Astrophysical calibration of the oscillator strengths of YJ-band absorption lines in classical Cepheids

Newly-developed spectrographs with increased resolving powers, particularly those covering the near-IR range, allow the characterization of more and more absorption lines in stellar spectra. This includes the identification and confirmation of absorption lines and the calibration of oscillator strengths. In this study, we provide empirical values of loggf based on abundances of classical Cepheids obtained with optical spectra in Luck (2018), in order to establish the consistency between optical and infrared abundance results. Using time-series spectra of classical Cepheids obtained with WINERED spectrograph (0.97-1.35 $\mu$ m, R ~28000, we demonstrate that we can determine the stellar parameters of the observed Cepheids, including effective temperature (Teff), surface gravity (logg), microturbulence, and metallicity. With the newly calibrated relations of line-depth ratios (LDRs), we can achieve accuracy and precision comparable to optical studies (Luck 2018), with uncertainties of 90K and 0.108 dex for Teff, and log g, respectively. Finally, we created a new atlas of absorption lines, featuring precise abundance measurements of various elements found in the atmosphere of Cepheids (including neutron-capture elements), with loggf values that have been astrophysically calibrated.


INTRODUCTION
Infrared spectroscopic observations provide valuable means for studying astronomical targets that are heavily obscured by interstellar extinction, which is a significant advantage over optical observations.The availability of mid-to-high resolution infrared spectra has led to the detection of both strong and weak absorption lines, The NIR regime, in particular, the YJ bands, is a relatively understudied region that has the potential to yield sig-nificant new insights into the chemical evolution of the Galaxy.
Despite the importance of basic spectral information, such as line positions and strengths in understanding the overall spectral scenario, the atomic information relevant to each spectral feature has not been widely covered in the literature (Ryde et al. 2019).The calibration of these parameters is crucial for further procedures, such as estimating abundances.The oscillator strength (log gf ) is a dimensionless quantity that measures the strength of a transition (absorption and emis-sion).In the NIR regime, accurately estimated log gf values are often lacking (see, e.g., Andreasen et al. 2016), and proper line identification is also frequently absent.
It is important to note that the definitive atomic parameters that are currently missing in the infrared bands must ultimately be obtained through laboratory astrophysics.However, in order to ensure consistency between optical and infrared chemical abundance results, we aim to obtain the empirical (or astrophysical) log gf values based on the optical abundance results for Cepheids presented in Luck (2018).Such empirical calibrations also help to reduce the internal errors caused by the choice of the atmosphere model and, ultimately, improve our abundance measurements.The sample of Luck (2018) constitutes the largest collection of Cepheids that have been homogeneously analyzed for different elements in the optical range, making it an ideal data set for this purpose.
Cepheid variables are not only valuable for distance estimates, but their spectra also exhibit a considerable number of absorption lines, making them useful tracers of chemical abundances in their host galaxies, particularly in the case of the Galaxy.In particular, Cepheids have been proven to be useful for studying the metallicity gradient of the Galactic disk e.g., (Genovali et al. 2014;Luck 2018).A key parameter in the spectroscopic analysis is the effective temperature, T eff .Once this intrinsic parameter is derived, the subsequent (but not less important) parameters, i.e., surface gravity (log g), microturbulent velocity (ξ), and metallicity ([Fe/H]) may be estimated with spectra.Besides the identification and calibration of absorption lines in Cepheids, we aim to exploit the capabilities of the unique high-resolution spectra with WINERED (Ikeda et al. 2022), by employing the empirical LDR method (Gray & Johanson 1991).We follow the prescription made in the NIR by Matsunaga et al. (2021) but extend it for the first time in the T eff and log g parameter space to cover the range of classical Cepheids.
This paper is organized as follows.In Section 2, we provide detailed information about the analyzed targets and the WINERED spectrograph, as well as the procedure employed to prepare the spectra for measurements.In Section 3, we discuss the capabilities of our data to provide precise and accurate stellar parameters (T eff and log g, primarily) based on our extension of the LDR method.Section 4 presents our method to identify and confirm the lines we will use for abundance determination, as well as the derivation of microturbulence (ξ) and the calibration of the oscillator strengths (log gf ).Based on the results in Section 4, we measured the abundances of Cepheids as we present in Section 5. We later discuss the scope and limitations of our results in Section 6.

