# Probing the nature of rotation in the Pleiades, Alpha Persei, and Hyades
clusters

C. J. Hao  Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210023, PR China  School of Astronomy and Space Science, University of Science
and Technology of China, Hefei 230026, PR China  Y. Xu  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210023, PR China  School of
Astronomy and Space Science, University of Science and Technology of China,
Hefei 230026, PR China  L. G. Hou  National Astronomical Observatories,
Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing
100101, PR China  S. B. Bian  Purple Mountain Observatory, Chinese Academy of
Sciences, Nanjing 210023, PR China  Z. H. Lin  Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210023, PR China  Y. J. Li  Purple
Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, PR China
Y. W. Dong  Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210023, PR China  School of Astronomy and Space Science, University of Science
and Technology of China, Hefei 230026, PR China  D. J. Liu  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210023, PR China  School of
Astronomy and Space Science, University of Science and Technology of China,
Hefei 230026, PR China

###### Abstract

Unraveling the internal kinematics of open clusters is crucial for
understanding their formation and evolution. However, there is a dearth of
research on this topic, primarily due to the lack of high-quality kinematic
data. Using the exquisite-precision astrometric parameters and radial
velocities provided by _Gaia_ data release 3, we investigate the internal
rotation in three of the most nearby and best-studied open clusters, namely
the Pleiades, Alpha Persei, and Hyades clusters. Statistical analyses of the
residual motions of the member stars clearly indicate the presence of three-
dimensional rotation in the three clusters. The mean rotation velocities of
the Pleiades, Alpha Persei, and Hyades clusters within their tidal radii are
estimated to be 0.24 $\pm$ 0.04, 0.43 $\pm$ 0.08, and 0.09 $\pm$ 0.03 km s-1,
respectively. Similar to the Praesepe cluster that we have studied before, the
rotation of the member stars within the tidal radii of these three open
clusters can be well interpreted by Newton’s theorem. No expansion or
contraction is detected in the three clusters either. Furthermore, we find
that the mean rotation velocity of open clusters may be positively correlated
with the cluster mass, and the rotation is likely to diminish as open clusters
age.

open clusters and associations: individual (Pleiades) – open clusters and
associations: individual (Alpha Persei) – open clusters and associations:
individual (Hyades) – stars: kinematics and dynamics – methods: statistical

##  1 Introduction

Most stars in the Milky Way are thought to form in star clusters (Lada & Lada,
2003; Bressert et al., 2010; Megeath et al., 2016). Open clusters (OCs),
gravitationally bound systems of tens to thousands of stars, are the rare
survivors of the embedded clusters gestated in molecular clouds (Elmegreen &
Clemens, 1985; Bastian & Goodwin, 2006), with ages spanning a broad range from
a few million years (Myr) to tens of billions of years. At present, the
formation, evolution, and eventual dispersal of OCs are still poorly
understood (e.g., Proszkow et al., 2009; Farias et al., 2018; Hao et al.,
2023), although thousands of Galactic OCs have been discovered (e.g., Hao et
al., 2022b; Castro-Ginard et al., 2022; Hunt & Reffert, 2023). In order to
better understand the formation and evolution of OCs, one of the key aspects
is to obtain a detailed characterization of their internal kinematics (see
Della Croce et al., 2023, and references therein). Hydrodynamical simulations
of the cluster formation and very early evolution suggest that rotation may be
common in the progenitors of OCs (e.g., Mapelli, 2017). However, unlike the
Galactic globular clusters, whose rotation has been studied extensively (e.g.,
van Leeuwen et al., 2000; Lanzoni et al., 2018; Leanza et al., 2022), our
knowledge about the internal rotation of OCs is sorely lacking.

To date, the signals of rotation have been reported in only about a dozen of
OCs. By identifying the correlation between the tangential velocities and the
parallaxes of member stars, Vereshchagin & Chupina (2013) and Vereshchagin et
al. (2013) explored the potential rotation in the Hyades and Pleiades
clusters, and speculated that their rotation was comparable. Similarly, Kamann
et al. (2019) and Healy et al. (2021) studied the tangential or radial
velocities of cluster members as a function of distances to the cluster
center, and they reported the presence of rotational signals in the Preasepe
and NGC 6791 and the absence of rotational signals in the Pleiades and NGC
6819. Based on the second data release of _Gaia_ (DR2, Gaia Collaboration et
al., 2016, 2018), Loktin & Popov (2020) analyzed the rotation in the Praesepe
through the residual proper motion or radial velocity methods. Recently, by
using the early data release 3 of _Gaia_ (EDR3, Gaia Collaboration et al.,
2021) to perform vector-field inference and create spatio-kinematic maps of
OCs, Guilherme-Garcia et al. (2023) reported the detection of rotation
patterns in eight clusters, with nine additional objects displaying possible
rotation signs. However, all these studies adopted the methods of projection
motion (proper motion or radial velocity) to study the rotation of OCs. The
three-dimensional (3D) rotation of OCs, especially their rotation nature,
remained elusive until the 3D rotation of Praesepe was revealed by Hao et al.
(2022a, hereafter, Paper I).

Observationally, studying the 3D rotation of OCs is challenging. One of the
main difficulties is obtaining a detailed characterization of the internal
stellar kinematics, which requires high-quality measurements of the proper
motions and radial velocities for the member stars. For the OCs with dozens to
hundreds of members assembled in crowded fields, the precise spatial
positions, and especially, the parallaxes of the member stars, are additional
difficulties when rotation is to be inferred. _Gaia_ DR3 (Gaia Collaboration
et al., 2023) opens the possibility of new studies of stellar cluster
dynamics. Considering the data quality of _Gaia_ DR3, we focus on the nearby
OCs that are rich in member stars, as _Gaia_ DR3 is able to provide high-
precision astrometric parameters and especially the radial velocities for a
sufficient number of their member stars. In addition, reliable membership
determination of an OC is also critical for revealing its internal rotation.
After cross-matching a series of large-scale surveys to complement the _Gaia_
catalog, Lodieu et al. (2019a, b) provided revised membership lists of four
nearby star-rich OCs based on comprehensive analyses, including the Pleiades,
Praesepe, Alpha Persei ($\alpha$ Per), and the Hyades. The ages of these four
OCs span tens to hundreds of million years. In Paper I, we have developed the
data-analyzing method and studied the Praesepe. In this work, we aim to assess
the 3D rotation of the Pleiades, $\alpha$ Per, and the Hyades with  _Gaia_
DR3, and then explore the possible relation between the rotation of OCs and
the cluster parameters (e.g., age, mass, and radius).

The Pleiades, also known as the Seven Sisters, are easily visible to the naked
eye in the constellation of Taurus (Lynga, 1981). The distance to the Pleiades
has been estimated by many methods, such as the trigonometric parallax (e.g.,
Gatewood et al., 2000; Soderblom et al., 2005), fitting a color-magnitude
diagram (e.g., An et al., 2007), and adoption of the eclipsing binaries as
standard candles (e.g., Southworth et al., 2005). Currently, the distance to
the Pleiades has been agreed to be the mean value of 130–140 pc (Melis et al.,
2014; Gaia Collaboration et al., 2016). The age of the Pleiades cluster is a
subject of debate, with estimates ranging from 70 to 80 Myr based on
comparisons of the color-magnitude diagram of the cluster with theoretical
isochrones of stellar evolution (e.g., Mermilliod, 1981; Vandenberg & Bridges,
1984), to 120–130 Myr derived from the lithium-depletion boundary method
(e.g., Basri et al., 1996; Dahm, 2015), and 132 Myr inferred from the white
dwarf in the cluster (Lodieu et al., 2019a). In addition, the Pleiades contain
more than 1000 member stars that share common proper motions of
($\mu_{\alpha^{*}}$, $\mu_{\delta}$) $\sim$ (19.5, –45.5) mas yr-1, a high
Galactic latitude ($b$ $\sim$ 24∘), and low reddening along the line of sight
of $E(B-V)$ = 0.03 mag (see Jones, 1981; Robichon et al., 1999; van Leeuwen,
2009). This cluster has a tidal radius of 11.6–19.5 pc, a core radius of
0.9–3.0 pc, and a cluster mass of 721–870 ${\rm M}_{\odot}$ (Pinfield et al.,
1998; Raboud & Mermilliod, 1998; Converse & Stahler, 2008, 2010).

