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A&A
Volume 697, May 2025
Article Number A167
Number of page(s) 17
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/202452401
Published online 16 May 2025

© The Authors 2025

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

Massive stars with initial masses in the range of ∼8 − 30 M⊙ become red supergiants (RSGs) during their late evolutionary phases (e.g., Levesque 2017, and references therein), following their departure from the main sequence (MS). The RSGs reside in the cool, luminous region of the Hertzsprung-Russell diagram (HRD). For stars within this mass range, mass loss during the MS phase, which is where they spend most of their lifetime, is generally minimal (although important for certain evolutionary outcomes, as discussed in Renzo et al. 2017). It is during the RSG phase that these stars experience substantial mass loss that profoundly influences their subsequent evolution and ultimate fate (e.g., Meynet & Maeder 2003; Meynet et al. 2015; Beasor et al. 2020).

Most RSGs are expected to end their lives as Type II core-collapse supernovae (SNe; e.g., Heger et al. 2003; Langer 2012). This has been confirmed observationally in pre-SN progenitor detections of nearby events (e.g., Smartt et al. 2009, 2015). However, some may directly collapse into black holes with or without a faint transient event (O’Connor & Ott 2011), a phenomenon with possible observed candidates (Adams et al. 2017; Beasor et al. 2024). This possibility has been proposed as the explanation of the “red supergiant problem”, that is, the lack of detections of luminous high-mass RSG progenitors (Smartt et al. 2009, although see a discussion about the low statistical significance of the issue in Davies & Beasor (2018). An alternative solution consists of strong winds that partially strip the H-rich envelope of luminous RSGs, which become hotter and bluer and avoid exploding as Type II SNe (e.g., Georgy et al. 2013; Meynet et al. 2015). Consequently, mass loss affects the evolution of a massive star and its position in the HRD, its final mass, the remnant formed during core collapse, and the observable characteristics of its eventual supernova explosion or black hole formation.

Although recent studies have contributed to constraining the mass-loss rate empirically (e.g., Beasor et al. 2020; Davies & Plez 2021; Decin et al. 2024; Yang et al. 2023; Antoniadis et al. 2024, 2025), or even theoretically (Kee et al. 2021; Vink & Sabhahit 2023; Fuller & Tsuna 2024), the driving mechanism and the rate of mass loss during the RSG phase remains uncertain (e.g., Yoon & Cantiello 2010; Smith 2014; Arroyo-Torres et al. 2015), with indications for episodic mass loss (e.g., Decin et al. 2006; Bruch et al. 2021; Dupree et al. 2022; Humphreys & Jones 2022; Cheng et al. 2024) also complicating the picture. To include the significant effect of mass loss during the RSG phase, stellar evolution models need to select from a wide range of predominantly empirical mass-loss rate prescriptions (Reimers 1975; de Jager et al. 1988; Nieuwenhuijzen & de Jager 1990; van Loon et al. 2005; Goldman et al. 2017; Beasor et al. 2020; Kee et al. 2021; Antoniadis et al. 2024) that vary by orders of magnitude. van Loon et al. (2005), Goldman et al. (2017), and Yang et al. (2023) predict higher mass-loss rates, mainly due to the assumption of radiatively driven wind being the mechanism of mass loss, in comparison with the weaker winds found when steady-state density distribution is considered (Beasor et al. 2020; Antoniadis et al. 2024, 2025). Most of these studies are based on the properties of the dust shell formed around the RSG. Different ranges of the assumed grain sizes can also affect the inferred mass-loss rate (Antoniadis et al. 2024). Furthermore, a uniform average gas-to-dust ratio is usually assumed, which can vary significantly within some galaxies, such as the Small Magellanic Cloud (SMC) (Clark et al. 2023). Recent alternative methods using gas and molecular diagnostics (Decin et al. 2024; González-Torà et al. 2024) also inferred a wide range of mass-loss rates. Incorporating accurate mass-loss prescriptions into stellar evolution models is essential for predicting the late stages of massive stars and their fate and understanding their roles in broader astrophysical processes.

Especially in low metallicity regimes, despite efforts to constrain RSG properties (e.g., Levesque et al. 2006; Davies et al. 2015; Patrick et al. 2017; Britavskiy et al. 2019; González-Torà et al. 2021; de Wit et al. 2023; Bonanos et al. 2024), the mass-loss rate during the RSG phase has been less constrained (with the exception of the study by van Loon et al. 2005), but there has been significant progress in the past decade (Goldman et al. 2017; Yang et al. 2023; Antoniadis et al. 2024, 2025). Yang et al. (2023, hereafter Y23) analyzed a comprehensive RSG sample in the SMC and discovered a notable “kink”, that is, a change in slope in the mass-loss rate as a function of luminosity where the winds significantly increase for RSGs with log(L/L⊙)≳4.6. This feature suggests that more luminous RSGs experience a higher mass-loss rate, potentially leading to earlier envelope stripping and shorter RSG lifetimes. The same feature was also found by Antoniadis et al. (2024) for the RSGs in the Large Magellanic Cloud, although at a slightly lower luminosity value. Motivated by the new empirical mass-loss rate relation presented by Y23, in this study we proceed to implement this and other mass-loss prescriptions in the literature in stellar evolution models of SMC metallicity to compare theoretical stellar populations with observations, investigating the impact of stellar mass loss on the life and the eventual death of RSGs.

The paper is structured as follows: In Sect. 2 we present the different RSG mass-loss rate prescriptions implemented in our stellar models, as well as the observational sample used for comparison. In Sect. 3 we show the effects of the RSG mass-loss rate on RSG evolution, their luminosity function, and their final fate. We estimate the possible effect of binaries, discuss the yellow-to-red supergiants ratio, the tension with observable constraints within the caveats of our study, and the feedback from RSGs into its environments in Sect. 4. We provide some concluding thoughts in Sect. 5.

2. Method

2.1. Stellar tracks and simulations of a red supergiant population with POSYDON

To investigate the effect of different RSG mass-loss rates in the evolution and final outcome of massive stars, we implemented the POSYDON1 framework (Fragos et al. 2023; Andrews et al. 2024), which is based on grids of detailed single- and binary-star models computed with the MESA 1D stellar evolution code (Paxton et al. 2011, 2013, 2015, 2018, 2019; Paxton 2021, version 11701) from zero-age MS up to carbon depletion in the core. Our results are based on single-star model grids spanning initial masses from 6 to 40 M⊙ with a logarithmic spacing of ∼0.0093. We ensured that we included all possible single massive star tracks that may pass through the RSG phase during their lifetime (e.g., Heger et al. 2003; Ekström et al. 2012) within our mass range. In principle, the RSG phase could be reached by stars of higher masses with different assumptions (Klencki et al. 2021), but we tested that this is avoided in models above 40 M⊙ in our set up due to the implemented luminous blue variable (LBV)-type mass-loss rate (discussed in Sect. 2.2). The default physical assumptions and numerical specifications follow those of POSYDON v2 stellar models Andrews et al. (2024), apart from the three extra different mass-loss rate prescriptions during the RSG phase, as we describe in Sect. 2.2. Although there is a scatter of metallicities across the SMC (Davies et al. 2015; Choudhury et al. 2018), we assumed an average metallicity of Z = 0.2 Z⊙ = 0.00284 ([Fe/H] ∼ −0.7), which is close to the values from previous stellar modeling of the SMC (Brott et al. 2011a; Georgy et al. 2013). The solar metallicity calibration is based on Asplund et al. (2009, Z⊙ = 0.0142).

A significant fraction of the RSGs in our sample are expected to be products of prior binary evolution (Sana et al. 2012; Moe & Di Stefano 2017) and are involved in scenarios of merging, mass accretion, or even partial stripping (e.g., de Mink et al. 2014; Justham et al. 2014; Eldridge et al. 2019). However, for simplicity, as well as in order to obtain a better handle on the differences among the various mass-loss rate assumptions explored, in this paper we focus on the single-star channel as a first step. We estimate the impact of a possible binary history of RSGs in Sect. 4.1.

We aim to prevent contamination of the RSG sample with asymptotic giant branch (AGB) stars, but distinguishing between these types of stars photometrically is challenging. Previously, Yang et al. (2023) implemented various observational cuts and criteria to minimize AGB contamination, while Yang et al. (2024), who analyzed a large spectroscopic sample, found the lower luminosity limit for RSGs in the Large Magellanic Cloud to be as low as log10(Lmin/L⊙)∼3.5. However, it remains uncertain whether the interior physics and the mass-loss rates of these lower-luminosity and lower-mass stars are analogous to more massive ones, and if they end up exploding as a core-collapse SN. Therefore, in this study, we chose to set a theoretically motivated limit. AGB and super-AGB stars originate from intermediate-mass stars, that eventually form a carbon-oxygen or (for initially slightly more massive cases) an oxygen-magnesium core that is not massive enough to collapse and instead forms a white dwarf (Poelarends et al. 2008; Doherty et al. 2015). To be conservative and avoid these scenarios altogether, we only accept tracks that eventually burn up all the heavy elements up to iron in their core, producing an iron-core-collapse (FeCC). We thus also reject progenitors of electron-capture SNe (ECSNe) from oxygen-magnesium cores close to the Chandrasekhar limit, with 1.37 < MCO, core, fin < 1.43 M⊙ (Tauris et al. 2015). This is dependent on the mass-loss prescription that we use, but corresponds to a minimum luminosity of log10(Lmin/L⊙)∼3.9.

