# The Rotation of Selected Globular Clusters and the Differential Rotation of
M3 in Multiple Populations from the SDSS-IV APOGEE-2 Survey

László Szigeti,1 Szabolcs Mészáros,1,2 Gyula M. Szabó,1 José G. Fernández-
Trincado3,4,5 Richard R. Lane,3,5 Roger E. Cohen,6
1ELTE Gothard Astrophysical Observatory, H-9704 Szombathely, Szent Imre Herceg
st. 112, Hungary
2Premium Postdoctoral Fellow of the Hungarian Academy of Sciences
3Instituto de Astronomía y Ciencias Planetarias, Universidad de Atacama,
Copayapu 485, Copiapó, Chile
4Institut Utinam, CNRS UMR 6213, Université Bourgogne-Franche-Comté, OSU THETA
Franche-Comté, Observatoire de Besançon,
BP 1615, 25010 Besançon Cedex, France
5Centro de Investigación en Astronomía, Universidad Bernardo O Higgins,
Avenida Viel 1497, Santiago, Chile
6Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In this paper, we analyze 10 globular clusters in order to measure their
rotational properties by using high precision radial velocity data from the
SDSS-IV APOGEE-2 survey. Out of the 10 clusters we were able to successfully
measure the rotation speed and position angle of the rotation axis for 9
clusters (M2, M3, M5, M12, M13, M15, M53, M92, M107). The comparison between
our results and previous ones shows a really good agreement within our
uncertainties. For four of the globular clusters, M3, M13, M5 and M15, we
separated the sample into two generation of stars using their [Al/Fe]
abundances and examined the kinematic features of these generations separately
from one another. In case of M3, we found significant difference between the
rotational properties of first and second populations, confirming for the
first time the predictions of several numerical simulations from the
literature. The other three clusters (M5, M13, M15) also show smaller
deviation between the two groups of stars, but those deviations are comparable
to our errors.

###### keywords:

globular clusters: general - stars: kinetics and dynamics, population II

††pubyear: 2021††pagerange: The Rotation of Selected Globular Clusters and the
Differential Rotation of M3 in Multiple Populations from the SDSS-IV APOGEE-2
Survey–The Rotation of Selected Globular Clusters and the Differential
Rotation of M3 in Multiple Populations from the SDSS-IV APOGEE-2 Survey

##  1 Introduction

The structure of globular clusters is shaped by complex interaction between
external (interaction with the host galaxy) and internal (for example,
relaxation, core collapse, mass segregation) forces. King (1966) successfully
described the star density of globular clusters with assuming no rotation.
Tiongco, Vesperini & Varri (2017) have suggested that the internal relaxing
processes dissipate all the angular momentum on a long term in every globular
cluster. However, growing number of studies Lane et al. (2010); Bellazzini et
al. (2012); Bianchini et al. (2013); Fabricius et al. (2014); Lardo et al.
(2015); Kimmig, et al. (2015); Boberg et al. (2017); Lee (2017); Cordero et
al. (2017); Kamann et al. (2018); Ferraro et al. (2018); Lanzoni et al.
(2018); Bianchini et al. (2018) present evidence that a significant amount of
internal rotation could still be observed in many globular clusters, however
the observed rotational strengths are only a fraction of the initial ones
(Kamann et al., 2018; Bianchini et al., 2018). These studies mostly utilize
recently recorded high quality radial velocity and proper motion data.

The presence of internal rotation could introduce some problems with the
theory of globular clusters formation and evolution. Some studies indicate
that the remaining rotation accelerates the evolution and shapes the
morphology of the cluster (Einsel & Spurzem, 1999; Bianchini et al., 2013).
Others suggest that the present day rotation could be a remnant of a strong
rotation during the early history of globular clusters (Vesperini, et al.,
2014; Mapelli, 2017).

Orbital motions may be isotropic even during the formation of GCs, for example
Lahén et al. (2020) have argued that the massive clusters are isotropic
already during their first 100 Myr after formation. Moreover, as Lane et al.
(2010) and Tiongco, Vesperini & Varri (2017) has shown, the interaction
between the cluster and the galactic tidal field combined within the internal
dynamics could produce complex kinematical features, e.g. radial variation in
the orientation of the rotational axis, and anomalous velocity dispersions.
The observational evidence of rotating globular clusters is based on high
precision radial velocity data (Bellazzini et al., 2012; Bianchini et al.,
2013; Lardo et al., 2015; Kimmig, et al., 2015; Boberg et al., 2017; Lee,
2017; Cordero et al., 2017; Ferraro et al., 2018; Lanzoni et al., 2018;
Cordoni et al., 2020a), integral field unit (IFU) spectrograph (Fabricius et
al., 2014; Kamann et al., 2018), and proper motion measurements (Massari et
al., 2013; Bellini et al., 2017; Bianchini et al., 2018; Mastrobuono-Battisti
& Perets, 2020).

The other aspect of the globular cluster rotation is the presence of multiple
generations of stars and their rotational properties. In the last decade, the
existence of two or more distinct generations of stars in most globular
clusters became well studied Carretta, et al. (2009a, b); Piotto, et al.
(2015); Milone, et al. (2017); Milone et al. (2018); Mészáros, et al. (2020),
however understanding the formation of multiple population is still an
astrophysical challenge. Multiple populations manifest in light-element
abundance variations, second generation stars (SG) are enhanced in N, Na, Ca
and Al and depleted in C, O and Mg, while the first generation stars (FG) are
the opposite. Most of the theories agree, that the second generation stars
formed out of the first generation’s ejecta mixed with the original
intracluster medium (Decressin, et al., 2007; Bekki, 2010; Denissenkov &
Hartwick, 2014; Bekki, 2017), but the exact process of this pollution is
currently not known, and many observed processes can not be explained with
this theory yet. There are other alternative explanations for this phenomena
and they are discussed in detail in Hénault-Brunet et al. (2015); Bastian &
Lardo (2018).

