1. Introduction
Asymptotic giant branch (AGB) stars are evolved stars born with low to intermediate masses,
$\sim 0.7$
–
$7\,\mathrm{M}_{\odot}$
, depending on metallicity. AGB stars are essential for the chemical enrichment of the Universe, as they synthesise a significant portion of the C, N, F, and about half of the nuclides heavier than iron (Kobayashi, Karakas, & Lugaro Reference Kobayashi, Karakas and Lugaro2020) through the slow-neutron capture process (s-process) (Clayton et al. Reference Clayton, Fowler, Hull and Zimmerman1961; Lugaro et al. Reference Lugaro, Pignatari, Reifarth and Wiescher2023). In the early Universe, at metallicities of
$Z \lesssim 0.0001$
, AGB stars also contributed significantly to the Galaxy dust budget (Valiante et al. Reference Valiante, Schneider, Bianchi and Andersen2009; Ventura et al. Reference Ventura2021; Yates et al. Reference Yates2024) and to the production of Mg (Fenner et al. Reference Fenner2003; Doherty et al. Reference Doherty2014).
The envelopes of AGB stars become enriched with heavy nuclides after the onset of repeated unstable shell He burning, known as thermal pulses. These thermal pulses drive structural change within the star, which allows for periodic episodes of stellar nucleosynthesis and convective mixing. Thermally pulsing AGB (TP-AGB) stars synthesise nuclides such as C and F through partial shell He burning. These elements are convectively mixed into the outer stellar envelope during third dredge-up events, which can occur after a thermal pulse. Depending on the metallicity, TP-AGB stars with masses
$\gtrsim 3\,\mathrm{M}_{\odot}$
may also experience temperatures at the bottom of their convective envelopes
$\gtrsim 50\,$
MK, which is sufficient for H-burning (Boothroyd, Sackmann, & Wasserburg Reference Boothroyd, Sackmann and Wasserburg1995; Karakas Reference Karakas2010). H-burning at the bottom of the convective envelope is known as hot-bottom burning. Hot-bottom burning allows AGB stars to contribute significantly to the Galaxy’s N budget. For detailed reviews on AGB evolution and nucleosynthesis, see Herwig (Reference Herwig2005) and Karakas & Lattanzio (Reference Karakas and Lattanzio2014).
The primary site of s-process nucleosynthesis in AGB stars is the He-rich intershell between the H and He-burning shells. During a third dredge-up event, protons are transported into the He-rich intershell. These protons fuse with
${}^{12}{\mathrm{C}}$
, which then produces the neutrons needed for the s-process via the
${}^{13}{\mathrm{C}}$
(
$\alpha$
,n)
${}^{16}{\mathrm{O}}$
reaction. In hot-bottom burning stars, H-burning during the third dredge-up prevents protons from mixing into the He-rich intershell (Goriely & Siess Reference Goriely and Siess2004). The s-process can also be active in the He-rich intershell during thermal pulses when temperatures reach
$\gt 300\,$
MK, using neutrons synthesised via the
${}^{22}{\mathrm{N}}$
e(
$\alpha,n$
)
${}^{25}{\mathrm{M}}$
g reaction (Karakas, Garc a-Hernández, & Lugaro Reference Karakas, Garca-Hernández and Lugaro2012; Lugaro et al. Reference Lugaro, Karakas, Stancliffe and Rijs2012).
Mass-loss through stellar winds allows AGB stars to eject their nuclides into the interstellar medium. The total amount of an element or isotope ejected by a star or population over its lifetime is known as the stellar yield (see Section 2.3). The stellar yields of AGB stars at
$Z=0.0001$
(or
$\text{[Fe/H]} \approx -2.2$
where
$\text{[Fe/H]} \approx \text{log}_\mathrm{10}[Z/Z_\mathrm{\odot}]$
), are not well constrained. Accurate yields at such low metallicities are essential to interpret the chemical signatures of ancient stars and reconstruct the enrichment history of the early Milky Way and its satellite galaxies. Detailed stellar models that evolve stars by directly solving the equations of stellar evolution (Herwig Reference Herwig2004; Karakas Reference Karakas2010; Cristallo et al. Reference Cristallo, Straniero, Piersanti and Gobrecht2015; Ritter et al. Reference Ritter2018; Choplin et al. Reference Choplin, Goriely, Siess and Martinet2025) differ in their treatments of convective mixing and mass-loss, resulting in large variations in their stellar yields. Because the only surviving stars from the early Universe are born with masses
$\lesssim 1\,\mathrm{M}_{\odot}$
, it is challenging to constrain stellar models across a range of initial masses at this metallicity.
In Galactic chemical evolution, the chemical contributions of AGB stars are often calculated using stellar yields from single-star models (Kobayashi et al. Reference Kobayashi, Karakas and Lugaro2020; Prantzos et al. Reference Prantzos, Abia, Cristallo, Limongi and Chieffi2020). However, observations of the remaining G, F, and K-type stars in the Galactic halo show that at least half of low- and intermediate-mass stars at
$Z\sim0.0001$
exist in binaries (Gao et al. Reference Gao2014; Yuan et al. Reference Yuan2015). Binary mechanisms such as Roche-lobe overflow (Eggleton Reference Eggleton1983), stellar wind accretion (Bondi & Hoyle Reference Bondi and Hoyle1944; Abate et al. Reference Abate, Pols, Izzard, Mohamed and de Mink2013), common envelope, and mergers, alter the evolutionary pathway of a star (Iben Reference Iben1991; De Marco & Izzard Reference De Marco and Izzard2017). This is evidenced by objects such as blue stragglers (Bailyn Reference Bailyn1995; Leigh et al. Reference Leigh2013), C-enhanced metal-poor stars (Beers & Christlieb Reference Beers and Christlieb2005; Frebel & Norris Reference Frebel and Norris2015; Sharma et al. Reference Sharma, Theuns, Frenk and Cooke2018), and He-core white dwarfs (Cool et al. Reference Cool, Grindlay, Cohn, Lugger and Bailyn1998; Serenelli et al. Reference Serenelli, Althaus, Rohrmann and Benvenuto2002).
The AGB is the final major nuclear-burning stage of low- and intermediate-mass stellar evolution and is the phase in which most heavy elements are synthesised. At solar metallicity, it was found that the disruption of a stellar companion can reduce the ejected amount of C and s-process elements from a stellar population by up to 25% (Osborn et al. Reference Osborn2025). Binary evolution can limit the ability of AGB stars to contribute to the chemical evolution of the Universe (Izzard Reference Izzard2004). Few Galactic chemical evolution models have used the stellar yields from low- and intermediate-mass synthetic binaries in their calculations (De Donder & Vanbeveren Reference De Donder and Vanbeveren2002, Reference De Donder and Vanbeveren2004; Sansom, Izzard, & Ocvirk Reference Sansom, Izzard and Ocvirk2009; Yates et al. Reference Yates2024), however they discuss only a few key elements.
In this work, we use the binary population synthesis code binary_c (Izzard et al. Reference Izzard, Tout, Karakas and Pols2004, Reference Izzard, Dray, Karakas, Lugaro and Tout2006, Reference Izzard, Glebbeek, Stancliffe and Pols2009, Reference Izzard2018; Izzard & Jermyn Reference Izzard and Jermyn2023; Hendriks & Izzard Reference Hendriks and Izzard2023) to model and calculate the elemental yield of all stable elements up to Bi (excluding Li, B, and Be) from low- and intermediate-mass stellar populations at metallicity
$Z=0.0001$
and quantify the impact introduced by binary evolution. Here we define low mass stars to have masses
$\sim 0.7$
–
$3\,\mathrm{M}_{\odot}$
and intermediate mass stars to have masses
$\sim 3$
–
$7\,\mathrm{M}_{\odot}$
. We evolve five stellar model sets using various wind mass-loss prescriptions on the TP-AGB to reflect the varying treatments used in detailed AGB models (Herwig Reference Herwig2004; Karakas Reference Karakas2010; Ritter et al. Reference Ritter2018). We also calculate delay-time distributions of the ejected C, N, F, Sr, Ba, and Pb from our stellar populations.
