# Exploring the stellar rotation of early-type stars in the LAMOST Medium-
Resolution Survey. II. Statistics

Weijia Sun Kavli Institute for Astronomy & Astrophysics and Department of
Astronomy, Peking University, Yi He Yuan Lu 5, Hai Dian District, Beijing
100871, China  Key Laboratory for Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District,
Beijing 100012, China  Xiao-Wei Duan Kavli Institute for Astronomy &
Astrophysics and Department of Astronomy, Peking University, Yi He Yuan Lu 5,
Hai Dian District, Beijing 100871, China  Key Laboratory for Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
20A Datun Road, Chaoyang District, Beijing 100012, China  Licai Deng Key
Laboratory for Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China
School of Astronomy and Space Science, University of the Chinese Academy of
Sciences, Huairou 101408, China  Department of Astronomy, China West Normal
University, Nanchong 637002, China  Richard de Grijs Department of Physics and
Astronomy, Macquarie University, Balaclava Road, Sydney, NSW 2109, Australia
Research Centre for Astronomy, Astrophysics and Astrophotonics, Macquarie
University, Balaclava Road, Sydney, NSW 2109, Australia

###### Abstract

Angular momentum is a key property regulating star formation and evolution.
However, the physics driving the distribution of the stellar rotation rates of
early-type main-sequence stars is as yet poorly understood. Using our catalog
of 40,034 early-type stars with homogeneous $v\sin i$ parameters, we review
the statistical properties of their stellar rotation rates. We discuss the
importance of possible contaminants, including binaries and chemically
peculiar stars. Upon correction for projection effects and rectification of
the error distribution, we derive the distributions of our sample’s equatorial
rotation velocities, which show a clear dependence on stellar mass. Stars with
masses less than $2.5\,\mathrm{M_{\odot}}$ exhibit a unimodal distribution,
with the peak velocity ratio increasing as stellar mass increases. A bimodal
rotation distribution, composed of two branches of slowly and rapidly rotating
stars, emerges for more massive stars ($M>2.5\,\mathrm{M_{\odot}}$). For stars
more massive than $3.0\,\mathrm{M_{\odot}}$, the gap between the bifurcated
branches becomes prominent. For the first time, we find that metal-poor ([M/H]
$<-0.2$ dex) stars only exhibit a single branch of slow rotators, while metal-
rich ([M/H] $>0.2$ dex) stars clearly show two branches. The difference could
be attributed to unexpectedly high spin-down rates and/or in part strong
magnetic fields in the metal-poor subsample.

Stellar rotation (1629), Astronomy data analysis (1858), Early-type stars
(430), Stellar properties (1624), Stellar evolution (1599)

††facilities: LAMOST††software: PARSEC (1.2S; Bressan et al., 2012), Astropy
(Astropy Collaboration et al., 2013), Matplotlib (Hunter, 2007), TOPCAT
(Taylor, 2005)

##  1 Introduction

Stellar rotation profoundly affects almost every aspect of stellar evolution
(Maeder & Meynet, 2000). It plays a crucial role in dynamo-driven magnetic
activity (Spruit, 2002), stellar winds (Matt & Pudritz, 2005), surface
abundances (Preston, 1974), chemical yields (Prantzos et al., 2018), internal
structure (Aerts et al., 2019), and external structure (Rivinius et al.,
2013). Consequently, understanding stellar rotation is essential for our
understanding of the origin and evolution of stellar angular momentum.
However, the connection between models and observations, although much
improved in the last few decades, still suffers from several shortcomings.

From a theoretical perspective, stellar rotation on the main sequence (MS) is
the result of the long-term evolution of angular momentum loss and
redistribution, regulated by intricate processes early in the star formation
phases, e.g. the fragmentation of rotating clouds (Boss & Bodenheimer, 1979),
magnetic braking (Mouschovias & Morton, 1985), bipolar overflows (Pudritz,
1985), and magnetic locking with the accretion disk (Lin et al., 2011). Once
the initial rotation rate at the zero-age MS (ZAMS) has been established, the
rotation evolution can be described by a stellar rotation model that
encompasses the physical prescriptions of meridional circulation, angular
momentum transport, and (radiative, supra-Eddington, and mechanical) mass loss
(Ekström et al., 2012). From an observational perspective, the observed
stellar rotation rates are usually entangled with projection effects, which
further complicates the use of observed rotation rates as benchmark for
stellar rotation models.

The distribution of stellar rotation rates can serve as a crucial probe of the
formation and evolution processes of stellar populations. A sharp decline in
rotation velocities of MS stars is found around $1.2\,\mathrm{M_{\odot}}$,
with more massive stars having mean rotation velocities of hundreds of
$\mathrm{km\,s^{-1}}$, while lower-mass stars have velocities on the order of
a few $\mathrm{km\,s^{-1}}$ (Kraft, 1970). This is mainly so, because more
massive stars lack deep convective envelopes or strong magnetic fields that
slow down stellar rotation. Many studies (e.g., Abt & Morrell, 1995; Penny,
1996; Huang & Gies, 2006; Huang et al., 2010) have been dedicated to
deconvolve the observational distribution and derive a true rotational
distribution by means of large population statistics. A bimodal distribution
of rotation rates is observed among young solar-mass stars in the Orion nebula
(Attridge & Herbst, 1992; Stassun et al., 1999; Barnes, 2003). The origin of
such a bimodality is not yet fully understood, but it is thought to be a
manifestation of star–disk interactions during pre-MS evolution (Bouvier et
al., 1993).

A similar bimodal distribution is also observed among massive stars
($M\gtrapprox 2.5\,\mathrm{M_{\odot}}$), although the underlying mechanism
might not be the same. Abt & Morrell (1995) found a bimodal distribution of
rotation velocities among A-type MS stars in the field, where the slowly
rotating population was mainly ascribed to chemically peculiar (CP) stars. A
large-scale study by Zorec & Royer (2012), who reported $v\sin i$ measurements
for 2014 B6- to F2-type field stars, found a substantial difference between
the rotation distributions of stars with masses below and above
$2.5\,\mathrm{M_{\odot}}$. The less massive population exhibits a unimodal
profile, while the more massive population is bimodal. They also revealed the
complex behavior of stellar rotation, with strong acceleration in the first
one-third of the MS evolutionary phase, followed by a mild spin-down. This
calls for more efforts to improve our understanding of stellar rotation, in
terms of both theory and observations.

