Asteroseismological Analysis of the Non-Blazhko RRab Star EPIC 248846335 in the LAMOST–Kepler/K2 Project

We conduct an asteroseismological analysis on the non-Blazhko ab-type RR Lyrae star EPIC 248846335, employing the Radial Stellar Pulsations module of the Modules for Experiments in Stellar Astrophysics based on the set of stellar parameters. The atmospheric parameters T eff = 6933 ± 70 K, log g = 3.35 ± 0.50, and [Fe/H] = −1.18 ± 0.14 are estimated from low-resolution spectra of LAMOST DR9. The luminosity L = 49.70−1.80+2.99 L ⊙ and mass M = 0.56 ± 0.07 M ⊙ are calculated, respectively, using the distance provided by Gaia and the metallicity estimated from the low-resolution spectra. The Fourier parameters of the light curves observed by K2 and radial velocity (RV) curves determined from the medium-resolution spectra of LAMOST DR10 are also calculated in this work. The period of the fundamental mode of the star and the residuals r of the Fourier parameters between the models and observations serve to select an optimal model, whose stellar parameters are T eff = 6700 ± 220 K, log g = 2.70, [Fe/H] = −1.20 ± 0.2, M = 0.59 ± 0.05 M ⊙, and L = 56.0 ± 4.2 L ⊙. The projection factors are constrained as 1.20 ± 0.02 and 1.59 ± 0.13 by the blue- and red-arm observed velocities with their corresponding RV curves derived from the best-fit model, respectively. The precise determination of stellar parameters in ab-type RR Lyrae stars is crucial for understanding the physical processes that occur during pulsation and for providing a deeper understanding of their period–luminosity relationship.