Observations and targets
We analyze multi-epoch spectra of eleven Cepheids in the solar neighborhood obtained with the WINERED spectrograph in its WIDE mode, which has a resolving power of R ∼ 28000.The observations were conducted from 2013 to 2016 at the 1.3 m Araki Telescope located at the Koyama Observatory of Kyoto Sangyo University in Japan.As given in Tables 1 and 2, our sample is divided into two groups, calibrators, and validators.The former consists of eight widely studied Cepheids for which there is a wealth of data sets available (photometric and spectroscopic).The latter consists of three Cepheids whose information in the literature is incomplete or scarce.For each target, we obtained multiepoch spectra, with better phase coverage for the calibrators.

Pipeline reduction
During the observations, the nodding technique was employed to subtract the background radiation from the sky and the ambient facility.Two types of nodding patterns were used, "ABBA" and "OSO" (object-skyobject).We decided the nodding pattern for each observation to avoid latent signals after exposure of bright targets.Calibration data include flat fielding images and ThAr lamp spectra for wavelength calibration.The raw data were reduced by the WARP pipeline, created and developed by Hamano, S.1 .Every science target has more than a single exposure, and for different echelle orders, a combined 1D, air wavelength calibrated spectrum was obtained.The pipeline also outputs information about the signal-to-noise ratio (S/N) for each target at three different regions of each echelle order.A WINERED spectrum in the WIDE mode is divided into 20 echelle orders, of which we use the 11 orders covering 9760-10890 Å and 11800-13190 Å which are not heavily affected by telluric contamination.

Telluric correction and continuum normalization
The analysis of infrared spectra is subject to telluric contamination, caused by the Earth's atmosphere.The telluric lines vary depending on factors such as the observatory location, weather conditions, and air mass.In order to accurately identify and study stellar features, it is necessary to remove telluric contamination from the observed spectra.In this study, the method developed in Sameshima et al. (2018) was employed to correct for the telluric contamination in the telluric standard stars observed in conjunction with the science targets.This correction was applied to every target and every order, with the exception of orders 53 and 54, corresponding to the spectral range covering λ : [10280−10680] where the telluric absorption in our atmosphere can be neglected.The resulting spectra were then normalized by means of the IRAF continuum task to account for any offset from unity.

Phased parameters from literature data
For the calibrators, we derived T eff , log g, and ξ at each phase from the interpolated Fourier curves based on the catalog of Luck (2018), which provides the measurement of these parameters with good phase sampling.We adopted the interpolated values for further analysis and calibrations.On the other hand, a different approach was applied to the validators by employing the Line-Depth Ratio (LDR) method.

