# Rank Optimization for MIMO systems with RIS: Simulation and Measurement

Shengguo Meng, Wankai Tang, Weicong Chen, Jifeng Lan,
Qun Yan Zhou, Yu Han, Xiao Li, and Shi Jin  Shengguo Meng, Wankai Tang,
Weicong Chen, Jifeng Lan, Yu Han, Xiao Li, and Shi Jin are with the National
Mobile Communications Research Laboratory, Southeast University, Nanjing
210096, China. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>jinshi@seu.edu.cn). Qun Yan Zhou is with the State Key Laboratory of
Millimeter Waves, Southeast University, Nanjing 210096, China. (e-mail:
qyzhou@seu.edu.cn).

###### Abstract

Reconfigurable intelligent surface (RIS) is a promising technology that can
reshape the electromagnetic environment in wireless networks, offering various
possibilities for enhancing wireless channels. Motivated by this, we
investigate the channel optimization for multiple-input multiple-output (MIMO)
systems assisted by RIS. In this paper, an efficient RIS optimization method
is proposed to enhance the effective rank of the MIMO channel for achievable
rate improvement. Numerical results are presented to verify the effectiveness
of RIS in improving MIMO channels. Additionally, we construct a 2$\times$2
RIS-assisted MIMO prototype to perform experimental measurements and validate
the performance of our proposed algorithm. The results reveal a significant
increase in effective rank and achievable rate for the RIS-assisted MIMO
channel compared to the MIMO channel without RIS.

###### Index Terms:

Reconfigurable intelligent surface, multiple-input multiple-output,
measurement, 6G.

##  I Introduction

In recent years, the increasing presence of various mobile communication
services such as video streaming and online gaming has sparked a significant
surge in the need for higher data rates. To meet the growing demands, various
technologies have been applied. Among them, multiple-input multiple-output
(MIMO) technology stands out as a solution to increase system capacity without
the need for additional frequency resources. However, as MIMO is widely
deployed, new challenges have arisen. In propagation environments
characterized by strong line-of-sight (LoS) path and limited scatterings, the
correlation in MIMO channels increases. This phenomenon weakens the
multiplexing of MIMO, resulting in a reduced capability to support multiple
data streams and subsequently diminishing the performance of communication
systems. Therefore, there is an urgent need for innovative solutions to
optimize the effective rank of MIMO channels to address these challenges. The
advent of reconfigurable intelligent surfaces (RIS), which can be integrated
into the channel, offers new opportunities for addressing this problem.

As one of the key potential technologies for the sixth-generation (6G)
networks, RIS can dynamically manipulate electromagnetic waves [1, 2, 3, 4,
5], thus holding the potential to customize wireless channels[6]. Some studies
have been conducted on leveraging RIS as part of the MIMO channels to improve
the achievable rate of the systems [7, 8, 9]. These related works demonstrate
the capability of RIS to improve achievable rates of MIMO systems in different
scenarios. Furthermore, RIS can be employed to improve the system performance
by directly enhancing the effective rank [10] or minimum singular value of the
MIMO channel [11]. However, the optimization algorithms proposed in the
aforementioned references are complex and may not be easily applicable to
practical systems, rendering a lack of relevant experimental measurements.

In this paper, we first develop an RIS-assisted MIMO channel model and then
propose an efficient RIS optimization method called the maximum cross-swapping
algorithm (MCA) to enhance the effective rank of the MIMO channel for
achievable rate improvement. Numerical simulations are conducted to verify the
effectiveness of RIS in improving the effective rank and the achievable rate.
Moreover, we set up an RIS-assisted 2$\times$2 MIMO communication prototype to
validate the performance of our proposed algorithm through experimental
measurements. The results show that compared to the MIMO channel without RIS,
the RIS-assisted MIMO channel, optimized using the MCA, achieves a notable
increase of 30.1% in the effective rank and 13.6% improvement in the
achievable rate.

Figure 1: The considered RIS-assisted MIMO communication system.

##  II System Model

We consider an RIS-assisted MIMO communication system as shown in Fig.1. The
transmitter and the receiver have $L$ and $Q$ antennas, respectively. The RIS
consists of $N$ unit cells. The entire MIMO channel between the transmitter
and the receiver is comprised of the RIS-assisted channel and the non-RIS-
assisted channel [11, 3], which can be modeled as

$\begin{aligned}
\mathbf{H}=&\sqrt{\alpha}\mathop{\underbrace{\mathbf{H}_{\mathrm{rx}}^{\mathrm{ris}}\mathbf{\Gamma
H}_{\mathrm{ris}}^{\mathrm{tx}}}}\limits_{\mathrm{RIS-assisted\ channel}}\\\
&+\sqrt{1-\alpha}\left(\underset{\mathrm{non-RIS-assisted\
channel}}{\underbrace{\sqrt{\frac{K}{1+K}}\mathbf{H}_{\mathrm{LoS}}+\sqrt{\frac{1}{1+K}}\mathbf{H}_{\mathrm{NLoS}}}}\right)\end{aligned},\vspace{-0.1cm}$
(1)

where $\mathbf{H}_{\mathrm{ris}}^{\mathrm{tx}}\in\mathbb{C}^{N\times L}$ and
$\mathbf{H}_{\mathrm{rx}}^{\mathrm{ris}}\in\mathbb{C}^{Q\times N}$ represent
the channel between the transmitter and the RIS, and the channel between the
RIS and the receiver, respectively. $\mathbf{\Gamma}\in\mathbb{C}^{N\times N}$
is a diagonal matrix composed of the reflection coefficients of each unit
cell. $\mathbf{H}_{\mathrm{LoS}}\in\mathbb{C}^{Q\times L}$ and
$\mathbf{H}_{\mathrm{NLoS}}\in\mathbb{C}^{Q\times L}$ represent the non-RIS-
assisted line of sight (LoS) and non-LoS (NLoS) paths between the transmitter
and the receiver, respectively. $\alpha\in(0,1)$ denotes the power ratio of
the RIS-assisted channel, i.e., the power ratio of
$\mathbf{H}_{\mathrm{rx}}^{\mathrm{ris}}\mathbf{\Gamma}\mathbf{H}_{\mathrm{ris}}^{\mathrm{tx}}$
in the overall channel $\mathbf{H}$. $K$ is the Rician factor of the non-RIS-
assisted channel.

Specifically, the channel between the transmitter and the RIS can be expressed
as

$\mathbf{H}_{\mathrm{ris}}^{\mathrm{tx}}=\left[\begin{matrix}h_{1,\mathrm{ris}}^{1,\mathrm{tx}}&h_{1,\mathrm{ris}}^{2,\mathrm{tx}}&\cdots&h_{1,\mathrm{ris}}^{L,\mathrm{tx}}\\\
h_{2,\mathrm{ris}}^{1,\mathrm{tx}}&h_{2,\mathrm{ris}}^{2,\mathrm{tx}}&\cdots&h_{2,\mathrm{ris}}^{L,\mathrm{tx}}\\\
\vdots&\vdots&\ddots&\vdots\\\
h_{N,\mathrm{ris}}^{1,\mathrm{tx}}&h_{N,\mathrm{ris}}^{2,\mathrm{tx}}&\cdots&h_{N,\mathrm{ris}}^{L,\mathrm{tx}}\\\
\end{matrix}\right],\vspace{-0.1cm}$ (2)

where $h_{n,\mathrm{ris}}^{l,\mathrm{tx}}$ is the channel coefficient between
the $l_{\mathrm{th}}(l=0,1,\dots,L)$ transmit antenna and the
$n_{\mathrm{th}}(n=0,1,\dots,N)$ unit cell $\mathrm{U}_{n}$. Considering the
wireless signal propagation loss between them, the channel coefficient can be
modeled as[12]

$h_{n,\mathrm{ris}}^{l,\mathrm{tx}}=\sqrt{\frac{G_{\mathrm{tx}}F\left(\theta_{n}^{l,\mathrm{AoA}},\varphi_{n}^{l,\mathrm{AoA}}\right)d_{\mathrm{x}}d_{\mathrm{z}}}{4\pi\left({d_{n,\mathrm{ris}}^{l,\mathrm{tx}}}\right)^{2}}}\times
e^{-j2\pi d_{n,\mathrm{ris}}^{l,\mathrm{tx}}/\lambda},\vspace{-0.2cm}$ (3)

where $G_{\mathrm{tx}}$ denotes the gain of each transmit antenna,
$\theta_{n}^{l,\mathrm{AoA}}$ and $\varphi_{n}^{l,\mathrm{AoA}}$ denote the
angle of arrival (AoA) from the $l_{th}$ transmit antenna to the unit cell
$\mathrm{U}_{n}$. $F(\theta,\varphi)$ is the normalized power radiation
pattern of each unit cell[14, 13]. $d_{\mathrm{x}}$ and $d_{\mathrm{z}}$
denote the width and length of each unit cell, respectively.
$d_{n,\mathrm{ris}}^{l,\mathrm{tx}}$ denotes the distance between the $l_{th}$
transmit antenna and the unit cell $\mathrm{U}_{n}$, and $\lambda$ represents
the wavelength. In analogy with $\mathbf{H}_{\mathrm{ris}}^{\mathrm{tx}}$,
$\mathbf{H}_{\mathrm{rx}}^{\mathrm{ris}}$ can be modeled.

Furthermore, the reflection matrix of the RIS $\mathbf{\Gamma}$ can be
represented as

$\mathbf{\Gamma}=\mathrm{diag}\left\\{e^{j\phi_{1}},e^{j\phi_{2}},\dots,e^{j\phi_{N}}\right\\},\vspace{-0.1cm}$
(4)

where $e^{j\phi_{n}}$ denotes the reflection coefficient of unit cell
$\mathrm{U}_{n}$. In practice, continuous control of the reflection phase for
each unit cell incurs significant hardware implementation costs. Thus, we
consider the discrete implementation, where the reflection phase is taken only
from a finite number of discrete values. We assign
$\boldsymbol{\phi}=\left[\phi_{1},\phi_{2},\dots,\phi_{N}\right]$ to be the
phase configuration vector of the RIS, satisfying
$\forall\phi_{n}\in\Xi_{b}=\left\\{0,\frac{2\pi}{2^{b}},\dots,\frac{2\pi\left(2^{b}-1\right)}{2^{b}}\right\\}$,
where $b$ denotes quantization bit of reflection phase of each unit cell.

