The use of double-mode RR Lyrae stars as robust distance and metallicity indicators

CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, China

Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China

School of Astronomy and Space Science, University of the Chinese Academy of Sciences, Beijing, 100049, China

Department of Astronomy, China West Normal University, Nanchong, 637009, China

These authors contributed equally: Xiaodian Chen, Jianxing Zhang.

RR Lyrae stars simultaneously pulsating at two different periods are classified as double-mode RR Lyr stars (RRd). The majority of them are classical RRd stars, characterized by the presence of the radial fundamental and first-overtone modes, where the first-overtone mode is usually the dominant mode. The default RRd stars in this paper are the classical RRd stars. We collected 1021 Galactic RRd stars from Gaia DR3[1], 2083 and 674 RRd stars belonging to the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC) from the Optical Gravitational Lensing Experiment (OGLE) database[2], and plotted the period–period ratio diagram (Petersen diagram[3]) in Fig. 1a and Fig. 1c. At each period, the RRd stars in different galaxies have a consistent period ratio distribution. Theory[4, 5] and observations[6, 7] suggest that metal-rich RRd stars have a shorter fundamental period ($P_{\rm F}$) and a smaller period ratio ($P_{\rm 1O}/P_{\rm F}$) compared to metal-poor RRd stars. We found that SMC RRd stars have longer mean periods and larger mean period ratios than the LMC and Galactic RRd stars (see Extended Data Fig. 1). Alternatively, the distribution of periods and period ratios of LMC and Galactic RRd stars is wider.

We cross-matched Gaia DR3’s RRd stars with all available spectroscopic observations and found 68 and 32 with metallicities from the Sloan Digital Sky Survey[8] (SDSS) and the Large Sky Area Multi-Object Fiber Spectroscopic Telescope[9] (LAMOST), respectively. Based on the Zwicky Transient Facility (ZTF) DR14 photometry, we additionally discovered twice the number of RRd stars that have spectral parameters. Our final sample contains 207 and 96 RRd stars with metallicity measurements from SDSS and LAMOST, respectively. These metallicities were measured from low-resolution spectra ($R\sim 2000$) with an external error of 0.13-0.19 dex. For either the SDSS or LAMOST metallicities, we found an intuitive linear relationship between the metallicity and the period ratio (Fig. 1b) or period (Fig. 1d). The best-fit period–period ratio–metallicity relations are shown in Equ. 1. $\sigma$ is the dispersion of the relations, while $R^{2}$ is the coefficient of determination. The use of logarithmic period ratios and periods facilitates comparison with theoretical results. We also adopted a mean period ratio of 0.745 and a mean period of 0.37 days as the zero point so that the intercept can directly present the metallicity. The relations for SDSS and LAMOST RRd stars are consistent with each other if uncertainties are taken into account. The main contribution of dispersion is the metallicity uncertainty of the low-resolution spectra. When we selected a better sample based on the internal error of metallicity, the dispersion of the period-period ratio-metallicity relation gradually decreases as the criterion becomes tighter. The best case is when limiting $\sigma_{\rm[Fe/H]}<0.04$ dex, based on 56 SDSS RRds, we obtained a dispersion of 0.13 dex. By error comparison (also see method), we found that the metallicity estimated using the period and the period ratio of RRd stars can be as accurate as the low-resolution spectra. The subsequent discussions in this paper are based on the SDSS RRd stars considering the larger number and the smaller metallicity uncertainty.

