# Stellar atmospheric parameters of $\sim$ 11,000 RR Lyrae stars from LAMOST
Spectra

Jiangtao Wang  CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, People’s Republic of China  Jianrong Shi  CAS
Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Beijing 100101, People’s Republic of China  School of Astronomy and Space
Science, University of Chinese Academy of Sciences, Beijing 100101, People’s
Republic of China  Jianning Fu  Institute for frontiers in astronomy and
astrophysics, Beijing Normal University, Beijing 102206, China  Department of
Astronomy, Beijing Normal University, Beijing 100875, People’s Republic of
China  Weikai Zong  Institute for frontiers in astronomy and astrophysics,
Beijing Normal University, Beijing 102206, China  Department of Astronomy,
Beijing Normal University, Beijing 100875, People’s Republic of China
Chunqian Li  CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, People’s Republic of China  School of Astronomy
and Space Science, University of Chinese Academy of Sciences, Beijing 100101,
People’s Republic of China

###### Abstract

Accurate determination of the stellar atmospheric parameters of RR Lyrae stars
(RRLs) requires short individual exposures of the spectra to mitigate
pulsation effects. We present improved template matching methods to determine
the stellar atmospheric parameters of RRLs from single-epoch spectra of LAMOST
(Large Sky Area Multi-Object Fiber Spectroscopic Telescope, also known as the
Guoshoujing telescope). We determine the radial velocities and stellar
atmospheric parameters (effective temperature: $T_{\rm eff}$, surface gravity:
$\log{g}$, and metallicity: [M/H]) of 10,486 and 1,027 RRLs from 42,729 low-
resolution spectra (LRS) and 7,064 medium-resolution spectra (MRS) of LAMOST,
respectively. Our results are in good agreement with the parameters of other
databases, where the external uncertainties of $T_{\rm eff}$, $\log{g}$, and
[M/H] for LRS/MRS are estimated to be 314/274 K, 0.42/0.29 dex, and 0.39/0.31
dex, respectively. We conclude with the variation characteristics of the
radial velocities ($RV$) and stellar atmospheric parameters for RRLs during
the pulsation phase. There is a significant difference of $28\pm 21$ km/s
between the peak-to-peak amplitude ($A_{\mathrm{ptp}}$) of $RV$ from H$\alpha$
line ($RV_{\rm H\alpha}$) and from metal lines ($RV_{\rm metal}$) for RRab,
whereas it is only $4\pm 17$ km/s for RRc. The $A_{\mathrm{ptp}}$ of $T_{\rm
eff}$ is $930\pm 456$ and $409\pm 375$ K for RRab and RRc, respectively. The
$\log{g}$ of RRab show mild variation of approximately $0.23\pm 0.42$ dex near
the phase of $\varphi=0.9$, while that of RRc almost remains constant. The
[M/H] of RRab and RRc show a minor variation of about $0.25\pm 0.50$ and
$0.28\pm 0.55$ dex, respectively, near the phase of $\varphi=0.9$. We expect
that the determined stellar atmospheric parameters would shed new light on the
study of stellar evolution and pulsation, the structure of the Milky Way, as
well as other research fields.

RR Lyrae variable stars; spectroscopy; Stellar atmospheric parameters

##  1 Introduction

The RR Lyrae stars (RRLs) are low-mass, old (with ages exceeding 10 Gyr), and
generically metal-poor (Walker, 1989; Savino et al., 2020; Mullen et al.,
2022). They reside at the intersection of the Classical Instability Strip (IS)
and the horizontal branch on the Hertzsprung-Russell diagram (HRD), having
evolved to stages with both helium core and hydrogen shell burning. Their
pulsations are driven by the $\kappa$ mechanism, which operates in the He II
partial ionization regions (Cox et al., 1973; Kiriakidis et al., 1992). RRLs
are large-amplitude variables with periods ranging from about 0.2 to 1.0 days
and pulsations amplitudes of about $0^{m}.2$ to $2^{m}.0$ in the visual band.
Based on the pulsation modes, RRLs are classified as RRab for the radial
fundamental mode, RRc for the first overtone radial mode, and RRd when both
modes are present (Soszyński et al., 2011).

Since RRLs have high brightness ($M_{\rm V}\sim 0.65$ mag, Muraveva et al.,
2018) and distinctive light curves, they can be easily identified. A large
number of RRLs have been discovered through time-domain photometric surveys in
recent years. In particular, the high-precision light curves provided by
Kepler (Koch et al., 2010) have revealed many new and unpredictable pulsation
phenomena in RRLs (Kolenberg et al., 2010b, 2011). For example, the discovery
of low-amplitude modes and period-doubling has enabled the study of nonlinear
and non-radial astroseismology (Molnár et al., 2012; Netzel & Smolec, 2022),
providing new insights into the physical mechanisms of the century-old
mysterious Blazhko effect (Szabó et al., 2010; Kolláth et al., 2011), which is
the periodic modulation of amplitude and/or phase in the light curves(Blažko,
1907).

The spectral observation of RRLs is quite limited compared with the photometry
surveys, and the radial velocities and stellar atmospheric parameters
determined through spectroscopy are invaluable for investigating the
structure, chemical and kinematic properties of the Milky Way (Ablimit & Zhao,
2017; Wang et al., 2022; Liu et al., 2022). However, the determination of
stellar atmospheric parameters for RRLs through spectra is a challenging task.

The large amplitude of expansion and contraction of RRLs lead to very drastic
variations in the stellar physical parameters during the pulsation cycle,
especially in effective temperature (Bono et al., 1995; For et al., 2011).
Consequently, large-aperture telescopes are required to obtain high signal-to-
noise (SNR) spectra for short exposures to weaken the pulsation effect,
resulting in high-resolution spectra (HRS) for only a small number of RRLs.
For example, Crestani et al. (2021) and Kovacs & Jurcsik (2023) have collected
only 162 and 269 samples of RRLs with HRS, respectively. Although the
metallicity can be estimated from Fourier parameters of the light curves
(Jurcsik & Kovacs, 1996; Mullen et al., 2021) and the $\Delta$S method from
the low-resolution spectrum (Preston, 1959; Wallerstein et al., 2012), there
is still a lack of the determination of $T_{\rm eff}$ and $\log{g}$.