Observational evidence have showed differences in spatial distribution between
FG and SG stars. For dynamically younger clusters, the SG is more concentrated
than the FG (Sollima et al., 2007; Bellini et al., 2009; Lardo et al., 2011;
Cordero et al., 2014; Boberg, Friel, & Vesperini, 2016; Lee, 2017; Gerber,
Friel, & Vesperini, 2020). On the other hand, the two generations are
completely mixed in globular clusters with more advanced evolutionary stages
(Dalessandro et al., 2014; Nardiello et al., 2015; Cordero et al., 2015;
Gerber, Friel, & Vesperini, 2018, 2020). In Dalessandro et al. (2019), the
link between concentration differences and evolutionary stage was explored in
detail based on observations and models. Other observational studies revealed
differences in the kinematics between the multiple generations (Richer et al.,
2013; Bellini et al., 2015, 2018; Cordero et al., 2017; Milone et al., 2018;
Libralato et al., 2019; Cordoni et al., 2020a, b), in other cases, the
multiple generation of stars share similar kinematic properties (Pancino et
al., 2007; Cordoni et al., 2020b) These literature sources used different
indicators of cluster kinematics, including rotation, velocity dispersion and
anisotropy to show the different kinematical properties of multiple
populations.

Our main purpose is to investigate the rotational properties of the selected
clusters using high precision and homogeneous radial velocity data. Our
secondary goal is to identify potential differences in the cluster rotation
properties between the multiple populations. For this study, we use state-of-
the-art data from the high-resolution spectroscopic survey Apache Point
Observatory Galactic Evolution Experiment (APOGEE) (Majewski, APOGEE Team &
APOGEE-2 Team, 2016). The APOGEE started as one component of the $3^{\rm rd}$
Sloan Digital Sky Survey (SDSS-III; Eisenstein et al. (2011)) and continues as
part of SDSS$-$IV (Blanton, et al., 2017) as
APOGEE-2111http://www.sdss.org/surveys/apogee-2/. The goal of APOGEE-2 is to
obtain high-resolution (R = 22500), high signal-to-noise, H-band spectra
($\lambda$ = 1.51$-$1.70$\mu$m) of more than 600,000 late-type stars in the
Milky Way by the end of 2020, and to determine chemical abundances of $\sim$26
elements in all observed stars. Most APOGEE targets are evolved red-giant
branch (RGB) and asymptotic giant branch (AGB) stars from all major Galactic
stellar populations.

##  2 Data and reduction

###  2.1 Target Selection and Radial Velocities

The data were gathered by the Sloan Foundation 2.5 m Richey-Chreritien
altitude-azimuth telescope (Gunn et al., 2006) at Apache Point Observatory.
The spectra were obtained via the APOGEE spectrograph (Wilson, et al., 2019)
with a resolution power of 22500. The stellar atmospheric parameters and
chemical abundances are calculated from these spectra with APOGEE Stellar
Parameters and Chemical Abundances Pipeline (ASPCAP) (García Pérez et al.,
2016). We use the list of stars compiled by Masseron, et al. (2019). The
target selection is explained in detail in Mészáros, et al. (2015), Masseron,
et al. (2019) and Mészáros, et al. (2020). In short, stars were selected based
on their radial velocity, distance from cluster center and metallicity. In
radial velocity, all studies required stars to be within three times the
velocity dispersion of the mean cluster velocity, which were taken from
Baumgardt et al. (2019).

We used the DR14 data release of APOGEE (Holtzman, et al., 2018) for our
study. Radial velocities are derived by the reduction pipeline (Nidever, et
al., 2015), while details can be found in that paper, we provide a brief
description of the algorithm. For almost all stars, observations are made in
multiple visits to improve S/N and to allow observations of faint objects,
which are then combined together to provide the final spectrum of the star.
The radial velocity is measured in multiple steps. First, we do an initial
measurement for each star from the actual individual spectrum by cross
correlating each spectrum with the best match in a template library. In the
second step, all of the visits are combined, and relative radial velocities of
each visit are iteratively refined by cross-correlating each visit spectrum
with the combined spectrum. The final absolute radial velocity is set by
cross-correlating the combined spectrum with the best match in a template
library. The peak of the cross-correlation function is fitted with a Gaussian
in order to determine the accurate spectral shift. Binary stars can distort
the rotational velocity profile if included in the sample. We removed all the
binaries from our sample using database from Price-Whelan et al. (2020). In
Price-Whelan et al. (2020), they identified nearly 20000 binary candidates
with high confidence in the 16th data release of APOGEE, out of these we have
found 65 stars in common with our targets, only $\approx~{}7\%$ of the total
number of stars.

The uncertainties of radial velocities depend on multiple factors, mainly the
characteristic of spectra, the resolution and the S/N ratio. For example, a
star with lots of deep and thin lines in the spectra had a more precise RV
than a star with wide and shallow lines. The typical uncertainty of the final
radial velocity for stars in this study is 0.1 km/s.