Due to an approximate linear correlation ($R^{2}=0.70$) between period and period ratio, the period–period ratio–metallicity relation can be simplified to period–metallicity relation or period ratio–metallicity relation. The dispersion of the linear relation between $\log P_{\rm F}$ and $\log P_{\rm 1O}/P_{\rm F}$ is $\sigma_{\log P_{\rm 1O}/P_{\rm F}}=0.0003$ (corresponding to a metallicity dispersion of 0.05 dex from Equ. 1). In terms of accuracy, the period–metallicity relation is closer to the period–period ratio–metallicity relation. Equ. 2 shows the determined period–metallicity relations for SDSS RRd stars, using the same sample as Equ. 1. These relations are important for the optimization of the theoretical model and the use of RRd stars for high-precision distance measurements. In Fig. 1, we compared our relations with the theoretical model of RRd stars[5]. Here we assumed that the metal abundance of the Sun is $Z=0.019$ and that all heavy elements vary by the same factor in different RRd stars. The black star symbols are 7 theoretical grid points (Zero-Age-Horizontal-Branch luminosity level) located near the observed sequences on the Petersen diagram. The general trends of theoretical grid points and observed period ratio–metallicity relations (Fig. 1b) and period–metallicity relations (Fig. 1d) are consistent. However, the uncertainty of the period ratio calculated by the nonlinear pulsation theory (see Fig. 1b) is 1-2 orders of magnitude larger than that of the observations ($\sigma_{P_{\rm 1O}/P_{\rm F}}=2\times 10^{-5}$). In Fig. 1c, we also compared the theoretical (black lines) and observed (colorful lines) equal-metallicity lines on the Petersen diagram. The lines for ${\rm[Fe/H]}=-1.5$ dex and $-1.8$ dex are in perfect agreement, but the theoretical line for ${\rm[Fe/H]}=-1.28$ dex corresponds to a smaller period ratio. We suspected that the possible reason for the discrepancy is the enhanced mass of the metal-rich RRd stars. Mass enhancement leads to an increase in the period ratio and a moderate decrease in the fundamental period[7, 10]. Our observed period–period ratio–metallicity relation can help to optimize the theoretical models, mainly at the metal-rich and metal-poor ends, and the accuracy of the period ratios. Once the theoretical model can predict the period–period ratio–metallicity relation well, it can also predict the period–luminosity relation (PLR) of RRd stars.

Fundamental-mode (RRab) and first overtone-mode (RRc) RR Lyr stars satisfy the period–metallicity–luminosity (PLZ) relations $M=a_{0}+a_{1}\log P+a_{2}{\rm[Fe/H]}$ in observations. In the optical band, these PLZ relations can be simplified to the metallicity–luminosity relations[11]. These relations can be predicted theoretically by combining the horizontal-branch evolutionary model and the pulsation model[5]. The two periods of RRd stars also satisfy the PLZ relation of RRab stars and RRc stars, respectively. Combining the period–period ratio–metallicity relation, we obtained $M=b_{0}+b_{1}\log P_{\rm 1O}+b_{2}\log{P_{\rm 1O}/P_{\rm F}}$. Due to the approximate linear correlation between $\log P_{\rm 1O}$ and $\log{P_{\rm 1O}/P_{\rm F}}$, the relation can be simplified to $M=c_{0}+c_{1}\log P_{\rm 1O}$ or $M=d_{0}+d_{1}\log P_{\rm F}$. The reasons why we prefer to use the PLR rather than the period–period ratio–luminosity relation are discussed in the Methods section. We used the LMC RRd stars from the OGLE database[12] to determine their PLRs. Since the distance of each RRd star with respect to the LMC mid-plane is non-negligible, using about 2000 RRd stars avoids the bias due to incompleteness. We adopted Wesenheit magnitude[13] to reduce the effect of extinction, i.e. $W_{VI}=I-1.55(V-I)$. As for the Gaia passbands, the Wesenheit magnitudes is $W_{G,BP,RP}=G-1.90(BP-RP)$[14]. The determined $M_{W}-\log P$ relations are shown in Equ. 3 and Fig. 2a. ${\rm DM_{LMC}}$ is the distance modulus of LMC. Since the primary period of an RRd star is the first-overtone period, it is preferable to use the first-overtone period to calculate the absolute magnitude, especially if the second period cannot be measured accurately. Nevertheless, the difference between the absolute magnitudes estimated by $P_{\rm F}$ and $P_{\rm 1O}$ is negligible for Gaia or OGLE RRd stars ($0.000\pm 0.001$ mag).

In addition to the theoretical derivation, there are direct observational evidences that can show that the PLR of RRd stars is independent of metallicity. The most direct evidence is that the PLR of RRd stars can be derived from the PLZ relation of RRab or RRc stars by simply using the period–metallicity relation of RRd stars to remove the metallicity dependence.