As to the non-RIS-assisted channel, the LoS path between the transmitter and
the receiver can be modeled as

$\mathbf{H}_{\mathrm{LoS}}=\left[\begin{matrix}h_{1,\mathrm{rx}}^{1,\mathrm{tx}}&h_{1,\mathrm{rx}}^{2,\mathrm{tx}}&\cdots&h_{1,\mathrm{rx}}^{L,\mathrm{tx}}\\\
h_{2,\mathrm{rx}}^{1,\mathrm{tx}}&h_{2,\mathrm{rx}}^{2,\mathrm{tx}}&\cdots&h_{2,\mathrm{rx}}^{L,\mathrm{tx}}\\\
\vdots&\vdots&\ddots&\vdots\\\
h_{Q,\mathrm{rx}}^{1,\mathrm{tx}}&h_{Q,\mathrm{rx}}^{2,\mathrm{tx}}&\cdots&h_{Q,\mathrm{rx}}^{L,\mathrm{tx}}\\\
\end{matrix}\right],$ (5)

where
$h_{q,\mathrm{rx}}^{l,\mathrm{tx}}=\sqrt{G_{\mathrm{tx}}G_{\mathrm{rx}}}\lambda/\left(4\pi
d_{q,\mathrm{rx}}^{l,\mathrm{tx}}\right)\times e^{-j2\pi
d_{q,\mathrm{rx}}^{l,\mathrm{tx}}/\lambda}$ denotes the channel coefficient of
the LoS path between the $l_{\mathrm{th}}$ transmit antenna and the
$q_{\mathrm{th}}(q=0,1,\dots,Q)$ receive antenna,
$d^{l,\mathrm{tx}}_{q,\mathrm{rx}}$ and $G_{\mathrm{rx}}$ are the distance
between them and the gain of each receive antenna, respectively. In addition,
the NLoS path can be represented by

$\mathbf{H}_{\mathrm{NLoS}}=\frac{\sqrt{G_{\mathrm{tx}}G_{\mathrm{rx}}}\lambda}{4\pi
d_{\mathrm{rx}}^{\mathrm{tx}}}\tilde{\mathbf{H}}_{\mathrm{NLoS}},\vspace{-0.2cm}$
(6)

where $d_{\mathrm{rx}}^{\mathrm{tx}}$ is the distance between the center of
the transmitter and the center of the receiver, and
$\tilde{\mathbf{H}}_{\mathrm{NLoS}}\in\mathbb{C}^{L\times Q}$ is the NLoS
component. Each element of $\tilde{\mathbf{H}}_{\mathrm{NLoS}}$ is an i.i.d.
complex Gaussian random variable with zero mean and unit variance.

##  III Optimization Method

In this section, we introduce an optimization method to improve the MIMO
channel. The effective rank is used as a metric to evaluate the orthogonality
among the MIMO subchannels[11]. According to the definition in [10], the
effective rank of the MIMO channel $\mathbf{H}$ between the transmitter and
the receiver can be expressed as

$\mathrm{erank}(\mathbf{H})=\exp\left(\sum_{i=1}^{\mathrm{rank}(\mathbf{H})}{-\sigma_{i}^{\prime}\ln\sigma_{i}^{\prime}}\right),\vspace{-0.1cm}$
(7)

where $\mathrm{rank}(\mathbf{H})$ is the rank of the MIMO channel, and
$\sigma_{i}^{\prime}={\sigma_{i}}/{\sum_{i}{\sigma_{i}}}$ is the normalized
singular value.

By incorporating RIS as a component of the MIMO channel, it is possible to
optimize the configuration of the RIS $\boldsymbol{\phi}$ to improve the
effective rank of the MIMO channel $\mathbf{H}$, thus improving the
orthogonality of the subchannels and overall system performance. Hence, the
optimization goal can be written as

$\displaystyle\underset{\boldsymbol{\phi}}{\mathrm{maximize}}\,\,\mathrm{erank}\left(\mathbf{H}\right)$
(8) $\displaystyle\,\,\mathrm{subject}\ \mathrm{to}\
\forall\phi_{n}\in\Xi_{b},\ n=1,2,...,N.$

We propose an efficient optimization method named MCA to optimize the phase
configuration of the RIS, which is described as follows.

###  III-1 Iterating through random configurations

In the MCA, we first generate a set of $T$ random RIS phase configurations,
i.e.,
$\varPsi=\left\\{\boldsymbol{\phi}_{1},\dots,\boldsymbol{\phi}_{T}\right\\}$.
Then, we configure the RIS according to these $T$ random configurations and
obtain the corresponding effective rank of the MIMO channel $\mathbf{H}$.
Furthermore, the $T$ random RIS configurations are sorted in ascending order
based on their individual effective ranks, and the top two configurations are
selected as the parent set $\varPsi_{\mathrm{parent}}$ to generate a new set
of RIS phase configurations for the next round. The corresponding parent set
is expressed as

$\varPsi_{\mathrm{parent}}=\left\\{\boldsymbol{\phi}_{\mathrm{max}},\boldsymbol{\phi}_{\mathrm{submax}}\right\\},\vspace{-0.1cm}$
(9)

where $\boldsymbol{\phi}_{\mathrm{max}}$ and
$\boldsymbol{\phi}_{\mathrm{submax}}$ represent the RIS configuration with the
highest and second-highest effective rank of the corresponding MIMO channel
$\mathbf{H}$ in the set $\varPsi$, respectively.

###  III-2 Cross-swapping the parent configurations

We generate offspring RIS phase configurations denoted as
$\varPsi_{\mathrm{son}}=\left\\{\boldsymbol{\phi}_{1}^{\mathrm{son}},\boldsymbol{\phi}_{2}^{\mathrm{son}},\dots,\boldsymbol{\phi}_{2^{N_{\mathrm{new}}}}^{\mathrm{son}}\right\\}$
by sequentially cross-swapping the phase designs of the first
$N_{\mathrm{new}}$ unit cells from the parent configurations
$\left\\{\boldsymbol{\phi}_{\mathrm{max}},\boldsymbol{\phi}_{\mathrm{submax}}\right\\}$.
The detailed procedure is outlined in Method 1.

Method 1 Cross-swapping Parent Configurations to Generate Offspring
Configurations.

1: $\mathbf{input:}\ N_{\mathrm{new}},\ \boldsymbol{\phi}_{\mathrm{max}},\
\boldsymbol{\phi}_{\mathrm{submax}}$

2: $\mathbf{for}\ i=1,2,\dots,2^{N_{\mathrm{new}}}\ \mathbf{do}$

3: convert $i$ to a binary number $i_{\mathrm{b}}$ with $N_{\mathrm{new}}$
bits

4: $\mathbf{for}\ j=1,2,\dots,N_{\mathrm{new}}\ \mathbf{do}$

5: $\mathbf{if}$ the $j_{\mathrm{th}}$ bit of $i_{\mathrm{b}}$ equals to 0

6: assign the $j_{\mathrm{th}}$ element of $\boldsymbol{\phi}_{\mathrm{max}}$
to the $j_{\mathrm{th}}$ element of $\boldsymbol{\phi}_{i}^{\mathrm{son}}$

7: $\mathbf{else}$ $\mathbf{if}$ the $j_{\mathrm{th}}$ bit of $i_{\mathrm{b}}$
equals to 1

8: assign the $j_{\mathrm{th}}$ element of
$\boldsymbol{\phi}_{\mathrm{submax}}$ to the $j_{\mathrm{th}}$ element of
$\boldsymbol{\phi}_{i}^{\mathrm{son}}$

9: $\mathbf{end\ if}$

10: assign the remaining elements of $\boldsymbol{\phi}_{\mathrm{max}}$ to the
remaining elements of $\boldsymbol{\phi}_{i}^{\mathrm{son}}$

11: $\mathbf{end}\ \mathbf{for}$

12: $\mathbf{end\ for}$

13: $\mathbf{output:}\ $offspring RIS configuration set
$\varPsi_{\mathrm{son}}=\left\\{\boldsymbol{\phi}_{1}^{\mathrm{son}},\boldsymbol{\phi}_{2}^{\mathrm{son}},\dots,\boldsymbol{\phi}_{2^{N_{\mathrm{new}}}}^{\mathrm{son}}\right\\}$

###  III-3 Obtaining the optimal offspring configuration

After applying each newly generated offspring configuration to the RIS, we can
calculate the corresponding effective rank of the MIMO channel $\mathbf{H}$.
The optimal offspring configuration, denoted as
$\boldsymbol{\phi}_{\mathrm{mca}}$, is determined as the one with the highest
effective rank among the set $\varPsi_{\mathrm{son}}$. The whole process of
the MCA is summarized in Algorithm 1.

Algorithm 1 Maximum Crosss-swapping Algorithm (MCA).

1: $\mathbf{input:}N,\ T,\ N_{\mathrm{new}},\ b$

2: generate a set $\varPsi$ containing $T$ random RIS configurations
$\left\\{\boldsymbol{\phi}_{t}\right\\}_{t=1}^{T}$ based on $\Xi_{b}$

3: calculate the effective rank of the MIMO channel $\mathbf{H}$ with the RIS
configured by each configuration $\boldsymbol{\phi}_{t}$

4: find the RIS configuration $\boldsymbol{\phi}_{\mathrm{max}}$ and
$\boldsymbol{\phi}_{\mathrm{submax}}$ with the highest and second-highest
effective rank of the corresponding MIMO channel $\mathbf{H}$ in the set
$\varPsi$, respectively.

5: generate offspring RIS configuration set $\varPsi_{\mathrm{son}}$ according
to Method 1

6: $\mathbf{for}\ i=1,2,\dots,2^{N_{\mathrm{new}}}\ \mathbf{do}$

7: calculate the corresponding effective rank of the MIMO channel $\mathbf{H}$
with the RIS configured by $\boldsymbol{\phi}_{i}^{\mathrm{son}}$

8: $\mathbf{end\ for}$

9: $\mathbf{output:}$
$\boldsymbol{\phi}_{\mathrm{mca}}=\underset{\boldsymbol{\phi}\in\varPsi_{\mathrm{son}}}{\mathrm{arg}\max}\,\,\mathrm{erank}\left(\mathbf{H}\right)$

##  IV Numerical Simulation

In this section, we evaluate the performance improvement of the RIS-assisted
MIMO channel through numerical simulations, utilizing the proposed
optimization method. As illustrated in Fig. 2, we consider the column-
controlled RIS, which can be treated as a uniform linear array (ULA). We set
the centers of the transmit antennas, the receive antennas, and the RIS all
positioned at the same height, thus reducing the system to two-dimension. The
AoA $\varphi_{n}^{l,\mathrm{AoA}}$ and AoD $\varphi_{q}^{n,\mathrm{AoD}}$ are
fixed at $\pi$ and $0$ in the x-y plane, respectively. In addition,
$d_{\mathrm{ris,x}}^{\mathrm{tx}}$ ($d_{\mathrm{rx,x}}^{\mathrm{ris}}$) and
$d_{\mathrm{ris,y}}^{\mathrm{tx}}$ ($d_{\mathrm{rx,y}}^{\mathrm{ris}}$)
represent the corresponding distance between the center of the transmitter
(receiver) and the center of the RIS in the x or y direction, respectively.
These distances are illustrated in Table I. Moreover, $d_{\mathrm{tx}}$ and
$d_{\mathrm{rx}}$ denote the transmit antenna spacing and the receive antenna
spacing, respectively. When the transmit antennas and receive antennas are
fixed, we can calculate $d_{n,\mathrm{ris}}^{l,\mathrm{tx}}$,
$d_{q,\mathrm{rx}}^{n,\mathrm{ris}}$, $d_{q,\mathrm{rx}}^{l,\mathrm{tx}}$,
$d^{\mathrm{tx}}_{\mathrm{rx}}$, $\theta_{n}^{l,\mathrm{AoA}}$, and
$\theta_{q}^{n,\mathrm{AoD}}$ based on the geometric relationship.