To create a population of RSGs, for each mass-loss assumption simulation, we randomly draw 105 initial masses between [6 − 40] M⊙, according to a canonical (Kroupa 2001) initial mass function (IMF). There is evidence for a slightly more top-heavy IMF in the Magellanic Clouds (Schneider et al. 2018), but it mostly affects stars above ∼30 M⊙ so we do not consider it. We linearly interpolate between the stellar tracks of different initial masses (as explained in Dotter 2016; Fragos et al. 2023). For each drawn mass we also pick a random time during its evolution between 8 and 70 Myr, to include all possible stellar lifetimes of single stars that may become RSGs (e.g., Ekström et al. 2012; Zapartas et al. 2017). As we want to compare with observational findings in the SMC from Y23 (discussed in Sect. 2.3), we weight the drawing according to a star formation history (SFH) of the SMC. We follow Rubele et al. (2015), which estimates three lookback time bins of slightly different star formation rates:

SFH SMC ( t ) / ( M ⊙ yr − 1 ) = { 0.052 , for t < 14 0.125 , for t ∈ [ 14 , 40 ] 0.06875 , for t ≥ 40 , $$ \begin{aligned} \mathrm {SFH}_{\mathrm {SMC}} (t) /(M_{\odot } \mathrm{yr}^{-1}) = {\left\{ \begin{array}{ll} 0.052,&\mathrm {\,\,for\,\,\,} t < 14 \\ 0.125,&\mathrm {\,\,for\,\,\,} t \in [14,40] \\ 0.06875,&\mathrm {\,\,for\,\,\,} t \ge 40, \\ \end{array}\right.} \end{aligned} $$(1)

where t is the lookback time in Myr. We consider a star to be in its RSG phase when its effective temperature reaches below Teff, max = 103.66 ≃ 4570 K (Meynet et al. 2015). We exclude drawn combinations of initial masses and ages that exceed the total lifetime of the stellar models, as such stars are considered to have already undergone collapse. We tested that our results converge for sample sizes that exceed 105 draws.

2.2. Mass-loss prescriptions

This study focuses on widely used and recent RSG mass-loss prescriptions, described below. These four prescriptions are derived employing different inference techniques, sample selections, and sizes, which in turn span a range of RSG mass-loss rates. The four grids of stellar tracks differ solely in the implemented prescription. We show in Fig. 1 the selected prescriptions as a function of RSG luminosity, with points depicting random draws of RSGs from our POSYDON stellar populations, with each color corresponding to a different mass-loss prescription. The concentration of drawn points at low luminosities reflects the weighting by the IMF and the longer RSG lifetimes for lower initial masses.

  • One selected RSG mass-loss prescription is from de Jager et al. (1988, from now on ṀJ88, green in Fig. 1), which is one of the most popular prescriptions used in stellar models for this evolutionary phase and is the default RSG mass-loss prescription used in POSYDON. The mass-loss rate of this empirical relation is based on the position at the HRD of around a dozen Galactic RSGs and is calculated for effective temperatures below 104 K,

    log 10 ( M ˙ / ( M ⊙ yr − 1 ) ) = 1.769 log 10 ( L / L ⊙ ) − 1.676 log 10 ( T eff / K ) − 8.158 . $$ \begin{aligned} \log _{10}(\dot{M}/(M_{\odot } \mathrm{yr}^{-1})) =&1.769\log _{10}(L/ L_{\odot }) \nonumber \\&-1.676\log _{10}(T_{\rm eff}/\mathrm{K}) - 8.158. \end{aligned} $$(2)

    The reason that some points of ṀJ88 do not fall exactly on top of the depicted lines of the theoretical prescription in Fig. 1 is that the prescription depends also on Teff, apart from luminosity. The drop of some drawn RSGs mass-loss rate luminosities below log10(L/L⊙)∼4.3 for RSG following ṀJ88 is because of the POSYDON implementation of Reimers (1975) mass-loss prescription (multiplied by a factor of 0.1, grey line) when the total stellar mass drops below 8 M⊙, which is considered a more viable option for- low and intermediate-mass stars.

  • We implemented the recent RSG mass-loss prescription Yang et al. (2023, from now on ṀY23), based on the largest sample of RSGs in the SMC, as shown in Eq. (3),

    log 10 ( M ˙ / ( M ⊙ yr − 1 ) ) = 20.93 log 10 ( L / L ⊙ ) − 5.26 [ log 10 ( L / L ⊙ ) ] 2 + 0.45 [ log 10 ( L / L ⊙ ) ] 3 − 34.56 . $$ \begin{aligned} \log _{10}(\dot{M}/(M_{\odot } \mathrm{yr}^{-1}))&= 20.93\log _{10}(L/ L_{\odot }) - 5.26[\log _{10}(L/ L_{\odot })]^2 \nonumber \\&+ 0.45[\log _{10}(L/ L_{\odot })]^3 - 34.56. \end{aligned} $$(3)

    Since ṀY23 does not have a dependence on the effective temperature, Teff, we need to define the region of its validity in the HRD. We thus applied in our models a linear transition from the default ṀJ88 scheme towards ṀY23 prescription between 6000 and 5000 K, testing that our results are not sensitive to the exact transition limits. Due to the dependence only on luminosity, all the drawn RSGs with ṀY23 fall by definition on top of the theoretical prescription in Fig. 1. On the same plot, we also depict with grey points the Y23 sample of RSGs, which ṀY23 was based on2. ṀY23 is only a factor of a few higher than ṀJ88 at the mid-range of RSG luminosities, 4.5 ≲ log10(L/L⊙)≲5.1, but even higher for lower or higher luminosities. ṀY23 is closer to the mass-loss rate implemented in the standard Geneva models (Ekström et al. 2012) multiplied by ten (“10xMdot”), as investigated in Meynet et al. (2015).

  • We have implemented the mass-loss rate prescription from Beasor et al. (2020, in its corrected form from Beasor et al. 2023; referred to hereafter as ṀB20), which is based on coeval samples of RSGs in Galactic clusters. The prescription functions in our study as an example of a low mass-loss rate during the RSG phase:

    log 10 ( M ˙ / ( M ⊙ yr − 1 ) ) = 4.8 log 10 ( L / L ⊙ ) − 0.23 log 10 ( M init / M ⊙ ) − 24.6 . $$ \begin{aligned} \log _{10}(\dot{M}/(M_{\odot } \mathrm{yr}^{-1}))&= 4.8\log _{10}(L/ L_{\odot }) \nonumber \\&- 0.23\log _{10}(M_{\rm init}/M_{\odot }) - 24.6. \end{aligned} $$(4)

    The ṀB20 prescription depends on the initial mass (Minit), based on the assumption that the RSGs in their cluster samples have similar initial masses. For convenience, we use a relation between initial mass, Minit, and average RSG luminosity, LRSG, avg, at various points throughout the study. The exact recalculation of the relation for our models is provided in Appendix A. Indeed, RSGs with ṀB20 follow very well the theoretical prescription, if we convolve it with our Minit − LRSG, avg best-fit relation of Eq. (A.1).

  • Finally, we have also applied the first attempt for a theoretical prescription of RSG mass loss by Kee et al. (2021, from now on ṀK21). This work assumes that turbulence is the dominant driving mechanism for mass loss, without the need of radiation pressure on dust grains, in line with previous studies (e.g., Josselin & Plez 2007). The mass-loss rate is analytically derived at a modified Parker radius, where turbulent and thermal pressure balance gravitational forces, providing a natural critical point to evaluate steady-state wind properties (for an abridged presentation of the derivation, see also Kee & Maestro Project 2024). The ṀK21 prescription is highly sensitive to the assumed turbulent velocity on the surface spanning potentially many orders of magnitudes (see their Fig. 3). We use vturb = 18.2 km s−1, which they find as the best-fit empirical value. In our calculation we assume a flux-weighted mean opacity of κ = 0.01 cm2/g). The model points in Fig. 1 occupy mostly the low luminosity regime, corresponding to initial masses around 7 − 8 M⊙. This is because for higher initial masses the stars quickly become stripped and only exist as RSGs for a short time. The strong dependence on the current mass during the RSG phase, (see their Fig. 2) creates a runaway effect of increasing mass-loss rate for a RSG which retains its position in the HRD (i.e., maintaining a constant stellar radius). This leads to a significant decrease in mass and, subsequently, surface gravity, which further intensifies the wind by orders of magnitude. As the star becomes progressively stripped, its hotter inner layers are exposed, ultimately leading to a blueward evolution. The theoretical curve of ṀK21 in Fig. 1 (red solid curve), when we convolve it with our best-fit Minit − LRSG, avg, Eq. (A.1), becomes similar to the one in Figure 8 of Kee et al. (2021). On the other hand, for fixed mass it becomes much steeper with luminosity (red dashed line in Fig. 1 for M = 7 M⊙). In practice, as we show in Sect. 3, the mass of a RSG with ṀK21 will decrease quickly and its mass-loss rate will keep increasing significantly.

We have not implemented a direct dependency on the metallicity in any of the considered RSG mass-loss prescriptions. No conclusive evidence of such a dependency has been found to date, with theoretical (Kee et al. 2021) and empirical efforts (e.g., Goldman et al. 2017; Antoniadis et al. 2025) pointing towards no or weak metallicity dependence. Therefore we refrain from adding an extra uncertainty into our analysis from a poorly constrained metallicity dependence of the RSG winds mass-loss rates. Nevertheless, the ṀY23 prescription has been calibrated on SMC data, and thus would not have required a metallicity adjustment.

thumbnail Fig. 1.