Introduction
RR Lyrae variables, with masses ranging from 0.5 to 0.8 M e , are large-amplitude pulsators.They are located at the intersection of the horizontal branch (HB) and the instability strip in the Hertzsprung-Russell (H-R) diagram, where they undergo helium core burning (Aerts et al. 2010).These stars pulsate due to the κ mechanism, driven by partial ionization of hydrogen and helium.They are radial pulsating variables with typical pulsation periods between 0.2 and 1 day.RR Lyrae stars display light variations of 0.3-1.7 mag in the V band and have effective temperatures ranging from 6100 to 7400 K, corresponding to spectral types A2 to F6.They can be categorized into the following types: RRab stars pulsating in the fundamental mode, RRc stars in the first overtone, and RRd stars in both modes (Bhardwaj et al. 2021b).The shorter-period RRc stars, occasionally referred to as RRe, represent the metal-rich extension of the RRc class (Bono et al. 1997).Due to their adherence to a precise period-luminosity-metallicity relation, particularly in near-infrared bands, RR Lyrae stars serve as critical tools for tracing and measuring distances to ancient stellar populations within the Milky Way and nearby galaxies (Bono et al. 2001;Catelan et al. 2004;Muraveva et al. 2015;Bhardwaj et al. 2021a).Moreover, the Blazhko effect, which is the periodic modulation of the light curves' amplitude and phase in RR Lyrae stars, has remained an interesting unsolved problem in astrophysics since its identification (Blažko 1907;Shapley 1916).
Models of RR Lyrae stars have been studied for a long time.A study by Bono et al. (2000) used full-amplitude, nonlinear, convective hydrodynamical models to investigate the behavior of the RRc variable star U Com.The study confirmed that the theoretical models accurately reflect the observed luminosity changes throughout the pulsation cycle.Marconi & Clementini (2005) applied nonlinear convective pulsation models to 14 Large Magellanic Cloud RR Lyrae stars, comprising an equal number of RRab and RRc stars (Bono et al. 2003;Marconi et al. 2003).This research evaluated the theoretical models and yielded a new independent distance estimate, which significantly impacted the calibration of the distance scale for RR Lyrae stars.Marconi & Degl'Innocenti (2007) successfully matched nonlinear pulsation models to the observed light curves of four RRc and two RRab stars in the Galactic globular cluster M3.This study demonstrated theoretical consistency with observed light-curve morphologies and intrinsic stellar parameters, in line with evolutionary expectations for the given metallicity.Smolec et al. (2013) employed nonlinear hydrodynamic pulsation models to explore the stellar parameters of OGLE-BLG-RRLYR-02793 (Pietrzyński et al. 2012), using light curves and radial velocity (RV) curves, although this object is not an RR Lyrae star.The radial pulsations of RR Lyrae stars offer a means to probe hydrodynamic processes through theoretical models, which can be benchmarked against observed light and RV curves to refine stellar parameters.However, acquiring complete RV curves is challenging, especially for fainter stars, due to the extensive telescope time required, as noted by Smolec et al. (2013).The LAMOST-Kepler/K2 surveys (LKS) have provided a wealth of multiepoch spectra for numerous Kepler/K2 targets (De Cat et al. 2015;Zong et al. 2018;Fu et al. 2020;Wang et al. 2020), enabling the extraction of atmospheric parameters and RV curves for RR Lyrae stars within the Kepler/K2 fields.Based on those resources, Wang et al. (2021) conducted asteroseismological analyses on the non-Blazhko ab-type star EZ Cnc (EPIC 212182292) using K2 light curves and RV data from LKS medium-resolution spectra (MRS).This analysis was performed with the Radial Stellar Pulsations (RSP) module of the Modules for Experiments in Stellar Astrophysics (MESA) suite (Paxton et al. 2011(Paxton et al. , 2013(Paxton et al. , 2015(Paxton et al. , 2018(Paxton et al. , 2019;;Jermyn et al. 2023), which simulates large-amplitude, self-excited pulsations as stars transit the instability strip on the H-R diagram.The study not only determined the stellar parameters for EZ Cnc but also estimated the projection factor (p = 1.22), a critical parameter in the Baade-Wesselink method for distance estimation (Nardetto et al. 2004;Karczmarek et al. 2017;Nardetto et al. 2017), which converts observed RV variations into pulsation velocities of the stellar photosphere.Notably, the p-factor may vary depending on the spectral lines used for RV measurements (Nardetto et al. 2017).According to Zhang et al. (2020), the MRS from LAMOST's blue arm predominantly targets the Mg Ib triplet, while the red-arm spectra capture the Hα line, thereby offering a unique opportunity to assess the p-factors using RV curves derived from both spectral regions.
In this work, we conduct an asteroseismological analysis of the non-Blazhko RRab-type star EPIC 248846335 (α 2000 = 10 h :48 m :11 650, δ 2000 = +11°48′44 08, Kp = 14.713 mag) to determine the values of the projection factors p and constrain the stellar parameters.We use the RSP module of MESA based on the light curves observed in the K2 field, and the RV curves derived from MRS of LAMOST DR10 with atmospheric parameters determined from the low-resolution surveys (LRS) of LAMOST DR9.In Section 2, we present the data collection and analysis.The numerical modeling and discussions are presented in Sections 3 and 4, respectively.Finally, Section 5 provides the conclusions of this paper.

Photometry
The target pixel file (TPF) of EPIC 248846335 obtained with a long-cadence observation of K2 was downloaded from the Mikulski Archive for Space Telescopes (MAST).All the K2 data used in this paper can be found in MAST (Huber et al. 2016).The package LightKurve is used to extract light curves from the TPF.To optimize the photometry of the star, several apertures with different pixel sizes are applied to the TPF.After extracting the photometry, the flux is converted to magnitude, and then the light curve is detrended by applying a third-order polynomial and adjusted to the Kp mean magnitude level given in MAST.The light curve of the star can be seen in Figure 1.