LMC RR Lyr stars from the OGLE database were used here. We took RRc stars as an example and determined their PLR as $M_{W_{VI}}=(-3.14\pm 0.03)\log P_{\rm 1O}+(16.66\pm 0.01)-{\rm DM_{LMC}},\sigma=0.136$ mag. The slope and intercept of RRc stars’ PLR are very different from those of RRd stars (see Equ. 3). We then added a metallicity-dependent term $+0.13({\rm[Fe/H]}+1.64)$ to the PLR of RRc stars, where 0.13 comes from theoretical calculations[5] and $-1.64\pm 0.20$ dex is the mean metallicity of LMC RRd stars (calculated by Equ. 1). Here we assumed that RRc stars and RRd stars have the same mean metallicity. In the Milky Way, the mean metallicities of RRab, RRc, and RRd stars are $-1.50\pm 0.37$ dex, $-1.62\pm 0.42$ dex, and $-1.71\pm 0.32$ dex, respectively, based on the Gaia DR3 RR Lyr sample and the SDSS metallicities. The mean metallicity differences between RRd stars and RRab, RRc stars are in the range of $0.1-0.2$ dex, and these differences have been considered since we used an error of 0.2 dex for the mean metallicity of RRd stars. This metallicity difference causes a $0.01-0.03$ mag change in the zero point of the PLR, but does not affect the slope. Combining the period–metallicity relation of RRd stars, we obtained the PLR as $M_{W_{VI}}=(-4.46\pm 0.07)\log P_{\rm 1O}+(16.08\pm 0.04)-{\rm DM_{LMC}}$, which is exactly consistent with the PLR of RRd stars. Similarly, we used RRab stars[5, 15] to obtain the PLR of the fundamental mode as $M_{W_{VI}}=(-4.58\pm 0.07)\log P_{\rm F}+(16.67\pm 0.04)-{\rm DM_{LMC}}$. If we considered the coefficient difference in the logarithmic periods between the two PLRs as the possible remaining metallicity dependence of the PLRs. The results are $\Delta M_{W_{VI}}=(0.003\pm 0.017){\rm[Fe/H]}$ and $\Delta M_{W_{VI}}=(0.006\pm 0.017){\rm[Fe/H]}$ for PLRs based on the first-overtone and fundamental period, which are negligible. These consistencies indicate that calculating the luminosity using the PLR of RRd stars is equivalent to calculating the luminosity using the PLZ relation of RRab or RRc stars. In turn, the period–metallicity relation of RRd stars can be determined by combining the PLZ relations of RRab or RRc stars with the PLR of RRd stars, even without knowing the metallicity of any RRd stars (see Methods). This means that RRd stars are RRab or RRc stars that satisfy the period–metallicity relation.

We estimated the metallicity of RRd stars from the period–period ratio–metallicity relation (Equ. 1) and checked the correlation between the magnitude residual of PLR and the metallicity by LMC, SMC, and Galactic RRd stars separately. Note that this metallicity is not a completely independent measurement, but it can be used to check whether the period ratio affects the PLR through metallicity. For LMC and SMC, we binned RRd stars in order of metallicity, with a bin size of 100. This bin size allows us to detect deviations as low as 0.01 mag. We then fixed the slope of the PLR and calculated the mean magnitude residual $\Delta W_{VI}=W_{VI}-W_{VI,{\rm PL}}$ in each bin separately. In both LMC and SMC, we found that the correlation between the magnitude residual of the PLR and the metallicity is less than 0.03 mag/dex (see Fig. 2b, c).

We estimated the PLR distances of 126 Galactic RRd stars with good Gaia DR3[16] parallaxes ($\varpi>0$, $\sigma_{\varpi}/\varpi<0.25$, $W_{G,BP,RP}<14$ mag and RUWE $<1.4$, $\varpi$ denotes the Gaia DR3 parallax) by assuming an LMC distance modulus of 18.48 mag[17]. The PLR distances were converted to parallaxes and we compared them with the Gaia DR3-corrected parallaxes $\varpi_{\rm corr}$[18]. The mean error of Gaia parallax for this sample is 12%. We found that the difference between the two parallaxes ${\rm zp}=\varpi-\varpi_{\rm PL}$ shows no correlation with metallicities (see Fig. 2d). The mean parallax offset is ${\rm zp}=8.3\pm 2.5\pm 2.6$ $\mu{\rm as}$. The statistical error is the standard deviation divided by the root of the sample size, while the systematic error is propagated from the distance uncertainty (1.1%) of LMC. This parallax offset agrees well with the result determined by classical Cepheids (${\rm zp}=14\pm 6$ $\mu{\rm as}$[19]), contact binaries (${\rm zp}=4.2\pm 1.9$ $\mu{\rm as}$[20]), and red giants (${\rm zp}=15\pm 5$ $\mu{\rm as}$[21]).