Mass-loss rate versus luminosity of the RSG wind prescriptions implemented. Points are random draws of RSGs from POSYDON stellar populations, following each prescription (see legend) and grey crosses are the Y23 observational sample. The top x-axis corresponds to the initial mass of a star to have LRSG, avg equal to the bottom axis, according to Eq. (A.1).

During a star’s blueward evolution, when its Teff > 104 K and its surface hydrogen mass fraction Xsurf drops below 0.4, we assume that the Wolf-Rayet line-driven winds kick in (Nugis & Lamers 2000, following its explicit metallicity dependence). We do not focus on their directly observable characteristics (Hainich et al. 2015) but rather on their mass-loss rates, which are stronger than the star’s prior winds during its MS evolution (Vink et al. 2000). For the latter we assume a metallicity dependence of (Z/Z⊙)0.68 (Vink et al. 2001). If a very luminous star enters inside the Humphreys-Davidson limit (HD; Humphreys & Davidson 1979), with log10(LRSG/L⊙)≳5.78 for RSGs, following Hurley et al. (2000), Belczynski et al. (2010, although more recent studies suggested a lower luminosity limit; e.g., Davies et al. 2018; McDonald et al. 2022), we change the mass-loss rate to a constant value of 10−4 M⊙ yr−1 (Belczynski et al. 2010), to simulate the regime of high but uncertain LBV-type mass loss. We do not implement the recent prescriptions including extra “eruptive” mass loss due to local super-Eddington layers at the stellar surface (Cheng et al. 2024).

2.3. Observational sample of red supergiants in the Small Magellanic Cloud

To test our population models, in Sect 3.2 we compare with the sample of RSGs in the SMC compiled by Y23, which is based on the catalogs from Yang et al. (2020) and Ren et al. (2021). The collection, after excluding potential AGB contaminants, consists of 2121 targets and comprises the largest, most complete sample of RSGs in the SMC so far. The ṀY23 prescription was derived based on this refined sample. The effective temperature Teff of the sources is inferred using the J − KS relation of Yang et al. (2023, Eq. (1)). For the dereddened colors, we assume a constant AV,CSM = 0.1, which is the typical value found for most RSGs in that study. We also add a foreground Galactic and SMC internal extinction of E(B − V) = 0.04 and 0.05 (Massey et al. 2007), respectively, combined with a canonical RV = 2.74 value for SMC (Gordon et al. 2003), (AJ/AV)2MASS = 0.243 and (AK/AV)2MASS = 0.078 (Wang & Chen 2019). We adopt a distance modulus of 18.95 mag to the SMC (Graczyk et al. 2014; Scowcroft et al. 2016). The luminosity of the sources has already been calculated in Yang et al. (2023) by integrating the spectral energy distribution (SED) of each target, which is our default luminosity estimate. Alternatively, we also calculate a luminosity from the Neugent et al. (2020) bolometric correction of the KS magnitude of each source. Depending on these two luminosity calculation methods, we find 930 and 990 sources, respectively, from the refined Y23 sample within our conservative limits for RSGs above Lmin and below Teff, max. As we demonstrate in Sect. 3.2, although we include both luminosity calculations, their differences are not significant.

3. Results

3.1. Mass lost and time spent during the red supergiant phase

To understand the evolution of massive stars subject to the different RSG mass-loss prescriptions, we study the time they spend and the amount of mass they lose during their life, including their RSG phase. In the top panels of Fig. 2 we see the evolution of a few indicative tracks on the HRD, colored by their evolutionary phase. Stars spend most of their time in the blue regime (log10(Teff/K) > 3.9) during their MS after which they expand quickly towards cooler temperatures. They pass through a short-lived “yellow” phase (3.9 > log10(Teff/K) > 3.66, following the definitions from Georgy et al. 2013; Meynet et al. 2015) before initiating helium core burning (yellow), and promptly become RSGs (log10(Teff/K) < 3.66).

thumbnail Fig. 2.

Evolution during different phases. Top: Indicative tracks of equal timesteps of 1kyr in the HRD with ṀY23 (left) and ṀJ88 (right), depicting the definition of each evolutionary phase. The initial mass is mentioned at the beginning of each track. The HD limit, where LBV-type winds kick in is also shown (grey dot-dashed line). Middle: Part of the stellar mass lost in each evolutionary phase, as a function of initial mass (evolution goes from top to bottom). We also show the value of the final helium core mass as a function of the initial (purple dotted line). Bottom: Fractional time spent in each evolutionary phase after the end of MS, as a function of initial mass (evolution goes from top to bottom).

Although the total, absolute, post-MS lifetime is almost independent of the mass loss during the RSG phase (as shown in Fig. B.1), the fraction of post-MS lifetime spent in each phase is sensitive to mass loss. Stars of Minit ≲ 15 following ṀY23 (left column), spend almost all of their post-MS life in the RSG phase (bottom panels of Fig. 2), until collapse. The fractional mass lost during that period becomes a minimum for stars around Minit ∼ 10 M⊙, being around 30% of the initial mass (middle-left panel of Fig. 2). For our POSYDON models and according to Eq. (A.1), this corresponds to log10(LRSG, avg/L⊙)∼4.6, which is the location of the “kink”, that is, the turning point of ṀY23 towards even higher mass-loss rates for increasing luminosity (Yang et al. 2023, also found in the RSG sample of the Large Magellanic Cloud; Antoniadis et al. 2024). Below this kink, ṀY23 is almost constant with luminosity, and combined with the prolonged RSG lifetime for lower initial masses, leads to a higher fraction of their mass lost for them before FeCC. On the other hand, the increasing mass-loss rate above the kink is the reason for the curve-like relative mass loss, shown in the middle-left panel of Fig. 2 for ṀY23. In contrast, for ṀJ88, the fraction of the initial mass lost to winds can be as low as ∼15% for stars around 10 M⊙. This fraction becomes negligible for Minit < 8 M⊙, where the Reimers (1975) prescription kicks in. Conversely, this fraction increases to ∼50% at higher initial masses.

For all RSG mass-loss prescriptions, stars below Minit ∼ 25 M⊙, lose most of their mass during their RSG phase. This remains true even if they transition back to the yellow or blue regions of the HRD and spend a significant amount of their remaining lifetime there. This occurs either because the RSG mass-loss prescriptions result in a reduced mass loss at higher effective temperatures (e.g., ṀJ88, ṀK21) or because line-driven winds are weaker in the bluer parts of the HRD (Vink et al. 2000). The earlier departure from the RSG phase for stars with higher initial masses, along with the longer duration they spend in the yellow or blue regions during helium core burning, results in a slight reduction in the fractional mass loss.

Red supergiants originating from initial masses of approximately 15 M⊙ or greater, following ṀY23, tend to evolve blueward. This happens because they lose a substantial portion of their envelope (as illustrated in the middle-left panel of Fig. 2), not because they undergo a “blue loop” which is a result of internal restructuring. They spend a considerable portion of their post-MS phase as helium-core burning yellow stars (orange points in Fig. 2) or once again in the blue region of the HRD, possibly becoming yellow again in their final stage after core helium depletion (brown points in Fig. 2). The transition where RSGs shift to the blue happens at initial masses as low as 15 M⊙ due to the higher mass-loss rate of ṀY23 above the turning point of the prescription, which increasingly diverges from ṀJ88 by more than a factor of two at log10(L/L⊙)∼5.2, and even more for more luminous RSGs. In contrast, for moderate mass-loss prescriptions, with a constant slope with luminosity, as in ṀJ88, the transition to blueward tracks occurs for stars with Minit ≳ 19 − 20 M⊙. Even then, massive stars spend a significant amount of their post-MS time in the yellow regime during helium core burning before going to the blue one (bottom-right panel).

RSGs of Minit ≳ 27 M⊙ get stripped and leave the RSG phase, even before helium core burning starts. They then go even more blueward, and when their surface hydrogen mass fraction drops below 0.4, the Wolf-Rayet mass loss kicks in. Stars of Minit = 27 − 40 M⊙ following ṀY23 reach that stage earlier in their evolution, spending 20–35% of their post-MS time as Wolf-Rayet stars.