In recent years, a multitude of extensive spectroscopic surveys have been
initiated, including RAVE (Radial Velocity Experiment, Steinmetz, 2003;
Steinmetz et al., 2020), BRAVA (The Bulge Radial Velocity Assay, Rich et al.,
2007; Kunder et al., 2020), SEGUE (Sloan Extension for Galactic Understanding
and Exploration, Yanny et al., 2009), APOGEE (Apache Point Observatory
Galactic Evolution Experiment, Majewski et al., 2017; Abdurro’uf et al.,
2022), GALAH (GALactic Archaeology with HERMES, De Silva et al., 2015; Buder
et al., 2021), and LAMOST (Large Sky Area Multi-Object Fiber Spectroscopic
Telescope, also known as the Guoshoujing telescope, Cui et al., 2012). These
surveys provide a valuable opportunity to obtain the stellar atmospheric
parameters for a large sample of RRLs. However, a significant limitation
arises when the parameter is derived from co-added spectra taken across
multiple individual exposures. This method can introduce phase contamination,
particularly when observations occur over different phases (For et al., 2011).
Also, the intrinsic swift variations in the atmospheres of RRLs challenge
conventional methods, which typically assume static atmospheres (Kolenberg et
al., 2010a). This is particularly evident in the distorted line profiles
observed at the peak luminosity phase (Gillet et al., 2019). For instance, as
shock waves traverse the atmosphere during specific phases, they can induce
broadening, doubling, or even the disappearance of H$\alpha$ or metal lines in
the majority of RRab (Preston, 2009; Gillet & Fokin, 2014; Yang et al., 2014;
Benkő et al., 2021; Preston et al., 2022). Such phenomena are also noted,
albeit less frequently, in RRc and RRd (Duan et al., 2021).

The LAMOST is capable of performing simultaneous spectroscopic observations of
4,000 targets (Cui et al., 2012). The LAMOST low-resolution regular survey,
which started in October 2012, classified observations into V-, B-, M-, and
F-plate modes based on the target magnitude, with exposure times of 10, 25,
30, and 30 minutes respectively, as detailed in Luo et al. (2015). In
September 2018, LAMOST began the medium-resolution survey during bright
nights, which adopts time-domain (TD) and non-time-domain (NT) modes for
distinct scientific goals, with a uniform exposure duration of 20 minutes (Liu
et al., 2020). For instance, the second phase of the LAMOST-$Kepler/K2$
project aims for at least 60 observations covering 50,000 interesting stars
within the $Kepler$ and $K$2 fields (Zong et al., 2020). At present, LAMOST
DR10111www.lamost.org has released more than 20 million spectra. Therefore, we
employ single-epoch of LRS and MRS from LAMOST to determine the stellar
atmospheric parameters of RRLs.

The structure of the paper is as follows. Section 2 presents a sample of RRLs
collected from multiple photometric survey projects and cross-matches these
with LAMOST spectra. The methodology used to determine the stellar atmospheric
parameters of RRLs from LAMOST spectra is introduced in Section 3. In Section
4, we analyze the internal and external uncertainty of the radial velocities
and stellar atmospheric parameters, as well as present the variations
characteristics of these parameters during the pulsation cycle. The properties
of stellar atmospheric parameters of RRLs and their application prospects are
discussed in Section 5. A brief summary is provided in Section 6.

##  2 Sources and Data

###  2.1 RR Lyrae sample

Over the past decade, a large number of RRLs have been found from the
explosive development of TD photometry surveys. Considering the areas of the
LAMOST observations (Dec. $\geq-10\arcdeg$), we collect RRLs samples from the
following four surveys.

  * •

$Gaia$: The $Gaia$ space telescope, launched successfully in December 2013,
aims to provide accurate positional, photometric, and astrometric parameters
for over a billion stars (Gaia Collaboration et al., 2016, 2021). The RRLs
catalog of $Gaia$ DR3 contains 270,905 RRLs, classified through the $Gaia$ DR3
multi-band ($G$, $BP$, and $RP$ band) light and radial velocity curves using
the SOS Cep&RRL pipeline (Clementini et al., 2022).

  * •

ASAS-SN: The ASAS-SN (The All-Sky Automated Survey for Supernovae, Jayasinghe
et al., 2018) is an all-sky optical survey project, that has confirmed about
1.5 million variable stars to date, of which $\sim$ 45,065 are RRLs (Christy
et al., 2023).

  * •

ZTF: The ZTF (The Zwicky Transient Facility, Bellm et al., 2019) is an optical
survey project focused on the Northern sky. Its primary scientific objectives
include the study of transient objects, stellar variability, and solar system
science. A total of 46,393 RRLs have been classified using the light curves
from ZTF DR2 (Chen et al., 2020).

  * •

PS1: The PS1 (The Pan-STARRS1 survey, Kaiser et al., 2010; Chambers et al.,
2016) is a multi-epoch and multi-color photometry project with a wide field of
view. The PS1 3$\pi$ survey is the largest sub-project within it, observing
the Northern sky with declinations greater than -30°. Approximately 239,044
RRLs candidates are obtained using data from the PS1 3$\pi$ survey. (Sesar et
al., 2017).

Although these four surveys provide a large sample of RRLs, there is an
overlap in targets between those catalogs. Therefore, we cross-matched these
catalogs with $Gaia$ DR3, and then obtained the total RRLs sample without
overlap by the $Gaiadr3.gaia\\_source$ ($Gaiaid$). The criterion for cross-
matching is to select the smallest coordinate separation ($\Delta d$) for the
same target within the range where $\Delta d\leq 1.0$ arcsec. We have
collected about 449,093 unique RRLs candidates, of which 174,030 are located
within the LAMOST field. We prioritize the completeness of the sample over the
purity of the RRLs sample, so that more RRLs spectra can be obtained from
LAMOST. Thus, the number of RRLs candidates is not constrained by the
probability of classification from the four catalogs. The probabilities from
the four catalogs for the sample of RRLs are provided in our results. Figure 1
illustrates the sky position of the 174,030 RRLs (gray dots).

Figure 1: The sky coverage of RRLs sample. The gray dots represent the RRLs
sample from five surveys located in the LAMOST field (Dec. $\geq-10\arcdeg$).
The blue dots and red crosse represent RRLs sample with LAMOST LRS and MRS,
respectively.

###  2.2 LAMOST Spectroscopy

The LRS (Resolution, $R\sim 1800$) published by LAMOST is a one-dimensional
spectrum with wavelength- and flux-calibration, covering the wavelength range
of ($3700\leq\lambda\leq 9000$ Å), and is co-added from the spectra observed
at the same night (Luo et al., 2015). Therefore, we chose the single-epoch
LRS, which is processed through the LAMOST 2D pipeline, as detailed in Bai et
al. (2021). The single-epoch LRS is a one-dimensional wavelength-calibrated
spectrum containing the blue- ($3700\leq\lambda\leq 5900$ Å) and red-arms
($5700\leq\lambda\leq 9000$ Å). The single-epoch MRS ($R\sim 7500$) contains
the blue- and red-arm with wavelength range covers of $4950\leq\lambda\leq
5350$ Å and $6300\leq\lambda\leq 6800$ Å, respectively (Luo et al., 2018; Liu
et al., 2020).

