Sébastien Martinet
email: sebastien.martinet@unige.ch

Georges Meynet

Devesh Nandal

Sylvia Ekström

Cyril Georgy

Lionel Haemmerlé

Raphael Hirschi

Norhasliza Yusof

Matthieu Gounelle

Vikram Dwarkadas

Very massive star winds as sources of the short-lived radioactive isotope 26 Al

Keywords: Stars: evolution, Stars: rotation, Stars:massive, Stars: abundances, Galaxy: abundances

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Introduction

26 Al holds a special position among the short-lived radioisotopes with lifetimes shorter or equal to about 1 Myr. It is, with 60 Fe, the only element emitting a gamma-ray line that has been observed as a diffusive emission coming from the disk of our Galaxy. The origin of the gamma-ray emission is the decay of the ground state of 26 Al with a half-life of 0.717 Myr [START_REF] Norris | Lunar and Planetary Science Conference Proceedings[END_REF]) into an excited state of 26 Mg, the de-excitation of which emits a 1.805 MeV photon.

The emission line was first detected by the HEAO-C satellite (Mahoney et al. 1982(Mahoney et al. , 1984)), then confirmed by balloon born experiments (Varendorff & Schoenfelder 1992) and later-on by the ACE satellite (Yanasak et al. 2001). The first map of the Milky Way in this line has been obtained by the CGRO Comptel (von Ballmoos et al. 1987;Chen et al. 1995). Subsequent new maps have been obtained by INTEGRAL (Diehl et al. 2003). Studying emission line intensity and adopting reasonable assumptions about the distribution of 26 Al in the Galaxy, a total amount between 1.7 and 3.5 M of 26 Al has been estimated to be present in the Milky Way today (Knödlseder 1999;Diehl et al. 2006;Wang et al. 2009), with current best estimate of 2 M (Pleintinger 2020). Interestingly, a comparison between the 1.8 MeV intensity map and other maps at different wavelengths showed that the best correspondence is obtained with the free-free emission (Knödlseder et al. 1999). This clearly points to ionized regions where hot stars are present, and thus favors short-lived massive stars as the main contributors to this 26 Al.

The amount of galactic 26 Al likely remains constant with time, since we can reasonably assume that over the whole Galaxy and on a timescale that covers a few 26 Al lifetimes, i.e. a few Myr, a stationary equilibrium is reached between the production and destruction rate of this isotope. Indeed, the change of the mass of 26 Al in the Galaxy, M 26 , can be written

dM 26 dt = P 26 - M 26 τ 26 ,
where P 26 is the rate of production of 26 Al and M 26 τ 26 its decay rate with τ 26 the decay constant. We see that when, for instance, M 26 is so low that the right-hand side is positive, then 26 Al will accumulate in the ISM. In the reverse case, M 26 is so high that the right-hand side is negative. Thus, on timescales of a few τ 26 , an equilibrium is reached between the production and the destruction rate of 26 Al. This implies that at any time, one can reasonably assume that M 26 ∼ P 26 τ 26 (see the question about the granularity of nucleosynthesis in Meyer & Clayton 2000). The 26 Al whose decay can be observed as a diffuse γ-ray emission in the Galaxy today needs to have been ejected into the interstellar medium in the last million years. Thus, the 1.8 MeV observation is a measure of the recent nucleosynthetic activity (of at least some specific sources) in our Galaxy (see e.g. the reviews by Diehl 2021).

To estimate P 26 , we must identify the sources of 26 Al. Massive stars can contribute to the 26 Al budget through their winds and at the moment of their explosion as a supernova. Grids of models predicting the quantities ejected by stellar winds (Prantzos & Casse 1986;Meynet et al. 1997;Palacios et al. 2005), by winds and supernovae (Limongi & Chieffi 2006), and more recently by winds in close binaries (Brinkman et al. 2019) have been published. Other sources have been explored such as supermassive stars (stars above 10 4 M Hillebrandt et al. 1987), Asymptotic Giant Branch (AGB) stars (Wasserburg et al. 2006;Mowlavi et al. 2005) and novae (Bennett et al. 2013). The AGB, however, are considered nowadays as likely minor contributors.

Based on the prevalence of massive young stars as 26 Al sources, models for specific young star forming regions like the Carina and the Cygnus regions for which maps can be obtained (Knoedlseder et al. 1996;Knödlseder et al. 2002;Bouchet et al. 2003), or more generally for starbursts regions have also been built (del Rio et al. 1996;Cerviño et al. 2000;Higdon et al. 2004;Rothschild et al. 2006;Voss et al. 2009;Lacki 2014;Krause et al. 2015), showing a reasonable agreement with observations. Of course, a still better constraint would be to see the emission of one source. Some trials have been done to detect a signal from the nearest Wolf-Rayet star γ Velorum (Oberlack et al. 2000) but without success. In case of a point-like source, models would however predict enough flux to be detectable. An explanation for the non-detection might be that due to the velocities of the stellar winds, the 26 Al is actually distributed rapidly enough in a large area around the star, which would then make the detection much harder (Mowlavi et al. 2005;Mowlavi & Meynet 2006). Indeed, the minimum level for detecting an emission is larger for a source diluted over a large area than for a point-like source.

