clusters
in this study. We also quantify the residual PM after fitting the model and
trace the kinematic outliers among clusters.

The paper is arranged as follows. In Section 2, we describe the data sets used
for our modeling. In Section 3, we describe the procedures of our kinematic
model. In Section 4, we present the estimated kinematic parameters of our data
sets, followed by the estimation of the residual PM. The discussions based on
our results are presented in Section 5, followed by Section 6, which
summarizes our work.

##  2 Data

The data sets of our modeling were derived from D24, which involved
parameterizing star clusters in MCs with Gaia DR3 data to estimate their age,
metallicity, distance modulus, and extinction. The study parameterized 1990
star clusters within the MCs, among which 1710 were in the LMC and 280 were in
the SMC. These clusters underwent thorough field star subtraction using nearby
regions, as explained in Section 2 of D24. For the 1710 LMC clusters, we used
the cleaned cluster regions and their associated field regions as data sets to
perform the kinematic model. First, we calculated the median of the PM and its
standard error for stars within each cluster and their associated field region
in both the RA and DEC directions. Then, we applied a PM cutoff of $<$ (median
PM$+$3$\sigma$) based on the PM distribution of field regions. We used it for
clusters and field regions both to eliminate a few outliers in the galaxy that
show relatively large PM. We retained 1705 clusters after this selection. The
primary modeling was performed using the clusters and their associated field
regions following this selection process. We further performed kinematic
modeling with data sets based on the control population, cluster richness,
cluster age, and spatial coverage of the LMC. The details of these data sets
are enumerated below.

  1. 1.

The control population for the model is obtained from the color-magnitude-
diagrams (CMDs) of field regions associated with each cluster. The choice of
control population was made to understand the kinematic nature of the cluster
and field population. We chose the young main-sequence (MS) stars within the
window of $G$ $<$ 19.5 mag and ($G_{BP}$-$G_{RP}$) $<$ 0.5, then red clump
(RC) stars based on the selection criteria from CMD of the LMC sources as
mentioned in Saroon & Subramanian (2022). Several field regions associated
with clusters do not have significant young MS stars in the CMD. We put a cut-
off of young MS stars $\geq$5 to further select the samples for the control
population. We are left with 1484 candidates in the four categories for
modeling, the clusters, nearby field regions, young MS stars, and RC stars.
Out of this final selection, we note that the $\sim$ 75% of the field regions
considered have young MS stars and RC stars with a strength of 20 and 50
stars, respectively.

  2. 2.

We note that the average number of members in each cluster in our study is
$\sim$ 40\. To investigate potential biases in the data sets, we conducted
modeling of groups based on the cluster members’ strength across a range from
10 to 60 members, with intervals of 10. The field regions associated with
these richness groups were also modeled to compare with the nature of
clusters. We created 12 models for the cluster groups and field groups based
on cluster richness.

  3. 3.

To understand the age dependency of the kinematic parameters, we performed
modeling with respect to clusters spanning from older to younger age groups.
We used age groups in ranges of $\log(t)$, Age group-1: $[\geq 9.10,<9.55]$,
Age group-2: $[\geq 8.65,<9.10]$, Age group-3: $[\geq 8.0,<8.65]$, and Age
group-4: $[\geq 6.55,<8.0]$ (as defined in D24). The associated nearby field
regions were also modeled with respect to each age group. It is to be noted
that the age-wise grouping is only valid for the clusters, and the associated
field regions are age-wise heterogeneous but analyzed for comparison.

  4. 4.

We investigated the variations in kinematic parameters concerning the spatial
extent of the LMC. Employing circular regions, we analyzed clusters and
associated field regions, ranging from 2∘ to 7∘ in increments of 0.5∘ from the
kinematic center on the sky plane. We used the kinematic center that we
calculated from the primary model encompassing clusters and field stars for
the radially separated groups.

Table 1 summarizes the data sets that are modeled. The subsequent section
covers the implementation of the kinematic model using the above data sets.

Table 1: The data sets used in the kinematic model of the LMC are summarized here. The length ($L_{d}$) and the number ($N_{d}$) of data sets are provided as well. Clusters and associated field regions are labeled based on the cluster richness in the 3rd row, such as C10 representing clusters with 10 or more stars, and so on. A total of 48 data sets are used in our study to perform the kinematic modeling. Data set | Ld | Nd
---|---|---
Clusters, Nearby field regions | 1705 | 2
(Primary sample) |  |
Clusters, Nearby field regions, | 1484 | 4
Young MS stars, RC stars |  |
(Control sample) |  |
C10,..,60, F10,..,60 | 1581, 973, 643, | 12
(groups based on cluster richness) | 455, 334, 257 |
Cluster age groups, | 316, 579, 551, 260 | 8
Nearby field regions |  |
Clusters, Nearby field regions | 372, 560, 551, 768, | 22
(groups based on spatial coverage, | 958, 1104, 1263, 1401, |
radii: 2∘ to 7∘, step size: 0.5∘) | 1604, 1667, 1688 |

##  3 Kinematic model of the LMC

We performed the kinematic model of the LMC corresponding to the observed
median PM for the data sets referenced in Section 2. The following subsections
provide a detailed explanation of the theoretical background we adopted in our
modeling and the Bayesian methodology employed to estimate the optimal
kinematic parameters for the data sets.

###  3.1 Analytical background of the modeling

The model PM for the candidates in the data sets is formulated based on the
equations outlined by van der Marel et al. (2002, hereafter, V02). These
equations define the directional vectors of local PM in the West and North
(MW, MN) in the sky plane using a series of 12 independent model parameters.
The parameters selected for our model encompass the inclination of the LMC
disc ($i$), the position angle of the line of nodes measured from West
($\theta$), dynamic centers ($\alpha_{0}$, $\delta_{0}$), the amplitude of the
tangential velocity of the LMC’s center of mass ($v_{t}$), the tangential
angle made by $v_{t}$ ($\theta_{t}$), scale radius ($R_{0}$), optimization
factor ($\eta$), and the amplitude of the rotational velocity ($v_{0}$). Our
modeling process is aimed at determining the most fitting values for these
nine kinematic parameters. It’s important to note that our modeling was
confined to the LMC’s sky plane due to the absence of line-of-sight velocity
components in our data sets.

The velocity of sources at any given point ($\alpha$, $\delta$) within the LMC
sky plane is primarily composed of three components. It consists of the
velocity of the center of mass (COM) of the LMC, the velocity component
associated with the precession and nutation of the galaxy, and the internal
rotation component of the galaxy. We assumed that there is no precession and
nutation of the LMC disc within the spatial coverage of the galaxy (less than
7∘) considered in this study (V02, N22). We assumed the RA and DEC of the
field regions associated with each cluster to have the same spatial
coordinates as the cluster centers. Also, we assumed a fixed distance of 49.59
kpc (P20) to the LMC center, D0 and a line-of-sight velocity of the COM,
$v_{sys}$, as 262.2 km s-1 (V02). The parametric form for the rotational
velocity in the disc plane was adopted from similar kinematic models used in
previous studies of the LMC (G21, N22).

The following subsection deals with the method and estimation of the best-
fitting kinematic parameters for the data sets summarized in Table 1.

###  3.2 Modeling procedure

We found the best-fitting 48 models and the associated kinematic parameters
for 48 data sets referenced in Table 1. We performed the fitting of the
parameters using the MCMC serial stretch move sampling algorithm introduced by
Goodman & Weare 2010. The code is developed and implemented in C language. A
similar MCMC approach is used in the studies by W20, C22 and N22. The observed
PM for the sources in data sets are along the RA and DEC directions, but the
model formulation is for the PM in the West and North directions. For that, we
took the negative of the source median PM in the RA direction. Now the model
PM (MW,m, MN,m) and the observed PM (MW,o, MN,o) can be used to construct the
log-likelihood function ($\ln\mathcal{L}$), which can be used with the
associated West and North direction standard errors of observed data sets
($\sigma_{W,o}$,$\sigma_{N,o}$) to sample the best fitting parameter with
MCMC. The equation for $\ln\mathcal{L}$ is given by,

$\displaystyle\ln\mathcal{L}$
$\displaystyle=-0.5\sum_{i=1}^{n}\left[\ln\left(2\pi\sigma_{W,o,i}^{2}\right)+\frac{\left(\mu_{W,o,i}-\mu_{W,m,i}\right)^{2}}{\sigma_{W,o,i}^{2}}\right.$
$\displaystyle\quad+\ln\left(2\pi\sigma_{N,o,i}^{2}\right)+\left.\frac{\left(\mu_{N,o,i}-\mu_{N,m,i}\right)^{2}}{\sigma_{N,o,i}^{2}}\right]$
(1)

The priors for the model were uniformly chosen, except for the rotation
velocity amplitude, $v_{0}$, for which we used a Gaussian prior of 76 km s-1
(van der Marel & Kallivayalil 2013; G21; N22). When working with modeling
datasets based on cluster age groups, we note that the spatial distribution of
clusters in the sky plane is not homogeneous. This results in the non-
convergence of $\eta$ in its posterior sampling distributions while performing
the MCMC with a uniform prior. To avoid this, we employed Gaussian weighting
on $\eta$ specifically for these age-based datasets, using the corresponding
value of the parameter estimated from the primary model encompassing all
cluster and field regions.

We executed 2000 steps of MCMC iteration involving 200 walkers evolving
sequentially at each step. From the sampled posterior values for the nine
parameters, we focused on the final 50%, calculating their median alongside
the 16th and 84th percentile errors for estimation. In the following section,
we present the comprehensive results obtained from the above modeling
procedures.

##  4 Results

This is the first 2D kinematic model of the LMC employing clusters and
neighboring field regions with Gaia DR3 data. The data coverage of the LMC
considered in this study is within $\sim$ 7∘ from the LMC center. The primary
modeling is performed with 1705 clusters and nearby field regions, estimating
nine best-fitting model parameters defining the kinematic properties of the
LMC. Figures 1 and 2 show the sampled posterior distribution of the parameters
for clusters and nearby field regions, respectively. We also performed the
parameter estimation for the rest of the data sets mentioned in Table 1.

Figure 1: The corner plot representing the sampled posterior distribution of
nine kinematic parameters for the primary cluster data set is shown here. The
vertical red lines represent the median values, and the black dashed lines
represent the 16th and 84th percentiles. Figure 2: The corner plot
representing the sampled posterior distribution of nine kinematic parameters
for the primary field data set is shown here. The vertical blue lines
represent the median values, and the black dashed lines represent the 16th and
84th percentiles.

In the following subsections, we provide the estimated kinematic parameters
obtained for all the modeled data sets. Additionally, we present the residual
PM derived from comparing the model and observed data.

###  4.1 Cluster and Field kinematics

Table 2 provides the estimated kinematic parameters for the primary model
involving clusters and field stars. The position angle of the line of nodes is
conventionally measured from the North of the galaxy, whereas the modeling we
performed involved measuring from the West. Therefore, we define the position
angle of the line of nodes measured from North (PA) as $\Theta$, which is
($\theta$-$90^{\circ}$) throughout the sections. The COM PM values in the West
and the North directions ($\mu_{W,com}$ and $\mu_{N,com}$) can be obtained
using the estimated tangential velocity vector estimated in our models (using
$v_{t}$ and $\theta_{t}$ and Equation 10 in V02). The orthographic projections
of the sky coordinate and PM vectors are performed using equations 1, 2, and
3, as mentioned in G21. Figure 3 shows the bulk motion and rotation of the LMC
obtained with clusters and field by subtracting the COM PM values estimated
from the model. Below we compare the notable kinematic properties of clusters
and field regions obtained from the modeling.