We employ the TOPCAT software (Taylor, 2005) to cross-match the coordinates
between the RRL sample from survey projects and the LRS and MRS datasets from
LAMOST DR10. The criterion for overlap is based on the closest target with a
coordinate separation of $\Delta d\leq 3.7$ arcseconds (Wang et al., 2020) and
SNR $\geq$ 10\. We identify potential contaminants for the RRLs spectra within
a range of $\Delta d\leq 3.7$ arcseconds based on the criterion of $G_{\rm
other}\leq G_{\rm RRLs}-1$, where $G_{\rm other}$ and $G_{\rm RRLs}$ represent
the magnitudes of potential contaminating targets and the RRLs sample,
respectively. The number of potential contaminants is provided in our results.

The position in the sky for RRLs sample with LRS and MRS is shown in Figure 1.
The distribution of $Gaia$ $G$ band magnitude for our 10,883 RRLs sample,
along with their 49,793 LAMOST spectra, is illustrated in Figure 2. The $G$
magnitude of the 42,729 LRS for 10,486 RRLs is mainly distributed in $G\sim
13^{m}-18^{m}$, while that of the 7,064 MRS of 1,027 RRLs is mainly
distributed in $G\sim 13^{m}-15^{m}$.

Figure 2: The $G$ magnitudes distribution with a bin width of $0.2^{m}$ for
our RRLs sample and their LAMOST spectra. The right panel shows the LAMOST
spectra and the left panel shows our RRLs sample.

##  3 Methodology

###  3.1 Synthetic spectra

The templates are created with three free stellar physical parameters, i.e.
$T_{\rm eff}$, $\log{g}$, and [M/H], and the ranges and intervals of these
parameters are shown in the first two columns of Table 1. The synthetic
spectra are generated by the iSpec software of the python framework (Blanco-
Cuaresma et al., 2014), where the calculations are performed using the
SPECTRUM code (Gray & Corbally, 1994) with the Kurucz stellar atmosphere
models (Kurucz, 2005), the atomic line-lists of the Vienna Atomic Line
Database (VALD3, Kupka et al., 2011) and the solar abundances from Asplund et
al. (2009) have been adopted. We assume a limb-darkening coefficient of 0.6,
and the enhancement of the $\alpha$ element is calculated using Equation 7.12
from Gray (2008). We calculated 5824 synthetic spectra for each resolution of
$R=1800$ and $R=7500$. The last two columns of Table 1 are the resolution and
wavelength range of synthetic spectra for the LAMOST LRS and MRS,
respectively.

Table 1: The properties of the template library. Parameters | range | step | LRS | MRS
---|---|---|---|---
$T_{\rm eff}$ (K) | 5500 - 8000 | 100 |  $R=1800$ $3600\leq\lambda\leq 9100$ Å |  $R=7500$ $4850\leq\lambda\leq 5450$ Å $6200\leq\lambda\leq 6900$ Å
$\log{g}$ (dex) | 1.4 - 4.0 | 0.2
[M/H] (dex) | -3.0 - 0.0 | 0.2

###  3.2 Spectra reduce and Radial velocities

The LRS and MRS are processed using LASPEC pipeline (Zhang et al., 2021),
which includes the removal of cosmic rays, normalization, and radial
velocities ($RV$) determination. The continuous of normalization is obtained
by least-squares fitting of a 6th-order polynomial. Figure 3 displays the blue
and red arms of the normalized LRS and MRS for RRab type EZ Cnc, the SNR of
the spectra are 100 and 50, respectively. The edge bands of blue- and red-arm
for LRS are not shown because of the high noise. The Balmer series and Mg Ib
triplet lines of the LRS and MRS are labeled with cyan lines and text.

In order to compare directly with the template, we determine the $RV$ of LRS
and MRS, respectively, through the cross-correlation function (CCF) in the
LASPEC pipeline. Duo to the intrinsic large amplitude expansion and
contraction of RRLs, stellar atmospheres display non-synchronous motions. As a
result, the velocities of lines originating from various atmospheric altitudes
differ significantly (Gillet et al., 2019). Consequently, we derive the $RV$
of both Balmer and metal lines separately. For LRS, we have determined
$RV_{\rm[l,H\alpha]}$, $RV_{\rm[l,H\beta]}$, $RV_{\rm[l,H\gamma]}$,
$RV_{\rm[l,H\delta]}$, and $RV_{\rm[l,metal]}$, which represent the $RV$ of
H$\alpha$, H$\beta$, H$\gamma$, and metal lines, from the segments of
$S_{\rm[l,H\alpha]}$ ($6450\leq\lambda\leq 6700$ Å), $S_{\rm[l,H\beta]}$
($4700\leq\lambda\leq 5000$ Å), $S_{\rm[l,H\gamma]}$ ($4250\leq\lambda\leq
4400$ Å), $S_{\rm[l,H\delta]}$ ($4000\leq\lambda\leq 4250$ Å), and
$S_{\rm[l,metal]}$ ($5000\leq\lambda\leq 5700$ Å), respectively. For MRS, we
have determined $RV_{\rm[m,H\alpha]}$ and $RV_{\rm[m,metal]}$, which represent
the $RV$ of H$\alpha$ and metal lines, from $S_{\rm[m,H\alpha]}$
($6500\leq\lambda\leq 6650$ Å) and $S_{\rm[m,b]}$ ($5000\leq\lambda\leq 5300$
Å) segements, respectively. The RRLs are generally metal-poor, so the strong
lines of the Mg Ib triplets primarily contribute to the $RV_{\rm metal}$
determined from the $S_{\rm[l,metal]}$ of LRS and the $S_{\rm[m,b]}$ of MRS,
as illustrated in Figure 3.

Figure 3:  The single exposure LRS and MRS of RRab EPIC 212182292. The top two
panels illustrate the segments from the red and blue arms of the LRS, and the
bottom two panels depict those from the MRS. The Balmer lines and Mg Ib
triplets of LRS and MRS are highlighted with cyan lines.

###  3.3 Stellar atmospheric parameters

In order to weaken the influence of pulsation, we determine the stellar
atmospheric parameters of RRLs using an improved template matching method,
described as follows:

  * •

Select segments

The Balmer lines are very sensitive to variations in temperature, and have
little dependence on variations in $\log{g}$, making them a good indicator of
$T_{\rm eff}$. The Mg Ib triplets, covered by LRS and MRS, are very sensitive
to variations in $\log{g}$, and their profiles depend little on pulsations.
Therefore, the $T_{\rm eff}$, $\log{g}$, and [M/H] are determined by the
Balmer series, Mg Ib triplet, and metal lines, respectively. For LRS, we
determine $T_{\rm eff}$ from the $S_{\rm[l,H\alpha]}$, $S_{\rm[l,H\beta]}$,
$S_{\rm[l,H\gamma]}$, and $S_{\rm[l,H\delta]}$, while $\log{g}$ and [M/H] from
the $S_{\rm[l,metal]}$. For MRS, we estimate $T_{\rm eff}$ from the
$S_{\rm[m,r]}$ ($6300\leq\lambda\leq 6800$ Å), while $\log{g}$ and [M/H] from
the $S_{\rm[m,b]}$.