This paper is structured as follows. Section 2 describes how we build our synthetic models and stellar populations with binary_c, including updates to the treatment of the CO core mass (Section 2.1) and the temperature at the base of the convective envelope (Section 2.2). In Section 3, we show the stellar yields from our stellar populations and describe the changes introduced by binary evolution. Section 4 discusses our results and the uncertainty in the evolution of stars at
$Z=0.0001$
and binary evolution. Finally, we highlight our conclusions in Section 5.
2. Binary population synthesis models
We use the binary population synthesis code binary_c version 2.2.4, the latest official release at the time of writing, to model our stellar populations. We use binary_c as it is the only binary population synthesis code that parameterises AGB stars with enough detail to also model AGB stellar nucleosynthesis (see Izzard et al. Reference Izzard, Dray, Karakas, Lugaro and Tout2006 for more details). This allows us to calculate the stellar yields directly from our modelled populations.
Our model parameters are presented in Table 1. We choose an initial single and primary star mass range of
$0.7$
–
$7\,\mathrm{M}_{\odot}$
as our single stars born with mass
$\lesssim 0.7\,\mathrm{M}_{\odot}$
do not evolve off the main sequence during the
$15\,$
Gyr of simulation time, and stars of masses
$\gtrsim 7 \,\mathrm{M}_{\odot}$
explode as supernovae and do not evolve through the TP-AGB. Initial chemical abundances lighter than
${}^{76}{\mathrm{G}}$
e are estimated from Kobayashi et al. (Reference Kobayashi, Karakas and Umeda2011) for
$Z=0.0001$
, and those including and heavier than
${}^{76}{\mathrm{G}}$
e are scaled from the Solar abundances (where
$Z = 0.0142$
) presented in Asplund et al. (Reference Asplund, Grevesse, Sauval and Scott2009) to
$Z=0.0001$
.
Table 1. Key stellar grid and model parameters shared by all model sets. Model parameters not listed here are set to the binary_c V2.2.4 default. A complete list of model parameters may be obtained upon request from the corresponding author.
Results from the Galactic chemical evolution models from Kobayashi et al. (Reference Kobayashi, Karakas and Umeda2011); Kobayashi et al. (Reference Kobayashi, Karakas and Lugaro2020) find that the stellar yields calculated from Karakas (Reference Karakas2010) match observations of N in the solar neighbourhood for
$\mathrm{[Fe/H]} \gt -1.5$
, where the contribution from AGB stars becomes dominant. Therefore, following the results from Karakas (Reference Karakas2010), we set hot-bottom burning to occur in stars with masses
$\gt3\,\mathrm{M}_{\odot}$
. Additionally, the stars modelled in Karakas (Reference Karakas2010) were evolved until their envelope masses reduced to
$\sim 0.1\,\mathrm{M}_{\odot}$
, where they continued to experience efficient third dredge-up, allowing the continued enrichment of heavy nuclides in the stellar envelope. Therefore, we set our binary_c models to terminate the third dredge-up at an envelope mass of
$0.1\,\mathrm{M}_{\odot}$
.
To model s-process nucleosynthesis in binary_c, we adopt the He-rich intershell abundance table described in Abate et al. (Reference Abate, Pols, Karakas and Izzard2015a), which is interpolated from the detailed models described in Lugaro et al. (Reference Lugaro, Karakas, Stancliffe and Rijs2012) and includes 320 isotopes. During a third dredge-up event, the depth to which protons are transported into the He-rich intershell is uncertain. The detailed models from Lugaro et al. (Reference Lugaro, Karakas, Stancliffe and Rijs2012) introduce a ‘partial mixing zone’, defining the depth protons penetrate the He-rich intershell. In Abate et al. (Reference Abate2015b), they found that a partial mixing zone mass of
$0.002\,\mathrm{M}_{\odot}$
at masses
$\leq 3\,\mathrm{M}_{\odot}$
best reproduced the observed surface abundances of C-enhanced metal-poor stars. At masses
$\gt3\,\mathrm{M}_{\odot}$
, we set the mass of the partial mixing zone to be zero as H burning during the third dredge-up inhibits protons being transported into the He-rich intershell (Goriely & Siess Reference Goriely and Siess2004).
Table 2. AGB stellar wind prescriptions of our five model sets.
To investigate the uncertainty introduced by stellar winds, we simulate stellar populations from five model sets evolved with various TP-AGB mass-loss prescriptions as described in Table 2. For each model set, we produce a grid of
$1\,000$
single and
$10^6$
binary star models sampled as described in Table 1. In model set VW, we apply the mass-loss prescription used in Karakas et al. (Reference Karakas, Lattanzio and Pols2002), which is from Vassiliadis & Wood (Reference Vassiliadis and Wood1993). The Vassiliadis & Wood (Reference Vassiliadis and Wood1993) mass-loss prescription is often used in other studies using binary_c, including Abate et al. (Reference Abate, Pols, Karakas and Izzard2015a). Some detailed models use the mass-loss prescription from Bloecker (Reference Bloecker1995) with
$\eta$
values varying between 0.01 and 0.1, often estimated by extrapolating from higher metallicities (Ventura, D’Antona, & Mazzitelli Reference Ventura, D’Antona and Mazzitelli2002; Herwig Reference Herwig2004; Ritter et al. Reference Ritter2018). Therefore, in model sets B01 and B02, we use mass-loss as described in Bloecker (Reference Bloecker1995) with
$\eta = 0.01$
and
$0.02$
, respectively. In Karakas (Reference Karakas2010), they use mass-loss as described in Vassiliadis & Wood (Reference Vassiliadis and Wood1993) for stars with masses
$\leq 3\,\mathrm{M}_{\odot}$
and Reimers (Reference Reimers1975) for masses
$\gt 3\,\mathrm{M}_{\odot}$
with
$\eta$
values ranging from 5 to 10. Therefore, in model sets
$VW\_B01$
and
$VW\_B02$
, we transition between the Vassiliadis & Wood (Reference Vassiliadis and Wood1993) and Bloecker (Reference Bloecker1995) mass-loss prescriptions. To facilitate the smooth transition between the TP-AGB mass-loss prescriptions at stellar mass around
$3\,\mathrm{M}_{\odot}$
, we use
\begin{equation} \dot M_\mathrm{TPAGB} = (1-f_\mathrm{l}) \dot M_\mathrm{VW93} + f_\mathrm{l} \dot M_\mathrm{B95} \text{,}\end{equation}
where
$\dot M_\mathrm{TPAGB}$
is the mass-loss during the TP-AGB,
$\dot M_\mathrm{VW93}$
is the mass-loss calculated using the Vassiliadis & Wood (Reference Vassiliadis and Wood1993) prescription,
$\dot M_\mathrm{B95}$
is the mass-loss calculated using Bloecker (Reference Bloecker1995), and
\begin{equation} f_\mathrm{l} = \frac{1}{1 + 0.0001^{M_\mathrm{1TP} - 3}},\end{equation}
where
$M_\mathrm{1TP}$
is the total stellar mass in
$\,\mathrm{M}_{\odot}$
at the first thermal pulse. We use the Bloecker (Reference Bloecker1995) prescription for
$M \gtrsim 3\,\mathrm{M}_{\odot}$
, instead of the Reimers (Reference Reimers1975) prescription like in Karakas (Reference Karakas2010), to avoid needing to also transition our mass-loss treatment near
$3.5\,\mathrm{M}_{\odot}$
and
$4.5\,\mathrm{M}_{\odot}$
to model how
$\eta$
changes like in Karakas (Reference Karakas2010).
2.1 CO core masses
In binary_c, the CO core mass during the early AGB phase (EAGB), prior to the TP-AGB, is calculated based on fits from Hurley et al. (Reference Hurley, Pols and Tout2000) to the models described in Pols et al. (Reference Pols, Schröder, Hurley, Tout and Eggleton1998). However, the CO core mass at the first thermal pulse is calculated using fits to Karakas et al. (Reference Karakas, Lattanzio and Pols2002). A key difference between these models is that Pols et al. (Reference Pols, Schröder, Hurley, Tout and Eggleton1998) calculate their models with convective overshoot, whereas Karakas et al. (Reference Karakas, Lattanzio and Pols2002) do not. This results in the CO cores at the beginning of the EAGB from Pols et al. (Reference Pols, Schröder, Hurley, Tout and Eggleton1998) being up to about
$0.4\,\mathrm{M}_{\odot}$
more massive for the same initial mass than those in Karakas et al. (Reference Karakas, Lattanzio and Pols2002). This causes numeric issues in binary_c when the CO core is more massive at the beginning of the EAGB than the predicted core mass at the first thermal pulse. At
$Z=0.0001$
, this occurs at masses between about 6–
$6.5\,\mathrm{M}_{\odot}$
, and binary_c responds by forcing the star to explode in a core-collapse supernova, despite the core lacking the mass to do so.