###  2.2 Method

GC name | N | [Fe/H] | Vhelio | Rh | davg | PA | PAerr | Arot | Aroterr
---|---|---|---|---|---|---|---|---|---
|  |  | [km/s] | [arcmin] | [arcmin] |  |  | [km/s] | [km/s]
M2 | 26 | -1.65 | -5.3 | 0.93 | 3.9 | 26 | 19 | 3.48 | 0.82
M3 | 145 | -1.5 | -148.6 | 1.12 | 6.3 | 164 | 13 | 1.19 | 0.28
M5 | 215 | -1.29 | 52.1 | 2.11 | 5.9 | 148 | 6 | 3.45 | 0.37
M12 | 65 | -1.37 | -42.1 | 2.16 | 5.8 | 56 | 93 | 0.24 | 0.19
M13 | 135 | -1.53 | -246.6 | 1.49 | 5.1 | 26 | 9 | 2.38 | 0.39
M15 | 138 | -2.37 | -107.5 | 1.06 | 4.8 | 120 | 11 | 2.38 | 0.44
M53 | 40 | -1.86 | -79.1 | 1.11 | 4.8 | 98 | 27 | 1.54 | 0.57
M71 | 28 | -0.78 | -22.9 | 1.65 | 2.7 | … | … | … | …
M92 | 72 | -2.31 | -121.6 | 1.09 | 4.2 | 154 | 14 | 2.06 | 0.58
M107 | 67 | -1.02 | -33.8 | 2.70 | 4.3 | 168 | 30 | 0.72 | 0.27
Table 1: The table contains the basic parameters of the targeted globular
clusters and the results of this study. The second column represents the
number of observed stars. The third and the fourth columns are the metallicity
and the clusters heliocentric radial velocity Harris (1996); Miocchi et al.
(2013). The fifth is the halflight radii Harris (1996). The sixth is the
average distance from the cluster centers in our samples. The seventh and
eighth columns represent the calculated position angle and its error, while
the last two columns are the rotational velocity and its uncertainties. The
position angles measured from North to East anti-clockwise direction.

In order to investigate the rotational velocity and the position angle of the
rotational axes of the cluster, we follow the same method as Cote, et al.
(1995); Bellazzini et al. (2012); Bianchini et al. (2013); Lardo et al.
(2015); Kimmig, et al. (2015); Boberg et al. (2017); Lee (2017); Lanzoni et
al. (2018). First, each cluster was split into two halves along the cluster
center. The position angle of separation was the independent variable of the
analysis, varied between PA = 0 and 180 (PA = 90 is toward East) in 2 degree
step-size.We ran the simulations with multiple step-sizes (e.g. 2, 5, 10, 20
degree) all providing the same end results within our uncertainties. At the
end we choose the 2 degrees for all of our clusters to appropriately sample
the densest areas. Next, the mean radial velocity of these sub-samples were
calculated and the difference between the two sub-samples mean velocity were
determined. If rotation is present in the system, the delta Vmean draws a
sinusoidal variation as a function of the position angle. The amplitude of the
function is twice the rotational velocity (because the amplitude is the
difference of the two hemispheres) and the min-max position (ideally the
difference is 180 degree) is the PA of the rotational axis. Thus, we caution
the reader that rotational velocity values printed in Table 2 and 3, and all
figures are twice as large as the real rotational velocity, in agreement to
what has been used in the literature. Our results are listed in Table 1.

###  2.3 Separating multiple populations in GCs

In this study, first (FG) and second generation (SG) stars are separated from
each other based on their [Al/Fe] abundances, following the suggested cuts by
Mészáros, et al. (2020) from APOGEE data. Previously, Mészáros, et al. (2015)
used an extreme-deconvolution code for fitting a distribution with multiple
Gaussian components to identify FG from SG stars in a mutli-dimensional
abundance space, they showed that almost all of the SG stars have
[Al/Fe]$>$0.3 dex. For this reason we use this simplified criteria to set
multiple populations apart using abundances from Masseron, et al. (2019). For
this analysis, we selected 4 clusters with the most observed stars, in which
we have enough samples to properly fit the rotational curve.

###  2.4 Error estimation

We checked the robustness of the method with a simple jackknife test. We
randomly dropped more and more stars from the sample and calculated the
rotational curves. The results indicate, we can get a good signal if we have
at least 20-30 stars to work with. Fewer stars than this is insufficient for a
robust measurement.

In order to calculate the final uncertainty, we randomly dropped 20 percent of
the stars (dropping more than this may result in fewer than 20 stars for some
clusters) and derived the position angle and the rotational velocity in the
sub-samples. We repeated this process 100 times, then the standard deviation
of the sinusoidal fit was calculated for the position angles and rotational
velocities for each of these sub-samples. The final uncertainties seen in
Figure 1 are the average of the differences between the original fit and the
sub-samples.

We were able to define the errors for all the selected clusters, however for
M12, the combination of the small number of observed stars, low rotational
signal, and the error estimation method produced a high uncertainty of the PA.
For M71 we could not get a clear sinusoidal signal from the data at hand.
Table 1 contains our derived results.

We tested the robustness and the stability of the derived rotational
amplitudes with a bootstrap analysis. For both populations in all clusters,
100 bootstrap distributions have been realised with redistributing the
measured velocities randomly (with re-sampling allowed) among the field stars.
These samples suffered a complete loss of any information on the rotation, and
contained a "null signal". The distributions were then evaluated following
exactly the same method as in the case of the observations, and we observed
the best-fit amplitude of the inferred rotating model. This amplitude was
considered as the upper limit of the rotation amplitudes if the clusters would
not rotate, and the measured amplitudes were just a product of numerical
fluctuations in the data distribution. The standard distribution of the
amplitudes in the bootstrap samples were in the range of 0.5–0.8 km/s, proving
that the detection of the rotation of all examined clusters is indeed
significant.