The cluster $\alpha$ Per, also known as Melotte 20 or Collinder 39 (Melotte,
1915; Collinder, 1931), is located in the northern constellation of Perseus.
It is composed of several blue B-type stars (Zuckerman et al., 2012). The
distance to this cluster is estimated to be between 170 and 190 pc (Robichon
et al., 1999; van Leeuwen, 2009; Majaess et al., 2011; Gaia Collaboration et
al., 2018). The trigonometric parallax and infrared color-magnitude diagram
fitting methods give consistent distances for $\alpha$ Per, making it a
benchmark on the cosmic distance ladder. In a comprehensive study, Lodieu et
al. (2019a) derived a mean distance of 177.7 $\pm$ 0.8 pc for the cluster.
Furthermore, the $\alpha$ Per has an estimated age of approximately 50–70 Myr
(Prosser et al., 1996; Makarov, 2006). Its extinction along the line of sight
is estimated to be $A_{V}$ = 0.30 mag, with a possible differential extinction
(Prosser, 1992). The $\alpha$ Per is located at a low Galactic latitude of
($b$ $\sim$ –7∘, e.g., Makarov, 2006; van Leeuwen, 2009), has a proper motion
of ($\mu_{\alpha^{*}}$, $\mu_{\delta}$) $\sim$ (23.1, –25.8) mas yr-1 (Lodieu
et al., 2019a), a core radius of 2.3 $\pm$ 0.3 pc, and a tidal radius of
$\sim$9.5 pc (Lodieu et al., 2019a).

The Hyades, regarded as a sister cluster of the Pleiades, is the nearest open
cluster and one of the best-studied star clusters (Messier, 1781). Using the
$Hipparcos$ dataset, Perryman et al. (1998) deduced the cluster distance to be
46.3 $\pm$ 0.3 pc, similar to the results obtained by using the $Hubble$ Space
Telescope measurements (McArthur et al., 2011) and infrared color-magnitude
diagram fitting (Majaess et al., 2011). Based on the _Gaia_ ’s trigonometric
parallax, the distance of the cluster has been updated to 47.0 $\pm$ 0.2 pc
(Lodieu et al., 2019a). The age of the Hyades cluster has been estimated by
different methods, such as the theoretical isochrone fitting method (625 $\pm$
50 Myr, Maeder & Mermilliod, 1981; Mermilliod, 1981), the cooling age of white
dwarf members (648 $\pm$ 45 Myr, De Gennaro et al., 2009), and the method of
the lithium-depletion boundary (650 $\pm$ 70 Myr, Lodieu et al., 2018). The
cluster consists of a roughly spherical group of hundreds of member stars
presenting the proper motions of 70–140 mas yr-1 on the sky (Perryman et al.,
1998; van Leeuwen, 2009; Lodieu et al., 2019a). The cluster has a core radius
of 2.5–3.0 pc and a tidal radius of 10 pc (see Perryman et al., 1998; Röser et
al., 2011). The reddening of the Hyades is low, with $E(B-V)$ $\leq$ 0.001 mag
(Taylor, 2006).

With the high-quality data from _Gaia_ DR3, the focus of this work is to
reveal the internal kinematics of the Pleiades, $\alpha$ Per, and the Hyades
clusters in detail. We first update the catalog of member stars of the three
OCs based on  _Gaia_ DR3, and then we extract those members with astrometric
parameters and radial velocities. Subsequently, we update the method used in
Paper I to investigate the rotation of the Pleiades, $\alpha$ Per, and the
Hyades. When a cluster does rotate, we determine its rotation axis, derive the
rotational velocities, and analyze the rotational characteristics of the
member stars. Finally, we discuss the possible relations between the internal
rotation of OCs and the cluster parameters.

##  2 Data

To study the internal kinematics of OCs, we adopt the _Gaia_ DR3 dataset. The
astrometric and photometric parameters in _Gaia_ DR3 are inherited from
_Gaia_ EDR3, which contains celestial positions (R.A., Decl.), parallaxes
($\varpi$), proper motions ($\mu_{\alpha^{*}}$, $\mu_{\delta}$), and three
photometric passbands ($G$, $G_{\rm BP}$, and $G_{\rm RP}$) of more than a
billion stellar sources. In addition, in comparison to _Gaia_ DR2, more than
33 million objects with new determinations of the mean radial velocities are
also included in _Gaia_ DR3. In this work, we adopt the radial velocity
measurements provided by Gaia DR3 to ensure the sample homogeneity and avoid
the potentially large systematic errors.

Table 1: Number of the member stars selected in different (sub) samples or different steps for the Pleiades, $\alpha$ Per, and the Hyades. |  Pleiades |  $\alpha$ Per |  Hyades
---|---|---|---
Initial sampleaafootnotemark:  |  2281 |  3162 |  1764
Step 1 |  66 |  182 |  87
Step 2 |  75 |  256 |  133
Step 3 |  77 |  894 |  907
Step 4 |  265 |  596 |  45
Samplebbfootnotemark:  |  1798 |  1234 |  592
Step 5 |  226 |  137 |  215
Step 6 |  1 |  0 |  1
Subsampleccfootnotemark:  |  225 |  137 |  214

The Pleiades, $\alpha$ Per, and the Hyades are three member-rich OCs in the
solar neighborhood that span a wide range of ages. We initiate the study by
adopting the membership lists of the Pleiades, Hyades, and $\alpha$ Per
clusters provided by Lodieu et al. (2019a, b). Lodieu et al. (2019a) combined
the _Gaia_ DR2 dataset with multiple large-scale public surveys from the
optical to mid-infrared wavelengths to provide the revised membership lists
for the Pleiades and $\alpha$ Per clusters. The selection criteria for the
member stars in Lodieu et al. (2019a) are initially designed to be as
inclusive as possible and become more selective in the later process. Finally,
they obtained 2281 member stars for the Pleiades and 3162 member stars for the
$\alpha$ Per. With the same method, Lodieu et al. (2019b) compiled a list of
member stars for the Hyades cluster, which includes a total of 1764 member
stars. We use _Gaia_ DR3 to update the aforementioned three membership lists.
The details are described below. The numbers of selected member stars in each
step are given in Table 1.

  * •

Step 1. Selection of the unmatched members after cross-matching the member
stars with the Gaia DR3 database through _Gaia_ ’s source_id.

  * •

Step 2. Removal of the members that lack astrometric ($\varpi$,
$\mu_{\alpha^{*}}$, $\mu_{\delta}$) or photometric ($G$, $G_{\rm BP}$, and
$G_{\rm RP}$) parameters.

  * •

Step 3. Rejection of those members that lie beyond three times of the tidal
radius ($r_{\rm td}$) from the cluster center. The distances of the members to
the cluster center are taken from Lodieu et al. (2019a, b).

  * •

Step 4. Selection of the members that have large parallax and/or proper motion
uncertainties based on the exclusion criteria of the typical uncertainties of
the faint stars ($G$ = 20) observed by _Gaia_ , i.e., eϖ $\geq$ 0.5 mas, and
e${}_{\mu_{\alpha^{*},\delta}}$ $\geq$ 0.6 mas yr-1 (Gaia Collaboration et
al., 2023). Member stars whose parallaxes and/or proper motions significantly
deviate (three times the standard deviation) from the overall sample are also
excluded.

  * •

Step 5. Selection of the members with radial velocity measurements and
corresponding uncertainties smaller than 2 km s-1, which is the same as Paper
I. Members whose radial velocities significantly deviate from the overall
sample are also deleted.

  * •

Step 6. Elimination of the nonsingle members that have not been correctly
handled in the _Gaia_ ’s astrometric solution according to the criterion of
ipd_gof_harmonic_amplitude $>$ 0.1 and ruwe $>$ 1.4 (Gaia Collaboration et
al., 2021).

##  3 Methods

The basic steps we develope to study the 3D rotation of OCs are described in
detail in Paper I. Here, we only give a brief description. First, we use the
astrometric parameters of the cluster members to determine their coordinates
($x_{\rm g}$, $y_{\rm g}$, $z_{\rm g}$) and velocities ($v_{x_{\rm g}}$,
$v_{y_{\rm g}}$, $v_{z_{\rm g}}$) in the Galactic Cartesian coordinate system
($O_{\rm g}$–$X_{\rm g}$$Y_{\rm g}$$Z_{\rm g}$), where the uncertainties are
estimated with a Monte Carlo method by taking into account the observational
errors. Second, the coordinates ($x_{\rm c}$, $y_{\rm c}$, $z_{\rm c}$) and
velocities ($v_{x_{\rm c}}$, $v_{y_{\rm c}}$, $v_{z_{\rm c}}$) of the members
in the cluster Cartesian coordinate system ($O_{\rm c}$–$X_{\rm c}$$Y_{\rm
c}$$Z_{\rm c}$) are converted from those in the $O_{\rm g}$–$X_{\rm g}$$Y_{\rm
g}$$Z_{\rm g}$ system. Third, based on the velocities ($v_{x_{\rm c}}$,
$v_{y_{\rm c}}$, $v_{z_{\rm c}}$) of the cluster members, we calculate the 3D
mean residual velocities and estimate their uncertainties with Monte Carlo
simulations in order to investigate their dependence on the position angles
(PA; i.e., $\alpha$, $\beta$, and $\gamma$ in Figure A1), as well as to check
the rotation of the cluster and determine the rotation axis $\vec{l}$.
Afterward, the coordinates ($x_{\rm r}$, $y_{\rm r}$, $z_{\rm r}$) and
velocities ($v_{x_{\rm r}}$, $v_{y_{\rm r}}$, $v_{z_{\rm r}}$) of the members
in the cluster rotation Cartesian coordinate system ($O_{\rm r}$–$X_{\rm
r}$$Y_{\rm r}$$Z_{\rm r}$, Figure A1) can be obtained according to the
rotation axis, which can then be transformed into the cylindrical coordinate
system $(r,\varphi,z)$ to reveal the internal rotation of an OC.