Construction of LDR relations
The LDR method is widely used for determining the effective temperature (T eff ) in spectroscopic analysis.Previous studies were primarily conducted on optical spectra (Gray & Brown 2001;Kovtyukh 2000, 2007, andreferences therein), while more recent works have been involved with infrared spectra (Fukue et al. 2015;Taniguchi et al. 2018;Jian et al. 2019).However, all of these previous studies were limited to certain temperature ranges which do not include warmer Cepheids (T eff > 6000 K).Furthermore, LDRs used in the previous works tend to exhibit dependency not only on T eff but also on log g and metallicity (Jian et al. 2019(Jian et al. , 2020)).In order to bypass such complexities, Taniguchi et al. (2021) proposed to use only Fe i lines to estimate T eff , followed by the work of Matsunaga et al. (2021), introducing the usage of neutral-ionized pairs to give the LDR relations with log g, setting the guideline of this study.
In this work, the LDR method is employed following the prescription stated in Matsunaga et al. (2021) by using Fe i, Fe ii, Ca i, and Ca ii within 9760-10860 Å and 11800-13190 Å corresponding to the WINERED echelle orders in the Y band (52nd-57th) and the J band (43rd-47th).From Matsunaga et al. (2021), we adopted the list of 97 lines for the four species (76 Fe i, 5 Fe ii, 11 Ca i and 5 Ca ii), but we searched for new line pairs that show good LDR relations.According to our preliminary test on our sample of Cepheids, the LDR relations presented in Matsunaga et al. (2021) do not work well in hotter targets (T eff > 6000 K) as some lines are too shallow at high temperatures.It is worth mentioning that other elements were found in the spectra such as Si and Ti displaying both neutral and ionized lines but the number of ionized lines is very limited.
In order to measure the depth of each of the 97 lines from Matsunaga et al. (2021), a Gaussian fit was performed on the section of five pixels around the line center (λ c ).The wavelength of the line minimum (λ 0 ) was then determined, and assuming that the continuum normalization was properly done, the depth was calculated as the "distance" from the continuum level (the unity) to the flux at λ 0 .We rejected measurements if the continuum level was not well normalized and the estimated depth was a negative number, or if the position of λ 0 was too far from λ c .This procedure was carried out using the ir ldr python package developed by Jian, M. 2 .The error in each depth measurement is linked to the signal-to-noise ratio (S/N) for the target (obj) and the telluric standard (tell), and the two errors were added in quadrature: (1) This equation was used for all the WINERED echelle orders except for the orders 53 and 54, for which no telluric correction was performed, as they cover the wavelength ranges corresponding to an atmospheric window free of telluric absorption.For those two orders, the error is simply e = e obj .We created a new list of line pairs that are effective in the Cepheids regime 5500 K < T eff < 6500 K, following the algorithm of pair selection in Matsunaga et al. (2021, Section 3.3).
To begin with the construction of our LDR relations, we use Fe i-Fe i pairs for estimating T eff , by pairing lines of low and high excitation potential (having a ∆EP ≥ 1 eV) gives higher sensitivity to T eff and their ratio can provide good diagnostics on temperature, reaching the utmost precision of ∼ 10 K (Gray & Johanson 1991).Thus, the LDR of each pair is defined as r = d low /d high , 2 ir ldr in Github i.e., the ratio of the depth of the line with a lower excitation potential (d low ) to the depth of the line with a higher excitation potential (d high ).We consider the four forms of the relation between r and T eff as described in Matsunaga et al. (2021, Section 3.4.1).We have 57 spectra of the calibrators in total, but the quality of the spectra shrank our sample to 51 (listed in table ??), as well as the validation of measured depths may reduce the number of useful depths and according to the number of ratios.We rejected the line pairs for which only 30 or fewer validated measurements of r were available and those for which the LDR relations had dispersion larger than 200 K.In addition, the range of T eff covered by the points used for the fitting must be larger than 1000 K.This last condition is considered in order to make each LDR relation useful for a wide range of targets.For each line pair, one of the four forms (T1)-(T4) was selected to give the most negligible dispersion.
Surface gravity (log g) is also key in deriving subsequent stellar parameters, and analogously to T eff there are multiple ways to estimate it.Its calculation is, however, not trivial.Some approaches rely on numerical models that are computationally expensive, and others depend on how stellar masses and radii can be estimated with photometry and interferometry.In order to estimate log g, we consider pairs of neutral and ionized lines, i.e., Fe i-Fe ii and Ca i-Ca ii pairs taken from the line list of Matsunaga et al. (2021).Their LDRs are defined as We selected the set of the LDR relations, among those which were not rejected, in order to include each line only in one line pair.We finally obtained 12 Fe i-Fe i, 5 Fe i-Fe ii, and 5 Ca i-Ca ii relations (Table 3).

Application of LDR relations
Using the relations we obtained in Section 3.2, we can determine T eff and log g as follows: (1) First, the LDR for each of the Fe i-Fe i pairs can be converted to an estimate of effective temperature, T i .We also estimate its error, e i , considering the error in the LDR and the scatter of the LDR relation (σ p in Table 3).Then, we calculate the weighted average and its standard error by where w i is a weight given by 1/e 2 i .
(2) We then estimate the LDR-based surface gravity by using the T LDR obtained above.Based on each LDR relation of a Fe i-Fe ii pair or a Ca i-Ca ii pair, we obtain an estimate of surface gravity, log g i , and its error.Then, we calculate the weighted mean (log g LDR ) and the error according to the formulae similar to Equations (2) and (3).
Figure 1 summarizes the results for T LDR and log g LDR we obtained for the calibrators together with the results for the validators.The derived T eff and log g for each phase are listed in the supplementary material.The deviations of T LDR and log g LDR from those expected from the literature time-series data show the standard deviations ∼ 90 K and ∼ 0.2 dex.Larger errors are found at higher temperatures for T LDR or at both higher and lower temperatures for log g LDR .Absorption lines of neutral atoms tend to get weak at higher temperatures, while lines of ions tend to get weak at lower temperatures within the range of interest.This trend is observed in the number of the line pairs in the bottom panels and explains the larger deviations and errors seen in the middle panels.The standard deviation of T LDR , ∼ 90 K, is larger than 42 K found by Matsunaga et al. (2021) for stars with 4800 ≤ T eff ≤ 6200 K and 1.35 ≤ log g ≤ 4.5, while the standard deviation of log g LDR ∼ 0.2 dex, is similar to 0.17 dex obtained in the aforementioned work.The larger standard deviation of T LDR can be ascribed to the fact that our sample includes significantly more stars with higher temperatures.The errors depend also on S/N .We will discuss this point in Section 6.