##  3 Discussion

###  3.1 Comparison with literature

| M2 |  | M3 |  | M5 |  | M12 |  | M13
---|---|---|---|---|---|---|---|---|---
| PA | Arot |  | PA | Arot |  | PA | Arot |  | PA | Arot |  | PA | Arot
Lane et al. (2010) | … | … |  | … | … |  | … | … |  | 40 | 0.15$\scriptsize\pm 0.8$ |  | … | …
Bellazzini et al. (2012) | … | … |  | … | … |  | 157 | 2.6$\scriptsize\pm 0.5$ |  | … | … |  | … | …
Fabricius et al. (2014) | … | … |  | 192 $\scriptsize\pm 11.8$ | … |  | 149$\scriptsize\pm 5.6$ | … |  | 89$\scriptsize\pm 19.3$ | … |  | 17$\scriptsize\pm 7.8$ | …
Kimmig, et al. (2015) | 53 | 4.7$\scriptsize\pm 1.0$ |  | … | 0.6$\scriptsize\pm 1.0$ |  | … | 2.1$\scriptsize\pm 0.7$ |  | … | 0.2$\scriptsize\pm 0.5$ |  | … | …
Lee (2017) | … | … |  | … | … |  | 128 | 3.36$\scriptsize\pm 0.7$ |  | … | … |  | … | …
Cordero et al. (2017) | … | … |  | … | … |  | … | … |  | … | … |  | 14$\scriptsize\pm 19$ | 2.7$\scriptsize\pm 0.9$
Kamann et al. (2018) | 41.7$\scriptsize\pm 2.7$ | … |  | … | … |  | 144$\scriptsize\pm 20.3$ | … |  | … | … |  | … | …
Ferraro et al. (2018) | … | … |  | 151 | 1.0 |  | … | … |  | … | … |  | … | …
Lanzoni et al. (2018) | … | … |  | … | … |  | 145 | 4.0 |  | … | … |  | … | …
Sollima et al. (2019) | 14$\scriptsize\pm 12.1$ | 3.01$\scriptsize\pm 0.7$ |  | … | 1.75$\scriptsize\pm 0.4$ |  | 132$\scriptsize\pm 6$ | 4.11$\scriptsize\pm 0.4$ |  | … | 0.93$\scriptsize\pm 0.4$ |  | 15$\scriptsize\pm 14.2$ | 1.53$\scriptsize\pm 0.6$
this work | $26\scriptsize\pm 19$ | $3.48\scriptsize\pm 0.8$ |  | $164\scriptsize\pm 15$ | $1.19\scriptsize\pm 0.3$ |  | $148\scriptsize\pm 6$ | $3.45\scriptsize\pm 0.4$ |  | $56\scriptsize\pm 93$ | $0.24\scriptsize\pm 0.2$ |  | $26\scriptsize\pm 9$ | $2.38\scriptsize\pm 0.4$
Table 2: Comparison with literature. Part 1.
Position angles and rotational amplitudes from earlier studies. Since different conventions were followed, we convert the published result to PA 90 = East, anti-clockwise system. Literature source which did not use the double rotational velocity were converted to our system. The first sub-column represent the position angle of the rotational axis and the second is the rotational velocity in [km/s]  | M15 |  | M53 |  | M71 |  | M92 |  | M107
---|---|---|---|---|---|---|---|---|---
| PA | Arot |  | PA | Arot |  | PA | Arot |  | PA | Arot |  | PA | Arot
Lane et al. (2009) | … | … |  | nf | nf |  | … | … |  | … | … |  | … | …
Bellazzini et al. (2012) | 110 | 3.8$\scriptsize\pm 0.5$ |  | … | … |  | 163 | 1.3$\scriptsize\pm 0.5$ |  | … | … |  | 84 | 2.9$\scriptsize\pm 1.0$
Bianchini et al. (2013) | 106$\scriptsize\pm 1$ | 2.84 |  | … | … |  | … | … |  | … | … |  | … | …
Fabricius et al. (2014) | … | … |  | 113$\scriptsize\pm 19.2$ | … |  | … | … |  | 99$\scriptsize\pm 12.0$ | … |  | … | …
Lardo et al. (2015) | 120 | 3.63$\scriptsize\pm 0.1$ |  | … | … |  | … | … |  | … | … |  | … | …
Kimmig, et al. (2015) | … | 2.5$\scriptsize\pm 0.8$ |  | … | 0.4$\scriptsize\pm 0.7$ |  | … | 0.4$\scriptsize\pm 0.8$ |  | … | 1.8$\scriptsize\pm 0.8$ |  | … | …
Boberg et al. (2017) | … | … |  | 74 | 2.8 |  | … | … |  | … | … |  | … | …
Kamann et al. (2018) | 151$\scriptsize\pm$10.4 | … |  | … | … |  | … | … |  | … | … |  | … | …
Ferraro et al. (2018) | … | … |  | … | … |  | … | … |  | … | … |  | 167 | 1.2
Sollima et al. (2019) | 128$\scriptsize\pm 28.8$ | 3.29$\scriptsize\pm 0.5$ |  | … | … |  | … | … |  | … | 1.46$\scriptsize\pm 0.6$ |  | … | …
this work | $120\scriptsize\pm 11$ | $2.38\scriptsize\pm 0.4$ |  | $98\scriptsize\pm 27$ | $1.54\scriptsize\pm 0.6$ |  | … | … |  | $154\scriptsize\pm 14$ | $2.06\scriptsize\pm 0.6$ |  | $168\scriptsize\pm 30$ | $0.72\scriptsize\pm 0.3$
Table 3: Comparison with literature. nf = not found evidence of rotation. Part
2

The latest results available in the literature were collected in Table 2 and
3, which contain the calculated position angles of the rotational axis for
clusters in common with our sample. These data also represented in Figure 3.
There are multiple conventions used in the literature for angle and direction
notations. We converted all these various approaches to the PA 90 = East
convention. The last row contains our values.

We detected systematic rotation in almost all of the targeted globular
clusters. We confidently could derive rotational velocity and position angles
for nine out of the ten selected clusters. All nine clusters have been studied
in the literature before, thus we are able to not only compare our results
with previous studies, but also homogenize the rotational velocity as our
radial velocities are from one homogeneous survey. We caution the reader that
the assumption of a constant rotation velocity as a function of distance to
the cluster centre is a significant simplification. The observations of Boberg
et al. (2017); Bianchini et al. (2018); Sollima et al. (2019) have shown that
the peak of the rotational curve is located approximately at the cluster half-
light radius, however this location is expected to change during the evolution
of the cluster (Tiongco, Vesperini & Varri, 2017). We listed the halflight-
radius and the average distance of our sample from the cluster centre in Table
1. In all cases, the average distance is at least 2-3 times larger than the
halflight-radius suggesting that our assumption of a constant rotation
velocity underestimates the magnitude of the rotational velocity.