Fourier Analysis
Fourier analysis serves as a potent instrument for exploring the pulsation characteristics of variable stars.The software package Period04 (Lenz & Breger 2005) is employed to perform multifrequency analysis, which applies Fourier transformation combined with least-squares fitting to the light curve, deducing the pulsation frequencies of the star.The main frequency is determined to be f 0 = 1.5627(10) day −1 , equating to a fundamental period P 0 = 0.6399(7) days.Fourier decomposition, first introduced by Simon & Lee (1981) to analyze Cepheid light curves, effectively characterizes the light-curve features of variable stars.This approach has since become prevalent in the investigation of RR Lyrae stars (Simon & Teays 1982;Simon & Clement 1993;Nemec et al. 2011;Mullen et al. 2021).The following Fourier sine series fit the light curve of the target star: where m(t) denotes the apparent Kp magnitude from K2 data, n is the number of harmonic terms, A 0 the mean Kp magnitude, and f 0 the fundamental frequency.The variable t corresponds to the time of K2 observations (BJD -2454833), with t 0 as the epoch of the first maximum.The coefficients A i and f i represent the amplitude and phase of the ith harmonic, respectively.Following Simon & Lee (1981), certain Fourier coefficients correlate directly with specific physical properties of pulsating stars, typically expressed as linear combinations or ratios of phases and amplitudes: where i = 2 or 3 for the fundamental mode of RR Lyrae stars (Simon & Lee 1981).Corrections for f 21 and f 31 may include integer multiples of 2π when necessary.The determined pulsation parameters with their corresponding uncertainties are listed in the second column of Table 1.The standard deviation of the residuals of the Fourier decomposition that applies to the light curve observed by K2 is σ LC = 0.008 mag.Adopting the methodology of Zong et al. (2023), we calculate the total amplitudes A tot and the rise times (RTs) of the light and RV curves, with the fitted parameters listed in Table 1.

Spectroscopy
We obtain 50 MRS for EPIC 248846335 from LAMOST DR10, each with a signal-to-noise ratio (S/N) greater than 3.0 in the i band.Those spectra are bifurcated into two wavelength ranges: the red arm covers 630-680 nm, and the blue arm spans 495-535 nm.We adopt the SLAM pipeline (Zhang et al. 2020) to extract RVs from the spectra of both arms.However, RV measurements may exhibit systematic discrepancies across different spectrographs and observation nights, potentially reaching several km s −1 , as documented by Liu et al. (2019) and Zong et al. (2020).These offsets can be eliminated via the comparison to constant stars (Liu et al. 2019;Zong et al. 2020), a technique integrated into the SLAM pipeline.Following this correction method, the computed RVs from the blue and red arms are listed in Tables 2 and 3 and shown in Figure 2, with panels (a) and (b) illustrating the blue-arm and red-arm RVs, respectively.We base the phase-folding and analysis of the RV curves on the more precise fundamental period derived from the light curve observed by K2.The pulsation parameters for the RV curves of both spectral arms are calculated using Equation (1) and listed in the third and fourth columns of Table 1.The standard deviations of the residuals from the Fourier fits to the RV curves are 5.92 km s −1 for the red arm and 2.43 km s −1 for the blue arm.Zhang et al. (2021) pointed out that this discrepancy may be attributed to the different levels of precision inherent in the red-and blue-arm MRS from LAMOST.Additionally, velocity curves derived from distinct spectral lines, which may reflect disparate kinematics even at identical phases, can account for variations in curve shapes and amplitudes, as suggested by Braga et al. (2021).
We have collected 92 single-exposure LRS with S/N exceeding 10.0 from LAMOST DR9 (Bai et al. 2021).The MRS survey by LAMOST aims to compile time-series spectra at medium resolution, with the acquisition of RVs for designated stars being a principal scientific objective (Liu et al. 2020;Zong et al. 2020).The LAMOST LRS seek to determine stellar parameters for a diverse array of targets across the Northern Hemisphere, specifically those with declinations above −10° (Luo et al. 2012(Luo et al. , 2015)); however, it excludes timedomain observations.An investigation by Liu et al. (2019) that used multiple MRS observations for nearly 1900 targets revealed that the RV scatter for stars with a standard deviation below 0.5 km s −1 was significantly lower-by a factor of 3 to 5 -than that for measurements obtained from LRS (Luo et al. 2015).In this paper, we utilize LRS data from LAMOST to analyze the atmospheric parameters of stars.Nonetheless, the determination of these parameters for RR Lyrae stars from spectral data is contentious.Studies have shown that both lowand high-resolution spectra can yield accurate stellar parameters at various phases, and the derived [Fe/H] abundances appear to be phase-independent (For et al. 2011;Crestani et al. 2021).However, Crestani et al. (2021) noted that the largeamplitude variations in RR Lyrae stars can systematically alter the effective temperature and luminosity, potentially affecting the determination of chemical abundances if spectra are taken at different phases.
It has been suggested by Kolenberg et al. (2010) that the most favorable phase for spectral analysis corresponds to the maximum radius of RR Lyrae stars, during which stellar parameters can be precisely determined using the equivalent width method, as implemented in the literature (Fossati et al. 2014;Wang et al. 2021).The changes in radius of a pulsating star can be inferred from the periodic RV variations using the  following equations: where P is the period, and V * represents the center-of-mass RV of the star, for which we adopt the mean values of the RV curves in this study.The factor p accounts for the geometrical projection and limb-darkening corrections.We used a value of p = 1.25, consistent with that adopted by Wang et al. (2021) and based on the investigation of Nardetto et al. (2017).The maximum variation in radius derived from the RVs of the red-arm MRS of LAMOST is ΔR(t) = 0.37 ± 0.02 R e observed at phase f max = 0.323 ± 0.003, and that from the RVs of the bluearm MRS is ΔR(t) = 0.52 ± 0.01 R e observed at phase f max = 0.320 ± 0.004.The maximum variations in radius occur at the same phase, within the uncertainties, for both the blue-and red-arm spectra.The atmospheric parameters of the star determined at phase f = 0.319 max corresponding to the maximum radius are T eff = 6933 ± 70 K, [Fe/H] = −1.18± 0.14, and  log g = 3.35 ± 0.50 using the template-matching method provided by J. Wang et al. (2024, in preparation).The variations in radius ΔR(t), derived from RVs and calculated using Equations (4) and (5), are displayed in Figures 2(c) and (d).