Dependency of log g on T eff and P
Looking at the (T eff , log g) obtained with the literature curves, for the calibrators sample, it can be noted that the two parameters are tightly correlated (Figure 2).Such a correlation can be expected given the fact that the mean stellar parameters of Cepheids fall within the Cepheid instability strip, i.e., their mean T eff , log g, and also the luminosity L should be (anti-) correlated.However, T eff at individual phases are not necessarily within the instability strip.Moreover, the variations of T eff and L do not follow the trend of the instability strip.A Cepheid becomes fainter when it gets cooler, while the instability strip makes fainter Cepheids warmer on average.Nevertheless, Figure 2 shows a tight correlation between T eff and log g at individual phases of the calibrator Cepheids.Fitting the 51 points available for 8 calibrators in total, we obtained the relation, with a scatter of 0.179 dex.
Furthermore, we found that including the log P term reduces the scatter.As illustrated in Figure 3, the residual around the T eff -log g without the log P relation shows the dependency on the period.With the log P with the scatter of 0.108 dex.This is smaller than the errors in the log g LDR presented above.It is hard to estimate log g with such high precision based solely on spectra (Mészáros et al. 2013).Although this relation is subject to systematic errors in Luck (2018), we can robustly obtain log g that would allow the abundance measurements to be consistent with the results in Luck (2018).The derived stellar parameters are found in detail in Appendix A.
Therefore, once T LDR gets estimated, we can use the relation (5) to estimate the surface gravity of classical Cepheids.Let us denote thus-estimated surface gravity as log g trend .For the precision of log g trend , we need to take into account the temperature error.A moderately large error of 100 K in T LDR would lead roughly to the error of 0.12 dex in log g trend combined with the scatter of the T eff -log g-log P relation.Using log g trend is, therefore, more accurate and robust than using log g LDR .

Line Selection
In this section, we discuss the line list to use for the abundance analysis.The first step is to select lines that are useful for measuring the abundances of different elements in the atmosphere of Cepheids.Concerning Fe i, we adopt the list of lines selected from VALD3 (Ryabchikova et al. 2015) and MB99 (Meléndez & Barbuy 1999) that were compiled by Kondo et al. (2019).We also include other elements, among these lines we have Si i, S i, Ca i, Ca ii, Fe ii, Zn i, Y ii, and Dy ii from VALD and MB99 line lists.Most of the lines were confirmed and validated by examining the time-series highresolution spectra we had available for this study.Figure 4 illustrates the absorption lines of different elements in our sample, the extended atlas of identified lines can be found in the supplementary online material.
For the species other than , we searched for candidate lines using synthetic spectra with MOOG (Sneden et al. 2012) considering a set of typical stellar parameters for Cepheids (T eff = 5900 K, log g = 2.0, [Fe/H] = 0.0).We employed the stellar atmosphere models from MARCS (Gustafsson et al. 2008) ([M/H] ∈ [−2.5, 0.5]) spherical geometry for 5 M ⊙ .The solar abundances in MARCS models are those of Grevesse et al. (2007), we converted them to the Asplund et al. (2009) scale.Potential candidates were then filtered according to the following criteria: • They have to be deep enough (d > 0.03) • Blending ratio (the fraction of the EW of contaminating lines, β 1 , defined in Kondo et al. (2019)) less than 0.3.
• Having a wide (T eff , log g) range, ∆T eff = 1000K, ∆log g = 1.0.The selection of lines resulted in 75 lines from VALD and 64 lines from MB99, and for the species other than Fe i.Including Fe i, the line list contains 105 lines from VALD and 90 lines from MB99.
We consider the line information obtained from VALD and MB99 separately.If the same line appears in both lists, we calibrate its log gf and measure the abundance using the line twice.In the analyses with each line list, we use each list for all metallic lines the atomic lines, including other species.