Before such a comparison can be made one must transform the results from the
literature to a common coordinate system (PA 90 = East, anti-clockwise). After
the transformation we are able to conclude while for some of the cluster we
have a good agreement within our uncertainties, other less studied clusters
with fewer stars show larger than expected discrepancy between studies. In the
next few sections we examine these differences for each cluster.

####  3.1.1 M5

M5 is a well observed cluster targeted by Bellazzini et al. (2012); Fabricius
et al. (2014); Kimmig, et al. (2015); Lee (2017); Kamann et al. (2018);
Lanzoni et al. (2018), therefore it is an excellent object to use it as a
standard to compare our results to, especially because these literature
sources used different measurement methods. The position angle varies between
144.3 to 157 degree in the literature, our result of 148 degree fits nicely in
this picture.

As mentioned before, our calculation technique is similar to many that studied
M5 (Bellazzini et al., 2012; Lee, 2017; Lanzoni et al., 2018), and by
comparing our results to these studies, we can find a good agreement in all
cases. Fabricius et al. (2014) and Kamann et al. (2018) used IFU spectrograph
for the analysis. The advantage of this method that it is possible to measure
crowded stellar fields in order to perform a detailed analysis of dispersion
fields and central rotation. In Bianchini et al. (2018) the rotational pattern
was derived from proper motion data (GAIA). Despite the different observations
and methodology we feel confident in our approach as it nicely reproduces the
rotation velocity and angle reported by the mentioned studies for M5. In
Sollima et al. (2019) the rotation of M5 (among other clusters) was
investigated via radial velocity component from VLT and Keck instruments and
proper motion data from GAIA 2nd data release. In Sollima et al. (2019) the
rotational velocity derived from all the three velocity components (taken into
consideration the inclination of the rotational axis) while we were able to
use only the line-of-sight part. Their derived results show a good agreement
with ours in case of PA and the difference in rotational velocity can be
explained by the fact that we observed line of sight velocity, while they were
able to determine the inclination.

####  3.1.2 M2

Kimmig, et al. (2015) derived the rotational velocity as A${}_{\rm
rot}=4.5~{}\rm km/s$, which is slightly larger than our value at $3.48\pm
0.8$. Considering our uncertainty we conclude that these differences are not
substantial. The position angle also differs from Kimmig, et al. (2015);
Kamann et al. (2018); Sollima et al. (2019), but our limited sample size cause
a 19 degree of uncertainty, which can explain the difference.

####  3.1.3 M3, M12 and M13

We have a good amount of observed stars in M3 and the derived position angles
are within errors to Ferraro et al. (2018). The difference from Fabricius et
al. (2014) can be explained by their relatively large errors and the different
analysis method applied.

Our derived result for the rotational properties of M12 have high uncertainty,
probably because either the amplitude of the rotation is too small, or the
inclination is close to 90 degrees, so that we look close to the direction of
rotational axis. We have a good agreement with the results Lane et al. (2010)
and considering the large uncertainty we agree well with Fabricius et al.
(2014) too.

Our results show a good agreement at the level of our uncertainties with
Fabricius et al. (2014), Cordero et al. (2017) and Sollima et al. (2019) for
M13. The rotational amplitude is also really similar to the one presented in
Cordero et al. (2017).

####  3.1.4 M15

In case of M15 we have a slight deviation in the PA from one of the previous
results, namely Kamann et al. (2018), however we are in a good agreement with
the other four (Bellazzini et al., 2012; Bianchini et al., 2013; Lardo et al.,
2015; Sollima et al., 2019). The source of the misalignment with Kamann et al.
(2018), which is 30 degrees, might be due to two different reasons. One is
that Kamann et al. (2018) used data from an IFU spectrograph and used Voronoi-
binned maps of the mean velocity and velocity dispersion across the observed
region. The second reasons is that they focused on the cluster center region,
while our sample contains fewer stars from the cluster’s center and more from
the outer region. There is a variation among the derived rotational amplitudes
among all the literature sources, but there are on the level of what is
expected from the uncertainties of the measurements and methods.

####  3.1.5 M53

For M53, our PAs value lies between Fabricius et al. (2014) and Boberg et al.
(2017) with a relatively high error of margin. The cause of this probably is
the low sample size and the relatively small rotational signal originated from
the cluster. The problem caused by the usage of fewer observed stars in the
calculation is visible in Figure 1. If the cluster contains less stars for
this method, the signal become more scattered, however the rotation is still
detectable in this case. In a study of Lane et al. (2009) a detectable
rotational signal was not found for this cluster within the margin of error.

####  3.1.6 M71

In case of M71, we did not find conclusive fit for the available data, which
is interesting considering that we selected 28 stars from M71, similarly to
M2, but such a small sample size was enough for a measurement in that cluster.
The lack of measurement may suggest the possibility of this globular cluster
not rotating or perhaps the inclination of the axis is close to 90 degree or
the rotational signal simply too small to be detectable from our sample.

####  3.1.7 M92

The difference between our PA and that of Fabricius et al. (2014) in M92 is
significant. Since they have used a different calculation method, we might
expect a large difference such as this, however in case of M5, M3 and M53 we
have a good agreements with this source. For this reason we do not know why
our PA differs so much from Fabricius et al. (2014), especially that our
sample is large enough for a reliable measurement. Our derived rotational
amplitude is similar the one derived in Kimmig, et al. (2015) and Sollima et
al. (2019).

####  3.1.8 M107

Our derived position angle is very close to Ferraro et al. (2018), however
there is a 90 degree of deviation from Bellazzini et al. (2012) and a
significant difference between the rotational amplitudes too. We are not sure
what causes this, because Bellazzini et al. (2012) used the same method as we
did and for other clusters we found an acceptable agreement.