We employ a similar solution to that used in Osborn et al. (Reference Osborn, Karakas, Kemp and Izzard2023). We refit the CO core masses at the beginning of the EAGB to those calculated in Karakas (Reference Karakas2010), reducing the CO core mass at the beginning of the EAGB. Our resulting fit for the CO core mass, in
$\,\mathrm{M}_{\odot}$
, at the beginning of the EAGB,
$M_\mathrm{CO, EAGB}$
, is
\begin{equation} \begin{split} &M_\mathrm{CO, EAGB} = \\ &(1-f_2)\left[ (8.24 \times 10^{-3}) M_\mathrm{EAGB}^2 + (2.83\times10^{-2}) M_\mathrm{EAGB} + 0.244 \right] \\ &+ f_2M_\mathrm{CO, Pols98}, \end{split}\end{equation}
where
$M_\mathrm{EAGB}$
is the total mass at the beginning of the EAGB in
$\,\mathrm{M}_{\odot}$
and
$M_\mathrm{CO, Pols98}$
is the CO core mass as estimated using the fit to Pols et al. (Reference Pols, Schröder, Hurley, Tout and Eggleton1998) and
\begin{equation} f_2 = \frac{1}{1 + 0.0001^{M_\mathrm{EAGB} - M_x}},\end{equation}
where
$M_x = 6\,\mathrm{M}_{\odot}$
. Equation (4) smooths the transition between our fit to the CO core masses calculated in Karakas (Reference Karakas2010) and those calculated in Pols et al. (Reference Pols, Schröder, Hurley, Tout and Eggleton1998) at
$M_x$
. The binary_c code uses a similar method to Equation (4) to transition between their fits to the CO core at the first thermal pulse, where they transition their fit to models described in Karakas et al. (Reference Karakas, Lattanzio and Pols2002) and Pols et al. (Reference Pols, Schröder, Hurley, Tout and Eggleton1998) at
$M_x = 7\,\mathrm{M}_{\odot}$
. For consistency, we set
$M_x = 6\,\mathrm{M}_{\odot}$
for the transition of the treatment CO core mass at the first thermal pulse. The new fits eliminate the exploding EAGB stars and results in
$Z=0.0001$
stars with masses
$\gtrsim6.2\,\mathrm{M}_{\odot}$
growing sufficiently massive cores to end their lives in a supernova.
The reduced CO core masses at the beginning of the EAGB results in the radii and luminosities of our stars suddenly decreasing between the final time step of the core He burning phase and the first time step of the EAGB. However, there is no significant impact on the overall stellar evolution and yields calculated from our models. The luminosities and radii of our stars modelled with
$M_\mathrm{CO, EAGB}$
fit to both Pols et al. (Reference Pols, Schröder, Hurley, Tout and Eggleton1998) and Karakas (Reference Karakas2010) finish the EAGB with near identical radii and luminosities. Note that binary_c does not model any stellar nucleosynthesis using the CO core mass during the EAGB.
2.2 Temperature at the base of the convective envelope
The treatment of hot-bottom burning in binary_c is detailed in Izzard et al. (Reference Izzard, Tout, Karakas and Pols2004, Reference Izzard, Dray, Karakas, Lugaro and Tout2006). In binary_c, the temperature at the base of the convective envelope,
$T_\mathrm{bce}$
, in Kelvin, is calculated using
\begin{equation} \begin{split} \mathrm{log_{10}}(T_\mathrm{bce}) = f_\mathrm{Trise} \times \mathrm{log_{10}}(T_\mathrm{bce, max}) \times f_\mathrm{ Tdrop}, \end{split}\end{equation}
where
$T_\mathrm{bce,max}$
is the maximum
$T_\mathrm{bce}$
calculated for the star (see Equation 37 of Izzard et al. Reference Izzard, Tout, Karakas and Pols2004),
$f_\mathrm{Trise}$
is the rise in temperature during the first few thermal pulses (see Equation 39 in Izzard et al. Reference Izzard, Tout, Karakas and Pols2004), and
\begin{equation} f_\mathrm{Tdrop} = (M_\mathrm{env}/M_\mathrm{env, 1TP})^{0.02},\end{equation}
where
$M_\mathrm{env}$
is the mass of stellar envelope and
$M_\mathrm{env, 1TP}$
is the mass of the stellar envelope at the first thermal pulse (see Equation 40 from Izzard et al. Reference Izzard, Tout, Karakas and Pols2004).
Figure 1 shows the results of Equation (6) compared to results from the stellar-models of initial masses
$\geq 3\,\mathrm{M}_{\odot}$
described in Karakas (Reference Karakas2010), which predict sufficient temperatures for hot-bottom burning. In binary_c, Equation (6) results in
$T_\mathrm{bce}$
cooling too quickly as
$M_\mathrm{env}/M_\mathrm{env,1TP}$
decreases compared to the stars modelled in Karakas (Reference Karakas2010). This results in hot-bottom burning elements such as N being under-produced.
Figure 1. We compare the fit for
$f_\mathrm{Tdrop}$
described by Equation (6) (Standard) to our new fit described by Equation (7) (New Fit), and the models presented in Karakas (Reference Karakas2010). We show
$f_\mathrm{ Tdrop}$
as a function of
$M_\mathrm{env}/M_\mathrm{env, 1TP}$
.
To improve the stellar yields of our stars modelled using binary_c to better fit the results of Karakas (Reference Karakas2010), we refit
$f_\mathrm{Tdrop}$
to
\begin{equation} \begin{split} f_\mathrm{Tdrop} = 1-\mathrm{exp}\left(-\frac{M_\mathrm{env}/M_\mathrm{env, 1TP}}{0.027} \right). \end{split}\end{equation}
Figure 1 shows that our new fit for
$f_\mathrm{Tdrop}$
results in
$T_\mathrm{bce}$
cooling more slowly with decreasing envelope compared to Karakas (Reference Karakas2010). Equation (7) has a root mean squared error value of
$5 \times 10^{-3}$
when considering
$M_\mathrm{env}/M_\mathrm{env, 1TP} \gt 0.3$
, which indicates a better fit to the data from Karakas (Reference Karakas2010) than Equation (6) which has a root mean squared error value of
$7 \times 10^{-3}$
. At
$M_\mathrm{env}/M_\mathrm{env, 1TP} \lt 0.3\,\mathrm{M}_{\odot}$
, Equation (6) is the better fit. However, this is not an issue since stars with
$M_\mathrm{env}/M_\mathrm{env,1TP} \lt 0.3$
have less than
$0.2\%$
of the TP-AGB phase remaining and
$M_\mathrm{env}/M_\mathrm{env,1TP}$
is declining rapidly.
2.3 Stellar and population yields
We calculate both stellar and population yields as described in Osborn et al. (Reference Osborn, Karakas, Kemp and Izzard2023, Reference Osborn2025), where we only consider the contribution of mass-loss due to stellar winds to the total yield. We calculate the total stellar yield using,
\begin{equation} y_{i} = \int ^{\tau} _0 X(i,t) \frac{\mathrm{d}M}{ \mathrm{d}t}\mathrm{d}t,\end{equation}
where
$y_{i}$
is the total stellar yield of element i in
$\,\mathrm{M}_{\odot}$
, X(i,t) is the surface mass fraction of species i,
$\tau$
is the lifetime of the star, and
$\frac{\mathrm{d}M}{ \mathrm{d}t}$
is the mass-loss rate from the stellar system noting it is always positive. We assume all short-lived radioactive isotopes have decayed. We do not decay the long-lived radioisotopes
${}^{48}{\mathrm{C}}$
a,
${}^{87}{\mathrm{R}}$
b,
${}^{96}{\mathrm{Z}}$
r,
${}^{113}{\mathrm{C}}$
d,
${}^{115}{\mathrm{I}}$
n,
${}^{144}{\mathrm{N}}$
d,
${}^{147}{\mathrm{S}}$
m,
${}^{148}{\mathrm{S}}$
m,
${}^{151}{\mathrm{E}}$
u,
${}^{176}{\mathrm{L}}$
u,
${}^{187}{\mathrm{R}}$
e,
${}^{186}{\mathrm{O}}$
s, and
${}^{209}{\mathrm{B}}$
i.