Line-by-line abundance
The abundance inferred from an individual line is estimated by iteratively searching for the set of free parameters, including the abundance of interest, line broadening, wavelength shift, and continuum normalization factor that results from the synthetic spectrum reproducing the observed spectrum with the minimal residual.We employ the OCTOMAN code (Taniguchi et al. in prep.), which performs such optimization based on the MPFIT algorithm by Takeda (1995).This algo-rithm has been utilized in various applications, including studies conducted with WINERED spectra (Kondo et al. 2019;Fukue et al. 2021).
The utilization of the OCTOMAN tool for estimating abundances may encounter difficulties in certain scenarios.For instance, when blended lines are present but not accurately represented in the synthetic spectra, the fitting process may be compromised, resulting in unreliable estimates of the parameters including the abundance.In such cases, it is appropriate to reject the measurements.For each absorption line, we measured the abundance as a function of microturbulence.We considered a grid with 24 different ξ values ranging from 1.4 to 6.0 km s −1 with the step of 0.2 km s −1 .Each run of OCTOMAN was performed for a fixed ξ ∈ [1.4 − 6.0] for each spectrum.
Due to the effect of line saturation, the abundance obtained in each OCTOMAN run depends on ξ.As illustrated in the example in Figure 5, different lines show different trends of abundance against ξ.Deep lines show higher dependency on ξ because they are more saturated, while weaker lines show little dependency.When many lines are strong and show the dependency on ξ, the averaged abundance itself depends on ξ, i.e., the estimates of abundance and ξ show degeneracy.In contrast, if the majority of the lines are weak and depend little on ξ, the final estimate of abundance is more-orless independent of the estimated ξ.
In some cases, the OCTOMAN failed to give reasonable abundances for a part of the ξ grid points.The rejection criteria for the OCTOMAN outputs consider the parameters, niter, fwhmv, rv, and cnorm.The OC-TOMAN measurements that passed the mentioned criteria are considered good when we have 19 or more grid points of ξ for each line.Also, when we encounter ξ − [X/H] curves displaying a significant upturn by more than 0.05 dex, that means, if we find that the value of [X/H] at the largest grid point of ξ is higher than the [X/H] at the smallest ξ by 0.05 dex or more, that curve will be rejected.Thus, we accept curves having a sufficient number of good measurements showing a flat or decreasing trend over ξ.Otherwise, we reject the measurements of a given line entirely if we get less than 20 measurements over the ξ grid.
The accepted curves for Fe i lines are used when we determine ξ in Section 5.1.Also, for other species, examining such trends is useful to see how the microturbulence affects the measured abundance.Moreover, we use a plot like Figure 5 to identify the absorption lines whose measurements for a particular spectrum need to be rejected.We calculate the mean and standard deviation, σ, at each ξ grid point.The solid black curve in  Because of the different dependency on ξ, some curves of good measurements (especially the lines showing strong dependency on ξ) may well get outside the range between the upper and lower limits at some ξ.However, we reject the absorption lines whose curve in the ξ-[X/H] diagram is outside the limits at all the ξ grid points.We make this rejection of the outlying curves once and re-calculate the mean and standard deviation at each ξ.