In Fig. 3 we plot the same information but for ṀB20 and ṀK21. Stars following the much weaker ṀB20 stay in the RSG phase (or just cross the log10(Teff/K) = 3.66 limit and become YSGs for Minit ∼ 28 − 34 M⊙) until the end of their evolution. They lose ≲15% of their total mass during their whole evolution, except for stars with initial masses ≳34 M⊙ that enter the HD limit where enhanced LBV-type winds begin to strip the star down to its helium core, pushing it towards the blue region. In contrast, the RSG mass-loss rate of models following ṀK21 is even stronger than in ṀY23 for the moderate luminosities typical of most RSGs. This is because ṀK21 induces a runaway effect: as a star enters the RSG phase, it experiences significant mass loss, which reduces its surface gravity and mass. Since ṀK21 is highly sensitive to these parameters, this leads to an even more intense wind, increasing by orders of magnitude, until the star is stripped enough to shift to the blue region, even for stars with an initial mass lower than 10 M⊙. In Fig. 4, we illustrate this runaway effect with an example system, showing that for a star of approximately ∼12.9 M⊙, its mass-loss rate increases from log10(ṀRSG/M⊙ yr−1) ≈ −5 upon entering the RSG phase to about ∼ − 3.5 as it becomes progressively stripped after just ∼7 × 104  yr, with only 6 M⊙ remaining when it leaves the RSG phase moving bluewards. In comparison, the mass-loss rate following the other prescriptions remains around log10(ṀRSG/M⊙ yr−1) ≈ −6, −5, with the star remaining in the RSG phase until the end of its evolution. This runaway behavior of ṀK21 has been found in other model tests too (Kee & Maestro Project 2024, Kee & de Koter, priv. comm.), although the exact minimum initial mass at which this occurs depends on factors that influence the RSG luminosity, such as convective overshooting during the MS, and on an even minor variation in the assumed turbulent velocity. On the other hand, for initial masses between 27 − 34 M⊙, the mass loss of ṀK21 when it enters the RSG regime is even lower than ṀJ88 (see Fig. 1 for log10(L/L⊙)∼5.6), preventing this runaway effect from occurring. In this part of the parameter space, the total mass lost with ṀK21 is relatively low, around 10–20% of the initial mass. However, for stars with even higher initial masses, they become sufficiently luminous to reach the HD limit, where their total mass lost increases again due to LBV-type winds.

thumbnail Fig. 3.

Same as Fig. 2 but for ṀB20 and ṀK21.

thumbnail Fig. 4.

The evolution of the mass-loss rate as a function of decreasing mass (from right to left) is shown for an example track with an initial mass of Minit = 12.92 M⊙, using the four different RSG mass-loss prescriptions (see legend). Points are equally spaced timesteps of 104 yr. The mass-loss rate increases when the Teff (colorbar) drops after the MS, with ṀK21 demonstrating the runaway effect of increasing mass-loss rate as the RSG mass decreases, becoming stripped and evolving bluewards in ∼7 × 104 yr.

3.2. Comparison with observations of red supergiants

To compare with observations, we simulate populations of massive single stars with POSYDON, using four grids of stellar tracks each implementing a different RSG wind. In Fig. 5 we show the expected position in the HRD of the population of RSGs. As discussed in Sect. 2.1, the contours are weighted by the IMF, the star formation history of SMC, as well as the time a RSG spends in each position in the HRD. Some indicative evolutionary tracks are also shown (one every eighth track on the grid). We also show the 930 RSG from the refined observational sample of Y23 (grey points) that are inside the limits of log10(Lmin/L⊙)∼3.9 and below log10(Teff, max/K) = 3.66.

thumbnail Fig. 5.

Hertzsprung-Russell diagram of the RSGs in the SMC. The contours show the probability density of the position of RSGs that lose mass according to ṀY23 and ṀJ88 (top panel) or ṀB20 and ṀK21 (bottom), with contour levels denoting the 68%, 95%, and 99% confidence regions. Representative stellar tracks for various initial masses are also displayed, with points corresponding to evenly spaced timesteps of 2 × 103 years. Grey points depict the observed RSGs from Yang et al. (2023) refined sample. We only show the results inside the conservative limits for a RSG of Lmin and Teff, max.

Although the mass lost from a star during its RSG phase varies significantly depending on the assumed prescriptions, RSGs will occupy a similar position in the HRD for a vast range of envelope masses (e.g., Justham et al. 2014; Farrell et al. 2020b). The key difference from the various RSG mass-loss prescriptions pertains to how long they remain as RSGs, as they become partially stripped of their H-rich envelope and evolve bluewards. The higher the mass-loss rate, the earlier in their life a RSG of the same initial mass will become stripped (see bottom row of Figs. 2 and 3). Furthermore, as the mass-loss rate is highly sensitive to the luminosity, we expect luminous RSGs (originating on average from initially more massive stars) to be more affected by the assumed wind prescription (Neugent et al. 2020). To depict the outcome of this effect on the population, and to further quantify the comparison with observations, we compute the luminosity function of RSGs in Fig. 6. The modeled and observed distributions of the effective temperatures and the mass-loss rates can be found in Fig. C.1.

thumbnail Fig. 6.

Theoretical and observed luminosity functions of RSGs from Y23. We limit our comparison to systems that eventually result in a FeCC, and for which log10(Lmin/L⊙) > Lmin, and Teff < Teff, max. The top x-axis shows the corresponding initial mass of models with an average RSG luminosity of the bottom x-axis, according to Eq. (A.1) following the best-fit parameters for ṀY23.

The luminosity function indirectly probes the (average) mass-loss rate of the RSGs during their lifetimes, not the instantaneous one (Massey et al. 2023), as it is affected by the total mass lost from a RSG and whether it is sufficient to go bluewards. This tool remains agnostic to the mechanisms of mass loss and thus can indirectly probe possible episodic mass-loss episodes throughout the RSG evolution. Thus, a stronger mass-loss rate is expected to lead to steeper luminosity functions of RSGs, because the luminous ones would leave the RSG phase even earlier. Indeed, we find that models with high RSG mass-loss rates, such as ṀK21 and even ṀY23 have a steeper decline towards high luminosities compared to ṀJ88 or ṀB20.

The observed number of the RSGs and the shape of the luminosity function are consistent with the predicted ones from ṀJ88, ṀY23 and ṀB20, for 4.6 ≲ log10L ≲ 5.0. On the high luminosity side, ṀY23 causes the predicted luminosity function to drop slightly earlier than the observed one, in contrast to the ṀJ88 and ṀK21 that result in a shallower decline, although we are in the regime of very low statistics. This is due to the “kink” in ṀY23 prescription with even higher mass-loss rates for the luminous RSGs, opposite to ṀB20. The luminosity function is depleted of luminous RSGs when we assume ṀK21 and vastly deviates from the others and the empirical one. Still, a few very luminous RSG around log10(L/L⊙)∼5.6 persist, originating from stars of Minit ∼ 27 M⊙ where a decrease in total mass lost is predicted, as discussed in Sect. 3.1.

We note that the luminosity functions in Fig. 6 are not normalized, instead they depict the expected (and observed) number of RSGs per luminosity bin. For all the models, apart from ṀK21, we find ∼1850 expected RSGs above Lmin. The predicted numbers are in tension with the 930 RSGs in the observed refined sample of Yang et al. (2023). This number can increase to 990, if we use the bolometric correction of the KS-magnitude for the luminosity estimate. In both cases, we assume a constant AV,CSM = 0.1 (an average of Yang et al. 2023 for all targets). Of course, our theoretical expectations of the total number of RSGs are sensitive to the exact shape of the IMF and the star formation history (SFH) of the SMC. The formation and interaction of binary systems would also play a role, and we further discuss this in Sec 4.1. Although the population following ṀK21 predicts ∼1231 RSGs, which seems more consistent with the observed number, as we see below this prescription cannot match the distribution of luminosities.

The observed RSGs are much fewer than predicted at low luminosities with log10(L/L⊙)≲4.6 (Fig. 6). The latter may be due to the decreasing completeness of the sample for lower RSG luminosities, even for log10(L/L⊙) > Lmin. Yang et al. (2020, 2023) point out that fainter RSGs are often missed by photometric surveys. Even when observed, they may have less reliable photometry and can be rejected as having poor quality. Furthermore, blended or unresolved sources may exist, especially in crowded regions. Y23 applied photometric quality criteria and cuts to avoid AGB contamination removing around 211 sources from their initial sample, which may include some rare dust-enshrouded RSGs (Beasor & Smith 2022), reddening their colors to resemble photometric AGBs.

We try to take into account possible bias against less luminous RSGs and any potential issues with the normalization of the luminosity function, by showing in Fig. 7 the ratio of luminous RSGs with log10(L/L⊙) > 5.0 divided by all RSGs above a base luminosity, Lbase, that we vary, as in Massey et al. (2023). For higher base luminosities, the ratio by definition increases and we expect the completeness of the observed sample to also increase. The observed ratio goes from a few percent to ∼15% for log10(Lbase/L⊙) = 4.5. ṀK21 has a sharp increase of the predicted ratio as we go to higher base luminosities, dominated by the few very luminous predicted sources in combination with the depletion of the moderate luminous ones, which are quickly stripped as they reach the RSG phase. Both ṀB20 and ṀJ88 seem to overestimate the number of luminous RSGs due to their weak mass loss, which retains them in this evolutionary phase without going bluewards. In contrast, ṀY23 appears to be consistent with this empirical ratio, especially for log10(Lbase/L⊙)≳4.5, where the predicted excess of low-luminosity RSGs is not taken into account.

thumbnail Fig. 7.

Ratio of luminous RSGs (log10(L/L⊙) > 5.0) to all RSGs above a variable base luminosity, Lbase, shown in the x-axis.