Figure 3: The observed PM plots for the clusters and field in the LMC sky plane are shown here. (a) Observed bulk motion of clusters; (b) Internal rotation PM of clusters; (c) Observed bulk motion of field; (b) Internal rotation PM of field. Table 2: The kinematic best-fitting parameters obtained for the LMC from the primary model of clusters and nearby field regions are tabulated below. Data | $i$ | $\Theta$ | $\alpha_{0}$ | $\delta_{0}$ | $v_{t}$ | $\theta_{t}$ | $R_{0}$ | $\eta$ | $v_{0}$
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (deg) | (deg) | (km s-1) | (deg) | (kpc) |  | (km s-1)
Clusters | 31.39${}^{\hskip 0.28436pt\scaleto{+0.73}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.74}{4.7pt}}$ | 122.22${}^{\hskip 0.28436pt\scaleto{+1.48}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.51}{4.7pt}}$ | 81.69${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | -69.59${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.51${}^{\hskip 0.28436pt\scaleto{+0.52}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.51}{4.7pt}}$ | 167.89${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.21}{4.7pt}}$ | 1.58${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.82${}^{\hskip 0.28436pt\scaleto{+0.16}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 83.97${}^{\hskip 0.28436pt\scaleto{+1.3}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.28}{4.7pt}}$
Nearby field | 30.81${}^{\hskip 0.28436pt\scaleto{+0.38}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.4}{4.7pt}}$ | 128.02${}^{\hskip 0.28436pt\scaleto{+0.79}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.81}{4.7pt}}$ | 80.57${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | -69.34${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.7${}^{\hskip 0.28436pt\scaleto{+0.34}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.34}{4.7pt}}$ | 169.46${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 2.43${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 2.36${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 84.15${}^{\hskip 0.28436pt\scaleto{+1.16}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.09}{4.7pt}}$

  * •

$\mu_{W,com}$ and $\mu_{N,com}$ were estimated as $1.849\pm 0.003$ mas yr-1
and $0.397\pm 0.007$ mas yr-1 for the clusters. Similarly, for the field
regions, the values are $1.851\pm 0.002$ mas yr-1 and $0.344\pm 0.004$ mas
yr-1. These values suggest no significant difference between the observed COM
PM between cluster and field.

  * •

We note an offset of 28$\pm$8 arcmin between the dynamic centers
($\alpha_{0}$, $\delta_{0}$) of the cluster and field.

  * •

Estimated $\Theta$ values of 122∘.22${}^{\hskip
0.28436pt\scaleto{+1.48}{4.7pt}}_{\hskip 1.13791pt\scaleto{-1.51}{4.7pt}}$ for
clusters and 128∘.02${}^{\hskip 0.28436pt\scaleto{+0.79}{4.7pt}}_{\hskip
1.13791pt\scaleto{-0.81}{4.7pt}}$ for field regions point to an offset of
5∘.8$\pm$1∘.7 between them, while the inclination, $i$, remains almost
similar.

  * •

The modeled rotational parameters, $R_{0}$ and $\eta_{0}$ appear to be larger
for field regions compared to clusters, while the $v_{0}$ remains almost
similar.

Two major notable differences in the kinematic properties obtained are, (1)
for the value of $R_{0}$, where clusters have a relatively low value when
compared to the field population, and (2) a significant offset in the dynamic
centers of cluster and field population. In the next sub-section, we look into
the control population we used in our model to understand the kinematic nature
of the cluster and field population.

###  4.2 Comparison with control population

Table 3 provides the estimated kinematic parameters for the control population
as mentioned in Section 2. Below we compare the cluster and field with the
young MS stars and RC stars to understand their kinematic nature.

Table 3: The kinematic best-fitting parameters obtained for the LMC based on the control sample (clusters, field population, young MS stars, RC stars) are tabulated below. Data | $i$ | $\Theta$ | $\alpha_{0}$ | $\delta_{0}$ | $v_{t}$ | $\theta_{t}$ | $R_{0}$ | $\eta$ | $v_{0}$
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (deg) | (deg) | (km s-1) | (deg) | (kpc) |  | (km s-1)
Clusters | 31.11${}^{\hskip 0.28436pt\scaleto{+0.82}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.81}{4.7pt}}$ | 120.27${}^{\hskip 0.28436pt\scaleto{+1.65}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.64}{4.7pt}}$ | 81.71${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.6${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 445.01${}^{\hskip 0.28436pt\scaleto{+0.58}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.59}{4.7pt}}$ | 167.86${}^{\hskip 0.28436pt\scaleto{+0.22}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 1.63${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.88${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 83.95${}^{\hskip 0.28436pt\scaleto{+1.3}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.3}{4.7pt}}$
Nearby field | 30.83${}^{\hskip 0.28436pt\scaleto{+0.4}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.41}{4.7pt}}$ | 127.7${}^{\hskip 0.28436pt\scaleto{+0.81}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.8}{4.7pt}}$ | 80.59${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | -69.35${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.93${}^{\hskip 0.28436pt\scaleto{+0.37}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.37}{4.7pt}}$ | 169.43${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | 2.45${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 2.35${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 84.56${}^{\hskip 0.28436pt\scaleto{+1.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.16}{4.7pt}}$
Young MS | 33.26${}^{\hskip 0.28436pt\scaleto{+0.56}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.57}{4.7pt}}$ | 115.45${}^{\hskip 0.28436pt\scaleto{+0.99}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.99}{4.7pt}}$ | 80.99${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | -69.64${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 446.65${}^{\hskip 0.28436pt\scaleto{+0.43}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.42}{4.7pt}}$ | 169.5${}^{\hskip 0.28436pt\scaleto{+0.15}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | 1.29${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | 1.5${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.09}{4.7pt}}$ | 90.92${}^{\hskip 0.28436pt\scaleto{+1.26}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.18}{4.7pt}}$
Red Clump | 31.1${}^{\hskip 0.28436pt\scaleto{+0.44}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.44}{4.7pt}}$ | 135.84${}^{\hskip 0.28436pt\scaleto{+0.91}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.89}{4.7pt}}$ | 80.68${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | -69.21${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 441.6${}^{\hskip 0.28436pt\scaleto{+0.48}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.46}{4.7pt}}$ | 169.01${}^{\hskip 0.28436pt\scaleto{+0.2}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.19}{4.7pt}}$ | 2.99${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 3.24${}^{\hskip 0.28436pt\scaleto{+0.28}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.24}{4.7pt}}$ | 81.39${}^{\hskip 0.28436pt\scaleto{+1.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.18}{4.7pt}}$

  * •

$\mu_{W,com}$ and $\mu_{N,com}$ appear to be similar in all the data sets of
the control population without any significant difference.

  * •

The value of $\alpha_{0}$ for the field and RC population are the same within
errors, whereas the value of $\delta_{0}$ shows a difference. The $\alpha_{0}$
for clusters shows a significant offset in comparison to other populations. In
general, the younger population tends to show a southern and eastward dynamic
center.

  * •

Estimated values of $\Theta$ = 115∘.45${}^{\hskip
0.28436pt\scaleto{+0.99}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.99}{4.7pt}}$ and
$i$ = 33∘.26${}^{\hskip 0.28436pt\scaleto{+0.56}{4.7pt}}_{\hskip
0.28436pt\scaleto{-0.57}{4.7pt}}$ for the young MS population appear distinct
from other data sets in the control population. RC population has the largest
value of $\Theta$ (135∘.84).

  * •

We note a significant shift in $R_{0}$, $\lvert\Delta R_{0}\rvert$ $\approx$
1.7 kpc between young MS and RC population. A corresponding change of $\eta$
is observed as well. We also estimate a larger value for $v_{0}$ =
90.92${}^{\hskip 0.28436pt\scaleto{+1.26}{4.7pt}}_{\hskip
0.28436pt\scaleto{-1.18}{4.7pt}}$ km s-1 for the young MS population in
comparison with the other data sets.

As observed in the previous comparison (subsection 4.1), the value of $\Theta$
appears to be changing across the control population. Notably, the clusters
and young MS population show a relatively small value in comparison to the
field and RC population. Specifically, the $R_{0}$ for the cluster and young
MS population falls within the range of 2 kpc, while the field and RC
population have values exceeding 2 kpc.

Figure 4: Spatial distribution of the rotational velocity (Vrot) of the LMC
for the clusters and the three control populations are depicted here.

The model parameters estimated for the cluster and the control samples are
used to obtain the values of rotational velocity (Vrot) of the LMC. Figure 4
shows the spatial distribution of Vrot among the clusters, field stars, Young
stars, and RC population. Though the overall appearance of the plots is
similar, we note that clusters (panel a) and the young MS stars (panel c)
appear to have similar spatial distributions of Vrot, whereas the field (panel
b) and the RC stars (panel d) appear to have similar Vrot distribution. We
also note that the elongated bar feature is more prominent in the Vrot maps of
clusters and young MS stars.

This suggests that the kinematic model of the LMC depends on the age of the
population. In the subsequent subsection, we look into the kinematic model of
the LMC with data sets of different ages, using the cluster age groups and
nearby field regions as mentioned in Section 2.

###  4.3 Age dependent kinematics of the LMC

The age-dependent variation of the kinematic parameters of the LMC was
estimated using the data sets mentioned in Section 2. Table 4 provides the
estimated kinematic parameters for the cluster and nearby field data sets
based on the cluster age groups. Below, we compare their kinematic properties.

