Atomic Absorption Line Diagnostics for the Physical Properties of Red Supergiants

Red supergiants (RSGs) represent a critical phase in massive stellar evolution. They are He-fusing evolved descendants of 10–25M_⊙ main-sequence stars, the end result of a nearly horizontal evolution across the Hertzsprung–Russell (H-R) diagram as their blue H-fusing predecessors leave the main sequence and cross the yellow void. They are the largest (in physical size) and coldest (∼3500–4500 K) members of the massive star population, representing a significant extreme in their evolution. These cool temperatures place them at the Hayashi limit for hydrostatic equilibrium (Hayashi & Hoshi 1961).

Effective temperature (T_eff) is, along with bolometric luminosity (M_bol), one of the two key physical properties needed to place a star on the H-R diagram, and it is the most critical physical property that must be determined for RSGs. At these cool temperatures, the bolometric corrections for standard UBVRI photometry are large (1–4 mag) and strongly dependent on T_eff (e.g., Massey & Olsen 2003; Levesque et al. 2005); as a result, accurately calculating the luminosity of an RSG requires a robust determination of the star’s T_eff.

The scarcity of nearby RSGs has limited the use of interferometric data in ascertaining an accurate T_eff scale (see, for example, Dyck et al. 1996). Alternatively, scales in the past have been determined by broadband colors of RSGs with known diameters (Johnson 1964, 1966; Lee 1970) or by bolometric corrections derived from infrared (IR) measurements under the assumption of a blackbody continuum (Flower 1975, 1977). However these methods are also limited because of the effects of line blanketing, which make color indices such as B − V highly sensitive to surface gravity (log g). More recently, Levesque et al. (2005, 2006) used the MARCS stellar atmosphere models to fit the strengths of the T_eff-sensitive TiO bands for K-type and M-type stars in the Milky Way and Magellanic Clouds, creating a T_eff scale significantly warmer than those of previous works (Humphreys & McElroy 1984; Massey & Olsen 2003) and one that shows good agreement with the predictions of stellar evolutionary tracks (including the metallicity dependence of the Hayashi limit). Davies et al. (2013) determined warmer T_eff values for RSGs using a broad spectral energy distribution (SED) fitting across the optical and near-infrared (near-IR). However, these results do not reproduce the correlation between spectral type and T_eff in RSGs or the metallicity dependence of RSG T_effs (see, for example, Levesque et al. 2006; Tabernero et al. 2018), and Davies et al. (2013) note that 3D models (as opposed to the 1D MARCS models) are required to properly account for wavelength-dependent optical depths in the extended atmospheres of RSGs that would otherwise lead to determining a warmer T_eff at longer wavelengths.

Several recent papers also have examined the potential utility of atomic lines in these cool stars’ spectra for determining T_eff and other physical properties. Dorda et al. (2016a) compared the widths of several atomic lines (including Fe i and Ti i features and the Ca ii triplet lines at 8498, 8542, and 8662 Å, hereafter CaT) observed in the spectra of a large sample of cool supergiants (CSGs; ranging from G0 to M7 in spectral type and thus encompassing the late-type yellow supergiant population as well as RSGs) in the Large and Small Magellanic Clouds; they found that the strength of the Ti i lines was strongly correlated with T_eff (though no similar correlation was seen for Fe i or CaT). This result was further supported by Dorda et al. (2016b), which successfully used a principal component analysis based on spectral features in the CaT region (the same region covered by the Gaia Radial Velocity Spectrometer) to automatically differentiate CSGs from other bright late-type stars. This is potentially a very exciting result, offering the possibility of determining T_eff for RSGs from data with relatively limited wavelength coverage (as opposed to existing methods which require optical+IR photometry or spectrophotometry with wide optical wavelength coverage). Tabernero et al. (2018) studied the T_eff scale of CSGs in different metallicity environments; while their method adopted atomic line fitting as a means of determining T_eff, they found a warmer and shallower scale than Levesque et al. (2006), with only a weak correlation in the Large Magellanic Cloud (LMC) and no correlation in the Small Magellanic Cloud (SMC).

However, the utility of using atomic line features for determining T_eff in RSGs is still unclear. While the correlation between Ti i and spectral type presented in Dorda et al. (2016a) is quite robust at earlier types (G and early K), the correlation is much weaker for the M-type stars in their sample, which represent a significant fraction of the RSG population. The potential dependence of these features on other physical properties is also a complicating factor. For example, recent observations of RSG J-band spectra in nearby galaxies have revealed that, while atomic absorption features such as Ti i, Fe i, and Si i are not strongly sensitive to T_eff, they serve as excellent probes of metallicity (e.g., Davies et al. 2010, 2015; Gazak et al. 2015; Patrick et al. 2015, 2016, 2017).