Significant progress in stellar rotation statistics is now achievable with
multi-object spectrographs acquiring data for large numbers of stars. Large
spectroscopic surveys such as the William Herschel Telescope Enhanced Area
Velocity Explorer (WEAVE, Dalton et al., 2012), the Galactic Archaeology with
High Efficiency and Resolution Multi-Element Spectrograph (GALAH, Buder et
al., 2021), the Radial Velocity Experiment (RAVE, Kunder et al., 2017), the
Apache Point Observatory Galactic Evolution Experiment (APOGEE, Majewski et
al., 2017), and the Large Sky Area Multi-Object Fiber Spectroscopic Telescope
(LAMOST, Cui et al., 2012; Liu et al., 2020) will continue to provide millions
of stellar spectra and, hence, allow to unveil an unprecedentedly detailed
view of stellar features. In Paper I (Sun et al., 2021), we compiled a catalog
containing more than 40,000 stars with effective temperatures of
$7000\,\mathrm{K}\leq T_{\mathrm{eff}}\leq 14,500\,\mathrm{K}$ from LAMOST
data release (DR) 7 medium-resolution spectra. With such a large and
homogeneous sample, we aim to review the rotation distribution of early-type
stars as function of their masses, ages, and metallicities.

This article is organized as follows. In Section 2, we analyze the
uncertainties in $v\sin i$ in more detail. The effects of possible
contamination and its influence on the derived rotation rates are discussed in
Section 3. Section 4 discusses our statistical approach to derive the true
rotational velocity distribution and we further apply this method to the
rotation rate distribution in Section 5. We also discuss the dependence of the
stellar rotation rates on metallicity, in Section 6. Finally, our conclusions
are summarized in Section 7.

##  2 Data

Our stellar sample was defined in Paper I, where we used a data-driven method
to estimate the stellar parameters and abundances (‘stellar labels’) of early-
type candidates selected from the H$\alpha$ and Mg i b line indices provided
by LAMOST DR7 medium-resolution spectra. Having adopted a selection based
mainly on effective temperatures ($7000\,\mathrm{K}\leq T_{\mathrm{eff}}\leq
14,500\,\mathrm{K}$) and data quality, we derived a final catalog composed of
40,034 stars. The precisions of our estimates are $\sim 75\,\mathrm{K}$, 0.06
dex, 0.05 dex, and $\sim 3.5\,\mathrm{km\,s^{-1}}$ for $T_{\mathrm{eff}}$,
$\log g$, [M/H], and $v\sin i$, respectively, for signal-to-noise ratios,
$\mathrm{SNR}>60$. However, this only represents the average behavior of our
estimates; the actual performance may vary (Sun et al., 2021). As regards the
observational uncertainties, they may originate from the line profiles of a
single exposure and/or from minor variations among multiple-epoch
observations.

Figure 1: Variations of $v\sin i$ estimates from single-epoch observations
with respect to the mean value. The data was analyzed using constant numbers
in each bin. Orange squares represent the 1$\sigma$ uncertainties, $\Delta\ln
v\sin i$, and red triangles represent the median value (bias). An inverse
proportional function (cyan dot-dashed line) is overplotted to fit the
standard deviations.

To investigate the dependence of the uncertainty on $v\sin i$, we used the
difference between the individual $v\sin i$ values from the single-epoch
observations and the associated mean value for the co-added spectra. There
ought to be only minor differences among the rotation velocities of single-
epoch exposures; their difference represents the uncertainty in our estimates.
Royer et al. (2002a) found that the error in $\ln(v\sin i)$ can be
approximated by a normal distribution, based on multi-epoch (48), high-SNR
($>250$) observations of Sirius. This is because fast rotation blends the
absorption lines and affects the accuracy of radial and rotation velocity
measurements. Therefore, the variation was calculated in logarithmic space
(see Figure 1). For $\ln(v\sin i)<4$ ($v\sin i<50\,\mathrm{km\,s^{-1}}$), the
standard deviation (orange squares) exhibits a dependence on the $v\sin i$
value. This is at odds with the result of Royer et al. (2002b, 2007), who
found a uniform error value on a logarithmic scale. This is due to the
differences between the methods used to estimate rotation velocities.

Because those earlier authors adopted the first zero of the Fourier transform
of the line profiles (Carroll, 1933; Gray, 1976) based on high-resolution
spectra ($R>20,000$) to determine $v\sin i$, their estimates are less affected
by resolution limitations. However, our data from the LAMOST MRS are
unavoidably further broadened due to the finite resolution power of the
instrument, which is known as instrumental broadening. The observed line
profiles are a convolution of the true profiles and the instrumental profile,
which makes it hard to estimate very slow rotation velocities and, hence,
results in larger uncertainties for slow compared with fast rotators.

Based on our estimates using a spectrograph with $R\sim 4000$ (Sun et al.,
2019), the detection limit for the MRS data ($R\sim 7500$) is around
$30\,\mathrm{km\,s^{-1}}$. For these slow rotators, the relative uncertainties
are comparable to the absolute values, and so such data should be used with
caution. We used an inverse proportional function to fit the standard
deviations. The median values of $\Delta\ln v\sin i$ are also presented as red
triangles in Figure 1. For stars with rotation velocities above the detection
limit, the relative bias—$\mathrm{Bias}(v\sin i)/v\sin i$—is smaller than 3%.

##  3 Contamination

The rotation behavior of our sample stars might not be solely the result of
their own evolution but it could also be affected by several external
processes. Therefore, possible polluters need to be discarded first. In this
section, we present two possible contaminants: binary and CP stars. We check
for the incidence of periodic variables and star cluster members in Section
3.3 and Section 3.4, respectively. We then discuss the different rotation
properties of these contaminants and the effects of possibly having missed
contaminants on the rotation distribution of ‘normal’ (single, non-CP, non-
variable, non-cluster member) stars. Following removal of all identifiable
contaminants, we are left with 33,655 normal stars (see Section 4.2).

Binaries, particularly close binaries, can drastically affect the rotation
properties of both of their components through tides, mass transfer, or even
mergers (de Mink et al., 2013). Tidal interactions tend to synchronize stellar
and orbital rotation, transferring the stellar spin angular momentum to the
orbital angular momentum of the binary system. Mass transfer from the donor
star can spin up the accreting star and cause it to approach the break-up
limit. Mergers may also change stellar rotation, although this is the least
well-understood process.