3.3. Red supergiants as Type II supernova progenitors

The different RSG mass-loss mechanisms will also affect the outcome of massive stars. In Fig. 8 we depict the region in the HRD that we expect the SN progenitors with Teff < 104 K to lie for the different RSG mass-loss assumptions. Our models reach technically up to carbon core depletion but we expect no significant change in the stellar global properties and thus in their HRD position during the remaining decades until the actual collapse, assuming no obscuration from extra pre-SN mass loss (e.g., Davies et al. 2022). The 2D histogram takes into account the weighted-with-SFHSMC IMF of the stellar progenitors, but in contrast to Fig. 5, it is not influenced by the time spent by them in their RSG phase. It only takes into account the final position of massive stars that are expected to lead to successful FeCC explosions according to the Patton & Sukhbold (2020) prescription (based on the progenitor’s carbon-oxygen core mass and central carbon 12C at the end of helium core depletion), which predicts some “windows” in the initial mass parameter space in which progenitors directly implode into a black hole. Thus, it excludes core-collapse events that do not result in observable transients due to black-hole implosion (shown with squares). In addition, as discussed in Sect. 2.1 we exclude from the analysis progenitors that are expected to produce ECSNe, although they are still shown in Fig. 8 with plus signs. The observed HRD positions of nearby Type II-P, II-L, and IIb SN progenitors are overplotted for comparison (Smartt et al. 2015). We also include a few indicative low-luminosity SN progenitors (from the sample shown in Farrell et al. 2020a), which are thought to arise from low-mass RSG progenitors (e.g., O’Neill et al. 2021; Valerin et al. 2022) and may be the progenitors of low-energetic ECSNe events (e.g., Lisakov et al. 2018).

thumbnail Fig. 8.

Hertzsprung-Russell diagram with the expected positions of the final state of the stars, before core-collapse. Successful FeCC explosions for each track (according to the SN prescription by Patton & Sukhbold 2020) are depicted with a star, implosions into a BH with probably no transient with a square, and ECSNe with plus signs. Contours represent the positions of the FeCC events only, with different colors representing the different RSG mass-loss prescriptions. We also show the position of the detected type II SN progenitors from the compilation of Farrell et al. (2020a).

Models following ṀB20 and ṀJ88 both form RSG progenitors that would be consistent with type II SN observations. For even higher luminosities, above log10(L/L⊙)∼5.25 they predict hotter progenitors, becoming even yellow supergiants (YSGs) for ṀJ88. However, most of them are presumed to not explode due to their high core masses, according to Patton & Sukhbold (2020). Thus, both prescriptions reproduce the lack of observed, luminous, high-mass RSG SN progenitors, introduced as the so-called “red supergiant problem”.

Simulations following ṀY23 are also consistent with the positions of observed progenitors up to log10(L/L⊙)∼5.2. The models diverge towards bluer final HRD positions compared to ṀJ88 and ṀB20, due to more severe stripping as we go to higher final luminosities. Still, most of them are expected to not explode according to Patton & Sukhbold (2020), predicting hotter candidates for failed FeCC compared to ṀJ88 and ṀB20. Thus, ṀY23 models are also consistent with the RSG problem. Few of these YSGs final models predicted by ṀY23 around that luminosity turn, are expected to explode, but have not been observed yet. Although the positions in the HRD of the exploding YSGs according to ṀY23 are close to the detected Type II-L SN2009kr progenitor, it is debatable whether this source is the core-collapse progenitor (Maund et al. 2015). The predicted temperature range of the YSG progenitors (∼4000–5200 K) may be consistent with the observed Type IIb progenitors, but their predicted luminosity of log10(L/L⊙)≳5.2 is 0.2 − 0.3 dex higher. We also note that predicted RSG progenitor positions, below log10(L/L⊙)∼5.1, are found slightly hotter by ≲200 K following ṀB20 compared to ṀJ88 and especially ṀY23. This is due to the higher envelope mass at the moment of collapse (see bottom figure 2 of Farrell et al. 2020a).

The high mass-loss stripping found in our models following ṀK21 is inconsistent with the formation of Type II-P and II-L SN progenitors, although it may lead to Type IIb SNe, due to the thin hydrogen-rich envelope these stars still have at the end of our simulations. However, the majority of Type IIb SNe may originate from alternative binary stripping scenarios (e.g., Nomoto et al. 1995; Claeys et al. 2011; Yoon et al. 2017; Sravan et al. 2020), as in the case of SN 2008ax (Folatelli et al. 2015) and SN 1993J (Podsiadlowski et al. 1993; Nomoto et al. 1993), for which candidate binary companions have been suggested (Maund et al. 2004; Fox et al. 2014). Possible single-star progenitors are expected to originate from a very small range of masses, as fine-tuning is needed for a thin hydrogen-rich envelope to be left, and even then it is not clear which is the fraction that undergoes direct collapse onto a black hole (Zapartas et al. 2021b).

4. Discussion

4.1. Possible binary history of red supergiants

In our analysis so far, we have overlooked the impact of binary interactions, even though most massive stars are born in binary or multiple systems, with a significant fraction of these binaries interacting during their lifetimes and initiating mass transfer (e.g., Sana et al. 2012; Almeida et al. 2017). As a result of mass transfer, stars can be stripped of their hydrogen envelopes (e.g., Paczyński 1971), which are then accreted by the companion stars, or the binary system may merge. In the former case, donor stars stripped through mass transfer might not experience a RSG phase (e.g., Götberg et al. 2017), impacting the overall shape and distribution of the RSG luminosity function. At the same time, the mass-gaining stars will form a more massive core and thus experience stronger winds due to their increased luminosity, affecting their subsequent evolution and stellar properties during the RSG phase. In the latter case, stellar mergers can also influence the RSG luminosity function as they can alter both the number of stars that undergo a RSG phase, and the evolution of the merger product (e.g., Chatzopoulos et al. 2020; Menon et al. 2024).

Indeed, it is now established that binary history can explain many observed features of the RSG population, including runaway RSGs (mass gainers ejected from a prior SN, Comerón & Figueras 2020) and the presence of “red stragglers” (Britavskiy et al. 2019; Wang et al. 2025). In addition, according to de Mink et al. (2014), about 15% of the observed massive blue stars are predicted to be merger products, with many of them expected to become RSG later in their evolution. This is further supported by studies that predict a significant fraction of binary products in the population of RSG that are Type II SN progenitors (Podsiadlowski et al. 1992; Zapartas et al. 2019), including mass-gaining stars, mergers of two MS stars (MS+MS), or mergers involving at least one post-MS star (post-MS merger, e.g., Blagorodnova et al. 2021).

Here, we aim to provide a rough estimate of how these different binary evolution scenarios influence the luminosity function of RSGs, by evolving a population of massive binary systems according to POSYDON v2 (Andrews et al. 2024). Knowing that RSG luminosity primarily depends on the helium core mass formed at the end of the MS phase (referred to as terminal-age MS (TAMS) and defined in our models when hydrogen central mass fraction drops below 1%; Farrell et al. 2020b; Schneider et al. 2024), we use the mass of each binary product at that point, MTAMS, as an indicator of its eventual post-MS helium core mass formed and thus its LRSG. This approach takes into account the increase in mass during the MS of the star (through MS+MS mergers or accretion onto MS stars scenarios), because a high level of mixing is expected for them due to the absence of a significant chemical boundary between the core and the formed envelope, and thus a higher helium core mass during its eventual RSG phase (e.g., Braun & Langer 1995; Glebbeek et al. 2013; Schneider et al. 2024). On the other hand, we ignore further mass accretion after TAMS, which is not expected to significantly increase the core mass any further. So, for RSGs resulting from post-MS merging, we treat the merger products as single stars with the same core mass as the giant component. We also exclude donor stars that become stripped of their envelopes after expanding beyond the TAMS and thus never become RSGs. Specifically, we exclude those that do not reach a minimum effective temperature of log10Teff, min = 3.66 at any point during their life. In this calculation, we neglect any potential binary stripping during the RSG phase. Our exploratory estimate eliminates the need for new stellar models that simulate the complex stellar structures expected in post-MS mergers. However, accurately incorporating merger models for the other two scenarios, which involve one or two evolved stars, requires a detailed simulation of merger products (e.g., Menon et al. 2024; Schneider et al. 2024). This is beyond the scope of this study and will be addressed in future work.

We assign to each remaining binary product a corresponding single-star initial mass using a fitting relation between MTAMS and Minit of POSYDON single-star models (Eq. (D.1)). In practice, we rearrange the IMF to approximate the effect of binarity on the luminosity distribution of its RSGs. To further weight each star with the SFH of the SMC, we assume that the time needed for the binary product to reach TAMS is also the time of its RSG phase (as the evolution from TAMS to the RSG phase is negligible and the duration of the RSG phase is short relative to the total stellar lifetime). The latter assumption is an acceptable approximation as the RSG timescale is much smaller than the bins in the SFH distribution we used as described in Sect. 2. Thus, a star that increased in mass during MS due to binary interaction (and still became RSG) would correspond to a higher initial mass IMF, but will be assigned to the weight of the SFH according to its TAMS time.

With the above calculation, we can effectively estimate the SFH-weighted IMF for future RSGs from binary products, relative to a population of RSGs from only single stars (Fig. 9). The peak of both distributions for 0.95 ≲ log10(Minit/M⊙)≲1.2 is due to the weighting of SMC SFH which boosts stellar sources with ages of 14–40 Myr (Sect. 2). Binary stars that reach TAMS and are expected to produce a RSG broaden the relative SFH-weighted IMF. This happens due to a boost at low initial masses of log10(Minit/M⊙)∼0.9 from actually intermediate-mass stars that, by accreting or merging, they become massive enough to explode. Most of these intermediate-mass progenitors are binary accretors that become massive enough at TAMS to become RSGs eventually. At the same time, mass accretion or MS+MS merging to higher masses before TAMS leads to a rearrangement of the relative IMF, slightly decreasing the contribution around the peak towards higher initial masses log10(Minit/M⊙)≳1.2. This is consistent with the slight shift of the default distribution of final core masses towards more massive ones compared to a canonical IMF, found in Zapartas et al. (2021a, Fig. 5).

thumbnail Fig. 9.