In our binary models, we calculate the stellar yield of the primary, secondary, and post-merger stars separately. Mass ejected via mass transfer, common envelopes, and mergers are also included in our stellar yield calculation and are treated as in Osborn et al. (Reference Osborn, Karakas, Kemp and Izzard2023). We assume that material ejected during a mass transfer or common envelope event originates from the donor star. During a stellar merger event, we assume the ejected material is a mix of both stellar envelopes, depending on the evolutionary phases of the stellar components. For example, if a giant star merges with a main-sequence star following a common envelope event, we assume the ejected material originates from the donating giant star (see Osborn et al. Reference Osborn, Karakas, Kemp and Izzard2023 for more details). We calculate the stellar yield contribution from stellar winds, mass-transfer, and stellar mergers over a total of
$15\,$
Gyr simulation time.
For Galactic chemical evolution, it is important to determine the net production or destruction of any given element. The net yield,
$y_{i, \mathrm{net}}$
, of a given species, i, is defined as,
\begin{equation} y_{i, \mathrm{net}} = \int ^{\tau} _0 \left[ X(i,t) - X(i,0) \right] \frac{\mathrm{d}M}{ \mathrm{d}t}\mathrm{d}t,\end{equation}
where X(i,0) is the surface mass fraction of species i at birth.
To express the total or net yield contribution of each model to a stellar population in units
$\,\mathrm{M}_{\odot}$
per
$\,\mathrm{M}_{\odot}$
of star-forming material (
$\,\mathrm{M}_{\odot}/\mathrm{M_{\odot,SFM}}$
), we apply a weighting factor
$w_j$
to each model j in our model sets where
\begin{equation} w_{j,\mathrm{s}} = (1-f_\mathrm{b}) \frac{w_\mathrm{m}}{n_\mathrm{s}} \frac{\pi_\mathrm{s}(\textbf{x}_{j})}{\xi_\mathrm{ s}(\textbf{x}_{j})},\end{equation}
for the single-star portion of the population and
\begin{equation} w_{j,\mathrm{b}} = f_\mathrm{b} \frac{w_\mathrm{m}}{n_\mathrm{b}} \frac{\pi_\mathrm{b}(\textbf{x}_{j})}{\xi_\mathrm{ b}(\textbf{x}_{j})},\end{equation}
for the binary star portion of the population where
$f_\mathrm{b}$
is the binary fraction of the stellar population,
$n_\mathrm{s}$
and
$n_\mathrm{b}$
are the number of models sampled for our single and binary grids respectively,
$\pi_\mathrm{s}(\textbf{x}_{j})$
and
$\pi_\mathrm{b}(\textbf{x}_{j})$
, respectively, describe the theoretical probability distributions of initial conditions of the observed single and binary populations, and
$\xi_\mathrm{ s}(\textbf{x}_{j})$
and
$\xi_\mathrm{b}(\textbf{x}_{j})$
are the probability distributions of our single and binary models, respectively, sampled in binary_c (Broekgaarden et al. Reference Broekgaarden2019; Kemp et al. Reference Kemp2021; Osborn et al. Reference Osborn2025), and
$w_\mathrm{m}$
is a mass normalisation term describing the average number of stellar systems forming per
$\,\mathrm{M}_{\odot}$
of star-forming material where,
\begin{equation} w_\mathrm{m} = \frac{\int^{M_\mathrm{1,max}}_{M_\mathrm{1,min}} \pi(M_\mathrm{1,0})\,\mathrm{d}M_\mathrm{ 1,0}}{\int^{M_\mathrm{1,max}}_{M_\mathrm{1,min}} M_\mathrm{1,0}\, \pi(M_\mathrm{1,0})\,\mathrm{d}M_\mathrm{1,0} + f_\mathrm{b}\int^{M_\mathrm{2,max}}_{M_\mathrm{2,min}} M_\mathrm{2,0}\, \pi(M_\mathrm{2,0})\,\mathrm{d}M_\mathrm{ 2,0}},\end{equation}
where
$M_\mathrm{1,0}$
is the initial mass of our single and binary primary stars born with a mass distribution
$\pi(M_\mathrm{1,0})$
normalised between
$M_\mathrm{1,min}$
and
$M_\mathrm{1,max}$
as described in Table 1,
$M_\mathrm{2,0}$
is the initial mass of our secondary stars born with a mass distribution
$\pi_\mathrm{b}(M_\mathrm{2,0})$
normalised between
$M_\mathrm{2,min}$
and
$M_\mathrm{2,max}$
as described in Table 1. The term
$\int^{M_\mathrm{1,max}}_{M_\mathrm{1,min}} \pi(M_\mathrm{1,0})\,\mathrm{d}M_\mathrm{1,0}$
describes the total number of stellar systems forming in our population,
$\int^{M_\mathrm{1,max}}_{M_\mathrm{1,min}} M_\mathrm{1,0}\, \pi(M_\mathrm{1,0})\,\mathrm{d}M_\mathrm{1,0}$
is the total mass of the combined single and binary primary stars in our population, and
$f_\mathrm{b}\int^{M_\mathrm{2,max}}_{M_\mathrm{ 2,min}} M_\mathrm{2,0}\, \pi(M_\mathrm{2,0})\,\mathrm{d}M_\mathrm{2,0}$
describes the contribution of the binary secondary stars to the total mass of our stellar population.
The birth distributions of stars within the Galactic halo are uncertain (Hallakoun & Maoz Reference Hallakoun and Maoz2021; van Oirschot et al. Reference van Oirschot2014). For our populations, we use the birth distributions for initial single star and primary mass, initial secondary mass, and initial orbital period as described in Osborn et al. (Reference Osborn2025), and summarised in Table 1.
We calculate the weighted total or net stellar yield
$y_\mathrm{pop, i}$
of a given species i, in units
$\,\mathrm{M}_{\odot}/\mathrm{ M_{\odot,SFM}}$
of our mixed stellar population using
\begin{equation} y_\mathrm{pop, i} = \sum^{n_\mathrm{s}}_{j=0} w_{j, \mathrm{s}} \times y_{i,j,\mathrm{s}} + \sum^{n_\mathrm{b}}_{j=0} w_{j, \mathrm{b}} \times (y_{i,j,\mathrm{b1}} + y_{i,j,\mathrm{b2}} + y_{i,j,\mathrm{b3}}),\end{equation}
where
$y_{i,j,\mathrm{s}}$
is the total or net stellar yield of element i from each single star model j in our model set and
${y_{i,j,\mathrm{b1}}}$
,
${y_{i,j,\mathrm{b2}}}$
, and
${y_{i,j,\mathrm{b3}}}$
are the total or net yields from our binary primary, secondary, and post-merger stars, respectively. In this work, we calculate the weighted total stellar yields of our populations with binary fractions ranging from 0 to 1 in increments of 0.1. The impact of stellar explosions, such as novae and supernovae, on the stellar and population yields is beyond the scope of this project. For stellar systems where an explosion occurs, we only use the contribution from stellar winds to the total yields.
3. Results
In this section, we first compare our single stars’ C and N total yields to detailed models. We then compare the weighted total yield of all stable elements between our single and binary star populations. Finally, we present delay-time distributions of the net C and N ejected by our single and binary populations.
3.1 Single star yields
Figure 2 shows the total C and N yields of our single star models compared to the yields calculated in Karakas (Reference Karakas2010), Ritter et al. (Reference Ritter2018), Cristallo et al. (Reference Cristallo, Straniero, Piersanti and Gobrecht2015), Herwig (Reference Herwig2004), who all use
$Z=0.0001$
, and Ventura et al. (Reference Ventura, D’Antona and Mazzitelli2002), who uses
$Z = 0.0002$
. We also include the total C and N yields ejected by our single star models where mass-loss is modelled using Bloecker (Reference Bloecker1995) with
$\eta = 0.1$
on the TP-AGB, which is notated as model set B10.
