Multiwavelength Properties of Miras

The Mira star (o Ceti) is historically the first identified variable star in modern astronomy. The discovery of its variability by David Fabricius at the end of the 16th century and its periodicity by Johannes Holwards in the 17th century has led to the birth of one of the most important branches of astronomical research—stellar variability.

The Mira-type stars are fundamental-mode asymptotic giant branch (AGB) pulsators belonging to a group of long-period variables (LPVs). The pulsation periods of Miras span a range between ∼100 days and 1000 days, or more. The Mira-type variables are relatively easy to detect due to large brightness variations in optical bands with ΔV > 2.5 mag (e.g., Payne-Gaposchkin 1951; Samus' et al. 2017), ΔI > 0.8 mag (e.g., Soszyński et al. 2005, 2009), and decreasing variability amplitude at longer wavelengths (ΔK > 0.4 mag; e.g., Whitelock et al. 2006). The upper boundary of brightness variations in the K band seems to be about 1 mag (Feast et al. 1982).

During their evolution through the tip of the AGB, Miras become very luminous (several thousand L_⊙). Due to their large luminosity, Mira-type stars are tracers of an old- and intermediate-age stellar populations in many galaxies, e.g., in the Large Magellanic Cloud (LMC; Soszyński et al. 2009), Small Magellanic Cloud (SMC; Soszyński et al. 2011), NGC 6822 (Whitelock et al. 2013), IC 1613 (Menzies et al. 2015), M33 (Yuan et al. 2017), Sgr dIG (Whitelock et al. 2018), NGC 4258 (Huang et al. 2018), NGC 3109 (Menzies et al. 2019), or NGC 1559 (Huang et al. 2020). Moreover, Miras obey well-defined period–luminosity relations (PLRs) in the near-infrared (NIR) and mid-infrared wavelengths (mid-IR; e.g., Ita & Matsunaga 2011; Yuan et al. 2018; Bhardwaj et al. 2019; Groenewegen et al. 2020; Iwanek et al. 2021), so they can be used as distance indicators.

The AGB stars are typically divided into the oxygen-rich (O-rich) and carbon-rich (C-rich) classes (see, e.g., Riebel et al. 2010). Such a division has been systematized by Soszyński et al. (2009) for the LMC LPVs. The authors divided LPVs into O-rich and C-rich stars based on the color–color (V − I versus J − K) diagram and the Wesenheit diagram (W_I versus W_JK ). Other authors proposed a similar division for the LPVs located in the Milky Way (Arnold et al. 2020) or nearby galaxies (e.g., Yuan et al. 2017, 2018; Huang et al. 2018).

Mira variables, as luminous high-amplitude AGB pulsators, are suspected to undergo the mass-loss phenomenon via the stellar wind (Perrin et al. 2020). Dying stars during the AGB phase eject a large amount of heavy elements created in a slow neutron-capture process (s-process), which leads to an enrichment of the interstellar medium with elements heavier than nickel. This phenomenon has been extensively studied due to its importance from the point of view of chemistry of galaxies (e.g., Whitelock et al. 1997; Battistini & Bensby 2016; De Beck et al. 2017; Höfner & Olofsson 2018; Kraemer et al. 2019; Li et al. 2019; Yu et al. 2021).

The variability of Miras, like other pulsators, is not strictly periodic. The long-term changes of the mean brightness (possibly caused by dust ejections), and cycle-to-cycle light and period variations have been observed in the past decades (e.g., Eddington & Plakidis 1929; Sterne & Campbell 1937; Lloyd 1989; Mattei 1997; Percy & Au 1999). Feast et al. (1984) discovered the variable dust obscuration around the C-rich Mira R Fornacis. A decade later, Winters et al. (1994) showed synthetic mid-IR light curves for C-rich stars, based on dynamical models of circumstellar dust shells. The authors showed models for R Fornacis along with data collected by Feast et al. (1984). The variability amplitude ratio between the KLM- and J-band light curves decreases with increasing wavelength. The variability amplitude ratios presented by Winters et al. (1994) are ∼0.61 and ∼0.42 for the K/J and L/J bands, respectively. From the theoretical light curves, it was shown that the variability amplitude ratio between the 25 μm and J-band light curves is approximately equal to 0.1. A phase shift between the KLM- and J-band light curves for R Fornacis was not reported.

Whitelock et al. (2003) noticed that pulsation amplitudes of Miras, as well as amplitudes of their long-term trends, are smaller at longer wavelengths. Lebzelter & Wood (2005) reported that the pulsation amplitude ratio between the K and V bands is equal to 0.2. Recently, Yuan et al. (2018) presented the variability amplitude ratio between the JHK_s bands and I band, separately for the O-rich (0.415, 0.435, and 0.409, respectively) and C-rich (0.771, 0.660, and 0.538, respectively) Miras. These ratios differ significantly for the two subclasses and are larger for the C-rich Miras.

Moreover, it is known that the variability maxima of Mira light curves are delayed in the near-IR with respect to the visual bands by about 0.1 periods (Pettit & Nicholson 1933; Smith et al. 2006). These phase lags were also studied by Yuan et al. (2018). In contrast to the variability amplitude ratio, the phase lag between the I band and JHK_s bands is larger for the O-rich Miras and increases with the increasing wavelength. The mean phase lags between the I band and JHK_s bands for the O-rich Miras are 0.126, 0.133, and 0.155, respectively. The phase lag observed in the C-rich Miras is the largest in the K_s band and is equal to 0.03 phase. Similar conclusions were made by Ita et al. (2021).