The CaT is widely cited as a potential tracer of luminosity class in cool stars due to its sensitivity to log g effects (e.g., Cenarro et al. 2001a, 2001b and references therein), and is also sensitive to metallicity (e.g., Armandroff & Da Costa 1991; Sakari & Wallerstein 2016). Non-local thermal equilibrium (non-LTE) effects in the atmospheres of these stars can also impact the equivalent widths (EWs) of some lines; Jennings & Levesque (2016) found that the Hα absorption feature in cool stars is also effective as a luminosity class diagnostic, a consequence of the density-dependent overpopulation of the metastable 2 s level and an effect that becomes stronger in the non-LTE conditions present in supergiant atmospheres. The Jennings & Levesque (2016) study of the CaT feature indicated that while the feature in early M-type stars had a clear relationship with luminosity class, as supported by the literature, this relationship also broke down in late-type (beyond M3-3.5) supergiants.

Previous work on the CaT has studied its effectiveness as a diagnostic for several physical parameters such as luminosity, log g, metallicity, and T_eff. The CaT is a near-IR feature and therefore is subject to contamination from multiple strong stellar features such as higher-order Paschen lines and the TiO absorption band at 8433 Å. Ginestat et al. (2016) studied the relationship between the EW of absorption features between 8380–8780 Å and spectral type, finding a positive correlation between CaT and luminosity for A- to M-type stars that was initially weak but began to increase for later types beginning at G0. Ginestat et al. (2016) also proposed that the weak Ca i, Ti i, and Fe i lines of the giants in their study may be due to low metallicity.

Erdelyi-Mendes & Barbuy (1990), hereafter EM&B, used synthetic stellar atmosphere models from Gustafsson et al. (1975) to generate synthetic CaT lines in order to examine their variation with T_eff, log g, and metallicity ([M/H]). EM&B found the CaT to be primarily dependent on [M/H] and log g. Their study indicates that the CaT is sensitive to metallicity for stars with [M/H] > −2.0 and sensitive to log g for giants with [M/H] > −1.0. EM&B also found that the relationship between CaT and temperature is only present in low log g populations, making it applicable to giants and supergiants rather than dwarf stars, and becomes more pronounced as metallicity increases. This is in agreement with the results of Smith & Drake (1990), but it should be noted that their work was restricted to stars between 4000 and 5500 K. When compared against T_eff, EM&B found only a weak relationship between the CaT triplet and T_eff, which they attributed to the increasing intensity of the 8433 Å TiO feature at cool temperatures. The increasing strength of this TiO band can lead to a decrease in the local continuum and a subsequent apparent weakening of the CaT (as noted by Ginestat et al. 2016, who used a local continuum definition for measuring the CaT in stars later than M2 in order to account for this effect); however, this particular TiO band is only prominent in the spectra of RSGs with relatively late spectral types (∼M4–M5; e.g., Levesque et al. 2005; Massey et al. 2017), an effect in agreement with the evolution of the CaT seen in Jennings & Levesque (2016).

Mallik (1996) analyzed the CaT features of 146 stars spanning from F7 to M4 to determine the dependence of CaT on luminosity, T_eff, and metallicity. They found a nonlinear relationship for luminosity that became more pronounced with increased metallicity, and that was more apparent in supergiants than in dwarfs, but did not find a relationship between CaT and T_eff across the full sample. Mallik also found that at low log g (0.0–2.0), the EW of the CaT in supergiants and giants decreased as log g increased. This correlation—which is counter to the typical expectation that lines will get stronger at higher log g due to increased collisional effects—has been explained as a continuum effect. An increase in the continuous absorption coefficient at higher log g (due to an increased electron density in stars where H⁻ is the dominant source of continuum opacity) leads to a lower apparent continuum level and subsequent weaker measurements of EW for the CaT.

Cenarro et al. (2001a) presented a new stellar library of the near-IR spectral region based on 706 stars with 2750 K <T_eff < 38,400 K, 0.0 < log g < 5.12, and metallicities of −3.45< [Fe/H] < +0.60. Based on these data they offer a newly defined index for measuring the strength of the CaT features, the CaT* index, developed with careful treatments of previously noted effects such as continuum definition and Paschen line contamination.

Collectively, the utility of the CaT feature has been extensively studied, but conclusions about its use as a diagnostic of log g, luminosity, and T_eff are conflicting and further complicated by the differing sample sizes and parameter spaces of previous works, with most samples of stars spanning from dwarfs to supergiants and covering a broad range of spectral types. In this work we specifically consider the utility of atomic absorption line features as potential T_eff diagnostics in the uniquely cool and low-density environments of M-type RSGs.

We present a study examining the strengths of Ca, Ti, and Fe absorption features in the spectra of M-type RSGs. Using echelle spectra of 25 Milky Way RSGs, 16 LMC RSGs, and 17 SMC RSGs, along with a series of RSG model atmosphere spectra (Section 2), we present the EWs of a large sample of atomic lines, including features of Fe i, Ti i, Ca i, and the CaT, and compare these EWs to the T_eff determinations of Levesque et al. (2005; Section 3). We find a strong positive and statistically significant correlation between CaT and T_eff for Milky Way RSGs, but no similar correlation in the Magellanic Cloud samples, and no relationship between the Ti i and Fe i features and T_eff as a function of metallicity (Section 4). We discuss the implications of these results for understanding the physical properties of RSGs as well as potential future work in this area (Section 4).