  * •

Maske the line cores of the Balmer series

The Kurucz stellar atmospheric model is not considered the chromosphere,
leading to inadequate line core fitting. Additionally, line profiles are
easily distorted near the phase of peak luminosity, for instance the emission
and doubling of the H$\alpha$ (Gillet & Fokin, 2014; Benkő et al., 2021).
Therefore, the core of Balmer series lines obtained by Gaussian fitting below
the half height is masked.

  * •

Initial and final stellar atmospheric parameters

In order to avoid the influence of $RV$ difference between different lines, we
shift the Balmer series and metal segments in the velocity space of their
corresponding lines before directly comparing them with the templates. The
difference between the observed and synthetic spectra is described by the sum
of squares of comparing residuals $\chi^{2}$. $\chi^{2}$ is defined as:

$\chi^{2}=\frac{1}{N-1}\sum^{N}_{i=1}\frac{(O_{i}-T_{i})^{2}}{\sigma^{2}_{i}}$
(1)

where $O_{i}$ and $T_{i}$ are the flux of the target and synthetic spectra at
the $i$th data point, respectively. $\sigma_{i}$ and $N$ are the flux error of
the $i$th data point and the total number of data points, respectively. The
determination of initial stellar atmospheric parameters is first to determine
$T_{\rm eff}$ through the segment of Balmer lines, and then to determine
$\log{g}$ and [M/H] through the segment of metal lines after fixing $T_{\rm
eff}$. The templates are then focused around the triple grid interval of the
initial stellar atmospheric parameters to simultaneously determine the final
$T_{\rm eff}$, $\log{g}$, and [M/H]. Each parameter is derived using a
weighted average based on an optimal template selected by $\chi^{2}\leq
a\cdot\chi^{2}_{\mathrm{min}}$. Where $\chi^{2}_{\mathrm{min}}$ is the minimum
$\chi^{2}$ and $a=1.01$ is a constant chosen to strike a balance between
computational efficiency and accuracy. The weights are calculated using the
formula $w=1-\frac{\chi^{2}-\chi^{2}_{\rm min}}{\chi^{2}_{\rm
max}-\chi^{2}}\times 0.5$, where, $\chi^{2}_{\rm max}$ is the maximum
$\chi^{2}$ in the optimal model (Xiang et al., 2015).

We determined the $RV$ of different lines and the stellar atmospheric
parameters for LRS and MRS, which are provided in Table 2 and 3. The full
tables can be downloaded at the LAMOST DR10 value-added catalogs
website222www.lamost.org.

Table 2: The radial velocities and stellar atmospheric parameters of LAMOST
LRS for 10,486 RRLs.

$Gaiaid$ | obsid | R.A. | DEC. | period | $\varphi$ | Type | HJD | SNRg | $RV_{\rm[l,metal]}$
---|---|---|---|---|---|---|---|---|---
|  |  |  | day |  |  | $-2450000.5$ | km/s |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
$RV_{\rm[l,H\alpha]}$ | $RV_{\rm[l,H\beta]}$ | $RV_{\rm[l,H\gamma]}$ | $RV_{\rm[l,H\delta]}$ |  $T_{\rm eff}$a |  $\log{g}$a | [M/H]a | Ref. | flag |
km/s | km/s | km/s | km/s | K | dex | dex |  |  |
(11) | (12) | (13) | (14) | (15) | (16) | (17) | (18) | (19) |
316067369162854528 | 159108049 | 01:38:06.34 | +33:00:02.41 | 0.652606 | 0.66 | RRab | 6560.7535 | 36.76 | -109$\pm$3
-89$\pm$2 | -103$\pm$2 | -110$\pm$3 | -120$\pm$4 | 6221.0$\pm$98 | 2.37$\pm$0.48 | -2.71$\pm$0.13 | $Gaia$ | [0.99,-3.72,0,0] |
3694931747383350272 | 203012066 | 12:20:28.08 | -01:33:13.73 | 0.592236 | 0.57 | RRab | 6659.8917 | 26.16 | 170$\pm$3
187$\pm$3 | 170$\pm$2 | 171$\pm$2 | 150$\pm$6 | 6369.0$\pm$95 | 2.41$\pm$0.50 | -1.00$\pm$0.11 | $Gaia$ | [0,0,0.99,0] |
4006725443696129024 | 508001157 | 12:09:49.73 | +27:33:13.37 | 0.343563 | 0.82 | RRc | 7756.9049 | 10.24 | -103$\pm$33
-197$\pm$7 | -171$\pm$4 | -176$\pm$14 | -156$\pm$10 | 7286.0$\pm$161 | 3.47$\pm$0.43 | -0.58$\pm$0.24 | $Gaia$ | [0,-10.51,0.93,0] |
… | … | … | … | … | … | … | … | … | …
… | … | … | … | … | … | … | … | … |

Note. — This table is available in its entirety in machine-readable form. a
The error represents the standard deviation of the parameters of the optimal
template obtained in template matching. The description of our catalog： (1)
$Gaiaid$: The $Gaia$ source names from $Gaia$ DR3 RRLs catalog; (2) obsid: The
unique identification number ID of the LAMOST spectrum; (3) R.A.: The input
right ascension (epoch J2000.0) to which the fiber was pointed (in
hh:mm:ss.ss); (4) Dec.: The input declination (epoch J2000.0) to which the
fiber was pointed (in dd:mm:ss.ss); (5) Period: The period of RRLs comes from
the catalog in Ref.; (6) $\varphi$: The phases derived from the $t_{0}$ and
period provided through the catalog in Ref.; (7) Type: The classifications of
RRLs provided by the catalog of Ref.; (8) HJD: The Heliocentric Julian Date
(HJD -2450000.5) at mid-exposure; (9) SNRg: The signal-to-noise ratio (SNR) of
the LRS in the SDSS g band, which is an indicator for the quality of the
spectrum; (10) $RV_{\rm[l,m]}$: The radial velocities of metal lines
determined from metal segment ($S_{\rm[l,metal]}$); (11)
$RV_{\rm[l,H\alpha]}$: The radial velocities of H$\alpha$ line determined from
H$\alpha$ segment ($S_{\rm[l,H\alpha]}$); (12) $RV_{\rm[l,H\beta]}$: The
radial velocities of H$\beta$ line determined from H$\beta$ segment
($S_{\rm[l,H\beta]}$); (13) $RV_{\rm[l,H\gamma]}$: The radial velocities of
H$\gamma$ line determined from H$\gamma$ segment ($S_{\rm[l,H\gamma]}$); (14)
$RV_{\rm[l,H\delta]}$: The radial velocities of H$\delta$ line determined from
H$\delta$ segment ($S_{\rm[l,H\delta]}$); (15) $T_{\rm eff}$: The effective
temperature; (16) $\log{g}$: The surface gravity; (17) [M/H]: The metallicity;
(18) Ref.: The catalog in which the period and $t_{0}$ are obtained, including
$Gaia$: Clementini et al. (2022), ASAS-SN: Christy et al. (2023), ZTF: Chen et
al. (2020), PS1: Sesar et al. (2017); (19) flag: A list containing the values
of four parameters. The first three values denote the probabilities of RRLs
provided by ASAS-SN, ZTF, and PS1 in the respective references, where a value
of 0 signifies that the target is not included in the reference catalog. The
last value indicates the count of contaminated targets within the $\Delta
d\leq 3.7$ arcsec.

