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A&A
Volume 676, August 2023
Article Number A84
Number of page(s) 16
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/202244770
Published online 10 August 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Red supergiant stars (RSGs) are an “extreme” stellar population that occupies the coolest and most luminous region of the Hertzsprung–Russell (HR) diagram. They have evolved past their main-sequence phase, where they spent most of their lifetime. As Population I stars with an age of about 8 − 20 Myr and moderately high initial masses (∼8 − 40 M⊙), they have low effective temperatures of Teff ∼ 3500 − 4500 K, high luminosities of ∼4000 − 400 000 L⊙, and large radii of ∼100 − 1500 R⊙ (Massey 1998; Massey & Olsen 2003; Levesque 2010, 2017; Neugent et al. 2020). Given all their distinct physical properties, RSGs play a critical role in massive star formation and evolution (Humphreys & Davidson 1979; Levesque et al. 2005; Ekström et al. 2013; Massey 2013; Davies et al. 2017).

There are several possibilities for the end fate of the RSGs. However, these pathways are highly dependent on the basic physical parameters, such as initial mass, metallicity, chemical composition, and more importantly (with more uncertainties) the mass-loss rate (MLR). The most common fate of RSGs is to explode as hydrogen-rich Type II-P core-collapse supernovae (CCSNe). More than half of the discovered CCSNe are Type II-P, and RSGs have been directly identified as progenitors in pre-explosion images of nearby events (Smartt 2009, 2015). Another end point for RSGs is a SN explosion at the blue end of the HR diagram, as it may evolve backwards and become a Wolf-Rayet star (WR) for a short time (Humphreys 2010; Ekström et al. 2012; Meynet et al. 2015; Davies & Beasor 2018). Recent studies suggest that some RSGs may directly collapse into a black hole without the fabulous SN explosion, that is, the so-called failed supernova (Kochanek et al. 2008; Adams et al. 2017). Moreover, in rare cases, episodic or eruptive mass-loss events in RSGs a few months or years before the SN explosion –which likely create a thick circumstellar envelope– may also lead to Type II-n SNe (Yoon & Cantiello 2010; Zhang et al. 2012; Smith et al. 2015). In any case, the final fate and the properties of the possible explosion of the RSGs must be largely impacted by their MLR. Apart from the final fate, the imprint of MLR on the shape of the wind bubbles around the stars and the chemical feedback in the surrounding environment will also be different. In that sense, stellar winds of RSGs drive the evolution of the host region and are therefore important in a more general way, and do not simply dictate the final fate (van Loon 2006; Javadi et al. 2013).

However, the most influential modes of mass loss are also the most uncertain (Smith 2014), as we are still far from clear about the dominant physical mechanisms of the mass loss of RSGs (e.g., episodic mass loss, stellar winds, luminosity, metallicity, binarity, and the roles of pulsation, convection, and rotation on them, etc.; MacGregor & Stencel 1992; Harper et al. 2001; Yoon & Cantiello 2010; Mauron & Josselin 2011; Beasor & Davies 2016). The commonly used mass-loss model for the RSGs is similar to that used for the asymptotic giant branch stars (AGBs; e.g., van Loon et al. 2006; Höfner 2008), because they have comparable physical properties (relatively high luminosity, low effective temperature, large radius, low surface gravity, etc.). The general physical picture is that, as the stellar atmosphere is convective and extended (gravitational binding is relatively weak at the outer boundary), pulsations (and/or convection) may lift gas to a distance (e.g., a few stellar radii) where a substantial amount of dust is able to condense with relatively low equilibrium temperature (e.g., ∼1000–1500 K). Dust-envelope expansion is then driven by radiation pressure on the dust grains, which collides with the surrounding gas and drags it along (Ivezic & Elitzur 1995; Willson 2000; Elitzur & Ivezić 2001; van Loon et al. 2005; Verhoelst et al. 2009; Smith 2014; Goldman et al. 2017; Höfner & Olofsson 2018).

In the context of the European Research Council project “Episodic Mass Loss in Evolved Massive Stars: Key to Understanding the Explosive early Universe” (ASSESS)1, the aim of which is to determine the role of episodic mass loss in the most massive stars, we set out to determine the MLRs of RSGs and their impact. In this paper, we present the analysis of MLR for the most comprehensive RSG sample in the SMC to date. The sample selection and data analysis are described in Sects. 2 and 3, respectively. The results and discussion are presented in Sects. 4 and 5. A summary is given in Sect. 6.

2. The RSG sample in the SMC

The initial RSGs sample used in this work was taken from Yang et al. (2020) and Ren et al. (2021), which included 1239 and 2138 RSG candidates in the SMC derived using stellar evolutionary tracks and H-band-related color–color diagram methods (both were constrained by Gaia astrometry; see more details in Yang et al. 2019, 2020; Ren et al. 2021), respectively. The two samples were cross-matched to remove duplication, which resulted in 2332 unique targets. Following Yang et al. (2019, 2021a), we retrieved data in 53 different bands including 2 ultraviolet (UV), 26 optical, and 26 infrared (IR) bands as listed in Table 1, ranging from the UV to the mid-IR. See Yang et al. (2019) for the content of the photometric catalogs we used.

Table 1.

Datasets for the SED fitting.

During the initial analysis, we discovered that previously used Spitzer Enhanced Imaging Products (SEIP) source list (including 2MASS, WISE, and Spitzer photometric data) might not be the best choice, while SAGE-SMC provided better photometry and quality assessment (Gordon et al. 2011). We therefore replaced the SEIP with SAGE-SMC data and removed blended targets according to the SAGE-SMC close source flag, which resulted in 2304 targets. Additionally, one more target was also removed due to lack of 2MASS data (visual inspection indicated that the target was contaminated by the nearby environment), because we normalized the spectral energy distribution (SED) at this wavelength. In total, the intermediate RSG sample contains 2303 sources. Meanwhile, the WISE data were also updated accordingly from the ALLWISE catalog, with the same photometric quality criteria (nb = 1, ext_flg = 0, w?cc_map=0 or w?flg=0) but different signal-to-noise ratios (S/Ns; S/NWISE1 > 5, S/NWISE2 > 5, S/NWISE3 > 7, and S/NWISE4 > 10) for each band. Moreover, in order to take advantage of the newly released data, we updated several datasets to their latest versions. For example, Gaia data were updated to EDR3 (Gaia Collaboration 2021), SkyMapper data to DR2 (Onken et al. 2019), NSC data to DR2 (Nidever et al. 2021), and VMC data to DR5. New data were also added to the data pool, for example MCPS and DENIS.

As we compiled the catalogs and checked the data, we also discovered several issues with respect to some datasets. For instance, the saturation flag in the VMC pipeline was not correctly applied, which resulted in several saturated stars marked as stellar instead of saturated objects (these data were discarded; Nicholas Cross, priv. comm.). The MCPS I-band data suffered from a poorly modeled point spread function (PSF; Zaritsky et al. 2002) and hence were discarded. Some targets fainter than about V = 15.5 mag in M2002 were discarded because they had large uncertainties. The high S/N criteria for WISE [12] and [22] band data were not sufficient in some cases (∼1%), which was identified by visual inspection of images. This is due to the low sensitivity and angular resolution at long wavelengths.

In the meantime, as indicated in Yang et al. (2020), there is a continuum with similarity and overlap between RSGs and AGBs in photometry, spectra, and light curves; the boundary between RSGs and AGBs is blurry with no perfect way to distinguish between them using current data (e.g., see discussion in Sect. 3.1 of Yang et al. 2020). Therefore, to be on the safe side, we removed the targets in the overlapping region between the two populations on the 2MASS color–magnitude diagram (CMD; see Fig. 1) by eye, which left 2121 targets in the final RSG sample. Figures 2, 1 and 3 show the spatial distribution, CMDs, and the normalized (at 2MASS KS-band) median SED of the final RSG sample.

thumbnail Fig. 1.

Gaia (upper left), 2MASS (upper right), IRAC (bottom left), and MIPS (bottom right) CMDs of the final RSG sample in the SMC. Targets in the overlapping region (green lines) between AGBs and RSGs on the 2MASS CMD are removed (open black circles). Other general stellar populations, e.g., AGB, RGB, TRGB, and BSG stars, are marked on the Gaia and 2MASS CMDs, but not on IRAC and MIPS CMDs due to the overlap between different populations (see Fig. 12 of Yang et al. 2021a). Background targets (gray dots) are from Yang et al. (2019).

thumbnail Fig. 2.

Spatial distribution of the final RSG sample (orange color) in the SMC. Targets removed from the initial sample are marked as open black circles (see text and Fig. 1 for details). Background targets (gray dots) are from Yang et al. (2019).

thumbnail Fig. 3.

Normalized (at 2MASS KS-band) median SED of the final RSG sample in the SMC. Due to the strict constraint of S/N, WISE [12] and [22] band data only include relatively bright targets with high fluxes.

With RV = 3.1 and E(B − V) = 0.033 (Schlafly & Finkbeiner 2011; Freedman et al. 2020), the average Galactic foreground extinction of the SMC was corrected in the optical and IR bands using the extinction law of Wang & Chen (2019), which obeyed the form of the model of Cardelli et al. (1989) with newly derived coefficients. Meanwhile, the foreground extinction in the far-UV (FUV) and near-UV (NUV) bands was corrected using AFUV/E(B − V) = 7.44 and ANUV/E(B − V) = 7.86 (Sun et al. 2021).

The luminosity of each individual target was calculated by integrating over the entire dereddened SED. Figure 4 shows the distribution of luminosity for the sample, spanning the range of log10(L/L⊙)≈3.5 − 5.5. In the majority of cases, this would yield an appropriate estimation of the luminosity with typical errors of less than 5%. However, it might underestimate the luminosities of targets with high MLRs because there are no data at longer wavelength other than 24 μm, or overestimate the luminosities of targets with hot binary companions because there were additional UV detections (see more details in discussion). A further comparison between our results and the theoretical models (e.g., Gilkis et al. 2021) reveals general agreement in the detected number of RSGs in the SMC, indicating the validity of our sample. We note that we followed the semi-empirical selection criteria (e.g., Boyer et al. 2011; Dalcanton et al. 2012; Melbourne et al. 2012; Yang et al. 2019; Neugent et al. 2020; Tantalo et al. 2022, etc.) to constrain our RSG sample as being brighter than the KS-band TRGB (KS-TRGB; see more details in Yang et al. 2020 and Ren et al. 2021). However, the theoretical threshold for the massive stars is around 8 M⊙ with log10(L/L⊙)≈4.0 (e.g., Woosley & Weaver 1986), which is in contradiction with the lower limit (log10(L/L⊙)≈3.5) of our sample. Therefore, we indicate that the low-luminosity targets (log10(L/L⊙)≲4.0) may not be true RSGs, but ∼6 − 8 M⊙ red helium-burning stars (but this does not impact our main results, as shown in the sections below). Moreover, inevitably, we may still have some small amount of contamination from the AGB population at the red and/or faint end of our sample (the Gaia and 2MASS CMDs in Fig. 1 show that the contamination from the general population of AGBs is avoided), which is the main reason why we remove the targets in the overlapping region between AGBs and RSGs, as mentioned above.

thumbnail Fig. 4.