The RSG echelle spectra used in these analyses were originally observed as part of a spectroscopic search for potential Thorne–Żytkow objects in the Milky Way and Magellanic Clouds (Levesque et al. 2014). The sample of 25 Milky Way stars was selected from the coldest RSGs identified in Levesque et al. (2005), all having spectral types of K5-M0I or later. The spectra were observed using the Astrophysics Research Consortium Echelle Spectrograph (Wang et al. 2003) on the Apache Point Observatory 3.5 m telescope on 2011 February 11 and 12 (UT). The observations were taken using the default 1.6 arcsec × 3.2 arcsec slit, along with quartz lamps and ThAr lamps after each individual exposure to achieve precise flat-field and wavelength caliberations for each star. The spectra were reduced using standard IRAF³ echelle routines, and each star’s spectrum was corrected for radial velocity (RV) effects using the wavelengths of the CaT triplet. Examples of our spectra and the CaT triplet are shown in Figure 1.

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Our Magellanic Cloud sample was drawn from late-type RSGs identified in Levesque et al. (2006) and supplemented by additional stars with broadband colors consistent with RSGs (for a complete discussion see Levesque et al. 2014). These stars were observed with the Magellan Inamori Kyocera Echelle (Bernstein et al. 2003) on the Magellan 6.5 m at Las Campanas Observatory during 2011 September 13–15. The spectra were taken using the 0.7 arcsec × 5 arcsec slit with 2 × 2 binning, slow readout, and the standard grating settings, and internal flats and ThAr lamps were observed for flat-fielding and wavelength calibration purposes. These data were reduced using a combination of standard IRAF echelle routines and the mtools package. The Thorne–Zytkow object candidate HV2112 is not included in our sample.

The physical properties of the observed stars in our sample that we adopt for our analyses—including T_eff, log g, and M_bol—are drawn from Levesque et al. (2005, 2006), and are based on fitting observed spectrophotometry of the RSGs with MARCS stellar atmosphere models and determining the best-fit model based on the strengths of the TiO absorption bands and the overall fit of the SED. This T_eff scale was chosen as it represents the physical properties for the RSGs determined based on the optical regime and optical absorption features, an appropriate choice for comparison with the optical atomic line features used in this work as it samples the same physical region of the RSG atmosphere (recalling these stars’ extended geometries and wavelength-dependent optical depths, as discussed above). These T_eff scales also show good agreement with stellar evolutionary models (including those that treat both single and binary evolution; Levesque 2018) and the effects of metallicity. For a complete list of the stars in our sample and their adopted physical properties, see Table 1.

Tabernero et al. (2018) recently published a new T_eff scale for CSGs in the LMC and SMC; however, this scale is based on atomic line fitting and the assumption we wish to test here, namely that these lines T_eff-sensitive. For late K- and M-type RSGs the Tabernero et al. (2018) scale is slightly warmer than the Levesque et al. (2006) scales as well as shallower (a weaker dependence on spectral type), but without knowing the dependence of the individual atomic features on T_eff in this very cool regime it is unclear whether this disagreement is due to a difference in method or a consequence of the lines’ behavior (for further discussion see Section 4.).

In addition to our sample of observed RSG spectra we also consider synthetic spectra produced by the MARCS stellar atmosphere models (e.g., Gustafsson et al. 2008). The spectra were generated for solar metallicity 15M_⊙ RSGs and adopt a spherical atmosphere geometry, a microturbulence parameter of 5 km s⁻¹, and log $g=-0.5,0.0,0.5,1.0$. The T_eff of the models used in this work range from 3400 to 4000 K in 100 K increments.

It should be noted that we restrict our use of the MARCS models to T_eff ≤ 4000 K, and do the same for our observed sample (one LMC star from the original Levesque et al. (2014) sample, LMC 169754, was cut from this work due to its relatively high T_eff of 4100 K). This was done specifically to restrict our study of atomic line T_eff diagnostics to the collision-dominated regime of cool star atmospheres (at T_eff ≳ 4000 K the effects of photoionization increasingly dominate the abundance of neutral Fe and Ti) and a regime where non-LTE effects are minimal; for further discussion see Section 4.

3.1. Atomic Lines and Equivalent Widths

3.2. Correlation Coefficients

We used the matplotlib Pylab software from SciPy to plot the measured EW data against the T_eff, M_bol, and log g of each star as determined by Levesque et al. (2005, 2006), with the EW as the dependent variable and the stellar parameters as the independent variables. For each variable pair we also calculated the Pearson’s r correlation coefficient and associated p-value for the sample, along with both linear (Ax+B) and second-degree polynomial (Ax²+Bx+C) functions of the best fit. Given our small sample sizes (ranging from 16 to 25 stars in a given host galaxy, which we treat separately due to metallicity effects) we adopt a conservative significance threshold of p < 0.01, rather than the more typical p < 0.05, to decrease our likelihood of incorrectly rejecting the null hypothesis. The Pearson correlation coefficients and best-fit function coefficients are summarized in Table 7. A similar analysis was also done for the MARCS stellar atmosphere model spectra, with the results summarized in Table 8.

