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

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.

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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 f₀ = 1.5627(10) day⁻¹, equating to a fundamental period P₀ = 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:

$$\begin{eqnarray}&&m(t)={A}_{0}+\displaystyle \sum _{i=1}^{n}{A}_{i}\sin [2\pi {{if}}_{0}(t-{t}_{0})+{\phi }_{i}],\end{eqnarray} \tag{ 1 }$$

where m(t) denotes the apparent Kp magnitude from K2 data, n is the number of harmonic terms, A₀ the mean Kp magnitude, and f₀ the fundamental frequency. The variable t corresponds to the time of K2 observations (BJD – 2454833), with t₀ as the epoch of the first maximum. The coefficients A_i and ϕ_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:

$$\begin{eqnarray}&&{\phi }_{i1}={\phi }_{i}-i{\phi }_{1},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{R}_{i1}=\displaystyle \frac{{A}_{i}}{{A}_{1}},\end{eqnarray} \tag{ 3 }$$

where i = 2 or 3 for the fundamental mode of RR Lyrae stars (Simon & Lee 1981). Corrections for ϕ₂₁ and ϕ₃₁ 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.

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).

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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, 2015); however, it excludes time-domain 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 low- and 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 large-amplitude 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:

$$\begin{eqnarray}&&\dot{R}=-p({V}_{r}(t)-{V}_{* })\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}R(t)={\int }_{0}^{P}\dot{R}\,{dt}\end{eqnarray} \tag{ 5 }$$

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_⊙ observed at phase ${\phi }_{\max }$ = 0.323 ± 0.003, and that from the RVs of the blue-arm MRS is ΔR(t) = 0.52 ± 0.01 R_⊙ observed at phase ${\phi }_{\max }$ = 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 ${\phi }_{\max }=0.319$ 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).

To calculate the bolometric luminosity of the star, the calibrated distance of the star provided by Gaia DR3 as d = ${6839.27}_{-789.22}^{+1327.24}$ pc is used. The formula given by Bailer-Jones et al. (2018) is adopted as follows:

$$\begin{eqnarray}&&M={M}_{G}+5(1-\mathrm{log}d)-{A}_{G}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&-2.5\mathrm{log}L=M+{\mathrm{BC}}_{G}({T}_{\mathrm{eff}})-{M}_{\mathrm{bol},\odot }\end{eqnarray} \tag{ 7 }$$

where M and M_G = 14.7 mag are the absolute magnitude and apparent magnitude in the G band of Gaia, respectively. The extinction coefficient A_G in Equation (6) is 0.1224 in the G band (Bailer-Jones et al. 2021). The parameter M_bol,⊙ is the bolometric magnitude of the Sun, which is defined by the IAU and its value is 4.74 mag (Mamajek et al. 2015). BC_G (T_eff) is the bolometric correction, which depends only on the effective temperature (Andrae et al. 2018). The bolometric luminosity estimated for this star is L = ${49.70}_{-1.80}^{+2.99}\,{L}_{\odot }$. The metallicity of the star is calculated by adopting the following equations (Bressan et al. 2012):

$$\begin{eqnarray}&&[\mathrm{Fe}/{\rm{H}}]=\mathrm{log}{(Z/X)-\mathrm{log}(Z/X)}_{\odot }\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&Y=0.2485+1.78Z\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&X+Y+Z=1\end{eqnarray} \tag{ 10 }$$

where the value of (Z/X)_⊙ 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)_⊙ 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_⊙ 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.

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_⊙ and M = 0.56 ± 0.02 M_⊙, respectively. To derive the optimal model of this star, we consider a wide range of luminosity and mass sets: 40 ≤ L/L_⊙ ≤ 65 and 0.35 ≤ M/M_⊙ ≤ 0.75. The resolution of the grid is set as ${{\rm{\Delta }}}_{M/{M}_{\odot }}=0.01$, ${{\rm{\Delta }}}_{{T}_{\mathrm{eff}}}$ = 50 K, ${{\rm{\Delta }}}_{L/{L}_{\odot }}=1,$ 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:

$$\begin{eqnarray}&&r=\sqrt{\sum \displaystyle \frac{{\left({p}_{i,\mathrm{mod}}-{p}_{i,\mathrm{obs}}\right)}^{2}}{{p}_{i,\mathrm{obs}}^{2}}}\end{eqnarray} \tag{ 11 }$$

where p_i represents one of the low-order Fourier parameters and amplitudes, p ∈ {RT, R₂₁, R₃₁, ϕ₂₁, ϕ₃₁}, ${p}_{i,\mathrm{mod}}$ 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.

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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. ${\sigma }_{\mathrm{mod},\mathrm{RVC}}$ and ${\sigma }_{\mathrm{mod},\mathrm{LC}}$ 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 $\tfrac{{\sigma }_{\mathrm{mod},\mathrm{RVC}}}{{\sigma }_{\mathrm{obs},\mathrm{RVC}}}$ ≤ 3 and $\tfrac{{\sigma }_{\mathrm{mod},\mathrm{LC}}}{{\sigma }_{\mathrm{obs},\mathrm{LC}}}$ ≤ 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.

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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:

$$\begin{eqnarray}&&\sigma ={d}^{2}/(S-{S}_{0}),\end{eqnarray} \tag{ 12 }$$

where σ is the uncertainty of the parameter, d is the step size of the parameter within the model grid, S₀ is the r² value of the best-fitting model (i.e., the minimum value), and S is the r² 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_⊙, T_eff = 6700 ± 220 K, [Fe/H] = −1.2 ± 0.2, and L = 56.0 ± 4.2 L_⊙. 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).