Multiwavelength Period–Luminosity and Period–Luminosity–Color Relations at Maximum Light for Mira Variables in the Magellanic Clouds

The GPR model fits are used to estimate VI-band magnitudes at the same epoch for OGLE-III Miras and to derive PL and PLC relations at mean and max-light. Figure 2 shows the I-band PL and PLC relations for Miras in the LMC. The max-light PLR for O-rich Miras in the LMC displays smaller scatter than the mean-light relation. C-rich Miras do not display any linear relation with their primary pulsation period in optical bands as they exhibit varying levels of circumstellar extinction (Ita & Matsunaga 2011). Adding the color-term results in a significant decrease in the scatter for both O- and C-rich Miras at mean and max-light. The results of the linear (single or two-slope model) and the quadratic regressions to the PL and PLC relations are listed in Table 2.

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The max-light PL and PLC relations exhibit scatter which is up to ∼30% smaller than their mean-light counterparts. The results for the max-light PLR also hold if we only take the minimum numerical magnitude of the light curve as the max-light magnitude in the I band. In this case, the dispersion in O-rich Mira PLR (N = 440, σ = 0.27 mag) is similar to the dispersion when we use our adopted definition (see Table 2). The residuals of Mira PLRs correlate with colors, as expected, but do not show any dependence on their periods. However, the residuals from the mean-light PL and PLC relations are larger than the max-light residuals at both extremes of the I-band amplitude distribution. At minimum light, the O-rich Miras fall well above the faint limit of OGLE-III, and their PLC relations exhibit a dispersion (N = 441, σ = 0.47 mag), almost three times the scatter in the max-light relations.

The period–color relations for Miras at mean- and max-light are shown in Figure 3. For long-period (P > 300 days) O-rich Miras, the period–color relation becomes shallower both at mean- and max-light, and the scatter in the relation over the entire period range is smaller at max-light. If the extinction corrections are applied, no significant changes are noted in the scatter in these relations. Assuming that the scatter in these relations is intrinsic, the temperature variations at max-light are significantly smaller than at mean-light for a group of Miras with similar periods: this results in the tighter PLC relation at the brightest epochs.

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Figure 4 displays the PL and PLC relations at max-light for Miras in the SMC. The number of O-rich Miras is very small in the SMC (see Boyer et al. 2011; Goldman et al. 2018) but they also exhibit significantly smaller dispersions in these relations at max-light when compared with their counterparts at mean-light. Note that the geometric extension and the line-of-sight depth of the SMC are much larger than for the LMC (Subramanian & Subramaniam 2015; Muraveva et al. 2018), which contributes to the larger scatter in PLRs. Therefore, single-slope models are fitted to the SMC PL and PLC relations, considering the dispersion and smaller number statistics. The color dependence of these PLRs for C-rich Miras is similar in both Magellanic Clouds. On the other hand, O-rich Miras in the SMC display a greater color dependence in the PLC relations than Miras in the LMC, albeit also with greater uncertainty. Note that O-rich stars produce dust less efficiently in low-metallicity environments, while C-rich Miras are as efficient as their higher-metallicity LMC counterparts (McDonald et al. 2010; Sloan et al. 2010, 2012, 2016). This difference in the dust properties for O-rich Miras due to metallicity differences can contribute to the changes in the color coefficient of the PLC relation but the effects of molecular absorption and the temperature variations in O-rich Miras may also depend on metallicity.

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To validate the statistical significance of the decrease in the dispersion of the PL and PLC relations at max-light, a parametric F-test is adopted to compare the variances in the mean- and max-light relations. The smaller scatter in the max-light relations may also be due to the smaller number of stars in the final fit, for example, due to iterative outlier clipping algorithm. Under the null hypothesis that the variances are similar in the mean- and max-light relations, the F-test results suggest that the reduction in the dispersion is statistically significant at the 95% confidence interval. The probability of acceptance of the null hypothesis is smaller than 0.05 dex in all cases.