Recent papers, that have simulated galactic-scale 26 Al maps (Pleintinger et al. 2019), have also explored the contribution of massive stars towards 26 Al and 60 Fe at the scale of the Galaxy (Wang et al. 2020). Furthermore, questions such as how 26 Al can trace metal losses through hot chimneys (Krause et al. 2021), or how it is distributed in superbubbles (Rodgers- Lee et al. 2019) have also been explored. All these works are based on 26 Al masses ejected by stars with initial masses between 25 and around 120 M . However, stars more massive than 120 M up to at least ∼ 300M likely exist (Crowther et al. 2010;Bestenlehner et al. 2020;Brands et al. 2022) and their extreme evolution and large mass loss rates could make them important contributor to the 26 Al production in the Galaxy. In the present paper, we explore the contribution of these very massive stars in enriching the interstellar medium in 26 Al through their winds. Although these stars are very rare (see Sect. 6.2), depending on the mass loss rates they exhibit, their contribution might be important.

The paper is organized as follows: in Section 2, we briefly indicate the physical ingredients used to compute the present models for very massive stars. The physics contributing to very massive stars being 26 Al potential sources is discussed in Section 3. The 26 Al masses ejected by the stellar winds of these stars and their dependence on initial mass, rotation. and metallicity is discussed in Section 4. A simple estimate of the very massive stars contribution to the total 26 Al mass budget in the Milky Way is given in Section 5. The main conclusions are synthesized in Section 6 with some possible links with other topics involving 26 Al.

Ingredients of the stellar models

We computed very massive stars models with GENEC (Eggenberger et al. 2008) with initial masses of 180, 250 and 300M for Z=0.006 and Z=0.014, without rotation and with a rotation rate of V/V c =0.4 where V c is the critical velocity1 . The nuclear network allows following the variation of abundances of 30 isotopes2 . In addition to the CNO cycles, the Ne-Na and Mg-Al chains are included. Note that the the isomeric ( 26 Al m ) and the ground state ( 26 Al g ) of 26 Al are considered as two different species. The nuclear reaction rate of 25 Mg(p,γ) 26 Al g,m is taken from Iliadis et al. (2001). Champagne et al. (1993) is used for 26 Al(p,γ) 27 Si and Caughlan & Fowler (1988) for 26 Al(n,α) 23 Na and 26 Al(n,p) 26 Mg. Note that the rates are the same as those used in the previous study on that topic by Palacios et al. (2005).

The present models differ mainly in three points (Martinet et al. in prep) with respect to the physics used in the grids by Ekström et al. (2012) and Yusof et al. (2022). First, we adopted the Ledoux criterion for convection instead of Schwarzschild and an overshoot of 0.2 H p instead of 0.1 H p . These changes were done since there are some indications that the Ledoux criterion might be more appropriate (Georgy et al. 2014;Kaiser et al. 2020) for these stars, and that an increase of the overshoot parameter is needed for stars larger than 8M (Martinet et al. 2021). Another change is that the present very massive star models have been computed with an equation of state accounting for electron-positron pair production (Timmes & Swesty 2000) 3 . These changes have, however, very little impact on the question discussed in the present paper. Indeed, the results depend on the structure of the models during the Main-Sequence (MS) phase. Changing from Schwarzschild to Ledoux during the MS phase does not change anything, since the mass of the convective core decreases with time, producing no gradient of chemical composition in the layer just above the convective core. A larger overshoot will tend to reduce the time between the beginning of the evolution and the first surface enrichment, in 26 Al. However, very massive stars have anyway very large convective cores and this time would be short even with a smaller overshoot. The change of the equation of state does not have any effect on the MS phase.

The radiative mass-loss rate adopted on the MS is from Vink et al. (2001); the domains not covered by this prescription (see Fig. 1 of Eggenberger et al. 2021) use the de Jager et al. (1988) rates. Gräfener & Hamann (2008) prescriptions are used in their domain of application, while [START_REF] Nugis | [END_REF] prescriptions are used everywhere else for the Wolf-Rayet phase. The Wolf-Rayet phase is assumed to begin when the model has an effective temperature larger than 10 000 K and a surface mass fraction of hydrogen at the surface below 0.3. The radiative massloss rate correction factor described in Maeder & Meynet (2000) is applied for rotating models. The dependence on metallicity is taken such that Ṁ(Z) = (Z/Z ) 0.7 Ṁ(Z), except during the red supergiant (RSG) phase, for which no dependence on the metallicity is used. This is done accordingly to van Loon et al. (2005) and Groenewegen (2012a,b) showing that the metallicity dependence for the mass loss rates of these stars do appear to be weak.