Table 7.  Correlation and Best-fit Coefficients–Observations

EW	R1:	P1:	Linear	Linear	Quadratic	Quadratic	Quadratic
	Teff versus EW	Teff versus EW	A × 10−3	B	A × 10−4	B × 10−2	C
Milky Way
Ca ii Sum	0.755	0.00001	14.2	−41.99	−3.6	28.11	−532.2
Ca i 6572.0	0.500	0.01	0.019	−0.3407	−0.09979	0.75	−13.81
Ti i Sum	−0.0115	0.9	−0.02	−1.755	0.0083	−0.0063	2.874
Ti i/Fe i Ratio	0.232	0.3	0.0005	−0.1956	−0.00271	24.76	−385.29
Fe i Sum	0.234	0.261	0.8	0.1800	−0.1071	7.953	−144.4
LMC
CaT Sum	0.320	0.3	370	−3.76	−1.57	11.8	−211.6
Ca i 6572.0	0.455	0.08	0.20	−0.246	−0.0691	0.520	−9.39
Ti i Sum	−0.508	0.04	−0.70	3.86	−0.00048	0.034	−60.09
Ti i/Fe i Ratio	−0.594	0.05	−1.2	4.927	−0.377	2.63	−44.98
Fe i Sum	0.712	0.002	2.00	−19.98	−0.691	5.73	−111.42
SMC
CaT Sum	0.344	0.2	4.0	−5.63	−0.1446	1.105	−0.0202
Ca i 6572.0	−0.0847	0.7	−0.02	0.412	0.06027	−0.0446	8.5192
Ti i Sum	−0.490	0.05	−0.6	3.35	0.2461	−0.1876	36.75
Ti i/Fe i Ratio	−0.465	0.06	−0.8	3.52	0.4702	−0.354	67.33
Fe i Sum	−0.3054	0.2	−1.4	9.56	0.5332	−0.407	81.91

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Table 8.  Correlation and Best-fit Coefficients—MARCS Models