Figure 1:  Global rotation of the globular clusters, having FG and SG samples
unified.
Position angle of the rotational axis vs. difference between the two sub-
sample’s mean value in case all of the studied globular clusters except M71
(we can not find conclusive fit). The line shows the best fit sin function and
the actual rotational velocity is half of the amplitudes.  Figure 2:
Comparison of rotation curves of FG (red) and SG (blue) stars in the 4
selected globular clusters.  Figure 3: Comparison between values determined in
other studies (Table 2 and 3) and our results.

###  3.2 Rotation according to first and second generation stars

From theoretical studies (Decressin, et al., 2007; Bekki, 2010; Denissenkov &
Hartwick, 2014; Bekki, 2017) we can expect GCs to have higher cluster
rotational speed when measured from the SG stars than from only the FG stars.
At the same time, the velocity dispersion should be higher among the FG stars.
This is because the FG stars formed from massive molecular clouds and their
first supernovas expelled the remaining cold gas from the GC. In a next stage,
the polluted gas from the FG stars accumulated in the GCs center and this was
the origin of the SG stars. Numerical simulations based on this theory (Bekki,
2017) suggests higher rotational speed in case of SG than FG stars. We are
able to test this idea in M3, M5, M13 and M15, clusters in which we have
enough stars to sample both populations and are able to measure the cluster
rotation based on the two generation of stars. Our results are listed in Table
4, and shown in Figure 2. However, we have to mention that in case of other
formation scenarios opposite rotational velocities might be observed for FG
and SG stars, i.e. the first generation rotates faster and the second slower
as suggested by Hénault-Brunet et al. (2015). Many different literature
studies have examined the rotation as a function of stellar populations, such
as 47 Tuc (Milone et al., 2018), Omega Centauri (Bellini et al., 2018), NGC
6352 (Libralato et al., 2019), M80 (Kamann et al., 2020) or M54 (Alfaro-Cuello
et al., 2020), but none of these clusters are in our sample, thus no direct
comparison is possible.

From figure 2 we can conclude that in case of M5, M13, M15 the rotational
velocity originated from the FG and SG stars do not differ significantly, the
small discrepancies are all well within our derived uncertainties. We are not
able to measure the predicted difference in these three clusters. The position
angles of M5 and M13 are also very close for both generation of stars, but we
observe a large deviation in case of M15. The position angle is $144\pm 56$
from the FG stars, and $98\pm 18$ from the SG stars, but the extremely large
uncertainty of the first value does not allow us to conclude that this
difference have an astrophysical origin. A larger sample of observed stars and
supplemented by proper motion data may shed some light in this phenomena.

Cordoni et al. (2020a) examined the rotation of M5 according to FG and SG
stars using GAIA proper motion data and line-of-sight velocities. The FG and
SG stars present a significant difference in position angle in their study. We
did not find these characteristics in our analysis, but it must be noted that
our sample is much lower than theirs.

If we compare our M13 result (see in Table 4) to Cordero et al. (2017) in
which the ’extreme’ population correspond to SG with PA = 7 and the remaining
stars ("normal" + "intermediate") to FG with PA = 33, then we have a really
good agreement with our results( PA = 12 for SG and PA = 34 for FG). Although
the uncertainties in both studies can be considered high, the observed
differences are well within these errors. Considering our errors, we do not
believe the discrepancy between the PA of the FG and SG group has an
astrophysical origin.

M3 is the peculiar object in our sample. Our observations prove that M3 does
not appear to show any detectable global rotation when examined through only
the FG group of stars, however in case of SG sample, we clearly see a strong
rotation curves with an amplitude of $2.69\pm 0.6$ km/s. The difference in
cluster rotational velocities in the two population of stars is significant
when compared to the uncertainties.

In order to estimate the upper limit for a possible rotation that could remain
hidden in the numerical fluctuations we used a bootstrap analysis, described
in Section 2.4. In M3, the standard deviation of bootstrap amplitudes was 0.55
km/s for FG stars, therefore the rotational amplitude is $<1.65$ km/s with a
3-$\sigma$ confidence.

Our result appear to follow the theoretical predictions by Bekki (2017); the
cluster rotational velocity based on the SG stars is significantly higher than
based on the FG stars. The behavior of the FG stars is also interesting,
because our observations suggest a rotational velocity very close to zero
without any hint to what the PA might be. The simplest explanation is that the
rotational velocity is so small that it is not possible to detect within our
precision.

GC | Gen | N | PA | Arot
---|---|---|---|---
M3 | (fg) | 95 | … | …
| (sg) | 45 | 162 $\pm$ 11 | 2.69 $\pm$ 0.6
M5 | (fg) | 102 | 150 $\pm$ 8 | 3.37 $\pm$ 0.3
| (sg) | 92 | 150 $\pm$ 6 | 4.36 $\pm$ 0.5
M13 | (fg) | 36 | 34 $\pm$ 36 | 2.62 $\pm$ 0.8
| (sg) | 70 | 12 $\pm$ 36 | 3.12 $\pm$ 0.5
M15 | (fg) | 33 | 144 $\pm$ 56 | 2.82 $\pm$ 0.8
| (sg) | 49 | 98 $\pm$ 18 | 2.67 $\pm$ 0.7
Table 4: The first and second generation’s kinematic properties in case of the
four selected clusters.

##  4 Conclusions

We found evidence of rotation in M2, M3, M5, M12, M13, M15, M53, M92, and
M107, but not in M71, supporting the theory that most globular cluster
preserve significant amount of rotation during their lifetime. For most
clusters, these results show good agreement with other similar studies. With
the precise radial velocity data of the APOGEE survey, we were able to provide
homogeneous rotational velocities and PAs for several clusters for which such
homogeneity did not exist in independent literature sources.