Calibrating the oscillator strengths
We calibrate the log gf values of absorption lines present in Cepheids using our spectra of the calibrators.We assume that the stellar parameters including the microturbulence (ξ) are known for each phase of spectroscopic observation.For the calibrators, as mentioned above, we obtained the stellar parameters (T eff , log g, ξ) at each phase of our observations with the WINERED by interpolating the curve of each parameter given by Luck (2018).For each combination of absorption line and spectrum, we run the OCTOMAN to get the abun-dance.The deviation ∆log gf of this abundance from the known abundance (Luck 2018) includes, together with other errors, the offset in log gf .
We find that the ∆log gf values obtained for deep lines tend to show large systematic offsets.This trend is clearer in the elements with both weak lines and very strong lines like Si i, more than Fe i. Figures 7-8 show all the available ∆log gf i values for the case of neutral Si.The ∆log gf i values tend to be positive at depths larger than 0.2 -0.25.This may be partly attributed to a systematic trend buried in the original log gf values in VALD and MB99.However, such a systematic trend of ∆log gf exceeding 0.5 dex for all strong lines is unexpected.It is instead understood as the limitation of using fixed microturbulence.As we discuss in more detail in Section 6, the microturbulence is expected to be significantly larger in the upper layers of the stellar atmosphere.Trying to calibrate log gf with underestimated ξ would lead to overestimates as we see here.We thus use the measurements for lines shallower than 0.2 in depth (the separation of the absorption core from the continuum level in each observed spectrum).In addition, among ∆log gf of the not-too-strong lines, we reject outliers with the 2 σ clipping according to the standard deviation.
Without including the measurements of lines whose depths were more than 0.2, we demand having more than 20 good measurements for each line in order to proceed with the calculations of new log gf .The accepted measurements made a toll of 97 lines for VALD and 86 lines for MB99.Subsequently, OCTOMAN was run for each of the accepted lines in order to obtain the abundance of each of the elements, let us denote it as [X/H], as a function of ξ.
For the accepted ξ − [X/H] curves, we estimate the [X/H] at the ξ given (expected) from the Luck (2018) dataset (Section 3), and thus, we can obtain the necessary shift to match the literature [X/H] lit values.
We denominate such an offset given by each spectrum (∆log gf i ), and calculate the mean of it from all the spectra that give accepted measurements for the particular line.In order to calculate the mean we need 20 or more ∆log gf i values.The final error of the calibrated log gf of this offset is obtained by calculating the standard deviation of the ∆log gf i .This calibration was effective enough for 42 VALD lines and 37 MB99 lines for Fe I, and for 48 VALD lines and 37 MB99 lines for the species different that Fe I.The results of this procedure are given in Table 7 and Table 8 given in Appendix B. In this section, we demonstrate the abundance analysis using the calibrated log gf but without using the stellar parameters estimated with the measurements by Luck (2018).Our analysis starts by estimating T LDR and log g trend as described in Section 3 and moves on to the determination of ξ before the abundance measurements.The basic idea for determining the microturbulence ξ is to find the condition that makes the abundances from lines of various strengths (non-or lesssaturated lines to saturated lines) consistent.Mucciarelli (2011) discussed some methods of determining ξ.We measure the abundance from each line as a function of ξ.Combining the results for lines with different strengths, we search for ξ with which the abundances show no dependency on line strength.We use the X index:

Determining microturbulence
as the indicator of line strength (Magain 1984;Gratton et al. 2006;Kondo et al. 2019).The equivalent width might be a good indicator, but measuring the equivalent width in the observed spectra is not a trivial task as other factors, such as blending, can easily disturb the results.We measure the line depth and use it for rejecting lines deeper than 0.2 in depth, but prefer to use the X indicator here because it is a theoretical value unaffected by observational errors.Then, at each ξ, we fit the linear relation,  [Fe/H] = aX + b, to the abundances ([Fe/H]) from individual lines as a function of the X index.The slope a changes, usually in a monotonic way, with ξ and we select the ξ, one of the ξ grid points with the 0.2 km s −1 step, that gives the slope closest to zero.Fukue et al. ( 2021) performed a simulation to find that determining ξ requires at least 20 absorption lines.We use only Fe i lines to estimate ξ.

Abundances with a given microturbulence
Once the microturbulence is obtained, it is relatively simple to derive the abundances making use of an established list of lines (with calibrated log gf ).For each species, we calculate the weighted mean of [X/H] i from N individual lines that were not rejected, using OC-TOMAN to calculate [X/H] i for individual lines and took the weighted mean, where the weights are given by w i = 1/e 2 i and e i is the error in the calibrated log gf .We consider the weighted standard deviation, as the error of [X/H].Using the standard deviation rather than the standard error (i.e., the standard deviation divided by √ N − 1) is commonly done and recommended by Jofré et al. (2019).
However, Equation ( 9) would underestimate the uncertainty of the averaged abundance when [X/H] i with a small number of lines get closer, than expected from the statistical errors to each other by coincidence.We consider another indicator of uncertainty, which is given by the error propagation.Then, we take the larger of e X,1 and e X,2 as the error e X of the derived [X/H].The results of this analysis can be found in Appendix C. When we have [X/H] estimated with multiple spectra of each Cepheid, we would like to take the mean of phase-by-phase measurements.We calculate the weighted mean using the formula like Equation ( 8) but with the weights that are determined with the e X of individual phases.The errors are also determined in the same way by taking the larger of the weighted standard deviation (Equation 9) and the propagated error (Equation 10)

Comparison with the abundances in Luck+18
Using the 46 VALD and 35 MB99 lines with log gf calibrated, we determined the abundances, [X/H], of 9 species.The results are given in Appendix (number).Figures 10 plot [X/H] and the deviations from the literature, Luck (2018), for the calibrators and the validators.For a few species, like Si i (Figure 10), we find very good agreements between our measurements and the literature values (σ ∼ 0.07 dex).Some species, however, show large scatters (σ > 0.15 dex).
Table 4 summarizes how well the abundances we derived agree with the literature values.The mean offsets from the abundance scale of Luck (2018) are not significant for both calibrators and validators.This agreement is expected, at least for the calibrators, because we calibrated log gf based on the abundances of Luck (2018).Yet, the standard deviation is large for some species, in particular, Ca ii and Zn i.The large SDs may be partly attributed to errors in Luck (2018).No significant offsets are found (compared to SD).Looking at Figure 9 for Ca i and Figure 10 for Si i, the abundances of some objects from Luck (2018) seem to show unexpected offsets from other objects; see, e.g., X Cyg for Zn i.