The aim of this paper is to comprehensively analyze the variability of Miras at a wide range of optical/near-IR/mid-IR wavelengths. This work contains two parts. The first one delivers analyses of variability based on the real data in up to 14 filters, while the second part provides analyses based on synthetic light curves derived from spectral energy distribution (SED) fitted to the real data in the 0.1–40 μm range. We derive the variability amplitude ratio and the phase lag as a function of wavelength for both the real and synthetic data.

In Section 2, we describe the data used in this work that include the sample of Miras and extinction correction procedure, and in Section 3 we describe the modeling and fitting of the light curves. We present and discuss the variability amplitude ratio and the phase lag as a function of wavelength in Section 4. In Section 5, we discuss the synthetic data that include the SED fitting and synthetic PLRs. Section 6 is devoted to the location and motions of Miras on the color–magnitude (CMD) and color–color (CCD) diagrams. The paper is summarized in Section 7, while in Section 8, we outline future areas of exploration.

2.1. Sample of Miras

2.2. Optical Photometry

2.3. NIR Photometry

2.4. Mid-IR Photometry

2.4.1. WISE Data

We cross-matched the sample of LMC Miras with the Wide Field Infrared Survey Explorer (WISE; Wright et al. 2010) databases. WISE is a 40 cm diameter infrared space telescope that observed the sky in four bands: W1 (λ_eff = 3.4 μm), W2 (λ_eff = 4.6 μm), W3 (λ_eff = 12 μm), and W4 (λ_eff = 22 μm). The main mission of the WISE telescope was to map the entire sky in these four mid-IR bands. This mission was completed in 2010. In early 2011, due to the depletion of the solid hydrogen cryostat, WISE was placed into hibernation mode. Reobservations began in 2013 as a part of the Near Earth Object WISE Reactivation Mission (NEOWISE-R; Mainzer et al. 2011,2014) and have been carried out in W1 and W2 bands to this day.

The WISE telescope observes each sky location every six months, and during one flyby, it collects several independent exposures, which span from one to a few days. The LMC area (near the South Ecliptic Pole) is observed much more often due to the polar trajectory of the WISE telescope. As a result, the light curves of the LMC stars are covered much more densely than in other locations.

We searched for objects around LMC Miras coordinates within 1''. We downloaded all available measurements in the W1, W2, W3, and W4 bands from the AllWISE Multiepoch Photometry Table, ⁵ and all measurements in the W1 and W2 bands from the NEOWISE-R Single Exposure (L1b) Source Table using NASA/IPAC Infrared Science Archive. ⁶ We found in the AllWISE table counterparts to 1592, 1594, 1616, and 1551 (out of 1663) Miras in the W1, W2, W3, and W4 bands, respectively. In the NEOWISE-R table, we found counterparts to 1632 and 1643 objects in the W1 and W2 bands, respectively.

For every single measurement in the database, the WISE team provided the reduced ${\chi }_{\mathrm{PSF}}^{2}$ from fitting the point-spread function (PSF) to objects detected in the collected images. We examined the distributions of ${\chi }_{\mathrm{PSF}}^{2}/\mathrm{dof}$ in each WISE band, both in AllWISE and NEOWISE-R data. We cleaned the WISE light curves from the worst-quality points, assuming that best-quality measurements have ${\chi }_{\mathrm{PSF}}^{2}/\mathrm{dof}\lt 5\times {\chi }_{\mathrm{PSF},\max }^{2}/\mathrm{dof}$, where ${\chi }_{\mathrm{PSF},\max }^{2}/\mathrm{dof}$ is the highest frequencies of the ${\chi }_{\mathrm{PSF}}^{2}/\mathrm{dof}$ distributions in each band. The ${\chi }_{\mathrm{PSF}}^{2}/\mathrm{dof}$ distributions are presented in Figure 1. We rejected in total 24.6%, 15.6%, 2.3%, and 1.7% observations in the W1, W2, W3, and W4 bands, respectively.

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The WISE light curves are divided into "epochs", containing several dozen measurements each. The W1, W2, W3, and W4 AllWISE light curves comprise of one or two epochs only, while the W1 and W2 NEOWISE-R light curves contain at least 12 such epochs, separated by roughly half a year. We calculated the weighted mean magnitude in each epoch, and we rejected measurements deviating more than 3σ from the mean. After the visual inspection, we decided to use the NEOWISE-R measurements in the W1 and W2 bands only because of two reasons: the time span was significantly longer and the number of data points was higher in the light curves retrieved from the NEOWISE-R table, and the internal scatter in the AllWISE and NEOWISE-R is different. We used the W3 and W4 light curves from the AllWISE database.

Finally, we were left with fully cleaned light curves. The median number of data points per light curve was 645, 703, 63, and 22 in W1, W2, W3, and W4 bands, respectively. Light curves with the greatest coverage contained over 2500 data points and over 200 data points for the W1/W2, and W3/W4 bands, respectively. The AllWISE observations were taken between 2010 February 8 and 2010 June 17, while the NEOWISE-R data were collected between 2010 December 13 and 2019 December 1.

We removed from further analysis stars that had fewer than 100 measurements in W1 and W2 bands and fewer than 3 data points in W3 and W4 bands. The final sample of Miras in the WISE bands contained 1311 stars (out of 1663).

2.4.2. Spitzer Data

The Spitzer Space Telescope is an 85 cm diameter telescope with three infrared instruments on board: the Infrared Array Camera (IRAC), Infrared Spectrograph (IRS), and Multiband Imaging Photometry for Spitzer (MIPS) (Werner et al. 2004). The LMC was observed by Spitzer in the Surveying the Agents of Galaxy's Evolution (SAGE; Meixner et al. 2006) survey using both the IRAC and MIPS instruments. The IRAC is equipped with four channels: [3.6] (λ_eff = 3.6 μm), [4.5] (λ_eff = 4.5 μm), [5.8] (λ_eff = 5.8 μm), [8.0] (λ_eff = 8.0 μm), while MIPS is equipped with three channels: [24.0] (λ_eff = 24 μm), [70.0] (λ_eff = 70 μm), [160.0] (λ_eff = 160 μm). In this paper, we use the [3.6], [4.5], [5.8], [8.0], and [24.0] channels, later also referred to as I1, I2, I3, I4, and M1, respectively.