Table 4: The kinematic best-fitting parameters obtained for the LMC based on the cluster age groups and nearby field regions are tabulated below. As mentioned in Section 2, the associated field population for each cluster age group is age-wise heterogeneous. Data | $i$ | $\Theta$ | $\alpha_{0}$ | $\delta_{0}$ | $v_{t}$ | $\theta_{t}$ | $R_{0}$ | $\eta$ | $v_{0}$
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (deg) | (deg) | (km s-1) | (deg) | (kpc) |  | (km s-1)
CAG-1 | 31.56${}^{\hskip 0.28436pt\scaleto{+1.22}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.3}{4.7pt}}$ | 145.25${}^{\hskip 0.28436pt\scaleto{+2.93}{4.7pt}}_{\hskip 0.28436pt\scaleto{-3.01}{4.7pt}}$ | 80.84${}^{\hskip 0.28436pt\scaleto{+0.38}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.38}{4.7pt}}$ | -69.19${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | 440.57${}^{\hskip 0.28436pt\scaleto{+1.49}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.52}{4.7pt}}$ | 169.02${}^{\hskip 0.28436pt\scaleto{+0.63}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.63}{4.7pt}}$ | 3.14${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 5.61${}^{\hskip 0.28436pt\scaleto{+1.76}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.4}{4.7pt}}$ | 78.63${}^{\hskip 0.28436pt\scaleto{+1.4}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.33}{4.7pt}}$
FAG-1 | 31.11${}^{\hskip 0.28436pt\scaleto{+0.78}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.79}{4.7pt}}$ | 137.44${}^{\hskip 0.28436pt\scaleto{+1.67}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.67}{4.7pt}}$ | 80.68${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.19}{4.7pt}}$ | -69.19${}^{\hskip 0.28436pt\scaleto{+0.06}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.06}{4.7pt}}$ | 439.83${}^{\hskip 0.28436pt\scaleto{+0.76}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.75}{4.7pt}}$ | 169.14${}^{\hskip 0.28436pt\scaleto{+0.33}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.32}{4.7pt}}$ | 2.84${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | 5.44${}^{\hskip 0.28436pt\scaleto{+0.97}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.81}{4.7pt}}$ | 77.21${}^{\hskip 0.28436pt\scaleto{+1.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.04}{4.7pt}}$
CAG-2 | 28.33${}^{\hskip 0.28436pt\scaleto{+1.33}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.45}{4.7pt}}$ | 133.87${}^{\hskip 0.28436pt\scaleto{+3.34}{4.7pt}}_{\hskip 0.28436pt\scaleto{-3.42}{4.7pt}}$ | 81.64${}^{\hskip 0.28436pt\scaleto{+0.27}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.27}{4.7pt}}$ | -69.6${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | 445.16${}^{\hskip 0.28436pt\scaleto{+1.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.11}{4.7pt}}$ | 168.04${}^{\hskip 0.28436pt\scaleto{+0.46}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.45}{4.7pt}}$ | 2.17${}^{\hskip 0.28436pt\scaleto{+0.2}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.21}{4.7pt}}$ | 2.52${}^{\hskip 0.28436pt\scaleto{+0.46}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.36}{4.7pt}}$ | 80.63${}^{\hskip 0.28436pt\scaleto{+1.52}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.51}{4.7pt}}$
FAG-2 | 30.79${}^{\hskip 0.28436pt\scaleto{+0.72}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.72}{4.7pt}}$ | 130.17${}^{\hskip 0.28436pt\scaleto{+1.61}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.63}{4.7pt}}$ | 80.7${}^{\hskip 0.28436pt\scaleto{+0.15}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | -69.25${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 442.59${}^{\hskip 0.28436pt\scaleto{+0.62}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.63}{4.7pt}}$ | 169.18${}^{\hskip 0.28436pt\scaleto{+0.27}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.27}{4.7pt}}$ | 2.52${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | 2.87${}^{\hskip 0.28436pt\scaleto{+0.29}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.25}{4.7pt}}$ | 81.03${}^{\hskip 0.28436pt\scaleto{+1.3}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.28}{4.7pt}}$
CAG-3 | 35.36${}^{\hskip 0.28436pt\scaleto{+1.2}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.31}{4.7pt}}$ | 111.11${}^{\hskip 0.28436pt\scaleto{+2.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-2.04}{4.7pt}}$ | 81.59${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.21}{4.7pt}}$ | -69.6${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 445.2${}^{\hskip 0.28436pt\scaleto{+0.9}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.9}{4.7pt}}$ | 168.31${}^{\hskip 0.28436pt\scaleto{+0.36}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.35}{4.7pt}}$ | 1.34${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 2.06${}^{\hskip 0.28436pt\scaleto{+0.34}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.27}{4.7pt}}$ | 81.31${}^{\hskip 0.28436pt\scaleto{+1.51}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.49}{4.7pt}}$
FAG-3 | 29.89${}^{\hskip 0.28436pt\scaleto{+0.72}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.73}{4.7pt}}$ | 123.5${}^{\hskip 0.28436pt\scaleto{+1.41}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.46}{4.7pt}}$ | 80.36${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | -69.34${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 443.01${}^{\hskip 0.28436pt\scaleto{+0.56}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.55}{4.7pt}}$ | 169.84${}^{\hskip 0.28436pt\scaleto{+0.22}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 2.28${}^{\hskip 0.28436pt\scaleto{+0.06}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | 2.34${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 81.88${}^{\hskip 0.28436pt\scaleto{+1.47}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.43}{4.7pt}}$
CAG-4 | 33.34${}^{\hskip 0.28436pt\scaleto{+1.87}{4.7pt}}_{\hskip 0.28436pt\scaleto{-2.04}{4.7pt}}$ | 118.39${}^{\hskip 0.28436pt\scaleto{+2.98}{4.7pt}}_{\hskip 0.28436pt\scaleto{-3.16}{4.7pt}}$ | 81.29${}^{\hskip 0.28436pt\scaleto{+0.26}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.26}{4.7pt}}$ | -69.68${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | 444.4${}^{\hskip 0.28436pt\scaleto{+1.65}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.59}{4.7pt}}$ | 168.33${}^{\hskip 0.28436pt\scaleto{+0.46}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.46}{4.7pt}}$ | 1.98${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.18}{4.7pt}}$ | 3.69${}^{\hskip 0.28436pt\scaleto{+0.85}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.64}{4.7pt}}$ | 79.6${}^{\hskip 0.28436pt\scaleto{+1.62}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.56}{4.7pt}}$
FAG-4 | 30.38${}^{\hskip 0.28436pt\scaleto{+1.06}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.1}{4.7pt}}$ | 125.54${}^{\hskip 0.28436pt\scaleto{+1.75}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.84}{4.7pt}}$ | 80.6${}^{\hskip 0.28436pt\scaleto{+0.2}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.2}{4.7pt}}$ | -69.57${}^{\hskip 0.28436pt\scaleto{+0.06}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.06}{4.7pt}}$ | 445.6${}^{\hskip 0.28436pt\scaleto{+1.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.13}{4.7pt}}$ | 169.5${}^{\hskip 0.28436pt\scaleto{+0.37}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.36}{4.7pt}}$ | 2.54${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 3.1${}^{\hskip 0.28436pt\scaleto{+0.4}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.35}{4.7pt}}$ | 81.96${}^{\hskip 0.28436pt\scaleto{+1.54}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.5}{4.7pt}}$

  * •

As noted in the previous subsections, the COM PM shows minimal variations
across different age groups. Even slight variations in the parameter $v_{t}$
estimated as in Table 4 for different age groups do not result in substantial
shifts in $\mu_{W,com}$ and $\mu_{N,com}$.

  * •

The clusters ranging in age from $\sim$ 100 Myr to 1.25 Gyr show a small
offset in the ($\alpha_{0}$, $\delta_{0}$) to the South-East with respect to
the older clusters, as evident from CAG-2 and CAG-3. Meanwhile, the nearby
field population in these age groups shows variation only in the South.

  * •

$\Theta$ attains its minimum value of 111∘.11${}^{\hskip
0.28436pt\scaleto{+2.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-2.04}{4.7pt}}$ for
the CAG-3, with field also showing a similar trend. Notably, the $i$
(35∘.36${}^{\hskip 0.28436pt\scaleto{+1.2}{4.7pt}}_{\hskip
0.28436pt\scaleto{-1.31}{4.7pt}}$) for CAG-3 is the largest when compared to
other age groups. In the case of the field regions, we do not detect any
significant shift in $i$, but a smaller shift in $\Theta$ is noted.

  * •

In the cluster data sets, the minimum value of $R_{0}$ is estimated to be
1.34${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip
0.28436pt\scaleto{-0.16}{4.7pt}}$ kpc for the CAG-3, while its maximum value
occurs at 3.14${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip
0.28436pt\scaleto{-0.22}{4.7pt}}$ kpc for the CAG-1. Similarly, in the field
data sets, there is a comparable pattern with the minimum and maximum values
of 2.28${}^{\hskip 0.28436pt\scaleto{+0.06}{4.7pt}}_{\hskip
0.28436pt\scaleto{-0.07}{4.7pt}}$ kpc and 2.84${}^{\hskip
0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ kpc
for FAG-3 and FAG-1, respectively. $\eta$ also shows a similar trend in the
age-based data sets.

  * •

The values of $v_{0}$ remain relatively consistent across all datasets, and
any observed changes are not significant, as they are within the margin of
errors.

As noted in the above comparison, there is a noticeable deviation of kinematic
parameters for the cluster age groups, in contrast to the datasets of the
field region associated with each age group. The comparison presented in this
section has shown that the dependence of kinematic parameters on the age of
the population is indeed present, and the variations are statistically
significant. In the following subsections, we further investigate the
dependence of estimated parameters on the cluster richness as mentioned in
Section 2.

###  4.4 Influence of cluster richness on the kinematic model

Table 5 shows the estimated kinematic parameters for the data sets based on
cluster richness, as outlined in Section 2. The cluster data sets and the
nearby field regions are modeled based on cluster richness, which helped in
checking for any kinematic changes in the model based on poor and rich
clusters.