The Gaia mission has provided one of the largest samples of LPVs (Mowlavi et al. 2018), which has enabled us to study PL and PLC relations with simultaneous multiband precise space-based photometry. This LPV sample is restricted to Miras with periods less than 670 days, owing to the observation duration of the Gaia data. Further, some of the epochs in the light curves that were flagged by the photometry and variability processing by the Gaia team are also excluded from this analysis. Although the magnitudes corresponding to most of those epochs fall on the light curves (Mowlavi et al. 2018), these were not included in our adopted criterion of the minimum number of 15 epochs per light curve. The mean- and max-light magnitudes in G, BP, and RP filters are obtained for epochs of simultaneous photometry in all three filters.

Figures 5 and 6 show the PL and PLC relations for Miras in the LMC based on G and RP band magnitudes and (BP − RP) colors, respectively. The results of the regression analysis are tabulated in Table 3. Similar to the traditional VI filters, PL and PLC relations at max-light display significantly smaller dispersions than at mean light in the Gaia filters as well. The scatter in the max-light PLC relations for O-rich Miras is comparable to that for the PLRs in the NIR JH bands in the LMC (Yuan et al. 2017). Figure 7 presents period–color relations for Miras using Gaia (BP − RP) colors. The O- and C-rich Miras display two distinct relations at mean-light, while the temperature variations are smaller at max-light. The max-light colors for C-rich Miras are very similar to O-rich Miras. The flatter color variation for longer-period O-rich Miras is also seen at mean-light similar to that in (V − I) color.

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Figure 8 displays the PL and PLC relations at max-light for Miras in the SMC. The decrease in the scatter in these relations is consistent with the results for Miras in the LMC. The color term for C-rich Miras is almost twice that for O-rich Miras in all optical colors. Assuming similar interstellar extinction, the effect of circumstellar extinction on C-rich Miras may be twice as large as that for O-rich Miras.

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Optical and NIR PL and PLC relations for long-period (P ≳ 300 days) O-rich Miras exhibit a steeper slope than the shorter period Miras (Feast et al. 1989; Ita & Matsunaga 2011). The excess luminosity in long-period O-rich Miras possibly occurs due to hot bottom burning (HBB, Sackmann & Boothroyd 1992; Whitelock et al. 2003) episodes in massive AGB stars (∼3−5M_⊙, for details, see Karakas 2014; Marigo et al. 2017). For AGB stars with initial masses between ∼3 and ∼5M_⊙, nuclear burning at the base of the convective envelope leads to luminosities that are higher than the core-mass–luminosity predictions. The HBB process also affects the surface chemistry of AGB stars, and the lower mass limit to initiate these episodes depends on metallicity (Marigo et al. 2013). The mass loss in AGB stars also changes as it evolves toward brighter luminosity and longer periods (Vassiliadis & Wood 1993; Goldman et al. 2017; Höfner & Olofsson 2018). Empirically, the exact period at which these effects may cause a change in the slope of the PLRs is not well quantified. Ita & Matsunaga (2011) found a kink in the PLRs around 400 days. A quadratic relation also seems to provide a robust fit to PLRs over the entire period range of O-rich Miras (Yuan et al. 2017).

The statistical significance of linear, nonlinear, or quadratic models is investigated using several methods: Schwarz Information Criterion (SIC), Random Walk (RW), Testimator (TM) and F-test (FT; see Kanbur et al. (2007) and Bhardwaj et al. (2016a) for mathematical details). In the case of a nonlinear regression model, a break period is varied between 200 and 500 days in steps of $\mathrm{log}P=0.01$ day. The SIC is used to compare the quadratic and two-slope regression model by maximizing the likelihood function and the model with the smallest SIC value is adopted as the preferred model. A break period is obtained from the two-slope regression model with the smallest SIC value. The RW method gives the probability of the acceptance of the null hypothesis that a nonlinear regression is the best model, and the break period corresponds to the maximum probability. The TM compares the slopes for different subsets of the data under consideration and provides a period range where the PLRs depart from linearity. The F-test compares the variances of the quadratic and nonlinear regression models under the null hypothesis that the two variances are equal. The period corresponding to the model with the largest F-value is adopted as the break period. Finally, a quadratic relation is compared with a linear relation for O-rich Miras with periods smaller than 400 days. The probability p(f) < 0.05 suggests that the two variances are not equal.