CP stars represent another possible contaminant type in our sample. They are
early-type MS stars exhibiting anomalous chemical abundances. According to
Preston (1974)’s classification, they can be divided into four subgroups:
metallic line (Am), magnetically peculiar (mAp), stars with enhanced Hg ii and
Mn ii (HgMn), and He-weak stars. Most CP stars exhibit slow rotation, but the
distribution may differ among subgroups. Abt & Morrell (1995) found that
virtually all Am and mAp stars have equatorial rotation velocities below
$120\,\mathrm{km\,s^{-1}}$.

###  3.1 Contamination by binary stars

If a binary system’s stellar rotation is modified by interactions between the
components, that should also affect the system’s radial velocity, $v_{R}$.
Variations in the latter could be identified through Doppler shifts in the
case of spectroscopic binaries. Ideally, such shifts should be periodic,
following the orbital period, which could be confirmed by means of multi-
epoch, high-cadence observations. However, only 1% of our sample objects have
been observed more than 10 times, while most ($>60$%) only have three epochs
of observations. Therefore, we used the variations in $v_{R}$ between single-
exposure spectra to tentatively infer binarity.

Relative variations were estimated following Xiong et al. (2021, in prep.). We
relied on relative variations to avoid systematic errors in the absolute
$v_{R}$ measures (Huang et al., 2018). In brief, we adopted the single-
exposure spectrum with the highest SNR as our spectral template and calculated
$\Delta v_{R}$ for the other spectra through cross-correlation. To
differentiate binaries from single stars, we introduced
$\sigma_{\mathrm{detect}}$ (Sana et al., 2013), i.e. the ratio of the relative
$\Delta v_{R}$ to its uncertainty. We considered a star to be a binary system
if any pair of single-exposure spectra met the following condition:

$\sigma_{\mathrm{detect}}=\frac{|v_{i}-v_{j}|}{\sqrt{\sigma^{2}_{i}+\sigma^{2}_{j}}}>C_{1}\quad\mathrm{and}\quad|v_{i}-v_{j}|>C_{2},$
(1)

where $|v_{i}-v_{j}|$ is the maximum difference between these measurements.
The latter criterion was introduced to exclude fake binary stars, i.e. stars
with intrinsic variability that may be caused by photospheric or wind effects.

We adopted ($C_{1}=4$, $C_{2}=16$) as our selection criteria for binary
systems. We detected 4955 binaries from a sample of 36,620 candidates
($N_{\mathrm{obs}}\geqslant 3$). The corresponding binary fraction for this
sample is $(13.5\pm 0.3)$%. This is lower than the binary fraction observed
for O-type (Sana et al., 2013) and early-B type stars (Dunstall et al., 2015;
Sana, 2017), but consistent with Guo et al. (2021)’s result for late-B and
early-A type stars. Note that a large fraction of binaries will remain
undetected. Guo et al. (2021) used a Monto Carlo method to correct for
observational biases and estimated that the intrinsic binary fraction for our
sample could be as high as about 40%. This type of analysis is not necessary
for the context in which we are looking for contaminants. Our method to detect
spectroscopic binaries is most sensitive to close binaries that could
influence the stellar rotation rates of their components. Any missing binaries
are mostly long-period systems, which are not expected to affect the rotation
rates of their individual components.

###  3.2 Contamination by CP Stars

Identification of CP stars relies on strong (or weak) absorption lines of
certain elements in their spectra. However, this is currently not feasible for
our sample. First, the wavelength coverage of the LAMOST MRS is rather narrow
and misses important lines that are used to identify anomalies (e.g., the Ca
ii H and K lines). However, a detailed analysis of the chemical abundances of
early-type stars based on our MRS spectra is beyond the scope of this study,
and we defer the identification of CP stars using the same data to future
papers.

Instead, we used the pre-compiled catalogs of Qin et al. (2019), Hümmerich et
al. (2020), and Paunzen et al. (2021) to identify Am, mAp, and HgMn stars (the
most frequent CP subgroups) in our sample. These three papers are all based on
previous LAMOST DRs, which hence enables us to better estimate the frequency
of CP-star occurrence. Qin et al. (2019) used a random forest machine-learning
algorithm trained on the sample from the study of Hou et al. (2015) to search
for Am stars from LAMOST DR5 high-SNR ($>50$) early-type LRS spectra and they
also used the equivalent width of the 4077 Å line, a blended line of Sr ii, Cr
ii, and Si ii, to label mAp candidates. We reselected our sample following
their selection criteria, and cross-matched our sample with their catalog. We
found 405 Am stars and 24 mAp candidates in a subsample of 8051 objects.
Hümmerich et al. (2020) and Paunzen et al. (2021) applied a different approach
based on MKCLASS to their DR4-based sample. We obtained 46 mAp and 12 HgMn
stars among 7676 candidates by cross-matching the DR4 data with their
catalogs. Note that the selection of the subsamples in the latter cases is not
the same as that described by Hümmerich et al. (2020) and Paunzen et al.
(2021), who used Gaia DR2 colors, as well as spectral types, to perform early-
type sample selection. We also cross-matched our sample with Renson & Manfroid
(2009), a catalog compiled from the literature. Ultimately, we identified 522
Am, 115 mAp, 13 HgMn, and 1 He-weak stars.

The apparent detection rate of Am stars in our subsample is around 5%, which
is similar to that reported by Qin et al. (2019). However, given that Am stars
are mostly found around $T_{\mathrm{eff}}\sim 7750\,\mathrm{K}$, the detection
rate based on a sample with a wide range of temparture does not properly
reflect the incidence. In the most likely temperature range, the frequency of
occurrence is more than 15%, but it is still lower than the values reported in
previous studies (Preston, 1974; Abt, 1981; Zorec & Royer, 2012). This is
probably due to Qin et al. (2019)’s strict screening, since they conducted
additional manual identification based on spectral features associated with
chemical peculiarity.

###  3.3 Contamination by periodic variables

Our comparison catalog of periodic variables was recently published by Chen et
al. (2020). The authors searched for and classified 781,602 variables into 11
main types based on the Zwicky Transient Facility (ZTF; Bellm et al., 2019)
DR2, which covers the northern sky. We also cross-matched our sample with a
combination of other catalogs, including the Wide-field Infrared Survey
Explorer (WISE, Chen et al., 2018), the Asteroid Terrestrial-impact Last Alert
System (ATLAS, Heinze et al., 2018), the All-Sky Automated Survey for
Supernovae (ASAS-SN, Jayasinghe et al., 2018), and the Catalina survey (Drake
et al., 2014). We found that our sample contained 325 periodic variables.