The system operates at $f=2.7$ GHz. Each column of the RIS can be seen as a
macro unit cell, whose width and length are $d_{\mathrm{x}}=0.05$ m and
$d_{\mathrm{z}}=1.6$ m, respectively. As shown in Fig. 2, the transmitter and
receiver are both equipped with 2 antennas, that is $L=Q=2$, with a transmit
and receive gain being $G_{\mathrm{tx}}=G_{\mathrm{rx}}=6$ dB. More specific
system parameters are listed in Table I.

Figure 2: Deployment of the considered RIS-assisted 2$\times$2 MIMO
communication system (top view).

When zero-forcing (ZF) equalization is applied to recover the data streams,
the achievable system rate can be expressed as [15]

$R_{\mathrm{ZF}}\left(\mathbf{H},\rho\right)=\sum_{i=1}^{\mathrm{rank}(\mathbf{H})}{\log_{2}\left(1+\frac{\rho}{\left[\left(\mathbf{H}^{H}\mathbf{H}\right)^{-1}\right]_{ii}}\right)},\vspace{-0.1cm}$
(10)

where $\rho$ is the transmit signal-to-noise ratio (SNR) and
$\left[\cdot\right]_{ii}$ denotes the $i_{\mathrm{th}}$ diagonal element of a
matrix. In the high SNR regime where water filling power allocation reduces to
equal power allocation, the capacity of the MIMO channel, which is introduced
as the upper bound of the simulation, can be approximated by [16]

$C\left(\mathbf{H},\rho\right)=\sum_{i=1}^{\mathrm{rank}(\mathbf{H})}{\log_{2}\left(1+{\sigma_{i}}^{2}\rho\right)}.\vspace{-0.2cm}$
(11)

In the following simulated cases, we set the transmit SNR $\rho=50$ dB and
Rician factor $K=20$ dB. All the results are averaged over 1000 Monte Carlo
channel realizations. We perform the MCA with $T=$160 and $N_{\mathrm{new}}=$
4.

TABLE I: PARAMETERS FOR THE NUMERICAL SIMULATION RIS Parameters | Operating Frequency $f$ | 2.7 GHz
---|---|---
Coding Type | 1-bit phase
Macro Unit Cell Number $N$ | 16/32/64
Macro Unit Cell Width $d_{\mathrm{x}}$ | 0.05 m
Macro Unit Cell Length $d_{\mathrm{z}}$ | 1.6 m
Distance Parameters | Transmitter to RIS $d_{\mathrm{ris,y}}^{\mathrm{tx}}$, $d_{\mathrm{ris,x}}^{\mathrm{tx}}$ | 1 m, 0.5 m
RIS to Receiver $d_{\mathrm{rx,y}}^{\mathrm{ris}}$, $d_{\mathrm{rx,x}}^{\mathrm{ris}}$ | 1 m, 0.5 m
Transmit Antenna Spacing $d_{\mathrm{tx}}$ | 0.1 m
Receive Antenna Spacing $d_{\mathrm{rx}}$ | 0.1 m
Figure 3: Effective rank versus power ratio of the RIS-assisted channel for
different numbers of unit cells. Figure 4: Comparison of the achievable rate
for the ZF receiver and the upper bound with increasing $\alpha$ when
considering different numbers of unit cells.

Fig. 3 demonstrates the effective rank versus power ratio of the RIS-assisted
channel for different numbers of unit cells. When the power ratio $\alpha$ =
0, the MIMO channel $\mathbf{H}$ is only composed of the non-RIS-assisted
channel, and the corresponding effective rank is 1.46. The reason for the
deficient effective rank is the lack of rich scattering in the propagation
environment between the transmitter and receiver. As $\alpha$ increases, the
corresponding power ratio of the RIS-assisted channel
$\mathbf{H}_{\mathrm{rx}}^{\mathrm{ris}}\mathbf{\Gamma}\mathbf{H}_{\mathrm{ris}}^{\mathrm{tx}}$
in the MIMO channel $\mathbf{H}$ rises. Consequently, the effective rank of
$\mathbf{H}$ rapidly approaches 2. The introduction of the RIS enhances the
scattering environment. This result reveals leveraging the RIS can improve the
effective rank of the MIMO channel.

We also evaluate the achievable rate of the ZF receiver and compare it with
the upper bound as $\alpha$ increases, while considering varying numbers of
unit cells. As shown in Fig. 4, when $N=$ 16, the achievable rate of the ZF
receiver increases first and then decreases. The upward trend is attributed to
the improvement of the effective rank of MIMO channel $\mathbf{H}$. The
subsequent downward trend occurs due to the limited number of unit cells, as
the gain provided by the RIS-assisted channel is insufficient to compensate
for the losses in the non-RIS-assisted channel. Thus, when power ratio
$\alpha$ of the RIS-assisted channel
$\mathbf{H}_{\mathrm{rx}}^{\mathrm{ris}}\mathbf{\Gamma}\mathbf{H}_{\mathrm{ris}}^{\mathrm{tx}}$
in the channel $\mathbf{H}$ increases, the corresponding achievable rate
actually decreases. However, when the RIS consists of more unit cells, i.e.,
$N=$ 32 or 64, the RIS-assisted channel has higher gains. As a result, the
achievable rate of the ZF receiver continues to increase. Increasing $N$
enhances the achievable rate of the ZF receiver. This effect becomes more
pronounced with larger $\alpha$.

Additionally, it can be observed that as the power ratio $\alpha$ increases,
the effective rank of the 2$\times$2 MIMO channel $\mathbf{H}$ approaches the
upper limit of 2 in the right region indicated by the black dashed line in
Fig. 3. This result implies that the two subchannels of $\mathbf{H}$ are
orthogonal to each other. Meanwhile, the achievable rate of the simple ZF
receiver approaches the upper bound, as shown in Fig. 4. Leveraging the RIS to
optimize the effective rank of the MIMO channel, we can enhance the MIMO
channel itself, simplify the receiver algorithm, and ultimately improve the
overall performance of the system.

##  V Experimental Measurement

In this section, we set up an RIS-assisted 2$\times$2 MIMO communication
prototype to conduct experimental measurements. As depicted in Fig. 5, the
transmitter and the receiver consist of two antennas and the software defined
radio (SDR) platform, respectively. The RIS is placed aside, with its center
aligned with the center of all antennas at the same height, thereby
establishing an RIS-assisted 2×2 MIMO communication system. The parameters of
the fabricated RIS and the corresponding distances are identical to those
utilized in the numerical simulation.

Figure 5: A photo of the RIS-assisted 2$\times$2 MIMO communication prototype.

When the configuration of RIS is completed, the receiver uses the least
squares (LS) method to estimate the overall MIMO channel $\mathbf{H}$.
Subsequently, the corresponding effective rank of the channel is calculated
based on equation (7) as a metric for the MCA to optimize the RIS
configuration. The achievable rate of the ZF receiver can be obtained by
summing up the achievable rates of each independent data stream after ZF
equalization, which can be expressed as

$R_{\mathrm{ZF}}^{\mathrm{mea}}\left(\mathbf{H},\rho_{i}^{\mathrm{ZF}}\right)=\sum_{i=1}^{\mathrm{rank}\left(\mathbf{H}\right)}{\log_{2}\left(1+\rho_{i}^{\mathrm{ZF}}\right)},\vspace{-0.1cm}$
(12)

where $\rho_{i}^{\mathrm{ZF}}$ is the SNR of the $i_{\mathrm{th}}$ data stream
after ZF equalization. The specific measurement method for each SNR
$\rho_{i}^{\mathrm{ZF}}$ can be found in Sections IV-D and V-B of reference
[12].

In order to modify the power ratio $\alpha$ of the RIS-assisted channel
$\mathbf{H}_{\mathrm{rx}}^{\mathrm{ris}}\mathbf{\Gamma}\mathbf{H}_{\mathrm{ris}}^{\mathrm{tx}}$
in the MIMO channel $\mathbf{H}$, we alter the orientation of the antennas
towards the RISs. As depicted in Fig. 6(a), when the transmitter directly
faces the receiver without any RIS, the corresponding $\alpha$ value is 0. In
Fig. 6(b) and Fig. 6(d), where the transmitter faces the receiver with the
RISs placed aside, the corresponding $\alpha$ slightly exceeds 0. As shown in
Fig. 6(c) and Fig. 6(e), when both the transmitter and the receiver face the
RISs, the corresponding $\alpha$ approaches 1. Furthermore, we evaluated the
impact of increasing the number of unit cells by comparing the results
obtained with one RIS and two RISs. Table II summarizes the corresponding
measurement results.

Figure 6: Layout of experimental scenarios: (a) Transmitter faces receiver
without RIS, (b) Transmitter faces receiver with one RIS aside, (c)
Transmitter and receiver both face one RIS, (d) Transmitter faces receiver
with two RISs aside, (e) Transmitter and receiver both face two RISs.

It can be observed that when using one RIS ($N$ = 16), the achievable rate of
the ZF receiver $R_{\mathrm{ZF}}^{\mathrm{mea}}$ increases from 17.7 to 19.4
bps/Hz at the beginning and then decreases to 18.9 bps/Hz as $\alpha$
increases. Meanwhile, the effective rank rises from 1.53 to 1.96. However,
when using two RISs ($N$ = 32), $R_{\mathrm{ZF}}^{\mathrm{mea}}$ continuously
increases by 13.6%, from 17.7 to 20.1 bps/Hz as $\alpha$ increases.
Simultaneously, the effective rank rises from 1.53 to 1.96, obtaining a 30.1%
increase. In scenarios where the RIS-assisted channel dominates $\mathbf{H}$,
as depicted in Fig. 6(c) and Fig. 6(e), adding one RIS (increasing $N$ from 16
to 32) leads to a 1.2 bps/Hz increase in $R_{\mathrm{ZF}}^{\mathrm{mea}}$. The
measurement results are consistent with the simulation results and verify that
RIS can be effectively used to optimize the effective rank of the channel and
improve the system performance.