Table 4 summarizes the results of the statistical tests to determine a break in the PL and PLC relations for O-rich Miras in the LMC. As these methods are sensitive to the dispersion of the underlying relation, optical PLC relations and NIR PLRs are used to determine the break period. Nonlinear regression is the best model, according to SIC. The break period varies between 2.35 < log P < 2.55 days at different wavelengths for all statistical methods. The F-test suggests that the variances of the quadratic and two-slope models are statistically similar. Further, the variances of a quadratic model and a linear relation for P < 400 days are also similar in most cases. Therefore, we adopt a break period of 300 days ($\sim \mathrm{log}P=2.48$) as the optimal period at which the change in the slope of PL and PLC relations occurs for O-rich Miras. The break period possibly shifts to longer periods (∼350 days, $\sim \mathrm{log}P=2.54$ days) at NIR wavelengths. NIR PLRs are discussed in detail in Section 5.

Table 4.  Results of the Statistical Tests to Find a Kink in the O-rich Mira PL and PLC Relations in the LMC

PL/PLC	SIC	RW	TM	FT	p(f)
⋯	$\mathrm{log}{P}_{b}$				⋯
I, (V − I)	2.37	2.42	2.35	2.35	0.51
IM, (VM − IM)	2.45	2.49	2.43	2.48	0.78
G, (BP − RP)	2.44	2.43	2.48	2.45	0.36
GM, (BPM − RPM)	2.40	2.39	2.48	2.46	0.88
RP, (BP − RP)	2.43	2.41	2.47	2.43	0.32
RPM, (BPM − RPM)	2.40	2.42	2.48	2.42	0.84
J	2.50	2.50	2.53	2.51	0.01
JM	2.55	2.47	2.48	2.54	0.23
K	2.52	2.51	2.49	2.53	0.19
KM	2.54	2.48	2.50	2.54	0.01

Note. SIC: Schwarz Information Criterion used to estimate the break period in a two-slope model. RW: Random walk, TM: The Testimator, FT: F-test p(f): Probability of acceptance of the null hypothesis that the variance of a quadratic and a linear relation for P < 400 days are equal.

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Optical photometry provides clear evidence that Miras display significantly smaller scatter in the PLRs at max-light than at mean-light. The pulsation properties and brightness variations of Miras are not strictly periodic and vary from cycle-to-cycle. Therefore, it is important to investigate the stability of max-light over different pulsation cycles. However, apart from OGLE I-band time-series observations, there is a lack of continuous photometry for Miras in the LMC, and even OGLE data do not necessarily cover the epochs of max-light during all pulsation cycles. High-precision photometry and subhour cadence from Kepler can be extremely useful to test the stability of max-light over different pulsation cycles. Bányai et al. (2013) analyzed variability in M giants from Kepler and identified a dozen long-period Mira-like variables.

We estimate the scatter in the max-light magnitudes and median magnitudes for Miras over different pulsation cycles. For OGLE Miras, the sample is restricted to those stars for which GPR predicts max-light magnitudes with a precision better than three times their median uncertainty, and the time coverage is equivalent to the period of each Mira. The top panel of Figure 9 displays the comparison of the scatter in the median- and max-light magnitudes for OGLE and Kepler Mira candidates. When the scatter in both sets of magnitudes is smaller than 0.4 mag, max-light magnitudes seem to be more stable for most O-rich Miras. C-rich Miras often exhibit long-term trends and thus display large scatter both at median- and max-light. The bottom panel of Figure 9 presents the variation in min-light magnitudes versus the scatter in median magnitudes. For a majority of the stars, min-light magnitudes display significantly larger scatter than median magnitudes over multiple pulsation cycles. The scatter in max-light magnitudes is systematically smaller than in min-light magnitudes with an average offset of 0.06 mag and the largest offset of 0.3 mag. High-precision photometry of Kepler Miras also displays smaller magnitude variations at max-light than at min-light. Similar magnitude variations are seen at max- and mean-light when the dispersion over multiple pulsation cycles in both sets of the magnitudes is large (>0.4 mag). The lack of accurate parallaxes for the Kepler Miras and solar-neighborhood Miras with high-cadence photometry precludes us from quantifying the impact of scatter in mean- and max-light magnitudes on the PLRs. Note that accurate Gaia parallaxes for these Miras are not available, presumably due to large astrometric errors that are caused by the motion of the stellar photocenter (Chiavassa et al. 2018). Miras also exhibit up to 40% variation in their stellar diameters (H band, Lacour et al. 2009), and angular diameters can be larger than the parallax measurements. Further, Miras display large magnitude and color variations along the pulsation cycle and chromaticity corrections are required at each epoch to estimate accurate parallaxes (Mowlavi et al. 2018). In addition to precise distances, extinction and metallicity measurements are also needed to calibrate PLRs for Miras and to investigate the impact of cycle-to-cycle variations in max-light magnitudes on their PLRs.