###  3.4 Contamination by cluster members

One of the LAMOST-II MRS survey’s science goals pertains to open clusters. The
survey’s target sample includes 18 open cluster-related fields, containing
several tens of open clusters (Liu et al., 2020). It has been reported that
the distribution of the rotation velocities of cluster members might be
different from that of field stars (Huang & Gies, 2006), in the sense that
cluster B-type stars contain fewer slow rotators relative to field B-type
stars. Although Huang et al. (2010) argued that the lower rotation rates of
the field stars are primarily the result of evolutionary spin-down changes, we
removed any cluster members to retain a clean sample of early-type field
stars. To do so, we adopted the cluster member catalog of Cantat-Gaudin &
Anders (2020), who applied a membership assignment procedure to the Gaia DR2
dataset and identified members of 1481 clusters. We found that our sample
included 726 cluster members, equivalent to less than 2% of the total sample.

###  3.5 $v\sin i$ contamination

Even though we have tried to filter out most contaminants, a fraction may have
been missed by our decontamination method, while the (poor) decontamination
completeness may compromise estimates of the rotation distribution. It is,
therefore, necessary to investigate the rotation properties of the
contaminants discussed in the previous sections.

Figure 2: $v\sin i$ distributions of (blue) normal stars, (cyan) binary stars,
(green) CP stars, (orange) periodic variables, and (red) cluster members.

In Figure 2, we present $v\sin i$ histograms for the normal stars as well as
the contaminants. Only CP stars (green) exhibit a significant deviation, in
the sense that they are mostly slow rotators. The other objects display
similar profiles as the normal stars (note that cluster members may exhibit a
slightly different $v\sin i$ distribution). As long as the $v\sin i$
contamination is similarly distributed as the $v\sin i$ distribution of the
normal stars, the overall rotation distribution will not be affected even if a
fraction of contaminants remains unidentified in our final sample. As for the
cluster subpopulation, we note that the completeness of this catalog should be
close to unity since it was derived using high-precision proper motion data
from Gaia.

Figure 3: $v\sin i$ as a function of stellar mass, $M$, for CP stars. The
corresponding distribution of normal stars is presented in the background. Am
and mAp stars are represented by blue and cyan circles, respectively. The
right panel shows histograms of the $v\sin i$ distribution of these two
subpopulations. The top panel shows the number fraction of CP stars in each
mass bin, from $1.6\,\mathrm{M_{\odot}}$ to $4.0\,\mathrm{M_{\odot}}$ with a
bin width of $0.2\,\mathrm{M_{\odot}}$.

CP stars tend to be slow rotators (e.g., Preston, 1974; Abt & Morrell, 1995).
Chemical peculiarities in Am and mAp stars are presumed to have been caused by
gravitational settling and radiative levitation, which can occur in stable
atmospheres associated with slow rotation rates. However, the existence of
slow rotators but without chemical anomalies challenges the connection between
their chemical peculiarities and stellar rotation. We confirm their special
rotation properties in Figure 3, where it is clear that most of CP stars have
small $v\sin i$ values, which is rather different from the behavior of the
normal stars in the $M$ versus $v\sin i$ diagram. Since most of the CP stars
detected are Am and mAp stars, only these two subtypes are shown. Am stars
(blue) span a narrow mass range from $1.5\,\mathrm{M_{\odot}}$ to
$2.0\,\mathrm{M_{\odot}}$; they are less massive than mAp stars. This is
consistent with the classification scheme of CP subtypes. In the right panel,
we present the normalized probability density function (PDF) of the $v\sin i$
values of Am and mAp stars. We note that 10–20% of these CP stars have
relatively large rotation velocities ($v\sin i>120\,\mathrm{km\,s^{-1}}$),
which was also noted by Netopil et al. (2017). The presence of such rapidly
rotating CP stars, if confirmed, would shed new light on the possible role of
stellar rotation in the context of CP star formation (Abt, 2009).

The top panel of Figure 3 shows the number fraction of CP stars
($N_{\mathrm{CP}}/N_{\mathrm{total}}$) in each mass bin from
$1.6\,\mathrm{M_{\odot}}$ to $4.0\,\mathrm{M_{\odot}}$ with a bin width of
$0.2\,\mathrm{M_{\odot}}$. The average CP fraction is around 1–2% across the
mass range, and any differences in CP fraction may be attributed to the
various subtypes of CP stars populating the different mass domains. However,
based on estimates of the frequency of CP stars by Preston (1974), Abt (1981),
and Zorec & Royer (2012), 15–30% of solar neighborhood stars ( covering
$7000\,\mathrm{K}<T_{\mathrm{eff}}<7800\,\mathrm{K}$) are CP stars. It thus
appears that a fraction of missing CP stars may remain undetected in our
sample, and they may bias the rotation distribution of the normal stars.
However, statistical corrections are difficult and imprecise, because the
observed incidence may depend on the temperature as well as the detection
method.

##  4 Distributions of equatorial rotation velocities

To obtain the equatorial rotation velocities, $v$, we need to correct the
projected rotation velocities $v\sin i$ for the effects of inclination, $i$.
For individual stars, this is only feasible in rare cases, e.g. for eclipsing
binaries (where strong tidal effects ensure that the rotation axis is parallel
to the orbital axis) and for stars with a rotational modulation signal in
their light curves, among others. However, we can correct for projection
effects statistically under the assumption of randomly oriented rotation axes
(Abt & Morrell, 1995; Royer et al., 2007; Zorec & Royer, 2012).

To estimate the distribution of the equatorial rotation velocities, we
followed Royer et al. (2007) and Zorec & Royer (2012), whose method can be
summarized in three steps:

  1. 1.

We first smooth the distribution using a Gaussian kernel. The selection of the
bandwidth of the kernel-smoothing window is done using Silverman’s rule of
thumb (Silverman, 1986) which minimises the mean integrated squared error

$h=\left(\frac{4\hat{\sigma}}{3n}\right)^{\frac{1}{5}}$ (2)

where $h$ is the bandwidth of the smoothing parameter, $\hat{\sigma}$ is the
standard deviation of the samples, and $n$ is the sample size.

  2. 2.

We then recover the PDF of the true projected rotation velocities,
$\theta\equiv\widehat{v\sin i}$, through rectification of the error law. As
our measurements are subject to random uncertainties, the observed $v\sin i$
represents the maximum of the probability distribution, whereas $\sigma_{v\sin
i}$ is the width of the distribution. Royer et al. (2007) assumed that the
error in $\ln(v\sin i)$ was normally distributed, with a mean of 0 and a
variance of $\left(\ln(1+\alpha)\right)^{2}$, where the coefficient
$\alpha=0.1$ is a constant. However, as shown in Figure 1, our estimates of
$v\sin i$ are susceptible to instrumental broadening for small $v\sin i$.
Therefore, $\alpha$ depends on the value of $v\sin i$, which must hence be
implemented into our calculation. The deconvolution is carried out using the
Lucy (1974) iterative technique.