Based on the above steps, we have made some updates in this work. First, the
method is improved by analyzing the mean residual velocity of member stars
within three times, one time, and half of the cluster’s tidal radius ($r_{\rm
td}$) as a function of PAs, respectively. This approach can enhance the
reliability of the rotation representation within the cluster and better
determine the rotation axis $\vec{l}$. Another improvement addresses the
difficulty of determining the cluster rotation axis in the $O_{\rm c}$–$X_{\rm
c}$$Y_{\rm c}$$Z_{\rm c}$ system when the rotational signals (sinusoidal
behavior) are not significant in all three dimensions simultaneously. To fix
this problem, we rotate the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$
system around its origin ($O_{\rm c}$) with a given angle. Concretely
speaking, with a set of PAs ($\alpha_{0}$, $\beta_{0}$, $\gamma_{0}$), we
transform the 3D coordinates and velocity components of cluster members from
the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$ system into the rotated
Cartesian coordinate system (signed as $O_{\rm r}^{\prime}$–$X_{\rm r}$$Y_{\rm
r}$$Z_{\rm r}$). Subsequently, we use the residual velocity method to
determine the rotation in the cluster and estimate the PAs ($\alpha^{\prime}$,
$\beta^{\prime}$, $\gamma^{\prime}$) of the rotation axis $\vec{l}$ in the
rotated system. Then, the vectors of the rotation axis $\vec{l}$ can be
converted from the $O_{\rm r}^{\prime}$–$X_{\rm r}$$Y_{\rm r}$$Z_{\rm r}$
system back into the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$ system via

$\begin{pmatrix}x_{\rm c}\\\ y_{\rm c}\\\ z_{\rm
c}\end{pmatrix}=\begin{pmatrix}{\rm cos}~{}\alpha_{1}&{\rm
cos}~{}\alpha_{2}&{\rm cos}~{}\alpha_{3}\\\ {\rm cos}~{}\beta_{1}&{\rm
cos}~{}\beta_{2}&{\rm cos}~{}\beta_{3}\\\ {\rm cos}~{}\gamma_{1}&{\rm
cos}~{}\gamma_{2}&{\rm cos}~{}\gamma_{3}\end{pmatrix}\begin{pmatrix}x_{\rm
r}\\\ y_{\rm r}\\\ z_{\rm r}\end{pmatrix},$ (1)

where $\alpha_{i}$, $\beta_{i}$, and $\gamma_{i}$ are the included angles
between the axes of the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$ system
and those of the $O_{\rm r}^{\prime}$–$X_{\rm r}$$Y_{\rm r}$$Z_{\rm r}$
system. When the vectors of $\vec{l}$ in the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm
c}$$Z_{\rm c}$ system are obtained, the PAs of $\vec{l}$ in the $O_{\rm
c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$ system ($\alpha$, $\beta$, $\gamma$) can
be determined. Then, we continue to perform the aforementioned procedures to
analyze the internal rotation of an OC.

In the same way as in Paper I, to elucidate the rotational nature of the
member stars of an OC, we compare their root-mean-square (RMS) rotational
velocities and the theoretical predictions from Newton’s theorems, with the
following equation:

$\displaystyle\begin{split}v_{\rm c}(r)=\left[GM_{\rm
t}\cdot\frac{r^{2}}{(r^{2}~{}+~{}r_{\rm
co}^{2})^{3/2}}\right]^{1/2}.\end{split}$ (2)

Here, $G$ = $4.3\times 10^{-3}~{}{\rm pc~{}M}_{\odot}^{-1}~{}({\rm
km~{}s}^{-1})^{2}$ is the gravitational constant, $r_{\rm co}$ is the core
radius, and $M_{\rm t}$ is the cluster mass inside the tidal radius.

##  4 Results

###  4.1 Rotation in Praesepe with Gaia DR3

In Paper I, we studied the rotational properties of the Praesepe cluster based
on  _Gaia_ EDR3. In comparison to EDR3, _Gaia_ DR3 provides high-precision
radial velocity measurements for more cluster members. We reanalyze the
Praesepe rotation with the new data of cluster members and the amended method
described above. We find that the features of the mean residual velocities of
the member stars within 3 $r_{\rm td}$, 1 $r_{\rm td}$, and 0.5 $r_{\rm td}$
of the Praesepe all confirm the 3D rotation in the cluster. In addition, the
rotational characteristics of member stars within the cluster tidal radius are
almost consistent with those in Paper I and can be well described by Newton’s
theorems. Moreover,, the new dataset still shows no expansion or contraction
in the Praesepe. As shown in Table 2, by comparison, the consistent parameters
confirm that the results obtained in Paper I are reliable, and the new data-
analyzing method updated in this work is feasible.

Table 2: Comparison of the rotational parameters of the Praesepe for the results obtained in this work and those in Paper I. |  This work |  Paper I
---|---|---
Position angles |  $\alpha$, $\beta$, $\gamma$ |  $\alpha$, $\beta$, $\gamma$
0.5 $r_{\rm td}$ |  $119^{\circ}$ $\pm$ $10^{\circ}$, $133^{\circ}$ $\pm$ $12^{\circ}$, $214^{\circ}$ $\pm$ $8^{\circ}$ |  –
1.0 $r_{\rm td}$ |  $128^{\circ}$ $\pm$ $4^{\circ}$, $132^{\circ}$ $\pm$ $7^{\circ}$, $212^{\circ}$ $\pm$ $5^{\circ}$ |  $121^{\circ}$ $\pm$ $5^{\circ}$, $118^{\circ}$ $\pm$ $17^{\circ}$, $211^{\circ}$ $\pm$ $8^{\circ}$
3.0 $r_{\rm td}$ |  $136^{\circ}$ $\pm$ $6^{\circ}$, $148^{\circ}$ $\pm$ $5^{\circ}$, $210^{\circ}$ $\pm$ $5^{\circ}$ |  $140^{\circ}$ $\pm$ $3^{\circ}$, $152^{\circ}$ $\pm$ $7^{\circ}$, $210^{\circ}$ $\pm$ $2^{\circ}$
$\vec{l}$ vs. GP |  $34^{\circ}$ $\pm$ $15^{\circ}$ |  $41^{\circ}$ $\pm$ $12^{\circ}$
Rotation velocity |  0.12 $\pm$ 0.05 km s-1 |  0.16 $\pm$ 0.05 km s-1
Core radius |  2.0 $\pm$ 0.4 pc |  1.6 $\pm$ 0.5 pc
Tidal mass |  609 $\pm$ 135 ${\rm M}_{\odot}$ |  537 $\pm$ 146 ${\rm M}_{\odot}$

###  4.2 The Pleiades cluster

Figure 1: Multidimensional distributions of all the member stars (gray) and
those with radial velocity measurements (red) for the Pleiades cluster. Panel
(a): the distribution in the sky; Panels (b) and (c): the relationship between
parallax and proper motions; Panel (d): the color-magnitude diagram; Panels
(e), (f), (g) and (h): the histograms of parallax, proper motions and radial
velocity, respectively.

Figure 1 shows the multidimensional distributions of astrometric parameters
(R.A., Decl., $\varpi$, $\mu_{\alpha^{*}}$, $\mu_{\delta}$) and color-
magnitude diagram of all the member stars (grey) and those with radial
velocities (red) for the Pleiades. The observed parameters of the member stars
can be approximated by a multidimensional normal distribution. In this
section, we describe the rotational properties of the Pleiades cluster and
explore the rotational characteristics of the cluster members.

Figure 2: Mean residual velocities as a function of the position angles (PAs)
for the Pleiades. The left, middle, and right panels show the $v_{x_{\rm c}}$
vs. $\alpha$, $v_{y_{\rm c}}$ vs. $\beta$, and $v_{z_{\rm c}}$ vs. $\gamma$
plots, respectively. The results for the member stars within three times of
the tidal radius (Panels a, b, c), within a tidal radius (Panels d, e, f), and
within half a tidal radius (Panels g, h, i) are shown from top to bottom,
respectively. The error bars (gray) and the best-fitting sine function (red)
are also shown in each panel.