Normalized SFH-weighted IMF for massive stars that reach TAMS and eventually become RSGs, for a single star population (grey) and from one where binarity is taken into account (orange). For binaries that at some point become RSGs, Minit is calculated based on their MTAMS (Eq. (D.1)), taking into account possible accretion (or merging) during their MS.

The impact of this relative change of the binary-corrected IMF of stars that eventually become RSGs, leads to a slightly different RSG luminosity function, seen in Fig. 10. The general shapes stay similar although the boost above log10(Minitial/M⊙)≳1.2 leads to a slight increase in the fraction of luminous RSGs for all assumed RSG mass-loss prescriptions (Fig. 11). This does not help in reconciling the discrepancy of the ratio according to ṀK21, ṀB20 and ṀJ88. ṀY23 still stays closer to the observed ratio for different Lbase values, especially at higher base luminosities (going to the right of this plot) where the ratio should be less biased to the completeness of the sample. At the same time, binarity cannot reconcile the discrepancy with observations at the low-luminosity regime.

thumbnail Fig. 10.

Same as Fig. 6 but with a binary-corrected relative IMF for RSG progenitors, as shown in Fig. 9.

thumbnail Fig. 11.

Same as Fig. 7 but with a binary-corrected relative IMF for RSG progenitors, as shown in Fig. 9.

4.2. Predictions for post-red supergiant stars

Large spectroscopic samples of RSGs and YSGs are scarce. Empirical estimates of the YSG-to-RSG ratio (YSG:RSG) typically rely on color-magnitude diagrams (e.g., Yang et al. 2021) and machine learning techniques applied on the larger, photometric samples (e.g., Maravelias et al. 2022; Dorn-Wallenstein et al. 2023). In a sample of 4000 evolved massive stars, Yang et al. (2021) estimated their YSG:RSG ∼17%. The machine-learning approach of Maravelias et al. (2022), trained on the colors of the stars, predicts a YSG:RSG of 2 − 8%. In our models, we find a YSG:RSG of ≲1%, ∼2.0%, ∼2.9% and ∼14% using ṀY23, ṀJ88, ṀB20 and ṀK21 respectively. The ratio due to the high level of stripping from ṀK21 is closer to the empirical values, although including the effects of stripping in binaries (an efficient scenario to form YSG Type IIb SN progenitors; Yoon et al. 2017; Sravan et al. 2018) or of episodic mass loss when reaching the ionization bump below log10(Teff/K)∼4 (Cheng et al. 2024) may drastically increase the theoretical ratio.

The stellar properties of YSGs that are formed after their RSG phase significantly differ from their pre-RSG counterparts (Dorn-Wallenstein et al. 2022; Humphreys et al. 2023), as a result of changes in the stellar core and subsequent mixing of material to the surface (for instance dredge ups), and mass loss through stellar winds affecting the mass and surface gravity. Assuming post-RSG YSGs are those showing short-period pulsations, Dorn-Wallenstein et al. (2022) suggest that 33% of the YSGs above log10(L/L⊙)∼5.0 are post-RSGs. Similarly, based on their expected infrared excess Gordon et al. (2016) found 30–40% of the YSG population to be post-RSG. From the evolutionary tracks, we find the ratio of post-RSG YSGs above log10(L/L⊙)∼5.0 compared to all YSG to be ∼16.7%, ∼5%, ≲1%, and ∼23.1% following ṀY23, ṀJ88, ṀB20 and ṀK21 respectively. Thus again, the degree of stripping predicted by the ṀY23 best explains the observed pre-to-post YSG ratio, neglecting again other mass-loss mechanisms such as binary stripping or episodic mass loss.

4.3. Tension with observations and caveats

The implemented RSG mass-loss prescriptions can reproduce various features of the tail of the luminosity function and the ratio of luminous RSGs, but are inconsistent with other aspects (Sect. 3.2). In general, the high mass-loss rate of ṀK21 appears inconsistent with the observed luminosity function, at least given the fixed set of the other stellar physical assumptions of POSYDON (discussed below). We note that severe stripping following ṀK21 would occur only for higher initial masses, for a different model-dependent Minit − LRSG relation (see the difference in the luminosities in our models compared to the ones used by Kee et al. (2021) in Fig. A1). Equivalently, a vturb value lower than 18.2 km/s would have resulted in weaker stripping, as the prescription is highly sensitive to this parameter.

The predicted mass-loss rates especially at high luminosities significantly affect the timespan of the RSG phase, thus the occurrence rate of luminous RSGs. The curved relation for RSG lifetime with initial mass for ṀY23 (bottom left panel of Fig. 2) originates due to the change of the slope in ṀY23 prescription with luminosity, with even higher mass-loss rates for the luminous RSGs (Fig. 1). This “kink” has been described in Y23, and is also found in the more recent Antoniadis et al. (2024) for the Large Magellanic Cloud. An effectively opposite change in slope is found in ṀB20, due to the dependence on initial mass, thus indirectly on RSG luminosity. The effect of this is that ṀJ88 and especially ṀB20 slightly overpredict the relative ratio of luminous RSGs (Fig. 7). These discrepancies seem to worsen when considering the effects of possible binary interactions of stars that eventually become RSGs (Sect. 4.1). A possible shallower IMF for SMC (e.g., Schneider et al. 2018) that increases the formation rate of massive stars would increase the ratio too, making the discrepancy even worse. On the other hand, ṀY23, which is based on the Y23 observed RSG sample that we compare with, seems to be the most consistent with the ratio of luminous RSGs. Interestingly, the lack of very luminous RSG due to ṀY23 stripping seems to also naturally reproduce the HD limit (Humphreys & Davidson 1979) that has been updated around log10(L/L⊙)∼5.5 for the Magellanic Clouds (Davies et al. 2018) and M31 (McDonald et al. 2022), without invoking an extra mass-loss mechanism (which in our models is triggered only at log10(L/L⊙)∼5.78, see Sect. 2.2). In addition, its high mass-loss rates would produce more Wolf-Rayet stars stripped by winds, consistent with recent observations at the SMC (Schootemeijer & Langer 2018; Schootemeijer et al. 2024).

In contrast, in the context of SN progenitors, the high mass-loss rate of ṀK21 faces difficulties in reproducing their position on the HRD due to the extreme stripping occurring even within a brief RSG phase. Conversely, ṀY23 can reproduce the observed SN progenitor’s positions. We note that a high mass-loss rate can be the cause of the high rate of observed events with low ejecta masses (Martinez et al. 2022). In addition, due to the stripping of luminous RSGs, ṀY23 predicts luminous YSG progenitors that are not observed in nearby SNe (Sect. 3.3). On the other hand, ṀJ88 and ṀB20 are consistent with these detections.

Consequently, we find no single prescription that can reproduce all the observational constraints that we considered at the same time. We speculate that a RSG prescription with low on-average mass-loss rates but with a fast increase of the mass-loss rates towards high luminosities, may be needed to reproduce both the RSG luminosity function and the detected Type II SN progenitors.

In the analysis above, we acknowledge the presence of various uncertain physical effects and assumptions in our work, as well as in prior studies, which ought to be kept in mind and whenever possible carefully investigated. The most significant of these factors are discussed below.

In the process of empirically inferring the mass-loss rates from fitting the infrared excess of the RSGs SEDs, various assumptions are made (which are not applicable for ṀK21 that is theoretically derived). ṀY23 assumed radiatively driven winds in DUSTY, where dust acts as the accelerating mechanism of mass loss. This assumption can lead to deriving higher mass-loss rates by two to three orders of magnitude than assuming a steady wind with a constant outflow velocity, where dust is considered a byproduct of the RSG mass loss (Antoniadis et al. 2024). Beasor et al. (2020) used steady-state winds which is one of the reasons that the rates are not as high as ṀY23. The variation in grain sizes in the dust shell models could also lead to different rates by a factor of up to 20–30 (Antoniadis et al. 2024).

In addition, when modeling a dust shell, spherical symmetry is usually assumed. However, dust around RSGs is clumped and asymmetric (Smith et al. 2001; Scicluna et al. 2015). Luminosity measurements could be affected by dense shells, resulting in higher or lower observed luminosities depending on clump orientation. Furthermore, photometric variability in RSGs can induce luminosity variations of up to 0.2 dex (Beasor et al. 2021) due to obscuration (Beasor & Smith 2022). These effects influence the dusty RSGs at the high luminosity end, affecting the tail of the luminosity function and ratio of luminous RSGs.

At the same time, the dust occurrence and its effect in the stellar SED probes only the recent mass-loss rate of RSGs, which could be considered instantaneous relative to the overall evolution of the star. In the case of abrupt and episodic mass-loss phenomena (Montargès et al. 2021; Munoz-Sanchez et al. 2024), the inferred rates would not be valid for the entire RSG lifetime, and implementing them as an average over the RSG lifetime overestimates the total mass lost.