TABLE II: RESULTS OF THE EXPERIMENTAL MEASUREMENT No. |  $\alpha$ and $N$ | $\mathrm{erank}(\mathbf{H})$ |  | $\rho_{1}^{\mathrm{ZF}}$
---
(dB)
| $\rho_{2}^{\mathrm{ZF}}$
---
(dB)
| $R_{\mathrm{ZF}}^{\mathrm{mea}}$
---
(bps/Hz)
a | $\alpha=0$ | 1.53 | 26.6 | 27.1 | 17.7
b |  |  $\alpha$ slightly exceed 0
---
$N$ = 16
1.70 | 27.9 | 30.2 | 19.4
c |  |  $\alpha$ approaches 1
---
$N$ = 16
1.96 | 28.1 | 28.8 | 18.9
d |  |  $\alpha$ slightly exceed 0
---
$N$ = 32
1.65 | 28.1 | 29.5 | 19.2
e |  |  $\alpha$ approaches 1
---
$N$ = 32
1.99 | 30.2 | 30.4 | 20.1

##  VI Conclusion

In this study, we proposed an efficient RIS optimization method to enhance the
effective rank of the MIMO channel for achievable rate improvement in RIS-
assisted systems. Through numerical simulations and experimental measurements,
we have confirmed that RISs can be utilized to improve the effective rank of
the MIMO channel as well as the achievable rate effectively. The results show
that the RIS-assisted MIMO channel optimized using MCA, compared to the
scenario without RIS, exhibited an increase of 30.1% in effective rank and a
13.6% achievable rate improvement. Based on these promising findings, our
study reveals the superior performance of RIS in optimizing MIMO channels,
which highlights the potential applications of RISs in future networks.

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# Multiple stellar populations at less evolved stages-III: a possible helium
spread in NGC 2210

Chengyuan Li (李程远)  School of Physics and Astronomy, Sun Yat-sen University,
Daxue Road, Zhuhai, 519082, China  CSST Science Center for the Guangdong-Hong
Kong-Macau Greater Bay Area, Zhuhai, 519082, China  Xin Ji (纪鑫)  Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China  School of Astronomy and Space
Science, University of Chinese Academy of Sciences, Beijing 100049, China
Long Wang (王龙)  School of Physics and Astronomy, Sun Yat-sen University, Daxue
Road, Zhuhai, 519082, China  CSST Science Center for the Guangdong-Hong Kong-
Macau Greater Bay Area, Zhuhai, 519082, China  Yue Wang (王悦)  Key Laboratory
of Optical Astronomy, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China  Baitian Tang (汤柏添)  School of Physics and
Astronomy, Sun Yat-sen University, Daxue Road, Zhuhai, 519082, China  CSST
Science Center for the Guangdong-Hong Kong-Macau Greater Bay Area, Zhuhai,
519082, China  Antonino P. Milone  Dipartimento di Fisica e Astronomia
“Galileo Galilei”, Univ. di Padova, Vicolo dell’Osservatorio 3, Padova,
IT-35122, Italy  Istituto Nazionale di Astrofisica - Osservatorio Astronomico
di Padova, Vicolo dell’Osservatorio 5, Padova, IT-35122, Italy  Yujiao Yang
(杨玉姣)  Key Laboratory of Space Astronomy and Technology, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China  Holger
Baumgardt  School of Mathematics and Physics, The University of Queensland,
St. Lucia, QLD 4072, Australia  Dengkai Jiang (姜登凯)  Yunnan Observatories,
Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming
650216, China

###### Abstract

Helium variations are common features of globular clusters (GCs) with multiple
stellar populations. All the formation scenarios predict that secondary
population stars are enhanced in helium but the exact helium content depends
on the polluters. Therefore, searching for helium variations in a star cluster
is a straightforward method to understand if it hosts multiple populations or
not, and constrain the formation scenario. Although this topic has been well
explored for Galactic GCs, GCs beyond the Milky Way are challenging to study
because of their large distances. This work studies the helium distribution of
GK-type main sequence dwarfs in an old ($\sim$12.5 Gyr) GC in the Large
Magellanic Cloud, NGC 2210, using the deep photometry observed by the Hubble
Space Telescope. We compare the observed morphology of the MS with that of
synthetic populations with different helium distributions. We confirm that NGC
2210 dwarfs have a helium spread, with an internal dispersion of
$\delta{Y}\sim$0.06–0.07. The fraction of helium enriched stars depends on the
$\delta{Y}$ distribution. A continuous $\delta{Y}$ distribution would indicate
that more than half of MS stars are helium enriched ($\sim$55%). If the
$\delta{Y}$ distribution is discrete (bimodal), a fraction of $\sim$30%
enriched stars is able to explain the observed morphology of the MS. We also
find that the He-enriched population stars are more centrally concentrated
than He-normal stars.

globular clusters: individual: NGC 2210 – Hertzsprung-Russell and C-M diagrams

††journal: ApJ

##  1 Introduction

In contrast to young open clusters and star-forming regions, which have been
proved to be simple stellar populations (SSPs. e.g., de Silva et al., 2009;
Bragaglia et al., 2012, 2014; Ting et al., 2012; Kos et al., 2021), most
globular clusters (GCs) and some intermediate-age clusters (older than $\geq$2
Gyr) are multiple stellar populations (MPs, e.g., Carretta et al., 2009a;
Milone et al., 2017a; Niederhofer et al., 2017; Li & de Grijs, 2019). The MPs
are characterized by star-to-star chemical variations in light elements such
as C, N, O, Na, Mg, Al (Carretta et al., 2009a; Marino et al., 2016; Pancino
et al., 2017; Dias et al., 2018), together with He (Piotto et al., 2007;
Milone et al., 2018).

The observed chemical pattern of MPs in GCs drives the debate on suitable
polluters. Although various scenarios were proposed to explain the presence of
MPs, none of these models can reproduce the exact pattern of abundances
observed in GCs (see Bastian et al., 2015). These models invoke interactive
binaries (de Mink et al., 2009; Jiang et al., 2014; Wang et al., 2020; Wei et
al., 2020; Renzini et al., 2022), fast-rotating massive stars (FRMS, Krause et
al., 2013), asymptotic giant branch (AGB) stars (D’Ercole et al., 2008;
D’Antona et al., 2016; Calura et al., 2019), very massive stars
($10^{2}$–$10^{3}$ $M_{\odot}$,VMS, Vink, 2018). Concerning nucleosynthesis,
all these models predict polluted stars should be He-enriched as He is the
direct product of H-burning. The helium abundance is also used as a proxy for
the chemical enrichment in numerical simulations (e.g., Howard et al., 2019).

Although a helium spread is expected in all GCs with MPs, direct helium
measurements ($Y$) are challenging, as most stars in GCs are too cold to
exhibit He lines. Because He absorption lines can only be detected among
horizontal branch (HB) stars (hotter than $\sim$9,000 K) in GCs111In addition,
HB stars used for He determination should not be hotter than $\sim$11,000 K to
avoid the Grundahl jump effect (Grundahl et al., 1999).(e.g., Marino et al.,
2014; Dupree et al., 2011; Dupree & Avrett, 2013). An alternative method is
based on the photometric investigation of GC stars. As helium is the second
most abundant element in stars, its variation has a notable impact on stellar
structure and evolution. Stellar evolutionary theory predicts that He-rich
stars will be hotter and brighter than normal stars at each stage, as He-rich
stars have a smaller radiative opacity and a higher mean molecular weight than
normal stars. In addition, He-rich stars evolve more rapidly than normal stars
as they have increased luminosities. At a given age, He-rich stars at the MS
turnoff (TO) stage will be less massive, populating a fainter MSTO boundary.
If both He-rich and normal stars experience the same mass loss content during
their red-giant branch (RGB) stage, they will end up with different masses
when evolve into the HB, thus different colors. The current stellar
evolutionary model also predicts that the RGB bump (RGBB) lifetime would be
shortened for He-enhanced stars (Bono et al., 2001; Di Cecco et al., 2010).
Indeed, helium distributions have been studied in some GCs through photometry
based on the morphologies of MS (e.g., Piotto et al., 2007), RGB (e.g., Milone
et al., 2017a), RGBB (e.g., Nataf et al., 2011; Lagioia et al., 2019), and HB
(e.g., Jang et al., 2019). Statistical studies based on Galactic GCs have
shown that both the maximum internal helium dispersion ($\delta{Y}$), and the
fraction of helium-enriched stars positively correlate with the total
clusters’ masses (Milone et al., 2017a, 2018). This correlation also applies
to extragalactic GCs (Chantereau et al., 2019, hereafter C19).

The method based on the HB could overestimate the internal helium variation if
it does not account for the mass loss effect (Tailo et al., 2020). Studying
the helium distribution of less-evolved stars such as MS stars would be more
reliable. This is difficult for extragalactic clusters because their large
distance requires ultra-deep photometry. A recent attempt was made by Li
(2021b), in which they have studied the helium distribution of MS stars for a
1.7 Gyr-old LMC cluster, NGC 1846. They find that NGC 1846 is consistent with
an SSP cluster and that its helium spread, if present, should be smaller than
2%. Another LMC cluster, NGC 1978, was studied by Ji et al. (2022), in which
they have analyzed the morphology of its RGBB. However, they can only
constrain its maximum helium spread to $\delta{Y}\leq 0.03$ (3%) due to the
limitation of the image quality. This is consistent with Milone et al. (2020a)
($\delta{Y}=0.002\pm 0.003$).

This work aims to study the helium distribution of MS dwarfs in an old GC, NGC
2210, in the LMC, which serves as a good comparison to our previously studied
younger LMC cluster, NGC 1846 (Li, 2021b). Since only clusters older than
$\sim$2 Gyr are known to harbor MPs (Bastian & Lardo, 2018), the old cluster
NGC 2210 is expected to have a significant helium spread, both in terms of the
maximum internal helium spread, and the fraction of He-enriched stars. This
work aims to examine this expectation. We present the data reduction and
method designation in Section 2, and the main results are in Section 3. A
discussion about our results is present in Section 4.