Table 3: The radial velocities and stellar atmospheric parameters of LAMOST
MRS for 1,027 RRLs.

$Gaia$ id | obsid | R.A. | Dec. | period | $\varphi$ | Type | HJD
---|---|---|---|---|---|---|---
|  |  |  | day |  |  | $-2450000.5$
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
SNRb | $RV_{\rm[m,metal]}$ | $RV_{\rm[m,H\alpha]}$ | $T_{\rm eff}$ | $\log{g}$ | [M/H] | Ref. | Flag
| km/s | km/s | K | dex | dex |  |
(9) | (10) | (11) | (12) | (13) | (14) | (15) | (16)
1376147275855922176 | 84338052 | 15:43:10.94 | +36:53:33.97 | 0.51198 | 0.46 | RRab | 8567.7684
31.31 | -58.8$\pm$0.2 | -66.1$\pm$0.4 | 6040.0$\pm$89 | 2.62$\pm$0.65 | -1.20$\pm$0.10 | $Gaia$ | [0,-45.71,0,0]
675464975453569664 | 84803006 | 07:54:08.77 | +24:10:42.63 | 0.53255 | 0.94 | RRab | 8890.6552
13.46 | -122.6$\pm$1.2 | -132.0$\pm$2.0 | 7445.0$\pm$50 | 3.23$\pm$0.17 | -1.24$\pm$0.09 | $Gaia$ | [0.99,0,0.75,0]
… | … | … | … | … | … | … | …
… | … | … | … | … | … | … | …

Note. — This table is available in its entirety in machine-readable form.

##  4 Uncertainty analysis

###  4.1 Internal uncertainty

Due to the intrinsic variability of the physical parameters of the RRLs, the
internal uncertainty ($\sigma_{\mathrm{in}}$) of these parameters is not
estimated by comparing results from multiple observations of targets.
Therefore, we estimate the $\sigma_{\mathrm{in}}$ of the $RV$ using the
Mentor-Carlo error calculated by LASPEC (Zhang et al., 2021). Furthermore, we
introduce artificial noise to the high-SNR spectra to the star with multiple
observations at the same phase. We then compare the parameters determined from
the spectra with different levels of artificial noise to estimate the
$\sigma_{\mathrm{in}}$ of the stellar atmospheric parameters.

The $\sigma_{\mathrm{in}}$ of $RV$ ($\sigma$($RV$)in) as a function of SNR is
illustrated in Figure 4. As predicted, the $\sigma$($RV$)in decreases with the
increase of SNR. The $\sigma_{\mathrm{in}}$ of $RV_{\rm[l,metal]}$,
$RV_{\rm[l,H\delta]}$, $RV_{\rm[l,H\gamma]}$, $RV_{\rm[l,H\delta]}$, and
$RV_{\rm[l,H\alpha]}$ are 11, 13, 10, 7, and 7 km/s, respectively, as
indicated by the fitted red curves at a SNR${}_{g}\sim$ 10 shown in Figure 4
(a)–(e). When SNRg $\geq$ 20, the $\sigma_{\mathrm{in}}$ of these $RV$
decreases to 5, 6, 5, 4, and 4 km/s. In Figure 4 (f) and (g), the
$\sigma_{\mathrm{in}}$ of $RV_{\rm[m,metal]}$ and $RV_{\rm[m,H\alpha]}$ are
1.7 and 2.8 km/s, respectively, for a SNRb of 10. When SNR${}_{b}\geq$ 50,
their $\sigma$($RV$)in decrease to 0.3 and 0.6 km/s, respectively, and remains
nearly constant. This result is consistent with that of Liu et al. (2019), who
found a precision of 0.3 km/s for $RV$ at SNR $\sim$ 50.

Figure 4: The internal uncertainty of radial velocities for LRS and MRS as a
function of SNR. The color represents the normalized number of parameters in
the boxes. The panels (a)–(e) are the internal uncertainty of
$RV_{\rm[l,metal]}$, $RV_{\rm[l,H\delta]}$, $RV_{\rm[l,H\gamma]}$,
$RV_{\rm[l,H\beta]}$, and $RV_{\rm[l,H\alpha]}$ determined from
$S_{\rm[l,metal]}$, $S_{\rm[l,H\delta]}$, $S_{\rm[l,H\gamma]}$,
$S_{\rm[l,H\beta]}$, and $S_{\rm[l,H\alpha]}$, respectively. The panels (f)
and (g) are the internal uncertainty of $RV_{\rm[m,metal]}$ and
$RV_{\rm[m,H\alpha]}$ determined from $S_{\rm[m,b]}$ and $S_{\rm[m,H\alpha]}$,
respectively. In each panel, the red line indicates the median within bins
with a SNR of 5. The accompanying text in each panel specifies the $\sigma$
for $RV$ SNR values of 10, 20, and 50, respectively.

We randomly add noise to the 134 LRS and 76 MRS selected based on a criterion
with a SNR higher than 100 and determine their stellar atmospheric parameters.
The internal uncertainty is derived using the equation
$\sigma_{\mathrm{in}}^{i}(P)=|(P_{i}-\bar{P}|$, where $i=1,2,...,n$ represents
the $i$th set of values of the parameter $P_{i}$, $n=20$ denotes the total
number of added noise sets for the same LRS/MRS, and $\bar{P}$ is the weighted
(SNR) average of $n$ sets of $P$. Figure 5 present the $\sigma_{\mathrm{in}}$
of $T_{\rm eff}$, $\log{g}$, and [M/H] as a function of SNR. The
$\sigma_{\mathrm{in}}$ of stellar atmospheric parameters for LRS and MRS
decrease with the increase of SNR. The red line represents the fitted curve
obtained from the reciprocal function of
$\sigma_{\mathrm{in}}=\frac{a}{\mathrm{SNR}}+b$, and the corresponding fitted
coefficients are presented in Table 4.1. The fitting results are valid with
SNR $\leq 50$, because the $\sigma_{\mathrm{in}}$ is constant when the SNR is
higher than 50. For SNR $\sim$ 10, the internal uncertainty of $T_{\rm eff}$,
$\log{g}$, and [M/H] for LRS/MRS are 189/109 K, 0.70/0.32 dex, and 0.52/0.33
dex, respectively, and they reach a stable level of 13/8 K, 0.07/0.03 dex, and
0.05/0.02 dex when SNR $\geq 50$.