Table 5: The kinematic best-fitting parameters obtained for the LMC based on the cluster richness and their nearby field regions are tabulated below. Data | $i$ | $\Theta$ | $\alpha_{0}$ | $\delta_{0}$ | $v_{t}$ | $\theta_{t}$ | $R_{0}$ | $\eta$ | $v_{0}$
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (deg) | (deg) | (km s-1) | (deg) | (kpc) |  | (km s-1)
C10 | 31.78${}^{\hskip 0.28436pt\scaleto{+0.72}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.75}{4.7pt}}$ | 122.19${}^{\hskip 0.28436pt\scaleto{+1.48}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.52}{4.7pt}}$ | 81.7${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.61${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.87${}^{\hskip 0.28436pt\scaleto{+0.55}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.54}{4.7pt}}$ | 167.87${}^{\hskip 0.28436pt\scaleto{+0.22}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 1.59${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.85${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 83.99${}^{\hskip 0.28436pt\scaleto{+1.3}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.24}{4.7pt}}$
F10 | 30.78${}^{\hskip 0.28436pt\scaleto{+0.4}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.41}{4.7pt}}$ | 127.85${}^{\hskip 0.28436pt\scaleto{+0.82}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.79}{4.7pt}}$ | 80.59${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | -69.35${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.82${}^{\hskip 0.28436pt\scaleto{+0.35}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.35}{4.7pt}}$ | 169.43${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | 2.41${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 2.34${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 84.15${}^{\hskip 0.28436pt\scaleto{+1.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.14}{4.7pt}}$
C20 | 31.76${}^{\hskip 0.28436pt\scaleto{+0.85}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.88}{4.7pt}}$ | 119.48${}^{\hskip 0.28436pt\scaleto{+1.73}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.73}{4.7pt}}$ | 81.85${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | -69.62${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.81${}^{\hskip 0.28436pt\scaleto{+0.58}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.61}{4.7pt}}$ | 167.59${}^{\hskip 0.28436pt\scaleto{+0.24}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.23}{4.7pt}}$ | 1.47${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | 1.81${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 82.89${}^{\hskip 0.28436pt\scaleto{+1.34}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.33}{4.7pt}}$
F20 | 30.18${}^{\hskip 0.28436pt\scaleto{+0.49}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.49}{4.7pt}}$ | 127.05${}^{\hskip 0.28436pt\scaleto{+0.94}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.97}{4.7pt}}$ | 80.59${}^{\hskip 0.28436pt\scaleto{+0.09}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.09}{4.7pt}}$ | -69.34${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.55${}^{\hskip 0.28436pt\scaleto{+0.4}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.4}{4.7pt}}$ | 169.43${}^{\hskip 0.28436pt\scaleto{+0.16}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 2.41${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 2.55${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 82.22${}^{\hskip 0.28436pt\scaleto{+1.19}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.15}{4.7pt}}$
C30 | 31.62${}^{\hskip 0.28436pt\scaleto{+0.94}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.98}{4.7pt}}$ | 119.72${}^{\hskip 0.28436pt\scaleto{+1.91}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.94}{4.7pt}}$ | 82.06${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | -69.63${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 445.06${}^{\hskip 0.28436pt\scaleto{+0.65}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.64}{4.7pt}}$ | 167.23${}^{\hskip 0.28436pt\scaleto{+0.24}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.24}{4.7pt}}$ | 1.31${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 1.69${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 82.59${}^{\hskip 0.28436pt\scaleto{+1.39}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.39}{4.7pt}}$
F30 | 29.86${}^{\hskip 0.28436pt\scaleto{+0.55}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.55}{4.7pt}}$ | 127.06${}^{\hskip 0.28436pt\scaleto{+1.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.13}{4.7pt}}$ | 80.72${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.1}{4.7pt}}$ | -69.33${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 442.4${}^{\hskip 0.28436pt\scaleto{+0.46}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.46}{4.7pt}}$ | 169.19${}^{\hskip 0.28436pt\scaleto{+0.19}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.18}{4.7pt}}$ | 2.42${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 2.58${}^{\hskip 0.28436pt\scaleto{+0.19}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.18}{4.7pt}}$ | 81.9${}^{\hskip 0.28436pt\scaleto{+1.27}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.24}{4.7pt}}$
C40 | 31.04${}^{\hskip 0.28436pt\scaleto{+1.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.08}{4.7pt}}$ | 118.5${}^{\hskip 0.28436pt\scaleto{+2.09}{4.7pt}}_{\hskip 0.28436pt\scaleto{-2.12}{4.7pt}}$ | 82.2${}^{\hskip 0.28436pt\scaleto{+0.16}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.17}{4.7pt}}$ | -69.61${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 445.07${}^{\hskip 0.28436pt\scaleto{+0.7}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.66}{4.7pt}}$ | 166.97${}^{\hskip 0.28436pt\scaleto{+0.28}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.28}{4.7pt}}$ | 1.34${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 1.68${}^{\hskip 0.28436pt\scaleto{+0.19}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.17}{4.7pt}}$ | 82.04${}^{\hskip 0.28436pt\scaleto{+1.47}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.42}{4.7pt}}$
F40 | 28.81${}^{\hskip 0.28436pt\scaleto{+0.63}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.64}{4.7pt}}$ | 126.27${}^{\hskip 0.28436pt\scaleto{+1.31}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.37}{4.7pt}}$ | 80.82${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | -69.33${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 442.28${}^{\hskip 0.28436pt\scaleto{+0.5}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.51}{4.7pt}}$ | 169.04${}^{\hskip 0.28436pt\scaleto{+0.2}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.2}{4.7pt}}$ | 2.35${}^{\hskip 0.28436pt\scaleto{+0.06}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.06}{4.7pt}}$ | 2.52${}^{\hskip 0.28436pt\scaleto{+0.2}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.19}{4.7pt}}$ | 80.96${}^{\hskip 0.28436pt\scaleto{+1.32}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.26}{4.7pt}}$
C50 | 31.05${}^{\hskip 0.28436pt\scaleto{+1.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.24}{4.7pt}}$ | 116.34${}^{\hskip 0.28436pt\scaleto{+2.27}{4.7pt}}_{\hskip 0.28436pt\scaleto{-2.3}{4.7pt}}$ | 82.36${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.17}{4.7pt}}$ | -69.65${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 445.58${}^{\hskip 0.28436pt\scaleto{+0.73}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.74}{4.7pt}}$ | 166.76${}^{\hskip 0.28436pt\scaleto{+0.29}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.29}{4.7pt}}$ | 1.15${}^{\hskip 0.28436pt\scaleto{+0.15}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 1.59${}^{\hskip 0.28436pt\scaleto{+0.19}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.17}{4.7pt}}$ | 81.29${}^{\hskip 0.28436pt\scaleto{+1.52}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.47}{4.7pt}}$
F50 | 28.94${}^{\hskip 0.28436pt\scaleto{+0.69}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.72}{4.7pt}}$ | 126.19${}^{\hskip 0.28436pt\scaleto{+1.5}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.53}{4.7pt}}$ | 81.06${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | -69.37${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 442.56${}^{\hskip 0.28436pt\scaleto{+0.51}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.53}{4.7pt}}$ | 168.66${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.21}{4.7pt}}$ | 2.18${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | 2.37${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.18}{4.7pt}}$ | 80.25${}^{\hskip 0.28436pt\scaleto{+1.38}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.31}{4.7pt}}$
C60 | 29.75${}^{\hskip 0.28436pt\scaleto{+1.35}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.4}{4.7pt}}$ | 116.94${}^{\hskip 0.28436pt\scaleto{+2.6}{4.7pt}}_{\hskip 0.28436pt\scaleto{-2.69}{4.7pt}}$ | 82.5${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.17}{4.7pt}}$ | -69.69${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 445.95${}^{\hskip 0.28436pt\scaleto{+0.77}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.75}{4.7pt}}$ | 166.49${}^{\hskip 0.28436pt\scaleto{+0.29}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.3}{4.7pt}}$ | 0.95${}^{\hskip 0.28436pt\scaleto{+0.16}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 1.43${}^{\hskip 0.28436pt\scaleto{+0.19}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 80.06${}^{\hskip 0.28436pt\scaleto{+1.56}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.58}{4.7pt}}$
F60 | 28.13${}^{\hskip 0.28436pt\scaleto{+0.77}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.8}{4.7pt}}$ | 127.28${}^{\hskip 0.28436pt\scaleto{+1.76}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.77}{4.7pt}}$ | 81.34${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | -69.42${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 442.91${}^{\hskip 0.28436pt\scaleto{+0.58}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.56}{4.7pt}}$ | 168.16${}^{\hskip 0.28436pt\scaleto{+0.22}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 2.07${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | 2.12${}^{\hskip 0.28436pt\scaleto{+0.19}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.17}{4.7pt}}$ | 80.47${}^{\hskip 0.28436pt\scaleto{+1.44}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.46}{4.7pt}}$
Figure 5: The variation of kinematic parameters based on cluster richness is
shown here. Estimated parameters for cluster groups (C10 to C60) and nearby
field groups (F10 to F60) are shown in red and blue colors from Panel (a) to
(f). (a) Variation in ($\alpha_{0}$, $\delta_{0}$); (b) Variation in $i$; (c)
Variation in $\Theta$; (d) Variation in ($\mu_{W,com}$, $\mu_{N,com}$); (e)
Variation in $R_{0}$; (f) Variation in $v_{0}$.

Figure 5 shows the variation of the kinematic parameters corresponding to
different cluster and field groups based on cluster richness. We note an
offset of dynamic centers from poor to rich clusters in the increasing RA
directions for both clusters and field populations. Meanwhile, the shift of
COM PM for poor to rich clusters is seen increasing in North and West
directions. Similarly, estimates for $\Phi$, $\Theta$, $R_{0}$, and $v_{0}$
show a decrease from C/F${10}$ to C/F${60}$ groups (as seen in Table 5).
However, a notable deviation becomes apparent in the kinematic parameters when
considering entire cluster groups compared to field groups. This once again
indicates that the kinematic model for clusters and the field population
indeed manifests distinct kinematic properties. We also note that the cluster
richness groups do not have similar spatial coverage. Overall, we do detect
variation in the COM PM, dynamic center, $i$, and $R_{0}$, which may partially
be due to the spatial coverage, and the variation in $\Theta$ is statistically
insignificant.

###  4.5 Impact of the LMC’s spatial coverage on kinematic properties

Table 6 shows the estimated kinematic parameters for the data sets based on
the spatial coverage of the LMC. The circular regions of radii increasing from
2∘ to 7∘ with a step size of 0.5∘ from the kinematic center (from the primary
model, see Table 2) of the LMC were used to perform the modeling.