We searched for objects within a 1'' radius around Miras coordinates. We downloaded all available measurements from the SAGE IRAC Epoch 1 and Epoch 2 Catalog using NASA/IPAC Infrared Science Archive (see footnote 7). We found 1601 (out of 1663) counterparts to our Mira sample. The measurements provided in the SAGE IRAC Epoch 1 and Epoch 2 Catalog do not contain the time of observations. Times for IRAC observations are available from the timestamp mosaic images. ⁷ The SAGE team provided an IDL program get_sage_jd.pro. This program requires IDL procedures from the IDL Astronomy User's Library. ⁸ Unfortunately, the timestamps for the MIPS observations are not publicly available.

Almost all SAGE IRAC light curves contain two epochs, collected from 2005 October to November. For further analysis, we used objects with two epochs in each IRAC band. This left us with 1470 Miras (out of 1663).

Additionally, we retrieved data from the SAGE MIPS 24 μm Epoch 1 and Epoch 2 Catalog. We found 1415 counterparts to our objects. As the observation time is not publicly available for MIPS measurements, we used these data in further analysis, treating them as a mean.

The main aim of this paper is to study the variability of Miras spanning a range of wavelengths and then to find the variability amplitude ratios and phase lags for these stars at a number of wavelengths.

3.1. Extinction Corrections

In this paper, we are interested in analyzing Miras located in the LMC, therefore their light is subject to absorption by interstellar dust. The most detailed reddening map of the LMC is based on LMC Red Clump (RC) stars and was published by Skowron et al. (2021, hereafter S21). The authors provided the reddening E(V − I) coefficients, along with the lower (σ_−,E(V−I)) and upper (σ_+,E(V−I)) uncertainties. For a given object, the reddening E(V − I) could be retrieved from the online form ⁹ by using their R.A. and decl. The extinction A_I could be calculated as

$$\begin{eqnarray}&&{A}_{I}\simeq 1.5\times E(V-I),\end{eqnarray} \tag{ 1 }$$

where the coefficient depends on the inner LMC dust characteristics and could vary between 1.1 and 1.7. We fixed this coefficient to 1.5 with the uncertainty equal to 0.2. Knowing that E(V − I) = A_V − A_I , the extinction A_V is

$$\begin{eqnarray}&&{A}_{V}\simeq 2.5\times E(V-I).\end{eqnarray} \tag{ 2 }$$

This is broadly consistent with coefficients of 1.505 for A_I and 2.742 for A_V derived for R_V = A_V /E(B − V) = 3.1 by Schlafly & Finkbeiner (2011). Both extinctions A_V and A_I can be transformed to other bands using relations published by Wang & Chen (2019).

As a cross-check, we compared the extinctions obtained with the S21 method to the extinction law derived by Cardelli et al. (1989). We first calculated the extinction in the K band A_K = 0.219 × E(V − I), using the reddening E(V − I) provided by Skowron et al. (2021), and then we transformed A_K to other bands using the relations published by Chen et al. (2018). This method will be referred to as C89. We then compared the extinction obtained using both methods for two IR bands: J (1.25 μm) and I4 (8.0 μm).

The extinction toward the LMC at short wavelengths is relatively small (the median value in J-band is A_J = 0.07 mag). Then, the influence of interstellar dust on stellar light decreases with increasing wavelength (see e.g., Cardelli et al. 1989). In the Spitzer I4 band, the extinction is one order of magnitude smaller than in the J band (the median value is A_I4 = 0.007 mag). In general, in the I4 band, the extinction is comparable to the level of a single photometric measurement uncertainty for most surveys. The difference between the S21 and C89 methods is less than 10%.

Throughout the paper, we used the S21 method to correct the magnitudes for the interstellar extinction, and both the stellar brightness and colors are extinction corrected and dereddened.

3.2. Template Light Curves

The variability of Miras is characterized by several components and is not strictly periodic, therefore their light curves cannot be adequately modeled with a pure periodic function (e.g., a sine wave). The main large-amplitude, cyclic variability of Miras is caused by pulsations (Kholopov et al. 1985). Mattei (1997) noticed cycle-to-cycle and long-term magnitude changes, which are associated with the mass loss and presence of circumstellar dust. The last, stochastic component of Miras' variability is related to supergranular convection in the envelopes of giants stars.

The semiparametric Gaussian process regression (GPR) model, which takes into account all these types of brightness changes, was proposed by He et al. (2016). The Miras light-curve signal g(t) can be decomposed into four parts:

$$\begin{eqnarray}&&g(t)=m+l(t)+q(t)+h(t),\end{eqnarray} \tag{ 3 }$$

where m is the mean magnitude, l(t) is a low-frequency trend across cycles (or slowly variable mean magnitude), q(t) is a periodic term, and h(t) is the high-frequency stochastic variability within each cycle. The l(t), q(t), and h(t) terms are modeled by the Gaussian process with different kernels (in particular squared exponential kernels and a periodic kernel). The use of the periodic kernel allows the brightness amplitude to change from cycle to cycle, which is a characteristic feature of Miras. Such a complex variability behavior of Miras is not reproducible by strictly periodic functions. The full code is distributed as an R (R Core Team 2020) package via GitHub (He et al. 2016). Using the above model and equipped with the high-quality, two-decade-long, and densely sampled OGLE I-band light curves, we derived template light curves for the selected LMC Miras. Both the dense sampling and high precision of OGLE I-band light curves lead to very smooth template light curves. Given that the GPR model is data-driven, the GPR models describe not only the I-band light curves exceptionally well, but also scaled and shifted template fits to other bands, in particular to the V-band light curves (in Figures 4 and 5). The synthetic light curve can be generated with any cadence.