Parameter Calculation
To calculate the bolometric luminosity of the star, the calibrated distance of the star provided by Gaia DR3 as d = -+ 6839.27789.22 1327.24pc is used.The formula given by Bailer-Jones et al. ( 2018) is adopted as follows: where M and M G = 14.7 mag are the absolute magnitude and apparent magnitude in the G band of Gaia, respectively. .The metallicity of the star is calculated by adopting the following equations (Bressan et al. 2012): where the value of (Z/X) e is 0.0207 (Caffau et al. 2011).X, Y, and Z are the hydrogen, helium, and metal abundances by mass fraction of the star, which we estimate as X = 0.748 ± 0.001, Y = 0.250 ± 0.001, and Z = 0.0010 ± 0.0003, respectively.The value of (Z/X) e in the RSP inlist provided by Asplund et al. (2009) is different from that of Caffau et al. (2011).The mass of the star is calculated as M = 0.56 ± 0.07 M e using Equation (22) of Jurcsik (1998), which is based on the HB models that indicate the dependence of the stellar mass on the metallicity within the instability strip proposed by Castellani et al. (1991).In this work, we only use the mass as the initial mass to construct the grid of models.

Model Construction and Selection
The convective code of RSP Module of MESA for stellar radial pulsations based on the time-dependent turbulent convection model (Kuhfuss 1986) was implemented by Smolec & Moskalik (2008).This model can effectively reproduce the light curves and RV curves of classical pulsating variables as it combines the convection and the pulsation driven by partial ionization.The turbulent energy and the kinetic energy are coupled to each other through coupling terms (Smolec & Moskalik 2008), which are controlled by eight convection parameters: the mixing length α, the eddy-viscous dissipation α m , the turbulent source α s , the convective flux α c , the turbulent dissipation α d , the turbulent pressure α p , the turbulent flux α t , and the radiative cooling γ r .According to Paxton et al. (2019), slightly different values of the convection parameters should be considered in constructing models for different types of stars, for instance, Cepheids, RR Lyrae stars, and other stellar systems.They suggested that α t ; 0.01, α m  1.0, and α  2 are useful initial choices from experience.The investigation of Kovács et al. (2023) revealed that varying convective parameters have distinct effects on the final RV and light curves as presented in their Figures 2 and 3.They pointed out that, among the parameters, α m of RSP has the most significant effect on the resulting RV and light curves, while other parameters have little effect.We adjust the values of α m and the other parameters following those recommended by Paxton et al. (2019).The value sets of these parameters are listed in Table 4 to produce optimal RV and light curves of the star, and the values of convective parameters are fixed to the four sets given in Table 4.
In this study, a grid of models is calculated using the RSP module of MESA with the stellar parameters.As suggested by Paxton et al. (2019), the initial input parameters mass, luminosity, effective temperature, hydrogen abundance (X), and metal abundance (Z) can be freely chosen and do not necessarily need to originate from a MESAstar model.Based on the atmospheric parameters determined from the LRS of LAMOST, the effective temperature of the star varies within the range 6100-7300 K, which falls within the typical range of effective temperature of RR Lyrae stars.The metallicity determined from those spectra ranges from −1.72 to −0.53.However, the observed phases of the LRS do not cover the entire pulsation cycle.Therefore, the set of metallicities used in this work is adjusted to −2.90 [Fe/H] 0.0.The metal abundance Z and hydrogen abundance X are calculated using Equations ( 8), ( 9), and (10), based on the corresponding values of [Fe/H].This method was also adopted by Wang et al. (2021) for determining the sets of Z and X in constructing their model for the non-Blazhko RRab star EZ Cnc (EPIC 212182292).The absolute luminosity and mass determined in this work are L = -+ 49.76 1.80 2.99 L e and M = 0.56 ± 0.02 M e , respectively.To derive the optimal model of this star, we consider a wide range of luminosity and mass sets: 40 L/L e 65 and 0.35 M/M e 0.75.The resolution of the grid is set as and Δ [Fe/H] = 0.1.An exemplary list is included in the Appendix of our paper.The absolute value Γ = 4.13 × 10 −6 is adopted to ensure that the models converge to a full-amplitude solution, as documented by Paxton et al. (2019).