For the rotation prescription, the models use the shear diffusion coefficient as given by Maeder (1997) and the horizontal diffusion coefficient from Zahn (1992).

26 Al production and wind ejection in very massive stars

The evolution as a function of the age for 60 and 250 M models are shown in the left and right panel of Fig. 1 respectively. As is well known, 26 Al is synthesized during the core-H burning phase by proton capture on 25 Mg. We see that in the 60 M , the abundance of 26 Al at the center reaches a maximum before the age of 1 Myr. The decrease that follows results from the fact that the β-decay of the radioisotope dominates over the production process when the abundance of 25 Mg decreases. We see that at the beginning of the core He-burning phase, there is a rapid drop in the central 26 Al mass fraction. 26 Al is destroyed at the beginning of the core He-burning phase mainly by neutron captures, initially released by the 13 C(α, n) 16 O reactions, and then mainly by 22 Ne(α,n) 25 Mg.

During the core H-burning phase, as a result either of mass loss alone (in non-rotating models) or of mass loss and rotationally induced mixing in the outer radiative zone, the surface, and hence the winds, become enriched in 26 Al. This surface enrichment lasts until products of core He-burning appear at the surface. From this stage on, the surface abundance of 26 Al rapidly decreases, reflecting the destruction of 26 Al when interior regions of the star processed by He-burning reactions are exposed at the surface by stellar winds.

Comparing the rotating and the non-rotating model for the 60 M , we see, as was already discussed in Palacios et al. (2005) that rotation favors the 26 Al wind enrichment through the following effects :

-Species synthesized in the core appear at the surface by rotational mixing in a timescale that is shorter than the time for mass loss to uncover layers whose composition has been changed by nuclear burning. Typically, looking at the left panel of Fig. 1, we see that the maximum abundance at the surface is reached at an age of ∼3.6 Myr in the nonrotating 60 M model, while the same surface mass fraction is reached at an age of less than 2 Myr in the corresponding rotating model. -Due to diffusion of hydrogen from the radiative envelope into the convective core, the convective core remains larger in the rotating model than in the non-rotating one. This also favors a rapid emergence of core H-burning products at the surface. -The core H-burning lifetime is increased by rotation from 3.5 to more than 4.5 Myr in the case of the 60 M model, which supports 26 Al wind ejection during that phase.

We can wonder whether the diffusion of some 25 Mg from the radiative envelope into the convective core contributes in some significant way to the increase of 26 Al produced in the rotating model. We note that the maximum value of the mass fraction of 26 Al at the center of the rotating 60 M is slightly above the maximum value reached in the non-rotating corresponding model. This may be in part due to that effect, but also due to the difference in the central temperatures between the two models (see the discussion in Sect. 4).

Looking now at the right panel of Fig. 1 that shows the case of a 250 M very massive star, we can note the following differences with respect to a more classical 60 M stellar model:

-26 Al appears at the surface significantly earlier, typically at ages of a few 0.1Myr. This comes mainly from the fact that convective cores in very massive stars occupies a much larger fraction of the total mass than in less massive stars. -The time difference between the reaching of the peak abundance at the center and at the surface is reduced, thereby further reducing the time for the decay of 26 Al between these two epochs. This favors larger ejected amounts. -The mass lost with a composition bearing the signatures of H-burning corresponds to 60% of the initial mass for the 250 M model, while it corresponds to 25% in the case of the rotating 60 M and less for the non-rotating one. This reflects how the mass loss increases with the initial mass. In the present models we likely underestimate the mass loss rates and thus the present predictions are likely on the conservative side (see the discussion in Sect. 6). -The impact of rotation, while non-negligible, is not as strong in the 250 M stellar model as in the 60 M one. This reflects the fact that mass loss by stellar winds and convection dominates in a more important manner the evolution of those stars compared to lower initial mass models.

Overall, rotation has a major effect on the increase of the 26 Al yields, as strong transport from rotational mixing brings large quantities of 26 Al to the surface earlier-on, enabling the 26 Al enriched envelope to be lost by the winds on longer timescales. Larger initial masses, while of course containing a larger reservoir of 25 Mg to produce 26 Al, will also have stronger mixing due to larger convective cores. This effect, combined with the larger mass loss events, will be predominant over the rotation effect when scaling to very high initial masses.