Table 4: The coefficients of the optimal fit to estimate the $\sigma_{\mathrm{in}}$ for the stellar atmospheric parameters.  | LRS |  | MRS
---|---|---|---
| $a$ | $b$ |  | $a$ | $b$
$T_{\rm eff}$ | 2202.8 | -31.2 |  | 1269.2 | -17.7
$\log{g}$ | 7.9 | -0.1 |  | 3.6 | 0.0
[M/H] | 6.0 | -0.1 |  | 3.8 | 0.0
Figure 5:  The internal uncertainty of $T_{\rm eff}$, $\log{g}$, and [M/H] is
determined from LRS ( left panels (a)-(c)) and MRS (right panels (d)-(f)) as a
function of SNR. The color represents the normalized number of parameters in
the boxes. The red curve represents the median of the best fit as a reciprocal
function of the SNR, and its values at SNR = 10, 20, and 50 are given
throughout the text.

###  4.2 Pulsation variation

To investigate the pulsation characteristic of stellar physical parameters for
RRLs, we only select those targets with at least three measurements of stellar
atmospheric parameters and an SNR $\geq$ 30 (with $g$ band for LRS or blue-arm
for MRS). The phase of RRLs sample is derived from the spectral observation
times (HJD), periods ($P$), and epoch zero ($t_{0}$), where the $t_{0}$ is
defined as at maximum luminosity.

Figure 6 shows the phase folding diagrams of $RV_{\rm H\alpha}$ and $RV_{\rm
metal}$ for RRab and RRc. These $RV$s have been corrected using the radial
velocity of the center of mass ($RV$0) from the $Gaia$ RRLs catalog
(Clementini et al., 2022). Within the dataset, there are 121 RRab and 25 RRc
in the LRS, and 98 RRab and 21 RRc in the MRS. The black line is the average
value of $RV$s at phases with 0.1 bins, and the error bars represent the 99%
confidence interval. In the top panel (a) and (b), it can be seen that the
average $A_{\mathrm{ptp}}$ value of $RV_{\rm H\alpha}$ is $78\pm 17$ km/s for
RRab, which is larger than $50\pm 13$ km/s of $RV_{\rm metal}$. For RRc, the
$A_{\mathrm{ptp}}$ of $RV_{\rm H\alpha}$ and $RV_{\rm metal}$ are $26\pm 13$
and $22\pm 11$ km/s, respectively (the bottom panel (c) and (d)). We find that
the $A_{\mathrm{ptp}}$ of $RV$ for RRc is smaller than that for RRab, because
the $A_{\mathrm{ptp}}$ of $RV$ is positively related to the amplitude of light
curves, and the RRab type stars have a larger average $A_{\mathrm{ptp}}$ of
light curves than that of RRc (Braga et al., 2021).

Figure 7 illustrates the variation of stellar atmospheric parameters as a
function of phase. In the top panel, the variation of $T_{\rm eff}$ during the
pulsation cycle is significant. According to the average value curves, for
RRab stars, the variation of $T_{\rm eff}$ ranges from $6264\pm 171$ to
$7194\pm 423$ K, with an average $A_{\mathrm{ptp}}$ of $930\pm 456$ K. For RRc
stars, the $T_{\rm eff}$ varies between $6924\pm 261$ and $7333\pm 256$ K,
with the average $A_{\mathrm{ptp}}$ being $409\pm 375$ K. In the middle panel,
the $\log{g}$ of RRab has a small amplitude variation of about $0.22\pm 0.42$
dex during the pulsation phase, while that of RRc remains almost constant. In
the lower panel, the [M/H] values of RRab and RRc show slight variations of
$0.25\pm 0.50$ and $0.28\pm 0.55$ dex, respectively, while the values for the
other phases remain almost consistent. In addition, there are many [M/H] of
RRab from MRS distributed near the metal-rich $-0.50$ dex, and remains nearly
constant during the pulsation cycle. Most of the data points are contributed
from Gaia DR3 689686230645254912 (EZ Cnc), Gaia DR3 255409438666669824, and
Gaia DR3 468529087451089024, with 57, 34, and 27 MRS, respectively.
Specifically, the weighted average value of [M/H] for EZ Cnc is $-0.46\pm
0.04$ dex, which is in good agreement with the metallicity value of
$Z=0.006\pm 0.002$ from astroseismology (Wang et al., 2021).

Figure 6: The variations of radial velocities during the pulsation cycle.
Where the left panels represent the $RV_{\rm H\alpha}$ while the right panels
are for the $RV_{\rm metal}$, and the top two panels show the $RV$ of RRab
($RV_{\mathrm{ab}}$), while the bottom panels are for RRc ($RV_{\mathrm{c}}$).
The blue and orange data correspond to LRS and MRS, respectively. The black
line is the mean within the phase of 0.1 bins, and the error bars indicate the
standard deviation. The center of mass radial velocity ($RV_{0}$) is obtained
from the RRLs catalog of $Gaia$ DR3 (Clementini et al., 2022).  Figure 7: The
variations of stellar atmospheric parameters during the pulsation cycle. The
left panels represent the LRS, while the right panels are for MRS. From the
top to the bottom panels, the variation in $T_{\rm eff}$, $\log{g}$, and [M/H]
with phase are shown. The method for determining mean values curves (black
line) and error bars is same with those employed in Figure 6.

###  4.3 Compare with other databases

To evaluate the external uncertainty ($\sigma_{\mathrm{out}}$) of the stellar
atmospheric parameters, we compare our results with those from other
databases, including $Gaia$, APOGEE, SEGUE, and Liu et al. (2019). Considering
that the stellar atmospheric parameters provided by the LAMOST
DR10333https://www.lamost.org/dr10/v1.0/doc/release-note and GALAH DR3 (Buder
et al., 2021) lacks coverage of Horizontal Branch stars, we do not compare
with them. Assuming a coordinate matching difference of less than 3.0 arcsecs
is more strict than that in Section 2.1, we establish the following criteria
for common targets to minimize dispersion attributable to phase differences.

  * •

SNR limitation: The SNRg of LRS and SNRb of MRS are higher than 20.