In our analysis, we compare the nonlinear periods derived from the models with the fundamental period determined from the observed light curve.An uncertainty of ΔP = 0.0007 days is applied for the fundamental period to ensure that the differences between the main period obtained from the K2 light curve and the periods derived from the models are smaller than this uncertainty value.This criterion helps us select the appropriate models from the grid.We obtain 40 models that meet the period criterion.
The light curves of the 40 models are generated using four different sets of convection parameters.The convection parameters are adjusted to generate optimal light curves and RV curves of the star.To maintain consistency between the model light curves and the K2 light curve, we convert the model light curves to the Kepler white band using the bolometric calibration coefficient (Lund 2019), which depends only on the effective temperature.The residuals r between the models and the observations in the Fourier parameter space (Smolec et al. 2013) are calculated using the following equation: where p i represents one of the low-order Fourier parameters and amplitudes, p ä {RT, R 21 , R 31 , f 21 , f 31 }, p i,mod refers to our models, and p i,obs refers to the observed curves.The smaller the value of r, the closer the observed K2 light curves and LAMOST RV curves are to those modeled with RSP and MESA, respectively.The distribution of r in the space of the stellar parameters for Sets A, B, C, and D is presented in Figure 3.
It should be noted that not all 40 models converge for each set of convection parameters.We obtain one model from each set with the smallest residual r value.The stellar parameters of the four models are listed in Table 5. s mod,RVC and s mod,LC are the standard errors of the residuals between the observed RV curves and light curves and their corresponding model-derived curves, respectively.Only the model derived from the convection parameters in Set A satisfies 3, suggesting that it is the optimal model of the star.Figures 4-6 show the comparison between the observed curves and that curves determined from the optimal model.
We estimate the uncertainties of the best-fitting model parameters using the prescription derived by Zhang et al. (1986), a method commonly adopted in the literature (Castanheira & Kepler 2008;Romero et al. 2012;Fu et al. 2019).The equation is as follows: where σ is the uncertainty of the parameter, d is the step size of the parameter within the model grid, S 0 is the r 2 value of the best-fitting model (i.e., the minimum value), and S is the r 2 value for the model with the prescribed change in the parameter by the amount r while keeping all other parameters fixed.The best-fitting model parameters and their uncertainties are M = 0.59 ± 0.05 M e , T eff = 6700 ± 220 K, [Fe/H] = −1.2± 0.2, and L = 56.0 ± 4.2 L e .The projection factors of the star are determined to be 1.20 ± 0.02 and 1.59 ± 0.13, which are constrained by the blue-and red-arm observed velocities and their corresponding RV curves derived from the structural profiles of the optimal model.We also calculate the light and RV curves of the optimal model considering different mesh numbers (e.g., RSP_nz = 150, RSP_nz_outer = 30, and RSP_nz = 200, RSP_nz_outer = 60) and time steps per pulsation cycle (RSP_target_steps_per_cycle=200 and RSP_target_steps_ per_cycle = 600).The results indicate that the light and RV curves of the models are not sensitive to these parameters, consistent with the findings of Paxton et al. (2019).