Histogram of the luminosity of the final RSG sample. We note that the low-luminosity targets (log10(L/L⊙)≲4.0) may not be true RSGs, but ∼6 − 8 M⊙ red helium-burning stars.

The effective temperatures (Teff) of targets were calculated using an algorithm from Yang et al. (2020) as

log 10 T eff = − 0.23 ( J − K S ) 0 + 3.82 , $$ \begin{aligned} \log _{10}T_{\rm eff}=-0.23(J-K_S)_0+3.82, \end{aligned} $$(1)

which converted the observed J − KS color to Teff with total extinction correction (including foreground extinction of the Milky Way, interstellar extinction of the SMC, and circumstellar extinction of the star) of AV = 0.1 mag (this value was justified by the fact that massive stars, like RSGs, are typically close to the star formation region because of their short evolutionary timescale, and RSGs produce dust by themselves). The typical uncertainty of Teff is ∼0.017 dex (Yang et al. 2020). However, the applied reddening correction was an average value for the whole SMC, and therefore targets within complicated environments or with high MLRs might be underestimated. Figure 5 shows the distribution of Teff for the sample, spanning the range of log10(Teff)≈3.5 − 3.75.

thumbnail Fig. 5.

Histogram of the Teff of the final RSG sample.

3. DUSTY grids and SED fitting

The dust particles in a circumstellar dust shell absorb and re-emit the radiation from the central stars, which results in changes to the output stellar spectra. There are several dust radiative transfer models available ranging from 1D to 3D. For example, van Loon et al. (2005), Goldman et al. (2017), Beasor et al. (2020), Humphreys et al. (2020), and Wang et al. (2021) used the 1D DUSTY code (Ivezic & Elitzur 1997) to estimate the MLRs of RSGs and AGBs in the Milky Way, LMC, M31, and M33. Groenewegen (2012) and Groenewegen & Sloan (2018) also used the modified DUSTY code “More of DUSTY” (MoD) to calculate MLRs of carbon- and oxygen-rich evolved stars in several Local Group galaxies. Verhoelst et al. (2009) used the 1D MODUST code (Bouwman et al. 2000, 2001) to assess the chemical composition and MLRs of a sample of Galactic RSGs. Sargent et al. (2010) and Riebel et al. (2012) used the Grid of Red supergiant and Asymptotic giant branch star ModelS (GRAMS; Sargent et al. 2011; Srinivasan et al. 2011) calculated by the 2D 2DUST code (Ueta & Meixner 2003) to compute the MLRs of RSGs and AGBs in the LMC. Liu et al. (2017) also used the 2DUST to investigate the chemical composite and MLR of a sample of O-rich evolved Galactic stars. Recently, Cannon et al. (2021) used the 3D MCMax3D code (Min et al. 2009) to characterize a dust clump of Antares. Similarly, Montargès et al. (2021) used 3D RADMC3D to investigate the effect of dust on the ‘Great Dimming’ of Betelgeuse.

3.1. The DUSTY library and grids

We used one of the most popular 1D codes, DUSTY (V42), to construct a grid that covers a wide range of dust parameters in order to match the observed SEDs. DUSTY solves the radiative transfer equation for a central source surrounded by a spherically symmetric dust shell at a certain optical depth (τ, at 0.55 μm in our case), with a range of dust condensation temperatures (Tin) that change at the inner boundary of the dust shell. The grid is produced by the radiatively driven wind model of DUSTY (assuming the dust shell extends to a distance of 104 times the inner radius). Full dynamics calculations of radiatively driven winds can be found in Ivezic & Elitzur (1995) and Elitzur & Ivezić (2001) and references therein, while an analytic solution takes the form of

η ( y ) ∝ 1 y 2 [ y y − 1 + ( v 1 / v e ) 2 ] 1 / 2 , $$ \begin{aligned} \eta ({ y})\propto \frac{1}{{ y}^2}\left[\frac{{ y}}{{ y}-1+({ v}_1/{ v}_{\rm e})^2} \right]^{1/2}, \end{aligned} $$(2)

shown here as a reference. The density distribution is described by the dimensionless profile η(y), which DUSTY normalizes according to ηdy = 1. Here, v1 and ve represent the initial and final velocities of the stellar wind, respectively (Elitzur & Ivezić 2001).

The central source used in the DUSTY modeling was taken from the MARCS stellar atmosphere model (15 M⊙; Gustafsson et al. 2008) at the metallicity of Z = −0.75 and a typical surface gravity of log g = 0, with Teff of 3300, 3400, 3500, 3600, 3700, 3800, 3900, 4000, 4250, and 4500 K. As the broad-band photometric data were not enough to distinguish subtle spectral features, and in order to reduce the computational time, we resampled the surface fluxes of MARCS model (ranging from 1300 to 200 000 Å with a constant resolving power of R = 20 000) to match the custom DUSTY wavelength grid in steps of 200 Å between 1000 and 10 000 Å, 400 Å between 10 000 and 25 000 Å, 1000 Å between 25 000 and 350 000 Å, and 150 000 Å between 350 000 and 3 000 000 Å. Wavelength ranges longer than 200 000 Å that were not covered by the MARCS model were extrapolated using a blackbody. Using a MARCS model is better than a simple blackbody because it also represents evolutionary effects on the stellar atmosphere, such as the “H-bump” (Yang et al. 2021b).

The chemical composition of dust species for the RSG wind was mainly oxygen-rich (identified by the wide, smooth, and featureless Si-O stretching 9.7 μm and O-Si-O bending 18 μm silicate features) as indicated in many previous works (van Loon et al. 2005; Groenewegen et al. 2009; Verhoelst et al. 2009; Sargent et al. 2011; Goldman et al. 2017; Wang et al. 2021), for which the optical constants of “astronomical silicate” from Draine & Lee (1984) were adopted with a dust bulk density of ρd = 3.3 g cm−3. The selection of this dust species is somewhat arbitrary (as astronomical silicate is a popular or general choice for many previous studies), because, as noted above, broad-band photometric data are not able to accurately distinguish different silicate features (see more details in the discussion section). Moreover, some RSGs have optically thin shells (τ ≈ 0.001) or even no dust at all, for which no discrimination can be made on the dust species and the actual MLR is very low in any case.

A simple power law with an exponential decay of Kim-Martin-Hendry distribution (KMH; n(a)∝a−qe−a/a0 with a > amin; Kim et al. 1994) was adopted for the distribution of dust-grain size with a power-law index of q = 3.5, a lower limit of amin = 0.005 μm, and the scale height of a0 = 1.0 μm (observationally, there is evidence indicating that the dust-grain sizes of RSG wind might be between 0.1 and 1 μm; Groenewegen et al. 2009; Smith et al. 2001; Scicluna et al. 2015). Such a distribution is better than the simplistic homogeneous sphere models of the Mathis-Rumpl-Nordsieck distribution (MRN; Mathis et al. 1977), because it avoids the sudden cut-off at large dust-grain size. We note that the derived MLR would still depend on the size of the dust and many studies used different size distributions (van Loon et al. 2005; Ohnaka et al. 2008; Verhoelst et al. 2009; Sargent et al. 2010; Liu et al. 2017; Beasor & Davies 2018). However, again, photometric data cannot be used to properly distinguish the differences.

The Tin at the inner boundary of the dust shell varied from 400 to 1200 K with a step of 100 K. The condensation temperature of dust species may depend on the L, Teff, metallicity, and other factors of the central source, or the chemical composition of dust itself. Previous studies found that dust might begin to form further from the photosphere with lower Tin, meaning that the condensation temperatures of silicate dust varied from around 400 K to 1500 K with typical values of ∼1000–1200 K (Gail et al. 1984, 2020; Gail & Sedlmayr 1999; van Loon et al. 2005; Ohnaka et al. 2008; Sargent et al. 2010). For simplicity, we also set the sublimation temperature of 1200 K for the O-rich dust.

The models were divided into two groups depending on the value of τ at 0.55 μm. The first group (“normal grid”; τ ≤ 1.0) corresponds to the normal targets with τ varying from 0.001 (optically thin) to 1.0 in 99 piecewise equally spaced logarithmic steps3. In particular, there are 30 steps between 0.001 and 0.1, and 70 steps between 0.1 and 1.0. The second group (“dusty grid”; τ > 1.0) corresponds to the very dusty targets with τ varying from 1.0 to 10.0 in 99 equally spaced logarithmic steps. The choice of such steps are addressed in the discussion section.

In total, there are 17 820 theoretical O-rich models generated by DUSTY, for which 8910 models are for the “normal grid” and 8910 models are for the “dusty grid”. Each model was convolved with the filter profiles4 of the 53 bands to derive the model flux at each wavelength.

3.2. Spectral energy distribution fitting

The best-fit model for each target was chosen by calculating a modified minimum chi-square ( χ m 2 $ \chi_{m}^2 $), after normalizing the SED to the 2MASS KS-band (this band was selected due to the fact that it was much less affected by the extinction and had much less variability compared to the optical bands, e.g., the V-band). The χ m 2 $ \chi_{m}^2 $ for each source was calculated as

χ m 2 = 1 N − p − 1 ∑ N [ f ( Obs , λ ) − f ( Model , λ ) ] 2 f ( Model , λ ) , $$ \begin{aligned} \chi _{m}^2=\frac{1}{N-p-1}\sum ^N\frac{[f(\mathrm{Obs}, \lambda )-f(\mathrm{Model}, \lambda )]^2}{f(\mathrm{Model}, \lambda )}, \end{aligned} $$(3)

for which

f ( Obs , λ ) = F ( Obs , λ ) F ( Obs , K S ) × C ( Obs , λ ) , $$ \begin{aligned} f(\mathrm{Obs}, \lambda )=\frac{F(\mathrm{Obs}, \lambda )}{F(\mathrm{Obs}, K_S)}\times C(\mathrm{Obs}, \lambda ), \end{aligned} $$(4)

f ( Model , λ ) = F ( Model , λ ) F ( Model , K S ) × C ( Obs , λ ) , $$ \begin{aligned} f(\mathrm{Model}, \lambda )=\frac{F(\mathrm{Model}, \lambda )}{F(\mathrm{Model}, K_S)}\times C(\mathrm{Obs}, \lambda ), \end{aligned} $$(5)

C ( Obs , λ ) = F ( Obs , K S ) F ( Obs , λ ) , $$ \begin{aligned} C(\mathrm{Obs}, \lambda )=\frac{F(\mathrm{Obs}, K_S)}{F(\mathrm{Obs}, \lambda )}, \end{aligned} $$(6)

where N and p are the numbers of data points and degrees of free parameters in the fitting, respectively. The derived χ m 2 $ \chi_{m}^2 $ for each individual target can therefore be directly compared (N is different for each target). The formula can also be rewritten as

χ m 2 = 1 N − p − 1 ∑ N [ 1 − f norm . ( Model ) / f norm . ( Obs ) ] 2 f norm . ( Model ) / f norm . ( Obs ) , $$ \begin{aligned} \chi _{m}^2=\frac{1}{N-p-1}\sum ^N\frac{[1-f_{\rm norm.}(\mathrm{Model})/f_{\rm norm.}(\mathrm{Obs})]^2}{f_{\rm norm.}(\mathrm{Model})/f_{\rm norm.}(\mathrm{Obs})}, \end{aligned} $$(7)

for which

f norm . = F ( λ ) F ( K S ) . $$ \begin{aligned} f_{\rm norm.}=\frac{F(\lambda )}{F(K_S)}. \end{aligned} $$(8)

It can be seen that, by multiplying with the constant C, the observed SED was converted into a flat spectrum with each wavelength having the same weight across the whole SED. The model with minimal χ m 2 $ \chi_{m}^2 $ then corresponds to the one that most closely resembles the flat spectrum. Meanwhile, the distribution of χ m 2 $ \chi_{m}^2 $ is more relevant to the goodness of fit than its absolute values (see Fig. 6). We note that this χ m 2 $ \chi_{m}^2 $ formula is a variant of the standard χ2 test (Pearson 1900),

χ 2 = ∑ ( O − E ) 2 E , $$ \begin{aligned} \chi ^2=\sum \frac{(O-E)^2}{E}, \end{aligned} $$(9)

thumbnail Fig. 6.