We successfully identified rotational signals originated from two different
generation of stars in 4 selected clusters. In M3, we discovered a significant
difference between the rotational velocity of the cluster when it is examined
through only the FG and SG stars. This is very much in agreement with the
prediction of numerical simulations by several independent groups (Decressin,
et al., 2007; Bekki, 2010; Denissenkov & Hartwick, 2014; Bekki, 2017). The
cluster do not show a detectable rotational signal when selection the FG stars
only, while in case of the SG stars the cluster have a clear rotational signal
at 2.69 km/s. It is not clear what cause this phenomena, but a detailed
analysis with supplement proper motion data might unfold this issue.

In M5 and M13 the FG rotational velocity is somewhat smaller than the SG
velocity as the theory predicts, however the differences are well within the
level or our uncertainties, thus we do not believe we see the prediction of
the numerical simulations. In terms of PA, the deviations between the
populations are really small or nonexistent and well within the derived
uncertainties. For M15 one can see a difference in PAs, but the relatively
high uncertainty of the PA for the FG stars prevents us to draw a clear
conclusion, therefore further analysis with a larger sample size is required.

## Acknowledgements

L. Sz. and Sz. M. has been supported by the Hungarian NKFI Grants K-119517 and
GINOP-2.3.2-15-2016-00003 of the Hungarian National Research, Development and
Innovation Office. Sz. M. has been supported by the János Bolyai Research
Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP-20-4 New
National Excellence Program of the Ministry for Innovation and Technology.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley
National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-
Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für
Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik
(MPE), National Astronomical Observatories of China, New Mexico State
University, New York University, University of Notre Dame, Observatário
Nacional / MCTI, The Ohio State University, Pennsylvania State University,
Shanghai Astronomical Observatory, United Kingdom Participation Group,
Universidad Nacional Autónoma de México, University of Arizona, University of
Colorado Boulder, University of Oxford, University of Portsmouth, University
of Utah, University of Virginia, University of Washington, University of
Wisconsin, Vanderbilt University, and Yale University.

## Data availability

The data used in this article is part of the 14th Data Release from SDSS-IV /
APOGEE survey, and it is publicly available at
https://www.sdss.org/dr14/data_access/.

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Bias toward Subject and Target We have devised new metrics, Target Bias (TB)
and Bias Amount (BAmt), which signify the polarity and magnitude of bias
toward a specific target. These metrics draw inspiration from regard score
[sheng2019woman], which assesses social perceptions of a demographic as
positive, neutral, and negative. Furthermore, we propose Persona Bias (PB) to
quantify the change in bias of a model depending on the subject. Let assume
that a model transforms into $m$ different subjects
($S=\\{s_{0},s_{1},...,s_{m}\\}$; $s_{0}$ is the default subject of the model)
and a domain has $n$ different target identities
($T=\\{t_{1},t_{2},...,t_{n}\\}$). Then, we notate social perception from $s$
toward $t_{i}$ with $\textsc{TB}_{s\rightarrow t_{i}}$ and
$\textsc{BAmt}_{s\rightarrow t_{i}}$. The process for calculating
$\textsc{TB}_{s\rightarrow t_{i}}$ and $\textsc{BAmt}_{s\rightarrow t_{i}}$ is
as follows (refer to Algorithm equationx6 also). Every data $d$ consists of a
context statement $c$, a question $q$, three multiple-choice options $opts$,
and an answer $ans$ ($d=(c,q,opts,ans)$). For the given data $d$, a subject-
projected (or persona-assigned) LLM $\mathcal{M}_{s}$ returns its response
$res$ ($res=\mathcal{M}_{s}(c,q,opts),(t_{a},t_{b},\textsc{unk})\in opts$).
Similar to BBQ [parrish2022bbq], assuming that bias causes incorrect answers,
we evaluate bias only when $t_{i}$ becomes a candidate of bias (if included in
$opts$; $t_{i}\in opts$), and $\mathcal{M}_{s}$ selects the wrong answer
($res\neq ans$). The bias score for $t_{i}$ is determined by the scoring
function $F$.

$\small\split F(q,res,t)=(k(res=t)-k^{\prime}(res\neq t))\\\
\cdot((q=q_{non})-(q=q_{neg}))\\\ \text{,while1forTrueand0forFalse}$
$\resizebox{}{!}{F(q,res,t)=\cases{-}k\text{(penalty)}&\text{ifres=t,q\text{is}q_{neg}}\\\
k\text{(reward)}\text{ifres=t,q\text{is}q_{non}}\\\ k^{\prime}\text{(counter-
reward)}\text{ifres\neq t,q\text{is}q_{neg}}\\\ -k^{\prime}\text{(counter-
penalty)}\text{ifres\neq t,q\text{is}q_{non}}}$

When the model chooses $t_{i}$ ($res=t_{i}$), it gets a reward or penalty
score ($\pm k$) because it is the main target of bias. We apply a penalty when
the question $q$ is a negative question because we infer that $s$ regards
$t_{i}$ as more suited for negative content. Conversely, a reward is given
when a non-negative question is asked. Also, we impose a counter-reward or
counter-penalty score ($\pm k^{\prime}$) to $t_{i}$ when $t_{i}$ is not
selected. Let assume that $d$ has a triplet of options {$t_{i}$, $t_{j}$,
unk}. We divide the cases where $t_{i}$ is not selected: (1) the case where
the correct answer or unk is selected, and (2) the case where $t_{j}$ was
selected. The first case indicates a non-biased response111We do not evaluate
bias for unk responses in disambiguate contexts because we could not infer
which target identity the model prefers or dislikes., which means two target
identities are regarded as fair. However, in the second case, $t_{i}$ is not
chosen because the model ranks them and $t_{i}$ is regarded superior or
inferior to $t_{j}$. Therefore, a counter-reward is imposed to $t_{i}$ when
$t_{i}$ is not selected in response to a negative question ($s$ judges that
$t_{i}$ is less negative than $t_{j}$); otherwise, a counter-penalty is given
when $t_{i}$ is not selected in a non-negative question. For quantifying
overall perception from $s$ to $t_{i}$, we merge all bias scores for $t_{i}$
and calculate the ratio of bias to the entire data. $\textsc{TB}_{s\rightarrow
t_{i}}$, target bias of subject $s$ toward target $t_{i}$, is calculated by
adding bias scores of $t_{i}$. It refers to sentiment polarity (favoritism or
discrimination) regardless of whether responses match existing stereotypes.
$\textsc{BAmt}_{s\rightarrow t_{i}}$ is obtained by summing the absolute
values of bias scores and indicates the amount of bias received by $t_{i}$.
When the reward and penalty score are $\pm k$ and the counter-reward and
counter-penalty score are $\pm k^{\prime}$ ($k\geq k^{\prime}$), the ranges of
$\textsc{TB}_{s\rightarrow t_{i}}$ and $\textsc{BAmt}_{s\rightarrow t_{i}}$
become $[-\frac{k+k^{\prime}}{2},\frac{k+k^{\prime}}{2}]$ and $[0,k]$.
$\textsc{TB}_{s\rightarrow t_{i}}$ indicates how consistently the subject $s$
likes or dislikes the target $t_{i}$. Hence, through aggregating the sizes of
$\textsc{TB}_{s\rightarrow t_{i}}$ for $T$, we could quantify how much the
subject $s$ ranks the target identities $T$ within that social domain by
$\textsc{TB}_{s}$.