Although our $v\sin i$ estimates are consistent with literature values up to
$\sim 400\,\mathrm{km\,s^{-1}}$, they could be affected by temperature and
gravity inhomogeneities introduced by rapid rotation. Gravity darkening
suppresses the equatorial region’s contribution to the rotational broadening,
which has been discussed extensively in the context of Be stars, i.e. B-type
stars exhibiting Balmer-line emission (e.g. Townsend et al., 2004).
Nonetheless, a correction for this underestimation is only necessary if the
rapid rotators represent a statistically significant fraction (Zorec & Royer,
2012).

  3. 3.

Following correction for the error law, the final step is to correct the
distribution for inclination effects. Under the assumption of a random
distribution of rotation axes, the PDF of the corrected projected rotation
velocities, $\Psi(\theta)$, becomes

$\Psi(\theta)=\int\gamma(v)P(\theta|v)\mathrm{d}v=\int\gamma(v)\frac{\theta}{v}\frac{H(v-\theta)}{\sqrt{v^{2}-\theta^{2}}}\mathrm{d}v,$
(3)

where $\gamma(v)$ is the true equatorial rotation velocity distribution and
$P(\theta|v)$ is the conditional probability. $H(x)$ represents the Heaviside
step function, which is 1 if $x\geq 1$ and otherwise 0.

Figure 4: Distributions of rotation velocities of normal stars in different
mass bins. (left) 1.8–$2.0\,\mathrm{M_{\odot}}$; (right)
2.6–$2.8\,\mathrm{M_{\odot}}$. The cyan histograms represent the observed
$v\sin i$ distribution using a bin width of $20\,\mathrm{km\,s^{-1}}$, while
the blue dotted lines are the corresponding smoothed distributions. The error-
rectified distributions are shown as the orange dashed lines. The final
products—projection-corrected equatorial velocity distributions—are shown as
the red solid lines.

We provide two examples for different mass ranges (from 1.8 to
$2.0\,\mathrm{M_{\odot}}$ and from 2.6 to $2.8\,\mathrm{M_{\odot}}$) to
illustrate the process of deconvolution: see Figure 4. The original observed
distribution of $v\sin i$ is shown as the cyan histograms with a bin width of
$20\,\mathrm{km\,s^{-1}}$. Its smoothed $v\sin i$ distribution, error-
corrected true projected rotation velocity ($\theta$) distribution, and
projection-rectified true equatorial rotation velocity ($v$) distribution are
represented by the blue dotted line, the orange dashed line, and the red solid
line, respectively.

These two mass ranges were selected to represent the typical situation in the
less massive ($M<2.5\,\mathrm{M_{\odot}}$) and more massive
($M>2.5\,\mathrm{M_{\odot}}$) regimes. Zorec & Royer (2012) found that less
massive stars have a unimodal equatorial velocity distribution, while more
massive stars trace a bimodal distribution. Our results are largely consistent
with previous results, except for several significant differences:

  * •

The velocity distribution of the less massive stars (Figure 4, left) shows a
peak velocity at $v\approx 110\,\mathrm{km\,s^{-1}}$, which is $\sim
40\,\mathrm{km\,s^{-1}}$ smaller than the peak velocity reported by Zorec &
Royer (2012) for the same mass range.

  * •

For the more massive stars, we found an excess of stars with $v\sin
i<100\,\mathrm{km\,s^{-1}}$, which results in a broad and rather flat
rectified distribution. This is different from the bimodal distribution found
by Abt & Morrell (1995) and Zorec & Royer (2012). The distribution in the
right panel of Figure 4 could be interpreted as showing two populations, with
one group representing slow rotators that share similar rotation velocities,
and a second group comprising fast rotators whose velocities are widely
spread. If this were correct, we could still claim a bimodal distribution for
our more massive sample. Note that the peak velocity of the slow rotators is
similar to that found by Zorec & Royer (2012). However, the number fraction of
slow rotators is rather distinctive in both papers. Zorec & Royer (2012) found
that the maximum PDF for slow rotators is smaller than that for fast rotators,
and the number fraction for the former population is less than 20%. However,
slow rotators provide a more distinct contribution to our sample.

  * •

Zorec & Royer (2012) claimed that the sum of two Maxwellian functions can fit
the rectified distributions. They used the derived Maxwellian functions to
quantify the typical velocities and number fractions of slow and fast
rotators. This was based on Deutsch (1967), who showed that the distribution
of $v$ can be described by Maxwellian functions. However, we note that this
practice may not always yield a reliable representation of the $v$
distribution, especially for the rapidly rotating population.

We suggest that the difference in peak velocity for the less massive stars and
the excess of slow rotators among more massive stars can be attributed to
missed contamination by CP stars. As discussed in Section 3.5, CP stars are
mainly slow rotators and we expected to have $\sim 15$% of such stars
misclassified as normal stars. Given these missing CP stars, the peak velocity
of the low-mass stars could be biased toward the lower end and a possibly
bimodal distribution could be hidden. This may also explain an excess of slow
rotators compared with previous studies.

However, recent papers have suggested the existence of a population of slowly
rotating normal early-type stars (Sikora et al., 2020; Qin et al., 2021). Most
of these studies are based on photometric variability caused by the
inhomogeneous structure of the stellar surface, which could be used to infer
the rotation period. Qin et al. (2021) searched for low-velocity rotating
normal (non-CP) A-type stars from the LAMOST–Kepler project and found a
significant fraction of slow rotators among normal A-type stars. Such a
contradiction is also reported for the 30 Doradus starburst region in the
Large Magellanic Cloud (LMC). The O-type stellar sample (Ramírez-Agudelo et
al., 2013) shows a similar rotation distribution as the right panel of Figure
4), exhibiting an excess of slow rotators, while the B-type stellar sample
(Dufton et al., 2013) present a bimodal population of very slow and fast
rotators, with few stars rotating at the low-velocity peak seen for the O-type
stars. These controversies may suggest that previous studies of the rotation
properties of early-type stars may have missed a significant number of slow
rotators.