Histogram of the χ m 2 $ \chi_{m}^2 $ distribution. The three vertical dashed lines indicate the 68th ( χ m 2 $ \chi_{m}^2 $ = 0.094), 90th ( χ m 2 $ \chi_{m}^2 $ = 0.203), and 95th ( χ m 2 $ \chi_{m}^2 $ = 0.371) percentiles, respectively.

where O and E represent the observed and expected (model) values, respectively. We did not take the normal approach of using the reduced χ2 (Andrae et al. 2010), which is widely used in astronomy as

χ r 2 = 1 p ∑ ( O − E σ ) 2 , $$ \begin{aligned} \chi _r^2 = \frac{1}{p}\sum \left(\frac{O-E}{\sigma }\right)^2, \end{aligned} $$(10)

where p and σ represent the number of degrees of freedom and errors, respectively. In the discussion section, we estimate the effect of our choice of fitting.

The final RSG sample was first fitted with the normal grids, and then the dusty ones (six targets) were again fitted with the dusty grids. After the automated fitting process, we also visually inspected each target (simultaneous inspection of SED fitting, optical to mid-IR images, and locations on the CMDs) to evaluate the fitting result. Figure 6 shows the histogram of the χ m 2 $ \chi_{m}^2 $ distribution from the normal grids. Three percentile values of 68th ( χ m 2 $ \chi_{m}^2 $ = 0.094), 90th ( χ m 2 $ \chi_{m}^2 $ = 0.203; presumably the beginning of relatively poor fitting), and 95th ( χ m 2 $ \chi_{m}^2 $ = 0.371) were shown in the diagram as references, respectively. Figure 7 shows the distribution of numbers of data points fitted for each target. On average, we had around 30 data points for each target, meaning that the shape of the SED was relatively well constrained.

thumbnail Fig. 7.

Histogram of numbers of data points fitted for each targets.

Figure 8 shows typical examples of the SED fitting for optically thin, normal, and very dusty targets (e.g., τ > 1.0) in the sample. In general, the very dusty targets are not well fitted, which is due to the lack of a dense grid at high optical depths, which would cost a large amount of computational time. Moreover, we also cross-matched our final sample with Spitzer/IRS Enhanced Products, which yields 25 matches. After visual inspection, we find that almost all (96%; 24/25) of the matched spectra in the sample show identical chemical composition compared to the results of the SED fitting, indicating that our SED fitting is working properly. However, due to the complication of mid-IR spectra, these percentages could vary. A few examples of IRS spectra are shown in Fig. 9.

thumbnail Fig. 8.

Examples of the typical SED fitting for optically thin (upper row), normal (middle row), and very dusty targets (bottom row). In each panel, the coordinate is marked on the upper left, and chemical composition, τ, and χ m 2 $ \chi_{m}^2 $ are marked on the upper right.

thumbnail Fig. 9.

Same as Fig. 8 but with examples of a zoomed-in region of the SED fitting with Spitzer/IRS spectra (gray) overlapped.

The resulting MLRs from DUSTY (for L = 104 L⊙, ψ = 200, and ρd = 3 g cm−1) were converted to real MLRs using the following prescription (Ivezic et al. 1999; van Loon 2000; Boyer et al. 2009; McDonald et al. 2011; Goldman et al. 2017), based on the scaling relation as,

M ˙ = M ˙ DUSTY ( L 10 4 ) 3 / 4 ( ψ 200 ) − 1 / 2 ( ρ d 3 ) 1 / 2 , $$ \begin{aligned} \dot{M}=\dot{M}_{\rm DUSTY}\left(\frac{L}{10^4}\right)^{3/4}\left(\frac{\psi }{200}\right)^{-1/2}\left(\frac{\rho _d}{3}\right)^{1/2}, \end{aligned} $$(11)

where ψ is the gas-to-dust ratio (assuming ψ ≈ 1000 in the SMC; e.g., Bouchet et al. 1985; Gordon et al. 2009; Roman-Duval et al. 2014), L is the stellar luminosity as derived in Sect. 2, and ρd is the bulk grain density mentioned in the previous section. The errors of the MLRs could be calculated by repeating the SED fitting and adding and subtracting known variabilities (standard deviation) before and after the 2MASS KS band, and vice versa, which naturally mimics the effect of variability in different wavelengths (see more details in discussion). However, we would like to indicate that except for some specific wavelength bands, not all wavelengths contain variability information. Hence, the errors of MLRs may have large uncertainties. A further discussion of the error on the MLR can be found in Sect. 5. Table 2 shows the full information of the sample with photometric datasets, number of data points, χ m 2 $ \chi_{m}^2 $, chemical composition, optical depth, L, Teff, the derived MLR, and so on.

Table 2.

Final sample of 2121 RSG star candidates in the SMC.

4. Mass-loss rate and physical properties of red supergiants

Figures 10 and 11 show the histograms of derived MLRs and τ of the final RSG sample. The majority of the targets have a typical MLR of ∼10−6 M⊙ yr−1 (τ ≲ 0.1), with a few outliers at the high-MLR end (about 10−4 ∼ 10−3 M⊙ yr−1 with τ ≳ 1.0). A total MLR of 6.16 × 10−3 M⊙ yr−1 is measured for the sample. We note that this result is a lower limit, as we may still miss some RSGs in the SMC. However, we believe that our sample is relatively close to complete compared to all the previous studies. Taking ψ ≈ 1000 as indicated before, this value can be converted to a dust-production rate (DPR) of ∼6 × 10−6 M⊙ yr−1. The dust-ejection rate by AGBs and RSGs in the SMC was estimated by several authors, such as Boyer et al. (2012), Matsuura et al. (2013), and Srinivasan et al. (2016). Boyer et al. (2012) calculated a total (AGBs and RSGs) DPR of (8.6 − 9.5)×10−7 M⊙ yr−1, with RSGs contributing the least (< 4%; ∼0.04 × 10−6 M⊙ yr−1). Matsuura et al. (2013) found a higher global DPR of ∼7 × 10−6 M⊙ yr−1, for which ∼3 × 10−6 M⊙ yr−1 came from O-rich AGBs and RSGs. Srinivasan et al. (2016) also derived a similar global AGB and RSG dust-injection rate of 1.3 × 10−6 M⊙ yr−1. Given the above, there is a general agreement between our result and the above studies within a factor of a few, except the one from Boyer et al. (2012). The discrepancies could be due to differences in sample sizes, the choice of ρd, ψ, and vexp, adopted optical constants, model assumptions, and so on, for which the MLR and DPR can vary significantly. In addition, we would also like to indicate that, previous studies derived the DPR for RSGs and AGBs together, in which AGBs were dominant. That is to say, the contribution of RSGs is even smaller, while the difference between our result and those studies can be larger (e.g., by up to one order of magnitude).

thumbnail Fig. 10.

Histogram of the derived MLRs of the final RSG sample.

thumbnail Fig. 11.

Histogram of the optical depth of the final RSG sample. For clarity, targets with large τ (e.g., > 1.0) are not shown in the diagram.

Moreover, a significant part of the MLR actually comes from a few dusty examples. About 34% of the MLR, which is ∼2.11 × 10−3 M⊙ yr−1, originates from seven targets with a MLR of higher than 10−4 M⊙ yr−1. About 31% of the MLR, which is ∼1.93 × 10−3 M⊙ yr−1, originates from 55 targets with 10−5 < MLR ≤ 10−4 M⊙ yr−1. Overall, about 3% of the relatively dusty targets have contributed about 65% of the MLR. Meanwhile, as mentioned in Sect. 3.2, there are six very dusty targets in our sample, which are not well fitted with the models. Their uncertainties are likely to be around 10−4 M⊙ yr−1, and so the total MLR and the contribution from the dusty targets may be even larger than our estimation.

Figure 12 shows the HR diagram and luminosity versus median absolute deviation (MAD) of the final RSG sample, respectively. For the HR diagrams, one prominent feature can be seen from the diagrams, that is, most of the relatively dusty targets (τ > 0.1) appear at the bright end, confirming the correlation between luminosity and MLR (Yang et al. 2018, 2020). The luminosity versus MAD diagram shows the similar trend that the MLR increases with increasing luminosity and variability (we derived MAD based on the WISE/NEOWISE time-series data in [3.4] band, for which MAD is a variability index that measures the robust variability of the object; see more details in Yang et al. 2018, 2019). Moreover, as there is a positive correlation between variability and luminosity for the RSGs (Kiss et al. 2006; Yang & Jiang 2011, 2012; Soraisam et al. 2018; Yang et al. 2018; Chatys et al. 2019), the MLR can be considered as a monotonous function of the luminosity.

thumbnail Fig. 12.

Hertzsprung-Russell (left) and luminosity versus MAD (right) diagrams for the final RSG sample. Targets are color-coded with the MLR, while targets with relatively large (> 0.1) and very large (> 1.0) τ are marked by the open squares and open circles in the diagram, respectively. The dashed lines indicate the luminosity of log10(L/L⊙) = 4.6. For clarity, only the general population (without the extreme outliers) of the sample is shown in the diagram. In the luminosity versus MAD diagram, there are a few targets showing large MAD with low luminosity and MLR, which are mostly targets crossing the instability strip (because of their large variability, their is Teff variable, meaning that sometimes they are RSGs and sometimes they are yellow supergiants).