Figure 2. Total stellar yield of C (top) and N (bottom) as a function of initial stellar mass. Here, we compare the results from detailed stellar evolution codes to those from our single stars models from our model sets as described in Table 2. All results from detailed stellar evolution codes are calculated with
$Z=0.0001$
, except for Ventura et al. (Reference Ventura, D’Antona and Mazzitelli2002) which uses
$Z=0.0002$
. Model set B10 describes our models where mass-loss on the TP-AGB is calculated using Bloecker (Reference Bloecker1995) with
$\eta = 0.1$
.
Despite differing treatments of mass-loss on the TP-AGB, Populations
$VW93\_B01$
and
$VW93\_B02$
agree reasonably well with the C and N yields from Karakas (Reference Karakas2010), as expected with our model calibrations. Our models disagree most with Ventura et al. (Reference Ventura, D’Antona and Mazzitelli2002) and Cristallo et al. (Reference Cristallo, Straniero, Piersanti and Gobrecht2015). The models from Cristallo et al. (Reference Cristallo, Straniero, Piersanti and Gobrecht2015) experience hot-bottom burning at masses
$\geq 5\,\mathrm{M}_{\odot}$
, which is more massive than our stars modelled in binary_c, reducing the total N output of their stars. C yields from the models described in Ventura et al. (Reference Ventura, D’Antona and Mazzitelli2002) are distinctly lower compared to the other models shown here. This is attributed to their relatively low third dredge-up efficiency of 0.3–0.5, compared to the
$\sim0.9$
in our models.
From model set B10, the high mass-loss rates introduced using
$\eta = 0.1$
result in the stellar envelopes of all single stars
$\lt1.8\,\mathrm{M}_{\odot}$
being ejected before experiencing five thermal pulses. For all other model sets, stars
$\gt0.9\,\mathrm{M}_{\odot}$
experience at least five thermal pulses. Models from Herwig (Reference Herwig2004), who use Bloecker (Reference Bloecker1995) with
$\eta = 0.1$
, do not model stars
$\lt2\,\mathrm{M}_{\odot}$
, and the C yields calculated for the
$2\,\mathrm{M}_{\odot}$
and
$3\,\mathrm{M}_{\odot}$
stars from Herwig (Reference Herwig2004) better agree with our model sets B01 and B02. Although set B10 reasonably reproduces the C and N yields from Herwig (Reference Herwig2004) for stars with masses
$\gtrsim 3.5\,\mathrm{M}_{\odot}$
, stars of mass
$\lt3.5\,\mathrm{M}_{\odot}$
make up the majority of a stellar population (Salpeter Reference Salpeter1955; Kroupa Reference Kroupa2001). We therefore exclude the model set B10 from further analysis.
3.2 Population yields including binaries
Before we discuss the population yields, it is important to understand how binary evolution changes the evolution of individual stars. Table 3 shows the formation rates of TP-AGB stars, including hot-bottom burning stars, and the total amount of material ejected by our single and binary populations from all model sets per unit of
$M_\mathrm{\odot, SFM}$
using Equations (10) and (11). We identify TP-AGB stars that experience at least five thermal pulses, and we identify hot-bottom burning stars with a total mass of at least
$3.25\,\mathrm{M}_{\odot}$
at the fifth thermal pulse. Our stellar models are set up to have hot-bottom burning at masses
$\gt 3\,\mathrm{M}_{\odot}$
, but we make a conservative estimate to account for binary evolution. For our binary-star population, we include the contribution from the binary primary, secondary, and post-merger stars in our calculations. We highlight that the results of our binary population include the combined effects of binary evolution and the redistribution of the star-forming mass of our single-star populations into our secondary stars. Due to the formation of the secondary stars, the stellar mass distribution of our binary-star populations are bottom-heavy compared to our single-star populations. To examine the impact of redistributing star-forming mass into our secondary stars, independent of binary evolution, Table 3 also includes results for our binary populations where the binary primary and secondary stars are treated as if they are single.
Table 3. Here we show the average of the total mass of material ejected by the single and binary populations calculated from our five model sets. We also show the average number of TP-AGB and hot-bottom burning TP-AGB stars forming in these populations. The population notated as ‘Binary
$^{*}$
’ shows the results of our binary populations where we treat the binary primary and secondary stars as single stars. The uncertainty is one standard deviation of the average.
Figure 3. Here we compare the population N yields of our single and binary star populations, calculated from model set B02. For our binary population, we show the contribution of the binary primary, secondary, and post-merger stars to the total population N yield. We show these results as a function of the initial single or binary-star mass. We bin the yield contribution of our secondary and post-merger stars by the initial mass of their binary primary stars. We stack the contributions from each component of the binary population, with their summation equalling the total population yield.
We find our binary population produces about 37% fewer TP-AGB stars per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
over the
${15\,}$
Gyr simulation time, including
$\sim 32\%$
fewer TP-AGB stars with hot-bottom burning than our single star population. Therefore, fewer stars are available to contribute C, N, and s-process elements to the interstellar medium. Our binary population also ejects about
$\sim5\%$
more material per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
than our single-star populations.
Table 3 shows that the impact of redistributing star-forming mass into our binary secondary stars has minimal impact on the total number of TP-AGB stars in our binary population, as the average TP-AGB formation rate agrees with our single-star populations within one standard deviation. Therefore, we attribute the 37% decrease in the formation of TP-AGB stars in our binary populations from our single-star populations to binary evolution. Table 3 also shows that the formation of hot-bottom burning TP-AGB stars is more sensitive to the redistribution of star-forming mass into our secondary stars, with 12% fewer hot-bottom burning TP-AGB stars forming than our single-star populations. This accounts for about 37% of the missing stars from our binary-star populations with binary evolution.
We now examine how including binaries in our population influences the yields. For example, in Figure 3 we compare the N yield from our single-star (binary fraction of 0) and binary-star (binary fraction of 1) populations. These yields are calculated from the model set B02 using Equation (13). The total population yield from our binary population is 25% lower compared to our single-star population. We can see from Figure 3 that there is an overall reduction in the N ejected by stellar systems with primary or single star mass
$\gtrsim 3\,\mathrm{M}_{\odot}$
, reflecting the reduction in the formation of hot-bottom burning stars due to binary evolution. However, binary systems with primary masses
$\lesssim 3\,\mathrm{M}_{\odot}$
overproduce N compared to our single star population. The additional N originates from our secondary stars, which accrete material through either mass-transfer or wind Roche-lobe overflow, and our post-merger objects. These stars enter the TP-AGB with masses
$\gtrsim 3\,\mathrm{M}_{\odot}$
, which allows the bottom of their convective envelopes to reach temperatures sufficient for hot-bottom burning.
Table 4 shows weighted population yields (see Equation 13) for C, N, and Pb from all of our stellar populations. At a binary fraction of 1.0, we find a 35–40% decrease in the ejected C, a 17–36% decrease in the ejected N, and a 36–41% decrease in the ejected Pb from all populations compared to our single star populations.
Here, we examine how the inclusion of binary stars influences the population yields of all studied elements. Figure 4 shows the average percentage deviation in the binary population yields from the single star population yields for our model sets (see Table 2). We show all elements with atomic numbers up to and including Bi, excluding Li, B, Be, and radioactive Tc and Pm. The change in Li ejected by our binary population compared to our single-star population varies from a 29% decrease (Population B01) to a 230% increase (Population VW93). Li yields calculated from stellar models are notoriously sensitive to the treatment of convective mixing and mass-loss (Ventura & D’Antona Reference Ventura and D’Antona2010; Lau et al. Reference Lau, Doherty, Gil-Pons and Lattanzio2012; Gao et al. Reference Gao2022), and modelling the Cameron-Fowler mechanism (Cameron & Truran Reference Cameron and Truran1977) requires a level of detail not captured by our synthetic models, so we conclude that our Li results are unreliable. We exclude B and Be as they are not included in our nuclear network. Tc and Pm have no stable isotopes, and we add their contributions to the yields of their daughter nuclei. As with the results of Table 3, we also show our results of our binary populations where we evolve the primary and secondary stars as single stars, effectively turning off binary evolution, to indicate the dependence of redistributing the star-forming material of our population into the secondary stars on our results.