Table 6: The kinematic best-fitting parameters obtained for the LMC based on the spatial coverage of the LMC with clusters and nearby field regions are tabulated below. Data | $i$ | $\Theta$ | $\alpha_{0}$ | $\delta_{0}$ | $v_{t}$ | $\theta_{t}$ | $R_{0}$ | $\eta$ | $v_{0}$
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (deg) | (deg) | (km s-1) | (deg) | (kpc) |  | (km s-1)
Cr[2.0] | 49.93${}^{\hskip 0.28436pt\scaleto{+1.36}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.41}{4.7pt}}$ | 104.79${}^{\hskip 0.28436pt\scaleto{+1.64}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.63}{4.7pt}}$ | 82.45${}^{\hskip 0.28436pt\scaleto{+0.3}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.29}{4.7pt}}$ | -69.89${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 457.7${}^{\hskip 0.28436pt\scaleto{+1.59}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.56}{4.7pt}}$ | 166.76${}^{\hskip 0.28436pt\scaleto{+0.53}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.56}{4.7pt}}$ | 1.41${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | 3.38${}^{\hskip 0.28436pt\scaleto{+0.99}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.67}{4.7pt}}$ | 75.84${}^{\hskip 0.28436pt\scaleto{+1.78}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.81}{4.7pt}}$
Fr[2.0] | 42.89${}^{\hskip 0.28436pt\scaleto{+0.77}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.8}{4.7pt}}$ | 110.07${}^{\hskip 0.28436pt\scaleto{+1.06}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.07}{4.7pt}}$ | 81.58${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.86${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 455.62${}^{\hskip 0.28436pt\scaleto{+0.82}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.76}{4.7pt}}$ | 167.66${}^{\hskip 0.28436pt\scaleto{+0.24}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.24}{4.7pt}}$ | 1.49${}^{\hskip 0.28436pt\scaleto{+0.09}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | 1.57${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | 77.5${}^{\hskip 0.28436pt\scaleto{+1.76}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.72}{4.7pt}}$
Cr[2.5] | 44.38${}^{\hskip 0.28436pt\scaleto{+1.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.17}{4.7pt}}$ | 106.01${}^{\hskip 0.28436pt\scaleto{+1.53}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.5}{4.7pt}}$ | 82.4${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.21}{4.7pt}}$ | -69.78${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 452.18${}^{\hskip 0.28436pt\scaleto{+0.96}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.96}{4.7pt}}$ | 166.59${}^{\hskip 0.28436pt\scaleto{+0.38}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.39}{4.7pt}}$ | 1.38${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 2.69${}^{\hskip 0.28436pt\scaleto{+0.52}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.41}{4.7pt}}$ | 76.94${}^{\hskip 0.28436pt\scaleto{+1.71}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.62}{4.7pt}}$
Fr[2.5] | 37.97${}^{\hskip 0.28436pt\scaleto{+0.64}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.66}{4.7pt}}$ | 115.03${}^{\hskip 0.28436pt\scaleto{+0.96}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.96}{4.7pt}}$ | 81.12${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | -69.79${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 453.23${}^{\hskip 0.28436pt\scaleto{+0.69}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.68}{4.7pt}}$ | 168.55${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.21}{4.7pt}}$ | 1.87${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | 1.93${}^{\hskip 0.28436pt\scaleto{+0.15}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 80.62${}^{\hskip 0.28436pt\scaleto{+1.62}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.64}{4.7pt}}$
Cr[3.0] | 39.51${}^{\hskip 0.28436pt\scaleto{+1.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.08}{4.7pt}}$ | 109.7${}^{\hskip 0.28436pt\scaleto{+1.53}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.55}{4.7pt}}$ | 82.16${}^{\hskip 0.28436pt\scaleto{+0.15}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | -69.71${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 448.62${}^{\hskip 0.28436pt\scaleto{+0.76}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.78}{4.7pt}}$ | 167.02${}^{\hskip 0.28436pt\scaleto{+0.27}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.27}{4.7pt}}$ | 1.32${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 2.19${}^{\hskip 0.28436pt\scaleto{+0.33}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.27}{4.7pt}}$ | 77.67${}^{\hskip 0.28436pt\scaleto{+1.61}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.58}{4.7pt}}$
Fr[3.0] | 33.93${}^{\hskip 0.28436pt\scaleto{+0.54}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.56}{4.7pt}}$ | 119.91${}^{\hskip 0.28436pt\scaleto{+0.92}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.91}{4.7pt}}$ | 80.74${}^{\hskip 0.28436pt\scaleto{+0.09}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.1}{4.7pt}}$ | -69.63${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 448.94${}^{\hskip 0.28436pt\scaleto{+0.49}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.49}{4.7pt}}$ | 169.2${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.17}{4.7pt}}$ | 2.04${}^{\hskip 0.28436pt\scaleto{+0.06}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.06}{4.7pt}}$ | 2.03${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | 81.53${}^{\hskip 0.28436pt\scaleto{+1.55}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.58}{4.7pt}}$
Cr[3.5] | 37.35${}^{\hskip 0.28436pt\scaleto{+0.94}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.99}{4.7pt}}$ | 113.34${}^{\hskip 0.28436pt\scaleto{+1.49}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.54}{4.7pt}}$ | 81.98${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.71${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 447.66${}^{\hskip 0.28436pt\scaleto{+0.71}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.68}{4.7pt}}$ | 167.43${}^{\hskip 0.28436pt\scaleto{+0.23}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.23}{4.7pt}}$ | 1.32${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 2.09${}^{\hskip 0.28436pt\scaleto{+0.27}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.23}{4.7pt}}$ | 78.5${}^{\hskip 0.28436pt\scaleto{+1.53}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.45}{4.7pt}}$
Fr[3.5] | 32.35${}^{\hskip 0.28436pt\scaleto{+0.49}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.5}{4.7pt}}$ | 122.78${}^{\hskip 0.28436pt\scaleto{+0.86}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.86}{4.7pt}}$ | 80.59${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | -69.5${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 445.93${}^{\hskip 0.28436pt\scaleto{+0.41}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.41}{4.7pt}}$ | 169.45${}^{\hskip 0.28436pt\scaleto{+0.15}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 2.15${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 2.13${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 81.67${}^{\hskip 0.28436pt\scaleto{+1.51}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.47}{4.7pt}}$
Cr[4.0] | 35.03${}^{\hskip 0.28436pt\scaleto{+0.9}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.96}{4.7pt}}$ | 115.7${}^{\hskip 0.28436pt\scaleto{+1.56}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.57}{4.7pt}}$ | 81.87${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.69${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 446.66${}^{\hskip 0.28436pt\scaleto{+0.66}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.66}{4.7pt}}$ | 167.61${}^{\hskip 0.28436pt\scaleto{+0.23}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 1.35${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.95${}^{\hskip 0.28436pt\scaleto{+0.22}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.2}{4.7pt}}$ | 79.76${}^{\hskip 0.28436pt\scaleto{+1.51}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.48}{4.7pt}}$
Fr[4.0] | 30.64${}^{\hskip 0.28436pt\scaleto{+0.46}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.49}{4.7pt}}$ | 125.48${}^{\hskip 0.28436pt\scaleto{+0.85}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.87}{4.7pt}}$ | 80.39${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | -69.38${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 443.67${}^{\hskip 0.28436pt\scaleto{+0.42}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.41}{4.7pt}}$ | 169.82${}^{\hskip 0.28436pt\scaleto{+0.15}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 2.37${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 2.29${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 83.74${}^{\hskip 0.28436pt\scaleto{+1.49}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.42}{4.7pt}}$
Cr[4.5] | 33.33${}^{\hskip 0.28436pt\scaleto{+0.88}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.91}{4.7pt}}$ | 118.37${}^{\hskip 0.28436pt\scaleto{+1.48}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.54}{4.7pt}}$ | 81.63${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.61${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.86${}^{\hskip 0.28436pt\scaleto{+0.63}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.64}{4.7pt}}$ | 167.99${}^{\hskip 0.28436pt\scaleto{+0.23}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.23}{4.7pt}}$ | 1.55${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.94${}^{\hskip 0.28436pt\scaleto{+0.19}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.17}{4.7pt}}$ | 82.72${}^{\hskip 0.28436pt\scaleto{+1.39}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.34}{4.7pt}}$
Fr[4.5] | 30.43${}^{\hskip 0.28436pt\scaleto{+0.45}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.45}{4.7pt}}$ | 126.41${}^{\hskip 0.28436pt\scaleto{+0.87}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.85}{4.7pt}}$ | 80.44${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | -69.35${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.91${}^{\hskip 0.28436pt\scaleto{+0.38}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.39}{4.7pt}}$ | 169.73${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 2.43${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 2.29${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 84.77${}^{\hskip 0.28436pt\scaleto{+1.43}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.36}{4.7pt}}$
Cr[5.0] | 32.34${}^{\hskip 0.28436pt\scaleto{+0.86}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.89}{4.7pt}}$ | 119.75${}^{\hskip 0.28436pt\scaleto{+1.53}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.5}{4.7pt}}$ | 81.61${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.6${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.66${}^{\hskip 0.28436pt\scaleto{+0.62}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.6}{4.7pt}}$ | 168.04${}^{\hskip 0.28436pt\scaleto{+0.22}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 1.6${}^{\hskip 0.28436pt\scaleto{+0.11}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.9${}^{\hskip 0.28436pt\scaleto{+0.18}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 83.67${}^{\hskip 0.28436pt\scaleto{+1.39}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.35}{4.7pt}}$
Fr[5.0] | 30.39${}^{\hskip 0.28436pt\scaleto{+0.42}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.44}{4.7pt}}$ | 127.44${}^{\hskip 0.28436pt\scaleto{+0.83}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.85}{4.7pt}}$ | 80.5${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | -69.34${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.65${}^{\hskip 0.28436pt\scaleto{+0.36}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.36}{4.7pt}}$ | 169.61${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | 2.45${}^{\hskip 0.28436pt\scaleto{+0.05}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.05}{4.7pt}}$ | 2.27${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 85.32${}^{\hskip 0.28436pt\scaleto{+1.32}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.31}{4.7pt}}$
Cr[5.5] | 31.68${}^{\hskip 0.28436pt\scaleto{+0.81}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.85}{4.7pt}}$ | 120.95${}^{\hskip 0.28436pt\scaleto{+1.52}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.52}{4.7pt}}$ | 81.64${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.61${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.8${}^{\hskip 0.28436pt\scaleto{+0.56}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.57}{4.7pt}}$ | 168.0${}^{\hskip 0.28436pt\scaleto{+0.22}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 1.58${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.1}{4.7pt}}$ | 1.86${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 83.55${}^{\hskip 0.28436pt\scaleto{+1.36}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.29}{4.7pt}}$
Fr[5.5] | 30.58${}^{\hskip 0.28436pt\scaleto{+0.41}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.42}{4.7pt}}$ | 127.76${}^{\hskip 0.28436pt\scaleto{+0.8}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.82}{4.7pt}}$ | 80.54${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$ | -69.35${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.79${}^{\hskip 0.28436pt\scaleto{+0.36}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.37}{4.7pt}}$ | 169.52${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | 2.45${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 2.28${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 85.28${}^{\hskip 0.28436pt\scaleto{+1.32}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.22}{4.7pt}}$
Cr[6.0] | 31.35${}^{\hskip 0.28436pt\scaleto{+0.76}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.78}{4.7pt}}$ | 121.69${}^{\hskip 0.28436pt\scaleto{+1.52}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.53}{4.7pt}}$ | 81.66${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | -69.61${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.79${}^{\hskip 0.28436pt\scaleto{+0.53}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.55}{4.7pt}}$ | 167.95${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.22}{4.7pt}}$ | 1.58${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.84${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 83.68${}^{\hskip 0.28436pt\scaleto{+1.32}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.25}{4.7pt}}$
Fr[6.0] | 30.66${}^{\hskip 0.28436pt\scaleto{+0.4}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.41}{4.7pt}}$ | 127.96${}^{\hskip 0.28436pt\scaleto{+0.78}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.8}{4.7pt}}$ | 80.57${}^{\hskip 0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | -69.34${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.68${}^{\hskip 0.28436pt\scaleto{+0.34}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.35}{4.7pt}}$ | 169.47${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.14}{4.7pt}}$ | 2.44${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 2.3${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 84.87${}^{\hskip 0.28436pt\scaleto{+1.26}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.19}{4.7pt}}$
Cr[6.5] | 31.61${}^{\hskip 0.28436pt\scaleto{+0.75}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.77}{4.7pt}}$ | 121.87${}^{\hskip 0.28436pt\scaleto{+1.54}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.54}{4.7pt}}$ | 81.65${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | -69.6${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.59${}^{\hskip 0.28436pt\scaleto{+0.54}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.53}{4.7pt}}$ | 167.96${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.21}{4.7pt}}$ | 1.57${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.87${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.16}{4.7pt}}$ | 83.35${}^{\hskip 0.28436pt\scaleto{+1.3}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.24}{4.7pt}}$
Fr[6.5] | 30.72${}^{\hskip 0.28436pt\scaleto{+0.39}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.4}{4.7pt}}$ | 127.99${}^{\hskip 0.28436pt\scaleto{+0.79}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.8}{4.7pt}}$ | 80.57${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | -69.34${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.66${}^{\hskip 0.28436pt\scaleto{+0.34}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.33}{4.7pt}}$ | 169.47${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 2.44${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 2.34${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 84.45${}^{\hskip 0.28436pt\scaleto{+1.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.15}{4.7pt}}$
Cr[7.0] | 31.54${}^{\hskip 0.28436pt\scaleto{+0.73}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.74}{4.7pt}}$ | 122.31${}^{\hskip 0.28436pt\scaleto{+1.47}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.52}{4.7pt}}$ | 81.69${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | -69.6${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | 444.55${}^{\hskip 0.28436pt\scaleto{+0.54}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.52}{4.7pt}}$ | 167.89${}^{\hskip 0.28436pt\scaleto{+0.21}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.21}{4.7pt}}$ | 1.58${}^{\hskip 0.28436pt\scaleto{+0.1}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.11}{4.7pt}}$ | 1.84${}^{\hskip 0.28436pt\scaleto{+0.16}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.15}{4.7pt}}$ | 83.9${}^{\hskip 0.28436pt\scaleto{+1.25}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.24}{4.7pt}}$
Fr[7.0] | 30.79${}^{\hskip 0.28436pt\scaleto{+0.39}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.39}{4.7pt}}$ | 128.04${}^{\hskip 0.28436pt\scaleto{+0.78}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.8}{4.7pt}}$ | 80.57${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | -69.34${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | 442.7${}^{\hskip 0.28436pt\scaleto{+0.34}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.34}{4.7pt}}$ | 169.46${}^{\hskip 0.28436pt\scaleto{+0.13}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 2.43${}^{\hskip 0.28436pt\scaleto{+0.04}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.04}{4.7pt}}$ | 2.36${}^{\hskip 0.28436pt\scaleto{+0.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.13}{4.7pt}}$ | 84.18${}^{\hskip 0.28436pt\scaleto{+1.14}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.1}{4.7pt}}$
Figure 6: The variation of kinematic parameters based on the spatial coverage
of the LMC is shown here. Estimated parameters for different spatial coverage
from 2∘ to 7∘ with a step size of 0.5∘ from the center (based on the primary
model) of the LMC are shown here. The corresponding cluster data sets (Cr=2 to
Cr=7) and nearby field data sets (Fr=2 to Fr=7)are shown in red and blue
colors from Panel (a) to (f). (a) Variation in ($\alpha_{0}$, $\delta_{0}$);
(b) Variation in $i$; (c) Variation in $\Theta$; (d) Variation in
($\mu_{W,com}$, $\mu_{N,com}$); (e) Variation in $R_{0}$; (f) Variation in
$v_{0}$.