Figure 13 shows the luminosity versus MLR diagram. The sample shows a “knee-like” shape with a notable turning point around log10(L/L⊙)≈4.6, where the enhanced mass loss seems to take place. This scenario can be understood straightforwardly by the radiatively driven wind mechanism of the RSGs (see also Sect. 1), that is, the pulsation may “throw” material from the outer layers of the stars to larger radii (well past the stellar photosphere) where the dust can be condensed. The dust envelope is then expanded by the radiation force on the dust grains, which collides with the surrounding gas and drags it along to form the stellar wind. As indicated by the luminosity versus MAD diagram, evident variability can be seen for the bright RSGs (≳log10(L/L⊙)≈4.6), for which the pulsation effectively develops (along with the increased radiation pressure at higher luminosity) to enhance the mass loss and further create the “knee-like” point. However, the exact mechanism of this turning point is still unknown. The degeneracy of luminosity, pulsation, low surface gravity, convection, and other factors is inevitable and may result in the appearance of such a catastrophic effect (Yang et al. 2018, 2020).

thumbnail Fig. 13.

Luminosity versus MLR diagrams for the final RSG sample. Targets are color-coded with τ, while targets with very large τ (>1.0) are marked by the open circles in the diagram. The dashed line indicates the luminosity of log10(L/L⊙) = 4.6. We note that the low-luminosity targets (log10(L/L⊙)≲4.0) may not be true RSGs, but ∼6 − 8 M⊙ red helium-burning stars.

Figure 14 shows the color versus MLR diagrams with three typical MLR indicators of J − [8.0], KS − [12], and [3.6]−[24]. All of them show the positive correlation between the redder color and larger MLR, ranging from 1 to 2.5 in J − [8.0], 0 to 1.75 in KS − [12], and 0 to 4 in [3.6]−[24], respectively.

thumbnail Fig. 14.

J − [8.0] (left), KS − [12] (middle), and [3.6]−[24] (right) versus MLR diagrams for the final RSG sample.

The left panel of Fig. 15 shows the final luminosity versus MLR diagram, where a third-order polynomial fitting is adopted as

log 10 ( M ˙ ) = 0.45 × [ log 10 ( L / L ⊙ ) ] 3 − 5.26 × [ log 10 ( L / L ⊙ ) ] 2 + 20.93 × [ log 10 ( L / L ⊙ ) ] − 34.56 . $$ \begin{aligned} \begin{split} \log _{10}(\dot{M}) =~~&0.45\times [\log _{10}(L/L_\odot )]^3-5.26\times [\log _{10}(L/L_\odot )]^2\\&+20.93\times [\log _{10}(L/L_\odot )]-34.56. \end{split} \end{aligned} $$(12)

thumbnail Fig. 15.

Derived MLR–L relation from this work (left) and comparison of the same relation between this and previous works (right). In the left panel, the very dusty targets (τ > 1.0) are marked in red. In the right panel, lines of the same color are variations of the same relation (as shown in Fig. 16).

Overall, except for the very dusty targets (τ > 1.0), the general population of the sample is well fitted with the third-order polynomial but this may underestimate the MLR at the extended faint end (e.g., log10(L/L⊙)≈3.0). However, as the MLR will indeed be very small at such luminosity, as can be predicted from the diagram, this effect should have no significant impact on our results. Moreover, as indicated in previous sections, both luminosities and MLRs have large uncertainties at the very bright end where a significant part of the contribution comes from, and therefore the relation may be less precisely determined.

5. Discussion

5.1. The complexity of mass-loss estimation based on the SED

First of all, we would like to indicate that the derived MLRs of RSGs based on current photometric datasets have large uncertainties. The reason for this is simple and straightforward: there is complexity in the error budgets and parameter spaces in both methodology and physics.

In the methodology, for example, the adopted magnitudes are derived based on the total system throughput (e.g., filters convolve with camera, telescope, and atmosphere for ground-based telescopes), which may have small uncertainties and change with time or external conditions (e.g., J-band is sensitive to the amount of precipitable water in the atmosphere5). In that sense, it may result in a few percent difference in flux (Philip Massey, priv. comm.). The difference between maximum and minimum flux for each individual target spans around 2-3 orders of magnitude (e.g., see Fig. 8), for which the weights are largely different (the typical solution for this problem is to fit the SED in logarithmic space, while we adopt a flat spectrum to solve it as described in Sect. 3.2). The photometric data used here are all derived from broad-band filters, meaning that some prominent spectral features may be smoothed out. For example, the 9.7 μm silicate feature, which covers about 8–13 μm, is not easily distinguished, except for a few targets (∼4.4%) detected by AKARI [S11] (λeff ≈ 10.2 μm with a range of ∼8.3–15.3 μm; see also Fig. 8), as it is right between the Spitzer [8.0] (λeff ≈ 7.6 μm with a range of ∼6.3–9.6 μm) and WISE [12] (λeff ≈ 11.6 μm with a range of ∼7.4–17.3 μm) bands (we note that the quantum efficiency drops fast at the edges of each filter). The photometric error is inhomogeneous across the SED, as it is usually about a few thousandths of a magnitude in the optical bands, but increases to ∼0.03–0.05 mag when wavelengths are longer than 10 μm (an important reason that we have not used reduced χ2, because it takes error as the weight and therefore the most important wavelength range of mid-IR will have relatively low weight; further discussion of the advantages and disadvantages of reduced χ2 can be found in Andrae et al. 2010). Similarly, the angular resolutions also show large inconsistencies between different wavelengths, ranging from subarcseconds to a dozen arcsec (e.g., the typical angular resolutions are about 4.5″, 2″, 0.9″, 0.9″, 5″, 6″, 2″, 12″, and 6″for GALEX, Gaia, NSC, VMC, 2MASS, WISE [3.6], Spitzer [8.0], WISE [22], and Spitzer [24], respectively). Hence, neighboring targets and environment may contaminate the measurement in low-resolution data, which is quite common in nearby and distant galaxies. Moreover, for wavelengths with available time-series information, the sampling of light curves is often sparse without the full coverage of the period (the typical periods of bright RSGs are about 300–800 d; Yang & Jiang 2011, 2012; Chatys et al. 2019; Ren et al. 2019). In our case, only the WISE [3.4] and [4.6] data have sufficiently long coverage to derive relatively accurate median magnitudes. Thus, the single-epoch/incomplete time-series measurements in some wavelengths may not represent the appropriate luminosities. Meanwhile, different datasets are taken at different times, ranging from the late 1990s to the present. Moreover, problems also come from the poor data quality, as mentioned in Sect. 2; that is, inaccurate measurements will influence the derived MLR (especially for the mid-IR data that dominate the MLR modelling). We tried our best to filter out the low-quality data but some cases may still be present. Different fitting methods or normalized wavelengths may also result in about 10%–30% differences, while the majority of them appear in the low optical depth region (τ < 0.1). Differences in the sample sizes (incomplete or biased samples) will also lead to different results. Finally, for the most important factor, the degeneracy of different models is inevitable and complex due to many parameters in the modelling. For example, the typically used, consecutive, equally spaced logarithmic steps of optical depth (e.g., 0.001 to 10.0 in 99 intervals) are too dense at the low optical depth (half of the steps will be less than 0.1), which means that adjacent models cannot be properly distinguished using only photometric data; this is also the reason why we set up our own steps for τ. Moreover, one may always find a best model with low χ2 for the target, as long as the numbers of parameters and generated models are sufficiently large; therefore, the solution may fall in a local minimum instead of the global minimum. In such cases, the choice of the correct solution cannot be justified, unless each parameter (or the dominant parameter) is precisely constrained (to a few percent; this is indeed very difficult for almost every case of SED fitting, not just for our study of MLRs).

In physics, for example, the emerging theoretical spectra are slightly different if we use different central illuminating sources, like blackbody or stellar atmosphere models (e.g., PHONEIX, ATLAS, or MARCS). The variability of RSGs may also cause some uncertainties. RSGs are semi-regular variables (considering also the contribution of convection). Typically, as the wavelength increases, the amplitude will gradually decrease, the light curve will progress towards a more symmetrical variation, and the phase of the maximum will shift toward later phases. In that sense, the phase and time lags in variability will change the shape of the SED accordingly (along with the pulsation, the convection may also play an important role in the mass loss of RSGs as they may help the release of materials from the outer layer of the star). In addition, it is also possible to have a small amount of contamination from the AGB population at the red and/or faint end of our sample, as mentioned in Sect. 2. The extinction is another issue, as we have only corrected the average foreground extinction towards the SMC. However, as the 3D structure and internal extinction of the SMC and the distances of the targets are not accurately determined, the internal extinction of the SMC may vary significantly from star to star, especially for RSGs that are supposedly close to the star formation regions. Moreover, the physical properties and driving mechanisms of the mass loss of RSGs are still poorly understood, not to mention the dependence of MLR on many factors. For example, for the chemical composition, one may use silicate from Ossenkopf et al. (1992) instead of Draine & Lee (1984). Additionally, there is evidence for polycyclic aromatic hydrocarbons (PAHs) or SiC existing in RSGs. This occurrence of carbon species in an O-rich environment is possible because of the dissociation of CO by the strong UV emission from the chromosphere (Beck et al. 1992; Höfner & Andersen 2007). The produced free carbon atoms may form carbon dust along with the expected O-rich silicate dust. Meanwhile, the UV photons from chromospheric emission may also excite the IR emission from PAHs or similar carbonaceous species (Allamandola et al. 1989). However, on the other hand, the diffuse ISM is also known to exhibit PAH emission features, which can be confused with the observed target. Therefore, the exact scenario of carbonaceous dust formation in the RSGs is still unclear (and a strict definition of “C-rich” RSG is still missing, because in almost all cases, the 9.7 μm silicate bump and PAH emission coexist) and we exclude carbon dust from our modeling. For the dust density and size distribution, one may use a steady-state density distribution of r−q with a power-law index of q ≤ 2 instead of a radiatively driven wind, or choose an average grain size instead of a size distribution. For ψ, traditionally an average value of 100–200 is assumed for the Milky Way, 400–500 for the LMC, and ∼1000 for the SMC. However, ψ may vary significantly between each individual source. Also, the wind-expansion velocity (vexp) highly depends on the luminosity of the central source, metallicity, and other factors (Goldman et al. 2017), and ρd may change upon the chemical composition and structure of the dust grain. For the metallicity, many studies often include both Milky Way and MC RSGs, spanning a range of metallicities. There is also an inherent uncertainty of ∼30% for the DUSTY radiatively driven wind model (Ivezic et al. 1999). Finally, some external factors, such as binarity, may also play an important role. A further discussion about modeling parameters can be found in Ivezic & Elitzur (1997).

One may argue that each of the mentioned issues create just a small error (e.g., a few percent). However, the errors are accumulated and propagated, meaning that the final error of MLR becomes large and complicated (potentially up to one order of magnitude or more; see also below). Therefore, the MLRs derived in this work (or any other works based on photometric data) can only be considered as a mixture of “snap shot” with large uncertainties. This effect is especially true for the bright RSGs with larger variabilities and MLRs, but is significantly mitigated for the faint RSGs with smaller variabilities and MLRs. While it is difficult to assess the MLR with a few percentage accuracy, our results may help to understand the underlying relation between MLR and other physical parameters and to put relatively stringent constraints on it. More details about the analysis will be presented in a future paper (Wen et al., in prep.).