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Mira spectra for Kepler stars are taken from the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) survey (Cui et al. 2012; Zhao et al. 2012). The LAMOST-Kepler project covered the original target fields at least twice. Therefore, all Mira candidates have one or more spectra, and a study as a function of the pulsation phase is possible wherever more than one spectrum is available. Figure 10 shows the light curves of two Kepler Mira candidates in the top panels and their LAMOST spectra in the bottom panels. The Kepler K_P and V-band light curves are taken from Bányai et al. (2013) and All-Sky Automated Survey for Supernovae (ASAS-SN) catalog of variable stars (Shappee et al. 2014; Jayasinghe et al. 2018). At least one spectrum close to min-, mean-, and max-light is available for these two Miras. All spectra display TiO molecular features, but no sequential vibrational bands of carbon-bearing molecules are seen. Therefore, both Miras are classified as O-rich Miras. The max-light spectra (close to the zero phase, which corresponds to the max of the K_P-band) for Kepler Miras exhibit strong Balmer line emission. Furthermore, both Miras display a Balmer increment at max-light and the strength of TiO bands increases from max- to min-light. Yao et al. (2017) showed that as an AGB star cools down from its warmest phase, the strength of TiO bands increases gradually and thus more flux is absorbed by lower-order Balmer lines (Merrill 1940).

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Spectro-interferometry studies of O-rich Miras show that stellar radius in continuum light changes from max- to min-light (Thompson et al. 2002; Wittkowski et al. 2016, 2018). While the continuum and water radii are anticorrelated with the visual light curve, the carbon monoxide (CO) bandhead diameter correlates well. The one-dimensional dynamic model atmospheres of Miras also predict an increase in the continuum radius between max and min-light, when the luminosity decreases, as well as an irregular variability in the outer molecular layers (see Ireland et al. 2011). As the continuum radius decreases and becomes smaller at max-light, the effective temperature increases and dominates the visual light curve. Unstable molecules, such as water vapor and TiO, are destroyed at maximum effective temperature and luminosity. More stable CO molecules instead become more excited while going from min- to max-light through the pulsation cycle. The formation of different molecules may vary in the extended atmosphere from cycle to cycle, which contributes to the stellar luminosity and in turn to the variability between different pulsation cycles (Wittkowski et al. 2018). At min-light, metallic oxides, such as TiO, form sufficiently to extend the visual photosphere almost twice its nominal size, and decrease the visual light significantly (Reid & Goldston 2002). Therefore, luminosity variations at max-light are significantly smaller than at mean- or min-light possibly due to smaller contributions from water vapor, TiO, and other unstable molecules at the warmest phase.

Recently, Yuan et al. (2017) derived PLRs in the JHK_s bands for Miras in the LMC based on the synoptic survey data from Macri et al. (2015). They generated NIR templates for OGLE Miras based on I-band light curves and derived robust mean-, max-, and min-light magnitudes from only three JHK_s epochs. While the templates do not necessarily reflect the true shapes of the infrared light curves, they provide a reasonable approximation to the JHK_s light-curve structure with respect to that in the I band. When templates are available, mean- and max-light magnitudes are adopted from Yuan et al. (2017). If not, max-light magnitudes from the original light curves are estimated as discussed in Section 2. The left panel of Figure 11 displays the mean and max-light PL and PLC relations for Miras in the LMC. At NIR wavelengths, both max- and mean-light PLRs show comparable scatter. The results of the regression analysis are tabulated in Table 5. Note that the mean-light NIR PLRs derived in this work are presented to compare with the max-light relations, and readers are referred to Yuan et al. (2017) for calibrated PLRs. In the case of PLC relations, a significant color dependence is found only in the J-band for O-rich Miras (Feast et al. 1989), and the color dependence decreases at longer wavelengths.