##  2 Methods and data reduction

The data used in this work was observed through the Advanced Camera for
Surveys (ACS) Wide Field Channel (WFC) of the Hubble Space Telescope (HST),
obtained through the Mikulski Archive for Space Telescope (MAST). The program
ID is GO-14164 (PI: Sarajedini). NGC 2210 was observed through the F606W and
F814W filters, with total exposure times of 4306 s and 6950 s, respectively.
We do not use another frame observed through the F336W filter of the WFC3
Ultraviolet and Visual channel in this GO program, because most GCs with MPs
have star-to-star nitrogen variations, which will produce a deep NH-absorption
line centered at $\sim$3370 Å. A point-spread-function (PSF) based photometry
was applied to all charge-transfer-efficiency corrected frames (the ‘_flc’ and
‘_drc’ frames), using the specific HST photometric package Dolphot2.0
(Dolphin., 2011a, b, 2013). Similar to Li (2021b), we filter the raw stellar
catalog to remove bad pixels, centrally saturated objects, extended sources,
cosmic rays and objects being contaminated by crowding.

The dust distribution in the ACS/WFC field of view (FoV,
202${}^{\prime\prime}\times$202′′, corresponding to 48.5 pc$\times$48.5 pc at
the distance of the LMC) may be inhomogeneous. We have used the method
designed in Milone et al. (2012) to minimize the effect caused by the dust
inhomogeneity – the differential reddening. We find a non-negligible
differential reddening across the whole FoV for NGC 2210, with a standard
deviation of $\sigma_{E(B-V)}$=0.008 mag, which will lead to an average color
variation of $\sigma_{({\rm F606W-F814W})}$=0.027 mag. In Fig.1, we exhibit
the differential reddening map for all stars observed in the NGC 2210 field.
In Fig.2 we show the color-magnitude diagrams (CMDs) before/after differential
reddening correction. We find that the observed colors of stars positively
correlate with their differential reddening, i.e., stars with negative
differential reddening are bluer than those with positive differential
reddening. We estimate that a residual of differential reddening of
$\sigma_{E(B-V)/40}$=0.0002 mag cannot be statistically removed222 We use the
40 nearest stars surrounding each individual star to correct for differential
reddening. To avoid a possible underestimation of the helium spread, we used
RGB rather than MS as the referenced population (see Milone et al., 2012)..
However, the assumption that there is a single referenced ridge line may not
be valid if a genuine spatial-dependent helium distribution is present, thus
introducing additional uncertainty. In particular, a helium enrichment would
mimic a negative differential reddening. Our method for reducing differential
reddening may also correct the color shift caused by helium spread, leading to
an underestimate of helium spread.

Figure 1: The differential reddening ($\Delta{E(B-V)}$) map for all stars in
the FoV of the NGC 2210, where $\Delta{E(B-V)}$ is the reddening difference
between the referenced star and the median reddening of all stars in the field
(color-coded). The standard deviation of the differential reddening is
$\sigma_{E(B-V)}$=0.008 mag. Figure 2: Left/right: CMDs of NGC 2210
before/after differential reddening correction.

Using the world coordinate system (WCS) parameters stored in the header of
‘_drc’ frames, we convert the observed pixel coordinates (X and Y) into the
equatorial coordinate system (the $\alpha_{\rm J2000}$ and $\delta_{\rm
J2000}$, the right ascension and declination). We directly adopt the center
coordinates, $\alpha_{\rm J2000}$=06h11m31.69s, $\delta_{\rm
J2000}$=$-69^{\circ}07^{\prime}18.37^{\prime\prime}$, as well as the half-mass
radius, $r_{\rm h}$=15.9${}^{+0.6}_{-0.2}$ arcsec (3.9 pc), for NGC 2210,
based on Ferraro et al. (2019). We divide our sample into two parts, stars
with a radius from the center, $r\leq 32$ arcsec (about 2$r_{\rm h}$) are
defined as cluster stars. Stars with $r\geq 110$ arcsec (about 7$r_{\rm h}$),
which is similar to the tidal radius ($107.1^{+7.6}_{-4.8}$ arcsec, through
the King model) determined in McLaughlin & van der Marel (2005), are defined
as referenced field stars. The spatial distribution and the CMD of NGC 2210
stars are presented in Fig.3, with cluster and referenced field stars being
color-coded (black and red).

Figure 3: Left: spatial distribution of stars in the NGC 2210 field. Right:
the CMD of all stars observed in the NGC 2210 field. Black and red dots are
selected cluster stars and referenced field stars, respectively. Only stars
within the magnitude range defined by dashed lines will be used for analysis.

In this work only MS stars within the magnitude range of 22.5
mag$<$F606W$<$24.5 mag are analyzed (stars between the dashed lines in Fig.3),
corresponding to a mass range from 0.60$M_{\odot}$ to 0.78$M_{\odot}$. We only
select these stars for three reasons: (1) These stars are all located well
below the MSTO region. We exclude stars near the TO region because a possible
helium spread may complicate the morphology of the TO region (see Section 1
for an explanation). (2) The photometric uncertainties for these stars are
small enough to study broadening caused by possible helium spread. (3) The
average completeness of these stars is sufficient for obtaining statistically
robust results, which are $\sim$65.8% for cluster stars and $\sim$82.6% for
referenced field stars (calculated through artificial star test, see below).

We used the Princeton-Goddard-PUC (PGPUC) stellar evolutionary code to
determine the cluster parameters through isochrone fitting (Valcarce et al.,
2012, 2013)333http://www2.astro.puc.cl/pgpuc/index.php. The PGPUC stellar
evolution code is focused mainly on the study of low-mass stars (i.e., stars
in GCs), which allows users to freely input the values of age, helium
abundance ($Y$), global metallicity ($Z$), solar scaled abundance of alpha
element ($[\alpha/{\rm Fe}]$) and Mass loss rate ($\eta$) to generate
different isochrones. Since the goal of our work is to study the helium
distribution of low-mass MS stars in NGC 2210, PGPUC is the most suitable tool
for modeling the observation. We determine the best fitting isochrone through
visual inspection, which is present in the left panel of Fig.4. The best-
fitting parameters are $\log({t/{\rm yr}})$=10.10 ($\sim$12.5 Gyr), $Y$=0.25,
[Fe/H]=$-$1.92 dex ($Z$=0.0002 with $Z_{\odot}$=0.0167 in PGPUC model),
($m-M)_{0}$=18.40 mag (47.86 kpc) and $E(B-V)$=0.06 mag ($A_{V}$=0.20 mag). We
have also assumed an enhanced [$\alpha$/Fe]=$+$0.30 dex (Wagner-Kaiser et al.,
2017) and a default mass loss rate of $\eta$=0.20. The latter is important for
stars at post-MS stages (e.g., HB), which does not affect our results since we
only concern MS dwarfs. Our best-fitting age, distance modulus and average
reddening are comparable to those determined in Wagner-Kaiser et al. (2017)
(our age of 12.5 Gyr vs. theirs of 11.63 Gyr${}^{+1.80}_{-1.12}$ Gyr; distance
modulus of ($m-M)_{0}$=18.40 mag vs. theirs of ($m-M)_{0}$=18.523$\pm$0.042
mag; $E(B-V)$=0.06 mag vs. theirs of $E(B-V)$=0.06–0.08 mag), but the adopted
metallicity is lower than the spectroscopic study of Mucciarelli et al. (2010)
(our metallicity of [Fe/H]=$-$1.92 dex vs. theirs of [Fe/H]=$-$1.65 dex).
However, we find even the best-fitting isochrone does not satisfactorily fit
the observation, which describes the ridge-line of the RGB but only fits the
blue and bright sides of the MS and the SGB. In this work, we want to generate
synthetic populations as close to the real observation as possible. We used
the MS ridge-line (MSRL) instead of the best-fitting isochrone to model
artificial stars. We calculate the MSRL below the TO region using the Gaussian
process and iterative trimming (ITGP) based robust regression method developed
by Li et al. (2020, 2021)444https://github.com/syrte/robustgp/. The MSRL is
shown in the right panel of Fig.4.

Figure 4: Left, the CMD of the NGC 2210 with the best-fitting PGPUC isochrone.
The best-fitting parameters are shown in the legend. Right, the same as the
left panel, with the calculated MSRL. Black dots are stars used for analysis
in this work.

Based on the MSRL, we evaluate the effect of He variation through the PGPUC
stellar evolution code. We calculate eleven isochrones with He-enrichments of
$\Delta{Y}$=0.01–0.12 ($Y$=0.26–0.37, with a step size of 0.01), i.e., a total
of twelve isochrones (including the $Y$=0.25 isochrone). The color deviations
for each isochrone to the standard isochrone ($Y$=0.25),
$\Delta($F606W$-$F814W), are added to the calculated MSRL. These curves are
loci of populations with $Y$=0.26–0.37. In panel $a$ of Fig.5, we show some
examples ($Y$=0.25,0.29,0.33,0.37) of these loci. We then generate synthetic
stellar populations using the technology of artificial stars (ASs). For each
population, we generate 2 million ASs in the corresponding magnitude range
following a Kroupa-like mass function. We totally generated 2.6$\times 10^{7}$
ASs. These ASs are produced using the appropriate PSF model. We perform the
same PSF photometry on these ASs. In order not to dramatically increase the
crowding. We repeat this procedure 260,000 times and each time we only input
100 ASs. The recovered ASs thus mimic a simulated observation with the same
photometric uncertainty (including noise added by cosmic rays, hot pixels,
crowding effect and other artifacts) to real stars. All ASs are homogeneously
distributed in the FoV of the observation. The artificial stellar catalog was
further reduced using the same procedures applied to real stars. Based on ASs,
we obtain the average completeness (the number ratio between the recovered ASs
after data reduction and the input ASs) for stars in the cluster and
referenced field regions ($\sim$65.8% and $\sim$82.6%).

We finally have 13 artificial stellar populations with $Y$=0.25–0.37, each
synthetic population is an SSP, which contains 2 million ASs with realistic
photometric uncertainty like the real observation. Because ASs with a flat
distribution suffers less crowding effect than the observation, if we directly
use the whole sample of ASs, we will underestimate the MS width of ASs, thus
overestimating the resulting helium abundance. Because of this, we only select
ASs in the cluster region to generate synthetic models. Because we have
applied the same data reduction to the artificial stellar catalog, they also
share the similar crowding like the observation. As a result, the selected AS
samples have a similar spatial distribution to the observation. From each
artificial stellar population, we select a subsample with the same luminosity
function and the total number of stars as the real observation as a
representation. A synthetic MP is a composition of these synthetic
populations. In this work, a series of synthetic MPs with different internal
helium spreads and fractions of He-rich stars will be used for quantitative
comparison, to determine the best-fitting property of stellar populations for
the observation. As an example, in panels $b$,$c$ and $d$ of Fig.5, we show
the observed MS, a synthetic SSP with $Y$=0.25, and a synthetic MPs with $Y$
ranges from 0.25 to 0.37, where each population with a certain He abundance
accounts for 1/13 of the total star number. Simply for a glance, we can see
that the observed MS is indeed wider than the synthetic SSP. Its morphology is
more consistent with the example MPs which has a helium dispersion of
$\Delta{Y}=0.12$ (in this toy model, we have assumed a flat distribution of
$Y$). Since a visual examination is unreliable, and a flat distribution of
$\Delta{Y}$ is possibly unphysical, in the next Section, we quantify the
similarity between models with different $\delta{Y}$ and $\Delta{Y}$
distributions, and the observation using statistical methods.