####  4.2.1 Rotation in the Pleiades

For the 225 member stars with radial velocities (see Table 1), the median
uncertainties of parallax, proper motion $\mu_{\alpha^{*}}$ and $\mu_{\delta}$
are 0.02 mas, 0.02 mas yr-1, and 0.02 mas yr-1, respectively. The median
uncertainty of radial velocity is 0.6 km s-1. Based on these member stars, the
means and standard deviations of the astrometric parameters of the Pleiades
cluster are determined as: (R.A., Decl.) = ($56.58^{\circ}$ $\pm$
$2.51^{\circ}$, $24.35^{\circ}$ $\pm$ $2.47^{\circ}$), $\varpi$ = $7.38$ $\pm$
$0.37$ mas, proper motions ($\mu_{\alpha^{*}}$, $\mu_{\delta}$) = ($20.04$
$\pm$ $1.80$, $-45.40$ $\pm$ $2.59$ ) mas yr-1, and radial velocity RV =
$5.57$ $\pm$ $0.77$ km s-1. These values are consistent with the previous
determinations (Liu et al., 1991; Mermilliod et al., 1997; Gao, 2019; Gaia
Collaboration et al., 2018; Lodieu et al., 2019a). With these parameters, we
derive the 3D coordinates and velocities of the Pleiades and its member stars
in the Galactic Cartesian coordinate system ($O_{\rm g}$–$X_{\rm g}$$Y_{\rm
g}$$Z_{\rm g}$), and then transform them to the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm
c}$$Z_{\rm c}$ system. After that, the mean residual velocities and
uncertainties of the member stars as a function of PAs are calculated. We
assign an $r_{\rm td}$ of 11.6 pc to the Pleiades cluster, which comes from
Lodieu et al. (2019a). Figure 2 (a–c), (d–f), and (g–i) present the mean
residual velocities as functions of PAs for the member stars within 3 $r_{\rm
td}$, 1 $r_{\rm td}$, and 0.5 $r_{\rm td}$, respectively. The error bars in
the figures are the uncertainties of the mean residual velocities. As shown in
Figure 2, the clear sinusoidal behaviors indicate the presence of 3D rotation
in the Pleiades cluster.

Table 3: Best-fit PAs of the Pleiades cluster. |  $\alpha$ |  $\beta$ |  $\gamma$
---|---|---|---
3.0 $r_{\rm td}$ |  $81^{\circ}$ $\pm$ $4^{\circ}$ |  $78^{\circ}$ $\pm$ $7^{\circ}$ |  $295^{\circ}$ $\pm$ $7^{\circ}$
1.0 $r_{\rm td}$ |  $102^{\circ}$ $\pm$ $8^{\circ}$ |  $59^{\circ}$ $\pm$ $11^{\circ}$ |  $337^{\circ}$ $\pm$ $7^{\circ}$
0.5 $r_{\rm td}$ |  $100^{\circ}$ $\pm$ $7^{\circ}$ |  $54^{\circ}$ $\pm$ $10^{\circ}$ |  $337^{\circ}$ $\pm$ $10^{\circ}$

Using the residual velocity method, the PAs ($\alpha$, $\beta$, $\gamma$) of
the rotation axis $\vec{l}$ of the Pleiades cluster in the $O_{\rm c}$–$X_{\rm
c}$$Y_{\rm c}$$Z_{\rm c}$ system can be determined. The best-fitting results
are shown in Table 3 and Figure 2, where the errors come from the Monte Carlo
simulations by considering the uncertainties of the mean residual velocities.
The PAs obtained by the member stars within 1 $r_{\rm td}$ are consistent with
those fitted by using the members within 0.5 $r_{\rm td}$. They satisfy the
relation of tan $\alpha$ $\cdot$ tan $\gamma$ = tan $\beta$ considering the
uncertainties. Nevertheless, the PAs obtained by the member stars of the
entire sample are different from the above results, although the three angles
also satisfy tan $\alpha$ $\cdot$ tan $\gamma$ = tan $\beta$ considering the
uncertainties. This phenomenon aligns with the findings in Paper I for the
Praesepe cluster, which may be due to the fact that the member stars located
outside the cluster’s tidal radius are not only dominated by the OC itself,
but also by the tidal force of the Milky Way.

We employ the best-fitting PAs for the member stars within 1 $r_{\rm td}$ to
determine the rotation axis $\vec{l}$ of the Pleiades cluster. The angle
between $\vec{l}$ and the Galactic plane is estimated to be $62^{\circ}\pm
13^{\circ}$, where the errors come from the uncertainties of the obtained PAs.
Considering the relationship of tan $\alpha$ $\cdot$ tan $\gamma$ = tan
$\beta$, we adopt ($\alpha$, $\beta$, $\gamma$) = ($102^{\circ}$,
$63^{\circ}$, $337^{\circ}$) to derive the vectors of the $X_{\rm r}$, $Y_{\rm
r}$, and $Z_{\rm r}$ axes in the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$
system, as described in Section 3. Then, the 3D coordinates and velocity
components of the member stars are transformed from the $O_{\rm c}$–$X_{\rm
c}$$Y_{\rm c}$$Z_{\rm c}$ system to the $O_{\rm r}$–$X_{\rm r}$$Y_{\rm
r}$$Z_{\rm r}$ system, and eventually converted to the 3D coordinates ($r$,
$\varphi$, $z$) and velocity components ($v_{r}$, $v_{\varphi}$, $v_{z}$) in
the cylindrical coordinate system. Based on the rotational velocities
($v_{\varphi}$) and the corresponding uncertainties of the member stars within
1 $r_{\rm td}$, the mean rotation velocity of the Pleiades cluster is
estimated to be 0.24 $\pm$ 0.04 km s-1, which is in general agreement with the
results illustrated in Figure 2.

Figure 3: Rotational velocity as a function of the distance to the cluster
center for the member stars within twice the tidal radius of the Pleiades.

####  4.2.2 Rotational properties of the member stars of Pleiades

Table 4: RMS rotational velocities of the member stars within the tidal radius of the Pleiades. $R_{i}$ |  $R_{o}$ |  $R_{m}$ |  $N$ |  RMS $v_{\varphi}$ |  $\epsilon_{{\rm RMS}\,v_{\varphi}}$
---|---|---|---|---|---
[pc] |  [pc] |  [pc] |  |  [km s-1] |  [km s-1]
(1) |  (2) |  (3) |  (4) |  (5) |  (6)
0.0 |  2.0 |  1.2 |  54 |  0.79 |  0.06
2.0 |  4.0 |  3.0 |  56 |  0.86 |  0.06
4.0 |  6.0 |  4.7 |  34 |  0.70 |  0.10
6.0 |  8.0 |  7.1 |  18 |  0.73 |  0.11
8.0 |  10.0 |  8.9 |  12 |  0.66 |  0.12
10.0 |  11.6 |  10.6 |  9 |  0.57 |  0.13
Figure 4: RMS rotational velocity as the function of distance to the center of
the Pleiades cluster. The black dots denote the observed RMS rotational
velocities of the member stars. The uncertainties are indicated by the error
bars. The gray shadowed area shows the theoretical values calculated by
Equation (2) after considering the possible ranges of the tidal mass and the
core radius of the Pleiades. The red line is the best-fitting curve for the
observed RMS rotational velocities.

Figure 3 presents the rotational velocities of the member stars of the
Pleiades as a function of their distances from the cluster center. Within the
tidal radius of the Pleiades (11.6 pc), the rotation velocities of the member
stars in the inner region are slightly higher than those in the outer region.
Similar to the Preasepe cluster (Paper I) and globular clusters (e.g., Lanzoni
et al., 2018; Leanza et al., 2022), not all member stars rotate in the same
direction. There are $\sim$1.3 times as many member stars with positive
rotational velocities as rotate with negative rotational velocities. This
situation may be attributed to the presence of mutual interference of stellar
feedback during the formation of the cluster progenitor. In Figure 3, the
rotation of the member stars located close to or outside the cluster’s tidal
radius seem to be disorganized, possibly because these member stars are
partially affected by the Galactic tidal force and are not purely governed by
the gravitational force of the cluster itself. In addition, within the tidal
radius of the Pleiades, several stars exhibit peculiar rotational velocities
that deviate from most of the cluster members, which may be passing stars and
not the genuine members of the Pleiades.

We compare the observations with the theoretical predictions to address
whether Newton’s theorems can describe the rotation of the member stars within
the tidal radius of the Pleiades cluster. Firstly, we calculate the skewness
of the astrometric parameters of the member stars within $r_{\rm td}$ to
investigate the symmetry of their density distributions. The skewness of the
Galactic longitude, Galactic latitude, and parallax are 0.1, –0.2 and 0.1,
respectively, which confirms the assumption of a spherically symmetric
distribution of the stars in the Pleiades. Further, we attempt to make a
comparison between the RMS rotational velocities derived from the cluster
members and the theoretical expectations of Newton’s theorems.

For the 196 member stars within the tidal radius of the Pleiades cluster, the
median absolute value of rotational velocities $v_{\varphi}$ is $\sim$0.5 km
s-1, and the standard deviation is $\sim$1.0 km s-1. To study the rotational
characteristics displayed by the member stars of the Pleiades, we divide them
into several bins based on their distances from the cluster center. A few
stars have peculiar rotational velocities that significantly deviate from
those of most member stars. We identify 13 such kind of stars by checking the
distribution of the rotational velocities of the member stars in each bin and
eliminate them. Then, we compute the RMS rotation velocity in each distance
bin and estimate the corresponding uncertainties using the Monte Carlo
simulations. The results are presented in Table 4 and Figure 4.