The 26 Al stellar yields

For each model, we can compute the quantity of 26 Al that is ejected by stellar winds, Y winds Al26 . This quantity is obtained computing the integral below

Y winds Al26 (M, Z, V) = τ(M,Z,V) 0 X S 26 (M, Z, V, t) Ṁ(M, Z, V, t)dt (1)
where τ(M, Z, V) is the lifetime of a star with an initial mass M, an initial metallicity Z and an initial rotation V, X S 26 and Ṁ are the mass fraction of 26 Al at the surface and the mass loss rates for the same model as a function of time. While the decay of 26 Al in the stellar interior is of course accounted for, we are not accounting for the 26 Al decay in the wind ejecta, because as explained in the introduction, we need to evaluate the production rate of 26 Al.

The stellar yields resulting from our different models can be found in Table A.1 in the appendix and are shown as a function of the initial mass, initial metallicity and rotation in Fig. 2. We see that the yields become larger than 10 -5 M for initial masses above about 25 M . In general, above this mass, the 26 Al increases with the mass, metallicity, and rotation. We can note the following interesting features in Fig. 2: -For initial masses equal to or above 60 M Y winds Al26 increases with the initial mass and the metallicity according to power laws. Typically, for solar metallicity models with rotation, Y winds Al26 ∝ M 1.93 , and for a rotating 120 M , Y winds Al26 ∝ Z 1.45 . -Above 60 M , the differences between rotating and nonrotating models decreases, reflecting the fact that, in the very high mass regime, models are dominated by mass loss and convection more than by rotation (at least for the rotational velocities considered here). Fig. 1. Evolution as a function of the stellar age in Myr of the total mass (M tot , see the continuous red curve), of the convective core mass (M cc , continuous blue curve) in solar mass units, of the central (X C 26 , dashed purple line) and surface 26 Al abundance (X S 26 , dashed-dotted purple line) in mass fractions for a non-rotating and a rotating stellar model at a metallicity Z=0.014. The initial rotation of the rotating models is 40% the critical velocity at the ZAMS. For clarity purpose, the curves showing the 26 Al mass fraction are multiplied by factors equal to 7.0×10 5 and 10 6 for respectively the central and surface values. The curve for the surface value is shifted so that its starting point at an age equal to 0 is 60 and 250 in the left and right panels respectively. The time-averaged convective core mass fraction during the MS phase is indicated. The light blue vertical line indicates the end of the core H-burning phase and the beginning of the core He-burning one.

-Below 60 M , we note that the behavior of the yields as a function of mass can be non-monotonous (see the nonrotating Z=0.006 models). Typically, at this metallicity, the yield of the 60 M model presents a local maximum. This is due to the specific combination of the effects of stellar winds and convection in this model. This combination at 60 M produces a longer phase than in higher initial mass models, during which layers processed by H-burning appear at the surface.

-The non-rotating models at Z=0.020 give similar yields as the rotating ones at Z=0.014 showing thus a degeneracy of the yields between increase in rotation and increase in metallicity.

-The rotating models at Z=0.006 give lower yields than the non-rotating models for masses below about 85 M . This is due to the fact that non-rotating Z=0.006 models from 40 to 85 M remain at lower effective temperature than rotating models during He-burning. This induces larger mass losses for these models and results in larger 26 Al yields.

Figure 3 compares the stellar yields obtained from the present models with and without rotation at Z=0.014 with models by different authors. Both Limongi & Chieffi (2018) and Brinkman et al. (2021) models displayed here are for single solar metallicity stars. For initial masses above 60 M , there is a general good agreement between the predictions of the different models for both rotating and non-rotating models. In the 30 to 60 M mass range, the non-rotating models of this work produce more 26 Al than Limongi & Chieffi (2018) and Brinkman et al. (2021) models, while the rotating models agree. Below 25 M , the present yields are larger than both Brinkman et al. (2021) and Limongi & Chieffi (2018) yields due to larger mass loss in RSG stars, except for the 12 to 15 M mass range, where a large difference is displayed by the Limongi & Chieffi (2018) models. This is due to the higher mass loss rates chosen for these models. They have implemented rates obtained from dust-driven wind during the RSG phase (see Chieffi & Limongi 2013). Except for this mass domain, models present an overall good agreement, with differences that remain at a moderate level, or at least at a level that cannot be discriminated by any observations at the moment.

Contribution of the winds of very massive stars to 26 Al in the Milky Way

In this section, we compute the quantity of 26 Al present in the Milky Way due to massive star stellar winds. Let us call Ψ(R) the Star Formation Rate (SFR) given in terms of the number of stars forming per unit surface, and per unit time at a given galactocentric radius, R, in the Galaxy. We assume that this function is constant in the last 10 Myr and that the Galaxy is axisymmetric. Note that our objective here is to estimate the impact of including very massive stars in the 26 Al due to stellar winds, rather than to provide a very detailed estimate based on a more complex model for the Galaxy. We normalize the initial mass function, Φ(M)dM, such that 300M 0.07M Φ(M)dM = 1.0.