##  5 Distributions of rotation rates

For a single star that is not affected by binary interactions, we can
generalize the formalism of the observed rotation velocity, $v$, by writing
out its dependence on mass and stellar age:

$v(M,\omega_{\mathrm{init}},t/t_{\mathrm{MS}}),$ (4)

where the stellar age is expressed as the relative age and the initial
rotation rate is given by
$\omega_{\mathrm{init}}=\Omega/\Omega_{\mathrm{crit}}$ (Georgy et al., 2013).
The parameter $\Omega$ is the angular velocity and $\Omega_{\mathrm{crit}}$ is
the critical velocity where the surface gravity can no longer maintain
equilibrium with the centrifugal motion. Since $v$ changes as a function of
time, and also depends on the stellar mass, the parameter $v$ itself is not
suitable for comparison among samples with different properties.

Figure 5: $v/v_{\mathrm{crit}}$ evolution (Georgy et al., 2013) for different
initial rotation rates (0.1, 0.3, 0.5, 0.6, 0.7, 0.8). Solid lines represent a
$2\,\mathrm{M_{\odot}}$ star of solar abundance, dotted lines represent a
$4\,\mathrm{M_{\odot}}$ star of solar abundance, while dashed lines represent
a $2\,\mathrm{M_{\odot}}$ star of $0.4\,\mathrm{Z_{\odot}}$. The region where
$v/v_{\mathrm{crit}}$ shows a strong dependence on relative age is marked in
gray. For $0.1\leq t/t_{\mathrm{MS}}\leq 0.9$, $v/v_{\mathrm{crit}}$ could be
used as a proxy for the initial rotation rate.

We suggest this dependence on mass and age can be alleviated by introducing
$v/v_{\mathrm{crit}}$ (e.g. Netopil et al., 2017). For slow rotators, this
ratio is almost constant, particularly for rigid rotators on the MS (Zorec &
Royer, 2012). We illustrate this by exploring the evolutionary models of the
Geneva group (Ekström et al., 2012; Georgy et al., 2013). In Figure 5, we show
the $v/v_{\mathrm{crit}}$ evolution for different initial rotation rates. The
ratio is almost constant for $0.1\leq t/t_{\mathrm{MS}}\leq 0.9$ if
$\omega<0.5$. For fast rotators, the ratio exhibits a mild increase with time.
This is to account for the enhanced ‘mechanical’ mass loss in the equatorial
region as a star reaches the critical velocity. Even though the
$v/v_{\mathrm{crit}}$ ratio unavoidably depends on evolution, it is still a
good proxy for the initial rotation rates and only weakly dependent on mass,
age, or metallicity. The earliest phase, $t/t_{\mathrm{MS}}\leq 0.1$,
corresponds to the time taken to reach a quasi-equilibrium state, while the
late phase at $t/t_{\mathrm{MS}}\geq 0.9$ represents the star’s deceleration
as it evolves off the MS.

Therefore, we apply a similar analysis as in Section 4 to the
$v/v_{\mathrm{crit}}$ ratio. Considering the significant $v/v_{\mathrm{crit}}$
variations that occur during the early and late phases of MS evolution, we
limit our analysis to stars characterized by $0.1\leq t/t_{\mathrm{MS}}\leq
0.9$, which covers 86% of our sample. The critical velocity was adopted from
Georgy et al. (2013) by interpolating the grid of masses and relative ages for
a given metallicity. Note that rapid rotation can increase the MS lifetime
(Georgy et al., 2013) by transporting fresh hydrogen into the core. Depending
on the initial rotation rate, the stellar lifetime could be extended by up to
30%. This was not considered in estimating the ages for our sample based on
the non-rotating stellar evolution models. Nevertheless, estimates of the
relative age, which represents the fraction of a star’s lifetime spent on the
MS, is not affected by this effect.

Figure 6: Distribution of $v/v_{\mathrm{crit}}$ for normal stars as a function
of stellar mass, color-coded by density in the one-dimensional normalized
distribution. Iso-density contours are shown as white curves.

An overview of the distribution of $v/v_{\mathrm{crit}}$ as a function of mass
for normal stars is displayed in Figure 6. Using the same color scale as
before, the figure is color-coded by the relevant number density pertaining to
the one-dimensional normalized distribution.

Figure 7: As Figure 6, but for a subsample with SNR $>50$ in the g band,
overlapping with LAMOST DR5. This subsample reproduces the parent sample of
Qin et al. (2019), Hümmerich et al. (2020), and Paunzen et al. (2021) and
reduces the impact of contamination by CP stars by generating a cleaner sample
than that used for Fig. 6.

For less massive stars ($M<2.5\,\mathrm{M_{\odot}}$), the distribution of
$v/v_{\mathrm{crit}}$ is unimodal, with a significant concentration around the
peak velocity, particularly in the mass range $1.8\,\mathrm{M_{\odot}}\leq
M\leq 2.0\,\mathrm{M_{\odot}}$. The velocity ratio exhibits a clear dependence
on stellar mass, in the sense that it increases from $\sim 0.3$ to $\sim 0.55$
as mass increases, which is consistent with Zorec & Royer (2012). However, a
lack of slow rotators ($v\leq 100\,\mathrm{km\,s^{-1}}$) previously reported
for this mass range is not seen in our data. Since Zorec & Royer (2012) found
that slow rotators are underrepresented even when they are not removed from
the sample, they claimed that this lack is unlikely due to contamination by CP
stars. Instead, they attributed it to sampling effects in the observed volume.
However, in our sample we note that there are fewer slow rotators following
our selection of a subsample with SNR $>50$ in the g band and overlapping with
LAMOST DR5 to reproduce the parent sample of Qin et al. (2019), Hümmerich et
al. (2020), and Paunzen et al. (2021). This way, we can reduce the impact of
contamination by CP stars by generating a cleaner sample than that used for
Figure 6. However, this relative absence of slow rotators only affects stars
with $v/v_{\mathrm{crit}}<0.2$ ($v\leq 70\,\mathrm{km\,s^{-1}}$); a
significant fraction of stars with $70\,\mathrm{km\,s^{-1}}\leq v\leq
100\,\mathrm{km\,s^{-1}}$ remains. Therefore, we conclude that, even if the
reported relative lack of stars is real, it must be affected by contamination
by CP stars but does not extend to $v\sim 100\,\mathrm{km\,s^{-1}}$.