log g	EW	R1:	P1:	Linear	Linear	Quadratic	Quadratic	Quadratic
		Teff versus EW	Teff versus EW	A × 10−2	B	A × 10−4	B	C
Milky Way
−0.5	CaT Sum	0.937	0.0002	0.34	−0.5974	−0.04081	−0.03412	−57.97
	Ca i 6575.5	0.780	0.0131	0.05918	−1.0271	−0.01515	0.01985	−22.32
-	Ti i Sum	0.0301	0.939	0.002432	2.2700	−0.02727	0.02053	−36.07
-	Ti i/Fe i Ratio	−0.979	0.000004	−0.04	2.2208	−0.000323	0.00199	−2.29
-	Fe i Sum	0.518	0.153	0.1361	3.7286	−0.07506	0.05782	−101.88
0.0	CaT Sum	0.869	0.0023	0.67	−7.875	−0.1269	0.1022	−186.3
-	Ca i 6575.5	0.834	0.0052	0.06	1.275	−0.01383	0.01105	−20.7247
-	Ti i Sum	0.347	0.3598	0.03	1.198	−0.02288	0.01746	−30.97
-	Ti i/Fe i Ratio	−0.980	0.000004	−0.04	2.0135	−0.002211	0.001278	−1.0956
-	Fe i Sum	0.604	0.0850	0.0016	224.74	−0.06842	0.05302	−93.95
0.5	CaT Sum	0.926	0.0003	0.3	−2.664	−0.03986	0.033005	−58.70
-	Ca i 6575.5	0.874	0.0048	0.07	−1.541	−0.0149	0.01908	−22.48
-	Ti i Sum	0.409	0.275	0.08988	−1.0564	0.05430	0.04175	−77.40
-	Ti i/Fe i Ratio	−0.974	0.001	−0.0004	1.913	−	−	−
-	Fe i Sum	0.785	0.0203	0.204	−0.4100	−0.06111	0.0481	−86.33
1.0	CaT Sum	0.818	0.0071	0.27	−1.7245	−0.06196	0.04931	−88.34
-	Ca i 6575.5	0.888	0.0014	0.07236	−1.6663	−0.01196	0.009721	−18.48
-	Ti i Sum	0.779	0.0134	0.08	−1.0700	−0.02192	0.01731	−31.89
-	Ti i/Fe i Ratio	−0.970	0.00007	−0.027	1.52	−0.00031	0.00199	−2.59
-	Fe i Sum	0.965	0.0001	0.216	−2.95	−0.0092	0.0234	−41.65
LMC
−0.5	CaT Sum	0.945	0.00039	58.3	−11.16	−0.00096	0.076	−138
	Ca i 6575.5	0.694	0.0561	1.000	−0.2009	0.0000231	−0.00159	2.867
	Ti i Sum	0.526	0.1805	2.4	1.325	0.00012	−0.0086	17.46
	Ti i/Fe i Ratio	0.357	0.386	0.00003	90.65	−0.000000989	0.000748	−0.4062
	Fe i Sum	0.222	0.5962	1.900	2.851	0.00025	−0.0179	35.68
0.0	CaT Sum	0.983	0.00001	50.70	−9.479	−0.000419	0.0357	−65.17
	Ca i 6575.5	−0.8156	0.01358	−4.3	−0.8156	0.0001309	−0.00989	19.33
	Ti i Sum	0.997	0.00186	0.000004	1.74	0.000000554	−0.00405	9.097
	Ti i/Fe i Ratio	−0.843	0.0086	−3.00	1.51	−0.000847	0.00589	−9.73
	Fe i Sum	0.604	0.079	0.01784	0.0009	0.5647	0.00000254	−0.000957
0.5	CaT Sum	0.926	0.00013	74.30	−18.65	−0.00000842	0.0136	29.83
	Ca i 6575.5	−0.8847	0.00356	−6.300	2.666	0.0000149	−0.0115	22.43
	Ti i Sum	−0.3407	0.409	−1.6	2.562	0.00002027	−0.0149	2946
	Ti i/Fe i Ratio	0.409	0.3142	−0.00005	1.133	0.0000000678	−0.00106	2.729
	Fe i Sum	0.672	0.0680	5.8	1.40	0.000295	−0.0210	40.60
1.0	CaT Sum	0.9917	0.000001	38.05	−6.991	−0.00001579	0.0153	−27.95
	Ca i 6575.5	−0.887	0.0033	−1.1	0.463	−0.00000115	−1.066
	Ti i Sum	−0.625	0.0974	−4.1	3.386	0.0000227	−0.0170	33.52
	Ti i/Fe i Ratio	−0.453	0.259	−0.3	1.119	−0.00000012	0.000059	0.955
	Fe i Sum	−0.663	0.0731	−4.9	4.719	0.0000257	−0.0192	38.77
SMC
−0.5	CaT Sum	0.927	0.00093	0.498	−4.926	−0.0999	0.0779	−137.46
	Ca i 6575.5	−0.994	0.000001	−0.015	0.630	0.000824	−0.000754	1.723
	Ti i Sum	−0.933	0.00071	−0.034	3.113	−0.000319	0.00199	−1.107
	Ti i/Fe i Ratio	−0.912	0.00158	−0.043	2.266	0.000659	−0.00524	11.00
	Fe i Sum	0.458	0.255	0.015	2.765	−0.000736	0.00552	−7.00
0.0	CaT Sum	0.974	0.00004	0.406	−4.0513	0.000436	0.0359	−61.862
	Ca i 6575.5	−0.892	0.00287	−0.0130	0.000216	−0.00170	3.415
	Ti i Sum	−0.7132	0.0469	−0.015	1.919	−0.000526	0.00367	−5.054
	Ti i/Fe i Ratio	−0.970	0.000071	−0.0304	1.357	−0.000169	0.000930	−0.886
	Fe i Sum	0.799	0.01732	0.0370	2.257	0.000926	−0.00639	14.543
0.5	CaT Sum	0.983	0.00001	0.342	−3.720	−0.0288	0.244	−41.93
	Ca i 6575.5	−0.9244	0.00102	−0.000104	0.569	0.000133	−0.00115	2.343
	Ti i Sum	0.879	0.00404	−3.300	2.852	0.000477	−0.00381	9.185
	Ti i/Fe i Ratio	0.391	0.330	0.0964	−23.157	0.906	0.670	−1225.411
	Fe i Sum	−0.972	0.00005	−4.600	3.862	0.000107	−0.00120	5.277
1.0	CaT Sum	0.967	0.00009	0.260	−2.204	−0.000179	0.0157	−25.924
	Ca i 6575.5	−0.887	0.0033	−0.011	0.463	−0.000115	0.00753	−1.0659
	Ti i Sum	−0.806	0.0157	−0.033	2.653	−0.000157	0.00739	−11.375
	Ti i/Fe i Ratio	−0.993	0.000001	−0.0402	1.725	−0.0000362	−0.000138	1.244
	Fe i Sum	−0.975	0.00004	−0.050	4.662	0.00000121	−0.000648	4.284

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Below we consider each spectral feature and its potential diagnostic utility.

CaT: in our Milky Way sample, the EW of the CaT feature (Figure 2) shows a strong and statistically significant positive correlation with T_eff, with Pearson’s r = 0.755 and p =0.00001. This is in good agreement with the MARCS stellar atmosphere models at Milky Way metallicity (Figure 8), which predict robust positive correlations between CaT and T_eff across the full range of supergiant surface gravities (e.g., Pearson’s r = 0.869 and p = 0.0023 for the log g = 0.0 models). However, no similar correlation is seen in the LMC and SMC samples, despite the LMC- and SMC-metallicity MARCS models also predicting strong positive correlations at all supergiant surface gravities (Figures 9 and 10).