5.2. Comparison with previous mass-loss rate prescriptions

We also compared our work with previous studies. Historically, many empirical relations have been used in the stellar evolutionary codes in order to estimate the MLR as a function of basic stellar parameters, such as L, Teff, mass (M), and radius (R). However, in the majority of cases, the dominant factor is the L (e.g., the Teff range of RSGs is relatively narrow, while R and M are L-dependent as shown below; Mauron & Josselin 2011), which is also relatively easily measured. Therefore, following previous studies, we simplified the MLR prescriptions.

The first widely used relation was the Reimers law (hereafter R75; Reimers 1975; Kudritzki & Reimers 1978):

M ˙ = 5.5 × 10 − 13 L R / M , $$ \begin{aligned} \dot{M} = 5.5\times 10^{-13}LR/M, \end{aligned} $$(13)

which was based on a small sample of stars including both red giants and RSGs (L, R, and M in solar units here and below). For RSGs, this relation can be simplified for an average temperature of Teff = 3750 K (same below when taking Teff into account) to

M ˙ = 3.26 × 10 − 12 ( L ) 1.17 , $$ \begin{aligned} \dot{M}=3.26\times 10^{-12}(L)^{1.17}, \end{aligned} $$(14)

as R/R⊙ = (L/L⊙)0.5 × (Teff/5770)−2 and L/L⊙ = f(M/M⊙)3 with f ≈ 15.5 (same below for the simplification; Mauron & Josselin 2011; Kee et al. 2021). We note that different studies might use different definitions of the M term (as well as the corresponding L term); for example, one might use initial mass, current mass, most expected mass, final mass, and so on. Using the same M–L relation for all of the MLR prescriptions may be slightly misleading, but sufficient for the simplification, considering the large uncertainties mentioned in the previous section.

About a decade later, de Jager et al. (1988) developed the famous MLR prescription for stars located over the whole HR diagram, including 15 Galactic RSGs, which was presented as a sum of Chebyshev polynomials (hereafter dJ88). Commonly used in stellar evolution codes, it is expressed as the first-order approximation,

M ˙ = 10 − 8.158 ( L ) 1.769 ( T eff ) − 1.676 . $$ \begin{aligned} \dot{M}=10^{-8.158}(L)^{1.769}(T_{\rm eff})^{-1.676}. \end{aligned} $$(15)

Meanwhile, a second formula was published later that also takes into account the stellar mass (hereafter NJ90; Nieuwenhuijzen & de Jager 1990), as

M ˙ = 10 − 14.02 ( L ) 1.24 ( M ) 0.16 ( R ) 0.81 . $$ \begin{aligned} \dot{M}=10^{-14.02}(L)^{1.24}(M)^{0.16}(R)^{0.81}. \end{aligned} $$(16)

Later, Feast (1992) found that there was a relatively close relationship between pulsation period (P), L, and MLR for 15 RSGs in the LMC based on the data from Reid et al. (1990), that is,

log 10 ( M ˙ ) = 1.32 × log 10 P − 8.17 , $$ \begin{aligned} \log _{10}(\dot{M})=1.32 \times \log _{10}P-8.17, \end{aligned} $$(17)

and

M bol = − 2.38 × log 10 P − 1.46 . $$ \begin{aligned} M_{\rm bol}=-2.38\times \log _{10}P-1.46. \end{aligned} $$(18)

In that sense, the relation between MLR and L (Feast law; hereafter F92) can be written as

M ˙ = 10 − 11.609 ( L ) 1.387 . $$ \begin{aligned} \dot{M}=10^{-11.609}(L)^{1.387}. \end{aligned} $$(19)

Meanwhile, a MLR prescription from Vanbeveren et al. (1998) was also based on the work of Reid et al. (1990) as (hereafter V98),

M ˙ = 10 − 8.7 ( L ) 0.8 . $$ \begin{aligned} \dot{M}=10^{-8.7}(L)^{0.8}. \end{aligned} $$(20)

Another relation was proposed by Salasnich et al. (1999), who adopted the Feast law but also took into account the possibility that the ψ varied with stellar luminosity. The Salasnich relation (hereafter S99) is

M ˙ = 10 − 14.5 ( L ) 2.1 . $$ \begin{aligned} \dot{M}=10^{-14.5}(L)^{2.1}. \end{aligned} $$(21)

Subsequently, van Loon et al. (2005) derived a well-known mass-loss law for oxygen-rich dust-enshrouded AGBs and RSGs in the LMC (hereafter vL05). The van Loon law is written as

M ˙ = 10 − 5.56 ( L 10 4 ) 1.05 ( T eff 3500 ) − 6.3 . $$ \begin{aligned} \dot{M}=10^{-5.56}\left(\frac{L}{10^4}\right)^{1.05}\left(\frac{T_{\rm eff}}{3500}\right)^{-6.3}. \end{aligned} $$(22)

More recently, Goldman et al. (2017; hereafter G17) and Beasor et al. (2020; hereafter B20) developed new MLR prescriptions for RSGs based on samples of AGBs and RSGs with OH masers in the LMC and the Milky Way, and RSGs in the clusters, respectively. The G17 relation is written as

M ˙ = 1.06 × 10 − 5 ( L 10 4 ) 0.9 ( P 500 d ) 0.75 ( ψ 200 ) − 0.03 , $$ \begin{aligned} \dot{M}=1.06\times 10^{-5}\left(\frac{L}{10^4}\right)^{0.9}\left(\frac{P}{500d}\right)^{0.75}\left(\frac{\psi }{200}\right)^{-0.03}, \end{aligned} $$(23)

where P is the pulsation period. The B20 relation is written as

M ˙ = 10 − 26.4 − 0.23 M ini ( L ) 4.8 , $$ \begin{aligned} \dot{M}=10^{-26.4-0.23M_{\rm ini}}(L)^{4.8}, \end{aligned} $$(24)

where Mini is the initial mass.

Wang et al. (2021) also developed a MLR prescription of RSGs in relatively metal-rich galaxies of M31 and M33 using DUSTY (hereafter W21). The prescription is applied for O-rich and C-rich RSGs, respectively, as

M ˙ = 10 − 8.31 ( L ) 0.83 ( M 31 , s i l i c a t e ) , M ˙ = 10 − 10.03 ( L ) 0.91 ( M 31 , c a r b o n ) , M ˙ = 10 − 8.64 ( L ) 0.92 ( M 33 , s i l i c a t e ) , M ˙ = 10 − 11.13 ( L ) 1.14 ( M 33 , c a r b o n ) . $$ \begin{aligned} \begin{aligned} \dot{M}=10^{-8.31}(L)^{0.83} \ (\mathrm M31, silicate), \\ \dot{M}=10^{-10.03}(L)^{0.91} \ (\mathrm M31, carbon), \\ \dot{M}=10^{-8.64}(L)^{0.92} \ (\mathrm M33, silicate), \\ \dot{M}=10^{-11.13}(L)^{1.14} \ (\mathrm M33, carbon). \end{aligned} \end{aligned} $$(25)

The right panel of Fig. 15 shows a comparison between our work and previous studies. The majority of the MLR prescriptions can be seen to have a similar trend but the scatter is large (within two orders of magnitude), except for a few cases. For clarity, we also show the comparison between our work and each previous individual study in Fig. 16. In general, the errors on MLR prescriptions are estimated based on parameters such as Teff, M, R, or P, and so on. However, in almost all cases, the errors are underestimated and inhomogeneous, which makes the comparison between different prescriptions difficult. We therefore adopted a 0.5 dex error for all prescriptions (including ours) as a reference shown in the diagrams. Moreover, as discussed in the previous section, a metallicity dependence and sample size may also contribute to the large scatter, as the comparison includes studies from both Milky Way and MC RSGs, which span a range of metallicities and sample sizes.

thumbnail Fig. 16.

Comparison of derived MLR–L relation between this and each previous individual study. A 0.5 dex error (the shaded region) is shown as a reference for all prescriptions (including ours).

As can be seen from the diagrams, the ones most resembling our relation are vL05 and F92, which almost overlap with our sample and relation over the whole luminosity range (3.5 ≲ log10(L/L⊙)≲5.3) with a slight overestimation around log10(L/L⊙)≈4.6. Meanwhile, at higher luminosity (log10(L/L⊙) > 5.3), our prescription seems more than an order of magnitude different than vL05 and F92. V98 and S99 are also very similar to our relation, for which V98 is more flattened with a slight overestimation at the medium and faint end of luminosity, while S99 is steeper with overestimation at the medium luminosity and underestimation at the faint end of luminosity. The classic R75, dJ88, and NJ90 are similar to each other within a few factors, but all underestimate at both the faint and bright ends, though a greater impact is found in the bright end with large MLRs. G17 and W21 (silicate) significantly overestimate rates at the medium and faint end of luminosity. However, G17 used OH/IR stars from the LMC, Galactic center, and Galactic bulge to derive MLRs of O-rich AGBs and RSGs, for which the OH/IR stars were targets with large MLRs and thick cirumstellar envelopes. Meanwhile, W21 derived the MLRs of RSGs in M31 and M33, where much higher metallicities might play an important role. Therefore, the overestimations of G17 and W21 are relatively reasonable. Finally, B20 is in good agreement with our relation at the very bright end (log10(L/L⊙)≳5.2 with initial mass of M = 15 M⊙; we indicate three typical initial masses of M = 8, 15, 25 M⊙ in the diagram) but largely underestimates the MLR at the medium and faint end of luminosity, which may be due to sparse sampling of initial masses of RSGs.

Overall, our prescription captures the change of slope in the MLR that other prescriptions cannot, because they are all based on a more limited sample. Therefore, our result may provide a better solution at the cool and luminous region (e.g., about 3500 ≤ Teff ≤ 5500 K and 3.5 ≤ log10(L/L⊙)≤5.5) of the HR diagram at low metallicity (the SMC level) compared to previous works, given that it is based on a much larger sample. In the meantime, we believe that our prescription could be extrapolated to a wider range (possibly covering a large part of the RHeBS); for example, ∼3000 ≤ Teff ≤ 6000 K and ∼3.0 ≤ log10(L/L⊙)≤6.0 (but above log10(L/L⊙) = 5.5 the extrapolation is more uncertain). We also ran some preliminary tests of our new prescriptions in MESA stellar evolution models (Paxton et al. 2011, 2013, 2015, 2018). The high MLR of massive luminous RSGs leaves them with only a thin envelope at the end of their life. This could in principle be a (partial) solution to the “RSG problem” (Smartt 2009), although further investigation is needed. Meanwhile, the widely used vL05 law and F92 law are the two most similar prescriptions to ours. Still, we would like to emphasize again that there are many uncertainties in the estimation of MLR for RSGs.