(2)

Thus we can interpret Φ(M)dM as the probability that a star of initial mass between M and M + dM is formed when stars with initial masses between 0.07 and 300 M are formed. We assume that this function does not vary with time, the SFR, neither with the metallicity, Z. Let us estimate the production rate of 26 Al by stellar winds of stars of a given age t, with an initial mass between M and M+dM, a initial metallicity Z and initial rotation V at a given galactocentric distance R, per unit surface. This quantity is given by Ψ(R) × Φ(M)dM × X S 26 (M, Z, V, t) Ṁ(M, Z, V, t). The contribution of such stars irrespective of their age is obtained by integrating this expression over a period that begins at a time t 0 -τ(M, Z, V) and t 0 thus over a period whose duration is equal to the total lifetime of the star considered. The contribution of stars of a given mass M, whatever their age, is given by Ψ(R) × Φ(M) × Y winds Al26 . The contribution of stars of different masses is then given by Ψ(R) Ȳwinds Al26 , where is the average mass of 26 Al ejected by stellar winds per star formed in a given initial mass interval. The rate of 26 Al production in a galactic ring with an internal radius R and width dR is given by 2πRΨ(R) Ȳwinds Al26 dR and the total mass in the Galaxy is obtained by integrating the above expression over the whole plane of the Milky Way. Let us call this quantity P 26 , the galactic production rate. Let us note that the different rings have different metallicities, since the Galaxy presents a metallicity gradient such that the metallicity tends to increase when the galactocentric distance decreases. The above integration needs to account for this, such that the yields corresponding to the metallicity of each ring depending on its distance to the galactic center are adopted. From this production rate, we can estimate the total amount of 26 Al in the Galaxy due to stellar winds, M 26 , assuming that this total amount does not depend on time. As seen in the introduction, it is equal to M 26 =P 26 τ 26 .

Ȳwinds Al26 = M Max M Min Y winds Al26 (M, Z, V)Φ(M)dM (
In Fig. 4, we show the result of such an integration adopting the star formation rate from Kubryk et al. (2015) 4 , slightly reduced to obtain a number of core-collapse supernovae of 2 per century in the Milky Way, a Salpeter's IMF with α=2.35 and a metallicity gradient taken from Hayden et al. (2014) (shown in the bottom panel of Fig. 4).

The top panel of Fig. 4 shows the 26 Al production in rings of 0.1 kpc width as a function of the galactocentric distance. We consider four different sets of stellar populations, with and without rotation, and with and without the inclusion of very massive stars (VMS) in the IMF. These four sets have been chosen to underline the effect of rotational mixing on the galactic production of 26 Al, and to explore the impact on 26 Al production when including only a few VMS in these populations. The choice of a maximum mass of 300M is motivated by the largest initial masses of VMS derived from observations of the Tarantula nebula (Bestenlehner et al. 2020;Brands et al. 2022) in the Large Magelanic Cloud (LMC). The resulting total amount of 26 Al produced by such populations ranges from 0.340M to 1.431M . This integrated quantities are displayed in the legend and give the total content of 26 Al in the Milky Way produced by the winds according to the stellar models. Only stars above 8M are used to compute the 26 Al lost by the winds. Indeed, lower mass stars have very limited mass loss and stars below 8-12M do not even produce 26 Al through the Ne-Na and Mg-Al chains due to their lower central temperature during core H-burning5 . Of course, the number of stars above 8M is computed through the SFR taking into account every star in the Milky Way model, hence normalized on the whole range from 0.07M to the maximum mass included (here either 120M or 300M ) as we have seen in Eq. 2. The extrapolation in metallicity has been made for each initial mass and are compatible with super solar mass loss rates. We can see that for every population models, the 26 Al peak production is around 4.5kpc. It is due to the combination between three factors: (1) the peak of SFR, leading to larger quantities of massive stars, producing more 26 Al; (2) the high metallicity of the inner part of the Milky Way, leading to higher yields as we have seen in Fig. 2 and(3) the surface covered by each bin, indeed while the bins are of 0.1kpc in width, the surface they cover increases with πR 2 . The combination of these three components explains the slight shift between the peak of SFR and the 26 Al production. The 26 Al production then diminishes more abruptly from 6kpc to the outer parts. This is directly linked to the change of trend in metallicity, dominating the yields.

The impact of rotation can be seen when comparing the red (non-rotating) and the yellow (V/V c =0.4) curves. As expected from Section 4, the higher yields produced by rotating models lead to a two times larger Galactic 26 Al production. The larger effect is seen around the peak of SFR, once again due to the larger number of massive stars, for which all rotating models produce more 26 Al at these super solar metallicities.