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In the Milky Way the CaT EW is positively correlated with log g and M_bol. No correlation is seen in the LMC data, however, the SMC sample shows a positive correlation between the CaT EW and M_bol.

Ca i: the MARCS models predict a correlation between the EW of the Ca i 6572.0 line and T_eff for all but the lowest surface gravities and highest metallicities (e.g., the log g =−0.5 models at Milky Way metallicity and the log g = −0.5 and log g = 0.0 models at LMC metallicity). However, this is not borne out by our observations (Figure 3); the LMC and SMC samples show no significant correlation between Ca i and T_eff, while the Milky Way sample shows a moderately strong positive correlation (r = 0.500) with a borderline p = 0.01. It is also worth noting that the MARCS models predict a positive correlation between Ca i and T_eff at Milky Way metallicity, but a negative correlation at the lower LMC and SMC metallicities, with the Ca i line getting weaker at higher T_eff.

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Ti i/Fe i Ratio: none of the observed data revealed significant correlations between the Ti i 8518.1/Fe i 8514.1 ratio (Figure 4) and RSG physical properties. By contrast, the MARCS models predict significant negative correlations between this ratio and T_eff for all log g values at Milky Way metallicity, while the LMC and SMC-metallicity models show significant negative correlations at particular values of log g (log g = 0.0 in the LMC and log g = −0.5, 0.0, and 1.0 in the SMC).

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Ti i Sum: none of the observed samples—at Milky Way, LMC, or SMC metallicity—show any evidence for statistically significant correlations between the sum of the Ti i line (Figure 5) EWs and any of the RSG physical properties.

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The Milky Way metallicity MARCS models predict no statistically significant correlation between the Ti i EW and T_eff. However, in the LMC the Ti i sum showed a strong positive correlation with T_eff for the log g = 0.0 models only (r =0.997, p = 0.00186), and the SMC models showed a strong negative correlation with T_eff at log g = −0.5 (r = −0.933, p =0.00071) and a strong positive correlation at log g = 0.5 (r = 0.879, p = 0.00404).

Fe i Sum: our observed Milky Way and SMC samples show no correlation between the sum of the Fe i line (Figure 6) EWs and any of the RSG physical properties, however, the LMC sample shows evidence of a positive correlation between the Fe i sum and T_eff. Both of these results are also at odds with the predictions of the MARCS models, which only predict a significant positive correlation between Fe i and T_eff for Milky Way supergiants with log g = 1.0 (r = 0.965, p = 0.0001) and a significant negative correlation between Fe i and T_eff for SMC supergiants with log g = 0.5 and 1.0 (r = −0.972, p = 0.00005 and r = −0.975, p = 0.00004, respectively).

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Ca ii 3D Plots: the CaT EW showed the most promise in our observed data as an atomic line diagnostic of T_eff, however, this feature is also well known as a potential diagnostic of log g and luminosity, calling the degeneracy of its T_eff correlation into question. To further examine this we created 3D plots for our observed data from all three host galaxies with T_eff, log g, and Ca ii EW (see Figure 7) as the respective x-, y-, and z-axes.⁴ These data were then fitted with linear and quadratic planes of the best fit. The full suite of correlation coefficients for the data in the 3D plots is given in Table 9, and equations for the linear and quadratic best fits are given in Table 10. The linear plane best-fit equation is $Z={Ax}+{By}+C$, while the quadratic plane fit is $Z={Ax}+{By}+{Cxy}+{{Dx}}^{2}+{{Ey}}^{2}+F$.

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Table 9.  3D CaT Correlation Coefficients

Galaxy	EW	R2:	P2:	R3:	P3:
		EW versus Logg	EW versus Logg	EW versus Mbol	EW versus Mbol
Milky Way	CaT Sum	0.665	0.00030	0.721	0.0002
Milky Way	Ca i 6572.0	0.558	0.0070	0.526	0.0119
Milky Way	Ti i Sum	0.162	0.472	0.1593	0.4789
Milky Way	Fe i Sum	−0.0700	0.757	−0.079	0.727
LMC	CaT Sum	0.443	0.0856	0.4471	0.0825
LMC	Ca i 6572.0	0.427	0.104	0.296	0.266
LMC	Ti i Sum	−0.149	0.5808	0.209	0.438
LMC	Fe i Sum	0.398	0.127	0.018	0.948
SMC	CaT Sum	0.5564	0.0204	0.631	0.0066
SMC	Ca i 6572.0	0.3601	0.156	0.436	0.0799
SMC	Ti i Sum	−0.2147	0.4079	−0.239	0.356
SMC	Fe i Sum	0.0866	0.7411	0.0866	0.741

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Table 10.  CaT 3D Best Fits

Galaxy	EW	Linear	Linear	Linear	Quad	Quad	Quad	Quad	Quad	Quad
		A × 10−2	B	C	A	B	C	D	E × 10−4	F
Milky Way	CaT Sum	0.8	2.927	20.46	−23.21	0.0358	−8.26	0.0027	−0.0869	−0.0741
LMC	CaT Sum	6572.0	0.308	3.50	−0.580	−219.29	0.128	73.02	−0.022	−0.180
SMC	CaT Sum	−0.02246	3.1687	10.5622	−1485.5	0.7967	−303.03	0.08187	−10.6	−11.340

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In light of the small sample sizes used in this work (25 stars in the Milky Way sample, 16 in the LMC, and 17 in the SMC) we are cautious about over-interpreting the statistical results drawn from our data. However, it is still interesting to examine areas where our observations and model results agree or diverge from each other and from past work, and to consider potential physical explanations for why this may be and the implications for future work.