Figure 5: Panel $a$: loci of synthetic populations with
$Y$=0.25,0.29,0.33,0.37. $b$: The observed MS. $c$: A synthetic SSP with
$Y$=0.25. $d$: A synthetic MPs with $Y$ ranges from 0.25 to 0.37.

##  3 Main Results

###  3.1 Helium spread among dwarfs

The minimum angular resolution of the HST at the F814W passband is $\sim$0.1
arcsec, corresponding to $\sim$5000 AU at the distance of the LMC, which is
larger than the separation of the widest binaries in the solar neighbourhood
(Duquennoy & Mayor, 1991). We can assume that all binaries in NGC 2210 are
unresolved. In the CMD, unresolved binaries will populate a brighter and
redder envelope to the MS which can be statistically estimated (e.g., Milone
et al., 2012; Li et al., 2013a). Although our observed sample must contain
some unresolved binaries, however, we do not find any significant unresolved
binary feature from the CMD (see Fig.4), which is different from what we found
for NGC 1846 (Li, 2021b). We find that it is difficult to define an
appropriate unresolved binary region like what we have done for NGC 1846. The
morphology of the MS, particularly its red side, is strongly affected by their
binary properties (the fraction, mass-ratio distribution) and line-of-sight
blending caused by crowding. These effects hamper an accurate estimation of
helium spread. Fortunately, the color of the He-enriched population will be
bluer than the bulk population, this behavior is opposite to unresolved
binaries and blending. For each star, we calculate their relative color
deviation to the MSRL, $\Delta({\rm F606W}-{\rm F814W})$. We have taken a
brief visual comparison between the color distributions of two SSPs
with/without binaries. Although the color distributions of these two stellar
populations are very different in the red sides of their MSs, their blue sides
are similar. We thus decide not to analyze stars lying in the red direction of
the MSRL, i.e., we only analyze stars with $\Delta({\rm F606W}-{\rm F814W})<$0
mag. In the top-left panel of Fig.6, we show the $\Delta({\rm F606W}-{\rm
F814W})$ distribution for all MS dwarfs. We find that the $\Delta({\rm
F606W}-{\rm F814W})$ distribution is not symmetric. The standard deviation of
the color difference is $\sigma_{\rm color}$=0.0323 mag, while the mean and
median photometric errors are $\mathrm{bar}\sigma_{\rm F606W}\approx 0.0086$
mag and $\mathrm{bar}\sigma_{\rm F814W}\approx 0.0084$ mag, respectively.
Their median errors are both 0.008 mag, with 97% measurements have their
photometric errors of $\sigma_{\rm F606W}\leq 0.016$ mag and $\sigma_{\rm
F814W}\leq 0.014$ mag, respectively. (see photometric error curves in Fig.7).
Clearly, photometric uncertainty cannot explain the observed color spread of
the MS solely. We also find a clear excess of ‘red stars’ with $\Delta({\rm
F606W}-{\rm F814W})>$0 mag. We determine the fraction of the excess of the
‘red stars’ is $\sim$2.3%, which is the fraction of unresolved binaries (equal
to a mass-ratio of $q=M_{1}/M_{2}\gtrsim 0.7$) and occasionally blending stars
in the line-of-light direction. The total binary fraction, if assuming a flat
mass-ratio distribution is $\sim$7.7%. If assuming a power-low mass-ratio
distribution, is $\sim$12.0%(e.g., Li et al., 2013a). Both are comparable to
those for Galactic GCs (5%–30%, with a flat mass-ratio distribution Milone et
al., 2012), but lower than younger LMC clusters ($\geq 50\%$, Li et al.,
2013a; Li, 2021b).

Figure 6: Top-left: the observed $\Delta({\rm F606W}-{\rm F814W})$
distribution (grey curve). The distribution for stars with $\Delta({\rm
F606W}-{\rm F814W})<$0 mag is indicated by the black curve. The black dashed
line represents the mirror of the black solid line (relative to the
$\Delta({\rm F606W}-{\rm F814W})=$0 mag). A clear excess of stars on the red
side of the MSRL appears, which indicates the contribution of unresolved
binaries. Top-right: the same as the top-left panel, with similar
distributions of synthetic MPs (red solid/dashed curves). Bottom panel: the
$\chi^{2}$ distribution as a function of $\Delta{Y}$ and the cubic fitting.
The implicated $\Delta{Y}$ for minimum $\chi^{2}$ is indicated by arrow.
Figure 7: The photometric error curves for F606W (top) and F814W (bottom)
passbands.

However, we emphasize that assuming no unresolved binary system would have a
negative color deviation is not true. Photometric uncertainty would scatter
some unresolved binaries (particularly those with low mass-ratios) to the blue
side of the MSRL, which is unavoidable in our analysis. In addition, there
must be some (low mass-ratio) binaries belonging to He-rich stellar
populations (if present) that are bluer than the MSRL. We think that the
number of He-rich binaries must be small. Because most primordial scenarios
assume the secondary stellar populations (thus He-rich) form in a more
centrally concentrated state than normal stars, which would lead to a more
severe dynamical disruption (Hong et al., 2016). Indeed, observations have
shown that 2P stars of various clusters have less binary fraction than 1P
stars (D’Orazi et al., 2010; Lucatello et al., 2015, but see Milone et al.
(2020b)). In this work we can only minimize (rather than exclude) the binary
contamination.

Using the same method described above, we analyze the relative color
distributions for synthetic MPs with different internal helium spreads. We
assume a flat $\delta{Y}$ distribution for all these MPs. For example, the
model with $\delta{Y}$=0.03 which contains four populations,
$Y$=0.25,0.26,0.27,0.28, would have each population occupy 25% number of
stars. For both the observation and the synthetic MPs, we only study their
distributions of stars with $\Delta({\rm F606W}-{\rm F814W})<$0 mag. The total
number of these stars is $\sim$5000\. We divide these stars into 20 color
bins, and the bin width is 0.005 mag. This bin size allows us to study their
helium distribution in more detail, and in each bin the number of stars is
high enough so that they are not strongly affected by statistical variations.
To obtain a preliminary comparison, we first analyzed the standard deviation
of the color distribution for an SSP, assuming that their color distribution
is Gaussian-like. Our result yields $\sigma_{\rm color}$=0.0257 mag. A helium
spread of $\delta{Y}$=0.06 ($\sigma_{\rm color}$=0.0317 mag) is required to
meet the observation ($\sigma_{\rm color}$=0.0323 mag).

We then use a $\chi^{2}$ minimization method to quantify the similarity
between models and the observation,

$\chi^{2}=\sum_{i}\frac{(N^{\rm obs}_{\rm i}-N^{\rm mod}_{\rm i})}{N^{\rm
obs}_{\rm i}}$ (1) $N^{\rm obs}_{\rm i}={N^{\rm c}_{\rm i}}-f\frac{A^{\rm
c}}{A^{\rm f}}N^{\rm f}_{\rm i}$ (2)

where $N^{\rm c}_{\rm i}$ and $N^{\rm f}_{\rm i}$ are the number of stars with
their relative colors dropped in the $i$-th bin. The subscript of ‘c’ and ‘f’
means these stars are in the cluster and referenced field regions,
respectively. In this work, the area of the cluster region is about 55.4% of
the referenced field region, which denotes $f$=0.554. $A^{\rm c}$ and $A^{\rm
f}$ are 0.658 and 0.826, which are the average completeness for cluster and
referenced field stars. $N^{\rm obs}_{\rm i}$ is thus the expected number of
stars observed in the cluster region, and with their relative colors belong to
the $i$-th bin. $N^{\rm mod}_{\rm i}$ is the corresponding number of ASs in
the model used for comparison. Finally, we want to examine if the result
$\chi^{2}$ correlates with the model internal helium spread, $\Delta{Y}$. We
plot their correlation in the bottom panel of Fig.6.

We find that the $\chi^{2}$ distribution exhibits a smooth trend with
$\delta{Y}$. The minimum $\chi^{2}$ occurs at $\delta{Y}=$0.05, with a
$\chi^{2}$=341. To avoid the effect of statistical noise, we used a cubic
curve to fit the $\chi^{2}$–$\delta{Y}$ correlation, which yields a local
minimum $\chi^{2}$ at $\delta{Y}=0.06$. In the top-right panel of Fig.6, we
exhibit comparisons between some models and the observation. Indeed, an SSP
($\delta{Y}$=0.0) does not produce the observed $\Delta({\rm F606W}-{\rm
F814W})$ distribution, while a MPs with $\delta{Y}$=0.05 exhibit a much better
fitting to the observation. For a better illustration, we have symmetrized the
$\Delta({\rm F606W}-{\rm F814W})$ distribution although we only analyze stars
with $\Delta({\rm F606W}-{\rm F814W})<$0 mag (in Fig.7).

The analysis indicates that NGC 2210 may indeed harbor He-rich population
stars. A disadvantage is that under the assumption of a flat $Y$ distribution,
all these toy models indicate that He-rich stars dominate the sample, which is
unrealistic. To derive a more realistic helium distribution, we have generated
a series of synthetic MPs with different $\delta{Y}$ and fractions of 2P stars
(stars with $\Delta{Y}>0$), $f_{\rm 2P}$. Among the 2P stars, their helium
distribution, $\delta{Y}$, is flat. For example, MPs with $\delta{Y}$=0.02 and
$f_{\rm 2P}$=30% would have 70% normal stars, and each He-rich population
($\delta{Y}$=0.01, 0.02) occupy a number fraction of 15%. As a result, we
totally generated 229 models, including one SSP model ($\delta{Y}$=0), and 228
MPs models with $\delta{Y}$=0.01–0.12 (in step of 0.01), and $f_{\rm
2P}=$5%–95% (in step of 5%). For each model, we calculate its corresponding
$\chi^{2}$. Finally we obtained a 2D-distribution of $\chi^{2}$ as a function
of $\delta{Y}$ and $f_{\rm 2P}$, we plot a contour of the $\chi^{2}$
distribution in Fig.8 (top panel). We find that if the fraction of 2P stars is
too low ($f_{\rm 2P}\leq 20\%$), it yields a $\delta{Y}\sim 0.08$, but the
$\chi^{2}$–$\delta{Y}$ distribution is very noisy. For $f_{\rm 2P}\geq 40\%$,
the $\chi^{2}$–$\delta{Y}$ correlation becomes smooth. They all report a best-
fitting $\delta{Y}$ ranges from 0.06–0.10. The minimum range of $\chi^{2}$
occurs at $\delta{Y}\sim$0.068–0.071 and $f_{\rm 2P}\sim$52%–61%(shadow region
in the top panel Fig.8), corresponding to a $\chi^{2}\leq$364.0, within this
region the variation of $\chi^{2}$ is dominated by noise. In summary, if we
assume a continuous $\delta{Y}$ distribution, NGC 2210 is likely to have $\sim
55\%$ He-rich stars, with an internal helium spread of
$\delta{Y}=0.069^{+0.002}_{-0.001}$.