With Equation (2), the theoretical rotational velocity profile provides a good
fit to the RMS rotation velocities of the cluster members. In addition, based
on previous estimates of the tidal mass and core radius (see Section 1), we
calculate the possible theoretical circular velocity curves for the Pleiades
cluster, as shown by the gray shaded area in Figure 4. It is clear that the
best-fit RMS rotational velocity curve is consistent with the theoretical
predictions. Therefore, we argue that Newton’s theorems is capable of
characterizing the rotational properties of the member stars within the tidal
radius of Pleiades. Based on the result in Table 4 and Equation (2), we
estimate the core radius and the mass within $r_{\rm td}$ to be 1.7 $\pm$ 0.3
pc and 806 $\pm$ 144 ${\rm M}_{\odot}$, respectively, where the errors are
estimated with the Monte Carlo simulations. These two values agree well with
previous results (see Section 1).

Figure 5: Radial velocity as a function of the distance to the cluster center,
for the member stars of the Pleiades. The mean values of the radial components
within 0.5 $r_{\rm td}$, 1 $r_{\rm td}$, and 3 $r_{\rm td}$ are close to zero,
as shown in the panel.

The Pleiades are about 100 Myr old (see Section 1). Whether the Pleiades
cluster is expanding or contracting can be examined by a statistical analysis
of the radial components $v_{r}$ of the member stars perpendicular to the
cluster rotation axis. In Figure 5, the mean radial components
$\overline{v_{r}}$ of the member stars within 0.5 $r_{\rm td}$, 1 $r_{\rm
td}$, and 3 $r_{\rm td}$ are 0.0 $\pm$ 0.05 km s-1, 0.1 $\pm$ 0.04 km s-1, and
–0.1 $\pm$ 0.04 km s-1, respectively. Here, the errors are obtained with the
Monte Carlo method by taking into account the uncertainties of ${v_{r}}$.
These results indicate the absence of expansion or contraction in the Pleiades
cluster, and this implies that the rotation of the member stars within $r_{\rm
td}$ can exhibit closed-loop motions. However, as shown in Figure 5, we note
the presence of non-negligible radial component acceleration and dispersion of
members in the Pleiades, although the dispersion is partially influenced by
the astrometric measurement uncertainties. Considering the absence of
expansion or contraction in the Pleiades, the system should possess additional
mass to provide the force that support the radial components acceleration of
the member stars. Hence, the derived mass of the Pleiades cluster represents a
lower limit.

Figure 6: Same as Figure 1, but for the $\alpha$ Per cluster.

###  4.3 Cluster $\alpha$ Per

Similar to Figure 1, we present the multidimensional distributions and color-
magnitude diagram for cluster $\alpha$ Per in Figure 6. In this section, we
show the rotational properties of cluster $\alpha$ Per, and analyze the
rotational characteristics of the cluster members.

Figure 7: Same as Figure 2, but for the $\alpha$ Per cluster.

####  4.3.1 Rotation in $\alpha$ Per

For the 137 member stars with radial velocities (see Table 1), the median
uncertainties of the parallax, proper motion $\mu_{\alpha^{*}}$ and
$\mu_{\delta}$, and for the radial velocity are 0.02 mas, 0.02 mas yr-1, 0.02
mas yr-1, and 0.9 km s-1, respectively. Based on the observational parameters
of these member stars, the means and standard deviations of the fundamental
astrometric parameters of the $\alpha$ Per cluster are determined as: (R.A.,
Decl.)=($51.87^{\circ}$ $\pm$ $3.65^{\circ}$, $48.93^{\circ}$ $\pm$
$2.47^{\circ}$), $\varpi$ = $5.81$ $\pm$ $0.30$ mas, ($\mu_{\alpha^{*}}$,
$\mu_{\delta}$) = ($23.11$ $\pm$ $1.87$, $-25.76$ $\pm$ $1.97$) mas yr-1, and
RV = $-0.43$ $\pm$ $4.10$ km s-1, which are in good agreement with previous
determinations (Stauffer et al., 1989; Prosser, 1994; Mermilliod et al., 2008;
Gaia Collaboration et al., 2018; Lodieu et al., 2019a). Similar to the
Pleiades, we obtain the 3D coordinates and motion of the $\alpha$ Per cluster
in the $O_{\rm g}$–$X_{\rm g}$$Y_{\rm g}$$Z_{\rm g}$ system and the 3D
coordinates and motions of its member stars in the $O_{\rm c}$–$X_{\rm
c}$$Y_{\rm c}$$Z_{\rm c}$ system.

The direct usage of the residual velocity method in the $O_{\rm c}$–$X_{\rm
c}$$Y_{\rm c}$$Z_{\rm c}$ system captures the rotation signals of the $\alpha$
Per cluster, but does not allow a good determination of the PAs ($\alpha$,
$\beta$, $\gamma$) of the rotation axis $\vec{l}$. In order to resolve this
matter, as described in Section 3, we rotate the $O_{\rm c}$–$X_{\rm
c}$$Y_{\rm c}$$Z_{\rm c}$ system around its origin with a set of angles
($\alpha_{0}$, $\beta_{0}$, $\gamma_{0}$) = ($40^{\circ}$, $40^{\circ}$,
$45^{\circ}$). The 3D coordinates and velocity components of the cluster
members are then transformed from the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm
c}$$Z_{\rm c}$ system to the rotated $O_{\rm r}^{\prime}$–$X_{\rm r}$$Y_{\rm
r}$$Z_{\rm r}$ system. After this, in the same way as for the Pleiades, we
estimate the residual velocities of the cluster members and uncertainties to
determine the PAs ($\alpha^{\prime}$, $\beta^{\prime}$, $\gamma^{\prime}$) of
the rotation axis $\vec{l}$ of the $\alpha$ Per in the rotated system. Figure
7 presents the mean residual velocities as functions of PAs
($\alpha^{\prime}$, $\beta^{\prime}$, $\gamma^{\prime}$) for the member stars
within 3 $r_{\rm td}$ (a–c), 1 $r_{\rm td}$ (d–f), and 0.5 $r_{\rm td}$ (g–i)
of the $\alpha$ Per cluster, where $r_{\rm td}=9.5$ pc is taken from Lodieu et
al. (2019a). The sinusoidal relations between the mean residual velocities and
PAs indicate the 3D rotation of the $\alpha$ Per cluster, although the result
in Figure 7(d) is not very significant.

The fitted PAs ($\alpha^{\prime}$, $\beta^{\prime}$, $\gamma^{\prime}$) of the
rotation axis $\vec{l}$ of the $\alpha$ Per cluster in the rotated system are
listed in Table 5 and marked in Figures 7. The PAs fitted by the members
within 0.5 $r_{\rm td}$ satisfy the relation of tan $\alpha^{\prime}$ $\cdot$
tan $\gamma^{\prime}$ = tan $\beta^{\prime}$ considering the uncertainties.
The PA $\alpha^{\prime}$ value cannot be well determined by the members within
1 $r_{\rm td}$, which suffer from large uncertainties. Alternatively, PA
$\alpha^{\prime}$ = $138^{\circ}$ can be determined from the tan
$\alpha^{\prime}$ $\cdot$ tan $\gamma^{\prime}$ = tan $\beta^{\prime}$
relation, combined with the well-fit $\beta^{\prime}$ and $\gamma^{\prime}$
values. In this way, the three PAs are compatible with those derived by the
members within 0.5 $r_{\rm td}$, and they can therefore be used to determine
the rotation axis $\vec{l}$. Compared with the Pleiades cluster, the stronger
rotation of the $\alpha$ Per cluster makes the PAs obtained by using the
member stars within 3 $r_{\rm td}$ almost consistent with the above results,
considering the uncertainties.

The PAs obtained by using the members within 1 $r_{\rm td}$ are employed to
determine the rotation axis $\vec{l}$ of the $\alpha$ Per cluster in the
rotated system ($O_{\rm r}^{\prime}$–$X_{\rm r}$$Y_{\rm r}$$Z_{\rm r}$), i.e.,
($\alpha^{\prime}$, $\beta^{\prime}$, $\gamma^{\prime}$) = ($138^{\circ}$,
$133^{\circ}$, $230^{\circ}$). Then, the PAs of ($\alpha$, $\beta$, $\gamma$)
= ($68^{\circ}$, $51^{\circ}$, $26^{\circ}$) for $\vec{l}$ in the $O_{\rm
c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$ system can be obtained by using Equation
(1). Thus, $\vec{l}$ is tilted at an angle of $48^{\circ}\pm 14^{\circ}$ with
respect to the Galactic plane. With the PAs, we derive the vectors of the
$X_{\rm r}$, $Y_{\rm r}$, and $Z_{\rm r}$ axes of the $O_{\rm r}$–$X_{\rm
r}$$Y_{\rm r}$$Z_{\rm r}$ system in the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm
c}$$Z_{\rm c}$ system. After transforming the 3D coordinates and velocity
components of the cluster members from the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm
c}$$Z_{\rm c}$ system to those in the $O_{\rm r}$–$X_{\rm r}$$Y_{\rm
r}$$Z_{\rm r}$ system, we also calculate the 3D coordinates ($r$, $\varphi$,
$z$) and velocity components ($v_{r}$, $v_{\varphi}$, $v_{z}$) of the cluster
members in the cylindrical coordinate system. As for the Pleiades, using the
rotation velocities ($v_{\varphi}$) and uncertainties of the member stars, we
estimate the mean rotation velocity of the $\alpha$ Per cluster within its
tidal radius to be 0.43 $\pm$ 0.08 km s-1, which is in consistent with the
results shown in Figures 7.