This trend between mass and $v/v_{\mathrm{crit}}$ continues to develop as
stars become more massive than $2.5\,\mathrm{M_{\odot}}$ and may continue up
to $\sim 3.0\,\mathrm{M_{\odot}}$, where a mild turnover/upper limit is seen.
The branch of slow rotators becomes prominent for $M>3.0\,\mathrm{M_{\odot}}$,
but the bimodality is already present for $M>2.5\,\mathrm{M_{\odot}}$. The
reason why the slowly rotating branch is not prominent in this figure is that
it covers an extended distribution across $v/v_{\mathrm{crit}}$. In fact,
Zorec & Royer (2012) found that a large fraction of stars have intermediate
rotation velocities compared with the slow or fast rotators, and it is not
until $M>3.0\,\mathrm{M_{\odot}}$ that a gap between the slowly and rapidly
rotating branches becomes apparent. The connecting structure is better visible
in Figure 7, where the slowly rotating branch has bifurcated from the rapidly
rotating branch at $M\sim 2.5\,\mathrm{M_{\odot}}$. For masses greater than
$\sim 3.25\,\mathrm{M_{\odot}}$, the gap between the slowly and rapidly
rotating branches is well-established.

As stellar masses increase to beyond $\sim 3.6\,\mathrm{M_{\odot}}$, our
analysis suffers from poor sampling in the $\log T$ vs $\log L$ parameter
space (Figure 5 Paper I). Massive MS stars in early evolutionary phases are
missing in our sample. Nevertheless, we can make an educated guess of their
properties based on the available information. If we assume that younger stars
behave similarly as their older counterparts, it appears that more stars
reside on the slowly rotating branch compared with stars with masses in the
range $2.5\,\mathrm{M_{\odot}}\leq M\leq 3.25\,\mathrm{M_{\odot}}$.

A similar bimodal distribution in rotation period has been observed for late-
type field dwarfs (e.g. McQuillan et al., 2013), mainly by the Kepler and K2
missions (Borucki et al., 2010). Gordon et al. (2021) detected a gap in the
rotation periods for $0.57\,\mathrm{M_{\odot}}<M<0.76\,\mathrm{M_{\odot}}$
based on a sample of $\sim$70,000 K2 targets and attributed it to a departure
from the Skumanich (1972) spin-down law. As a young star with its envelope
initially decoupled from its core loses angular momentum due to magnetic
braking, it could slow or even halt its spin-down if the core and envelope
begin to recouple at a later time. Despite the differences between our and
their samples in terms of the mass regime covered, such a mechanism might also
occur in early-type stars (e.g. Zorec & Royer, 2012).

##  6 Dependence on metallicity

Other than on mass, stellar rotation also depends on metallicity (e.g.,
Ekström et al., 2008). The meridional circulation cells that carry angular
momentum from the inner regions to the surface become less efficient at lower
metallicity. In addition, stellar winds also cause angular momentum losses.
Wind strengths are greater for higher metallicities. The models of Georgy et
al. (2013) and Szécsi et al. (2015) suggest that normally evolving stars with
low mass loss (e.g., low metallicities) may increase their rotation during MS
evolution and that the rotation distribution of early-type MS stars in lower-
metallicity environments might be different from that in higher-metallicity
environments. However, thus far, such differences have not been observed
conclusively. Studies of rotational velocities of O- and B-type stars in the
LMC and the Small Magellanic Cloud (SMC) (e.g., Penny et al., 2004; Penny &
Gies, 2009; Dufton et al., 2019) found little compelling evidence for
differences with metallicity. Hunter et al. (2008) found that massive stars
(with masses greater than $8\,\mathrm{M_{\odot}}$) in the SMC rotate faster
than those in the solar neighborhood. (There is no significant difference
between the distribution of rotation rates in the Galaxy and the LMC.)
However, the analysis was hindered by field star contamination (similar to the
situation in Section 3.4) because of the complicated composition of their
sample, which includes target stars projected toward young clusters and a
fraction of field stars.

To address the possible metallicity dependence, we selected metal-poor
($\mathrm{[M/H]}<-0.2\,\mathrm{dex}$) and metal-rich
($\mathrm{[M/H]}>0.2\,\mathrm{dex}$) subsamples. Each comprises more than 5000
normal stars to ensure sufficient sample sizes for statistical analysis. Using
the same procedures as discussed in Section 5, we analyzed their rotation
distributions as a function of stellar mass.

Figure 8: As Figure 6, but for different metallicity ranges. (top) Metal-poor
($\mathrm{[M/H]}<-0.2\,\mathrm{dex}$); (bottom) Metal-rich
($\mathrm{[M/H]}>0.2\,\mathrm{dex}$).

The top and bottom panels of Figure 8 show the rotational profiles for metal-
poor and metal-rich stars, respectively. There is a significant difference
between both distributions, in the sense that metal-poor stars only exhibit a
single branch of slow rotators, with $v/v_{\mathrm{crit}}$ centered at
$0.3$–$0.4$, while metal-rich stars clearly show two branches of the stellar
velocity ratio. The rapidly rotating branch is similar to the feature we saw
in Figures 6 and 7, but only for $M\geq 2\,\mathrm{M_{\odot}}$. The velocity
ratio of the less massive, metal-rich stars peaks at $v/v_{\mathrm{crit}}\sim
0.2$. This is much smaller than previously observed for our entire sample.
Moreover, the slowly rotating branch is also located at lower velocity ratios
than that of the metal-poor subsample.

The shape and location of the velocity distribution pertaining to the metal-
rich subsample could be affected by slowly rotating CP stars. Since most CP
stars are metal-rich (e.g. Qin et al., 2019), there is a larger fraction of
missing CP stars in this metal-rich subsample compared with both the entire
sample and the metal-poor subsample, where contamination is expected to be
minimal. Netopil et al. (2017) found a linear correlation between the masses
and rotation rates of mAp stars, with $v/v_{\mathrm{crit}}$ increasing from
0.08 to 0.17 as mass increases from $\sim 1.8\,\mathrm{M_{\odot}}$ to $\sim
4.5\,\mathrm{M_{\odot}}$ ($\sim 30\%$ scatter). This could be the reason for
the low rotation rate of the slowly rotating branch as seen for the metal-rich
subsample. Note that the incidence of CP stars is not large enough to
completely explain the rotation characteristics for $M\leq
2\,\mathrm{M_{\odot}}$. Thus, among less massive stars (which are most
numerous), the rotation rates of normal metal-rich stars are lower than those
for the metal-poor stars. The contribution of the rapidly rotating branch
becomes obvious for $M\geq 2.5\,\mathrm{M_{\odot}}$.

