# Observational predictions for Thorne–Żytkow objects

R. Farmer,1 M. Renzo,2 Y. Götberg,3 E. Bellinger,1,4 S. Justham,1,5,6,7 and
S.E. de Mink,1,7
1Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741
Garching, Germany
2Center for Computational Astrophysics, Flatiron Institute, New York, NY
10010, USA
3The Observatories of the Carnegie Institution for Science, 813 Santa Barbara
Street, Pasadena, CA 91101, USA
4Stellar Astrophysics Centre, Aarhus University, 8000 Aarhus, Denmark
5School of Astronomy & Space Science, University of the Chinese Academy of
Sciences, Beijing 100012, China
6National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China
7Anton Pannekoek Institute for Astronomy and GRAPPA, University of Amsterdam,
NL-1090 GE Amsterdam, The Netherlands E-mail: rfarmer@mpa-
garching.mpg.deHubble Fellow

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Thorne–Żytkow objects (TŻO) are potential end products of the merger of a
neutron star with a non-degenerate star. In this work, we have computed the
first grid of evolutionary models of TŻOs with the MESA stellar evolution
code. With these models, we predict several observational properties of TŻOs,
including their surface temperatures and luminosities, pulsation periods, and
nucleosynthetic products. We expand the range of possible TŻO solutions to
cover $3.45\lesssim\rm{\log\left(T_{eff}/K\right)}\lesssim 3.65$ and
$4.85\lesssim\rm{\log\left(L/L_{\odot}\right)}\lesssim 5.5$. Due to the much
higher densities our TŻOs reach compared to previous models, if TŻOs form we
expect them to be stable over a larger mass range than previously predicted,
without exhibiting a gap in their mass distribution. Using the GYRE stellar
pulsation code we show that TŻOs should have fundamental pulsation periods of
1000–2000 days, and period ratios of $\approx$0.2–0.3. Models computed with a
large 399 isotope fully-coupled nuclear network show a nucleosynthetic signal
that is different to previously predicted. We propose a new nucleosynthetic
signal to determine a star’s status as a TŻO: the isotopologues
$\mathrm{{}^{44}Ti}\rm{O}_{2}$ and $\mathrm{{}^{44}Ti}\rm{O}$, which will have
a shift in their spectral features as compared to stable titanium-containing
molecules. We find that in the local Universe ($\sim$SMC metallicities and
above) TŻOs show little heavy metal enrichment, potentially explaining the
difficulty in finding TŻOs to-date.

###### keywords:

stars: evolution – stars: abundances – stars: interiors – stars: variables

††pubyear: 2023††pagerange: Observational predictions for Thorne–Żytkow
objects–D

##  1 Introduction

Thorne–Żytkow objects (TŻOs) are the hypothetical unique product of the merger
of a neutron star (NS) with a non-degenerate star leading to the formation of
a single object (Thorne & Zytkow, 1975, 1977). Depending on the mass of the
combined star, it can be supported either via nuclear burning on/or near the
surface of the NS and by accretion onto the NS (Eich et al., 1989; Biehle,
1991).

A number of potential TŻOs candidates have been suggested: U Aqr (Vanture et
al., 1999), HV 2112 (Levesque et al., 2014), HV 11417 (Beasor et al., 2018),
and VX Sgr (Tabernero et al., 2021), though they are not without controversy
(Coe & Pightling, 1998; Vanture et al., 1999; Tout et al., 2014; Maccarone &
de Mink, 2016; Tabernero et al., 2021). Objects have been also been proposed
that may form a TŻO (TIC 470710327 Eisner et al. 2022), or may be the remnants
of a TŻO (1E161348-5055 Liu et al. 2015). The issue with identifying a TŻO is
distinguishing it from other similar stars, such as asymptotic giant branch
(AGB) or super-AGB (SAGB) stars (Biehle, 1991, 1994; O’Grady et al., 2020,
2023). TŻOs are expected to appear as cool red supergiants (RSG) (Cannon et
al., 1992) (hereafter C92), though many challenges remain in observing and
modelling RSGs complicating their analysis (Levesque et al., 2005; Davies et
al., 2018). Observations to date have relied on indirect measurements, such as
their predicted unique nucleosynthetic signatures (Biehle, 1994; van Paradijs
et al., 1995). Future gravitational wave observations may provide an
opportunity to detect either the formation of the TŻO (Nazin & Postnov, 1995;
Morán-Fraile et al., 2023), a rotating NS inside a TŻO (DeMarchi et al.,
2021), or a post-TŻO black hole (BH) (Cholis et al., 2022).

TŻOs are expected to be fully convective down to a “knee” where it then
transitions to a flat temperature profile until the material reaches the
surface of the NS (Eich et al., 1989). At the knee, nuclear-processed material
will be mixed outwards into the convective envelope, while fresh hydrogen is
mixed into the region below the knee. Material may also be burnt at the base
of the convective envelope. It is expected that the material is burnt via the
interrupted rapid proton process (irp-process, Cannon 1993 hereafter C93),
where rapid proton captures (Wallace & Woosley, 1981; van Wormer et al., 1994;
Fisker et al., 2008) are interrupted when the material is mixed outwards by
convection. This material then beta decays before being mixed back into the
inner regions where additional proton captures can occur.

This irp-process can lead to the production of heavy elements such as
Rubidium, Strontium, Yttrium, and Molybdenum (C93). It is also expected that
the TŻOs will be enriched in $\mathrm{{}^{40}Ca}$ (Biehle, 1991, 1994), which
in normal stars is difficult to produce and mix to the surface (Tout et al.,
2014). Lithium is also expected to be produced via
$\mathrm{{}^{3}He}$($\alpha,\gamma$)$\mathrm{{}^{7}Be}$($e^{-},\nu$)$\mathrm{{}^{7}Li}$
(Cameron, 1955). In non-TŻO stars the $\mathrm{{}^{7}Li}$ would be destroyed
by high temperatures inside the star, but in a fully convective star the
$\mathrm{{}^{7}Be}$ can be mixed outwards to cooler regions before it captures
an electron (Podsiadlowski et al., 1995). However, the nucleosynthetic signal
is not unique, there are also TŻO imposters: SAGB stars (Kuchner et al.,
2002), or stars polluted by the winds of a SAGB star and are now an AGB
themselves (Maccarone & de Mink, 2016), which may have similar nucleosynthetic
signals to a TŻO.

There are a number of potential formation mechanisms of TŻOs: engulfment of
the NS during a common envelope (Taam et al., 1978; Terman et al., 1995;
Ablimit et al., 2022); the NS receiving a kick such that it has a direct
impact with the companion (Leonard et al., 1994); or a dynamical merger either
in a dense stellar cluster (Ray et al., 1987) or a triple system (Eisner et
al., 2022). The formation of a TŻO is likely to produce a transient event
(Hirai & Podsiadlowski, 2022). Finally, TŻOs are expected to die either when
they run out of rp-seed material to burn forcing the material near the NS to
heat up and enter the pair-instability region. This causes a runaway increase
in the neutrino losses, causing the accretion onto the NS to no longer be
Eddington limited, and the NS collapses into a BH (Podsiadlowski et al.,
1995). This may possibly form a transient event (Moriya, 2018; Moriya &
Blinnikov, 2021). Alternatively, they may eject their envelope via wind mass
loss, leaving behind a “bare” NS (Bisnovatyi-Kogan & Lamzin, 1984).

The number of TŻOs in the Galaxy will depend on the rate of mergers, the
fraction of mergers that successfully produce a TŻO, and the lifetime of the
resulting TŻO. Podsiadlowski et al. (1995) estimated a birth rate of
$>10^{-4}\,\rm{yr}^{-1}$ from common-envelope evolution, and $\sim
10^{-4}\,\rm{yr}^{-1}$ from NS kicks. Renzo et al. (2019) calculated the rate
of collision from a NS being kicked during a SN with a companion to be
$\approx 10^{-4}$ the core-collapse SN rate. While Ablimit et al. (2022)
calculated formation rates of the merger of a ONeMg WD with a non-degenerate
companion, where the core collapses and forms a NS, as between $\sim
10^{-5}$–$10^{-4}\,\rm{yr}^{-1}$. Population synthesis calculations have shown
an expected merger rate of a NS with a giant star as high as $\approx 1$ per
100 core-collapse supernovae, depending on the assumed physics of the common
envelope merger (Grichener, 2023). N-body simulations of globular clusters
show dynamical mergers of a NS with main sequence star at a rate of the order
1 per $\sim 4000$ NS containing binaries (Kremer et al., 2020). The actual
formation rate of TŻOs will however depend on the probability that the merger
is successful and does not lead to a transient or complete disruption of the
system (Schrøder et al., 2020).