The impact of including very massive stars can be seen when comparing the red (IMF going from 8 to 120M ) and the blue (8 to 300M ) curves. The inclusion of the VMS into the stel-lar population leads to a 120-150% increase of the 26 Al galaxy production. The larger effect is seen once again around the peak of SFR, where VMS have a higher probability to be produced. This means that a few VMS are sufficient to increase the 26 Al production significantly.

Finally, the combined effect of rotation and including the VMS leads to a four times larger 26 Al production compared to models neither accounting for rotation nor VMS. These results show that despite the fact that very massive stars are very rare (see Sect. 6.2 and Fig. 6), their effect is still significant due to their large yields. Only a few VMS can have an important impact on the 26 Al production at the Galactic scale. This underlines the need to improve our knowledge about their frequency at various metallicities.

Discussion and conclusions

Impact of changing physical ingredients of the models

The evolution of very massive star models depends mainly on the mass loss rates. In the present work, we likely underestimate the mass loss rates due to uncertainties on Eddington mass loss rates (Vink et al. 2011;Bestenlehner et al. 2014;Vink 2018) and hence the 26 Al yields might also be underestimated. Increasing the mass loss rates shortens the period preceding the time when layers having belonged to the convective core appear at the surface and increases the quantity of mass lost enriched in 26 8-500M V/V c =0.4 26 Al = 2.601 +0.149 -0.138 M 8-500M Non-rot 26 Al = 1.965 +0.118 -0.11 M 8-300M V/V c =0.4 26 Al = 1.431 +0.085 -0.078 M 8-300M Non-rot 26 Al = 0.838 +0.048 -0.044 M 8-120M V/V c =0.4 26 Al = 0.662 +0.038 -0.035 M 8-120M Non-rot 26 Al = 0.34 +0.015 -0.014 M Fig. 5. Effect of the upper-mass limit on the 26 Al production in rings of 0.1 kpc width as a function of the galactocentric distance.

The same methodology is used as in Fig. 4. The 26 Al yields above 300M are extrapolated.

that stage on. From Fig. 1, looking at the 250 M model, we see that the first effect, i.e. shortening the phase before the surface is 26 Al enriched, will have little impact. Indeed, this phase is already very short. With the present mass loss rates, starting from the time when the surface is enriched in 26 Al, more than 100 M are lost through winds. Increasing the amount of mass lost will increase the 26 Al yield, and will thus increase the contribution of VMS on the global 26 Al budget. The impact of rotational mixing is also very significant, as a population of rotating massive stars produces twice the amount of 26 Al than a population of non-rotating stars would in the Milky Way. An increase of the rotation rate would result in even higher 26 Al yields due to an even stronger transport from rotational mixing, bringing even more 26 Al to the surface early-on. Of course, less efficient transport of the chemical species will act the other way round and would decrease the quantities of 26 Al ejected by stellar winds. Convection also plays an important role in the transport of 26 Al to the surface. With an increase of the core size (e.g as suggested by Martinet et al. 2021;Scott et al. 2021), the transport of 26 Al would also be enhanced (as we have seen Fig. 1), resulting in larger 26 Al yields. We have seen that increasing the initial mass increases the mass loss and the convective core size for VMS. It means that increasing the upper mass limit for VMS (e.g 300 to 500 M ) would also result in an even larger increase of the 26 Al galactic production. This can be seen in Fig. 5, where the 26 Al production in the galaxy is displayed for an IMF with an upper-mass limit up to 500 M . The yields used for stars with initial masses over 300 M have been extrapolated and are compatible with the mass loss obtained in the 500 M models of Yusof et al. (2013). Pushing the uppermass limit to 500 M increases the total 26 Al output from winds by a factor of two in comparison to 300 M and would account for a very large fraction of the galactic 26 Al.

Some changes in the nuclear reaction rates may of course affect the present results. The interested reader can look at the paper by Iliadis et al. (2011), in which a detailed study of the dependence of the 26 Al masses synthesized by massive stars on the nuclear reaction rates has been performed. For the 26 Al produced in the H-burning convective core, an important reaction is the one synthesizing 26 Al from 25 Mg by the 25 Mg(p,γ) 26 Al reaction. A higher reaction rates will increase the quantity of 26 Al ejected by the winds, and the contrary for a lower rate. The me-dian rate by Iliadis et al. (2010) for the 25 Mg(p,γ) 26 Al is at most 20% smaller than the rate used in this work (taken from Iliadis et al. 2001) for the typical temperatures in the H-burning cores of massive stars. Such a reduction will not have a dramatic impact on the yields, lowering them slightly if all other parameters are unchanged. Very recently, the 26 Al(n,α) reaction rate has been updated by Lederer-Woods et al. (2021), but this reaction does not impact the phase during which most of the 26 Al is produced and ejected and thus it should not have a strong effect on the present results.