Figure 6 shows the radial variation of the kinematic parameters in the case of
clusters and nearby field regions. Within the radii considered in this study,
there is a notable offset in the amplitude of PM, of the order of $\sim$ 0.06
mas yr-1 in ($\mu_{W,com}$, $\mu_{N,com}$) between the inner and outer
cluster/field regions. Similarly, we note a positional offset of $\sim$ 0.4∘
for clusters and $\sim$ 0.6∘ for the field between the estimated values of
$\alpha_{0}$ and $\delta_{0}$. The value of $i$ steeply decreases from the
inner to outer radii, from $\sim$ 50∘ to $\sim$ 32∘ in the case of clusters,
and $\sim$ 43∘ to $\sim$ 31∘ in the case of field population. Meanwhile, the
value of $\Theta$ increases from inner to outer radii, from $\sim$ 108∘ to
$\sim$ 122∘ in the case of clusters, and $\sim$ 110∘ to $\sim$ 128∘ in the
case of field population. The rotational parameters, $R_{0}$ and $v_{0}$ after
5∘ show convergence to the values estimated as in the primary model (see Table
2). Therefore, the estimated parameters show a significant radial dependence
for both clusters and field populations.

###  4.6 Residual PM of the LMC: Cluster vs Field

The residual PM value for the clusters and nearby field populations is found
by subtracting the net modeled PM from the net observed PM values. The
residual PM vectors for clusters and field regions are plotted in the X-Y
plane as shown in Figure 7, panels (a) and (b), respectively. Clusters show
larger residual PM in the spatial plot when compared with the field
population. The spatial residual plot for clusters shows relatively large
residuals in the bar region and in the northern LMC, whereas such large
residuals are not found in the corresponding plot for the field population.
The residual PM amplitudes ($|$Residual PM$|$) of both the clusters and the
nearby field are used to generate the probability distribution plot as in
Figure 7, panel (c) depicting the distribution of their values. We obtained an
RMS value of 0.146$\pm$0.002 mas yr-1 for clusters and 0.069$\pm$0.001 mas
yr-1 for the nearby field population. The RMS distribution of clusters shows a
broader profile, while the field shows a narrower one.

Figure 7: Residual PM of clusters and field population. (a) Spatial plot of
residual PM vectors for the clusters; (b) Spatial distribution of residual PM
vectors for the field population; (c) The Gaussian KDE showing the
distribution of the $|$Residual PM$|$.

##  5 Discussion

In this study, we performed the kinematic modeling of the LMC using 1705 star
clusters and nearby field regions for the first time. We also created an
extensive additional 46 models using two control samples (young MS stars and
RC stars), different cluster age groups and nearby fields, cluster richness
groups, and samples with varying spatial coverage of the LMC. The clusters
were cleaned using field star contamination (D24) and found less than 2% of MW
source contamination in each cluster while cross-matching with the MW/ LMC
source catalog by J23. The median value of the PM is used for the study for
both clusters and fields and hence is unlikely to be affected by the
foreground MW contamination.

In the following subsections, we discuss the important results we obtained
from the modeling. We compare the estimated kinematic parameters with the
previous studies, analyze the rotation curve of the LMC for the control
samples, trace the spatial variation of residual PM across the LMC, analyze
the kinematic outliers from the model, and comment on the kinematic variation
in parameters based on cluster age.

###  5.1 Comparison of kinematic properties with previous studies

The estimated kinematic parameters of the LMC are compared with those from
previous studies, mainly in which either the authors estimated the kinematic
centers and COM PM values or adopted them. First we compare the estimations of
($\alpha_{0}$, $\delta_{0}$), ($\mu_{W,com}$, $\mu_{N,com}$), $i$, and
$\Theta$ from studies by N22, C22, G21, W20, and van der Marel & Kallivayalil
(2014, hereafter V14). The parameters estimated are tabulated in Table 7 along
with estimations from this study for comparison.

Table 7: Comparison of the estimated kinematic parameters with previous studies. The first two rows list the estimations from this study. $\alpha_{0}$ | $\delta_{0}$ | $\mu_{W,\text{com}}$ | $\mu_{N,\text{com}}$ | $i$ | $\Theta$ | Reference
---|---|---|---|---|---|---
(deg) | (deg) | (deg) | (deg) | (deg) | (deg) |
81.69${}^{\hskip 0.28436pt\scaleto{+0.12}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.12}{4.7pt}}$ | -69.59${}^{\hskip 0.28436pt\scaleto{+0.03}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.03}{4.7pt}}$ | -1.849${}^{\scaleto{+0.003}{4.7pt}}_{\scaleto{-0.003}{4.7pt}}$ | 0.397 ${}^{\scaleto{+0.007}{4.7pt}}_{\scaleto{-0.007}{4.7pt}}$ | 31.39${}^{\hskip 0.28436pt\scaleto{+0.73}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.74}{4.7pt}}$ | 122.22${}^{\hskip 0.28436pt\scaleto{+1.48}{4.7pt}}_{\hskip 0.28436pt\scaleto{-1.51}{4.7pt}}$ | Clusters
80.57${}^{\hskip 0.28436pt\scaleto{+0.07}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.07}{4.7pt}}$ | -69.34${}^{\hskip 0.28436pt\scaleto{+0.02}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.02}{4.7pt}}$ | -1.851${}^{\scaleto{+0.002}{4.7pt}}_{\scaleto{-0.002}{4.7pt}}$ | 0.344 ${}^{\scaleto{+0.004}{4.7pt}}_{\scaleto{-0.004}{4.7pt}}$ | 30.81${}^{\hskip 0.28436pt\scaleto{+0.38}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.40}{4.7pt}}$ | 128.02${}^{\hskip 0.28436pt\scaleto{+0.79}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.81}{4.7pt}}$ | Field
$79.95^{\scaleto{+0.22}{4.7pt}}_{\scaleto{-0.23}{4.7pt}}$ | $-69.31^{\scaleto{+0.12}{4.7pt}}_{\scaleto{-0.11}{4.7pt}}$ | $-1.867^{\scaleto{+0.008}{4.7pt}}_{\scaleto{-0.008}{4.7pt}}$ | $0.314^{\scaleto{+0.014}{4.7pt}}_{\scaleto{-0.014}{4.7pt}}$ | $33.5^{\scaleto{+1.2}{4.7pt}}_{\scaleto{-1.3}{4.7pt}}$ | $129.8^{\scaleto{+1.9}{4.7pt}}_{\scaleto{-1.9}{4.7pt}}$ | N22 111Estimated using VMC data
$81.07$ | $-69.41$ | $-1.862^{\scaleto{+0.002}{4.7pt}}_{\scaleto{-0.002}{4.7pt}}$ | $0.383^{\scaleto{+0.002}{4.7pt}}_{\scaleto{-0.002}{4.7pt}}$ | $28.7^{\scaleto{+1.4}{4.7pt}}_{\scaleto{-1.5}{4.7pt}}$ | $126.0^{\scaleto{+2.5}{4.7pt}}_{\scaleto{-2.6}{4.7pt}}$ | N22 222Estimated with fixed ($\alpha_{0}$, $\delta_{0}$) based on G21, using VMC data
$80.443$ | $-69.272$ | $-1.859$ | $0.375$ | $23.396^{\scaleto{+0.493}{4.7pt}}_{\scaleto{-0.501}{4.7pt}}$ | $138.856^{\scaleto{+1.360}{4.7pt}}_{\scaleto{-1.370}{4.7pt}}$ | C22 333Estimated with fixed ($\alpha_{0}$, $\delta_{0}$) and COM PM, using RC population with Gaia EDR3
$81.07$ | $-69.41$ | $-1.847$ | $0.371$ | $33.28$ | $130.97$ | G21 444Estimated with Gaia EDR3
$81.28$ | $-69.78$ | $-1.858$ | $0.385$ | $34.08$ | $129.92$ | G21 555Estimated with fixed ($\alpha_{0}$, $\delta_{0}$), using Gaia EDR3
$80.90\pm 0.29$ | $-68.74\pm 0.12$ | $-1.878\pm 0.007$ | $0.293\pm 0.018$ | $25.6\pm 1.1$ | $135.6\pm 3.3$ | W20 666Estimated using carbon stars, with SkyMapper DR1
$81.23\pm 0.04$ | $-69.00\pm 0.02$ | $-1.824\pm 0.001$ | $0.355\pm 0.002$ | $26.1\pm 0.1$ | $134.1\pm 0.4$ | W20 777Estimated using RGB stars, with SkyMapper DR1
$80.98\pm 0.07$ | $-69.69\pm 0.02$ | $-1.860\pm 0.002$ | $0.359\pm 0.004$ | $29.4\pm 0.4$ | $152.0\pm 1.0$ | W20 888Estimated using Young stars, with SkyMapper DR1
$78.76\pm 0.52$ | $-69.19\pm 0.25$ | $-1.910\pm 0.020$ | $0.229\pm 0.047$ | $39.6\pm 4.5$ | $147.4\pm 10.0$ | V14 999Estimated with the third epoch of Hubble Space Telescope (HST) data, using PM.
$79.88\pm 0.83$ | $-69.59\pm 0.25$ | $-1.895\pm 0.024$ | $0.287\pm 0.054$ | $34.0\pm 7.0$ | $139.1\pm 4.1$ | V14 101010Estimated with the third epoch of HST data, using PM + vLOS of old stars.
$80.05\pm 0.34$ | $-69.30\pm 0.12$ | $-1.891\pm 0.018$ | $0.328\pm 0.025$ | $26.2\pm 5.9$ | $154.5\pm 2.1$ | V14 111111Estimated with the third epoch of HST data, using PM + vLOS of young stars.
Figure 8: The parameter space of the estimated ($\alpha_{0}$, $\delta_{0}$)
and ($\mu_{W,com}$, $\mu_{N,com}$) are compared with the reference studies
(Table 7) are provided in panels (a) and (b), respectively. The corresponding
estimated parameters for the clusters (CLS), field (FLS), young MS stars
(YMS), and RC stars (RCS) are marked with red dots along with labels. The
reference studies are marked with black dots along with labels.