Our observed spectra show a strong positive correlation (r = 0.755, p = 0.00001) between CaT and T_eff for Milky Way RSGs, in agreement with the predictions of the MARCS stellar atmosphere models. However, it is difficult to discern whether this is primarily a consequence of T_eff or log g effects on the CaT absorption features and surrounding continuum. In the Milky Way there is a significant correlation (r = 0.549, p = 0.0082) between T_eff and log g, with cooler stars having lower surface gravities (an unsurprising consequence of the effect that a decreasing T_eff and constant or increasing M_bol will have on the stellar radii), and both log g and M_bol are positively correlated with the CaT sum (see Table 9). Figure 7 compares the CaT EW, T_eff, and log g of our Milky Way stars in 3D space, along with the best quadratic plane fit to the sample, but—as also noted by previous work—it is unclear which physical property is primarily responsible for driving the evolution of CaT in RSG spectra.

We also do not see any correlation between CaT and T_eff in either the LMC or SMC observations. While this suggests that metallicity may also play a role in the evolution of the CaT with stellar properties (in agreement with previous work that found a metallicity dependence in the CaT EW for supergiants (e.g., Armandroff & Da Costa 1991; Mallik 1996; Ginestat et al. 2016), this is at odds with the predictions of the MARCS models, which predict a strong correlation between CaT and T_eff at all of the model metallicities. The models do, however, predict an expected overall decrease in the strength of the CaT with metallicity. Figures 7–9 compare the MARCS models and observed data, highlighting the decrease in EW with metallicity as well as comparing the EWs predicted by the models to those observed in the data. Note that Dorda et al. (2016b) also directly compare LMC- and SMC-metallicity MARCS model EWs to T_eff (though their data span a broader T_eff range of 3300–4500 K to better encompass the warmer F-, G-, and K-type supergiants in their observed sample) and find similar results.

Our observed data also blend a range of surface gravities that sample the lower end of the RSG surface gravities (the mean log g of the Milky Way, LMC, and SMC samples is 0.15, −0.275, and −0.247, respectively) while the models with different log g are considered separately. If we combine the results from the MARCS models across all surface gravities, the correlations between CaT and T_eff get weaker in the Milky Way (r = 0.467, p = 0.007) and LMC (r = 0.712, p = 0.0004), while the SMC data fails to satisfy our p < 0.01 significance threshold (r = 0.356, p = 0.046). Considering these results, it is possible that decreased metallicity combined with a mix of surface gravities in our observed samples could contribute to the lack of statistically robust correlation between CaT and T_eff in the LMC and SMC data.

Quadratic best fits to the CaT EW, T_eff, and log g data in 3D space show a stronger relationship between the three parameters for the lowest T_eff and log g values. The CaT EW is consistently high at high log g values (≥0.6) for all values of T_eff, but decreases nearly linearly with log g at low T_eff. Above T_eff ≥ 3750 K, the relationship between EW and log g becomes more complex. By comparison, both the LMC and SMC quadratic best fits in 3D show a concave shape, with the evolution of CaT as a function of log g and T_eff that is hard to quantify and not well fit by a linear relation (for example, both high T_eff + low log g and low T_eff + high log g correspond to CaT EW minimums).

The relationships are simplified (but also more poorly fit) in the linear plane best fits to the data, given in Table 10. In this case the Milky Way and LMC results broadly align with predictions from Ginestat et al. (2016) asserting that the intensity of the CaT is correlated with log g, but the SMC does not, suggesting a more complex relationship for this sample. Erdelyi-Mendes & Barbuy (1990) concluded that at high metallicity Ca ii was inversely related to log g; as the SMC has the lowest metallicity of our sample it suggests that this difference could in part be attributable to changes in this relationship as a function of metallicity.

The Dorda et al. (2016a) sample consisted of early-G to M3 stars, with a small sample of later-type M stars, as it was thought that the TiO band would significantly erode the continuum in the latest M-type stars and make reliable EW measurements difficult beginning at a spectral type of M0. However, as found both by Ginestat et al. (2016) and this work, the stronger TiO band at later spectral types (corresponding to cool T_eff; e.g., Levesque et al. 2005; Tabernero et al. 2018) reliably corresponds to a decrease in the local continuum, and a subsequent apparent weakening of the CaT, in agreement with the evolution of the CaT seen at warmer T_eff. This effect therefore improves rather than weakens the utility of the CaT as a T_eff diagnostic. For a direct comparison between the models results and the Milky Way, LMC, and SMC results see Figures 8–10.