Synthesis of the main results and future perspectives

We computed the yields of 26 Al ejected by stellar winds for massive and very massive stars with and without rotation at three different metallicities. We showed that their impact on the global budget of 26 Al is significant. This underlines the need to search for such objects and obtain data on their frequency at different metallicities/environments.

Using a simple galactic model to compute the total 26 Al mass due to the winds of massive and very massive stars, we obtain that the Galaxy globally contains around 120 000 stars with masses between 8 and 300M , and 500 with initial masses between 120 and 300M . The variation of these numbers as a function of the galactocentric distance is shown in Fig. 6 (top panel). The lower panel displays the expected number of VMS in a disk around the sun as a function of the radius of the disk. For example, we expect about 2.5 VMS in a sphere of 1 kpc radius around the Sun. These are of course very rough estimates. Figure 1 in Meynet (1994) gives the maximum distance at which a given mass of 26 Al can be detected as a point source by INTEGRAL. If one considers the mass ejected by our rotating model of 180 M and assume that half of the total yield of 1.3e-03 M is still emitting γ-ray emission at 1.8 MeV and that this amount is sufficiently concentrated that it would appear as a point source, then the maximum distance at which such a star would be detectable is 1.25 kpc. This is a very optimistic value. Indeed, stochastic effects, the fact that such a source might not appear as point-like or (that the source) presents a smaller amount of non-decayed 26 Al than assumed here, would decrease the distance at which the star could be detectable. At the moment, the most convincing candidate for very massive stars have been observed in the LMC (Crowther et al. 2010;Bestenlehner et al. 2020). They are way too far (roughly 50 kpc) for being observed at 1.8 MeV. An interesting candidate in the Galaxy is Westerhout 49-2 (Wu et al. 2016), that has an estimated mass of 250M , but lies at 11.1 kpc, still far beyond INTEGRAL sensitivity range. One of the closest candidates would be WR 93 (HD 157504), a 120 M star (Rate & Crowther 2020). Its distance of 1.76 kpc is however beyond the maximum distance at which such a source might be detectable. This discussion indicates that while VMS might contribute significantly to the enrichment, it is still compatible with the fact that no point-like source of 26 Al has been detected so far. While the detection of such a source seems plausible in terms of instrument sensitivity, it would necessitate highly favorable circumstances that seem, for the time being, unlikely.

In addition to the question of the origin of the galactic 26 Al, this isotope is also much discussed in two other contexts. There is ample evidence in meteorites for the presence of live 26 Al in the cloud that gave birth to the solar system 4. 56 Gyr ago (e.g. et al. 1976;Park et al. 2017). Because of its short halflife, 26 Al can help us probe the Solar System astrophysical environment at its birth (e.g. Adams & Laughlin 2001). To account for 26 Al elevated abundance in the nascent Solar System compared to the 26 Al/ 27 Al ratio found in Calcium-aluminium-rich inclusions (Jacobsen et al. 2008), in-situ irradiation of the proto-planetary disk gas and dust has been proposed (Lee et al. 1977;Gounelle et 2001) but is now abandoned after some nuclear cross-sections have been remeasured (Fitoussi et al. 2008). Late delivery by supernovae and AGB stars has long been considered as a possibility to account for solar System 26 Al. Supernovae are now often discarded because they would yield a 60 Fe/ 26 Al ratio at least one order of magnitude larger than the one observed in the early solar system (Gounelle & Meibom 2008). The probability of associating an AGB star with a star-forming region is very small (Kastner & Myers 1994). At present, the most promising models are the ones involving the winds of massive stars (Arnould et al. 1997;Gounelle & Meynet 2012). A setting whereby a massive star injects 26 Al in a dense shell it itself generated (Gounelle & Meynet 2012;Dwarkadas et al. 2017) seems to be a likely model (Gounelle 2015).

Another aspect in which 26 Al plays an important role is the question of the sources of pristine circumstellar grains, i.e. of grains formed around stars that travel through space and are finally locked in meteorites. Traces of 26 Mg due to 26 Al present at the time of formation of these pristine circumstellar grains, along with the measurements of the abundances of other isotopes, provide a clue about the nature of stars that produce such grains. (see e.g. Zinner et al. 1991;Dauphas & Chaussidon 2011). What proportion of these grains might be produced around VMS, or might the 26 Al injected into the proto-solar nebula comes from a very massive star, are interesting questions for future prospects.

Table A.1. 26 Al wind yields (calculated using Eq. 1) in M units. The VMS models at Z=0.006 and Z=0.014 are from Martinet et al. (in prep.) probing initial masses of 180, 250 and 300M , while the models at Z=0.020 are from Yusof et al. (2022), with VMS models computed only for 150 and 300M . More details on the massive stars models at Z=0.006 can be found in Eggenberger et al. (2021) and in Ekström et al. (2012) for the ones at Z=0.014. Dashes indicate where the models where not computed.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Masses of 26 Al ejected by stellar winds during the total stellar lifetime of massive stars with different initial masses, initial metallicities and rotation rates.