As a distance tracer, we used both the LMC distance and Gaia parallaxes to optimize the PLR of RRd stars. The PLR can then be used to determine the distances of distant galaxies or dwarf galaxies. The Gaia parallaxes can also provide an independent constraint for the PLR zero point of RRd stars. Based on the method of ref.[19], we used nonlinear least squares to fit Equ. 4. We fixed the slope of PLR $a_{0}=-3.557\pm 0.225$ (see Equ. 3) and determined $a_{1}=M_{W_{G,BP,RP}}(P_{\rm 1O}=0.37{\rm d})=-0.388\pm 0.051$ mag and ${\rm zp}=13.4\pm 5.8$ $\mu{\rm as}$. Based on the RRd stars with different parallaxes, the degeneracy of these two parameters is largely broken. Based on this PLR, the determined distance modulus of the LMC is $18.530\pm 0.051$ mag. By calculating the weighted average of the PLR zero points determined based on the Gaia parallax and the LMC distance ($M_{W_{G,BP,RP}}(P_{\rm 1O}=0.37{\rm d})=-0.338\pm 0.024$ mag), we obtained the final zero points of PLRs as $M_{W_{G,BP,RP}}(P_{\rm 1O}=0.37{\rm d})=-0.348\pm 0.022$ mag and $M_{W_{VI}}(P_{\rm 1O}=0.37{\rm d})=-0.496\pm 0.022$ mag (1.0% distance uncertainty). Based on 617 SMC RRd stars, the determined distance modulus and average metallicity are ${\rm DM(SMC)}=18.913\pm 0.007\pm 0.022$ mag ($60.62\pm 0.64$ kpc) and ${\rm[Fe/H]}=-1.87\pm 0.19$ dex. Due to the existence of non-negligible irregular spatial structure of SMC[23, 22], the actual error of the average distance will be larger. The metallicity dispersion of SMC RRd stars is smaller than that of the LMC (0.21) and the Milky Way (0.24).

Compared to RRab stars, distance measurements based on RRd stars are no longer affected by metallicities. The difficult-to-measure metallicity is replaced by an easy-to-measure period. In addition, RRd stars can provide the metallicity distribution of the galaxy’s old populations to help RRab stars’ distance measurements. This is crucial because for most galaxies with RRab stars, we cannot directly measure the metallicities of RRab stars. RRab stars’ metallicities are also difficult to infer from the average metallicity of the host galaxy. Although the metallicity of RRab star can be obtained indirectly from the light-curve shape through the parameter $\phi_{31}$[24], this requires high-precision and high-sampling data, otherwise the prevalence of light-curve modulation in RRab will make $\phi_{31}$ measurements inaccurate. In the Gaia DR3 RR Lyr sample, the $\phi_{31}$-based metallicity error (only propagation error) is nearly 50 times larger than the $(P_{\rm 1O},P_{\rm 1O}/P_{\rm F})$-based metallicity error for the same number of epochs (see Extended Data Fig. 2). Besides, optimizing the $L({\rm RRab})=f(P,\phi_{31})$ relationship still requires more work because the value of $\phi_{31}$ is different in different bands. In contrast, RRd stars require only periods to estimate distances, which is not only convenient but also reduces systematic biases. The metallicity effect introduces a systematic error of about 1.8% to the RRab stars-based distance measurements[25]. The metallicity effect and the PLR zero point are the two most significant components of the systematic error in distance measurements of RR Lyr stars and Cepheids. We calculated the root sum square of these two errors and call it the base systematic error. The base systematic error is the lower limit error of the tracer in distance measurement, so it can be used as a criterion to evaluate the goodness of the distance tracer. When using RRab and RRd stars to measure the distances of galaxies or dwarf galaxies, the base systematic errors are $\sqrt{1.8\%^{2}+1.0\%^{2}}=2.1\%$ and $\sqrt{0.0\%^{2}+1.0\%^{2}}=1.0\%$. Compared to classical Cepheids, distance measurements based on RRd stars avoid the effects of metallicity and binarity. The fraction of RR Lyr star in a binary system is as low as 7%[26]. Cepheid pulsations in both fundamental and first-overtone modes was also found to fulfill a period ratio–metallicity relation[27]. However, the smaller number (95 in LMC) and the larger dispersion on the Petersen diagram limit the application of these Cepheids in distance measurements. The dispersion of the period ratios based on the quadratic curve fit is 0.0004 (RRd stars) and 0.0043 (Cepheids with F/1O modes).