5.3. Other interesting factors

Additionally, we derived the relation between the KS band and bolometric magnitudes as

m bol = 0.90 × K S + 3.43 , $$ \begin{aligned} m_{\rm bol}=0.90\times K_S+3.43, \end{aligned} $$(26)

shown in Fig. 17. Compared to previous studies, our relation is flatter at the faint end, but very similar to Davies et al. (2013) at the bright end. Meanwhile, both ours and that of Davies et al. (2013) are different from Josselin et al. (2000). All the differences are most likely due to the sample sizes and/or metallicities, for which our work indicates an improved relation based on a much larger sample.

thumbnail Fig. 17.

KS versus bolometric magnitudes diagram. Our work indicates an improved relation based on a larger sample compared to previous studies.

Furthermore, we detected UV flux in nine targets from the final sample (see examples in Fig. 18), which could be an indicator of OB-type companions of binary RSGs as mentioned in Neugent et al. (2018, 2019, 2020), Neugent (2021), and Patrick et al. (2022). The large difference of possible binary fraction (e.g., < 1% for our case and > 15% in other studies) is most likely due to the low resolution and sensitivity of GALEX. Moreover, even without the UV detection (again, because of the limitation of GALEX data), some targets (∼7% in the final sample) also show abnormal optical excess or depression in the blue end similar to the UV-detected targets (see examples in Fig. 19), which may be the indication of the existence of a binary companion.

thumbnail Fig. 18.

Examples of the targets with UV detection, which likely indicate a hot binary companion.

thumbnail Fig. 19.

Examples of the targets with unusual optical excess or depression, which may indicate a binary companion.

6. Summary

In order to investigate the MLR of RSGs at relatively low metallicity, we assembled an initial RSG sample of 2303 targets taken from Yang et al. (2020) and Ren et al. (2021) in the SMC with 53 different bands of photometric data. The sample was further constrained to a final sample of 2121 targets by removing the targets in the overlapping region between AGBs and RSGs on the 2MASS CMD.

We used the radiatively driven wind model of the 1D radiative transfer code DUSTY to create a grid covering a wide range of dust parameters in order to match the observed SEDs from our sample. In all, 17 820 O-rich theoretical models are generated by DUSTY, such that “normal” and “dusty” grids are half-and-half. We first select the best model for each individual target by calculating the minimum χ m 2 $ \chi_{m}^2 $ from the normal grids. The dusty targets (six targets) were then fitted again with the dusty grids. The resulting MLRs from DUSTY were then converted to real MLRs based on the scaling relation.

For our final RSG sample (assuming our sample is relatively close to complete), a total MLR of 6.16 × 10−3 M⊙ yr−1 is measured, while a typical MLR of ∼10−6 M⊙ yr−1 is witnessed for the majority of the RSGs. Our result of the total MLR can be converted into a DPR of ∼6 × 10−6 M⊙ yr−1, which is in general agreement with previous studies by a few factors, while discrepancies could be caused by many factors, such as sample sizes, adopted optical constants, model assumptions, and so on. Meanwhile, our findings also indicate that a few dusty targets contributed a significant part of the MLR; for example, about 3% of the dusty targets has contributed about 65% of the MLR. The HR and luminosity versus MAD diagrams of the final sample indicate a positive relation between luminosity and MLR. Meanwhile, the luminosity versus MLR diagrams show a “knee-like” shape with enhanced mass loss taking place around log10(L/L⊙)≈4.6, which may be due to the degeneracy of luminosity, pulsation, low surface gravity, convection, and other factors.

The complexity of mass-loss estimation based on the SED is discussed from almost all aspects covering both methodology and physics. This complexity indicates that MLRs derived based on photometric data can be only considered as a mixture of “snap shot” with large uncertainties.

We derive our MLR relation using a third-order polynomial fit of the final sample. We also compare our results with many previous empirical MLR prescriptions, finding that most prescriptions show a similar trend within two orders of magnitude. Specifically, van Loon et al. (2005) and Feast (1992) are the two prescriptions that most resemble ours. The classic Reimers (1975) and de Jager et al. (1988; Nieuwenhuijzen & de Jager 1990) relations lead to underestimates of the MLR at both faint and bright ends. Beasor et al. (2020) is in good agreement with our relation at the very bright end but largely underestimates the MLR at the medium and faint end of luminosity. In general, based on a much larger sample, our MLR prescription captures the change of the slope of the MLR, which the other studies cannot, and provides a more accurate relation at the cool and luminous region of the HR diagram at low metallicity compared to previous works.

Finally, a small fraction of our sample is detected in the UV band, which could be an indication of OB-type companions of binary RSGs. Similarly, some targets without the UV detection also show abnormal optical excess or depression, which may also indicate the existence of a binary companion.

3

There is a bug in memory management in DUSTY V4 that prevents more than 100 steps of τ; Zeljko Ivezic, priv. comm.

Acknowledgments

We would like to thank the anonymous referee for many constructive comments and suggestions. This study has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 772086). B.W.J., M.Y. and S.W. gratefully acknowledge support from the National Natural Science Foundation of China (Grant No.12133002 and 12003046). H.T. is supported by Beijing Natural Science Foundation No. 1214028. J.M.L. is supported by the PhD research startup foundation of Hebei Normal University (Grant No. L2022B07). We acknowledge the science research grants from the China Manned Space Project with No.CMS-CSST-2021-A08. This work is also supported by National Key R&D Program of China No. 2019YFA0405501. We thank Philip Massey and Alistair Walker for providing valuable comments on the total system throughput, Zeljko Ivezic for finding the DUSTY V4 bug, Nicholas Cross for identifying the VMC pipeline bug, Marta Sewilo for the discussion of Spitzer data quality flag, Jing Tang and Tianmeng Zhang for helpful discussion on the evolution of massive stars and supernova. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology. It is funded by the National Aeronautics and Space Administration. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of the Spanish Virtual Observatory (https://svo.cab.inta-csic.es) project funded by MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00 (Rodrigo et al. 2012; Rodrigo & Solano 2020). This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI : 10.26093/cds/vizier). The original description of the VizieR service was published in Ochsenbein et al. (2000). This research has made use of the Tool for OPerations on Catalogues And Tables (TOPCAT; Taylor 2005).