CONCLUDING REMARKS
We have successfully established a list of elements in neutral and ionized state of nine species (including Fe) that are useful for measuring the abundances of classical Cepheids.We calibrated the log gf of these lines by comparing our measurements with the abundances of eight calibrators in Luck (2018), and thus-calibrated log gf gives the abundances of the validators consistent with those in Luck (2018).The precision for a few species is as high as the abundance analysis with optical spectra.A few points are worth further consideration.
(1) The method described in this work for obtaining effective temperatures using the LDR relations leads to high precision (≤ 100 K) except for some cases with high T eff (≥ 6000 K) and/or low S/N (≤ 150).This gives a rough guide to future observers; YJ -band spectra with S/N ∼ 100 or higher would allow precise abundance measurements, while warmer Cepheids would require higher S/N.
(2) The impact of microturbulence on the strength of each line is known to be affected by the line strength.In the determination of log gf of a given species, a question arises as to whether it is appropriate to use ξ obtained solely with Fe i lines.We have observed so that the utilization of stronger lines, such as those of Silicon or Strontium, may also impact the resulting ξ, providing evidence that ξ changes along the atmosphere of Cepheids at different τ .This phenomenon was initially  reported by Takeda & Sadakane (1997) in an HB star, and later on in RR Lyrae by Takeda et al. (2006); Kolenberg et al. (2010);Fossati et al. (2014).This suggests that ξ increases with height in the atmosphere of RR Lyrae stars, as reported in Fossati et al. (2014).Additionally, it is recommended that a specific ξ be applied to lines with specific strength to avoid erroneous abundance estimates.Considering the depth-dependent ξ could make it possible to include such strong lines, but we have no set of atmosphere models with depthdependent ξ for Cepheids in which the turbulent velocity field may well be different from static stars.Therefore, in this paper, we rejected lines stronger than 0.2 and focused on weaker lines and we suggest to avoid such strong lines in the future abundance analyses.
(3) Phosphorus, an odd-Z (Z = 15) element, whose isotope belongs to the same group as Nitrogen is believed to have been produced during the C and Ne burning.It is one of the most fundamental life-bearing elements dominated by macromolecules composed mainly of Carbon that are present in every single organism formed by cells.Each contains membranes and inner organelles whose primary component is a molecule denominated Phospholipid (a.k.a.phosphatides) which is a class of lipid whose molecule is formed by a hydrophilic head where the phosphate group is located as well as two hydrophobic tails.Moreover, from Biology, we know that when DNA is synthesized, a molecule called adenosine triphosphate (ATP) brings energy to the whole DNA structure, by keeping it together, as a superglue.Phosphorus, chemically-wise, has been surprisingly skipped for years.An example of this is that prior 2011, with the study of Caffau et al. (2007), P was barely analyzed in stars.A possible cause for this sort of dodging was raised back in the early 1930's by Struve (1930) who found that no neutral P was available in spectra of stars whose spectral type ranged from F-K. Though, it was acknowledged that Phosphorus ions might be visible in other kinds of stars or other regimes (Maas et al. 2020;Naghma et al. 2018), such as P ii, P ii, P iii, P iv observable in UV spectra.A handful of studies have been done on stars after the study of Caffau et al. (2007) which first suggested using the high-excitation P i lines in multiplet at 1051-1068 nm.Different authors have focused their attention on various other targets of interest: such as Hubrig et al. (2009) obtained P i abundance of HB stars in globular clusters.Meléndez et al. (2009) investigated the P i abundances of a handful of solar twins.Sbordone et al. (2009) suggested that Sulphur could have been produced by proton capture on P, leading to a debate on how possible it is to create the amount of S found there as a product of P when P was never measured on stars in that globular cluster.The work on stars in our galaxy was first reported with the P abundance on 22 MS (main sequence) stars Caffau et al. (2005), stars in the galactic disk Caffau et al. (2011), and dwarf stars (Caffau et al. 2016).Discussions on biological and geological implications can be found, e.g., in Hinkel et al. (2020).We detected more lines of P i present in our NIR spectra observed with WINERED whose abundances were also measured.None of the aforementioned studies have included Cepheids.Our study on the P abundance is probably the first to be performed on stars as hot and variable as Cepheids.We confirmed ten P i lines in our spectra of which 9 had their log gf values calibrated.
(4) From the works of Luck ( 2018) on hundreds of Cepheids, which itself included our sample of Calibrators and Validators, the abundances of a handful of elements were provided.Our estimates of chemical abundances are in agreement with the abundance gradient found by Luck (2018) and prove that we can use spectra in the Y J bands to take the study of the Galactic Chemistry to the next level by including the interstellarhidden Cepheids that can be observed in the infrared only.