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Moving beyond the CaT feature, it is interesting to note that there is no correlation in any of our observed data between Ca i or the Ti/Fe ratio and any of the RSG physical properties, at odds with what the models predict. Beyond that, most of the observed and model samples predict no correlation between the sum of the Ti i or Fe i absorption features and the physical properties of RSGs, with a few noted exceptions. For example, the observed LMC spectra show a positive correlation between the Fe i sum and T_eff. However, this is at odds with the predictions of the MARCS stellar evolution models, which only predict correlations between Fe i and T_eff for the highest log g models (notably higher than the average log g of our observed RSGs), and is almost certainly a consequence of small number statistics in our sixteen-star LMC sample. Still, since Dorda et al. (2016a) predict a strong correlation between Ti i and Fe i and spectral type for CSGs—which we would expend to extend to a correlation between these features and T_eff and be observable even in a small sample—it is worth considering some of the physical phenomena that may impact the formation and evolution of these lines in RSG spectra.

Our MARCS model results do not highlight Ti i and Fe i as robust diagnostic lines for M-type RSGs, however, these models also assume LTE. How might non-LTE conditions affect these predictions? Bergemann et al. (2012) studied individual RSGs to determine the impact of non-LTE on Ti i and Fe i spectral features. They found that the significance of non-LTE corrections was dependent on T_eff, metallicity, and log g, noting that for both Ti i and Fe i, non-LTE corrections in order to align results with observations were lower, or near zero, at lower temperatures, −3400 K ≥ T_eff ≤ 3800 K. At higher temperatures, the formation of Fe i lines remains largely unaffected by non-LTE, while the Ti i line does show some variation, with Ti i EWs underestimated by LTE as compared to non-LTE models. However, for M-type RSGs non-LTE effects on these lines do not play a significant role and as such cannot be considered a variable for the disagreements between the observed data and the models.

As these are all neutral lines, it is also worth considering the excitation potential of these features. Both Dorda et al. (2016a) and our models predict a relationship between Fe i and T_eff and between Ti i and T_eff, with Dorda et al. (2016a) arguing that this is due to their low excitation potentials (6.82 eV for Ti i, 7.87 eV for Fe i), thus rendering the neutral abundances of these elements particularly sensitive to T_eff. However, while this reasoning is robust for warmer stars it breaks down for M-type RSGs. At these low temperatures (≲4000 K) photoionization is no longer the primary means of producing Ti ii and Fe ii, and instead collision becomes the dominant means of excitation (for more discussion see Bergemann et al. 2012). This decouples the Ti i/Ti ii and Fe i/Fe ii fractions from T_eff in the cool and low-density atmospheres of RSGs. Taken as a whole, the Ti i and Fe i absorption features are not effective diagnostics of T_eff for the coolest RSGs (a result in agreement with Figures 7 and 8 of Dorda et al. (2016b), which show a weak correlation between these features and spectral type for the M0–M3 stars in their sample).

The Ca i absorption feature remains a puzzle. The MARCS models predict a positive correlation between Ca i EW and T_eff at Milky Way metallicity, but a negative correlation for SMC metallicities and the higher surface gravity LMC models. The latter is what would naively be expected based simply on the evolution of the CaT absorption feature: as the Ca ii abundance (and the CaT EW) increases, we would expect a corresponding decrease in the Ca i abundance and EW. It is unclear why this is predicted at lower metallicities (and higher surface gravities) but not at solar metallicity. It is possible that at higher metallicities the Ca i abundance is high enough to saturate, resulting in a nonlinear evolution of the Ca i absorption feature at higher metallicities. As in the case of Ti and Fe, the relative contributions from photoionization and collisional excitation could also play a role. Finally, none of these expected correlations appear in our observed data, suggesting that additional effects (including the impact of non-LTE) could further complicate the formation and evolution of the Ca i with T_eff.

While these results are based on only a small sample of M-type RSGs, it is nevertheless important to consider whether these or other atomic lines can be used to directly infer the stars’ physical properties. To take just one example, spectra from Gaia span only a narrow wavelength range (∼8450–8750 Å), but this critical regime includes the CaT absorption feature as well as all of the Ti i absorption lines and seven of the Fe i absorption lines included in this work. Identifying—or excluding—useful T_eff diagnostics in this wavelength regime represents a potentially powerful tool for leveraging the wealth of potential RSG data available in current and future Gaia data releases (which extends throughout the Milky Way and to the LMC and SMC), and may make it possible to greatly improve the accuracy of the physical properties determined for these stars.

We would like to thank Trevor Dorn-Wallenstein, Philip Massey, and George Wallerstein for useful discussions regarding this research, as well as the staff of Apache Point Observatory and Las Campanas Observatory for their support in acquiring the observed spectra used in this work. These efforts were supported in part by a fellowship from the Alfred P. Sloan Foundation.

Software: IPython (Pérez & Granger 2007), SciPy (Jones et al. 2001), NumPy (Van Der Walt et al. 2011), IRAF (Tody 1986, 1993), Matplotlib (Hunter 2007).