References

  1. Adams, S. M., Kochanek, C. S., Gerke, J. R., et al. 2017, MNRAS, 468, 4968 [Google Scholar]
  2. Allamandola, L. J., Tielens, A. G. G. M., & Barker, J. R. 1989, ApJS, 71, 733 [Google Scholar]
  3. Andrae, R., Schulze-Hartung, T., & Melchior, P. 2010, ArXiv e-prints [arXiv:1012.3754] [Google Scholar]
  4. Beasor, E. R., & Davies, B. 2016, MNRAS, 463, 1269 [NASA ADS] [CrossRef] [Google Scholar]
  5. Beasor, E. R., & Davies, B. 2018, MNRAS, 475, 55 [NASA ADS] [CrossRef] [Google Scholar]
  6. Beasor, E. R., Davies, B., Smith, N., et al. 2020, MNRAS, 492, 5994 [Google Scholar]
  7. Beck, H. K. B., Gail, H.-P., Henkel, R., et al. 1992, A&A, 265, 626 [NASA ADS] [Google Scholar]
  8. Bessell, M., Bloxham, G., Schmidt, B., et al. 2011, PASP, 123, 789 [Google Scholar]
  9. Bianchi, L., Shiao, B., & Thilker, D. 2017, ApJS, 230, 24 [Google Scholar]
  10. Bouchet, P., Lequeux, J., Maurice, E., et al. 1985, A&A, 149, 330 [NASA ADS] [Google Scholar]
  11. Bouwman, J., de Koter, A., van den Ancker, M. E., et al. 2000, A&A, 360, 213 [NASA ADS] [Google Scholar]
  12. Bouwman, J., Meeus, G., de Koter, A., et al. 2001, A&A, 375, 950 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  13. Boyer, M. L., McDonald, I., van Loon, J. T., et al. 2009, ApJ, 705, 746 [CrossRef] [Google Scholar]
  14. Boyer, M. L., Srinivasan, S., van Loon, J. T., et al. 2011, AJ, 142, 103 [Google Scholar]
  15. Boyer, M. L., Srinivasan, S., Riebel, D., et al. 2012, ApJ, 748, 40 [Google Scholar]
  16. Brunner, M., Maercker, M., Mecina, M., et al. 2018, A&A, 614, A17 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  17. Cannon, E., Montargès, M., de Koter, A., et al. 2021, MNRAS, 502, 369 [NASA ADS] [CrossRef] [Google Scholar]
  18. Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245 [Google Scholar]
  19. Cioni, M.-R. L., Clementini, G., Girardi, L., et al. 2011, A&A, 527, A116 [CrossRef] [EDP Sciences] [Google Scholar]
  20. Chatys, F. W., Bedding, T. R., Murphy, S. J., et al. 2019, MNRAS, 487, 4832 [CrossRef] [Google Scholar]
  21. Dalcanton, J. J., Williams, B. F., Melbourne, J. L., et al. 2012, ApJS, 198, 6 [Google Scholar]
  22. Davies, B., & Beasor, E. R. 2018, MNRAS, 474, 2116 [NASA ADS] [CrossRef] [Google Scholar]
  23. Davies, B., Kudritzki, R.-P., Plez, B., et al. 2013, ApJ, 767, 3 [Google Scholar]
  24. Davies, B., Kudritzki, R.-P., Lardo, C., et al. 2017, ApJ, 847, 112 [Google Scholar]
  25. de Jager, C., Nieuwenhuijzen, H., & van der Hucht, K. A. 1988, A&AS, 72, 259 [NASA ADS] [Google Scholar]
  26. Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89 [NASA ADS] [CrossRef] [Google Scholar]
  27. Dye, S., Lawrence, A., Read, M. A., et al. 2018, MNRAS, 473, 5113 [Google Scholar]
  28. Ekström, S., Georgy, C., Eggenberger, P., et al. 2012, A&A, 537, A146 [Google Scholar]
  29. Ekström, S., Georgy, C., Meynet, G., Groh, J., & Granada, A. 2013, EAS Publ. Ser., 60, 31 [Google Scholar]
  30. Elitzur, M., & Ivezić, Ž. 2001, MNRAS, 327, 403 [NASA ADS] [CrossRef] [Google Scholar]
  31. Epchtein, N., de Batz, B., Capoani, L., et al. 1997, The Messenger, 87, 27 [NASA ADS] [Google Scholar]
  32. Feast, M. W. 1992, Instabilities in Evolved Super- and Hypergiants, 18 [Google Scholar]
  33. Freedman, W. L., Madore, B. F., Hoyt, T., et al. 2020, ApJ, 891, 57 [Google Scholar]
  34. Gaia Collaboration (Prusti, T., et al.) 2016, A&A, 595, A1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  35. Gaia Collaboration (Brown, A. G. A., et al.) 2018, A&A, 616, A1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  36. Gaia Collaboration (Brown, A. G. A., et al.) 2021, A&A, 649, A1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  37. Gail, H.-P., & Sedlmayr, E. 1999, A&A, 347, 594 [Google Scholar]
  38. Gail, H.-P., Keller, R., & Sedlmayr, E. 1984, A&A, 133, 320 [NASA ADS] [Google Scholar]
  39. Gail, H.-P., Tamanai, A., Pucci, A., et al. 2020, A&A, 644, A139 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  40. Gilkis, A., Shenar, T., Ramachandran, V., et al. 2021, MNRAS, 503, 1884 [NASA ADS] [CrossRef] [Google Scholar]
  41. Goldman, S. R., van Loon, J. T., Zijlstra, A. A., et al. 2017, MNRAS, 465, 403 [Google Scholar]
  42. Gordon, K. D., Bot, C., Muller, E., et al. 2009, ApJ, 690, L76 [NASA ADS] [CrossRef] [Google Scholar]
  43. Gordon, K. D., Meixner, M., Meade, M. R., et al. 2011, AJ, 142, 102 [NASA ADS] [CrossRef] [Google Scholar]
  44. Groenewegen, M. A. T. 2012, A&A, 543, A36 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  45. Groenewegen, M. A. T., & Sloan, G. C. 2018, A&A, 609, A114 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  46. Groenewegen, M. A. T., de Jong, T., & Gaballe, T. R. 1994, A&A, 287, 163 [NASA ADS] [Google Scholar]
  47. Groenewegen, M. A. T., Sloan, G. C., Soszyński, I., & Petersen, E. A. 2009, A&A, 506, 1277 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  48. Gustafsson, B., Edvardsson, B., Eriksson, K., et al. 2008, A&A, 486, 951 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  49. Hanner, M. 1988, In NASA Washington, Infrared Observations of Comets Halley and Wilson and Properties of the Grains, 22 [Google Scholar]
  50. Harper, G. M., Brown, A., & Lim, J. 2001, ApJ, 551, 1073 [NASA ADS] [CrossRef] [Google Scholar]
  51. Höfner, S. 2008, A&A, 491, L1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  52. Höfner, S., & Andersen, A. C. 2007, A&A, 465, L39 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  53. Höfner, S., & Olofsson, H. 2018, A&ARv, 26, 1 [Google Scholar]
  54. Humphreys, R. M. 2010, Hot and Cool: Bridging Gaps in Massive Star Evolution, 425, 247 [NASA ADS] [Google Scholar]
  55. Humphreys, R. M., & Davidson, K. 1979, ApJ, 232, 409 [Google Scholar]
  56. Humphreys, R. M., Helmel, G., Jones, T. J., et al. 2020, AJ, 160, 145 [NASA ADS] [CrossRef] [Google Scholar]
  57. Ita, Y., Onaka, T., Tanabé, T., et al. 2010, PASJ, 62, 273 [NASA ADS] [Google Scholar]
  58. Ivezic, Z., & Elitzur, M. 1995, ApJ, 445, 415 [NASA ADS] [CrossRef] [Google Scholar]
  59. Ivezic, Z., & Elitzur, M. 1997, MNRAS, 287, 799 [Google Scholar]
  60. Ivezic, Z., Nenkova, M., & Elitzur, M. 1999, User Manual for DUSTY [Google Scholar]
  61. Javadi, A., van Loon, J. T., Khosroshahi, H., et al. 2013, MNRAS, 432, 2824 [NASA ADS] [CrossRef] [Google Scholar]
  62. Jones, O. C., Woods, P. M., Kemper, F., et al. 2017, MNRAS, 470, 3250 [NASA ADS] [CrossRef] [Google Scholar]
  63. Josselin, E., Blommaert, J. A. D. L., Groenewegen, M. A. T., Omont, A., & Li, F. L. 2000, A&A, 357, 225 [NASA ADS] [Google Scholar]
  64. Kato, D., Nagashima, C., Nagayama, T., et al. 2007, PASJ, 59, 615 [NASA ADS] [Google Scholar]
  65. Kee, N. D., Sundqvist, J. O., Decin, L., et al. 2021, A&A, 646, A180 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  66. Keller, S. C., Schmidt, B. P., Bessell, M. S., et al. 2007, PASA, 24, 1 [NASA ADS] [CrossRef] [Google Scholar]
  67. Kim, S.-H., Martin, P. G., & Hendry, P. D. 1994, ApJ, 422, 164 [Google Scholar]
  68. Kiss, L. L., Szabó, G. M., & Bedding, T. R. 2006, MNRAS, 372, 1721 [Google Scholar]
  69. Kochanek, C. S., Beacom, J. F., Kistler, M. D., et al. 2008, ApJ, 684, 1336 [NASA ADS] [CrossRef] [Google Scholar]
  70. Kudritzki, R. P., & Reimers, D. 1978, A&A, 70, 227 [Google Scholar]
  71. Levesque, E. M. 2010, New A Rv., 54, 1 [NASA ADS] [CrossRef] [Google Scholar]
  72. Levesque, E. M. 2017, in Astrophysics of Red Supergiants (Bristol, UK: IOP Publishing) [Google Scholar]
  73. Levesque, E. M., Massey, P., Olsen, K. A. G., et al. 2005, ApJ, 628, 973 [Google Scholar]
  74. Liu, J., Jiang, B. W., Li, A., et al. 2017, MNRAS, 466, 1963 [CrossRef] [Google Scholar]
  75. MacGregor, K. B., & Stencel, R. E. 1992, ApJ, 397, 644 [NASA ADS] [CrossRef] [Google Scholar]
  76. Massey, P. 1998, ApJ, 501, 153 [Google Scholar]
  77. Massey, P. 2002, ApJS, 141, 81 [NASA ADS] [CrossRef] [Google Scholar]
  78. Massey, P. 2013, New A Rev., 57, 14 [NASA ADS] [CrossRef] [Google Scholar]
  79. Massey, P., & Olsen, K. A. G. 2003, AJ, 126, 2867 [NASA ADS] [CrossRef] [Google Scholar]
  80. Massey, P., Plez, B., Levesque, E. M., et al. 2005, ApJ, 634, 1286 [NASA ADS] [CrossRef] [Google Scholar]
  81. Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, ApJ, 217, 425 [Google Scholar]
  82. Matsuura, M., Woods, P. M., & Owen, P. J. 2013, MNRAS, 429, 2527 [NASA ADS] [CrossRef] [Google Scholar]
  83. Mauron, N., & Josselin, E. 2011, A&A, 526, A156 [CrossRef] [EDP Sciences] [Google Scholar]
  84. McDonald, I., Boyer, M. L., van Loon, J. T., et al. 2011, ApJ, 730, 71 [CrossRef] [Google Scholar]
  85. Montargès, M., Cannon, E., Lagadec, E., et al. 2021, Nature, 594, 365 [Google Scholar]
  86. Morrissey, P., Conrow, T., Barlow, T. A., et al. 2007, ApJS, 173, 682 [Google Scholar]
  87. Melbourne, J., Williams, B. F., Dalcanton, J. J., et al. 2012, ApJ, 748, 47 [NASA ADS] [CrossRef] [Google Scholar]
  88. Meynet, G., Chomienne, V., Ekström, S., et al. 2015, A&A, 575, A60 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  89. Min, M., Dullemond, C. P., Dominik, C., et al. 2009, A&A, 497, 155 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  90. Murakami, H., Baba, H., Barthel, P., et al. 2007, PASJ, 59, S369 [CrossRef] [Google Scholar]
  91. Nanni, A., Bressan, A., Marigo, P., et al. 2013, MNRAS, 434, 2390 [NASA ADS] [CrossRef] [Google Scholar]
  92. Neugent, K. F. 2021, ApJ, 908, 87 [NASA ADS] [CrossRef] [Google Scholar]
  93. Neugent, K. F., Levesque, E. M., & Massey, P. 2018, AJ, 156, 225 [CrossRef] [Google Scholar]
  94. Neugent, K. F., Levesque, E. M., Massey, P., et al. 2019, ApJ, 875, 124 [NASA ADS] [CrossRef] [Google Scholar]
  95. Neugent, K. F., Levesque, E. M., Massey, P., et al. 2020, ApJ, 900, 118 [CrossRef] [Google Scholar]
  96. Nidever, D. L., Dey, A., Olsen, K., et al. 2018, AJ, 156, 131 [NASA ADS] [CrossRef] [Google Scholar]
  97. Nidever, D. L., Dey, A., Fasbender, K., et al. 2021, AJ, 161, 192 [Google Scholar]
  98. Nieuwenhuijzen, H., & de Jager, C. 1990, A&A, 231, 134 [NASA ADS] [Google Scholar]
  99. Ochsenbein, F., Bauer, P., & Marcout, J. 2000, A&AS, 143, 23 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  100. Ohnaka, K., Driebe, T., Hofmann, K.-H., et al. 2008, A&A, 484, 371 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  101. Onaka, T., Matsuhara, H., Wada, T., et al. 2007, PASJ, 59, S401 [Google Scholar]
  102. Onken, C. A., Wolf, C., Bessell, M. S., et al. 2019, PASA, 36, e033 [Google Scholar]
  103. Ossenkopf, V., Henning, T., & Mathis, J. S. 1992, A&A, 261, 567 [NASA ADS] [Google Scholar]
  104. Patrick, L. R., Thilker, D., Lennon, D. J., et al. 2022, MNRAS, 513, 5847 [Google Scholar]
  105. Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3 [Google Scholar]
  106. Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4 [Google Scholar]
  107. Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15 [Google Scholar]
  108. Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34 [NASA ADS] [CrossRef] [Google Scholar]
  109. Pearson, K. 1900, Philos. Mag. Ser., 50, 11 [Google Scholar]
  110. Reid, N., Tinney, C., & Mould, J. 1990, ApJ, 348, 98 [NASA ADS] [CrossRef] [Google Scholar]
  111. Reimers, D. 1975, Mem. Soc. R. Scie. Liege, 8, 369 [NASA ADS] [Google Scholar]
  112. Ren, Y., Jiang, B.-W., Yang, M., & Gao, J. 2019, ApJS, 241, 35 [NASA ADS] [CrossRef] [Google Scholar]
  113. Ren, Y., Jiang, B., Yang, M., et al. 2021, ApJ, 923, 232 [NASA ADS] [CrossRef] [Google Scholar]
  114. Riebel, D., Srinivasan, S., Sargent, B., et al. 2012, ApJ, 753, 71 [NASA ADS] [CrossRef] [Google Scholar]
  115. Rodrigo, C., Solano, E., & Bayo, A. 2012, IVOA Working Draft [Google Scholar]
  116. Rodrigo, C., & Solano, E. 2020, XIV.0 Scientific Meeting (virtual) of the Spanish Astronomical Society, 182 [Google Scholar]
  117. Roman-Duval, J., Gordon, K. D., Meixner, M., et al. 2014, ApJ, 797, 86 [NASA ADS] [CrossRef] [Google Scholar]
  118. Ruffle, P. M. E., Kemper, F., Jones, O. C., et al. 2015, MNRAS, 451, 3504 [NASA ADS] [CrossRef] [Google Scholar]
  119. Salasnich, B., Bressan, A., & Chiosi, C. 1999, A&A, 342, 131 [NASA ADS] [Google Scholar]
  120. Sargent, B. A., Srinivasan, S., Meixner, M., et al. 2010, ApJ, 716, 878 [NASA ADS] [CrossRef] [Google Scholar]
  121. Sargent, B. A., Srinivasan, S., & Meixner, M. 2011, ApJ, 728, 93 [NASA ADS] [CrossRef] [Google Scholar]
  122. Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103 [Google Scholar]
  123. Scicluna, P., Siebenmorgen, R., Wesson, R., et al. 2015, A&A, 584, L10 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  124. Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163 [NASA ADS] [CrossRef] [Google Scholar]
  125. Smartt, S. J. 2009, ARA&A, 47, 63 [NASA ADS] [CrossRef] [Google Scholar]
  126. Smartt, S. J. 2015, PASA, 32, e016 [NASA ADS] [CrossRef] [Google Scholar]
  127. Smith, N. 2014, ARA&A, 52, 487 [NASA ADS] [CrossRef] [Google Scholar]
  128. Smith, N., Humphreys, R. M., Davidson, K., et al. 2001, AJ, 121, 1111 [CrossRef] [Google Scholar]
  129. Smith, N., Mauerhan, J. C., Cenko, S. B., et al. 2015, MNRAS, 449, 1876 [Google Scholar]
  130. Soraisam, M. D., Bildsten, L., Drout, M. R., et al. 2018, ApJ, 859, 73 [NASA ADS] [CrossRef] [Google Scholar]
  131. Srinivasan, S., Sargent, B. A., & Meixner, M. 2011, A&A, 532, A54 [CrossRef] [EDP Sciences] [Google Scholar]
  132. Srinivasan, S., Boyer, M. L., Kemper, F., et al. 2016, MNRAS, 457, 2814 [NASA ADS] [CrossRef] [Google Scholar]
  133. Sun, M., Jiang, B., Yuan, H., et al. 2021, ApJS, 254, 38 [NASA ADS] [CrossRef] [Google Scholar]
  134. Sylvester, R. J., Barlow, M. J., & Skinner, C. J. 1994, MNRAS, 266, 640 [NASA ADS] [CrossRef] [Google Scholar]
  135. Sylvester, R. J., Skinner, C. J., & Barlow, M. J. 1998, MNRAS, 301, 1083 [Google Scholar]
  136. Tantalo, M., Dall’Ora, M., Bono, G., et al. 2022, ApJ, 933, 197 [NASA ADS] [CrossRef] [Google Scholar]
  137. Taylor, M. B. 2005, ASP Conf. Ser., 347, 29 [Google Scholar]
  138. Udalski, A., Szymanski, M., Kaluzny, J., Kubiak, M., & Mateo, M. 1992, Acta. Astron., 42, 253 [NASA ADS] [Google Scholar]
  139. Ueta, T., & Meixner, M. 2003, ApJ, 586, 1338 [NASA ADS] [CrossRef] [Google Scholar]
  140. Vanbeveren, D., De Loore, C., & Van Rensbergen, W. 1998, A&A Rv., 9, 63 [NASA ADS] [Google Scholar]
  141. van Loon, J. T. 2000, A&A, 354, 125 [NASA ADS] [Google Scholar]
  142. van Loon, J. T. 2006, Stellar Evolution at Low Metallicity: Mass Loss, Explosions, Cosmology, 353, 211 [NASA ADS] [Google Scholar]
  143. van Loon, J. T., Cioni, M.-R. L., Zijlstra, A. A., & Loup, C. 2005, A&A, 438, 273 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  144. van Loon, J. T., Marshall, J. R., Cohen, M., et al. 2006, A&A, 447, 971 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  145. Verhoelst, T., van der Zypen, N., Hony, S., et al. 2009, A&A, 498, 127 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  146. Wang, S., & Chen, X. 2019, ApJ, 877, 116 [Google Scholar]
  147. Wang, T., Jiang, B., Ren, Y., et al. 2021, ApJ, 912, 112 [NASA ADS] [CrossRef] [Google Scholar]
  148. Werner, M. W., Roellig, T. L., Low, F. J., et al. 2004, ApJS, 154, 1 [NASA ADS] [CrossRef] [Google Scholar]
  149. Willson, L. A. 2000, ARA&A, 38, 573 [NASA ADS] [CrossRef] [Google Scholar]
  150. Wolf, C., Onken, C. A., Luvaul, L. C., et al. 2018, PASA, 35, e010 [Google Scholar]
  151. Woosley, S. E., & Weaver, T. A. 1986, ARA&A, 24, 205 [NASA ADS] [CrossRef] [Google Scholar]
  152. Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868 [Google Scholar]
  153. Yang, M., & Jiang, B. W. 2011, ApJ, 727, 53 [NASA ADS] [CrossRef] [Google Scholar]
  154. Yang, M., & Jiang, B. W. 2012, ApJ, 754, 35 [NASA ADS] [CrossRef] [Google Scholar]
  155. Yang, M., Bonanos, A. Z., Jiang, B.-W., et al. 2018, A&A, 616, A175 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  156. Yang, M., Bonanos, A. Z., Jiang, B.-W., et al. 2019, A&A, 629, A91 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  157. Yang, M., Bonanos, A. Z., Jiang, B.-W., et al. 2020, A&A, 639, A116 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  158. Yang, M., Bonanos, A. Z., Jiang, B., et al. 2021a, A&A, 646, A141 [EDP Sciences] [Google Scholar]
  159. Yang, M., Bonanos, A. Z., Jiang, B., et al. 2021b, A&A, 647, A167 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  160. Yoon, S.-C., & Cantiello, M. 2010, ApJ, 717, L62 [NASA ADS] [CrossRef] [Google Scholar]
  161. Zaritsky, D., Harris, J., Thompson, I. B., et al. 2002, AJ, 123, 855 [NASA ADS] [CrossRef] [Google Scholar]
  162. Zhang, T., Wang, X., Wu, C., et al. 2012, AJ, 144, 131 [NASA ADS] [CrossRef] [Google Scholar]