Table 4. Total population yields for all elements at binary fractions ranging from 0 to 1 for all model sets. Here, we show our results for C, N, and Pb. Tables showing the net and total stellar yields of all stable elements up to and including Bi, excluding Li, B, and Be, are available online.
Figure 4. Here we show the average of the percentage change in the total elemental yields of our binary star populations from our single star populations from our five model sets. For the data labelled ‘Binary Population’, we are comparing our populations with a binary fraction of 1 to populations with a binary fraction of 0. The error bars indicate one standard deviation of the average, highlighting the variation introduced by our choice of mass-loss on the TP-AGB. For the data labelled ‘Binary Population (P+S Isolated Evolution)’, we are showing the average and one standard deviation of our results where we evolve the stellar components of our binary-star population as if they are single.
Figure 4 shows that binary evolution has a high impact on the production of C, F, and Ne, with our binary-star populations producing
$\lesssim40\%$
less than our single-star populations. C, F, and
${}^{22}{\mathrm{N}}$
e are synthesised in the He-burning shells of TP-AGB stars, and the third dredge-up mixes these products into the stellar envelope. The reduction of C, F, and Ne is mainly attributed to binary evolution preventing the formation of TP-AGB stars. The decrease in TP-AGB systems also reduces the chemical yield of the s-process elements by about 35–40%. The weighted yields of the iron peak elements slightly increase in our binary population, but this is due to the increase in ejected material per M
$_\mathrm{\odot,SFM}$
from our binary stars, as shown in Table 3, rather than any increase in elemental production.
Elements synthesised through hot-bottom burning, such as N and Na, are underproduced by our binary populations compared to our single-star populations. Our choice of mass-loss prescription drastically alters the lifetimes of intermediate-mass TP-AGB stars, hence the large uncertainty on the yields of hot-bottom burning elements. For example, a single
$5\,\mathrm{M}_{\odot}$
star modelled using mass-loss on the TP-AGB described with the Vassiliadis & Wood (Reference Vassiliadis and Wood1993) prescription exists on the TP-AGB for
$1.0 \times 10^6$
years, but the star modelled using the Bloecker (Reference Bloecker1995) prescription with
$\eta = 0.02$
exists for only
$3.4 \times 10^5$
years, which is a 66% decrease. Additionally, the effect of allocating star-forming material to our secondary stars introduces a comparable decrease in the production of hot-bottom burning elements as binary evolution. This is especially apparent for the yields of Al, Si, and Ni where the average yields between our binary populations with and without binary evolution agree within one standard deviation. Our single-star models show that the production of Al, Si, and Ni, peaks in stars of masses
$4-6\,\mathrm{M}_{\odot}$
, which is the mass range most heavily impacted by our redistribution of star-forming mass into our secondary stars. However, low- and intermediate-mass stars do not contribute significantly to the Al, Si, and Ni in the Galaxy (Kobayashi et al. Reference Kobayashi, Karakas and Lugaro2020).
The uncertainty in the lifetime of the TP-AGB also introduces uncertainty in the yields of the elements of the first s-process peak, such as Sr and Y. Models predict that in hot-bottom burning stars, the s-process is active during thermal pulses using neutrons produced via the
${}^{22}{\mathrm{N}}$
e(
$\alpha,n$
)
${}^{25}{\mathrm{M}}$
g reaction. A single
$5\,\mathrm{M}_{\odot}$
star modelled using mass-loss on the TP-AGB as described in Vassiliadis & Wood (Reference Vassiliadis and Wood1993) experiences 158 third dredge-up events, but only experiences 42 when modelled using the Bloecker (Reference Bloecker1995) prescription with
$\eta = 0.02$
.
3.3 Delay-time distributions
It is important to the field of Galactic chemical evolution that we investigate how binaries influence the elemental production as a function of time. Figure 5 shows the average net C and N ejected by our populations at binary fractions 0 and 1, in the first 1 Gyr following a burst of star formation. Note that these results reflect the combined effect of binary evolution and the redistribution of star-forming material of our population into the secondary stars. Tables showing the net C, N, F, Sr, Ba, and Pb ejected during the first
$5\,$
Gyr after formation for populations of binary fractions varying from 0 to 1 for all model sets are available online.
Figure 5. Here, we show the average net C and N yield of our stellar populations as a function of time. We are comparing our populations where the binary fraction is 0 (single star population) and 1 (binary star population) following a single burst of star formation. We show our results up to
$1\,$
Gyr after formation, and we bin with a
$100\,$
Myr time-step. The histograms are transparent and overlapping. The error bars indicate one standard deviation in the average population yield, calculated from our five model sets.
Throughout the first Gyr, the introduction of binaries results in a consistent underproduction of C. For N, there is an underproduction in the first
$\sim 300$
Myr as our binary populations produce fewer hot-bottom burning stars through binary evolution and the formation of secondary stars (see Table 3). However, between 300–700 Myr, our binary populations overproduce N by over an order of magnitude. After
$700\,$
Myr our binary populations continue to overproduce N by a factor of at least 2. The overproduction is mainly attributed to binary systems with initial primary mass
$\lt 3\,\mathrm{M}_{\odot}$
(see Figure 3). Stellar mergers and mass transfer between the stars in these systems allow their stars to gain sufficient mass for hot-bottom burning.
Figure 5 shows the uncertainty introduced by our choice of mass-loss on the TP-AGB. At simulation times between
$100\text{--}600\,{\rm Myr}$
, the underproduction of C introduced by binaries is significant to at least two standard deviations. In the case of N, our binary populations overproduce N compared to our single star populations at simulation times
$\gtrsim 300\,$
Myr after formation, significant to at least two standard deviations. These results indicate that binary evolution can significantly impact the yield outputs of C and N as a function of time.
4. Discussion
Here, we compare our results to previous studies of populations of AGB stars in binaries. We then discuss the uncertainty in our intermediate-mass and binary models, our choice to exclude novae and supernovae from our population yields, and the limitations of comparing our results to observations.
4.1 Comparison with previous work
Galactic chemical evolution models that explore the impact of binary evolution include De Donder & Vanbeveren (Reference De Donder and Vanbeveren2002); De Donder & Vanbeveren (Reference De Donder and Vanbeveren2004) and Yates et al. (Reference Yates2024). Here, we discuss their conclusions regarding their stellar wind contribution from low and intermediate-mass stars in comparison to what we find from our results.
We first start with a comparison to De Donder & Vanbeveren (Reference De Donder and Vanbeveren2002) as they include the contribution from low- and intermediate-mass stars, noting they only do so for models of
$Z \ge 0.001$
. De Donder & Vanbeveren (Reference De Donder and Vanbeveren2002) find that the inclusion of intermediate-mass binary stars results in a reduction in the C yielded from their populations compared to their single-star populations, and binarity has a negligible impact on the yields of their low-mass stars. This disagrees with our populations, as low-mass stars are the primary source of C in our populations. The low-mass stars modelled in De Donder & Vanbeveren (Reference De Donder and Vanbeveren2002) are reported to only contribute He to the interstellar medium, and therefore likely do not experience any third dredge-up. Additionally, the models used in De Donder & Vanbeveren (Reference De Donder and Vanbeveren2002) do not model hot-bottom burning, and their models do not reproduce the N abundances observed in the Solar neighbourhood for
$\text{[Fe/H]} \lt -1$
. The study presented in De Donder & Vanbeveren (Reference De Donder and Vanbeveren2004) update their low- and intermediate-mass stellar yields based on the models calculated in van den Hoek & Groenewegen (Reference van den Hoek and Groenewegen1997) to include hot-bottom burning stars, and they still find that binary evolution reduces the C contribution from their low- and intermediate-mass populations. Although the C now originates from low-mass stars rather than intermediate-mass stars as in De Donder & Vanbeveren (Reference De Donder and Vanbeveren2002).