The first models of TŻOs presented in Thorne & Zytkow (1975, 1977) were
equilibrium models. Here the TŻO was split into three regions: an outer region
which encompasses the envelope down to the knee; a middle region between the
knee and the surface of the NS; and an inner region for the NS itself. These
models were not evolved in time; instead, static solutions where found based
on the assumed NS and envelope properties. Biehle (1991) improved on the
static models by including a simplified model of rp-burning, instead of just
assuming CNO burning. Finally, C92 & C93 created a set of evolutionary models
where the NS was modelled by altering the EOS such that the electrons become
increasingly degenerate in the vicinity of the NS. However, these models
ignored wind mass loss and ignored changes in the composition.

In Section 2 we show how we build a TŻO in MESA and discuss our default model.
In Section 3 we show the evolution of a grid of TŻO models. In Section 4 we
make predictions for the expected pulsation signal. In Section 5 we explore
the nucleosynthetic signal from TŻOs. We discuss the suitability of our model
assumptions in Section 6, and show the possible final fates of TŻOs in Section
7. Finally, we discuss our results in Section 8 and conclude in Section 9.

##  2 Building a TŻO model

Realistic modelling of the formation of a TŻO is a complex multi-dimensional
problem involving the merger of a NS with the core of another star. In this
work, we ignore both the NS and the merger process. Doing so, we construct a
spherically symmetric post-merger structure of the envelope of the TŻO that is
then allowed to evolve. There are three phases to creating our models: 1)
producing an initial seed model, 2) modifying the inner boundary of our
models, and 3) adjusting the global parameters of the model to match a chosen
set of starting conditions. Our inlists and models are available at
https://doi.org/10.5281/zenodo.4534425. An example model is also available in
MESA’s star/test_suite as of version a7c411b1.

###  2.1 Seed model

Using MESA r22.11.1 (Paxton et al., 2011; Paxton et al., 2013, 2015, 2018,
2019; Jermyn et al., 2023) we evolve a 20$\,\rm{M}_{\odot}$ star from the pre-
main sequence until midway through the main sequence (MS). This model is used
to initialise the stellar structure equations for all TŻOs, independent of
their final mass and composition. The precise choice of physics does not
matter here, as the formation process will wipe away any knowledge the model
had of its pre-TŻO state.

###  2.2 Adjusting the inner boundary

With our initial stellar model, we then model the NS by adjusting the inner
boundaries of the stellar model. We gradually increase the inner mass, radius,
and luminosity at the inner boundary of the model using MESA’s relax_core
method. This gradually changes the inner boundary while smoothly increasing
the core density and core energy generation rate. By changing the inner mass
boundary we do not change the total mass of the star, thus after forming a
TŻO, we assume the total mass remains constant.

###  2.3 Starting conditions

With the base TŻO formed, we then further alter the model to match our
required starting conditions. We use MESA’s relax_mass to add (remove) mass
via wind accretion (loss) to alter the total mass of the TŻO. We then change
the initial metal composition $\rm{Z_{init}}$, assuming a solar-scaled
composition, and the initial helium fraction $\rm{Y_{init}}$with
relax_composition. This allows us to build a TŻO with an arbitrary initial
mass and composition, which can be used to approximate the post-merger
structure for different formation scenarios.

We allow for arbitrary helium compositions, as the merger between a NS and a
star may occur at any point in the star’s lifetime (ignoring selection effects
in terms of which stars are likely to successfully merge and form a TŻO). If a
TŻO forms from a NS kick or a dynamical merger then the age of the companion
star is a free parameter111Though this will be biased by the initial mass
ratio and previous binary evolution., while mergers from a common envelope
will likely, but not necessarily, involve the companion star having evolved
beyond the MS. The mixing of the helium produced in the centre of a MS/post-MS
star into the envelope will raise the helium mass fraction of the TŻO envelope
compared to a canonical RSG. Mass loss from the envelope of the star during
the merger will likely preferentially remove H-rich material and thus increase
the average helium content of the TŻO. Thus when the star is fully mixed the
average helium content over the entire star will increase. In the limit of
when the entire hydrogen envelope is removed, a pure helium TŻO could be
formed if it survives the merger process. However, whether a TZO may form from
the merger of a NS with an evolved star or not is even more uncertain than TŻO
formation with main sequence stars (Papish et al., 2015; Metzger, 2022).

At this point, we can also set other physics options, such as the choice of
the nuclear network, accretion rate onto the NS, wind mass loss efficiency, or
the mixing length. These are discussed further in Appendix A. Although orbital
angular momentum is involved in the merger process, the amount retained post-
merger is still an open question (e.g., Schneider et al., 2019). Thus, for
simplicity, in this work we consider only non-rotating models.

###  2.4 Default TŻO model

Here we describe our default TŻO model and the choices for uncertain physical
and numerical quantities we have made. The effect of varying these choices is
explored further in Appendix A. We assume a default NS mass of $M_{\rm
NS}=1.4\,\,\rm{M}_{\odot}$. The radius of a NS is still uncertain and depends
on the chosen NS equation of state (EOS) (Steiner et al., 2010; Miller et al.,
2019). A realistic assumption would be to assume
$R_{\rm{NS}}=10$–$20\,\mathrm{k}\mathrm{m}$ for the radius of the NS and use
that as the inner boundary of our model222The inner boundary includes both the
NS and the “middle” radiative burning region. However this radiative region is
geometrically thin ($\sim 100$m) and contains little mass ($\sim
10^{-8}\,\rm{M}_{\odot}$) (Thorne & Zytkow, 1977).. However, we have found
that to be numerically difficult to model. Thus we move the inner boundary out
to a radius of $\approx 650\,\mathrm{k}\mathrm{m}$. This implies an average
core density of
$\langle\rho_{c}\rangle=10^{9.3}\,\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3}$,
instead of $\langle\rho_{c}\rangle\approx
10^{14}\,\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3}$ for a
$10\,\mathrm{k}\mathrm{m}$ NS.

To compensate for the fact that we do not fully calculate the structure of the
TŻO down to the surface of the NS, we inject energy at the inner boundary of
our model. This energy injection approximates the missing energy generated
from accretion and nuclear burning below the knee in the “middle” region.
These assumptions are tested in Section 6. We parameterise this inner energy
injection based on the Eddington luminosity onto the NS as:

$L_{\rm{knee}}=\epsilon_{L}L_{\rm Edd}$ (1)

where the Eddington luminosity, $L_{\rm Edd}$, is:

$L_{\rm Edd}=\frac{4\pi cGM_{\rm NS}}{\kappa_{c}}$ (2)

with $G$ the standard gravitational constant, $M_{\rm NS}$ is the mass of the
neutron star, $c$ is the speed of light, and $\kappa_{c}$ of the opacity of
the material at the inner boundary of the model. $L_{\rm{knee}}$ is allowed to
evolve with time as the mass of the NS and the opacity at the inner boundary
of the TŻO changes. Finally, $\epsilon_{L}$ is an efficiency factor, for which
by default we use $\epsilon_{L}=1.0$. We use $L_{*}$ to denote the outer
surface luminosity of our models.

By parameterising the energy in this way we can control how deep below the
knee we model. The point where the local luminosity $L>L_{\rm Edd}$ is the
point where radiation can no longer carry the energy and the envelope becomes
convective. Thus when $\epsilon_{L}=1.0$ the inner boundary of the structure
we are calculating corresponds to the knee, which is also the base of the
convection zone (C93). Smaller values of $\epsilon_{L}$ include more of the
material below the knee in the calculation but come at a greater computational
cost. By injecting energy in this way we are assuming that the material below
the knee is actually able to generate that much energy. It is possible that
some of our models would not be able to do this, in which case the
corresponding TŻO may not form. We are also assuming that the material below
the knee is not mixed into the envelope, which may change the nucleosynthetic
signatures.

We assume the NS grows at a rate set by the Eddington accretion rate on the
NS:

$\dot{M}_{\rm Edd}=\epsilon_{\dot{M}}\frac{L_{\rm Edd}}{c^{2}}$ (3)

where $\epsilon_{\dot{M}}$is a scale factor which by default we set as
$\epsilon_{\dot{M}}=1.0$. This leads to typical accretion rates of $\sim
10^{-8}\,\rm{M}_{\odot}\,\rm{yr}^{-1}$ for $\epsilon_{\dot{M}}=1$.

Our default model assumes an initial metallicity of $\rm{Z_{init}}=10^{-4}$
and initial helium fraction $\rm{Y_{init}}=0.28$. Our default TŻO has an
initial total mass of $\rm{M_{init}}=5\,\,\rm{M}_{\odot}$, which includes a
default NS $M_{\rm NS}=1.4\,\,\rm{M}_{\odot}$, thus having an envelope mass of
3.6$\,\rm{M}_{\odot}$. After 100 years of evolution, we enable MESA’s
hydrodynamic capabilities (Paxton et al., 2015). This wait is to allow the
star to return to gravothermal equilibrium after the TŻO formation process. We
also explore a series of model grids of TŻOs with masses between
$\rm{M_{init}}=5$–20$\,\rm{M}_{\odot}$, $\rm{Z_{init}}=10^{-5}$–0.03, and
$\rm{Y_{init}}=0.28$–0.65.