The mass-loss rate prescriptions are implemented in POSYDON stellar models of MESA, so the results are affected by the assumptions in the physical processes during their evolution (explained in detail in Fragos et al. 2023; Andrews et al. 2024). Notably, the single-star models used are assumed to be non-rotating, avoiding any extra rotationally induced mixing throughout the evolution. At the same time, exponential convective core overshooting has been assumed, calibrated according to Brott et al. (2011b), which is tailored specifically for massive stars. This assumed overshooting is consistent with the investigation of mixing processes for massive stars in the SMC from Schootemeijer et al. (2019), Gilkis et al. (2021), Sabhahit et al. (2021) (although our assumed semiconvection parameter αsc = 0.1 is on lower than the suggested constraints in those studies). At the same time, this core overshooting is on the high side compared to other stellar evolution tracks for RSGs, e.g., GENEC (Ekström et al. 2012) and MIST (Choi et al. 2016), leading to high helium core masses and thus RSG luminosities in our study (e.g., Farrell et al. 2020b), and consequently in high average mass-loss rates for the same initial mass. The positions of the SN progenitors, as well as those expected to collapse into BHs are consistent with the recent investigation of how core overshooting affects the final fate of massive stars (Temaj et al. 2024).

At the same time, the surface temperatures of the convective envelopes of RSGs are governed by their evolution on the Hayashi tracks (unless they experience a blue loop to a radiative envelope or stripping of the envelope to go to the blue). The exact RSG structure for a given luminosity is dependent on various physical assumptions, most importantly the treatment of convective instability through mixing length theory (Meynet et al. 2015). Thus, the effective temperatures of RSGs in our results should be sensitive to the adopted mixing length parameter, αMLT = 1.93 (Fragos et al. 2023).

It is important to note that the observed SN progenitor sample includes stars of varying metallicities with generally higher metallicity than solely SMC-like. Additionally, the binary history of a significant fraction of Type II SN progenitors could affect their final position on the HRD (Podsiadlowski et al. 1992; Zapartas et al. 2019; Menon et al. 2019, 2024; Schneider et al. 2024). Furthermore, possible obscuration of the SN progenitor due to pre-SN episodic mass loss (Arnett & Meakin 2011; Fuller 2017) almost synchronized with the SN explosion may also contribute to the observed discrepancies (Davies et al. 2022). Evidence of such mechanisms has been observed, for example in the recent case of SN2023ixf (Jacobson-Galán et al. 2023; Kilpatrick et al. 2023; Bostroem et al. 2023). The termination of our models at core carbon depletion, with less than a few or tens of years until core collapse, does not account for any potential episodic mass loss that may occur during this remaining time. In any case, this is an uncertain mechanism with no well-constrained prescriptions regarding the probability of such events and the resulting mass-loss rate.

4.4. Feedback

Massive stars play a crucial role in the evolution of galaxies by providing mechanical feedback and chemically enriching the interstellar medium (e.g., Barkana & Loeb 2001). For stars below Minit ∼ 25 M⊙ the ejected mass before the explosion is dominated by the RSG phase (Figs. 2 and 3). In Fig. 12 we calculate the total mass ejected from a stellar population as described in Sect. 2.1, that is, weighted by the IMF and following the SFH of SMC. Different RSG wind prescriptions modify the feedback in several ways, influencing not only the total mass ejected during their pre-SN evolution, but also the initial masses that contribute most significantly and thus the timing of the feedback too as it depends on the evolutionary timescales of these masses. For example, following the high mass-loss rate prescriptions of ṀY23 and ṀK21, the most important contributors in the SMC are the stars of Minit ∼ 8 − 15 M⊙ range, as these are favoured by the IMF. However, when implementing weaker wind prescriptions such as ṀB20 and ṀJ88, the dominant contribution shifts to stars with initial masses of ∼15 − 20 M⊙ models. This shift occurs because lower-mass stars have smaller fractional mass loss according to these prescriptions (Figs. 2 and 3), and more massive are rare due to the IMF.

thumbnail Fig. 12.

Total mass lost during the stellar pre-SN lifetime, per unit initial mass bin, weighted with the IMF and following the SMC SFH.

The SN explosions are another important source of feedback, which can drive outflows on galactic scales (e.g., Shapiro & Field 1976; Hopkins et al. 2011), and may trigger further star formation (Krumholz & McKee 2005). ṀK21 but also ṀY23 predict smaller final hydrogen-rich envelope masses, diminishing the amount of mass expected to be ejected during the SN itself. This implies that, from a feedback perspective, the question of which stars will explode becomes less critical when we assume stronger RSG wind prescriptions, as most of the mass is ejected before the SN occurs.

5. Summary and conclusions

In this work, we investigate the impact of different red supergiant (RSG) mass-loss on the evolutionary pathways and observable properties of massive stars in the Small Magellanic Cloud (SMC), and their eventual roles as supernova (SN) progenitors. We achieved this by implementing in grids of single-stellar tracks within the POSYDON framework, various RSG mass-loss prescriptions, inferred with different techniques and samples, and resulting in a wide range of mass-loss rates.

We find that RSG mass-loss prescriptions significantly affect the mass lost and the duration of the RSG phase. Higher mass-loss rates, such as in Kee et al. (2021) or even Yang et al. (2023) for the luminous ones, result in earlier envelope stripping and reduced RSG lifetimes, whereas lower rates, such as in Beasor et al. (2020), prolong the RSG phase. The reason is that higher mass-loss rates cause RSGs to transition to a hotter phase at lower initial masses, down to 15 M⊙ for Yang et al. (2023). These stripped stars also avoid becoming progenitors of Type II SNe, influencing their final position in the Hertzsprung-Russell diagram and determining whether they will explode or implode directly as a black hole.

Our findings indicate that none of the considered mass-loss prescriptions are fully consistent with all observational constraints simultaneously, highlighting the empirical complexities and theoretical uncertainties inherent in modeling RSGs. All the incorporated prescriptions have difficulty reconciling the observed distribution of RSG luminosities across the entire spectrum, particularly at lower luminosities, although this may be sensitive to observational biases. The high mass-loss rates suggested by Kee et al. (2021) result in intensive stripping of the stellar envelope, leading to a predicted dearth of luminous RSGs and an excess of YSG SN progenitors, which are not commonly observed as progenitors of nearby supernovae. Conversely, the de Jager et al. (1988) and Beasor et al. (2020) prescriptions slightly overestimate the number of luminous RSGs. Given a crude estimate of the effect of binarity, the above discrepancy would increase due to even more luminous binary products that will become RSGs. The empirical prescription of Yang et al. (2023) seems more consistent with the observed luminosity function, naturally reproducing the updated Humphreys-Davidson limit of log10(L/L⊙)∼5.5 for the Magellanic Clouds (Davies et al. 2018) and M31 (McDonald et al. 2022). This highlights the possible importance of a turning point to increased mass-loss rates for luminous RSGs, as suggested also by Vink & Sabhahit (2023) and Antoniadis et al. (2024). In contrast, the stripping of the highly luminous RSGs according to Yang et al. (2023), which predicts YSG SN progenitors, seems inconsistent with detected ones.

In conclusion, our study highlights the importance of constraining RSG mass loss with more complete observational samples especially at lower luminosities, and a clear investigation of the model uncertainties (e.g., Antoniadis et al. 2024), the dependence on other parameters such as the gas-to-dust ratio, the outflow speed (e.g., Goldman et al. 2017) or the stellar mass (Beasor et al. 2020). Further observational constraints (e.g., Decin et al. 2024) are crucial for refining the important theoretical attempts to model RSG mass loss (Kee et al. 2021; Vink & Sabhahit 2023; Fuller & Tsuna 2024) and eventually understanding the causing mechanism. Finally, studies of the effect of possible binary-induced or episodic mass loss (Cheng et al. 2024), during the RSG phase or before the SN, would enhance our insight into the late stages of massive stellar evolution.

2

As noted in Antoniadis et al. (2024), there is a discrepancy between the equation published in Y23 and the correct value which would yield higher mass-loss rates by 0.7 dex, but we followed the published version for consistency.

Acknowledgments

The authors thank Ming Yang for helpful discussions and for providing the data of the 2023 work he led, prior to publication. They also thank Dylan Kee, Alex de Koter, Jakub Klencki, and Evangelia Christodoulou for useful discussions. EZ and DS acknowledge support from the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “3rd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers” (Project No: 7933). EZ, SdW, KA, GMS, AB, GM acknowledge funding support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (“ASSESS”, Grant agreement No. 772086). The POSYDON project is supported primarily by two sources: the Swiss National Science Foundation (PI Fragos, project numbers PP00P2_211006 and CRSII5_213497) and the Gordon and Betty Moore Foundation (PI Kalogera, grant award GBMF8477). MB acknowledges support from the Boninchi Foundation. KAR is also supported by the Riedel Family Fellowship and thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant No. 1829740, the Brinson Foundation, and the Moore Foundation; their participation in the program has benefited this work. KK is supported by a fellowship program at the Institute of Space Sciences (ICE-CSIC) funded by the program Unidad de Excelencia María de Maeztu CEX2020-001058-M. JJA acknowledges support for Program number (JWST-AR-04369.001-A) provided through a grant from the STScI under NASA contract NAS5-03127. ZX acknowledges support from the China Scholarship Council (CSC).

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Appendix A: Average red supergiant luminosity as a function of initial mass

In Fig. A.1 we show the luminosity of stars during their post-MS evolution. Each streak of vertical dots represents the luminosity during the RSG phase for a given Minit in 100 points equally distributed in time. We also show the luminosity of that model at TAMS, just before becoming a RSG. The luminosity shows an increase from TAMS to around ≳0.5 dex when it reaches the RSG phase and then quickly increases a bit. The star keeps an almost constant luminosity during the longer timescale of core helium burning, increasing again only after helium depletion (for initially lower mass stars) or after being stripped and ending their RSG phase (for initially higher mass stars). During that phase, the RSG may change its temperature (even doing a blue loop), and thus part of its evolution with Teff > Tmin will not be included.

