A Study of the Pulsation Properties of 57 Non-Blazhko Ab-type RR Lyrae Stars with Homogeneous Metallicities from the LAMOST–Kepler/K2 Survey

The Fourier coefficient of A₁ is approximately proportional to the total amplitude of A_tot, since it is the dominant component of A_tot. In order to illustrate this relationship between A₁ and A_tot, we use a cubic and a linear equation to fit the two coefficients A₁ and A_tot following Nemec et al. (2011), who found that the dependence between the two coefficients might be cubic using the 19 non-Blazhko RRab stars observed by Kepler, and Skarka (2014), who analyzed 176 non-Blazhko RRab stars from ASAS and WASP surveys found that the relation might be linear for the two coefficients, respectively. The cubic fitting is as follows:

while the linear fitting is as follows:

When the cubic curve (as shown in the top panel of Figure 3) is subtracted from the A_tot–A₁ diagram, the residuals show no significant deviations from the average value of zero with the rms of 0.0125 mag (as shown in the bottom panel of Figure 3). But for the linear fitting, although the residuals show no significant variations (presented in the bottom panel of Figure 4) after subtracting the linear trending (presented in the top panel of Figure 4) from the A_tot–A₁ diagram, the value of the rms is 0.013 mag, which is slightly larger than that of the cubic fitting. It is worth mentioning that the data points away from 3σ are not considered for these two kinds of fittings as shown in the top panels of the two figures.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Skarka (2014) and Nemec et al. (2011) investigated the relation between the two Fourier coefficients ϕ₂₁ and ϕ₃₁ of the non-Blazhko RRab stars, and they both pointed out that the two coefficients of the stars follow a linear relation. In this work, we perform a linear fitting for the two Fourier coefficients ϕ₂₁ and ϕ₃₁ and the fitting equation is as follows:

The fitting is shown in the top panel of Figure 5, and the residuals are plotted in the bottom panel of the figure, with an rms of 0.082 rad. The data points beyond 3σ away from the trending are not considered in the fitting.

Zoom In Zoom Out Reset image size

Figure 6 shows how the RT correlates with the parameters of the light curves, including the fundamental period, total amplitude A_tot, the amplitude ratio R₃₁, and the phase difference ϕ₂₁. Panel (a) shows the relationship between the RT and the main pulsation period. It can be seen that the longer period probably corresponds to a higher RT for most of the stars. This is the first time that this tendency is noticed for Kepler and K2 non-Blazhko RRab stars. The data points of the total amplitudes A_tot versus RT are scattered as shown in panel (b). R₃₁ and RT might follow a roughly linear trend as shown in panel (c). The distribution of RT versus ϕ₂₁ is scattered as shown in panel (d).

Zoom In Zoom Out Reset image size

A series of studies (Jurcsik & Kovacs 1996; Kovacs & Jurcsik 1996; Kovács & Walker 2001; Jurcsik et al. 2009; Nemec et al. 2013; Ngeow et al. 2016; Plachy et al. 2016; Iorio & Belokurov 2021; Mullen et al. 2021) have been conducted to investigate the period–ϕ₃₁–[Fe/H] relation. In this study, we use the fundamental periods P and the phase differences ϕ₃₁ of the 54 non-Blazhko RRab stars with the metal abundances [Fe/H] of these stars provided by the LAMOST–Kepler/K2 survey to give a new calibration of the relation. This relation can be fitted by the following equation:

where and are the mean values of the fundamental periods and phase differences ϕ₃₁ of the 54 non-Blazhko stars studied in this work, respectively. We adopt the least squares method, which is implemented by the SciPy package (Virtanen et al. 2020) to estimate the best-fitting coefficients with their corresponding standard errors, for which a = −3.650 (1), b = 0.848 (4), and c = −3.992 (1). Figure 7 illustrates the distribution of the [Fe/H] in the ϕ₃₁ versus the plane of the period of those RRLs.

Zoom In Zoom Out Reset image size

In this study, we find that the values of standard errors of cubic fit and linear fit for the relation of the amplitude of the primary frequency A₁ and the total amplitude A_tot of the 54 non-Blazhko RRab stars are σ₁ = 0.013 and σ₂ = 0.0125, respectively, which means that they follow either a cubic or linear relation. We do not find any relation for RT with the fundamental period, total amplitude A_tot, and phase difference ϕ₂₁ of the stars studied in this work. However, Skarka (2014) suggested that the RT follows a linear relation with those parameters for the non-Blazhko RRab stars but does not for the Blazhko RRab stars, which is not consistent with our result. This might be due to that the precision of the photometry from Kepler and K2 is much higher than that of the photometric data in the previous study. As far as the relation between ϕ₂₁ and ϕ₃₁ of the stars, we find that they follow a linear relation. But Skarka (2014) found that the dependence of the two coefficients of ϕ₂₁ and ϕ₃₁ is not very strong.