We conservatively stop our models once the velocity of the surface layers
exceeds 10% of the escape velocity. At this stage the models undergo RSG
pulsations, where there are large-amplitude surface pulsations which cause
spiral patterns (Heger et al., 1997) in a Hertzsprung-Russell diagram (HRD).
This can change the stellar radius by factors of $\approx 2$ over timescales
of years. At this point the envelope begins expanding and contracting
supersonically, and can reach $\sim 40\%$ of the escape velocity before we can
longer follow the evolution. These pulsations are resolvable in models of RSGs
in MESA when the timestep becomes much shorter ($\sim$days) than the pulsation
timescale ($\sim$years) (Paxton et al., 2013). The timestep becomes this small
due to changes in the nuclear burning (see Section 4.2). While the evolution
can be continued beyond this point by suppressing the hydrodynamics either
globally or only in the cool outer envelope, the results become numerically
unstable. More work is needed to determine how to couple the instabilities in
the nuclear burning with the pulsational instabilities in the envelope.

We evolve the majority of our models using MESA’s approx21.net nuclear
network. This contains the main PP, CNO, and alpha capture reactions up to
$\mathrm{{}^{56}Fe}$. This is a computational convenience. While TŻOs are
expected to undergo irp-burning, most of the energy is generated via CNO
burning. We also test a large 399 nuclear network for a limited set of models.
This network was built by first adding all stable isotopes up to
$\mathrm{Ru}$, then adding an additional neutron to the heaviest stable
isotope for each element included. We then added all proton-rich isotopes with
half-lives $>1$s. Finally, there was some hand tuning to make sure there were
sufficient beta-decay pathways for all isotopes, as well as adding the light
isotopes needed for PP, CNO, and the CNO breakout reactions. This network is
available in the online Zenodo material.

We assume a mixing-length alpha parameter of $\rm{\alpha_{mlt}}=1.8$, and use
MESA’s time-dependent convection (TDC) mixing treatment (Jermyn et al., 2023).
TDC has been shown to improve numerical stability during dynamical phases of
evolution while reducing to the standard Cox-MLT prescription (Cox & Giuli,
1968) over longer timescales (Jermyn et al., 2023). In models that have
material below the base of the convection zone (when $\epsilon_{L}<1.0$), we
assume that there is an additional weak mixing process occurring, with a
diffusion coefficient of $10^{6}\,\rm{cm^{2}/s}$ in the material below the
knee. This helps to prevent compositional gradients from building up near the
surface of the NS (Piro & Bildsten, 2007; Keek et al., 2009). We do not
include convective overshoot, semiconvective, or thermohaline mixing, as the
star is almost fully convective.

Our EOS for the stellar component of the TŻO is a combination of freeEOS
(Irwin, 2004), HELM (Timmes & Swesty, 2000), and Skye (Jermyn et al., 2021).
We make no assumptions about the NS EOS, as we can not make models with
$10\lesssim R_{\rm NS}/\mathrm{k}\mathrm{m}\lesssim 20$, where NS radii are
expected to be. We use the wind prescription of van Loon et al. (2005), which
is based on observations of cool, dusty AGB & RSG stars, with a wind scaling
factor of $\eta_{\rm{\rm VL}}=1.0$. This leads to typical wind mass loss rates
of $\rm{\dot{M}}\approx 10^{-5}$–$10^{-4}\,\rm{M}_{\odot}\,\rm{yr}^{-1}$.

MESA does not include any general relativistic (GR) correction factors. For
this work, we add a GR correction factor to correct the continuity equation
(Thorne, 1977; Ayasli & Joss, 1982). This prescription adjusts the
gravitational constant, $G$, as a function of the mass coordinate inside the
star. All masses in this work are baryonic. Other MESA choices are specified
in Appendix C and all options can be found in the online Zenodo inlists.

Our models have on average $\approx 1700$–$2000$ mesh points and we
artificially cap the maximum timestep to be $\delta t=2\times 10^{8}\,\rm{s}$
($\approx 6$ years). While models can take longer timesteps, this comes at the
cost of an increased number of timesteps that needed to be rejected and taken
again with a smaller $\delta t$ as MESA could not find a valid solution that
satisfies our required numerical constraints. Variations in spatial and
temporal resolution of by factors of two smaller/larger lead to changes of
order $\Delta\log\left(\rm{L_{*}/L_{\odot}}\right)\approx 0.01\,\rm{dex}$.

##  3 Structure and evolution of a TŻO

Figure 1: Hertzsprung-Russell diagram (HRD) of the TŻO as a function of their
initial mass, at fixed initial composition. Models are evolved with our
default assumptions and a $M_{\rm NS}=1.4\,\,\rm{M}_{\odot}$. Colours indicate
the initial mass of the TŻOs. The direction of evolution is always towards
decreasing luminosity. The grey boxes show the observational constraints on HV
2112 (Levesque et al., 2014) and VX Sgr (Tabernero et al., 2021). The
horizontal dashed grey line shows the quoted luminosity of HV 11417 (Beasor et
al., 2018), which has a variability amplitude of $\Delta m=1.86$ (Soszyński et
al., 2009). The grey lines show the models of C92, where evolution is always
to increasing luminosity. Arrows mark the midpoint of the TŻOs lifetime.
Figure 2: Top-panel: Temperature-density profiles our of our default TŻO
models as a function of initial mass, at fixed initial composition. Models are
evolved with our default assumptions and a $M_{\rm NS}=1.4\,\,\rm{M}_{\odot}$.
Colours indicate the initial mass of the TŻOs. Dash-dotted lines show Model A
and C from figures 6 and 7 of C92. Bottom-panel: The opacity as a function of
the density inside the star, dash-dotted lines are from tables 1 and 2 of C92.
Models shown are $\approx 40,000$ years post TŻO formation. The red line marks
the edge of the pair-instability region where electron-positron production
becomes significant.

Figure 1 shows the evolution of the surface temperature and luminosity of our
default TŻO models. Figure 1 shows the models once they are at least 100 years
old (when we turn on the hydrodynamics), where they start from a zero-age TŻO
(ZATŻO) line. The endpoint of the models will be discussed further in Section
7, as shown it is when the evolution reaches 1000 years before we can no
longer follow the evolution. We can see that models with higher initial masses
have higher initial luminosities and temperatures compared to lower-mass
models. Higher-mass models also evolve to lower final surface temperatures and
live longer.

Contrary to the models of C92 our models always evolve to lower luminosities.
The total luminosity of our TŻOs is a combination of the injected energy,
$L_{\rm{knee}}$, and nuclear burning above the knee. The nuclear burning above
the knee provides $\approx 1\%$–$10\%$ of the total luminosity, and this
fraction decreases with time as the knee cools. Thus the evolution is driven
by changes in $L_{\rm{knee}}$, where $L_{\rm{knee}}\propto M_{\rm
NS}/\kappa_{c}$. The accretion rate on to the NS is
$10^{-9}$–$10^{-8}\,\rm{M}_{\odot}\,\rm{yr}^{-1}$, which combined with a
lifetime of $\approx 100,000$ years means the NS can only gain
$10^{-4}$–$10^{-3}\,\,\rm{M}_{\odot}$. Thus the evolution of our TŻOs can be
approximated as entirely driven by the increase in the opacity of material at
the knee, which lowers $L_{\rm{knee}}$.

The opacity for the material at the knee is provided by Compton scattering
(Poutanen, 2017) and thus depends on the number of free electrons per nucleon.
Three factors will drive the evolution of the number of free electrons: H
burning into He will decrease the number of free electrons, proton captures in
rp-burning will decrease the number of nucleons, and $e^{\pm}$ production will
increase the number of free electrons. If we turn off composition changes from
nuclear burning (by setting dxdt_nuc_factor=0) while preserving the energy
generated from nuclear burning, we find $L_{\rm{knee}}$ still decreases.
However, it does this at a much faster rate than when the composition is
allowed to change. Therefore, nuclear burning slows the decrease in
$L_{\rm{knee}}$ but can not stop it. Thus, the changes in $L_{\rm{knee}}$ are
driven by the production of $e^{\pm}$ pairs, while not entering the pair-
instability region.