3.3. Fitting Templates to the NIR and Mid-IR Data

Fitting the well-sampled I-band templates to the sparsely sampled near-IR and mid-IR data is key to a reliable calculation of the true mean magnitudes of Miras in these bands. The procedure turned out, however, to be nontrivial due to insufficient sampling or data span of some of the data sets.

In the first method (method 1), we used a simple fitting of the I-band template to other bands by using a χ² minimization method. We assumed that the shape of light curves is the same in each band, and the variability is a simple scaled, shifted in magnitude, and shifted in phase version (hence three fitted parameters) of the I-band template. This methodology worked well for the V, YJK_s, and W1 and W2 bands, as the data span and phase coverage were sufficient. The resulting variability amplitude ratios and phase lags are presented in Figures 2 and 3, respectively, as blue points for the O-rich Miras and red points for the C-rich Miras. The uncertainties of both blue and red points presented in Figures 2 and 3 are calculated from the interquartile range (IQR) as 1σ = 0.741 × IQR for all obtained variability amplitude ratios and phase lags in this method. We do not find correlations between the amplitude ratios or phase lags and Mira pulsation periods in the mid-IR bands.

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Because bands I1, I2, I3, I4, W3, and W4 contained typically two points only spanning a very narrow phase range, the resulting individual fits were unreliable (degenerate). We therefore tested a method, where we simultaneously fitted all light curves with their respective models, all of them shifted by the same phase lag and variability amplitude ratio, but individual magnitude shifts (method 2). The minimization was over the sum of χ² from individual fits. In the case of M1 observations, the fitting procedure is not possible due to the lack of observations epochs.

Another method (method 3) we used to study the sparse I1, I2, I3, I4, W3, and W4 bands was a method described in Section 6 of Soszyński et al. (2005). In short, each light curve consisted of two measurements. For a given pair of the variability amplitude ratio and phase lag, we modified the template light curve and shifted it in magnitude so the first measurement ended up exactly on the modified template light curve and then again we modified the template light curve and shifted it in magnitude so the second measurement ended up exactly on the modified template light curve. We searched for a pair of the variability amplitude ratio/phase-lag parameters, where the difference between the two modified template light curves was the smallest.

Light curves in the OGLE VI bands; VMC YJK_s bands; Spitzer I1, I2, I3, and I4 bands; and WISE W1, W2, W3, and W4 bands are presented in Figures 4 and 5 for O- and C-rich Miras, respectively, along with I-band templates, template components, and fitted templates to the near-IR and mid-IR data.

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In Figures 2 and 3, we present the variability amplitude ratio R and phase lag ϕ as a function of wavelength. These figures present both the real (dots/stars) and synthetic data (bands), the latter described in Section 5.1.

Let us concentrate on the blue (O-rich Miras) and red (C-rich Miras) points in these figures. Both the blue and red data points were calculated with the χ² fitting (method 1) of the template I-band light curves to the V, Y, J, K_s, W1, and W2 light curves. Therefore, we explore the variability amplitude ratio R and phase lag ϕ with respect to the I-band light curves, where R = 1 and ϕ = 0 are for the I band. The negative (positive) ϕ means that the analyzed light curve or band leads (lags) the I-band light curve.

From Figure 2, it is clear that the variability amplitude ratio R decreases with the increasing wavelength. For the O-rich Miras (top panel), the V-band variability amplitude is approximately 2.3 times greater than the amplitude in the I band, while the W1 and W2 variability amplitude is approximately four times smaller than that in the I band. We observe a similar dependence of the variability amplitude ratio for the C-rich Miras (bottom panel), albeit with a somewhat smaller amplitude (with approximately 1.6 greater variability in the V band) at short wavelengths as compared to the O-rich Miras.

The phase-lag picture is not as striking as the one for the variability amplitude ratio. From both panels in Figure 3, a weak increase of the phase lag ϕ with increasing wavelength is noticeable. For both the O-rich (top panel) and C-rich (bottom panel) Miras, the V band leads the I band, while at longer wavelengths it appears that a positive phase lag is preferred, albeit with a rather choppy increase.

The gray points in Figures 2 and 3 were derived with method 2 or 3, with the uncertainties calculated as 1σ = 0.741 × IQR. Due to the narrow time span of the Spitzer and long-wavelength WISE data, effectively probing typically a small fraction of the phase, we conclude these measurements of the variability amplitude ratio R and phase lag ϕ are unreliable. They are presented here for completeness.

The magnitudes of a star observed in multiple filters may be converted to "average in-band" (or "in-filter") fluxes λF_λ (in units of erg s⁻¹ or L_⊙), where F_λ is the spectral flux density, being the flux per unit wavelength. To form the SED of that star, the zero-magnitude fluxes and transmission properties of filters must be known. To model such an observed or synthetic SED one may use the Planck function(s) that need to be converted into in-band average fluxes F = ∫W(λ)λ F_λ d λ by using the filter transmissions W(λ). To find a best-fitting SED model, we used a standard χ² minimization procedure, where the model parameters were either two or four parameters: the amplitude(s) and temperature(s) of the Planck function(s). The median uncertainty of the stellar temperature is 6%, while for the dust it is 10%.