Discussion
Cassisi & Pietrinferni (2021) pointed out that modeling the evolution of RR Lyrae stars is not a trivial task, and includes difficulties related to uncertainties in modeling the helium flash and mass loss on the red giant branch.We adopt the updated HB models from the Bag of Stellar Tracks and Isochrones (BaSTI) project (Hidalgo et al. 2018) to calculate the properties of HB models (M = 0.61 M e , M = 0.62 M e , M = 0.63 M e , and M = 0.64 M e ) using a chemical composition of Z = 0.001 and Y = 0.246, which is similar to that derived from the optimal model, with the input parameter α = 1.5 and the mass loss efficiency η = 0.4 (Reimers 1975).The comparison between the pulsation modeling results and the evolutionary tracks is presented in the H-R diagram (Figure 7), where the optimal model is located in the middle of the instability strip.The comparison between the positions of the optimal model and the evolutionary tracks indicates discrepancies between the masses and luminosities derived from pulsation modeling and those associated with the evolutionary tracks situated in analogous regions of the instability strip.Nemec et al. (2011), who studied 19 non-Blazhko RRab stars using Kepler photometry, revealed a discrepancy between the masses and luminosities derived from evolutionary tracks and those obtained from pulsation calculations.As they documented, it is not clear whether the luminosity and mass derived from the evolutionary tracks or those calculated from the pulsation code are correct.In a study conducted by Wang et al. (2021), a comparison between results obtained from evolutionary tracks and pulsation modeling revealed a similar discrepancy in masses and luminosities.
Netzel & Smolec (2022) suggested that there is no direct mass determination for any known RR Lyrae stars since no RR Lyrae star is known to be in an eclipsing binary system.The most promising candidate for an RR Lyrae star in a binary system turned out to be a star with a significantly smaller mass and formed through a different evolutionary channel (Pietrzyński et al. 2012).The search for RR Lyrae stars in binary systems is ongoing and has resulted in several candidates (Hajdu et al. 2018).Unfortunately, binary systems detected using the light-time effect will not yield dynamical masses for RR Lyrae stars.However, Netzel & Smolec (2022) also suggested that in the absence of direct mass determination for  RR Lyrae from the sodium and Hα absorption lines corresponding to the deep layers of the photosphere and upper atmosphere, respectively.As previously mentioned, the RV curves derived from the blue-and red-arm MRS of LAMOST are based on the Mg Ib triplet and Hα lines (Zhang et al. 2020), respectively.The amplitudes of the RVs derived from the redand blue-arm MRS are 66.10 (±5.26) km s −1 and 43.53 (±2.01) km s −1 , respectively, as listed in Table 1 in our paper.The former amplitude is 51% larger than the latter, which is consistent with the results of Braga et al. (2021).In their study, the RV amplitudes derived from Hα are 24%-52% larger than the amplitudes derived from the Mg Ib triplet, as listed in Tables 8 and 9 of their paper.Bono et al. (2020) suggested that the amplitude difference of RV curves is caused by the physical conditions under which the spectral lines form.According to their findings, a smaller optical depth corresponds to a larger RV amplitude.Braga et al. (2021) suggested that RR Lyrae stars are pulsating stars, and different lines may exhibit distinct kinematics even when observed at the same phase.As a result, the velocity curves derived from different lines can display varying shapes and amplitudes.