  * •

Difference of phase constraints: We define $\Delta RV=\Delta\varphi\cdot
A_{RV}^{\mathrm{metal}}$ to convert the phase difference ($\Delta\varphi$)
between common targets into their $RV$ difference ($\Delta RV$) to address the
lack of observation times or phases in other databases, where
$A_{RV}^{\mathrm{metal}}$ denotes the $A_{\mathrm{ptp}}$ of $RV_{\rm metal}$.
To ensure that the phase difference is sufficiently small, while the number of
common targets is sufficiently large, we set $\Delta\varphi=0.2$. The
corresponding $\Delta RV$ values for RRab and RRc are approximated as 10 and 5
km/s, respectively.

  * •

Weakening pulsation effects: The phase was restricted to a range of
$0.2\leq\varphi\leq 0.8$ to avoid anomalous variations in stellar atmospheric
parameters near the 0.9 phase.

When compared to the other database, we adopt the weighted average of the
stellar atmospheric parameters. The comparison of $T_{\rm eff}$ is shown in
Figure 8. In the left panel, the common targets of our LRS sample with $Gaia$,
APOGEE, and SEGUE are 288, 7, and 104, respectively. The data points cluster
closely around the bisector line, and the mean ($\mu$) and standard deviation
($\sigma_{\mathrm{out}}$) values of residuals are $-81$ and 207 K,
respectively, indicating a good consistency between our results and other
datasets. In the right panel, the $T_{\rm eff}$ for MRS shows a good consist
with these from $Gaia$ and APOGEE, and the $\mu$ and $\sigma_{\mathrm{out}}$
values are -46 and 142 K, respectively.

Figure 8: The comparison of $T_{\rm eff}$ with the other datasets. The blue,
orange, and green data correspond to $Gaia$, APOGEE, and SEGUE, respectively,
in legend. The numbers indicate the count of common targets. The residuals of
$T_{\rm eff}$ between other datasets and our values are shown at the bottom,
along with their mean ($\mu$) and standard deviation ($\sigma$) in the text.
The black solid and dashed lines indicate the bisector and zero position,
respectively. The left panel displays $T_{\rm eff}$ of LRS, and the right
panel for MRS.

Figure 9 presents a comparison of $\log{g}$. For common targets of LRS, the
$\mu$ and $\sigma_{\mathrm{out}}$ of the residuals are $-0.06$ and 0.21 dex,
respectively. The comparison of MRS shows a good agreement, with $\mu$ and
$\sigma_{\mathrm{out}}$ of $-0.13$ and 0.16 dex, respectively. The $\log{g}$
may be underestimated for low surface gravity conditions of $\log{g}$ $<2.2$
dex, while overestimated for high surface gravity conditions of $\log{g}$
$>3.5$ dex.

Figure 9: The comparison of $\log{g}$ with the other datasets. The colors of
data and legends are the same as in Figure 8 but for $\log{g}$.

The comparison of [M/H] is illustrated in Figure 10. The $\mu$ of residues for
LRS and MRS are 0.02 and $-0.13$ dex, respectively, and
$\sigma_{\mathrm{out}}$ are 0.24 and 0.18 dex. For the LRS, the residuals show
a linear trend similar to that of $\log{g}$, which indicates that the [M/H]
from LRS is underestimated for the metal-poor region of [M/H] $<-2.50$ dex,
while overestimated for the slightly metal-rich region of [M/H] $>-1.00$ dex.

Figure 10: The comparison of [M/H] with the other datasets. The red points
represent the data from Liu et al. (2019), and the other colors and legends
are the same as in Figure 8 but for [M/H].

##  5 Discussion

In section 4.2, we analyze the characteristics of the $RV$ and stellar
atmosphere parameters during the pulsation cycle. The differences of
$A_{\mathrm{ptp}}$ between $RV_{\rm H\alpha}$ and $RV_{\rm metal}$ is $28\pm
21$ km/s for RRab, while it is $4\pm 17$ km/s for RRc. The differences in
$A_{\mathrm{ptp}}$ of $RV$ between the RRab and RRc from the H$\alpha$ and
metal lines are $52\pm 21$ and $28\pm 17$ km/s respectively. Similar
differences have been presented in the results of the $RV$ curves of 36 RRLs
obtained from the HRS by Braga et al. (2021). The difference of the
$A_{\mathrm{ptp}}$ of $RV$ for different lines indicates asynchronous movement
within the atmospheres of RRLs at varying altitudes. This suggests that the
Doppler effect of the segments needs to be corrected by the $RV$ of their
corresponding lines, which is a key step in the improved template matching
method.

There are differences between the stellar atmospheric parameters of the RRab
and RRc stars. For RRab, the $T_{\rm eff}$ ranges from $6264\pm 171$ to
$7194\pm 423$ K with an amplitude of $930\pm 456$ K, while for RRc, it ranges
from $6924\pm 261$ to $7333\pm 256$ K with an amplitude of $409\pm 375$ K. The
$\log{g}$ of RRab experiences a slight upward and then downward variation
around the phase of $\varphi=0.9$, while that of RRc remains almost constant.
This is because $\log{g}$ represents the effective gravity ($g=g_{M}+g_{r}$)
resulting from the combination of mass gravity ($g_{M}=\mathrm{G}M/r^{2}$) and
radius acceleration ($g_{r}=dRV(t)/dt$) , where $\mathrm{G}$ is the
gravitational constant, $t$ represents time, and $M$ and $r$ are the mass and
radius of the star, respectively (Chadid & Preston, 2013). As a result, the
$\log{g}$ of RRab, which shows larger amplitude compared to RRc, increases
more discernibly near the phase of $\varphi=0.9$ where the $r$ is minimized,
and $dRV(t)$ largest (Chadid & Preston, 2013). This implies that the position
of individual RRLs within the HRD undergoes significant shifts due to the
variation of $T_{\rm eff}$. In the phase folding diagrams of stellar
atmospheric parameters obtained by the HRS for 11 RRab (For et al., 2011) and
19 RRc stars (Sneden et al., 2017), as displayed in Magurno et al. (2019), the
characterization of the variations in $T_{\rm eff}$ and $\log{g}$ is very
similar to our results. In addition, based on the stellar atmospheric
parameters near the phase of $\varphi=0.5$, we obtain average $T_{\rm eff}$
and $\log{g}$ of $6264\pm 171$ K and $2.48\pm 0.29$ dex for RRab, while
$6924\pm 261$ K and $2.99\pm 0.29$ dex for RRc, respectively. This
characteristic was already noted by Magurno et al. (2019), indicating that
RRab is located closer to the cooler and darker red edge of the IS from
Marconi et al. (2015) compared to RRc.