Our results differ from those of C92 due to differences in the chosen stopping
criteria. Our models stop when supersonic pulsational instabilities form in
the envelope, preventing MESA from continuing the evolution. The end condition
for C92 is when the NS grows to the Oppenheimer-Volkoff (OV) (Oppenheimer &
Volkoff, 1939) mass limit for the their assumed EOS, which for C92 is
2$\,\rm{M}_{\odot}$. As the NS mass increases, $L_{\rm{knee}}$ will increase,
increasing the total luminosity of the star in agreement with the results of
C92. However, our models stop significantly earlier than C92 such that the NS
only gains $10^{-4}$–$10^{-3}\,\,\rm{M}_{\odot}$, which does not cause
$L_{\rm{knee}}$ to increase by a significant amount. We note in passing that
C92 states they can take single timesteps of $\approx 10^{5}$ years, which is
comparable to the entire lifetime of many of our models which is between
50,000 and 200,000 years for the TŻOs shown in figure 1.

We can replicate the results of C92 by increasing the accretion rate onto the
NS. If the accretion rate onto the NS is high enough that the mass of the NS
can grow significantly (at least $\approx 0.1\,\,\rm{M}_{\odot}$) then $L_{\rm
Edd}$ will increase in time. This causes the TŻO to become more luminous and
more closely follow the tracks of C92, though still at higher surface
temperatures and lower surface luminosities. This requires accretion rates of
between $10^{3}$–$10^{4}\,\dot{M}_{\rm Edd}$ (See Appendix A). The remaining
differences between our models can likely be attributed to changes in the
microphysics (EOS, opacities, and nuclear reaction rates) and choice of
metallicity.

Figure 2 shows the temperature-density profile and opacity-density profile
inside our TŻOs arbitrarily at $\approx 40,0000$ years after TŻO formation. As
the initial mass of the model increases the density also increases and the
models reach higher temperatures at the base of the convection zone. Our
models do not form a knee, due to the energy injection at the inner boundary
of the models, so we are only modelling material above the knee. The dash-
dotted lines show models A & C of C92. We can see that our models are
significantly denser (by a factor $\approx 100$) than previously predicted,
though this is still a factor $\approx 100$ less dense than a typical AGB/RSG.

None of our models ever evolve significantly into the pair-instability region
(where $\Gamma_{1}<4/3$). Instead, models evolve around the edge of the pair-
instability region (See Section 6). Note that the temperature and density of
the knee evolves with time, but always stays outside of the pair-instability
region.

The bottom panel of Figure 2 shows the opacity of our models in colour, and
models A & C of C92 with grey dash-dotted lines. When
$\log(\rho/\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3})\leq 0.0$ our models have
slightly higher opacities, as they are cooler for the same density. Once
$\log(\rho/\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3})>0.0$ our models have
higher opacities than model A, but lower opacities than those of model C
(which enters the pair-instability region and so $e^{\pm}$ production
dominates the opacity).

In C93 they find a set of “high” and “low” mass solutions, which depends on
how the energy is generated at the knee, and the production of $e^{\pm}$ due
to models entering the pair-instability region which changes the required
$L_{\rm{knee}}$ as the opacity increases (these are also the “giant” and
“supergiant” models of Thorne & Zytkow 1977). C93 finds a gap where, for
certain masses, models are unable to produce sufficient energy to keep the
envelope convective. Our models do not exhibit this “luminosity gap” (which
can be mapped into a mass gap) in the mass distribution of TŻOs. This is due
to the higher density of our models, such that they do not enter the pair-
instability region. Therefore our models would not be limited by the same
mechanism as proposed by C93 and thus we should not expect a split into “high”
and “low” mass solutions, even if we did model the knee. C92 also finds valid
solutions for all masses, where C93 attributes this difference due to changes
in the nuclear reaction rates used.

###  3.1 Composition effects

Figure 3 shows the average metal fraction of our TŻOs as a function of time
since the TŻO formed. These models start at $Z=10^{-4}$ but can increase their
total metal fractions by factors of 10-100 within $\approx 10,000$–100,000
years. Higher initial mass models evolve to higher total metal fractions,
reaching near solar values for their total metallicity, though this is a
significantly non-solar scaled composition. They also evolve faster to higher
metallicities than the low-mass models and live longer. The ages shown are
only lower limits on the lifetime of a TŻO (see Section 7), but likely lead to
lifetimes of $\approx 10^{4}$–$10^{5}$ years.

Figure 3: The evolution with time of the average metal fraction (Z) as a
function of the initial mass, at fixed initial composition. Models are evolved
with our default assumptions and $M_{\rm NS}=1.4\,\,\rm{M}_{\odot}$. Colours
indicate the initial mass of the TŻOs. The final ages of these models are only
lower limits to the lifetime of a TŻO.

(a) Initial metallicity

(b) Initial helium fraction

Figure 4: HRD for variations in the initial metallicity (panel a) and initial
helium fraction (panel b). Solid lines denote a $5\,\,\rm{M}_{\odot}{}$ TŻO,
while dashed lines denote a $10\,\,\rm{M}_{\odot}{}$ TŻO. Grey lines and grey
boxes have the same meaning as in Figure 1.

Figure 4(a) shows the HRD of TŻOs with varying $\rm{Z_{init}}$ with initial
masses of 5 and 10$\,\rm{M}_{\odot}$. The TŻOs can be broken into two groups
of behaviour, a low initial metallicity and a high initial metallicity
behaviour. The exact initial metallicity at which this behaviour changes
depends on the initial mass, for the 5$\,\rm{M}_{\odot}$ models this is at
$\rm{Z_{init}}\approx 3\times 10^{-4}$ while for the 10$\,\rm{M}_{\odot}$
models this occurs at $\rm{Z_{init}}\approx 10^{-3}$. This limit is
approximately the final metallicity shown for each mass in Figure 3 (though
the match is not exact). Effectively the low-metallicity models evolve up to
the mass dependent critical $Z$, with the evolution converging to the same
final surface temperature and luminosity once they reach the critical $Z$. In
contrast, the models that start with metallicities above this critical limit
follow their own $Z$-dependent evolutionary tracks. We can also see in Figure
4(a) that the high-metallicity models evolve along similar tracks independent
of the initial mass (except for their starting luminosities). We also note
that as the initial metallicity increases the 5$\,\rm{M}_{\odot}$ models are
more likely to become numerically unstable.

As the metallicity increase (either due to a higher initial metallicity or due
to metals produced from nuclear burning), the opacity at the knee increases
(Xin et al., 2022). This decreases $L_{\rm{knee}}$ and thus decreases the
temperature at the knee. This decreases the amount of heavy metal burning as
the metallicity increases, though the rate of CNO burning increases as the
metallicity increases due to the increased amount of CNO material. Thus the
nuclear burning luminosity is greater in the high metallicity models, but the
rate of production of all metals is lower. Hence at LMC and SMC metallicites
TŻOs may show little metal enrichment. The high metallicity models can have
lifetimes up to a factor $\approx 2$ more than shown in Figure 3.

Figure 4(b) shows variations in the initial helium fraction $\rm{Y_{init}}$,
for a 5 and 10$\,\rm{M}_{\odot}$ TŻO at $\rm{Z_{init}}=10^{-4}$. As the
initial helium fraction increases, the models become more luminous and have
higher surface temperatures. The variation in the initial helium fraction may
be expected due to variations in the evolutionary state of the companion when
the TŻO merger takes place. As $\rm{Y_{init}}$ increases the final luminosity
they reach increases and is at higher surface temperatures. These changes are
due to the knee decreasing in temperature as $\rm{Y_{init}}$ increases. This
temperature decrease also decreases the rate of metal production.

The upper limit of $\rm{Y_{init}}=0.65$ in Figure 4(b) is due to numerical
issues, which prevents the modelling of TŻOs with $\rm{Y_{init}}=0.65$–0.95.
While there are differences between H-rich and He-rich TŻOs, they both occupy
similar regions of a HRD, appearing as RSGs. It is interesting to speculate
that, if more He-rich TŻOs can form, there may exist a population of H-poor
RSGs. While we are unaware of any He-lines detectable in a cool RSG
atmosphere, H-lines are readily detectable. Thus weak or no detectable H-lines
in a RSG spectrum could indicate a very He-enriched TŻO.

###  3.2 Comparison with proposed TŻO candidates

Here we briefly compare observed TŻO candidates with our models at both our
default composition (Fig. 1) and for alternative compositions (Fig. 4). We
emphasize that our default model has $\rm{Z_{init}}=10^{-4}$, i.e. much lower
than that of the SMC or LMC.

HV 2112’s temperature and luminosity can be well approximated by models using
with an initial mass between
$5\lesssim\rm{M_{init}}/\,\rm{M}_{\odot}{}\lesssim 8$ when adopting our
default composition. HV 2112 can also be well fitted by our
5$\,\rm{M}_{\odot}$ TŻO models with $\rm{Z_{init}}\approx 10^{-3}$, though for
models at this metallicity we would not expect to see any metal enrichment
(see Section 3.1). Thus while HV 2112 was initially proposed as a TŻO
candidate due to its unusual chemical composition, _we believe this unusual
composition now rules out HV 2112 being a TŻO_.