However, such a high fraction of 2P stars is surprising. According to
primordial scenarios, the primordial population (1P) stars form earlier than
the chemically enriched population (2P) stars with a more extended
configuration. As a result, the 1P stars are more easily stripped by the
external galactic tidal field (e.g., D’Ercole et al., 2008). The number ratio
between the 2P and 1P stars would be lower for GCs in a weaker external tidal
field (the LMC) than those in the Galaxy. A possible explanation is that the
$\delta{Y}$ distribution is discrete (such as NGC 2808) rather than
continuous. We therefore generated another set of models, in these models, MPs
have a bimodal distribution of $\delta{Y}$. For example, a model with 30% He-
rich stars and $\delta{Y}$=0.02 only contains two populations, i.e., 70%
normal stars ($\delta{Y}$=0.00) and 30% He-rich stars ($\delta{Y}$=0.02).
Again, under the adoption of bimodal distributions for MPs, we plot the
$\chi^{2}$ distribution as a function of $\delta{Y}$ and $f_{\rm 2P}$, which
is present in the bottom panel of Fig.8. This time, the lowest $\chi^{2}$
region occurs at $\delta{Y}$=0.068–0.074 and $f_{\rm 2P}$=26%–34% (shadow
region in the bottom panel Fig.8). We find that MPs with a bimodal
distribution of $\delta{Y}$ can better reproduce the observation, as they
return a lower $\chi^{2}$ ($\lesssim$332) than the case of a flat $\delta{Y}$
distribution ($\lesssim 364$). We suggest that a small fraction of 2P stars is
more reasonable. Indeed, studies have shown that chemically enriched
populations in some LMC clusters only occupy a small fraction of 10%–20%
(Hollyhead et al., 2019; Dondoglio et al., 2021). We conclude that NGC 2210
most likely harbors $\sim 30\%$ He-enriched stars, with a maximum helium
spread of $\delta{Y}=0.071\pm 0.003$.

Figure 8: The contour of $\chi^{2}$ distribution as a function of helium
spread ($\delta{Y}$) and fractions of 2P stars ($f_{\rm 2P}$). In the top
panel, model MPs have a continuous $\delta{Y}$ distribution. In the bottom
panel, model MPs have a bimodal $\delta{Y}$ distribution.

Our results indicate that NGC 2210 is very different to NGC 1846, the latter
is likely an SSP cluster ($\delta{Y}<$0.02). However, the detection of a
helium variation does not indicate that the He-rich stars do belong to the 2P.
In addition to He-enhancement, if we strictly define the 2P stars are those
with Na, C, N, O variations, some 1P stars (stars without these specific
chemical patterns) are found to have helium spread (e.g., Milone et al.,
2017a) as well, although the reason remains unclear yet. How to determine if
the derived helium spread is an internal spread of 1P stars or if it indicates
the presence of MPs? One way is to examine their radial distributions. If both
He-normal and He-rich stars are 1P stars, they should be fully mixed in
spatial. Otherwise 1P and 2P stars may exhibit different central
concentrations, according to primordial scenarios. In this work, we cannot
determine if a specific star is He-enriched or normal. Alternatively, we
compared each star’s color deviation to their photometric uncertainty. Stars
with a color deviation $|\Delta({\rm F606W-F814W})|$ larger than three times
the expected color uncertainty are defined as He-rich stars. Using this
criterion, 33.7% stars (1547 of 4557) are defined as He-rich stars. This is
consistent with the indication derived by bimodal $\delta{Y}$ distribution
models ($\sim$30%). If the observed MS dwarfs are SSP, we would expect only
$14\pm 4$ stars (0.3%) meet this criterion.

We divide both the normal and He-rich stars into seven radial bins ranging
from the cluster center to a radius of 7 pc ($\sim$2$r_{\rm hm}$), with a bin
size of 1 pc, the latter is roughly the core size ($r_{\rm c}$) of NGC 2210
(Ferraro et al., 2019). We study the radial profile of the number ratio
between He-rich and normal stars. If He-rich stars belong to a secondary
stellar population formed in the cluster center after the formation of 1P
stars, the number ratio radial profile should exhibit a decreasing trend from
the cluster center to its outskirt. In the right panel of Fig.9 we exhibit
this number ratio radial profile. We cannot tell any radial difference between
the He-rich and normal stars within 2$r_{\rm c}$. We find a significant
decreasing trend in the range of 2–7 pc, indicating that the He-rich
population has a more compact configuration than normal stars. Since He-rich
and normal stars are all in the same magnitude range, this radial difference
cannot be explained by their completeness difference in the radial direction.
In addition, He-rich dwarfs are less massive than normal dwarfs with similar
luminosities, which further strengthens the implication that they must be
initially much more compact than normal stars.

Figure 9: Left, the CMD of He-rich (red dots) and normal (blue dots) stars.
Right, the number ratio radial distribution between He-rich and normal stars.
The red dashed and dash-dotted lines represent the positions of the core and
half-mass radius, respectively. Associated error bars are Poisson-like.

Milone et al. (2017a) have derived a clear correlation between the internal
maximum helium spread and cluster’s present-day mass. We want to examine if
NGC 2210 also follows the trend. We compare our results with Galactic GCs
(Milone et al., 2017a), LMC clusters (Li, 2021b; Ji et al., 2022), other SMC
clusters with internal helium spread studied in literatures (Chantereau et
al., 2019; Lagioia et al., 2019, hereafter C19/L19). This result is shown in
the top panel of Fig.10. We find that although the helium spread of NGC 2210
is relatively higher than its Galactic counterparts, it is consistent with the
same correlation. If we only consider LMC clusters, this correlation is likely
steeper than that for Galactic GCs.

The cluster initial mass should be a more appropriate parameter that decide
the property of MPs than their present-day masses. For Galactic GCs, Baumgardt
et al. (2019) have integrated their orbits backward in time to derive the
cluster initial masses, taking the effects of dynamical friction and mass-loss
of stars into consideration. Using the same method, we have calculated the
initial mass of NGC 2210 using N-body simulations, which yields 5.1$\times
10^{5}$ $M_{\odot}$. We find this cluster have lost very limited mass through
dynamical effects because of its high mass and the weak tidal field of the
LMC. The difference between the present-day and the initial mass is almost
entirely due to stellar evolution mass loss. As a result, the present-day
number ratio between 1P and 2P stars of NGC 2210 should be almost identical to
its initial value. In the bottom panel, we present the helium spread–initial
mass relationship for Galactic GCs (grey dots) and NGC 2210 (the red star). It
turns out that NGC 2210 indeed harbor a higher internal helium spread than its
Galactic counterparts with similar initial masses. It remains unclear if this
would indicate that LMC GCs would exhibit a different helium spread–initial
mass correlation, studies of more LMC samples are required. The initial masses
for LMC/SMC clusters will be present in a forthcoming article.

Figure 10: Top: the internal helium spreads and the clusters present-day
masses diagram, $\delta{Y}$–$\log(M/M_{\odot})$. Red circles and a pentagram
(NGC 2210, this work) are LMC clusters. Small grey dots are Milky Way GCs.
Dark/light grey circles are SMC clusters. For NGC 1846 we have used its
expected total mass at $\sim$10 Gyr. Bottom: the correlation between the
internal helium spreads and the clusters initial masses, for Milky Way GCs and
NGC 2210.

###  3.2 Comparing with evolved giants

Using the same method, we have derived the helium spread among red-giant stars
of NGC 2210. The sample we used is red-giant (RG) stars lying significantly
above the bottom of the RGB (F606W$\sim$21.28 mag) and below the RGB bump
(F606W$\sim$17.56 mag), in the range of 19.08 mag$\leq$F606W$\leq$19.98 mag.
We constrain the sample RG stars with a color range of 0.65
mag$\leq$F606W$-$F814W$\leq$0.75 mag. The selections of the magnitude and
color ranges are arbitrary. We confirm that the calculated ridge line of the
selected RGB part is close to the best-fitting isochrone. We exhibit the
selected RG stars and the best-fitting isochrone in Fig.11. However, NGC 2210
exhibits many blue straggler stars (BSSs). A significant fraction of these
BSSs may lie in a mass-transferring binary system, where another binary
component is likely a sub-giant (SG) or a RG star. These BSS-SG/RG binaries
would be distributed in a region between the RGB and the BSS locus, partially
overlapping with the He-rich RGB. We generate a large number of artificial
BSS-SG/RG binaries and plot them in the CMD of the cluster. For each BSS-SG/RG
binary, the BSS is randomly selected from the BSS locus, which is described by
a 1 Gyr-old isochrone (with other parameters identical to the best-fitting
isochrone). The SG/RG star is selected from the best-fitting isochrone. We
find that the region of the BSS-SG/RG binaries exhibits a clear top boundary,
which gradually decreases from the region close to the TP-AGB toward the TO
region of the BSS. Between this boundary and the HB region, there are no
stars. This ideally describes the observation. Given that single stars in the
Hertzsprung gap region evolve rapidly, we conclude that most observed stars in
this region are unresolved BSS-SG/RG binaries. Some of these binaries will
strongly contaminate the He-rich RG population (if the RG component dominates
the flux of the binary system). Because of this, we expect that the helium
spread derived from the width of the RGB would be overestimated if we cannot
rule out the binary contamination.

Figure 11: RG stars (orange dots) were selected to study the helium spread.
The blue dots are simulated BSS-SG/RG binaries (unresolved). The solid blue
curve describes the BSS locus. The red curve is the best-fitting isochrone.

Indeed, our analysis report that, if we assume that helium spread fully
accounts for the width of the RGB, the internal helium spread will reach
$\delta{Y}$=0.12, the upper limit of the model we used. Binaries play an
important role in the broadening of the RGB. The fraction of He-rich stars is
22%–33%, which is consistent with the result derived from MS stars (Fig.12)

Figure 12: The same as Fig.8, but for RG stars. The helium distribution of the
model RGB is bimodal.