As the Fourier decomposition coefficients were derived from light curves observed in the Kepler white band of the 54 non-Blazhko RRab, we convert them into V mag using formula (2) of Nemec et al. (2011). Figure 8 shows the correlations of properties of the non-Blazhko RRab stars in this study with the 177 RRab stars located in several Galactic and Large Magellanic Cloud (LMC) GCs (Kovács & Walker 2001). Panel (a) shows the logP–ϕ₃₁ diagram. The stars in the GCs with poor and intermediate metallicities clearly define two edges, respectively. The metal-poor subsample consists of 19 stars, whose metallicities are in the range of −1.70 to −1.99 dex with an average value of −1.8 dex (Kovács & Walker 2001; Nemec et al. 2011). The other subsample stars are the intermediate-metallicity stars containing 39 members, whose metallicities are between −0.97 and −1.23 dex with an average value of −1.1 dex. It is clear that most non-Blazhko RRab stars in the LAMOST–Kepler/K2 fields have metal abundances between −1.80 and −1.10 dex, except for two Kepler stars and five K2 stars with metallicities higher than −1.1 dex. Panel (b) shows that the distribution of the Fourier coefficients R₂₁ and R₃₁ of the 54 non-Blazhko RRab stars in the Kepler/K2 survey are similar to the RRab stars in GCs. For the stars studied in this work, we find that some of them with low R₃₁ values have relatively high R₂₁ values. We also find that most stars appear in the upper right-hand side of panel (b) and most of them have high metallicities. Panel (c) shows that the stars studied in this work do not differ from the stars in GCs. We also note that the metallicities may not show any significant effect in this panel. Panel (d) shows the agreement between the phase parameters of the 54 non-Blazhko RRab stars and GC RRab stars, which supports that the phase parameters of and might follow a linear relation. However, panel (d) also demonstrates very little dependence on the metallicities.

Zoom In Zoom Out Reset image size

As the cluster RRLs adopted in this work cover roughly 1 dex in the intermediate metal regime, we collected the data of the reference stars from a large sample of high-resolution spectroscopic surveys of RRLs (For 2011; Nemec et al. 2013; Govea et al. 2014; Chadid et al. 2017; Sneden et al. 2017; Magurno et al. 2019; Crestani et al. 2021; Gilligan et al. 2021), with the metallicities of RRLs ranging from −3.0 dex to solar or super-solar iron abundance based on those high-resolution spectra. We cross match the catalog of those studies with our data for the reference stars. We derived seven common stars from the study of Nemec et al. (2013) but no common stars from other literature (For 2011; Govea et al. 2014; Chadid et al. 2017; Sneden et al. 2017; Magurno et al. 2019; Crestani et al. 2021; Gilligan et al. 2021) and those stars are all in the Kepler field. But we note that our database of the metallicities of RRLs studied in this work is a subsample of Liu et al. (2020), who collected the data of the reference stars with reliable estimates of metallicity either from high-resolution spectroscopy with the metallicity in the range of −2.95 to −0.59 dex (Clementini et al. 1995; For 2011; Kinman et al. 2012; Nemec et al. 2013; Govea et al. 2014; Pancino et al. 2015) or as a member star of GCs (Harris 2010) in the metallicity range of −2.37 to −1.29 dex. They finally obtained 47 stars in common, which formed their reference star sample. The scale of metallicity adopted by them was the one established by Carretta et al. (2009), which was derived from the old scale of metallicity (Zinn & West 1984). They found that the values of metallicities of RRLs estimated from the low-resolution spectra of LAMOST DR6 agree well with those of the compiled reference stars, with a negligible offset of -0.04 dex and a standard deviation of 0.22 dex. They also found that the dispersion is comparable to that yielded by multi-epoch observations.

We also compare the Fourier decomposition coefficients R₂₁, R₃₁, , and with those derived for the RRLs in the central regions of the LMC. The latter is determined from the OGLE Collection of Variable Stars by Soszyński et al. (2016) and transformed from the I to V band using equations provided by Morgan et al. (1998). It is clear that the coefficients of the 54 non-Blazhko RRab stars determined in this work agree with the coefficients of the RRab stars well and differ from other types of RRLs a s shown in Figure 9.