All Tables

Table 1.

Datasets for the SED fitting.

In the text
Table 2.

Final sample of 2121 RSG star candidates in the SMC.

In the text

All Figures

thumbnail Fig. 1.

Gaia (upper left), 2MASS (upper right), IRAC (bottom left), and MIPS (bottom right) CMDs of the final RSG sample in the SMC. Targets in the overlapping region (green lines) between AGBs and RSGs on the 2MASS CMD are removed (open black circles). Other general stellar populations, e.g., AGB, RGB, TRGB, and BSG stars, are marked on the Gaia and 2MASS CMDs, but not on IRAC and MIPS CMDs due to the overlap between different populations (see Fig. 12 of Yang et al. 2021a). Background targets (gray dots) are from Yang et al. (2019).
In the text
thumbnail Fig. 2.

Spatial distribution of the final RSG sample (orange color) in the SMC. Targets removed from the initial sample are marked as open black circles (see text and Fig. 1 for details). Background targets (gray dots) are from Yang et al. (2019).
In the text
thumbnail Fig. 3.

Normalized (at 2MASS KS-band) median SED of the final RSG sample in the SMC. Due to the strict constraint of S/N, WISE [12] and [22] band data only include relatively bright targets with high fluxes.
In the text
thumbnail Fig. 4.

Histogram of the luminosity of the final RSG sample. We note that the low-luminosity targets (log10(L/L⊙)≲4.0) may not be true RSGs, but ∼6 − 8 M⊙ red helium-burning stars.
In the text
thumbnail Fig. 5.

Histogram of the Teff of the final RSG sample.
In the text
thumbnail Fig. 6.

Histogram of the χ m 2 $ \chi_{m}^2 $ distribution. The three vertical dashed lines indicate the 68th ( χ m 2 $ \chi_{m}^2 $ = 0.094), 90th ( χ m 2 $ \chi_{m}^2 $ = 0.203), and 95th ( χ m 2 $ \chi_{m}^2 $ = 0.371) percentiles, respectively.
In the text
thumbnail Fig. 7.

Histogram of numbers of data points fitted for each targets.
In the text
thumbnail Fig. 8.

Examples of the typical SED fitting for optically thin (upper row), normal (middle row), and very dusty targets (bottom row). In each panel, the coordinate is marked on the upper left, and chemical composition, τ, and χ m 2 $ \chi_{m}^2 $ are marked on the upper right.
In the text
thumbnail Fig. 9.

Same as Fig. 8 but with examples of a zoomed-in region of the SED fitting with Spitzer/IRS spectra (gray) overlapped.
In the text
thumbnail Fig. 10.

Histogram of the derived MLRs of the final RSG sample.
In the text
thumbnail Fig. 11.

Histogram of the optical depth of the final RSG sample. For clarity, targets with large τ (e.g., > 1.0) are not shown in the diagram.
In the text
thumbnail Fig. 12.

Hertzsprung-Russell (left) and luminosity versus MAD (right) diagrams for the final RSG sample. Targets are color-coded with the MLR, while targets with relatively large (> 0.1) and very large (> 1.0) τ are marked by the open squares and open circles in the diagram, respectively. The dashed lines indicate the luminosity of log10(L/L⊙) = 4.6. For clarity, only the general population (without the extreme outliers) of the sample is shown in the diagram. In the luminosity versus MAD diagram, there are a few targets showing large MAD with low luminosity and MLR, which are mostly targets crossing the instability strip (because of their large variability, their is Teff variable, meaning that sometimes they are RSGs and sometimes they are yellow supergiants).
In the text
thumbnail Fig. 13.

Luminosity versus MLR diagrams for the final RSG sample. Targets are color-coded with τ, while targets with very large τ (>1.0) are marked by the open circles in the diagram. The dashed line indicates the luminosity of log10(L/L⊙) = 4.6. We note that the low-luminosity targets (log10(L/L⊙)≲4.0) may not be true RSGs, but ∼6 − 8 M⊙ red helium-burning stars.
In the text
thumbnail Fig. 14.

J − [8.0] (left), KS − [12] (middle), and [3.6]−[24] (right) versus MLR diagrams for the final RSG sample.
In the text
thumbnail Fig. 15.

Derived MLR–L relation from this work (left) and comparison of the same relation between this and previous works (right). In the left panel, the very dusty targets (τ > 1.0) are marked in red. In the right panel, lines of the same color are variations of the same relation (as shown in Fig. 16).
In the text
thumbnail Fig. 16.

Comparison of derived MLR–L relation between this and each previous individual study. A 0.5 dex error (the shaded region) is shown as a reference for all prescriptions (including ours).
In the text
thumbnail Fig. 17.

KS versus bolometric magnitudes diagram. Our work indicates an improved relation based on a larger sample compared to previous studies.
In the text
thumbnail Fig. 18.

Examples of the targets with UV detection, which likely indicate a hot binary companion.
In the text
thumbnail Fig. 19.

Examples of the targets with unusual optical excess or depression, which may indicate a binary companion.
In the text

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