VX Sgr’s position in the HRD can be matched with a TŻO model of mass
$\rm{M_{init}}\gtrsim 8\,\,\rm{M}_{\odot}$ at our default metallicity, and is
also consistent with our $10\,\,\rm{M}_{\odot}$ models with modified
composition.

HV 11417 has a luminosity which is lower than any of our default-composition
models in Fig. 1. However, lower-mass NSs can allow models to drop to the
luminosity of HV 11417 with our default composition (see Appendix A), and
several of the higher-metallicity $5\,\,\rm{M}_{\odot}$ models shown in Fig. 4
reach the luminosity of HV 11417 even for our canonical NS mass.

##  4 Pulsations

Figure 5: Pulsation properties of our TŻOs with default model assumptions.
Panel a: the evolution of the pulsation period as a function of time for the
fundamental (solid), first (dashed), and second overtones (dot-dashed). Panel
b: Petersen diagram showing the ratio of the first overtone to the fundamental
(solid), and the second to the fundamental (dashed). Panel c: The pulsation
period as a function of the surface luminosity, lines have the same meaning as
panel a. Colours indicate the initial mass of the TŻO. Grey dashed lines in
panel a mark the observed pulsation periods of HV 2112 (Soszyński et al.,
2011).

###  4.1 Hydrostatic pulsations

RSGs may pulsate due to the $\kappa$-mechanism in hydrogen ionisation zones,
with periods between $\approx 100$–1000 days (Fox & Wood, 1982; Heger et al.,
1997). Longer period pulsations may also occur, but observations can be
hindered by a lack of long baseline photometry (Soraisam et al., 2018).
Observations of RSG pulsations are complicated by irregular photometric
variability, presumed to be caused by the interaction of large convection
cells (which can have sizes of order the stellar radius) and the pulsation
modes (Kiss et al., 2006).

Using the GYRE v6.0 (Townsend & Teitler, 2013; Townsend et al., 2018) stellar
oscillation code, we compute the adiabatic pulsations for our TŻOs with
$\rm{M_{init}}=5$–$20\,\,\rm{M}_{\odot}$ for the radial ($l=0$) modes. Figure
5 shows the time evolution of the pulsation period, Figure 5 shows a Petersen
diagram (Petersen, 1973) of the period ratios, while Figure 5 shows the
period-luminosity relationship.

In 5 we can see that the fundamental mode is always greater than 1000 days,
and this increases with time. The first and second overtones are $\approx 500$
and $250$ days respectively. The highest-mass models start with lower periods
than the low-mass models. While exploring other parameters, we find that the
period of the fundamental mode is predominately sensitive to $\rm{Y_{init}}$,
potentially providing a way to constrain the helium composition of a TŻO.

Figure 5 shows a Petersen diagram of the period ratios compared with the
fundamental period. We can see that in general the period ratio is small
($\approx 0.2$–0.3), decreases as the period increases (and thus decreases
with time), as well as decreasing with initial mass. OGLE observations of the
SMC and LMC (Soszyński et al., 2004), show most objects with
$\log(\rm{P/days})>3$, have period ratios $\approx 0.1$. Therefore our TŻOs
exist in a region of parameter space distinct from most other objects,
potentially providing an easy method to search for TŻO candidates.

Finally, Figure 5 shows the period-luminosity relationship for our TŻOs. We
can see a tight relationship between their pulsation period and luminosity.
Thus TŻOs may prove useful for determining distances if they can be detected
and identified.

HV 2112 has a measured set of pulsation periods from OGLE of 165.59, 604.4,
and 887.3 days (Soszyński et al., 2011). From Figure 5 it is difficult to
match the models to the measured periods of HV 2112. To match the 604.4 and
887.3 day periods will require the first and second overtones to increase
their periods by a factor of $\approx 2$.

Variations in other model parameters (Appendix A) lead to changes in the
predicted pulsation periods. However, no parameter variation we have tested is
able to fit the measured periods of HV 2112, over the time we have evolved our
TŻOs for (though our grid is not exhaustive). The pulsation periods are
predominately set by the initial mass, the mass of the NS,
$\rm{\alpha_{mlt}}$, and the initial helium fraction, while not being
sensitive to the initial metallicity. The fundamental mode is most strongly
affected by $\rm{Y_{init}}$, with $\rm{Y_{init}}$=0.65 able to have $P>10^{5}$
days, while the first overtone is more strongly affected by
$\rm{\alpha_{mlt}}$.

The closest match in our (non-comprehensive) grid is from the
5$\,\rm{M}_{\odot}$ TŻO with $\rm{\alpha_{mlt}}=3.0$, with a first overtone
period of $\approx 750$ days. Based on 3D models values of
$\rm{\alpha_{mlt}}=3$–4 have been proposed for RSG envelopes (Goldberg et al.,
2022). We speculate a slightly more massive TŻO with a relatively high
$\rm{\alpha_{mlt}}$ would be able to better fit the pulsation periods of HV
2112. If HV 2112 were a TŻO with a high $\rm{\alpha_{mlt}}$, then we would
predict that we have observed the first, second, and third overtones, and that
the fundamental mode is currently undetected with a period in the 1500–3000
day range.

VX Sgr has a measured pulsation period of 757 days, with a possible longer
period at 28,279 days (Tabernero et al., 2021). Though Tabernero et al. (2021)
caution the 28,279 days is comparable to the total time span that VX Sgr has
been observed for, and thus may be an artifact of the observing period. There
are also a number of shorter pulsations that vary in period and amplitude,
around $~{}600$ days. The lack of a detected period between 1000-3000 days
rules out most of our TŻO models. The only models that can match the 28,279
day period (assuming that it is real), is our most helium enriched model with
$\rm{Y_{init}}=0.65$. This model has a fundamental mode that increase with age
and can be between $P\approx 10^{3}$–$10^{5}$ days. Thus if VX Sgr where a TŻO
it would imply the merger occurred with an evolved star after significant
amounts of helium had been produced in the companion.

HV 11417 has measured pulsation periods of 214, 793, and 1092 days (Soszyński
et al., 2009, 2011). Based on Figure 5, the 1092 day period would imply it was
a very young ($\lesssim 2000$ year) and massive ($\sim 20\,\rm{M}_{\odot}$)
TŻO. However models in this mass and age range would be inconsistent with the
reported luminosity of HV 11417 by $\sim 0.6$ dex. Thus we conclude HV 11417
is inconsistent with our a TŻO models.

###  4.2 Oscillations in the nuclear burning rate

Figure 6 shows the total nuclear energy integrated over our default
5$\,\rm{M}_{\odot}$ TŻO for the
$\rm{\mathrm{{}^{12}C}(p,\gamma)\mathrm{{}^{13}N}}$ and
$\rm{\mathrm{{}^{14}N}(p,\gamma)\mathrm{{}^{15}O}}$ reactions. These two rates
constitute 85% of the nuclear energy generated in our models (with the
approx21.net) network. As the approx21.net network approximates CNO burning,
$\rm{\mathrm{{}^{12}C}(p,\gamma)\mathrm{{}^{13}N}}$ is a proxy for the chain
$\rm{\mathrm{{}^{12}C}(p,\gamma)\mathrm{{}^{13}N}(,e^{+}\nu)\mathrm{{}^{13}C}(p,\gamma)\mathrm{{}^{14}N}}$,
while $\rm{\mathrm{{}^{14}N}(p,\gamma)\mathrm{{}^{15}O}}$ proxies the chain
$\rm{\mathrm{{}^{14}N}(p,\gamma)\mathrm{{}^{15}O}(,e^{+}\nu)\mathrm{{}^{15}N}}$.

When the TŻOs are $\lesssim 24,000$ years old the variations in the reaction
rates are small and approximately the size of the linewidth in Figure 6. Once
the TŻO reaches $\approx 24,000$ years the reaction rates begin showing a
medium level variability, with a short period variability of a few years
embedded in a longer $\approx 100$ year period cycle (which is shown in the
insert). This leads to a variation in the surface luminosity of
$\Delta\rm{\log\left(L/L_{\odot}\right)}\approx 0.001$ dex. Then when the TŻO
reaches $\approx 50,000$ years, the model undergoes large variations in the
reaction rates. The energy generation rate increases by a factor $\approx 15$
to $\approx 10^{38}\,\rm{erg/s}$ and the surface luminosity can vary by
$\Delta\rm{\log\left(L/L_{\odot}\right)}\approx 0.5$ dex. This is when we can
no longer follow the evolution with hydrodynamics included in the model,

We can locally suppress the hydrodynamics in the envelope by using
velocity_logt_lower_bound=5, which turns off the hydrodynamics in zones where
the local temperature is $\rm{\log\left(T_{eff}/K\right)}<5.0$ which allows
the evolution to continue for another $\approx 40,000$ years. As the TŻO
evolves at this point the nuclear energy generation rates drops, returning to
its pre-large amplitude burning values, before once again becoming numerically
unstable and the evolution ceases.