Zoom In Zoom Out Reset image size

We compare the metallicities [Fe/H] of the stars studied in this work calculated using Equation (7) with those derived by relationships as documented in the literature (Jurcsik & Kovacs 1996; Nemec et al. 2013; Martinez-Vazquez et al. 2016; Iorio & Belokurov 2021; Mullen et al. 2021). For consistency, the metallicities of those investigations are converted to the often-used scale of Carretta et al. (2009). The result of this comparison is shown in Figure 10; the abscissa of all subgraphs refers to the metallicities of the stars calculated using Equation (7), and the ordinate of all subgraphs refers to the metallicities of the stars derived with the relations in the literature. Panel (a) shows a comparison of the metallicities between ours and those of Jurcsik & Kovacs (1996), whose relation was derived using photometric data of 81 RRab field stars in the V band with metallicities based on the high-dispersion spectroscopy scale of Jurcsik (1995). For consistency, we first convert the metallicities derived with their relation to the Carretta et al. (2009) scale using the formula provided by Kollath et al. (2011): [Fe/H]_C09 = 1.001[Fe/H]_JK96 −0.112. We then convert the Fourier coefficient ϕ₃₁ in the K p system to the V band with formula (2) of Nemec et al. (2011). We find that the comparing result between the two relations is obviously different within the calibration range of (−2.1 dex ≤ [Fe/H] ≤ 0.7 dex) of Jurcsik & Kovacs (1996) (red horizontal lines), particularly in the metal-rich regime. This might be due to that the photometric data of Jurcsik & Kovacs (1996) were collected from heterogeneous observations at various sites and either lacked phase coverage or had excessive noise that caused a failure of the Fourier fit in Jurcsik & Kovacs (1996). Nemec et al. (2013) derived a quadratic period–ϕ₃₁–[Fe/H] relation using 19 RRab stars in the Kepler field with accurate metallicities measurements. The comparison of the metallicities between ours and those of Nemec et al. (2013) is shown in panel (b). The Fourier coefficients ϕ₃₁ of the 57 stars are in the same K p system as that of Nemec et al. (2013). Furthermore, the scale of metallicity adopted by Nemec et al. (2011) and ours is of the same scale as that in Carretta et al. (2009). Note that the scatter is obviously large for either the higher metallicity or lower metallicity in their range (−1.5 dex ≤ [Fe/H] ≤ 0.03 dex). Mullen et al. (2021) suggested that this might be caused by the higher-order term of the relationship given by Nemec et al. (2013), and they had only one RRab star with [Fe/H] ≤ −2.0 dex resulting in the scarcity of calibrators in their sample for low [Fe/H]. Martinez-Vazquez et al. (2016) gave a new calibration of the period–ϕ₃₁–[Fe/H] relation based on a sample of 381 RRab stars in the GCs and eight RRab field stars in order to extend the metallicities range of their samples. The metallicities of their sample were on the Carretta et al. (2009) scale, and we also convert the K p system ϕ₃₁ value to the V-band system using formula (2) in Nemec et al. (2011). Our metallicities compared with those of Martinez-Vazquez et al. (2016) show an obvious scatter within the entire calibration range in Martinez-Vazquez et al. (2016), as presented in panel (c). This phenomenon might be due to that the sample of 381 RRab stars was binned by period; they computed the mean period, ϕ₃₁, and V-band amplitude, which means that the calibration provided by Martinez-Vazquez et al. (2016) was based on the average instead of the individual properties. Iorio & Belokurov (2021) derived a G-band period–ϕ₃₁–[Fe/H] relation based on the light curves of 84 RRab stars in Gaia DR2 with known spectroscopic metallicities. For this comparison, we first use formula (2) of Kollath et al. (2011) to convert the ϕ₃₁ value in the Kp system to the V-band system, then convert it to that in the G-band system using formula (6) of Clementini et al. (2016). What is more, an additional π offset should be subtracted from ϕ₃₁ to set the coefficients on the same scale as Iorio & Belokurov (2021) as suggested by Mullen et al. (2021). However, the metallicity abundances adopted by Iorio & Belokurov (2021) were on the scale used in Zinn & West (1984). We convert the metallicity abundances of Iorio & Belokurov (2021) to the Carretta et al. (2009) scale using the formula [Fe/H]_C09 =1.105[Fe/H]_ZW84 + 0.160. The comparison of the metallicities between ours and those of Iorio & Belokurov (2021) exhibit a general agreement within the entire range of metallicity (−2.53 dex ≤ [Fe/H] ≤ 0.33 dex), with an rms of 0.123 dex, as shown in panel (d). However, Mullen et al. (2021) pointed out that the relation given by Iorio & Belokurov (2021) tends to overestimate the metallicity at the metal-poor end and underestimate the metallicity at the metal-rich end. Panel (e) shows the comparison of metallicities between ours and those of Mullen et al. (2021), who used 1980 RRab stars with metallicity in the range of −3.0 dex ≤ [Fe/H] ≤ 0.4 dex. We first convert the ϕ₃₁ value in the Kp system to the V-band system using formula (2) in Nemec et al. (2011). A small shift of 0.08 dex is considered in converting to the often-used Carretta et al. (2009) scale of [Fe/H] as suggested by Mullen et al. (2021). The result exhibits an obvious scatter between the two relations for the entire range of metallicities. This might be due to that the sample of RRab stars in Mullen et al. (2021) is significantly larger than ours.

Zoom In Zoom Out Reset image size