Because PGPUC does not calculate HB phase, we used the Modules for Experiments
in Stellar Astrophysics (MESA, Paxton et al., 2011) to examine if our result
also fits the HB morphology. We use the MESA to calculate three 12.5 Gyr-old
HB loci with $Y=$0.26, 0.30, 0.34, respectively. They thus briefly describe
the morphology of HB with $\delta{Y}\sim 0.08$. The adopted metallicity is
[Fe/H]=$-$1.92 dex (the same as the best-fitting PGPUC isochrone). The most
important parameter controlling the HB morphology is the mass loss rate.
During the RGB phase, it is described by Reimer’s mass loss rate ($\eta_{R}$,
Reimers, 1975, 1977). The mass loss rate for RG stars varies from cluster to
cluster, covering a range of $\eta_{R}<0.2$ to $\eta_{R}>0.6$ (Tailo et al.,
2020). Because the mass loss rate in our model is a free parameter, the helium
spread among HB stars is uncertain. To make a qualitative comparison, we first
conservatively set a $\eta_{R}=0.2$ to our model. In this case, the simulated
HB with $\delta{Y}$=0.08 exhibits a more extended morphology than the
observation. Under the adoption of $\eta_{R}=0.2$, a $\delta{Y}$=0.04
($Y$=0.26–0.30) is sufficient to explain the length of the observed HB. We
then calculated another two model sets with $\eta_{R}=0.1$ and 0.05. We find
that once we adopt $\eta_{R}=0.05$, the simulated HB population with a helium
spread of $\delta{Y}=0.04-0.08$ fits the observation better. We finally
constructed a stellar HB population with three helium abundances ($Y=$0.26,
0.30, 0.34), with each sub-population containing stars with different mass
loss rate $\eta_{R}=0.05-0.20$. We present the fitting to the observation in
Fig.13.

Figure 13: The CMD of the HB, overlap with the simulated HB populations with
different helium abundances and mass loss rates.

##  4 Discussion and Conclusion

Before we discuss the physical implications of our results, we first examine
if there is no helium spread, at what value an additional differential
reddening is required to fully account for the width of the MS, by comparing
the synthetic stellar population with different differential reddening with
the de-reddened observation. Our analysis reports an additional differential
reddening of $\sigma_{E(B-V)}=0.004$ mag is required to explain the observed
MS. This is about 20 times the expected differential reddening residual. We
confirmed that the signature of such an additional reddening is significantly
enough to be revealed through our de-reddening method, if present. Therefore,
we conclude that the broadening of the width cannot be fully explained by
differential reddening.

Another effect that would contribute to the width of the MS is metallicity
spread. Decreasing the metallicity would reduce the stellar atmospherical
opacity, leading to a decrease in cooling efficiency, thus, an increase in the
stellar surface temperature. For this reason, stars with lower metallicity
will look bluer than normal stars at each evolutionary stage, populating a
bluer MS. We then generate a series of isochrones with different
metallicities, and compare their loci with the $Y=0.33$ isochrone, the latter
corresponds to the $\Delta{Y}=0.07$ stellar population locus. However, we find
that even if we generate an isochrone with $Z$=0.00001 (the lower metallicity
limit of the PGPUC model), the color difference between this isochrone and the
best-fitting isochrone ($Z$=0.00016) cannot describe the width of the MS.
Since we only concern at what value a metallicity spread can describe the
width of the MS, we release the upper limit of the metallicity. Although in
that case, the isochrone may not be able to describe the CMD well. We find
that a metallicity spread from $Z$=0.0002 to $Z$ = 0.001
([Fe/H]=$-1.92$$\sim$$-1.22$ dex) is required to fully account for the
observed width of the MS. Such a metallicity spread would produce a very wide
SGB and RGB. The inconsistency between the model isochrones and the
observation can be easily derived visually. We thus exclude the presence of a
dramatic metallicity spread among NGC 2210 members. We also confirm that a
spread of $[\alpha/{\rm Fe}]$=0.00–0.30 dex (the maximum input range allowed
for the PGPUC model) has a negligible contribution to the width of the MS.

A similar analysis reports that RGB stars may contain $\sim$30% He-rich stars,
which agrees with our analysis for MS dwarfs. But it implies a higher helium
spread content, $\delta{Y}$–0.12. As we have illustrated, this is likely due
to the contamination of BSS/RG binaries. The morphology of the HB, is very
short, however. Our simulation indicates that under the fixed mass loss rate
of $\eta_{R}=0.2$, only a helium spread of $\delta{Y}$=0.04 is sufficient to
explain the observed HB, which is lower than the value derived from MS dwarfs
($\delta{Y}=0.06-0.07$). A lower mass loss rate down to $\eta_{R}=0.05$ would
fit the observation better with a higher helium spread up to $\delta{Y}$=0.08.
However, this would indicate that RGB stars in NGC 2210 experienced less mass
loss than Galactic GCs (Tailo et al., 2020). Speculation is that the parameter
that controls the HB color extension is environment-dependent. The tidal
fields of their host galaxies affect the mass loss during their RGB phase. We
highlight that another LMC GC, Hodge 11, exhibits an extended HB (Gilligan et
al., 2019, their figure 19). An accurate determination of internal helium
variation for this cluster would be crucial.

Since analyses of RGB and HB members are affected by binaries and mass loss
rates from star to star, both are very uncertain. We are now back to
discussing the results derived from MS dwarfs, as the helium spread inferred
by MS dwarfs is indicative of the helium contents of the clouds from which
multiple populations formed. In summary, in this work, our main conclusions
are,

  * 1.

NGC 2210 does exhibit a helium spread of $\delta{Y}\sim 0.06$–0.07. The number
ratio of He-rich stars to the whole population is about $\sim$30%, if assume
that the $\delta{Y}$ distribution is bimodal. Otherwise it would be more than
half, 55%, if the $\delta{Y}$ distribution is continuous.

  * 2.

He-rich stars are more centrally concentrated than normal stars, indicating
that the detected helium spread is not an internal spread among 1P stars, but
for two stellar populations formed in different initial configurations.

  * 3.

The internal helium spread, $\delta{Y}$, of NGC 2210 is consistent with the
correlation between the helium spread and the clusters’ present-day mass for
Galactic GCs. If we only consider LMC clusters, this correlation is even
steeper.

In a previous study/search on multiple populations in NGC2210, Gilligan et al.
(2019) detected a broadened MS and estimated that the second population
includes 20$\pm$5% of stars. The fraction of second-population stars derived
in this work is higher than theirs, if we assume a bimodal helium distribution
(26%–34%). In addition, we inferred the internal helium variation of NGC2210
by assuming that the MS color broadening is mostly due to an helium scatter.
This is a reasonable hypothesis because most elemental absorption lines
concentrated in the UV band, their effects on the color of F606W$-$F814W are
negligible (see figure 5 of Milone et al., 2018, as an example).

For LMC clusters, the positive correlation between the internal helium spread,
$\Delta{Y}$, and the present-day GC mass (Li, 2021b), is similar to that for
Galactic GCs (Milone et al., 2018). If the $\delta{Y}$ distribution is (close
to) bimodal, the 2P fraction of NGC 2210 would be smaller than its Galactic
counterparts with comparable masses (Milone & Marino, 2022, their figure 8).
This would support scenarios where GCs preferentially lose their 1P stars, as
1P stars of LMC GCs experience weaker tidal stripping than Milky Way clusters.
The fact that He-rich stars are more centrally concentrated than He-normal
stars in NGC 2210 is in qualitatively agreement with the prediction from the
main scenarios on the formation of multiple populations (e.g., Krause et al.,
2013; D’Antona et al., 2016; Calura et al., 2019; Gieles et al., 2018; Wang et
al., 2020). After the gas expulsion, both 1P and 2P stars escape during the
long-term evolution by the galactic tide, and a large amount of time (up to
$\sim$20$t_{\rm rh}$) is needed to fully mix the 1P and 2P stars (e.g.,
Vesperini et al., 2013). According to McLaughlin & van der Marel (2005), the
$t_{\rm rh}$ for NGC 2210 is 1.0–1.2 Gyr ($\log{t_{\rm th}}$=9.01–0.06, model
dependent). If the half-mass relaxation timescale does not significantly
change during its evolution, the dynamical age of NGC 2210 is 10–12$t_{\rm
rh}$. At this dynamical age, we would expect that at least the part with
$r\leq{r_{\rm rh}}$ will already be fully mixed, which is inconsistent with
our observation. We speculate that this is because McLaughlin & van der Marel
(2005) used a simplified model to calculate the $r_{\rm rh}$ in which the
average stellar mass in NGC 2210 is assumed $M_{\star}$=0.5$M_{\odot}$, which
introduces an additional uncertainty. Using the same method of Baumgardt et
al. (2019), our calculation yields a longer half-mass relaxation time of
$t_{\rm rh}\sim$3.2 Gyr for NGC 2210, indicating that NGC 2210 is only 3–4
$t_{\rm rh}$ old. This is in good agreement with our observation as only stars
within the core radius are fully mixed.

The difference between NGC 2210 and NGC 1846, where the latter exhibit a
minimum helium spread, may be controlled by the difference in their masses or
ages, as NGC 1846 is younger and, less massive (at the age of $\sim$10 Gyr)
than NGC 2210. To determine which parameter plays the critical role, further
studies focusing on younger and more massive LMC clusters (i.e., NGC 1850) in
terms of their helium distributions are required. Again, this would require
high precision photometry focusing on their MS as young clusters usually do
not have well-populated RGB and HB.

If the He-rich stars detected in NGC 2210 represent 2P stars, they may exhibit
common patterns of MPs, i.e., Na-O anti-correlation, C-N anti-correlation. A
spectroscopic analysis of these stars is not possible because these stars are
too faint. Alternatively, utilizing UV-optical photometry may statistically
examine whether or not these stars are different in C, N, O abundances (e.g.,
Li et al., 2021a). Again, this requires deep photometry which will consume
lots of HST time. The next-generation Chinese Space Station Telescope (CSST)
with similar parameters to the HST, can take over this task with a larger FoV
(Li et al., 2022).

C. L. is supported by the National Key R&D Program of China (2020YFC2201400).
D.J. acknowledge support from the National Natural Science Foundation of China
(Nos 12073070, 11733008). C.L. and L.W. acknowledge support from the one-
hundred-talent project of Sun Yat-sen University and the National Natural
Science Foundation of China through grant 12073090 and 12233013. B.T.
gratefully acknowledges support from the National Natural Science Foundation
of China under grant No. U1931102, and the Natural Science Foundation of
Guangdong Province under grant No. 2022A1515010732.This work is also supported
by the China Manned Space Project with NO.CMS-CSST-2021-A08, CMS-
CSST-2021-B03, National Key R&D Program of China with No. 2021YFA1600403 and
CAS ‘Light of West China’ Program. Y.W. acknowledges the support by the
Special Research Assistant Foundation Project of Chinese Academy of Sciences.

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