We used RRd stars to study four globular clusters: IC 4499, M15, M3 and M68, a dwarf galaxy: Sculptor as examples. IC 4499, M15, and Sculptor are the three targets in the Gaia DR3 RR Lyr catalog with more than five RRd stars, with numbers of 11, 6, and 40, respectively. M3 and M68 contain 7 and 9 RRd stars with periods and $VI$ mean magnitude information from the literature[29, 28]. The determined distances and metallicities based on RRd stars agree well with previous works[30, 25, 31] in $1\sigma$ and $2\sigma$ (see Fig. 3), except for M15’s metallicity ($2.9\sigma$) if we considered a metallicity uncertainty of 0.05 dex in literature. Our overall distance uncertainty is at the level of $2-3\%$ for globular clusters. The distance error of these four globular clusters determined by the different works is typically around 5%, while the best distance error is around 2-3%[32]. The independent distances provided by RRd stars can help to optimize the average distances obtained by combining different methods. The advantage is that it has a base systematic error of only 1%. When combined with other distances (especially photometric distances), more components of the error can be eliminated. The metallicity scatters of RRd stars in globular clusters and the dwarf galaxy Sculptor are about 0.05-0.08 dex and 0.16 dex. Note that dispersion smaller than 0.16 dex may be unrealistic, and here it is certain that the dispersion of the period and the period ratio of RRd stars in globular clusters is much smaller than in dwarf galaxies.

For Sculptor, there are two groups of RRd stars with ${\rm[Fe/H]}=-1.59\pm 0.06$ dex and ${\rm[Fe/H]}=-1.96\pm 0.09$ dex, respectively. This is consistent with the bimodal distribution of metallicity inferred based on the horizontal branching discontinuity and the double red giant branch bumps[31]. The average metallicities of Sculptor based on 107 RR Lyr stars[33] are $-1.83\pm 0.26$ dex (on the scale of Ref.[34]) and $-1.64\pm 0.27$ dex (on the scale of Ref.[35]), the latter being more consistent with our average metallicity $-1.67\pm 0.16$ dex. There are 6 RRd stars among these RR Lyr stars. We found that our metallicity measurements are also consistent with the literature, with deviations of $-0.13\pm 0.25$ dex and $0.07\pm 0.26$ dex for two different scales. The determined distance modulus of Sculptor is $19.51\pm 0.03\pm 0.02$ mag ($79.86\pm 1.46$ kpc). We find the mean metallicities of IC 4499 and M15 are ${\rm[Fe/H]}=-1.60\pm 0.05$ dex and ${\rm[Fe/H]}=-2.10\pm 0.08$ dex. Our M15 metallicity is 0.27 dex higher than the literature, but the deviation is still within $3\sigma$. The reason for the overestimation may be that the metallicity ${\rm[Fe/H]}=-2.37$ has reached the metal-poor end of our period–period ratio–metallicity relation. The determined distance moduli are ${\rm DM(IC4499)}=16.36\pm 0.05\pm 0.03$ mag ($18.66\pm 0.49$ kpc) and ${\rm DM(M15)}=15.04\pm 0.06\pm 0.02$ mag ($10.19\pm 0.32$ kpc).

The period ratios of RRd stars in M3 and M68 are not accurate enough to be used to determine metallicities. We adopted the first-overtone period and converted it to the period ratio based on an empirical quadratic function (Equ. 5 in Methods). Metallicity is then determined by Equ. 1. Metallicities of M3 and M68 are ${\rm[Fe/H]}=-1.53\pm 0.06$ dex and ${\rm[Fe/H]}=-2.06\pm 0.07$ dex. Note that four M3 RRd stars are not classical RRd stars and we excluded them in determining metallicity and distance (see Methods and discussion in Ref.[28]). The determined distance moduli are ${\rm DM(M3)}=15.00\pm 0.08\pm 0.02$ mag ($9.99\pm 0.37$ kpc) and ${\rm DM(M68)}=15.04\pm 0.04\pm 0.02$ mag ($10.20\pm 0.21$ kpc).

The percentage of RRd in RR Lyr stars is 10%, 5% and $\sim 3\%$ in SMC, LMC and the Milky Way halo, respectively. With the continued observations of the ZTF, OGLE and Gaia, the number of RRd stars in the Milky Way will increase by $\sim 4000$ in the next few years. To date, there are at least 22 galaxies or dwarf galaxies with more than 100 RR Lyr stars[36], all of which are suitable for distance (1-2% accuracy) and metallicity measurements using RRd stars. With RRd star’s PLRs determined for the Dark Energy Camera and the Hubble Space Telescope, RRd stars can trace distances and metallicities to 300 kpc and 1 Mpc. With future’s China Space Station Telescope (CSST) and the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST), RRd stars will provide an independent distance ladder for the near-field universe to examine the distance ladder based on the classical Cepheid[37] or the tip of the red giant branch[38].