Table 5: Best-fit PAs of the $\alpha$ Per cluster. |  $\alpha^{\prime}$ |  $\beta^{\prime}$ |  $\gamma^{\prime}$
---|---|---|---
3.0 $r_{\rm td}$ |  $146^{\circ}$ $\pm$ $5^{\circ}$ |  $121^{\circ}$ $\pm$ $4^{\circ}$ |  $229^{\circ}$ $\pm$ $10^{\circ}$
1.0 $r_{\rm td}$ |  $25^{\circ}$ $\pm$ $125^{\circ}$ |  $133^{\circ}$ $\pm$ $10^{\circ}$ |  $230^{\circ}$ $\pm$ $11^{\circ}$
0.5 $r_{\rm td}$ |  $138^{\circ}$ $\pm$ $7^{\circ}$ |  $131^{\circ}$ $\pm$ $31^{\circ}$ |  $251^{\circ}$ $\pm$ $20^{\circ}$

####  4.3.2 Rotational properties of the stars in $\alpha$ Per

Figure 8: Rotational velocity as a function of distance to the cluster center, for the member stars within two times of the tidal radius of the $\alpha$ Per. Figure 9: RMS rotational velocity as a function of distance to the center of the $\alpha$ Per cluster. The black dots denote the observed RMS rotational velocities of the member stars, with uncertainties indicated by error bars. The red line is the best-fitting curve for the observed RMS rotational velocities according to the Newton’s theorems. Table 6: RMS rotational velocities of the member stars within the tidal radius of the $\alpha$ Per cluster. $R_{i}$ |  $R_{o}$ |  $R_{m}$ |  $N$ |  RMS $v_{\varphi}$ |  $\epsilon_{{\rm RMS}\,v_{\varphi}}$
---|---|---|---|---|---
[pc] |  [pc] |  [pc] |  |  [km s-1] |  [km s-1]
(1) |  (2) |  (3) |  (4) |  (5) |  (6)
0.0 |  2.0 |  1.3 |  15 |  0.70 |  0.11
2.0 |  4.0 |  3.0 |  17 |  0.90 |  0.12
4.0 |  6.0 |  4.9 |  21 |  0.89 |  0.10
6.0 |  8.0 |  7.1 |  21 |  0.86 |  0.11
8.0 |  9.5 |  8.9 |  3 |  0.58 |  0.24

The rotational velocities of the member stars as a function of their distances
to the cluster center are presented in Figure 8. Similar to the Preasepe
(Paper I) and the Pleiades (this paper), not all of the member stars of the
$\alpha$ Per cluster rotate in the same direction. The ratio of the number of
members with positive $v_{\varphi}$ values to those with negative values is
$\sim$2.3, which is higher than that of the Pleiades. Due to the tidal force
of the Milky Way, the rotation of the member stars located beyond the cluster
tidal radius (9.5 pc) seems to be disordered, as shown in Figure 8. Moreover,
in a few stars, the peculiar rotational velocities deviate from those of most
cluster members within the tidal radius of the $\alpha$ Per.

Understanding the rotational patterns of the member stars within the tidal
radius of the $\alpha$ Per cluster is an intriguing issue. To this end, we
compare our observations with the theoretical predictions of Newton’s
theorems, following the same analysis procedure as for the Pleiades. Within
the tidal radius of the $\alpha$ Per, the skewness of the Galactic longitudes,
Galactic latitudes, and parallaxes of member stars are 0.2, 0.2, and –0.4,
respectively, indicating that the assumption of the spherically symmetric
distribution in the cluster is reasonable. Next, we try to use Equation (2) to
describe the RMS rotational velocity derived from the member stars of the
$\alpha$ Per cluster.

Within the tidal radius of the $\alpha$ Per cluster, there are 85 member stars
in the sample. The median absolute value of their rotation velocities
$v_{\varphi}$ is $\sim$0.6 km s-1, with a standard deviation of $\sim$1.4 km
s-1. For these member stars, we study their rotational characteristics by
dividing them into several bins according to their distances from the cluster
center. Eight stars with peculiar rotational velocities deviating from the
overall velocity are eliminated. Similar to the Pleiades, using the remaining
77 member stars, we calculate the RMS rotation velocity and uncertainty in
each distance bin. The results are given in Table 6 and Figure 9. The RMS
rotational velocities of the member stars can be well fit by the velocity
profile of circular motion expected by Newton’s theorems, suggesting that the
rotation of the member stars within the tidal radius of the $\alpha$ Per can
be elucidated by Newton’s theorems. The core radius and the mass within the
tidal radius of the $\alpha$ Per cluster is 2.3 $\pm$ 0.5 pc and 1186 $\pm$
312 ${\rm M}_{\odot}$, respectively, derived by fitting the RMS velocities
listed in Table 6 with Equation (2), with the same technique for the Pleiades.
We find that the best-fitting core radius matches the result of 2.3 $\pm$ 0.3
pc reported by Lodieu et al. (2019a), but is significantly smaller than the
early estimate ($\sim$4 pc, see Artyukhina, 1972; Kharchenko et al., 2005).

Figure 10: Same as Figure 5, but for the $\alpha$ Per cluster.

The age of the $\alpha$ Per cluster has been estimated to be approximately
50–70 Myr (see Section 1). As presented in Figure 10, the mean radial
components $\overline{v_{r}}$ of the member stars within 0.5 $r_{\rm td}$, 1
$r_{\rm td}$, and 3 $r_{\rm td}$ are 0.0 $\pm$ 0.09, 0.1 $\pm$ 0.07, and 0.1
$\pm$ 0.06 km s-1, respectively, using the same approach as for the Pleiades.
In this context, we posit that the $\alpha$ Per cluster shows no obvious signs
of expansion or contraction, suggesting that the rotation of the member stars
within the tidal radius exhibits closed-loop motions. Additionally, ass for
the Pleiades, the above estimated mass of the $\alpha$ Per is expected to be a
lower limit, since the non-negligible radial component acceleration and
dispersion (Figure 10) of the cluster members imply that the cluster ought to
have additional mass.

###  4.4 The Hyades cluster

In Figure 11, we display the multidimensional distributions and color-
magnitude diagram for the Hyades cluster. In this section, we picture the
rotation in the Hyades cluster, and investigate the rotational characteristics
of the member stars.

Figure 11: Same as Figure 1, but for the Hyades cluster.

####  4.4.1 Rotation in Hyades

For the subsample of 214 member stars with radial velocities (see Table 1),
the median uncertainties of parallax, proper motion $\mu_{\alpha^{*}}$ and
$\mu_{\delta}$ are 0.02 mas, 0.03 mas yr-1, and 0.02 mas yr-1, respectively.
The median uncertainty of the radial velocity is 0.3 km s-1. From these member
stars, we estimate the means and standard deviations of the basic parameters
of the Hyades cluster to be: (R.A., Decl.)=($67.41^{\circ}$ $\pm$
$6.50^{\circ}$, $17.18^{\circ}$ $\pm$ $5.69^{\circ}$), $\varpi$ = $20.99$
$\pm$ $2.61$ mas, ($\mu_{\alpha^{*}}$, $\mu_{\delta}$) = ($102.68$ $\pm$
$27.97$, $-27.25$ $\pm$ $16.30$) mas yr-1, and RV = $39.39$ $\pm$ $3.10$ km
s-1, in concordance with previous results (Perryman et al., 1998; van Leeuwen,
2009; Majaess et al., 2011; Lodieu et al., 2019b). With these parameters, we
calculate the center, 3D motions, and corresponding uncertainties of the
Hyades cluster in the Galactic $O_{\rm g}$–$X_{\rm g}$$Y_{\rm g}$$Z_{\rm g}$
system. For each member of the Hyades, we first calculate its 3D coordinates,
3D motions, and corresponding uncertainties in the Galactic coordinate system,
and we then transform them into the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm
c}$ system.