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While the LMC NIR data for Miras are limited in terms of temporal coverage, Ita et al. (2018) provided NIR time series for a small region of the SMC spanning over a decade of observing epochs. Their catalog consists of 14 O-rich and 76 C-rich Miras that have OGLE-like light curves in the JHK_s bands. The right panel of Figure 11 shows the PL and PLC relations for Miras in the SMC. The PLRs display evidence of a smaller dispersion at max-light than at mean-light despite the statistically small sample of Miras. The color coefficients of PLC relations for C-rich Miras are similar for both Magellanic Clouds and significant even at longer wavelengths.

O-rich Mira PLRs at NIR wavelengths can be used as potential distance indicators. While the long-period Miras are affected by the circumstellar dust extinction, the extinction and metallicity effects are small at NIR wavelengths. Assuming that the effects of circumstellar dust are similar at both mean and max-light, we used O-rich Mira PLRs in the LMC and SMC to estimate the relative distance between the Magellanic Clouds. The top panel of Figure 12 presents the optical PLC relations and the bottom panel displays NIR K_s-band PLRs at max-light for O-rich Miras in the Magellanic Clouds. The zero-point offset between the Clouds is estimated with respect to fixed LMC relations. Table 6 lists the results for the relative distance modulus between the LMC and the SMC estimated using both mean- and max-light relations. The extinction-corrected distance estimates based on max-light relations exhibit smaller statistical uncertainties by virtue of the smaller dispersion in max-light relations. A mean relative distance modulus, Δμ = 0.48 ± 0.08 mag, is obtained between the Magellanic Clouds using O-rich Miras. The distance estimate using max-light PLRs are consistent with the relative distance moduli based on classical Cepheids, Miras, and other distance indicators (Matsunaga 2012; de Grijs et al. 2014; de Grijs & Bono 2015; Bhardwaj et al. 2016b).

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Matsunaga et al. (2009) carried out a NIR survey of LPVs toward the Galactic center and identified 1364 variables, including several Mira candidates. Due to significantly high extinction, J-band light curves are not available for several variables while K-band magnitudes approach the saturation limit for the brightest LPVs. We selected a subsample of Miras with the following criteria: P ≤ 350 days, max-light magnitudes ≥9 mag in K, amplitudes greater than 0.5 mag in K. These criteria led to a sample of 148 Miras with both H- and K-band measurements, which is used in the subsequent analysis. We assume that most of these Miras are O-rich as C-rich Miras are rare toward the central region of the Galaxy (Matsunaga et al. 2017).

Figure 13 displays the PLC relation for Miras toward the Galactic center. The scatter is comparable in both mean- and max-light relations. Matsunaga et al. (2009) adopted total-to-selective absorption ratios (e.g., A_H/A_K) given by the reddening law of Nishiyama et al. (2006) to simultaneously estimate distance and extinction. Similar to their analysis, we used Equations (8) and (9) from Matsunaga et al. (2009) to estimate the distance and K-band extinction for Miras with both max- and mean-light PLRs. The calibrator NIR data for Miras in the LMC are taken from Yuan et al. (2017) and the PLRs are restricted to Miras with P ≤ 350 days. Mira PLRs are calibrated using the most precise ∼1% distance to the LMC (Pietrzyński et al. 2019). The middle panel of Figure 13 shows the histogram of distances to individual Miras toward the Galactic center. Our distance estimates using mean-light are similar to that of Matsunaga et al. (2009) and the peak of the distribution is consistent with the ∼0.3% geometric distance (∼8.18 kpc) to the Galactic center (Gravity Collaboration et al. 2019), and with measurements based on RR Lyrae and Type II Cepheids (Dékány et al. 2013; Bhardwaj et al. 2017). However, the peak of the distribution using max-light PLRs shows an offset toward larger distances, but is consistent within the uncertainties. The bottom panel of Figure 13 shows the K-band extinction estimated using mean and max-light PLRs. The typical values and the range of A_K are consistent between the two approaches and similar to the results of Matsunaga et al. (2009).

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