Figure 8 shows the distribution of ($\alpha_{0}$, $\delta_{0}$) and
($\mu_{W,com}$, $\mu_{N,com}$) estimations from the literature studies as
mentioned above, along with the estimations from this study. We can notice a
significant variation in the values of ($\alpha_{0}$, $\delta_{0}$) from
various studies. Our estimations (shown in red) are located well within the
range of other estimations, except the $\alpha_{0}$ for the cluster
population. Most clusters ($\sim$ 75%) used for the modeling in this study are
of ages younger than 1 Gyr (D24). Due to this, the estimated values of
($\alpha_{0}$, $\delta_{0}$) for clusters are closer to the estimation by W20
using the young stellar population. Meanwhile, the estimation from the young
MS population is much closer to the W20 estimations. The field population and
RC population are closely in agreement with the recent studies by G21, V14,
N22, and C22. The offset of ($\alpha_{0}$, $\delta_{0}$) in the older and
younger stellar population was previously noted by W20.

The $\mu_{W,com}$ and $\mu_{N,com}$ values for the clusters and field
population also align with the recent studies by N22, C22, G21, and W20 in a
similar manner. Notably, the clusters show a relatively larger PM in the North
direction when compared to the field. However, the kinematic centers and COM
PM are distinct for the clusters and field population.

The values of $i$ and $\Theta$ estimated in this study are compared with the
previous studies (see Table 7) and are more or less in agreement. However, the
kinematic parameters such as $i$ and $\Theta$ are known to vary with respect
to the coverage of the LMC (N22). The variations, as seen from the table, also
contribute to the coverage. In this study, for the first time, we studied the
variation of kinematic parameters as a function of the spatial coverage of the
LMC using star clusters, as described in the subsection 4.5. The inner regions
of the LMC show a larger inclination with respect to outer regions, which is
already noted in the study by Saroon & Subramanian (2022) using RC population.

The following section discusses the rotational parameters ($v_{0}$, $R_{0}$)
and rotation curves of the LMC for various populations.

###  5.2 Rotation of the LMC

The rotation curves for the clusters, nearby fields, young MS stars, and RC
stars from the control sample (as mentioned in subsection 4.2) in the LMC
plane, after deprojection are shown in Figure 9 (a. Clusters, b. Nearby field,
c. young MS stars, d. RC stars). The running average for the data points is
made with a bin resolution of 0.25 kpc, and shown in red. The fitted model is
shown with the blue curve. The rotational velocity amplitude ($v_{0}$) does
not show a significant variation across the clusters and field population. The
estimated $v_{0}$ for clusters, fields, and RC stars are consistent with the
estimates by W20. The young MS stars in our study show a slightly larger
$v_{0}$ ($\sim$ 91 km s-1) when compared to the RC stars. However, it is
consistent with the observed $v_{0}$($\sim$ 90 km s-1) for the younger
population studied by N22.

In the cluster age-dependent kinematic model (subsection 4.3), the variation
of $R_{0}$ for cluster age groups is provided. The age group CAG-3 (log(age) =
8.0 - 8.65) and young MS stars from the control population show reduced values
of $R_{0}$ at 1.34${}^{\hskip 0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip
0.28436pt\scaleto{-0.16}{4.7pt}}$ kpc and 1.29${}^{\hskip
0.28436pt\scaleto{+0.08}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.08}{4.7pt}}$
kpc, respectively. This is suggestive of a steeper rise of the rotational
velocity when compared to the older population. The decreased $R_{0}$ for
these age groups may imply a redistribution of mass, leading to a higher mass
density in the central regions. Also, in the youngest group CAG-4 (log(age) =
6.55 - 8.0), the cluster density in the galaxy shifts towards the North-East
regions (D24). In CAG-4, there is a shift in the value of $R_{0}$ to a
slightly higher value of 1.98${}^{\hskip
0.28436pt\scaleto{+0.17}{4.7pt}}_{\hskip 0.28436pt\scaleto{-0.18}{4.7pt}}$
kpc. The field population near clusters is dominated by the older population,
hence the $R_{0}$ value remains high.

Panel (a) of Figure 10 shows all the four model velocity profiles fitted to
the control population provided in Figure 9. The $v_{0}$ for the control
population is almost identical ($\sim$ 81 to 84 kms-1), whereas the value for
the young MS stars remains slightly higher ($\sim$ 91 kms-1). It is clear from
this figure that the clusters and young MS stars show a steeper rise of the
Vrot suggesting an increased mass density in the inner regions. A similar
trend of small $R_{0}$ in the case of the younger population and larger
$R_{0}$ in the case of the older population is noted in the study by W20. The
variation in the value of $R_{0}$ with respect to population may be due to the
effect of the evolution of the bar and its activity with time, resulting in
the redistribution of mass in the inner regions.

Notably, the dispersion of the rotation velocity profile resides at $\sim$ 23
km s-1 and $\sim$ 11 km s-1 up to 6 kpc in the case of clusters and field
population, respectively. In the case of the young MS population, there is a
larger $\sigma_{rot}$ of $\sim$ 22 km s-1 up to 2 kpc, then a slight decline
and further rises after 4 kpc. Meanwhile, the RC population has a
$\sigma_{rot}$ of $\sim$ 20 km s-1 within 1 kpc and slowly declines and levels
off at $\sim$ 12 km s-1 afterward. The larger dispersion values in the central
2 kpc might be due to the non-circular motions due to the bar.

Panel (b) of Figure 10 shows the radial variation of the Vrot/$\sigma_{rot}$
ratio for the four control population. As all of the population have similar
Vrot (except slightly higher value for the young MS stars), the higher values
observed for the field stars suggest a relatively low $\sigma_{rot}$. The RC
population has a similar profile, though with a slightly lower ratio
(suggesting a relatively large $\sigma_{rot}$). The young MS stars have a more
or less similar profile till $\sim$ 3 kpc, and we note a lower value of the
ratio beyond this radius (suggesting a relatively large $\sigma_{rot}$). The
cluster has a shallow profile and lowest value for the ratio, suggestive of
the highest value of $\sigma_{rot}$ at all radii. Overall, the low value of
$\sigma_{rot}$ the field population points to the minimal disk heating for
this population. However, the ratios suggest that young clusters and young
stars have relatively large velocity dispersion, which is suggestive of
relatively recent perturbation(s) in the LMC disk. This may point to the
heating of the gas in the disk (resulting in the formation of stars with
similar kinematics), with minimal heating of the stellar disk. The heating may
be due to internal perturbations, such as the bar and spiral arms, or external
perturbation, which is the interaction with the SMC.

The rotational velocity maps shown in Figure 4 closely match with the velocity
maps from G21. Additionally, the variation in slope between evolved and
younger populations observed in Figure 10 is consistent with G21. However,
there is a discrepancy in the rotation velocity dispersion profiles when
compared to G21.

Figure 9: The rotation curve of the LMC based on the parameter estimation for
the control sample (Clusters, Nearby field, young MS stars, and RC stars) are
plotted in panels (a) to (d). The magnitude of the rotational velocity in the
LMC plane (Vrot) is shown with black dots in each panel. The red curve
represents the running average over the observed Vrot with a bin size of 0.25
kpc, and the blue curve shows the best-fitting model depending on the
rotational parameters ($v_{0}$, $R_{0}$, $\eta$) estimated for each data set
(see Table 3). Figure 10: Variataion of $R_{0}$ and $V_{rot}$/$\sigma_{rot}$
for the control population (Clusters, Nearby field, young MS stars, and RC
stars). (a) Modeled velocity profiles showing the variation in $R_{0}$ across
the control sample; (b) Variation of $V_{rot}$/$\sigma_{rot}$ across the
control sample, estimated using a bin size of 0.4 kpc in radii.

###  5.3 Spatial variation of residual PM: Clusters and Field

In order to trace regions with large variations in the residual PM, the mean
$|$Residual PM$|$ is traced for different radii (1 to 6∘ with a bin size of 1∘
using annular regions) as shown in Figure 11. We observe a similar trend in
radial variation of mean $|$Residual PM$|$ for both the clusters and field
population, though the values are significantly different. For inner radii
less than 2∘, the mean residual PM of the LMC is larger for both populations,
likely to be due to the presence of the bar. Beyond 2∘, we note a decline in
mean residual PM, and it increases after 4∘. This could be due to the presence
of spiral arms in the galaxy, where dense star formation is noted. However,
there is a clear signature of larger residual PM for clusters when compared to
the field population.

Figure 11: The radial variation of mean $|$Residual PM$|$ in the sky plane.
Annular regions with a bin size of 1∘ are used. Mean $|$Residual PM$|$ of
clusters and field population are marked with red and blue dots, respectively.

The residual PM RMS profiles in Figure 7 can be compared with the study by C22
based on the numerical simulations by B12 with a population older than 1 Gyr.
Figure 8 in C22 has provided the predicted $log\text{RMS}_{|\text{Residual
PM}|}$ as a function of the impact parameter (P in kpc) and impact timing (T
in Myr) of the recent LMC-SMC interaction. We adopted the value of T as 149
Myr, based on the significant CF peak in the SMC as observed in Figure 8b of
D24 to estimate the likely value of P. We note the assumed value of T is also
in good agreement with the estimate by Zivick et al. (2018), who suggests a
recent direct collision between the LMC and the SMC at 147 Myr ago. The field
$log\text{RMS}_{|\text{Residual PM}|}$ of $-$1.611 and $-$1.219 from the
primary model and control sample support the Future 40 and Future 60 models
provided in C22. We estimate that the impact parameter, P, must reside at less
than 3 kpc, and most likely match with the impact parameter of 2 kpc as per
B12 simulations. However, the RC stars in the control sample show a larger
residual (0.08 mas yr-1) compared to the estimates of C22 (0.058 mas yr-1). We
note that our model involves treating the COM PM and kinematic centers as free
parameters, unlike C22. Hence a direct comparison of our results with C22 may
be unrealistic.

The clusters and young stars show larger disc heating, with
$log\text{RMS}_{|\text{Residual PM}|}$ of -0.853 and -0.957. This suggests
that the young population is more perturbed, unlike the RC stars and other
field stars. Also, the clusters, irrespective of age, show larger residual PM.
The kinematic deviant clusters are analyzed in the next section.

###  5.4 Kinematically deviant clusters: Tracing the regions of larger
residual PM

The residual PM distribution of clusters, as shown in Figure 7a suggests that
some clusters have a large residual PM. We traced clusters with significantly
large residuals ($>$3$\sigma$ of $|$Residual PM$|$) and identified the region
with the highest density of such clusters, as illustrated in Figure 12(a) with
the 2D Gaussian KDE. We find a significant density of clusters in the North-
West at $\sim$ 1-2∘ with large residual PM. The bar region extending from
North-West to South-East of the LMC shows relatively large residuals in PM.

We note several outlier clusters in the observed vrot profile (Figure 9) of
the LMC. The clusters falling outside the 3$\sigma$ margin from the running
average of the vrot profile are treated as outliers. Their locations are
traced, and their internal PM is plotted in Figure 12b.

The residual PM vectors are plotted in Figure 12c for clusters with the
relative angular difference ($\theta_{r}$) between the observed and modeled
internal motion greater than 45∘ and 90∘. The red vectors indicate clusters
with $\theta_{r}>$ 90∘, suggestive of counter-rotation. These are dominantly
seen in the North-West of the galaxy. The deviant clusters also trace the non-
circular motion of the bar.

The North-West region with the largest residual PM is located between a radial
distance of 1-2∘. The same location also shows the presence of counter-
rotating clusters. This region has PM deviation both in angle (panel c) and in
value (panel a), pointing to the region experiencing an event of perturbation.
As this region is located at an impact distance similar to that estimated by
C22 and in this study, we speculate that this may be the area that experienced
the largest impact due to the recent LMC-SMC collision. We also note that
there could also be other reasons, such as perturbations that could occur at
the ends of the bar, but a similar disturbance is not seen at the eastern end
of the bar. Hence bar perturbation may not be the reason for this disturbance.