We fit the time-averaged LRSG, avg as function of Minit, using the same equation as in Kee et al. (2021):

L RSG , avg = f · ( M init / M ⊙ ) α $$ \begin{aligned} L_{\rm RSG,avg}=f\cdot (M_{\rm init}/M_{\odot })^{\alpha } \end{aligned} $$(A.1)

The relation they use in that work has α = 3, with the factor f ranging from 12.5 for the GENEVA models (Ekström et al. 2012) to 18.5 for the default MESA models (Paxton et al. 2011), eventually picking an average value. We show these relations in Fig. A.1. In this study, following ṀJ88, we fit the on average higher luminosity values during the RSG phase, resulting in a shallower slope of 2.27 and a much higher f = 208.6.

thumbnail Fig. A.1.

Luminosity during the RSG phase as a function of the initial mass of the star. The points are distributed in 100 equal timesteps that span the whole time range of each model during its RSG phase, colored by the central helium mass fraction. For reference, we also show the luminosity at TAMS and various fittings from this or previous works, of the form of Eq. A.1.

Interestingly, we found the relation not to be highly dependent on the RSG mass-loss prescription applied, especially the slope α which takes the value of 2.25, 2.26, 2.18, and f factor becoming 225, 214.9, 294.1 for the ṀY23, ṀB20, ṀK21, respectively. This is probably caused by the fact that luminosities are more tightly correlated to the helium core mass of the star, which is formed at TAMS and relatively unaffected by the amount of mass of the surrounding envelope (Justham et al. 2014; Farrell et al. 2020b). Of course, it is still sensitive to the model’s uncertain physical assumptions, including convective overshooting and rotational mixing.

We confirm the loose correlation of the initial or current total mass of a RSG with its luminosity, but here we want to correspond a time-averaged luminosity during the RSG phase for a given initial mass. However, as discussed in Sect. 3.1 the time spent during the phase is highly influenced by the wind mass-loss prescription.

Appendix B: Absolute red supergiant lifetime

In Fig. B.1 we present the absolute total time spent during the RSG phase as a function of initial mass for different mass-loss prescriptions. For low initial masses, the RSG phase constitutes almost the entire post-MS lifetime, but for higher initial masses the stripping of the stars reduces the time spent on the cool side of the HRD, as also illustrated in Fig. 2 and 3).

thumbnail Fig. B.1.

Absolute time spent during the RSG phase, as a function of initial mass of our mdoels, for different implemented mass-loss prescriptions. We also show the total post-MS lifetime (dotted line), found almost independent of the mass loss during the RSG phase.

Appendix C: Temperature and mass loss of red supergiants

In Fig. C.1 we show the effective temperature and the mass-loss rate of RSGs for the different prescriptions, compared with the RSG sample of Y23, with the limits as discussed in Sect. 2.3. Teff, RSG of the observed sample is inferred from J − KS color of the sample, using relation from Yang et al. (2020) and Britavskiy et al. (2019).

It is also interesting that models following ṀY23 have a more peaked mass-loss rate distribution than the observationally inferred one of Y23, where this prescription was based. This is mainly because the prescription is only luminosity dependent and thus cannot fully capture the spread of the inferred mass-loss rates of the sample for a given luminosity (as depicted in Fig. 1).

thumbnail Fig. C.1.

Normalized distributions of a) effective temperatures b) mass-loss rates of the POSYDON populations of RSGs, for all the different mass-loss prescriptions, compared against the observationally inferred values.

Appendix D: Relation between initial and terminal-age main-sequence mass

In Fig. D.1 we present the relation between initial and mass at TAMS for our POSYDON models which was used in Sect. 4.1 to take into account binary products in the luminosity function of RSGs. We choose MTAMS as a proxy of the core mass and luminosity during the RSG phase for binary products, and we extract the following fitting relation with Minit :

M init / M ⊙ = 0.391 + 0.923 · ( M TAMS / M ⊙ ) + 0.0034 · ( M TAMS / M ⊙ ) 2 $$ \begin{aligned} M_{\rm init}/{M}_{\odot } = 0.391 + 0.923\cdot (M_{\rm TAMS}/{M}_{\odot }) + 0.0034\cdot (M_{\rm TAMS}/{M}_{\odot })^2 \end{aligned} $$(D.1)

thumbnail Fig. D.1.

Relation between initial and mass at the TAMS for our POSYDON models.

All Figures

thumbnail Fig. 1.

Mass-loss rate versus luminosity of the RSG wind prescriptions implemented. Points are random draws of RSGs from POSYDON stellar populations, following each prescription (see legend) and grey crosses are the Y23 observational sample. The top x-axis corresponds to the initial mass of a star to have LRSG, avg equal to the bottom axis, according to Eq. (A.1).
In the text
thumbnail Fig. 2.

Evolution during different phases. Top: Indicative tracks of equal timesteps of 1kyr in the HRD with ṀY23 (left) and ṀJ88 (right), depicting the definition of each evolutionary phase. The initial mass is mentioned at the beginning of each track. The HD limit, where LBV-type winds kick in is also shown (grey dot-dashed line). Middle: Part of the stellar mass lost in each evolutionary phase, as a function of initial mass (evolution goes from top to bottom). We also show the value of the final helium core mass as a function of the initial (purple dotted line). Bottom: Fractional time spent in each evolutionary phase after the end of MS, as a function of initial mass (evolution goes from top to bottom).
In the text
thumbnail Fig. 3.

Same as Fig. 2 but for ṀB20 and ṀK21.
In the text
thumbnail Fig. 4.

The evolution of the mass-loss rate as a function of decreasing mass (from right to left) is shown for an example track with an initial mass of Minit = 12.92 M⊙, using the four different RSG mass-loss prescriptions (see legend). Points are equally spaced timesteps of 104 yr. The mass-loss rate increases when the Teff (colorbar) drops after the MS, with ṀK21 demonstrating the runaway effect of increasing mass-loss rate as the RSG mass decreases, becoming stripped and evolving bluewards in ∼7 × 104 yr.
In the text
thumbnail Fig. 5.

Hertzsprung-Russell diagram of the RSGs in the SMC. The contours show the probability density of the position of RSGs that lose mass according to ṀY23 and ṀJ88 (top panel) or ṀB20 and ṀK21 (bottom), with contour levels denoting the 68%, 95%, and 99% confidence regions. Representative stellar tracks for various initial masses are also displayed, with points corresponding to evenly spaced timesteps of 2 × 103 years. Grey points depict the observed RSGs from Yang et al. (2023) refined sample. We only show the results inside the conservative limits for a RSG of Lmin and Teff, max.
In the text
thumbnail Fig. 6.

Theoretical and observed luminosity functions of RSGs from Y23. We limit our comparison to systems that eventually result in a FeCC, and for which log10(Lmin/L⊙) > Lmin, and Teff < Teff, max. The top x-axis shows the corresponding initial mass of models with an average RSG luminosity of the bottom x-axis, according to Eq. (A.1) following the best-fit parameters for ṀY23.
In the text
thumbnail Fig. 7.

Ratio of luminous RSGs (log10(L/L⊙) > 5.0) to all RSGs above a variable base luminosity, Lbase, shown in the x-axis.
In the text
thumbnail Fig. 8.

Hertzsprung-Russell diagram with the expected positions of the final state of the stars, before core-collapse. Successful FeCC explosions for each track (according to the SN prescription by Patton & Sukhbold 2020) are depicted with a star, implosions into a BH with probably no transient with a square, and ECSNe with plus signs. Contours represent the positions of the FeCC events only, with different colors representing the different RSG mass-loss prescriptions. We also show the position of the detected type II SN progenitors from the compilation of Farrell et al. (2020a).
In the text
thumbnail Fig. 9.

Normalized SFH-weighted IMF for massive stars that reach TAMS and eventually become RSGs, for a single star population (grey) and from one where binarity is taken into account (orange). For binaries that at some point become RSGs, Minit is calculated based on their MTAMS (Eq. (D.1)), taking into account possible accretion (or merging) during their MS.
In the text
thumbnail Fig. 10.

Same as Fig. 6 but with a binary-corrected relative IMF for RSG progenitors, as shown in Fig. 9.
In the text
thumbnail Fig. 11.

Same as Fig. 7 but with a binary-corrected relative IMF for RSG progenitors, as shown in Fig. 9.
In the text
thumbnail Fig. 12.

Total mass lost during the stellar pre-SN lifetime, per unit initial mass bin, weighted with the IMF and following the SMC SFH.
In the text
thumbnail Fig. A.1.

Luminosity during the RSG phase as a function of the initial mass of the star. The points are distributed in 100 equal timesteps that span the whole time range of each model during its RSG phase, colored by the central helium mass fraction. For reference, we also show the luminosity at TAMS and various fittings from this or previous works, of the form of Eq. A.1.
In the text
thumbnail Fig. B.1.

Absolute time spent during the RSG phase, as a function of initial mass of our mdoels, for different implemented mass-loss prescriptions. We also show the total post-MS lifetime (dotted line), found almost independent of the mass loss during the RSG phase.
In the text
thumbnail Fig. C.1.

Normalized distributions of a) effective temperatures b) mass-loss rates of the POSYDON populations of RSGs, for all the different mass-loss prescriptions, compared against the observationally inferred values.
In the text
thumbnail Fig. D.1.

Relation between initial and mass at the TAMS for our POSYDON models.
In the text

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