The lack of a rapidly rotating branch for the metal-poor subsample is
intriguing. One possible explanation is that this feature is determined within
the first few Myr of a star’s lifetime through ‘disk locking’ (e.g. Bouvier et
al., 1993). Magnetic interactions between a star and its accretion disk can
remove angular momentum and form intrinsically slow rotators (Lin et al.,
2011), and the longer disk lifetime results in slow rotation velocities on the
MS (Bastian et al., 2020). However, Yasui et al. (2009, 2010) found that the
near-infrared disk fraction of low-metallicity stars declines rapidly on
timescales $<1\,\mathrm{Myr}$, much faster than the $\sim 6\,\mathrm{Myr}$
observed for solar abundance. If this were to also apply to our metal-poor
subsample, and the initial rotation rates are determined by this mechanism,
the metal-poor subsample should rotate faster than their metal-rich
counterparts. This may explain the difference in velocity ratio in the slowly
rotating branch between the metal-poor and metal-rich subsamples, but not the
absence of a rapidly rotating branch.

Another possible mechanism to slow down stellar rotation at birth is magnetic
braking (Meynet et al., 2011). The coupling of wind material and magnetic
field lines that extend well beyond the stellar surface exerts a torque on the
stellar surface layers and thus slows down rotation. Unlike magnetic fields in
late-type stars, which are thought to be generated through dynamo action,
magnetic fields in early-type stars are likely of fossil origin (Donati &
Landstreet, 2009). Sanyal et al. (2017) proposed that high-mass ($M\geq
25\,\mathrm{M_{\odot}}$) stars develop envelope inflation during their
lifetime on the MS, which is favorably correlated with metallicity. Therefore,
stronger fossil fields are expected for massive stars with lower metallicity
than for higher-metallicity stars under the fundamental physics law of flux
conservation, where the surface magnetic field strength is expected to
decrease as the stellar radius expands. If confirmed, magnetic braking could
be more prominent in the metal-poor subsample. However, the incidence rate of
magnetic fields in early-type MS stars is around 10% within the solar
neighborhood (Sikora et al., 2019), and thus the lack of a rapidly rotating
branch cannot be explained exclusively by magnetic braking.

Figure 9: Histograms of the relative ages of (cyan) all normal stars, (blue)
the metal-poor subsample ($\mathrm{[M/H]}<-0.2\,\mathrm{dex}$), and (red) the
metal-rich subsample ($\mathrm{[M/H]}>0.2\,\mathrm{dex}$).

Instead, the lack of a rapidly rotating branch could also be related to the
fact that the metal-poor subsample is generally much older than the full
sample (see Figure 9). If the overall spin down of metal-poor stars is
stronger than predicted by stellar evolutionary models and more (specific)
angular momentum is lost during the MS phase, the rapidly rotating branch of
the older metal-poor subsample may have been moderated and may even have
merged with the slowly rotating branch. This would make it appear as a single
population with slightly higher rotation rates than characteristic of the
slowly rotating branch.

##  7 Conclusions

Using the catalog of Paper I, in this paper we have studied the statistical
characteristics of stellar rotation in the MS evolutionary phase. The
distribution of the equatorial rotational velocities has been estimated by
rectifying the error distribution and correcting for projection effects. We
also introduced the ratio $v/v_{\mathrm{crit}}$ to alleviate the dependence of
the rotational velocity on mass and age.

Our main results are the following:

  * •

The velocities of the less massive stars ($M<2.5\,\mathrm{M_{\odot}}$) exhibit
a unimodal distribution, with their peak velocity ratio increasing from 0.3 to
0.5 as stellar mass increases.

  * •

There is no clear sign of a lack of slow rotators for $M\sim
1.9\,\mathrm{M_{\odot}}$. Even if contamination by CP stars is better
corrected for, the gap is smaller than previously reported.

  * •

A bimodal rotation distribution emerges as stars increase in mass
($M>2.5\,\mathrm{M_{\odot}}$). The gap between slow and fast rotation becomes
prominent for $M>3.0\,\mathrm{M_{\odot}}$. Compared with the less massive
stars, the rapidly rotating branch exhibits a milder increase with stellar
mass and a turnover at $M\sim 3.0\,\mathrm{M_{\odot}}$.

  * •

For stars more massive than $3.6\,\mathrm{M_{\odot}}$, more stars reside on
the slowly rotating branch compared with stars with masses in the range
$2.5\,\mathrm{M_{\odot}}\leq M\leq 3.25\,\mathrm{M_{\odot}}$.

  * •

Metal-poor ([M/H] $<-0.2$ dex) stars only exhibit a single branch of slow
rotators, with $v/v_{\mathrm{crit}}$ centered at 0.3–0.4, while metal-rich
([M/H] $>0.2$ dex) stars clearly show two branches. The difference could be
attributed to unexpectedly high spin-down rates in the metal-poor subsample.
Strong magnetic fields associated with stars with low metallicity could partly
explain the result.

  * •

The bulk of the metal-poor stars are characterized by higher rotation rates
than the metal-rich stars.

  * •

Roughly 10% of CP (Am and mAp) stars exhibit abnormally high rotational
velocities.

Our understanding of early-type stellar rotation could be improved in two
ways: (1) currently, sample decontamination largely depends on external
catalogs, whose homogeneity and completeness are difficult to assess. This
caveat is partially caused by the insufficiently high SNR, limited number of
observation epochs, and narrow wavelength coverage of our current
spectroscopic survey. In the future, armed with more observations collected by
LAMOST-MRS, combined with LAMOST-LRS, we can improve the catalog and apply a
more consistent decontamination approach. (2) The description of stellar
rotation ($v/v_{\mathrm{crit}}$) still depends on the stellar evolutionary
models adopted. Our sample’s homogeneity and size allow us to quantitatively
study the rotation distribution as a function of stellar mass, age, and
metallicity. For instance, a variate-controlled sample could be used to test
the characteristics of stellar evolution on the MS, which in turn could help
refine and determine the free parameters remaining in stellar rotation models.
Further detailed analysis of the present stellar sample and models of rotating
stars will be pursued in forthcoming papers.

L. D. acknowledges research support from the National Natural Science
Foundation of China through grants 11633005, 11473037, and U1631102. The
Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic
Telescope; LAMOST) is a National Major Scientific Project built by the Chinese
Academy of Sciences. Funding for the project has been provided by the National
Development and Reform Commission, China. LAMOST is operated and managed by
the National Astronomical Observatories, Chinese Academy of Sciences. This
work has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC;
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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