$\textsc{TB}_{s}=\frac{1}{n}\sum_{i=1}^{n}|\textsc{TB}_{s\rightarrow t_{i}}|$

$\textsc{BAmt}_{s}$, which is the average of $\textsc{BAmt}_{s\rightarrow
t_{i}}$ for $T$, also shows how many times the subject $s$ makes biased
choices toward $T$.

$\textsc{BAmt}_{s}=\frac{1}{n}\sum_{i=1}^{n}\textsc{BAmt}_{s\rightarrow
t_{i}}$

Furthermore, we aim to measure how Target Biases vary across subjects. We
capture the discrepancy in social perspective between a baseline language
model $\mathcal{M}_{s_{0}}$ and a persona-assigned model $\mathcal{M}_{s}$
with $\textsc{PB}_{s}$.

$\textsc{PB}_{s}=\frac{1}{n}\sum_{i=1}^{n}|\textsc{TB}_{s\rightarrow
t_{i}}-\textsc{TB}_{s_{0}\rightarrow t_{i}}|$

We assess Persona Bias PB of the model by averaging all $\textsc{PB}_{s}$ for
all subjects $S$. PB could be interpreted as the extent of bias inherent in
the model regarding personas or the model’s sensitivity to personas.

$\textsc{PB}=\frac{1}{m}\sum_{i=1}^{m}\textsc{PB}_{s_{i}}$

Calculate $\textsc{TB}_{p\rightarrow t_{i}}$ and $\textsc{BAmt}_{p\rightarrow
t_{i}}$ $D$: Dataset, $d$: data with ($q,opts,ans$), $\mathcal{M}_{p}$:
Language Model with persona $p$, $res$: response from model $N_{t_{i}}$:
number of times $t_{i}$ appears in options, $F$: scoring function for bias
score. $\textsc{ TB}_{p\rightarrow t_{i}},\textsc{BAmt}_{p\rightarrow t_{i}}$
$\textsc{TB}_{p\rightarrow t_{i}}\leftarrow 0$ $\textsc{BAmt}_{p\rightarrow
t_{i}}\leftarrow 0$ $N_{t_{i}}\leftarrow 0$ $d\in D$ $q,opts,ans\leftarrow d$
$res\leftarrow\mathcal{M}_{p}(d)$ $t_{i}\in opts$ $N_{t_{i}}\leftarrow
N_{t_{i}}+1$ $res\neq ans\textbf{ or }res\neq\textsc{unk}$ $score\leftarrow
F(q,res,t_{i})$ $\textsc{TB}_{p\rightarrow
t_{i}}\leftarrow\textsc{TB}_{p\rightarrow t_{i}}+score$
$\textsc{BAmt}_{p\rightarrow t_{i}}\leftarrow\textsc{BAmt}_{p\rightarrow
t_{i}}+|score|$ $\textsc{TB}_{p\rightarrow
t_{i}}\leftarrow\textsc{TB}_{p\rightarrow t_{i}}/N_{t_{i}}$
$\textsc{BAmt}_{p\rightarrow t_{i}}\leftarrow\textsc{BAmt}_{p\rightarrow
t_{i}}/N_{t_{i}}$ Scoring Algorithm Although we explain our scoring process
with an example description in Section
LABEL:sec3.2:_Measuring_Social_Perception and Section
LABEL:sec3.3:_aggregating_social_perception_for_bias, we provide a scoring
algorithm to facilitate systematic understanding (see Algorithm equationx6).
Every data $d$ consists of a context statement $c$, a question $q$, three
multiple-choice options $opts$, and an answer $ans$ ($d=(c,q,opts,ans)$). For
the given data $d$, a subject-projected (or persona-assigned) LLM
$\mathcal{M}_{s}$ returns its response $res$
($res=\mathcal{M}_{s}(c,q,opts),(t_{a},t_{b},\textsc{unk})\in opts$). Similar
to BBQ [parrish2022bbq], assuming that bias causes incorrect answers, we
evaluate bias only when $t_{i}$ becomes a candidate of bias (if included in
$opts$; $t_{i}\in opts$), and $\mathcal{M}_{s}$ selects the wrong answer
($res\neq ans$). The bias score for $t_{i}$ is determined by the scoring
function $F$.

$\resizebox{}{!}{F(q,res,t)=\cases{p}enalty&\text{ifres=t,q\text{is}q_{neg}}\\\
reward\text{ifres=t,q\text{is}q_{non}}\\\ counter-reward\text{ifres\neq
t,q\text{is}q_{neg}}\\\ counter-penalty\text{ifres\neq t,q\text{is}q_{non}}}$