According to the statistical results, the amplitude difference between
$RV_{\rm H\alpha}$ and $RV_{\rm metal}$ in RRab is more than twice that of
RRc. This significant difference suggests that complex movements are more
easily generated in the interiors of atmospheres for RRab, which can affect
their line profiles. Consequently, this influences the determination of
$\log{g}$ and [M/H]. For example, Gillet et al. (2019) investigated RR Lyr of
RRab type showing the Blazhko effect and concluded that the intensity of shock
excitation at the phase of $\varphi\sim 0.90-0.943$ is positively correlated
with the modulated $A_{\mathrm{ptp}}$ of light curves, i.e., the
$A_{\mathrm{ptp}}$ of $RV$. Therefore, in Figure 7, the [M/H] of RRab and RRc
shows a mild decrease of about $0.29\pm 0.49$ and $0.26\pm 0.52$ dex near the
phase of $\varphi=0.9$, respectively.

We propose to convert the phase difference constraints between our sample and
other databases into radial velocity difference constraints. The results of
this comparison indicate that our stellar atmosphere parameters from MRS show
a good agreement with other databases, indicating the reliability of results
from MRS. However, the residuals of the comparisons between our $\log{g}$ and
[M/H] from LRS and those from other databases present a linear trend,
suggesting that [M/H] and $\log{g}$ from LRS have systematic uncertainties.

##  6 summary

We collect a sample of about 449,093 RRLs from the catalogs of the $Gaia$,
ASAS-SN, ZTF, and PS1 survey projects, and 174,030 RRLs are in the region of
the LAMOST field. Through cross-matching with LAMOST DR10, we obtained 42,729
LRS and 7,064 MRS.

We first determine the $RV$s of H$\alpha$, H$\beta$, H$\gamma$, H$\delta$, and
metal lines for LRS, while the $RV$s of H$\alpha$ and metal lines for MRS. We
then optimize the template-matching method to estimate the stellar atmospheric
parameters of the RRLs, as described in Section 3.3. The spectral information,
periods, phase, and determined $RV$s and stellar atmospheric parameters of our
RRLs sample are presented in Tables 2 and 3.

We find that the internal uncertainty of $T_{\rm eff}$, $\log{g}$, and [M/H]
decreases with increasing SNR, and stabilizes when after SNR $\geq$ 50\. For
comparisons with other databases, we only chose stellar atmospheric parameters
with the differences of $RV_{\rm metal}$ less than 10/5 km/s and the phases of
0.2 $\leq\varphi\leq$ 0.8 to avoid uncertainties due to the pulsation. We
conclude that the systematic differences of $T_{\rm eff}$, $\log{g}$, and
[M/H] are $-81/-46$ K, $-0.06/-0.13$ dex, and $0.02/-0.13$ dex for LRS/MRS,
respectively, while the $\sigma_{\mathrm{out}}$ are 207/142 K, 0.21/0.16 dex.
and 0.24/0.18 dex. The final uncertainties ($\sigma$) of the stellar
atmospheric parameters are estimated by combining the $\sigma_{\mathrm{in}}$
and $\sigma_{\mathrm{out}}$, which are calculated using the formula
$\sigma=\sqrt{\sigma_{\mathrm{in}}^{2}+\sigma_{\mathrm{out}}^{2}}$. Where
$\sigma_{\mathrm{in}}$ is a function of SNR at SNR $\leq$ 50, the fitted
coefficients are displayed in Table 4.1.

We summarize the variation characteristics of the stellar physical parameters
of RRLs during the pulsation cycle.

  * •

The $A_{\mathrm{ptp}}$ of $RV_{\rm H\alpha}$ variation is greater than that of
$RV_{\rm metal}$, and this difference is significantly greater for RRab than
that for RRc. The $A_{\mathrm{ptp}}$ of $RV_{\rm H\alpha}$ and $RV_{\rm
metal}$ are $78\pm 17$ and $50\pm 13$ km/s for RRab, and $26\pm 13$ and $22\pm
11$ km/s for RRc, respectively.

  * •

There is a significant difference in the $T_{\rm eff}$ variation of RRab and
RRc during the pulsation cycle. The $T_{\rm eff}$ variation ranges from
$6264\pm 171$ to $7194\pm 423$ K with an $A_{\mathrm{ptp}}$ of $930\pm 456$ K
for RRab, while $6924\pm 261$ to $7333\pm 256$ K with an $A_{\mathrm{ptp}}$ of
$409\pm 375$ K for RRc.

  * •

The $\log{g}$ of RRab has a small variation near the phase $\varphi=0.9$,
while that of RRc is difficult to identify. Similar to the $\log{g}$ of RRab,
the [M/H] shows a slight variation near the 0.9 phase, with values of $0.25\pm
0.50$ and $0.28\pm 0.55$ dex for RRab and RRc, respectively.

In conclusion, we present two catalogs of stellar physical parameters,
containing about 11,000 RRLs. The stellar atmospheric parameters of the large
RRLs sample are very helpful for various fields of astronomical research.
According to the metallicity-luminosity relationship, the distance can be
determined by the precise metallicities (Muraveva et al., 2018). The accuracy
of the near-infrared period-luminosity relationship can be enhanced through
[M/H] calibration (Marconi et al., 2015; Muhie et al., 2021; Li et al., 2023),
which is beneficial for calibrating the zero point of Gaia distances and
compensating for the accuracy of Gaia distance for distant targets (Garofalo
et al., 2022). Through the variation of $T_{\rm eff}$ and $\log{g}$ with a
phase, the red and blue edges of the IS can be constrained, and further
exploration of the effects of [M/H] on the IS edge can be conducted
(Fiorentino et al., 2015; Luo et al., 2021). In particular, for some RRLs,
they have $RV_{\rm H\alpha}$ and $RV_{\rm metal}$ in almost all folding
phases, which will help us to understand the kinematic and dynamical
information of different envelopes of the internal stellar atmosphere (Gillet
et al., 2019), and will provide observational conditions on the physical
mechanism of the Blazhko effect (Chadid & Preston, 2013), also will assist in
asteroseismology studies in combined with light curves (Wang et al., 2021).

Our research is supported by the National Natural Science Foundation of China
under grant Nos. 12090040, 12090042, 12090044, 11833006, 11833002, and the
National Key R&D Program of China No. 2019YFA0405502. JTW acknowledges support
from the LAMOST FELLOWSHIP fund. The LAMOST FELLOWSHIP is supported by Special
Funding for Advanced Users, budgeted and administered by the Center for
Astronomical Mega-science, Chinese Academy of Sciences (CAMS-CAS). We
acknowledge the science research grants from the China Manned Space Project.
This work is supported by the Cultivation Project for LAMOST Scientific Payoff
and Research Achievement of CAMS-CAS. Guoshoujing Telescope (LAMOST) is a
National Major Scientific Project built by CAS. Funding for the project has
been provided by the National Development and Reform Commission. LAMOST is
operated and managed by the National Astronomical Observatories, CAS.

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