Figure 6: The nuclear energy released, integrated over the TŻO for the two
most energetic reactions in our default 5$\,\rm{M}_{\odot}$ TŻO evolved with
the approx21.net. These two reactions provide $\approx 85\%$ of the star’s
nuclear energy. The inset shows a zoom-in where the reaction rates begin to
undergo medium oscillations. These oscillations are still within the TŻO’s
hydrostatic phase of evolution. At $\approx 50,000$ years the nuclear energy
generation rate increases rapidly and exceeds the plot limits while becoming
hydrodynamic. Text labels denote the approximate boundaries between the
different amplitudes of the pulsations.

###  4.3 Hydrodynamic pulsations

Convective envelopes are known to be able to have dynamical instabilities that
can lead to mass loss (Paczyński, 1969; Tuchman et al., 1978, 1979). Given a
suitable excitation these dynamical instabilities can lead to mass loss rates
of $\sim 10^{-3}\rm{M}_{\odot}\,\rm{yr}^{-1}$ through pulsational mass loss
events (Clayton et al., 2017). Here we show what happens to our TŻOs as they
undergo RSG pulsations.

Figure 7: The surface temperature and luminosity for a our default
5$\,\rm{M}_{\odot}$ TŻO during the dynamical pulsations. This plot shows
$\approx 180$ years of evolution. Figure 8: The radial velocity of the surface
layers of our $5\,\rm{M}_{\odot}{}$ default TŻO over time. The right-hand axis
shows the approximate fraction of the escape velocity that the material
reaches. The x-axis is the time until we can no longer follow the evolution
with MESA.

Figure 7 shows the HRD of our 5$\,\rm{M}_{\odot}$ default TŻO during this
dynamical phase. Note this is only a representative plot, the exact shape of
the spiral and number of cycles we can follow depends sensitively on the input
physics and numerical resolution. We can see that as the star evolves the
change in surface temperature and luminosity increases with each additional
cycle.

Figure 8 shows the radial velocity of the surface layers of our default
5$\,\rm{M}_{\odot}$ TŻO. As the TŻO evolves the surface layers can reach
velocities of $\approx\,10\,{\mathrm{km\ s^{-1}}}$, and during the final
contraction phase becomes supersonic. These velocities are of a similar
magnitude to those found in Yoon & Cantiello (2010) for the evolution of
massive stars with pulsation-driven superwinds. This suggests that we should
expect pulsation-driven mass loss in a TŻO. Yoon & Cantiello (2010) finds the
mass loss rate may increase up to $10^{-2}\,\rm{M}_{\odot}\,\rm{yr}^{-1}$.
These RSG pulsations are resolved when the timestep of our models drops
significantly below the pulsation period. This occurs in our TŻOs when the
nuclear burning rate spikes, as seen in Figure 6, but can occur at earlier
times if we artificially enforce a short timestep. If the mass loss rates are
at the upper end of those predicted by Yoon & Cantiello (2010) then this
implies a lifetime of $\approx$ 100–1000 years. However, our wind mass loss is
based on that of van Loon et al. (2005) which is based on the observations of
RSG’s and thus has this mass-loss built into the time averaged mass-loss
rates. More work is needed to understand the wind/pulsational mass loss rates
in RSG’s, as this will set the lifetime of a TŻO.

##  5 Nucleosynthesis

Figure 9 shows the surface composition relative to the initial composition for
TŻOs evolved with a fully coupled 399 isotope nuclear network at $\approx
10,000$ years after formation. Figure 9(a) shows the composition as
$\rm{Y_{init}}$ is varied, for a fixed $\rm{M_{init}}=5\,\rm{M}_{\odot}$ and
$\rm{Z_{init}}=10^{-4}$. Figure 9(b) shows the composition as $\rm{M_{init}}$
is varied for fixed $\rm{Y_{init}}=0.28$ and $\rm{Z_{init}}=10^{-3}$. A HRD
for the models in Figure 9(a) can be found in Appendix A. We include all
isotopes produced in our models, without taking into account any radioactive
decay. A comparison with the results of C93 can be found in Appendix B. We
note, as a word of caution, that this abundance pattern is sensitive to the
choice of the initial composition, as the initial metals act as seed nuclei
for the rp-burning.

For the $\rm{Z_{init}}=10^{-4}$ we find that for $Z<20$, there is no
enhancement relative to the initial composition. For $21\leq Z\leq 25$ (Sc to
Mn) there is an enhancement that increases as $\rm{Y_{init}}$ increases, while
both Fe and Ca are not enhanced. For higher atomic numbers the enhancement
relative to the starting composition becomes larger, up to $\approx 10^{5}$
their starting values. As $\rm{Y_{init}}$ increases the most enhanced element
decreases in atomic number, for $\rm{Y_{init}}=0.28$ it is Br,
$\rm{Y_{init}}=0.4$ it is As, and $\rm{Y_{init}}=0.6$ it is Ga. After this
peak element, there is a rapid decline in the production of heavier elements.
Mo is only enhanced in our $\rm{Y_{init}}$=0.28 model and is thus not a good
element for determining a TŻO status.

These differences are due to the knee temperature decreasing as
$\rm{Y_{init}}$ increases. This lowers the maximum mass of an element that can
be produced in the irp-process. We reconfirm the previous findings of Biehle
(1991) and show that we can have significant enhancement of the elements Rb,
Sr, Y, Zr and Mo but this depends sensitivity on the initial He composition.
We find that models with very high initial helium fractions can lack Rb. Thus
there could be a population of helium-rich TŻOs without Rb, possibly
explaining the difficulty in confirming the detection of a TŻO with the Rb
mass fractions alone.

For the $\rm{Z_{init}}=10^{-3}$ models we see almost no enhancement of metals
at all. There is a slight enhancement of N due to CNO burning as well as an
enhancement of S–Ar for the $\rm{M_{init}}=5\,\rm{M}_{\odot}$ model, and
enhancements in Ca–Mn for the $\rm{M_{init}}=10\,\rm{M}_{\odot}$ and
$15\,\rm{M}_{\odot}$ models. However there is no production of elements with
$Z>25$. This is due to the higher opacity at the knee, lowering the knee
luminosity, which lowers the knee temperature. Thus, at metallicities
comparable to the SMC, LMC, or MW, TŻOs are unlikely to be distinguishable
from non-TŻO stars based on their surface composition alone of elements such
as Rb or Mo.

Table 1 shows the surface mass-fraction ratios for a selected set of elements
for the $\rm{Z_{init}}=10^{-4}$ models. Also shown is a comparison with Model
A of C93. We can see that the estimates for Ni/Fe are similar for all initial
helium fractions. Estimates for Rb/Ni and Rb/Fe depend on the initial helium
fraction but can be consistent with C93 for $0.28<\rm{Y_{init}}<0.4$ (C93
assumes $\rm{Y_{init}}$=0.32). Finally, Mo/Fe is extremely different, with our
models predicting very little absolute Mo, with the valued decreasing as
$\rm{Y_{init}}$ increases. Other ratios including K/Ca and Ca/Fe are a factor
10 smaller than C93.

(a) Variations in $\rm{Y_{init}}$

(b) Variations in $\rm{M_{init}}$

Figure 9: The surface composition relative to the initial composition at
10,000 years post-TŻO formation. Left panel: The composition as a function of
of the proton number Z for a 5$\,\rm{M}_{\odot}$ TŻO and
$\rm{Z_{init}}=10^{-4}$ with $\rm{Y_{init}}=0.28$ (red square),
$\rm{Y_{init}}=0.4$ (orange triangle), $\rm{Y_{init}}=0.6$ (blue star). Right
panel: The composition as a function of of the proton number Z for
$\rm{Y_{init}}=0.28$ and $\rm{Z_{init}}=10^{-3}$, with
$\rm{M_{init}}=5\,\rm{M}_{\odot}{}$ (red square),
$\rm{M_{init}}=10\,\rm{M}_{\odot}{}$ (orange triangle), and
$\rm{M_{init}}=15\,\rm{M}_{\odot}{}$ (blue star). All models were evolved with
a fully coupled 399 isotope nuclear network. Vertical lines mark elements that
may be useful for detecting TŻOs. Note the change in the y-scale between
panels. Data tables are available in the online Zenodo material with the time
evolution of the composition.  Figure 10: The total isotopic flow rate due to
nuclear reactions for a selected set of isotopes in our default
5$\,\rm{M}_{\odot}$ TŻO evolved with an 399 isotope nuclear network. The
coloured lines show the logarithm of the net number of reactions per second
for each isotope, showing only the most significant reaction, taking into
account both forward and reverse reactions and summed over the entire TŻO.
Arrows show the direction of flow. The model shown is $\approx 10,000$ years
post-TŻO formation. Isotopes with no arrows have total reaction rates less
than the lower limit of $10^{10}$ reactions/s. The atomic mass is quoted for
each isotope in each box.