The work from Yates et al. (Reference Yates2024) build their stellar populations using models from binary_c. They define a ‘wind group’ which describes the combined contribution by stellar winds, Roche-lobe overflow, Thorne-Zytkow objects (Thorne & Zytkow Reference Thorne and Zytkow1977; Levesque et al. Reference Levesque, Massey, Zytkow and Morrell2014), and common envelopes to the chemical enrichment of the Galaxy. A notable result from their ‘wind group’ at
$Z=0.0001$
is that they find common envelopes boost all elemental yields by about 3-4 orders of magnitude
$\sim 4$
–
$64\,$
Myr after formation. They only report on the H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe. In our binary populations, we find an overproduction of about 4 orders of magnitude of the total C and N ejected
$\sim 40$
–
$50\,$
Myr after formation. However, the contribution to the total C and N from our binary population 40–
$50\,$
Myr after formation are on the order of
$10^{-9}$
and
$10^{-10} \,\mathrm{M}_{\odot}/\mathrm{M_{\odot,SFM}}$
, respectively, which is insignificant compared to the total C and N yield of our stellar populations.
There are multiple potential explanations for the discrepancy. The binary_c models evolved for Yates et al. (Reference Yates2024) include stars with initial masses up to
$120\,\mathrm{M}_{\odot}$
, whereas we only include stars up to
$7\,\mathrm{M}_{\odot}$
. Common envelopes and Roche-lobe overflow events with massive stars will contribute to the total yield of the ‘wind group’. Also, their treatment of the common envelope is almost identical to ours, except for their choice to use a constant binding energy efficiency parameter
$\lambda_\mathrm{CE} = 0.5$
, whereas we use a variable
$\lambda_\mathrm{CE}$
dependent on the stellar mass and radius as described in Dewi & Tauris (Reference Dewi and Tauris2000).
4.2 Uncertainty in intermediate-mass models
Mass-loss through stellar winds on the TP-AGB introduces significant uncertainty to the lifetimes and hence the yields of hot-bottom burning stars. Models from Doherty et al. (Reference Doherty2014) and Gil-Pons et al. (Reference Gil-Pons2021) indicate this uncertainty increases with decreasing metallicity. Observations of intermediate-mass hot-bottom burning stars are vital to constrain our models. Unfortunately, the only stars observed with sufficient resolution at
$Z=0.0001$
exist within the Galactic halo. Stars born within the Galactic halo are
$\sim 10\,$
Gyr in age, with only stars of mass
$\lesssim 0.8\,\mathrm{M}_{\odot}$
currently surviving. Previous studies (Izzard et al. Reference Izzard, Glebbeek, Stancliffe and Pols2009; Pols et al. Reference Pols, Izzard, Stancliffe and Glebbeek2012) have used observations of N-enhanced metal-poor stars to constrain their models and identify three objects with
$-2.8 \leq \text{[Fe/H]} \leq -1.8$
. However, they do not consider the possibility of contamination of N-enhanced objects born in globular clusters within the Galactic halo following a merger (Horta et al. Reference Horta2021; Kim et al. Reference Kim, Lee, Beers and Kim2023). Observations of white dwarfs in the Galactic halo may be another option (Romero, Campos, & Kepler Reference Romero, Campos and Kepler2015); however, there are only a few known observations of massive white dwarfs (
$\gtrsim 0.8\,\mathrm{M}_{\odot}$
) in the Galactic halo (Torres et al. Reference Torres, Rebassa-Mansergas, Camisassa and Raddi2021).
Since binary_c models are based on fits to single stars, incorrect assumptions are likely made when evolving stars within binaries. For example, our binary_c models will allow stars to extend their lifetimes on the TP-AGB and extend hot-bottom burning if they accrete additional material after evolving off the main sequence. In this scenario, stars enter the TP-AGB with a relatively low-mass core compared to their total mass, reducing their stellar radii and mass-loss rates.
Models described in Osborn et al. (Reference Osborn, Karakas, Kemp and Izzard2023) show that if material is accreted during or before core He burning, the star might not evolve onto the AGB with a low-mass core. Instead, they found the core grows to mass similar as predicted for a single star of the new total mass during core He burning, and the star evolves like a single star on the AGB without any major extension of the AGB lifetime like predicted from our models. Note that the work of Osborn et al. (Reference Osborn, Karakas, Kemp and Izzard2023) only evolve two detailed stellar models to explore post-merger hot-bottom burning stars, they do not explore sufficient parameter space for us to implement this into binary_c.
Detailed binary-star models are necessary to address the incorrect single-star assumptions applied to stellar evolution within binaries. Next-generation binary population synthesis models such as MINT (Mirouh et al. Reference Mirouh, Hendriks, Dykes, Moe and Izzard2023), METTISE (Agrawal et al. Reference Agrawal, Hurley, Stevenson, Szécsi and Flynn2020), and POSYDON (Fragos et al. Reference Fragos2023) evolve their models based on fits to binary detailed models; however, they currently do not model AGB stellar evolution and nucleosynthesis.
4.3 Uncertainty introduced by binary evolution
Throughout this paper, we have discussed the uncertainty introduced by mass-loss on the TP-AGB, but not from binary effects such as mass transfer and common envelopes. One of the most poorly constrained binary mechanisms is the evolution of a common envelope system (Ivanova et al. Reference Ivanova2013). Our models utilise the common envelope prescription described in Webbink (Reference Webbink1984), Tout et al. (Reference Tout, Aarseth, Pols and Eggleton1997), where energy from the stellar orbit is transferred to the common envelope with an efficiency
$\alpha_\mathrm{CE}$
.
$\alpha_\mathrm{CE}$
influences whether or not a common envelope system results in a stellar merger or the ejection of the common envelope. Many binary population synthesis codes adopt this formalism (Hurley, Tout, & Pols Reference Hurley, Tout and Pols2002; Izzard et al. Reference Izzard, Tout, Karakas and Pols2004; Riley et al. Reference Riley2022; Fragos et al. Reference Fragos2023). In this study, we have used the default
$\alpha_\mathrm{CE} = 1$
. However, this might not be accurate for all stellar systems (Politano Reference Politano2004; Iaconi & De Marco Reference Iaconi and De Marco2019; Hirai & Mandel Reference Hirai and Mandel2022). Since the outcomes of a common envelope event have vastly different consequences on the subsequent stellar evolution and nucleosynthesis of the involved stars, it is important to quantify the impact of
$\alpha_\mathrm{CE}$
.
The TP-AGB formation rate ranges from
$0.151 \pm 0.006$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
for
$\alpha_\mathrm{CE} = 0.1$
to
$0.113 \pm 0.07$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
for
$\alpha_\mathrm{CE} = 5$
. In the case of hot-bottom burning TP-AGB stars, the rates range from
$(1.76 \pm 0.03) \times 10^{-2}$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
when
$\alpha_\mathrm{CE} = 0.1$
to
$(1.14 \pm 0.03) \times 10^{-2}$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
when
$\alpha_\mathrm{CE} = 5$
. When
$\alpha_\mathrm{ CE} = 0.1$
, our binary populations average a
$(16 \pm 1)\%$
reduction in the ejected C and a
$(16 \pm 11)\%$
increase in the ejected N compared to our single star populations. When
$\alpha_\mathrm{CE} = 5$
we find a
$(47 \pm 1)\%$
reduction in the ejected C and a
$(41 \pm 5)\%$
reduction in the ejected N. Our choice of
$\alpha_\mathrm{CE}$
introduces more uncertainty to the amount of C and N ejected by our stellar population than our choice of mass-loss prescription on the TP-AGB.
Observations of C-enhanced metal-poor stars might help constrain our treatment of
$\alpha_\mathrm{CE}$
and binary evolution in general. However, previous studies exploring binary mechanics have shown that populations modelled using binary_c do not reproduce all their observed frequencies and abundances (Izzard et al. Reference Izzard, Glebbeek, Stancliffe and Pols2009; Abate et al. Reference Abate2015b). Advancements in the treatment of binary mechanisms, such as stellar wind accretion (Saladino & Pols Reference Saladino and Pols2019), mass transfer (Temmink et al. Reference Temmink, Pols, Justham, Istrate and Toonen2023), and common envelope evolution (González-Bolvar et al. Reference González-Bolvar, De Marco, Lau, Hirai and Price2022; Hirai & Mandel Reference Hirai and Mandel2022), may help improve our models. However, observational surveys estimate the fraction of C-enhanced metal-poor stars in the metal-poor stellar population (
$\text{[Fe/H]} = -2$
) to be about 10–30% (Lucatello et al. Reference Lucatello2006; Lee et al. Reference Lee2013; Placco et al. Reference Placco, Frebel, Beers and Stancliffe2014). Observational surveys of C-enhanced metal-poor stars do not always agree with one another due to selection effects, uncertainties, and biases in the spectral analysis (Arentsen et al. Reference Arentsen2022), which limits our ability to reliably constrain our models. Presently, binary evolution remains a significant source of uncertainty for the chemical output of a stellar population.