Detailed analyses of SEDs showed that mean magnitudes in both the W3 and W4 bands were not reliable for many individual Miras. The W3- and especially W4-band measurements appeared as outliers in the SED models, while the M1-band (24 μm) measurements seemed to fit rather well in many cases (measurements in W4 at 22 μm and M1 at 24 μm often disagreed by as much as an order of magnitude in luminosity).

In our sample of 1663 Miras, there was a subsample of 140 sources (29 O-rich and 111 C-rich Miras, hereafter the golden sample) with complete and high-quality V, I, Y, J, K_s, W1, and W2 light curves, for which we simultaneously measured the variability amplitude ratios, phase lags, and magnitude shifts very precisely (with method 1). Having the derived model parameters, we were able to shift and scale the I-band templates to the remaining bands.

Equipped with complete information about their extinction-corrected brightness in the V, I, Y, J, K_s, W1, and W2 bands for the duration of multiple periods and with hundreds of model epochs per period, we converted these magnitudes to fluxes (λF_λ in L_⊙) assuming the LMC distance of d_LMC = 49.59 kpc (Pietrzyński et al. 2019) and the respective filter parameters/transmissions. Then, for each star, we created 6000 synthetic SEDs spanning 6000 days (one SED per day). We modeled these SEDs with either a single-Planck (Figure 6) or double-Planck (Figure 7) function, where the double-Planck function means a sum of two Planck functions: the hotter one for the star and the cooler one (most likely) for the dust. SEDs for all considered O-rich Miras did not require contribution from the dust, while such a contribution was frequently necessary for the C-rich Miras.

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5.1. Variability Amplitude Ratio and Phase Lag as a Function of Wavelength

5.2. Bolometric Luminosities

From fitting a single- or a double-Planck function to the SED, we obtained both the amplitude and temperature of each Planck function. It was then straightforward to calculate the total bolometric luminosity for each Mira, as well as the separate bolometric luminosities for the star and dust. We integrated the synthetic SEDs in the 0–100 μm range. In Table 3, we provide the bolometric luminosity for both the star and dust, and the combined luminosity.

Table 3. Bolometric Luminosities of Miras from the Golden Sample

ID	Type	Lbol (L⊙)	${L}_{\mathrm{bol}}^{\mathrm{star}}$ (L⊙)	${L}_{\mathrm{bol}}^{\mathrm{dust}}$ (L⊙)	fstar	fdust
OGLE-LMC-LPV-08058	C	32139 ± 752	9882	22257	0.307	0.693
OGLE-LMC-LPV-08424	C	46070 ± 1837	3934	42136	0.085	0.915
OGLE-LMC-LPV-09268	C	20767 ± 1859	17849	2918	0.860	0.140
OGLE-LMC-LPV-10812	C	18032 ± 446	3656	14376	0.203	0.797
OGLE-LMC-LPV-12829	C	24235 ± 1051	24235	0	1.000	0.000
OGLE-LMC-LPV-13563	C	20241 ± 960	9696	10545	0.479	0.521
OGLE-LMC-LPV-13815	C	15053 ± 383	3102	11951	0.206	0.794
OGLE-LMC-LPV-14860	C	15281 ± 508	7882	7399	0.516	0.484
OGLE-LMC-LPV-15353	C	14580 ± 802	10564	4016	0.725	0.275
OGLE-LMC-LPV-16684	C	1904 ± 32	522	1382	0.274	0.726
OGLE-LMC-LPV-82526	O	3648 ± 244	3648	0	1.000	0.000

Note. The reported star ID and the O/C type are from the catalog of Soszyński et al. (2009). Table rows are sorted by the star ID. L_bol is the total bolometric luminosity presented with its uncertainty, ${L}_{\mathrm{bol}}^{\mathrm{star}}$ and ${L}_{\mathrm{bol}}^{\mathrm{dust}}$ are the bolometric luminosity of the star and dust, while ${f}^{\mathrm{star}}={L}_{\mathrm{bol}}^{\mathrm{star}}/{L}_{\mathrm{bol}}$ and ${f}^{\mathrm{dust}}={L}_{\mathrm{bol}}^{\mathrm{dust}}/{L}_{\mathrm{bol}}$ are fractions of the bolometric luminosity of the star and dust to the total bolometric luminosity (and f^star + f^dust = 1).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

None of the O-rich Miras required contribution from the dust to the SED, while it was frequently necessary for the C-rich stars. The O-rich Miras have bolometric luminosity in a range of 3600–29,200 L_⊙ with a median value of 13,500 L_⊙. The bolometric luminosity for the C-rich Miras span a range of 800–50,400 L_⊙ with a median value of 19,200 L_⊙. The second (dust) component is present in 73% of the C-rich Miras, where the bolometric luminosity of the dust spans a range of 300–42,100 L_⊙ with the median value of 12,500 L_⊙. We do not find a correlation between the bolometric luminosity and the temperature.

5.3. Synthetic PLRs

Equipped with 140 Miras in the selected golden sample, each with a mean SED as well as a series of 6000 synthetic SEDs as a function of time, we investigated the dependence of the PLR slope on the wavelength, but also the temperature and color changes as a function of time.

We used these SEDs to create 42 synthetic PLRs in frequently used filters in major present/future surveys. They include V, I from OGLE; Y, J, and K_s from VMC; G, G_bp , and G_rp from Gaia; u, g, r, i, z, and y from the Legacy Survey of Space and Time (LSST); F110W, F140W, and F160W from the Hubble Space Telescope (HST); J129, H158, and F184 from the Nancy Grace Roman Space Telescope (formerly WFIRST); F200W, F277W, F356W, F444W, F560, F770W, F1000, F1130W, F1280, F1500W, F1800W, F2100W, and F2550W from the James Webb Space Telescope (JWST); W1, W2, W3, and W4 (3.4, 4.6, 12, 22 μm) from WISE; and I1, I2, I3, I4, and M1 (3.6, 4.5, 5.8, 8.0, and 24 μm) from IRAC/MIPS Spitzer. The transmission curves for all mentioned bands are presented in Figure 8. Magnitudes in all filters are provided in the Vega magnitude system with the exception of the LSST filters, where the magnitudes are provided in the AB magnitude system.