Conclusions
In this work, we conduct an asteroseismological investigation of the non-Blazhko RRab star EPIC 248846335 using homogeneous MRS in the red and blue arms collected by the LAMOST-Kepler/K2 project, along with photometric data provided by the Kepler space telescope.The RV curves of this star are obtained from the red-and blue-arm spectra of LAMOST DR10.The Fourier decomposition method is applied to the light curve and RV curves to determine the pulsation parameters of the star.The stellar atmospheric parameters, including the effective temperature T eff = 6933 ± 70 K, surface gravity log g = 3.35 ± 0.50, and metallicity [Fe/H] = −1.18± 0.14, are estimated from the single-exposure LRS of LAMOST DR9.The stellar mass M = 0.56 ± 0.07 M e is also calculated based on the value of [Fe/H].We determine the absolute luminosity L = -+ 49.70 1.80 2.99 L e of the star using the distance provided by Gaia DR2.
A series of time-independent convection grid models are constructed based on the estimated stellar parameters using the RSP module of MESA.The fundamental period of the star and the residuals r of the Fourier parameters between the models and observations serve to select the optimal model.The stellar parameters of the optimal model are determined as follows: T eff = 6700 ± 220 K, log g = 2.70, [Fe/H] = −1.20 ± 0.2, M = 0.59 ± 0.05 M e , and L = 56.0 ± 4.2 L e .The values of the projection factors of the star are constrained to be 1.20 ± 0.02 and 1.59 ± 0.13 by the blue-and red-arm observed velocities with their corresponding RV curves derived from the best-fit model.It is hoped that in the future a larger number of precise light curves and spectra of RR Lyrae stars will be obtained, which would place new constraints on the hydrodynamic models constructed for these stars and help improve our understanding of the evolution of RR Lyrae stars.

Figure 2 .
Figure 2. The radial velocity curves and their corresponding variations in radius.(a), (b) The radial velocities determined from the blue-and red-arm MRS of LAMOST DR9 are marked in blue and red, respectively.(c), (d) The variations in radius calculated based on the radial velocities estimated from the blue-and red-arm MRS using Equation (5), respectively.

Figure 3 .
Figure 3. Determination of the best-fit model (residual r) of Sets A, B, C, and D. For better visibility, in panels (a), (c), (e), and (g), 1/r 2 is plotted as a function of mass (m e ) and T eff with the corresponding color scale for the four different sets of convective parameters; in panels (b), (d), (f), and (h), 1/r 2 is plotted as a function of metallicity ([Fe/H]) and luminosity (L e ) with the corresponding color scale for the four different sets of convective parameters.

Table 1
Fourier Decomposition Parameters of the Light Curves and Radial Velocity Curves of EPIC 248846335 Note.Column (1): ID of the parameters; Column (2): Fourier decomposition parameters of the light curves; Columns (3) and (4): Fourier decomposition parameters of RV curves derived from the blue-and red-arm MRS of LAMOST, respectively.

Table 2
RVs of the Target Star Measured from the Blue-arm MRS of LAMOST DR10 The data are ordered by barycentric Julian date (BJD).

Table 3
RVs of the Target Star Measured from the Red-arm MRS of LAMOST DR10

Table 4
The Four Sets of Convection Parameters

Table 5
Properties of the Best Models for EPIC 248846335 in the Four Sets of Convection Parameters