As in the case of the $\alpha$ Per, we can determine the rotation signals in
the Hyades by using the residual velocity method in the $O_{\rm c}$–$X_{\rm
c}$$Y_{\rm c}$$Z_{\rm c}$ system, but we can not pin down the PAs ($\alpha$,
$\beta$, $\gamma$) of the rotation axis $\vec{l}$. Using a set of PAs
($\alpha_{0}$, $\beta_{0}$, $\gamma_{0}$) = ($10^{\circ}$, $10^{\circ}$,
$45^{\circ}$), the 3D coordinates and velocity components of the cluster
members are transformed from the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$
system to the rotated $O_{\rm r}^{\prime}$–$X_{\rm r}$$Y_{\rm r}$$Z_{\rm r}$
system. We then determine the PAs of the rotation axis $\vec{l}$
($\alpha^{\prime}$, $\beta^{\prime}$, $\gamma^{\prime}$) in the rotated system
by using the same method as for the aforementioned clusters. In Figure 12, we
display the mean residual velocities as functions of PAs, using the member
stars within 3 $r_{\rm td}$ (a–c), 1 $r_{\rm td}$ (d–f), and 0.5 $r_{\rm td}$
(g–i) of the Hyades cluster. According to Lodieu et al. (2019b), the Hyades
cluster has a tidal radius of 9.0 pc. The sinusoidal behavior of the mean
residual velocities as a function of PAs is an excellent indication of the 3D
rotation of the Hyades cluster, although the result in Figure 12(a) is not
very significant.

Figure 12: Same as Figure 2, but for the Hyades cluster.

Table 7 and Figure 12 summarize the fitted PAs ($\alpha^{\prime}$,
$\beta^{\prime}$, $\gamma^{\prime}$) of the rotation axis $\vec{l}$ of the
Hyades in the rotated system. Like the Pleiades, the PAs obtained by using the
member stars within the tidal radius of the Hyades are concordant with those
obtained by using the members within half of the tidal radius, and they all
satisfy the relation of tan $\alpha$ $\cdot$ tan $\gamma$ = tan $\beta$
considering the uncertainties. In contrast, the PAs derived by using the
member stars within three times of the tidal radius are different from the
above results, although they also satisfy tan $\alpha$ $\cdot$ tan $\gamma$ =
tan $\beta$ considering the uncertainties. This phenomenon has also been shown
for the Praesepe and Pleiades clusters.

For the Hyades cluster, the PAs obtained by the members within the tidal
radius are exploited to determine the rotation axis $\vec{l}$. Considering the
relation tan $\alpha^{\prime}$ $\cdot$ tan $\gamma^{\prime}$ = tan
$\beta^{\prime}$, we adopt the PAs of ($\alpha^{\prime}$, $\beta^{\prime}$,
$\gamma^{\prime}$) = ($87^{\circ}$, $90^{\circ}$, $96^{\circ}$) to obtain the
vectors of the rotation axis $\vec{l}$ in the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm
c}$$Z_{\rm c}$ system, and we then use Equation (1) to determine the PAs of
the rotation axis $\vec{l}$ in the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm
c}$ system to be ($\alpha$, $\beta$, $\gamma$) = ($10^{\circ}$, $11^{\circ}$,
$48^{\circ}$). We find that the rotation axis of the Hyades cluster is nearly
parallel to the Galactic plane, with an estimated angle of $7^{\circ}\pm
7^{\circ}$. The vectors of the $X_{\rm r}$, $Y_{\rm r}$, and $Z_{\rm r}$ axes
of the $O_{\rm r}$–$X_{\rm r}$$Y_{\rm r}$$Z_{\rm r}$ system in the $O_{\rm
c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$ system can be given by the PAs
($\alpha$, $\beta$, $\gamma$). Subsequently, we transform the 3D coordinates
and velocity components of the cluster members from the $O_{\rm c}$–$X_{\rm
c}$$Y_{\rm c}$$Z_{\rm c}$ system to the $O_{\rm r}$–$X_{\rm r}$$Y_{\rm
r}$$Z_{\rm r}$ system, and we finally obtain the 3D coordinates ($r$,
$\varphi$, $z$) and velocity components ($v_{r}$, $v_{\varphi}$, $v_{z}$) of
the cluster members in the cylindrical coordinate system. Using the same
technique as the previous two clusters, averaging the rotation velocities
($v_{\varphi}$) of the member stars yields a rotation velocity of 0.09 $\pm$
0.03 km s-1 for the Hyades cluster, in agreement with the results shown in
Figures 12.

Table 7: Best-fit PAs of the Hyades cluster. |  $\alpha^{\prime}$ |  $\beta^{\prime}$ |  $\gamma^{\prime}$
---|---|---|---
3.0 $r_{\rm td}$ |  $176^{\circ}$ $\pm$ $26^{\circ}$ |  $53^{\circ}$ $\pm$ $23^{\circ}$ |  $312^{\circ}$ $\pm$ $28^{\circ}$
1.0 $r_{\rm td}$ |  $87^{\circ}$ $\pm$ $14^{\circ}$ |  $96^{\circ}$ $\pm$ $20^{\circ}$ |  $96^{\circ}$ $\pm$ $17^{\circ}$
0.5 $r_{\rm td}$ |  $85^{\circ}$ $\pm$ $6^{\circ}$ |  $96^{\circ}$ $\pm$ $36^{\circ}$ |  $82^{\circ}$ $\pm$ $27^{\circ}$

####  4.4.2 Rotational properties of stars in Hyades

Figure 13 shows the variation in the rotational velocities of the member stars
of the Hyades cluster vary with their distance to the cluster center. Similar
to the properties of the former two clusters, not all member stars rotate in
the same direction. The number of the member stars with positive rotational
velocities is 1.4 times of those with negative rotational velocities. Compared
with the Pleiades and the $\alpha$ Per cluster, there are fewer member stars
have peculiar rotational velocities that deviate from that of most cluster
members within the tidal radius of the Hyades. In addition, the rotation of
the member stars located near or beyond the tidal radius does not appear to be
disorganized, as seen from Figure 13.

Figure 13: Rotational velocity as a function of distance to the cluster center
for the member stars within twice of the tidal radius of the Hyades.

In the same way as for the Pleiades and $\alpha$ Per, the skewness of the
astrometric parameters of the member stars within the tidal radius of the
Hyades cluster are calculated to assess the symmetry of the cluster density
distribution. The skewness of the Galactic longitude, Galactic latitude, and
parallax are 0.1, 0.1, and 0.0, respectively, indicating that it makes sense
for us to assume a spherically symmetric distribution in the Hyades.

Figure 14: Same as Figure 9, but for the Hyades cluster. Table 8: RMS rotational velocities of the members stars within the tidal radius of the Hyades cluster. $R_{i}$ |  $R_{o}$ |  $R_{m}$ |  $N$ |  RMS $v_{\varphi}$ |  $\epsilon_{{\rm RMS}\,v_{\varphi}}$
---|---|---|---|---|---
[pc] |  [pc] |  [pc] |  |  [km s-1] |  [km s-1]
(1) |  (2) |  (3) |  (4) |  (5) |  (6)
0.0 |  2.0 |  1.2 |  36 |  0.52 |  0.05
2.0 |  4.0 |  2.9 |  44 |  0.61 |  0.06
4.0 |  6.0 |  5.0 |  46 |  0.62 |  0.05
6.0 |  8.0 |  6.9 |  21 |  0.59 |  0.08
8.0 |  9.5 |  8.3 |  9 |  0.50 |  0.09
Figure 15: Same as Figure 5, but for the Hyades cluster.

In the sample, 163 member stars lie within the tidal radius of the Hyades
cluster. The median absolute value of their rotational velocities
$v_{\varphi}$ is $\sim$0.5 km s-1 with a standard deviation of $\sim$0.5 km
s-1. Similar to the analysis for the Pleiades and the $\alpha$ Per cluster, we
first eliminate 7 stars with peculiar rotational velocities after inspecting
the distribution of rotational velocities of the member stars. Then, using the
remaining 156 member stars, we calculate the RMS rotation velocity and
uncertainty for the members within each distance bin. The results are given in
Table 8 and Figure 14. Using Equation (2), the RMS rotational velocities as a
function of the distances to the cluster center are well fit by a theoretical
circular motion velocity curve. Therefore, it may be feasible to describe the
rotation of the member stars within the tidal radius of the Hyades using
Newton’s theorems. Furthermore, with the same method as for the above two
clusters, after using Equation (2) to fit the RMS velocities listed in Table
8, we infer that the core radius and mass within the tidal radius of the
Hyades cluster are 2.1 $\pm$ 0.3 pc and 569 $\pm$ 108 ${\rm M}_{\odot}$,
respectively.

We also examine whether the Hyades cluster, an OC with an age of about 650 Myr
(see Section 1), is expanding or contracting by statistically analyzing the
radial components $v_{r}$ of the member stars. As indicated in Figure 15, the
mean radial components $\overline{v_{r}}$ of the member stars within 0.5
$r_{\rm td}$, 1 $r_{\rm td}$, and 3 $r_{\rm td}$ are 0.0 $\pm$ 0.04, –0.1
$\pm$ 0.03, and –0.1 $\pm$ 0.03 km s-1, respectively. Similar to the Pleiades
and the $\alpha$ Per cluster, the mean radial components are also close to
zero. These results suggest that there is no significant expansion or
contraction in the Hyades, and this implies that the rotation of the cluster
members within the tidal radius may exhibit closed-loop motions. Figure 15
also shows that the member stars of the Hyades cluster show non-negligible
acceleration and dispersion in their radial components, indicating that there
should be additional mass in the system to account for the force supporting
the radial component acceleration. Therefore, the estimated mass in this work
is a lower limit for the Hyades.