Figure 10 shows how the isotopic composition will change over time, by showing
the reaction flow for the dominant reaction for each isotope. Diagonal lines
to the lower right are due to beta decays. There are insufficient free
neutrons for the $(n,p)$ reactions to dominate as both $\mathrm{{}^{13}C}$ and
$\mathrm{{}^{22}Ne}$ have mass fractions $X\lesssim 10^{-17}$ at the base of
the envelope. Vertical lines indicate a proton capture, while horizontal lines
denote neutron captures. $\alpha$-captures are included in the network but are
rarely significant in this mass range. Isotopes with no lines are included in
the network but have all their reaction rates $<10^{10}$ reactions/s.

Isotopes are predominately decaying via beta-decays faster than proton
captures can increase the atomic number of the isotope. Thus the flow is
towards more neutron-rich isotopes, rather than more proton-rich isotopes.
This prevents significant rp-burning from occurring and causes the limited
amount of Mo shown in Figure 9(a). The maximum atomic number that is reached,
before beta-decays outpace the proton captures will depend on the peak
temperature reached by the TŻO at the base of the convection zone (Fisker et
al., 2008). Thus, given Figure 2, the peak atomic number reached depends on
the initial mass as well.

We have tested our choice of nuclear network in Appendix B. Changing our
nuclear network to one that closely matches that of C93, has little effect on
the results (see Appendix B). Though by extending to higher atomic numbers we
can see a similar turn over in the mass fraction pattern in C93 as we see,
except for C93 this occurs at higher atomic numbers.

Element | $\rm{Y_{init}}=0.28$ | $\rm{Y_{init}}=0.40$ | $\rm{Y_{init}}=0.60$ | C93
---|---|---|---|---
ratio |  |  |  | Model A
Rb/Ni | 2.18E+00 | 3.00E-03 | 9.94E-06 | 1.82E-01
Rb/Fe | 3.38E-01 | 1.07E-03 | 8.04E-06 | 1.48E-01
Li/Ca | 1.55E-04 | 1.49E-04 | 1.23E-04 |
Li/K | 2.67E-03 | 2.67E-03 | 2.64E-03 |
Mo/Fe | 3.99E-04 | 4.44E-06 | 3.36E-06 | 1.23E-01
Ni/Fe | 1.55E-01 | 3.57E-01 | 8.08E-01 | 8.13E-01
K/Ca | 5.80E-02 | 5.59E-02 | 4.64E-02 | 1.58E-01
Ca/Fe | 4.91E-02 | 4.58E-02 | 4.23E-02 | 8.91E-01
Rb/Zr | 1.76E+01 | 5.99E+01 | 5.97E-01 | 4.17E+00
Table 1: Ratio of surface mass fractions for selected elements for our
5$\,\rm{M}_{\odot}$ TŻOs, evolved with the large 399 isotope nuclear network,
at 10,000 years after TŻO formation at $\rm{Z_{init}}=10^{-4}$. The final
column contains the Model A data from C93 which only provides mass fractions
for Carbon and heavier elements and thus lacks Lithium for comparison with.

Neutrino losses from our TŻOs are $\approx 10^{37}\rm{erg/s}$. This is
dominated by the losses due to beta-decays, while the thermal neutrino losses
are negligible. This is only a lower limit on the neutrino flux, as there may
be additional neutrino emissions from the material below the knee which we do
not model. This neutrino flux is comparable to a solar-mass star at the tip of
the red giant branch (Farag et al., 2020), and is unlikely to be detectable
with current detectors (Patton et al., 2017a; Patton et al., 2017b).

###  5.1 ${}^{44}\rm{Ti}\rm{O}_{2}$ and ${}^{44}\rm{Ti}\rm{O}$

While the predicted unique nucleosynthetic signal of a TŻO has been used
previously to make claims for the detection (or not) of a TŻO, it is not
without controversy (Tout et al., 2014). Thus we propose a new signal which
provides a more constraining nucleosynthetic signal, namely the detection of
molecules of TiO2 and TiO containing the radioactive isotope
$\mathrm{{}^{44}Ti}$.

$\mathrm{{}^{44}Ti}$ has a half life of $\approx 60$ years (Audi et al., 2003;
Ahmad et al., 2006) and is usually found in the ejecta of core-collapse
supernovae (Iyudin et al., 1994). Typical core-collapse supernovae have ejecta
of $10^{-5}$–$10^{-4}\,\,\rm{M}_{\odot}$ of $\mathrm{{}^{44}Ti}$ rich material
(Magkotsios et al., 2010). This suggests that the detection of
$\mathrm{{}^{44}Ti}$ in a TŻO could be the result of contamination from the
supernovae that formed the NS initially. However, given its short half-life,
unless we detect a TŻO shortly after the birth of the NS (when the SN remnant
should still be visible) then the detection of $\mathrm{{}^{44}Ti}$ requires a
continuous production site.

In non-TŻO stars, $\mathrm{{}^{44}Ti}$ is normally produced deep in the
stellar core, via $\mathrm{{}^{40}Ca}(\alpha,\gamma)\mathrm{{}^{44}Ti}$ during
the late stages of stellar evolution and during explosive burning episodes
(Timmes et al., 1996). Coupled with this, is that we need to be able to
efficiently mix the $\mathrm{{}^{44}Ti}$ to the surface of the star, which is
difficult to achieve unless the convective envelope penetrates deep into the
core which is unexpected when
$\mathrm{{}^{40}Ca}(\alpha,\gamma)\mathrm{{}^{44}Ti}$ is active.

For reference we computed a $\rm{M_{init}}=5\,\rm{M}_{\odot}$ AGB star,
$\rm{M_{init}}=8\,\rm{M}_{\odot}$ SAGB star and
$\rm{M_{init}}=10\,\rm{M}_{\odot}$ massive star with $\rm{Z_{init}}=0.00142$
($\sim Z_{\odot}/10$). For computational reasons we used a truncated version
of our 399 isotope nuclear network. We took our 399 network and removed all
elements heavier than Fe, as we are only interested in the Ti isotopes, which
brings the total isotope count down to 172.

The 5$\,\rm{M}_{\odot}$ model is evolved until $\approx 40$ thermal pulses
while the 8 and $10\,\rm{M}_{\odot}$ models are evolved up to carbon ignition.
We find the maximum surface mass fraction of
$\mathrm{{}^{44}Ti}$/$\mathrm{{}^{48}Ti}$ remains effectively zero (and always
lower than the numerical tolerance imposed on the nuclear network solver)
throughout the evolution.

However, distinguishing different isotopes directly in a spectrum can be
challenging. Thus we propose looking for isotopologues of $\rm{Ti}\rm{O}_{2}$
and $\rm{Ti}\rm{O}$. Our TŻOs almost always have surface temperatures $\log\
(T_{\rm eff}/K)<3.6$, where $\rm{Ti}\rm{O}_{2}$ and $\rm{Ti}\rm{O}$ molecules
are expected to form (Pavlenko et al., 2020), thus there should be a
significant amount of $\rm{Ti}\rm{O}_{2}$ and $\rm{Ti}\rm{O}$ in the
atmospheres of the TŻOs.

The most common Ti isotope is $\mathrm{{}^{48}Ti}$, with contributions from
$\mathrm{{}^{46-50}Ti}$ (Asplund et al., 2009). Detecting the different
isotopologues has been achieved for these isotopes, due to a shift in the
molecular lines as the mass of the molecules $\rm{Ti}\rm{O}_{2}$ and
$\rm{Ti}\rm{O}$ change with the different isotopic compositions (Breier et
al., 2019). The size of this shift is relative to the change in the molecular
mass between the isotopologues (Herzberg, 1950). Thus a molecule containing
$\mathrm{{}^{44}Ti}$ will have a larger shift in its molecular lines than the
already detectable isotopologues containing $\mathrm{{}^{46-50}Ti}$ (Pavlenko
et al., 2020; Serindag et al., 2021). It may also be possible to use
millimetre/submillimetre observations to detect $\mathrm{{}^{44}Ti}\rm{O}$
given the detection of other TiO isotopologues (Kamiński et al., 2013;
Lincowski et al., 2016).

Figure 11: The ratio of the surface mass fraction of $\mathrm{{}^{44}Ti}$ to
surface $\mathrm{{}^{48}Ti}$ over the TŻOs lifetime. In blue models with
$\rm{M_{init}}=5\,\rm{M}_{\odot}$, and $\rm{Z_{init}}=10^{-4}$ where solid
line has $\rm{Y_{init}}=0.28$, dot-dashed $\rm{Y_{init}}=0.4$, and dashed
$\rm{Y_{init}}=0.6$. In green models with $\rm{Z_{init}}=10^{-3}$ and
$\rm{Y_{init}}=0.28$, where the sold line has
$\rm{M_{init}}=10\,\rm{M}_{\odot}$, and dashed has
$\rm{M_{init}}=15\,\rm{M}_{\odot}$. Models were stopped at $\approx 25,000$
years.