Figure 12: The clusters of larger residual PM are plotted here. (a) Spatial 2D
Gaussian KDE of cluster locations with $>3\sigma$ of $|$Residual PM$|$; (b)
Internal PM vectors of clusters $>3\sigma$ and $<3\sigma$ in Vrot profile of
Figure 9(a); (c) Residual PM vectors of clusters with model and observed
internal PM vectors differing by $\theta_{r}>$45∘ and $\theta_{r}>$90∘

.

###  5.5 Kinematic variation of parameters in the younger clusters

The kinematic model of the LMC for different age groups is studied as in
subsection 4.3. We note the kinematic parameters of the LMC, mainly the $i$,
$\Theta$, and $R_{0}$, show variation in the younger age group CAG-3 (100-440
Myr). To further investigate the change in kinematic parameters, we split the
$C_{AG-3}$ into two groups (one from 100 to 200 Myr and the other from 200 to
440 Myr) and performed the kinematic modeling. The variation of $i$, $\Theta$,
and $R_{0}$ are shown in Figure 13.

The age group from 100 to 200 Myr shows the largest inclination ($\sim$ 39∘)
compared to other age groups. Meanwhile, we note the smallest $\Theta$ value
($\sim$ 105∘) in this age range when compared to other age ranges. The cluster
density shifted towards central regions of the LMC in the age group CAG-3 and
towards North-East in CAG-4 (D24). We note this effect in the variation of
$R_{0}$ to the inner radii of the galaxy, as shown in Figure 13c. The value of
$R_{0}$ reaches 1.1 kpc for clusters of age between 100 to 200 Myr. A similar
effect is seen in the nearby young population, where the value reaches 0.5
kpc. Meanwhile, the field population shows smaller $i$ and lesser variation in
$R_{0}$ compared to other age groups. The kinematic model of cluster groups
based on richness also shows this trend of variation in $R_{0}$. The younger
and richer clusters with more than 60 members are located more toward the
central regions of the galaxy as well. We note $R_{0}$ smaller than 1 kpc at
this cluster group. In summary, we note that the $i$, $\Theta$, and $R_{0}$
values for clusters and young MS stars in the age range 100-200 Myr are
distinct from the other groups. A possible reason for this deviation is
discussed in subsection 5.7.

###  5.6 Kinematic signatures of the LMC bar

The spatial PM plots of clusters and field population, as shown in Figure 3,
do not bring out the bar feature explicitly. The residual PM plots, as shown
in Figure 7, on the other hand, show the bar feature, though more prominent in
the case of clusters. The radial gradient of the residual PM (Figure 11, panel
a) shows a large gradient in the bar region, pointing to residuals resulting
from kinematics related to the bar. The residual PM plots shown in Figure 12
(panels a and c) show the possible presence of non-circular motions in the bar
region. A significant reduction in the value of $R_{0}$ for clusters and young
MS stars with respect to RC stars (see Figure 10a) is probably related to the
mass distribution within the inner 2 kpc, where the bar is present. The fact
that the younger population reaches the maximum value of Vrot in a shorter
radius when compared to the older population points to a redistribution of
mass, most likely due to the evolution of the bar.

In summary, this study has traced three kinematic signatures of the LMC bar,
relatively large residual PM in the inner 2∘ radius for both clusters and
field population, the presence of non-circular motion among star clusters, and
a decrease of $R_{0}$ as a result of the possible evolution of the bar.
Recently, J23 also found that the dynamics of the inner disk are dominated by
the bar.

###  5.7 Effect of the recent LMC-SMC interaction on the LMC disk

The panels (a) and (c) of Figure 12, show that though there are kinematically
deviant clusters spread through the LMC disk, there is a specific location
where the deviation is maximum. Panel (c) indicates that the same location
also shows the presence of cluster motion in all directions, suggesting a
possible specific external disturbance. It is therefore important to identify
the source of the disturbance identified in this study.

It is possible that this disturbance is caused by the LMC-SMC collision at
$\sim$ 150 Myr ago as the radial distance of this region from the center also
matches the value of the impact factor of the LMC-SMC collision as derived by
C22. The age-dating of the disturbance, as shown in Figure 13 suggests a
significant shift in inclination $i$, PA, and $R_{0}$ for clusters in the age
range 100-200 Myr. The kinematic parameters of clusters younger than this
period are more similar to the overall disk properties. This is an indication
that the clusters formed during this period have significantly different
kinematic parameters. We speculate that this is the first evidence of direct
collision in the LMC disk and the spatial and temporal kinematic disturbance,
matching with the predicted time and location of the LMC-SMC collision. We
also speculate that the interaction mainly affected the gas, and the clusters
and stars born from the disturbed gas bear the signature of the perturbation.

Figure 13: The variation of $i$, $\Theta$, and $R_{0}$ with respect to the
cluster age groups (subsection 5.5) are shown here. Red, Blue, and Green
markers are used for clusters, fields, and young MS stars, respectively. (a)
Variation of $i$; (b) Variation of $\Theta$ and (c) Variation of $R_{0}$ are
shown for 5 age groups. The age groups are valid only for clusters (Section
2). The field and the young MS stars are age-wise heterogeneous but located
near the clusters within each age group.

##  6 Summary

We summarise the results and conclusions from this comprehensive study of the
LMC disk kinematics using Gaia DR3 data.

  1. 1.

We performed the kinematic model of the LMC that corresponds to the observed
median PM using 1705 star clusters and field regions. This is a comprehensive
study with a total of 48 data sets analyzed to create 48 models to study the
dependence of the kinematic model on various parameters.

  2. 2.

The model PM for the tracers in the data sets is formulated based on the
equations outlined by V02. The model parameters estimated include inclination
of the LMC disc ($i$), the position angle of the line of nodes ($\Theta$),
dynamic centers ($\alpha_{0}$, $\delta_{0}$), the amplitude of the tangential
velocity of the LMC’s center of mass ($v_{t}$), the tangential angle made by
$v_{t}$ ($\theta_{t}$), scale radius ($R_{0}$), optimization factor ($\eta$),
and the amplitude of the rotational velocity ($v_{0}$). We assumed a fixed
distance to the LMC center and a line-of-sight velocity of the COM, $v_{sys}$.
The fitting of the parameters was performed by an MCMC technique, and model
parameters were estimated for all datasets listed in Table 1.

  3. 3.

This is the first 2D kinematic model of the LMC employing clusters and
neighboring field regions with Gaia DR3 data. The data coverage of the LMC
considered in this study is within $\sim$ 7∘ from the LMC center. The
parameters estimated in this study show good agreement with estimations in the
literature when the comparison is made between similar populations.

  4. 4.

There is no significant difference between the observed COM PM between cluster
and field. We note an offset of 28$\pm$8 arcmin between the dynamic centers
($\alpha_{0}$, $\delta_{0}$) of the cluster and field. Estimated $\Theta$
values of 122∘.22${}^{\hskip 0.28436pt\scaleto{+1.48}{4.7pt}}_{\hskip
1.13791pt\scaleto{-1.51}{4.7pt}}$ for clusters and 128∘.02${}^{\hskip
0.28436pt\scaleto{+0.79}{4.7pt}}_{\hskip 1.13791pt\scaleto{-0.81}{4.7pt}}$ for
field regions point to an offset of offset of 5∘.8$\pm$1.7 between them, while
the inclination, $i$, remains almost similar. The modeled rotational
parameters, $R_{0}$ and $\eta_{0}$ appear to be larger for field regions
compared to clusters, while the $v_{0}$ remains almost similar.

  5. 5.

We also estimated the kinematic model parameters for two control populations,
RC stars and young MS stars. We find that the younger population tends to show
a southern and eastward dynamic center and a relatively smaller value of
$R_{0}$. We also note a varying value of $\Theta$ across the control
population.

  6. 6.

This study establishes that the kinematic model of the LMC disk varies with
the age of the cluster population used for the estimation, in contrast to the
surrounding field population.

  7. 7.

The estimated parameters show a significant radial dependence for both cluster
and field population. The value of $i$ steeply decreases from the inner to
outer radii, from $\sim$ 50∘ to $\sim$ 32∘ for clusters, and $\sim$ 43∘ to
$\sim$ 31∘ for field population. The value of $\Theta$ increases from inner to
outer radii, from $\sim$ 108∘ to $\sim$ 122∘ for clusters, and $\sim$ 110∘ to
$\sim$ 128∘ for field population. The rotational parameters, $R_{0}$ and
$v_{0}$ after 5∘ show convergence to the values estimated from the full
sample.

  8. 8.

Clusters show larger PM residuals when compared with the field population. The
RMS distribution of the residual PM for clusters shows a broader profile,
while the corresponding distribution for the field shows a narrower one.

  9. 9.

The rotational velocity amplitude ($v_{0}$) for clusters, fields, and RC stars
($\sim$ 81 to 84 km s-1) are consistent with the estimates by W20. The young
MS stars in our study show a slightly larger $v_{0}$ ($\sim$ 91 km s-1) when
compared to the RC stars, as also noted by N22.

  10. 10.

The modeled rotational parameters, $R_{0}$ and $\eta_{0}$ appear to be larger
for field population when compared to clusters, while the $v_{0}$ remains
almost similar. We note a significant shift in $R_{0}$ of $\approx$ 1.7 kpc
between young MS and RC population. The variation in the value of $R_{0}$ with
respect to population may be due to the redistribution of mass in the inner
regions, and it may be due to the evolution of the bar and its activity over
time.

  11. 11.

The dispersion of the rotation velocity is found to be $\sim$ 23 km s-1 for
clusters, and is likely to have contributions from the bar and spiral arms.
The value of $\sim$ 11 km s-1 for field population is relatively low and
suggests insignificant stellar disk heating. Young MS stars show a relatively
large velocity dispersion, similar to clusters.

  12. 12.

The residual PM for the cluster and the field decreases from the center up to
3∘, then increases beyond 4∘. The increased value in the inner 2∘ region is
likely to be the effect of the bar. We also detect evidence for non-circular
motion in the bar region among the clusters.

  13. 13.

Location of kinematically deviant clusters show an increased density in the
North-West side. We also trace the presence of a number of counter-rotating
clusters, mainly in the same region.

  14. 14.

We detect a specific kinematic perturbation between 2-3∘ from the center in
the North-West direction. We also note a significant shift in the $i$,
$\Theta$, and $R_{0}$ of clusters in the age range of 100-200 Myr. This
spatial and temporal disturbance matches with the impact factor and the time
of the LMC-SMC collision as estimated by C22 and this study. Could this be the
evidence for the recent LMC-SMC collision?

##  7 Acknowledgements

We thank the referee for valuable suggestions that helped to improve the
manuscript. Annapurni Subramaniam acknowledges support from the Science and
Engineering Research Board (SERB) Power fellowship. This work made use of the
optical data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), which was processed by the Gaia Data
Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). The funding for the DPAC
has been provided by national institutions, particularly the institutions
participating in the Gaia Multilateral Agreement.

Software: ASTROPY (Astropy Collaboration et al., 2013), SCIPY (Virtanen et
al., 2020), MATPLOTLIB (Hunter, 2007), NUMPY (Harris et al., 2020), CORNER
(Foreman-Mackey, 2016)

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