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For each star, we calculated its synthetic, calibrated (extinction-free) magnitudes in all 42 bands. We then fitted a linear relation in the form ${m}_{\lambda }={a}_{0}+{a}_{1}\times (\mathrm{log}P-2.3)$ in each filter to either O-rich (in the range $2.0\lt \mathrm{log}P\lt 2.65$) or C-rich (in the range $2.3\lt \mathrm{log}P\lt 2.76$) Miras. During the fitting procedure, we applied the σ-clipping procedure rejecting outliers deviating more than 3σ from the fit. The synthetic magnitudes as well as the best-fit PLRs are presented in Figures 9–12. From these figures, it is clear that the PLR slope changes from having the positive sign in the u band to having a negative sign at near- and mid-IR wavelengths. In Tables 5 and 6, we present the model parameters for the synthetic PLRs.

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From the synthetic SEDs and without using particular filters, we also calculated the PLR slope as a function of wavelength in a range of 0.1–40 μm (Table 4). Indeed, the PLR slope changes smoothly from a positive slope in the UV to a negative slope in the near- and mid-IR wavelengths. In Figure 13, we present the relation between the synthetic slope and wavelength for the O-rich Miras (left panel) and the C-rich Miras (right panel). The shaded band in both panels corresponds to the uncertainty in the slope measurement from the synthetic magnitudes. In both panels, we also presented measurements of the PLR slope from the literature, which includes Feast et al. (1989), Soszyński et al. (2007), Riebel et al. (2010), Ita & Matsunaga (2011), Yuan et al. (2017), Bhardwaj et al. (2019), and Iwanek et al. (2021).

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Table 4. Synthetic PLR Slope a₁ Derived from the SED Analysis, as a Function of Wavelength, for the O- and C-rich Miras

λ (μm)	a1
	O-rich Miras	C-rich Miras
0.1	42.892 ± 13.814	47.341 ± 11.189
0.2	18.591 ± 6.690	23.555 ± 5.113
0.3	10.490 ± 4.320	15.627 ± 3.129
0.4	6.440 ± 3.138	11.663 ± 2.177
0.5	4.010 ± 2.431	9.288 ± 1.647
0.6	2.390 ± 1.964	7.714 ± 1.332
0.7	1.235 ± 1.632	6.602 ± 1.144
0.8	0.370 ± 1.387	5.774 ± 1.032
0.9	−0.301 ± 1.199	5.114 ± 0.963
1.0	−0.835 ± 1.052	4.534 ± 0.914
40.0	−4.529 ± 0.419	−9.607 ± 0.586

Note. The PLR slope a₁ from the linear fit in a form ${m}_{\lambda }={a}_{0}+{a}_{1}\,\times (\mathrm{log}P-2.3)$.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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For the O-rich Miras (left panel of Figure 13), the vast majority of measurements reported in the literature follow our synthetic PLR slope-wavelength relation. Note, however, that our synthetic relation serves here as a guide only, because our data are limited to a small number of 140 stars, spanning a period range of $\mathrm{log}P\approx 2.0$–2.8, and well described by a linear relation. On the other hand, a large sample of the O-rich Miras shows a clear PLR relation that is not linear but rather parabolic (Iwanek et al. 2021).

Because SEDs of the C-rich Miras are typically well described by two Planck components, the hot one and the cooler one (as in Figure 7), this combination may somewhat impact the PLR slope–wavelength relation. In particular, the cooler component that is sparsely present in the O-rich Miras in the near-IR and mid-IR, for the C-rich Miras it is significant at these wavelengths (so adds typically a significant fraction of light in these bands). From the right panel of Figure 13, we can see that measurements reported in Feast et al. (1989), Soszyński et al. (2007), and Riebel et al. (2010) show a clear departure from the synthetic PLR slope-wavelength relation. On the other hand, measurements from Iwanek et al. (2021) seem to follow that relation closely.

The list of 140 Miras from the golden sample, which includes their coordinates, surface chemistry classification, pulsation periods, temperatures, and 42 synthetic, calibrated (extinction-free) magnitudes from the existing and future sky surveys are presented in Table 7.

As described in Section 2, our full sample of Miras consists of 1663 stars with light curves in the OGLE survey, 595 counterparts in the VMC data set, and 1311 identifications in the WISE survey. We analyzed the location of these stars in the OGLE, VMC, and WISE CMDs and CCDs.

In both panels of Figure 14 as the gray 2D histograms, we present field stars where for I < 18 mag we used the OGLE data set and for the I ≥ 18 mag we used the HST F555W and F814W data. Because of that, we can clearly see a density change above the red clump stars with (V − I, I) ≈ (1.0, 18.48) mag. In the left panel of Figure 14, with blue (red) dots we present the O-rich (C-rich) Miras. It is clear that on average in the I band, the O-rich Miras appear to be brighter than the C-rich Miras. In the right panel of Figure 14, we show the motion on the CMD (loops) of two Miras during multiple pulsation cycles: a bolometrically faint, 5200 L_⊙ O-rich Mira (blue, Mira OGLE ID: OGLE-LMC-LPV-36521) and a bolometrically luminous, 46,000 L_⊙ C-rich Mira (red, Mira OGLE ID: OGLE-LMC-LPV-08424). The red loops after each period seem to be somewhat shifted along the long axis as this star significantly changes in mean brightness with time.