Fig. 4 .

 4 Fig. 4. Top panel: 26 Al production in rings of 0.1 kpc width as a function of the galactocentric distance. The production is displayed for a Salpeter's IMF (α=2.35) with different maximum mass and with models with and without rotation. The results are obtained taking into account the SFR and the metallicity radial gradient in the Milky Way. The dashed lines show the result when considering the uncertainties on the metallicity displayed in the bottom panel. The integrated quantities over the whole Milky Way of 26 Al are given in the legend. Middle panel: Star formation rate as a function of the galactocentric distance from Kubryk et al. (2015). Bottom panel: Metallicity as a function of the galactocentric distance from Hayden et al. (2014).

Fig. 6 .

 6 Fig.6. Top panel: Number of massive stars (8-120 M ) and VMS (120-300 M ) in rings of 0.1 kpc width as a function of the galactocentric distance. We use here a Salpeter's IMF with α=2.35 and the SFR used in the middle panel of Fig.4fromKubryk et al. (2015). Bottom panel: Number of VMS in a disk around the sun as a function of the maximal distance to the sun, following the same methodology.

  Mass Non-rotating V/V c =0.4 Non-rotating V/V c =0.4 Non-rotating V/V c =0.4

		Z=0.006	Z=0.014	Z=0.020
	12 M	4.63E-13	3.85E-11	1.01E-09	4.62E-09	2.09E-09	1.46E-08
	15 M	1.94E-12	7.73E-10	1.48E-10	3.09E-08	1.12E-09	1.00E-07
	20 M	1.12E-09	4.87E-07	5.57E-07	4.46E-06	1.32E-06	5.42E-06
	25 M	1.89E-06	1.06E-05	9.50E-06	9.67E-06	1.09E-05	2.02E-05
	32 M	1.67E-05	1.24E-05	2.04E-05	3.08E-05	2.72E-05	4.25E-05
	40 M	2.32E-05	1.53E-05	3.82E-05	3.66E-05	4.03E-05	6.73E-05
	60 M	7.70E-05	2.24E-05	5.33E-05	1.05E-04	1.12E-04	2.27E-04
	85 M	4.25E-05	5.17E-05	1.67E-04	3.21E-04	2.78E-04	5.68E-04
	120 M	9.57E-05	1.39E-04	3.64E-04	7.09E-04	6.50E-04	1.26E-03
	150 M	-	-	-	-	1.17E-03	1.96E-03
	180 M	1.70E-04	3.37E-04	8.37E-04	1.30E-03	-	-
	250 M	3.67E-04	6.51E-04	1.93E-03	2.63E-03	-	-
	300 M	5.39E-04	8.97E-04	3.03E-03	3.71E-03	5.78E-03	6.43E-03

The critical velocity is the velocity such that the centrifugal force at the equator balances the gravity there. Its expression is taken as indicated by expression 6 inEkström et al. (2008).

2 These isotopes are 1

H,3,4 He,12,13,14 C, 14,15 N, 16,17,18 O, 18,19 F, 20,21,22 Ne, 23 Na,24,25,26 Mg,26,27 Al, 28 Si, 32 S, 36 Ar, 40 Ca, 44 Ti, 48 Cr,52 Fe and 56 Ni.3 The models at Z=0.020 have been computed with the same equation of state as inEkström et al. (2012)

To obtain from this star formation rate, given in solar mass per pc -2 and per Gyrs (see middle panel of Fig.4), a star formation rate given in number of stars per pc -2 and per Gyrs, we need to divide by the average mass of a star when stars are formed over the whole mass range between 0.07 and 300 M .

Note that AGB stars can reach high temperature in their H-burning shell and thus can contribute to the production of this element. However, this is another channel of production that is not discussed here.

Acknowledgements. SM has received support from the SNS project number 200020-205154. GM, SE, CG and DN have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 833925, project STAREX). RH acknowledges support from the World Premier International Research Centre Initiative (WPI Initiative, MEXT, Japan), STFC UK, the European Union's Horizon 2020 research and innovation program under grant agreement No 101008324 (ChETEC-INFRA) and the IReNA AccelNet Network of Networks, supported by the National Science Foundation under Grant No. OISE-1927130. NY acknowledged the support from Fundamental Research Grant scheme grant number FP042-2021 under Ministry of Higher Education Malaysia. VVD's work is supported by National Science Foundation award 1911061 to the University of Chicago (PI V. Dwarkadas)

Appendix A: Table Table A.1 presents the stellar 26 Al yields from winds obtained in this work as a function of the initial mass and metallicity, for both non-rotating and rotating at V/V c =0.4 models.