4.4 Excluding the yield contribution from novae and supernovae
Throughout our work, we have excluded the contribution of supernovae and novae from our stellar yields. Our focus on the evolution of AGB stars motivated this choice. However, low- and intermediate-mass stars are required for explosions such as Type Ia supernovae, which contribute significantly to the iron-peak elements (Iwamoto et al. Reference Iwamoto1999; Kobayashi et al. Reference Kobayashi, Umeda, Nomoto, Tominaga and Ohkubo2006; Keegans et al. Reference Keegans2023; Cavichia et al. Reference Cavichia2024). Additionally, mergers and mass accretion may lead to stars born of intermediate-mass to gain sufficient material to explode in a core-collapse or electron-capture supernova.
At a binary fraction of 0, our populations have an average electron-capture supernova rate of
$(1.50 \pm 0.08)\times 10^{-3}$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
and core-collapse supernova rate of up to
$2 \times 10^{-4}$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
. These originate from stars of initial mass between about 6.2–7
$\,\mathrm{M}_{\odot}$
, with
$7\,\mathrm{M}_{\odot}$
being the maximum initial mass we model in our stellar populations. At a binary fraction of 1, the average electron capture supernova rate decreases to
$(1.09 \pm 0.02) \times 10^{-3}$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
and the core-collapse supernova rate increases to
$(2.0 \pm 0.1) \times 10^{-3}$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
. We also find an average type 1a supernova rate of
$(1.48 \pm 0.05) \times 10^{-4}$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
from our binary star populations. However, we do not construct our models with the goal of measuring supernova rates and therefore, do not consider these rates reliable.
Previous studies have explored the rates of novae (Kemp et al. Reference Kemp2022) and supernovae (Ruiter et al. Reference Ruiter2011; Zapartas et al. Reference Zapartas2017), including their yield contribution (Izzard & Tout Reference Izzard and Tout2003; Izzard Reference Izzard2004; Yates et al. Reference Yates2024; Kemp et al. Reference Kemp2024). The omission of novae and supernovae will likely not introduce significant uncertainty to the yields of key elements such as C, N, F, and s-process elements due to the dominance of production within AGB stars (Kobayashi et al. Reference Kobayashi, Karakas and Lugaro2020), but given sufficient frequency they will impact elements such as O, Na, Mg, Al, and the iron peak elements.
4.5 Comparing the results of our delay time distributions to observed populations
Our delay-time distributions in Figure 5 show, for example, our binary populations overproduce N by over an order of magnitude during the period
$\sim 300$
–
$700\,$
Myr after formation, compared to our single star populations. To verify these results and those for the other elements we study, we need to compare our predictions with the abundances observed in stellar populations.
Predictions from galactic chemical evolution models are mostly compared to the abundances of unevolved low-mass field stars (De Donder & Vanbeveren Reference De Donder and Vanbeveren2002; Valentini et al. Reference Valentini2019; Kobayashi et al. Reference Kobayashi, Karakas and Lugaro2020; Molero et al. Reference Molero2025). A similar comparison using the ejecta from our models, however, is not informative. The surface compositions of field stars in the Galactic halo (Fulbright Reference Fulbright2002; Venn et al. Reference Venn2004) are mostly representative of their abundances at birth, and their stellar ages are not well enough defined to disentangle the individual generations of stars. For each simulated population, we do not attempt to calculate how the ejected material mixes with the material in the interstellar medium, nor do we calculate the composition of the following generation of stars.
A comparison to metal-poor globular clusters would also not be informative. Their ages are relatively well resolved (Valcin et al. Reference Valcin, Bernal, Jimenez, Verde and Wandelt2020), and multiple stellar populations can be identified (Ziliotto et al. Reference Ziliotto2023; Howell et al. Reference Howell, Campbell, Stello and De Silva2024). However, globular cluster stars are chemically anomalous compared to Galactic stars of the same metallicity (Hendricks et al. Reference Hendricks2014; Gratton et al. Reference Gratton2019). They also have such high stellar densities that dynamical interactions become significant, resulting in their current-day binary fraction to be
$\lesssim 10\%$
(Ivanova et al. Reference Ivanova, Belczynski, Fregeau and Rasio2005).
In its current state, we are unable to directly compare our calculated delay time distributions to observed stellar populations, such as the Galactic Halo. Our delay time distributions are not designed to infer the ages of a given observed stellar population. They are designed to provide an estimate of how elements such as N can be expelled into the Galaxy as a function of time, owing to stellar and binary evolution. The most informative step would be to use our yields within a Galactic chemical evolution code and evolve the abundances as a function of time, for comparison to field stars in different stellar populations. That work, however, is beyond the scope of this study.
5. Conclusion
We have used the binary population synthesis code binary_c to stellar populations from five model sets with various mass-loss prescriptions on the TP-AGB at
$Z=0.0001$
. We have found that for our populations with a binary fraction of 1, the formation rate of TP-AGB stars reduces by about 37% compared to our populations calculated with a binary fraction of 0. This correlates with our binary populations ejecting about 38% less C and about 35–40% less s-process elements than our single-star populations. Our binary populations also produce about 32% fewer hot-bottom burning stars. Our choice of mass-loss prescription introduces significant uncertainty to the chemical output of our hot-bottom burning models. However, we find an overproduction of N over an order of magnitude in our binary star population
$\sim 300$
–
$700\,$
Myr after formation. The role of wind uncertainty is far less significant on our lower mass stars (
$\lesssim 3\,\mathrm{M}_{\odot}$
).
Binary evolution adds significant uncertainty to our models. Our treatment of common envelope evolution varies the formation rate of TP-AGB stars in our binary population from about
$0.113$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
to
$0.151$
per
$\mathrm{M}_{\odot, \mathrm{SFM}}$
, introducing a significant variation to the C and N yields. Future work will refine the treatment of mass transfer and common envelope events in our models and explore how they influence the population yields in detail.
Acknowledgements
ZO acknowledges this research was supported by an Australian Government Research Training Program (RTP) Scholarship. DK acknowledges funding support from the Australian Research Council Discovery Project DP240101150. AJK acknowledges financial support from the Flemish Government under the long-term structural Methusalem funding programme by means of the project SOUL: Stellar evolution in full glory, grant METH/24/012 at KU Leuven. RGI is funded by STFC grants ST/Y002350/1, ST/L003910/1, and ST/R000603/1 as part of the BRIDGCE UK network. CK acknowledges funding from the UK Science and Technology Facilities Council through grant ST/Y001443/1.
This work was performed on the OzSTAR national facility at Swinburne University of Technology. The OzSTAR programme receives funding in part from the Astronomy National Collaborative Research Infrastructure Strategy (NCRIS) allocation provided by the Australian Government, and from the Victorian Higher Education State Investment Fund (VHESIF) provided by the Victorian Government.
We used the package from https://github.com/keflavich/imf to generate the initial mass distribution of our single and binary primary stars.
We thank the anonymous referee for their detailed suggestions, which helped us to improve our paper.
For the purpose of Open Access, the author has applied a Creative Commons Attribution (CC BY) public copyright licence to any Author Accepted Manuscript version arising from this submission.
Data availability statement
Tables presenting the total and net yields of all populations and the delay-time distributions of C, N, F, Sr, Ba, and Pb are available online at https://doi.org/10.57891/4j74-wj84. Additional data can be made available upon reasonable request to the corresponding authors.
The official release of binary_c version 2.2.4 is available at https://gitlab.com/binary_c/binary_c/-/tree/releases/2.2.4?ref_type=heads.
Open access