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In Figure 15, we present the location of Miras in the CMDs (top row) and CCDs (bottom row) for the VMC survey with the Y, J, and K_s filters. The field stars are, again, presented as 2D gray histograms, while the 595 matched Miras are presented in the left column and the motion of the two Miras is presented in the right column. Both the O- and C-rich Miras are very luminous in the K_s band, with the absolute magnitude K_s ≈ −7 mag. Because the O-rich Miras are less affected by dust, they lie on the top of the locus of the field stars. The C-rich Miras, on the other hand, are affected by the dust (that adds IR light), and their location is shifted toward redder J − K_s colors. In the top-right panel of Figure 15, we present the motion of the two aforementioned Miras: a bolometrically faint, 5200 L_⊙ O-rich Mira (blue, Mira OGLE ID: OGLE-LMC-LPV-36521) and a bolometrically luminous, 46,000 L_⊙ C-rich Mira (red, Mira OGLE ID: OGLE-LMC-LPV-08424). The track of the red C-rich Mira appears to be a straight line because the phase lag between the J- and K_s-band light curves is tiny, 0.8 days or 0.0016 in phase.

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In Figure 16, we present the location of Miras in the CMDs (top panel) and motions on CMD of the two fore-mentioned Miras: a bolometrically faint, 5200 L_⊙ O-rich Mira (blue, Mira OGLE ID: OGLE-LMC-LPV-36521) and a bolometrically luminous, 46,000 L_⊙ C-rich Mira (red, Mira OGLE ID: OGLE-LMC-LPV-08424; middle panel), and CCDs (bottom panel) for the WISE survey. In the top panel, the red C-rich Miras extend significantly from the stellar locus toward red W1 − W2 colors, while the O-rich Miras appear to be on average fainter and bluer than the C-rich Miras. In the bottom panel of Figure 16, the red points (C-rich Miras) are scattered along the diagonal, which corresponds to the decreasing blackbody temperature toward the top-right corner. The coolest Miras with colors in the vicinity of (W2–W3, W1–W2) ≈ (2.5, 1.2) enter the color area occupied by active galactic nuclei (e.g., Figure 2 in Nikutta et al. 2014).

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In this paper, we studied the variability of 1663 Mira-type stars in the LMC at a wide range of wavelengths (0.5–24 μm) and timescales (up to 25 yr). We modeled the OGLE I-band data, with the median of 1310 epochs and up to a 25 yr span, using a GPR model that included the periodic component, slowly changing mean component, and a high-frequency stochastic component. We then fitted this high-quality model to light curves in 12 bands (V, Y, J, K_s, Spitzer IRAC 3.6–8.0 μm, and WISE 3.5–22 μm data).

The fitting procedure provided us with the variability amplitude ratio R (Figure 2) and phase lag ϕ (Figure 3) of other bands compared to the I band. For both the O-rich and C-rich Miras the variability amplitude ratio decreases with increasing wavelength. The phase lag, on the other hand, seems to weakly increase with increasing wavelength, where the V-band light curves seem to lead the I band, with the near-IR and mid-IR light curves lagging.

We selected a subsample of 140 Mira stars (the golden sample) with high-quality light curves in the V, I, Y, J, K_s, W1, and W2 bands. For each star, we scaled the high-quality I-band model to these bands and created 6000 SEDs spanning 6000 days. Each SED was modeled with either a single- or double-Planck function. With a large number of SED models as a function of time, we studied the temperature and flux changes as a function of time. As a rule of thumb the O-rich Miras are well described by a single-Planck function (hence a lack of dust presence), while the C-rich Miras do require two Planck functions, which we interpret as the presence of both the star with a higher temperature and the dust component with a lower temperature. The SEDs of O-rich Miras peak at about 1 μm with a typical (i.e., median) temperature of 2900 K and the C-rich Miras show the SED peak closer to 2 μm with the stellar temperature of 2300 K and the dust temperature of 1000 K.

The median bolometric luminosity of the golden sample Miras is 16,000 L_⊙, with a range from 800 L_⊙ (for the C-rich Mira OGLE-LMC-LPV-38689 with P = 450.69 days) to nearly 50,400 L_⊙ (for the C-rich Mira OGLE-LMC-LPV-37168 with P = 555.60 days).

From synthetic SED-based light curves, we derived the synthetic variability amplitude ratio R (Table 1) and phase lag ϕ (Table 2) as a function of wavelength (0.1–40 μm) for the O-rich and C-rich Miras in the golden sample, shown as color bands in Figures 2 and 3.

For the golden sample stars, we also calculated synthetic PLRs for 42 bands (Tables 5 and 6, Figures 9–12) using filters from major current or future surveys (Figure 8) and the mean SEDs for our golden sample. The calibrated absolute and observed magnitude zero points (a₀) for both O-rich and C-rich Miras at periods of 200 days ($\mathrm{log}P=2.3$) and PLR slopes (a₁) are provided in Tables 5 and 6, respectively. The 42 filters include LSST (u, g, r, i, z, y), Gaia (G, G_bp , G_rp ), OGLE (V, I), VMC (Y, J, K_s), HST (F110W, F140W, F160W), the Nancy Grace Roman Space Telescope (formerly WFIRST; J129, H158, F184), JWST (F200W, F277W, F356W, F444W, F560, F770W, F1000, F1130W, F1280, F1500W, F1800W, F2100W, F2550W), Spitzer (3.6, 4.5, 5.8, 8.0, 24 μm), and WISE (3.4, 4.6, 12, 22 μm). In Table 7 we present the list of Miras from the golden sample, which includes the Miras' coordinates, surface chemistry classification, pulsation periods, temperatures and 42 synthetic, calibrated (extinction-free) magnitudes in all analyzed bands.