ACKNOWLEDGMENTS
We are grateful to the staff of the Koyama Astronomical Observatory for their support during our observation.This study is supported by JSPS KAKENHI No.18H01248 and No.19KK0080.The WINERED spectrograph was developed by the University of Tokyo and the Laboratory of Infrared High-resolution Spectroscopy (LiH), Kyoto Sangyo University under the financial support of KAKENHI (numbers 16684001, 20340042, and 21840052) and the MEXT Supported Program for the Strategic Research Foundation at Private Universities (numbers S0801061 and S1411028).SE acknowledges the financial support of Millenium Nucleus ERIS NCN2021 017, and the ANID Millennium Institute of Astrophysics MAS (ICN)12.009.DT is financially supported by JSPS Research Fellowship for Young Scientists and accompanying Grant-in-Aid for JSPS Fellows, DT (No. 21J11555).DT also acknowledges financial support from Masason Foundation.

Figure 1 .
Figure 1.Performance of our LDR method for T eff and log g

Figure 3 .
Figure 3. T eff -log P -log g trend, displaying the dispersion when the pulsation period P is included (right) or not (left) in the relation.T mid = 5800.

Figure 5
Figure5indicates the curve of mean, while the dashed curves indicate the upper and lower limits (the mean ±2 σ).Because of the different dependency on ξ, some curves of good measurements (especially the lines showing strong dependency on ξ) may well get outside the range between the upper and lower limits at some ξ.However, we reject the absorption lines whose curve in the ξ-[X/H] diagram is outside the limits at all the ξ grid points.We make this rejection of the outlying curves once and re-calculate the mean and standard deviation at each ξ.

Figure 4 .
Figure 4. Species found in order 53 (Y band) for three calibrators, SU Cas, δ Cep, and X Cyg.Lines in gray correspond to those found in Fukue et al. (2021), while those in red correspond to those newly found in this study.

Figure 5 .
Figure 5. Fe i abundances as a function of microturbulence.The case of η Aql.

Figure 7 .
Figure 7. ∆log gf i for Si i lines from VALD.

Figure 8 .
Figure 8. ∆log gf i for Si i lines from MB99.

Figure 9 .
Figure 9.The derived abundances (upper) and the deviations from the literature values (lower) for Ca i.

Figure 10 .
Figure 10.The derived abundances (upper) and the deviations from the literature values (lower) for Si i.

Table 5 .
Stellar Parameters derived for the calibrators sample.NT LDR log gLDR e log g LDR N log g LDR log g trend e log g trend LDR NT LDR log gLDR e log g LDR N log g LDR log g trend e log g trend

Table 6 .
Stellar Parameters derived for the validators sample.NT LDR log gLDR e log g LDR N log g LDR log g trend e log g trend CALIBRATED LOG GF FOR FE I AND SPECIES LINES Here, we include the results of our method for calculating the oscillator strengths (log gf values) by following the procedure stated in Section 4.3.

Table 7 .
Calibration of log gf of Fe i lines

Table 7 .
Calibration of log gf of Fe i lines-continued.

Table 8 .
Calibration of log gf of lines other than Fe I

Table 8 .
Calibration of log gf of lines other than Fe I-continued.
C. DERIVED ABUNDANCES FOR THE CALIBRATORS AND VALIDATORS SAMPLE.

Table 9 .
Derived microturbulence and Fe i abundance

Table 9 .
Derived microturbulence and Fe i abundance-continued.

Table 10 .
Abundances other than Fe I (Part 1)

Table 12 .
Abundances other than Fe I (Part 3 -continued)