##  5 Discussion

Table 9: Parameters of the Praesepe, Pleiades, $\alpha$ Per, and the Hyades. | Rotational velocity |  $\vec{l}$ vs. GP | $Dist.$ | Mass | Age |  $r_{\rm td}$ ($r_{\rm co}$) | $|Z|$
---|---|---|---|---|---|---|---
| [km s-1] | [degree] | [pc] | ${\rm M}_{\odot}$ | [Myr] | [pc] | [pc]
Praesepe | 0.12 $\pm$ 0.05 |  $34$ $\pm$ $15$ | 178 $\pm$ 0.8[1] |  $\sim$510[3] | 590–700[1,7] | 10.7(2.6)[1] | 101 $\pm$ 2.2[1]
Pleiades | 0.24 $\pm$ 0.04 |  $62$ $\pm$ $13$ | 135 $\pm$ 0.4[1] |  $\sim$820[4] | 110–160[1,7] | 11.6(2.0)[1] | 55 $\pm$ 0.2[1]
$\alpha$ Per | 0.43 $\pm$ 0.08 |  $48$ $\pm$ $14$ | 187 $\pm$ 3.9[1] |  $\sim$920[5] | 50–70[8,9] | 9.5(2.3)[1] | 20 $\pm$ 0.5[1]
Hyades | 0.09 $\pm$ 0.03 |  $7$ $\pm$ $7$ | 47 $\pm$ 0.2[2] |  $\sim$280[6] | 650–780[2,10] | 9.0(3.1)[2] | 17 $\pm$ 0.1[2]

Figure 16: Panels (a), (b), (c), and (d): rotational velocities of the
Praesepe, Pleiades, $\alpha$ Per, and the Hyades clusters as functions of
their masses, ages, radii ($r_{\rm td}$, $r_{\rm co}$), and $z$-scale heights
($Z$) from the Galactic plane. Panels (e), (f), (g), and (h): the same as
Panels (a-d), but for the angles of the cluster rotation axes ($\vec{l}$) with
respect to the Galactic plane.

Based on _Gaia_ DR3, we investigate the 3D internal rotation of four nearby
member-rich OCs and estimate their rotation axes and mean rotational
velocities. Interestingly, Newton’s theorem can well interpret the rotation of
member stars within the tidal radii of these OCs. In addition, no indication
of expansion or contraction has been detected in the four OCs, in agreement
with the result of Della Croce et al. (2023), who reported that the clusters
older than 30 Myr do not experience expansion and are mostly compatible with
equilibrium configurations. Although the sample size is small, Praesepe, the
Pleiades, $\alpha$ Per, and the Hyades are currently the only four OCs with a
determined 3D rotation, which makes it feasible to explore the possible
relations between the rotation of OCs and other cluster parameters. In Table
9, we compile the distances, masses, ages, radii, and $z$-scale heights of
these four OCs from the literature. The cluster rotational velocities and
rotation axes given by this work are also listed. Clearly, the rotational
parameters of the OCs have no obvious relation with the cluster distances from
the Sun. The results for the other cluster parameters are given in Figure 16.

Figure 16(a-d) present the variation in the rotational velocities with other
cluster parameters. It is shown in Figure 16(a) that OCs with higher masses
tend to have greater rotational velocities, implying that the rotational
velocity of OCs may be positively correlated with their masses. In fact, the
Galactic globular clusters, as the stellar populations that are much more
massive than OCs, are observed to have internal rotation velocities of several
km s-1 (e.g., Lanzoni et al., 2018; Leanza et al., 2022), which is one order
of magnitude higher than these four OCs. In Figure 16(b), it is interesting to
note that young OCs have higher rotation velocities than old OCs, which
suggests that the rotational velocity of OCs may gradually decrease as they
age. If this is the case, the present-day rotation of OCs is likely to
originate from their predecessor clusters, whose rotation is found to be
common in hydrodynamical simulations (e.g., Mapelli, 2017). With the available
data, we do not find any clear relation between the rotational velocities and
the tidal radii or the $z$-scale heights of OCs.

Figure 16(e-h) present the included angles between the cluster rotation axes
and the Galactic plane as a function of cluster parameters. In Figure 16(e),
we find that the included angles of the massive OCs appear to be larger than
those of the low-mass OCs. In comparison with old OCs, the younger OCs seem to
have larger included angles, as shown in Figure 16(f). The above two relations
are not significant when the errors of the included angles are taken into
account. Based on these four clusters, no clear correlations can be identified
between the directions of the rotation axes and the cluster radii or the
cluster $z$-scale heights, as shown in Figure 16(g) and (h).

So far, we have only studied the 3D rotation of four star-rich OCs within 200
pc from the Sun. The sample size is very limited. The above results need to be
further confirmed by the rotational characteristics of more OCs. An accurate
detection of 3D rotation in more distant star-rich OCs (such as those clusters
located within 500 pc) requires high-precision astrometry and improved
methods. More efforts will be undertaken to reveal the internal 3D rotation of
more OCs, which will help us to improve our understanding of the formation and
evolution of OCs in the Milky Way.

##  6 Summary

We report significant detections of rotation in the Pleiades, $\alpha$ Per,
and Hyades clusters based on the high-quality data of _Gaia_ DR3. In order to
unveil the internal kinematics of star clusters, we have slightly amended the
method developed in our previous work (Paper I). The analyses on the motions
of the member stars not only revealed 3D rotation of the Pleiades, $\alpha$
Per, and the Hyades, but also determined their rotation axes in the Milky Way,
with angles of $62^{\circ}\pm 13^{\circ}$, $48^{\circ}\pm 14^{\circ}$, and
$7^{\circ}\pm 7^{\circ}$ relative to the Galactic plane, respectively. We also
unveil the rotational properties of the member stars in the three OCs and
suggest that the theorems of Newtonian mechanics can well characterize the
rotational velocities of the member stars within each cluster tidal radius. In
addition, the statistics of current observations suggest that the three
clusters do not expand or contract.

Combining the Praesepe cluster analyzed in Paper I with the three OCs
investigated in this work, we find a possible relation between the rotational
velocity and the age and mass of the cluster. Massive OCs tend to have
significant internal rotation, and the rotation velocity gradually decreases
from young to old OCs. However, these results need to be further confirmed
when the rotational features of more clusters are revealed. To date, the
internal kinematics of almost all known OCs remain elusive, leaving a major
gap in the study of star clusters in the Milky Way. The method reported in
this work takes a step in the direction of unveiling the detailed kinematic
properties of OCs. The 3D rotations of OCs has been measured based on the
high-precision parallaxes, proper motions, radial velocities from the _Gaia_
DR3 dataset. With the improvement of astrometric accuracy and precision in the
upcoming _Gaia_ Data Releases, it is expected that there will be a wealth of
yet-to-be-revealed interesting kinematic properties of OCs, deserving our
attention and exploration.

We thank the anonymous referee for the instructive comments and suggestions
that greatly helped us to improve the paper. This work was funded by the NSFC
grant 11933011, 11988101, National SKA Program of China (grant No.
2022SKA0120103), and the Key Laboratory for Radio Astronomy. L.G.H.
acknowledges the support from the Youth Innovation Promotion Association CAS.
Y.J.L. acknowledges support from the NSFC grant 12203104 and the Natural
Science Foundation of Jiangsu Province (grant No. BK20210999). We used data
from the European Space Agency mission Gaia (http://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC; see
http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for DPAC has been
provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement. Figure A1 presents the
definitions of the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$ and the
$O_{\rm r}$–$X_{\rm r}$$Y_{\rm r}$$Z_{\rm r}$ systems.

Figure A1: Definitions of the $O_{\rm c}$–$X_{\rm c}$$Y_{\rm c}$$Z_{\rm c}$
(black) system and the $O_{\rm r}$–$X_{\rm r}$$Y_{\rm r}$$Z_{\rm r}$ (blue)
system (from Paper I). The cluster rotation axis $\vec{l}$ (red) is in
accordance with the $Z_{\rm r}$ axis. The projections of $\vec{l}$ in the
$Y_{\rm c}$–$Z_{\rm c}$, $X_{\rm c}$–$Z_{\rm c}$, and $X_{\rm c}$–$Y_{\rm c}$
planes are $l_{yz}$, $l_{xz}$, and $l_{xy}$ (purple), respectively. The
projection of the $X_{\rm r}$–axis in the $X_{\rm c}$–$Y_{\rm c}$ plane is
consistent with $l_{xy}$. $\alpha$, $\beta$ and $\gamma$ indicate the included
angles between $l_{yz}$ and $Y_{\rm c}$, $l_{xz}$ and $X_{\rm c}$, and
$l_{xy}$ and $X_{\rm c}$, respectively.

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