We also studied the change of the PLR slope (a₁) as a function of wavelength from 0.1 to 40 μm (Table 4, Figure 13). The PLR slope for both O-rich and C-rich Miras strongly decreases with the increasing wavelength, with roughly zero slope in optical and plateauing in mid-IR. We therefore confirm findings from preceding papers that PLR slopes in optical are generally flat (e.g., Bhardwaj et al. 2019) and that they are sloped in near-IR and mid-IR (e.g., Feast et al. 1989; Soszyński et al. 2007; Riebel et al. 2010; Ita & Matsunaga 2011; Yuan et al. 2017; Bhardwaj et al. 2019; Iwanek et al. 2021).

Finally, we present the locations and motions as a function of time, phase, or temperature for the O-rich and C-rich Miras on both CMDs and CCDs for the OGLE, VMC, and WISE surveys.

Miras are very bright stars in the near- and mid-IR, with an approximate absolute magnitude of −6 mag/−7 mag at about 1–2 μm (at the SED peak). This means that Miras can be observed to at least multimegaparsec distances in the universe.

We will now discuss the usability of Miras as a distance indicator in the JWST NIRCam F200W and F444W (2.0 and 4.4 μm) filters. As our model galaxy, we will use M33 with the distance modulus 24.57 ± 0.05 mag or ${820}_{-19}^{+20}$ kpc (Conn et al. 2012), with about 1° in apparent size in the sky.

As estimated from our analyses, the O-rich Miras have the absolute magnitude at P = 200 days ($\mathrm{log}P=2.3$) of −6.55 ± 0.04 mag in the F200W filter and −7.42 ± 0.04 mag in the F444W filter, while the C-rich Miras have −6.96 ± 0.01 mag and −7.49 ± 0.01 mag, respectively (Tables 5 and 6). At the M33 distance they will have the observed brightness at P = 200 days ($\mathrm{log}P=2.3$) of 18.02 mag and 17.15 mag for the O-rich Miras, and 17.61 mag and 17.08 mag for the C-rich Miras in the F200W and F444W filters, respectively.

We used the Yuan et al. (2017) catalog of 1848 Miras in M33 to derive the nearest projected distance between them. The caveat in our rough estimation here is that this sample is approximately 80%–90% complete with a 12% impurity. Also note, there will be many IR sources in a galaxy that are not Miras, which will certainly lead to severe blending in the images. Using the difference images analysis method (Alard & Lupton 1998; Wozniak 2000), however, the problem of blending may be alleviated as on the difference images all non-variable sources vanish and what is left are only variable and/or moving sources.

The point-spread function (PSF) FWHM in the F200W filter is 0066 or 2.14 pixel and 0145 or 2.302 pixels in F444W. As a rough guide, we calculated separations between all pairs of Miras from Yuan et al. (2017) and found a minimum value of 012 or 0.05 kpc. For the conservative estimation, we assume we can resolve stars separated by twice the FWHM (029 in F444W). We put our model M33 galaxy at a distance, where the typical separation between Miras is equal to our conservative resolution of 029. This is about 42 times farther away than its present distance (so approximately 34 Mpc), which would make these Miras fainter by 8.12 mag, meaning the observed magnitude of approximately 25 mag. This may be too faint of a magnitude for a reasonable JWST observing program.

Let us discuss this issue from the observed magnitude perspective. We assume that we would like to detect Miras with >0.3 mag variability amplitude at 2–4 μm (>0.8 mag in the I band times the variability amplitude ratio of 0.4) at a distance of 10 Mpc (distance modulus 30 mag). To detect 0.3 mag variability amplitude reliably, we require the photometric uncertainties below 0.1 mag or the signal-to-ratio above 25. We used the JWST Exposure Time Calculator (Pontoppidan et al. 2016) to find that in a 12 minute integration we should reach the signal-to-noise ratio of 25 for a 23 mag Mira in F200W, and 4 minute exposure time in F444W to reach the signal-to-noise ratio of 30.

As a rule of thumb, a JWST observing program for a 10 Mpc galaxy would require about 30 observations spanning about 2 yr (∼2 pulsation periods) to reliably measure periods of Miras, their mean magnitudes, and colors to separate between the O-rich and C-rich Miras. This would mean a total of 6-to-7-hour JWST observing program in both F200W and F444W filters.

The mock M33 galaxy at a distance of 10 Mpc means a 12 times greater distance than the real one, which also means 12 smaller apparent size in the sky. Our mock galaxy would have a size of about 5' in the sky. The field of view of NIRCam is 2.1 × 21, significantly less than the size of this mock galaxy. Given the high expected interest in the JWST time, however, a monitoring program of a single pointing for every such galaxy will have to suffice.

We are grateful to the anonymous referee for many inspiring comments at all stages of the review process, in particular for comments to the paper dedicated to PLRs for Miras in the WISE and Spitzer bands (Iwanek et al. 2021). Remarks raised by the referee led us to explore the subject of multiwavelength variability of Miras (leading, in turn, to writing this paper). P.I. is partially supported by the Kartezjusz program no. POWR.03.02.00-00-I001/16-00 founded by the National Centre for Research and Development, Poland. This work has been supported by the National Science Centre, Poland, via grants OPUS 2018/31/B/ST9/00334 to SK and MAESTRO 2016/22/A/ST9/00009 to IS.

This publication makes use of data products from WISE, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration (NASA). This work is based in part on archival data obtained with the Spitzer Space Telescope, which was operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.

Software: TOPCAT (Taylor 2005), R (R Core Team 2020), varStar (He et al. 2016), TATRY code (Schwarzenberg-Czerny 1996).