In our TŻO models $\mathrm{{}^{44}Ti}$ is produced in the CaScTi333While
MESA’s approx21 net does contain $\mathrm{{}^{44}Ti}$, it is only produced via
$\mathrm{{}^{40}Ca}(\alpha,\gamma)\mathrm{{}^{44}Ti}$. Thus the approx21 show
significantly smaller amounts of $\mathrm{{}^{44}Ti}$ than when using our
large nuclear network.  cycle via
$\mathrm{{}^{43}Sc}\left(p,\gamma\right)\mathrm{{}^{44}Ti}$ (Fisker et al.,
2008). The $\mathrm{{}^{44}Ti}$ is then mixed outwards before it can capture
another proton. Figure 11 shows the surface mass fraction of
$\mathrm{{}^{44}Ti}$/$\mathrm{{}^{48}Ti}$. Depending on $\rm{Y_{init}}$,
$\rm{Z_{init}}$, and $\rm{M_{init}}$ the ratios starts between
$10^{-4}$–$10^{-2}$ and decreases with time. On long timescales, we can see
that the ratio tends towards $X(\mathrm{{}^{44}Ti}/\mathrm{{}^{48}Ti})\approx
10^{-4}$. This is due to increasing amounts of $\mathrm{{}^{48}Ti}$ (other
stable Ti isotopes are also increasing with time), while the absolute amount
of $\mathrm{{}^{44}Ti}$ remains approximately constant. For the TŻOs with
$\rm{Z_{init}}=10^{-3}$ the ratio decreases with $\rm{M_{init}}$. A
5$\,\rm{M}_{\odot}$ TŻO with $\rm{Z_{init}}=10^{-3}$ did not show any
enrichment of $\mathrm{{}^{44}Ti}$. These ratio’s are much greater than the
values found for the 5, 8, and 10$\,\rm{M}_{\odot}$ non-TŻO stars.

$\rm{M_{init}}$ | $\rm{Y_{init}}$ | $\rm{Z_{init}}$ | $\mathrm{{}^{44}Ti}$ | $\mathrm{{}^{46}Ti}$ | $\mathrm{{}^{47}Ti}$ | $\mathrm{{}^{48}Ti}$ | $\mathrm{{}^{49}Ti}$ | $\mathrm{{}^{50}Ti}$
---|---|---|---|---|---|---|---|---
5 | 0.28 | $10^{-4}$ | 3.85E-12 | 2.10E-09 | 2.76E-09 | 2.02E-08 | 6.50E-09 | 9.71E-10
5 | 0.40 | $10^{-4}$ | 8.39E-11 | 1.03E-08 | 2.12E-08 | 1.01E-07 | 6.88E-08 | 9.96E-10
5 | 0.60 | $10^{-4}$ | 1.31E-11 | 3.38E-09 | 5.66E-09 | 3.35E-08 | 1.71E-08 | 9.77E-10
5 | 0.28 | $10^{-3}$ | 5.56E-16 | 1.43E-08 | 1.32E-08 | 1.34E-07 | 1.00E-08 | 9.86E-09
10 | 0.28 | $10^{-3}$ | 5.46E-11 | 4.45E-08 | 4.57E-08 | 2.02E-07 | 6.47E-08 | 1.07E-08
15 | 0.28 | $10^{-3}$ | 1.51E-10 | 3.96E-08 | 4.95E-08 | 2.55E-07 | 1.08E-07 | 1.02E-08
Sun |  | 2.23E-07 | 2.08E-07 | 2.15E-06 | 1.64E-07 | 1.64E-07
5 |  | $Z_{\odot}/10$ | 3.05E-26 | 2.02E-08 | 1.85E-08 | 1.88E-07 | 1.44E-08 | 1.39E-08
8 |  | $Z_{\odot}/10$ | 2.16E-98 | 2.05E-08 | 1.89E-08 | 1.91E-07 | 1.43E-08 | 1.41E-08
10 |  | $Z_{\odot}/10$ | 2.57E-98 | 2.05E-08 | 1.89E-08 | 1.91E-07 | 1.43E-08 | 1.41E-08
Table 2: Surface mass fractions for different Ti isotopes for our large
nuclear network models. The composition is taken at $\approx 10,000$ years
post TŻO formation. The 5, 8, and 10$\,\rm{M}_{\odot}$ non-TŻO stars are
measured at the time of maximum surface $\mathrm{{}^{44}Ti}$. Solar values
from Grevesse & Sauval (1998).

Table 2 shows the absolute mass fraction fractions of the Ti isotopes in our
large nuclear network models. We can see that the amount of all Ti isotopes
slightly increases with increasing initial helium fraction. There is however a
factor $\sim 10$ increase in $\mathrm{{}^{44}Ti}$ as $\rm{Y_{init}}$
increases. Given the small absolute mass of $\mathrm{{}^{44}Ti}$, which is
much less than that typically seen in supernovae ejecta (Magkotsios et al.,
2010), it may be difficult to directly detect the gamma-ray from the decay of
$\mathrm{{}^{44}Ti}$.

We note though that before this becomes a viable method it is likely we will
need improved theoretical models of the molecular lines of
$\mathrm{{}^{44}Ti}\rm{O}_{2}$ and $\mathrm{{}^{44}Ti}\rm{O}$, as to date
experiments have concentrated on the more common and stable isotopes of
titanium, namely 46–50 (Brünken et al., 2008; Breier et al., 2019; McKemmish
et al., 2019; Witsch et al., 2021).

##  6 Suitability of model assumptions

(a) Central density

(b) Central luminosity

Figure 12:  Panel a: The evolution of the surface luminosity as a function of
the assumed average core density. Light colours denote higher core densities
and smaller assumed NS radii. Panel b: The evolution of the surface luminosity
as a function of the efficiency factor $\epsilon_{L}$. Light colours denote
higher efficiencies and thus the energy injected into the inner boundary is
closer to $L_{\rm Edd}$. Evolution was arbitrarily stopped when either the
model reached 100,000 years, 100,000 timesteps, or when MESA could no longer
follow the evolution.  Figure 13: Top Panel: The temperature-density
(T-$\rho$) profile for the models shown in Figure 12(b) at a time $\approx
10,000$ years post TŻO formation. The $\epsilon_{L}=0$ model did not reach
10,000 years and is thus not shown. Dash-dotted lines are the models of C92
while the red region is the pair-instability region.

The models presented in this work depend on two key approximations, that make
the TŻO models computationally feasible. Without these approximations, these
models would not be computationally viable. Both approximations deal with the
fact that we can not model down to the surface of the NS while simultaneously
including the RSG envelope. First, we assume a $\log\
\langle\rho_{c}/\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3}\rangle=9.3$, which
implies an effective $\rm{R_{NS}}\approx 600\,\mathrm{k}\mathrm{m}$ and
secondly we inject additional energy at the base of our models. We will now
test those assumptions, as far as possible, to determine their effect on our
results. For this test, we turn off the hydrodynamics, as it can cause
numerical convergence issues.

Figure 12 shows the total luminosity of the TŻO as a function of the age since
formation, for variations in the core density (12(a)), and the injected energy
(12(b)). Firstly we can see that both sets of models evolve to lower
luminosities over time. Figure 12(a) compares models with
$\log\langle\rho_{c}/\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3}\rangle$ between
$10^{9}$–$10^{12}\,\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3}$ (this equates to
effective NS radii between $\approx 90$–$900\,\mathrm{k}\mathrm{m}$). It shows
that as the core density increases the total luminosity increases. This is due
to a lower opacity at the knee, causing $L_{\rm{knee}}$ to increase. Figure
12(b) also shows that as the injected energy decreases, the surface luminosity
also decreases. Thus, surprisingly, over the range of parameters explored
here, the two approximations act with different signs but similar magnitudes,
in terms of the total luminosity. Therefore they approximately cancel out.
Comparing the surface metal mass fraction, the low $\epsilon_{L}$ models
converge to values about a factor 2 greater than the $\epsilon_{L}=1$, while
the highest core density models have a factor 2 lower surface metal mass
fractions. Again, our two assumptions are approximately cancelling each other
out.

The large luminosity jumps leading to the models then evolving on a different
luminosity track are due to the lack of hydrodynamics. In models with
hydrodynamics included, those jumps occur at the time when the energy
generated by nuclear burning spikes and the models would normally have been
stopped due to large amplitude surface pulsations being resolved in the
models.

In Figure 13 we show the temperature-density profiles inside the TŻOs as a
function of $\epsilon_{L}$. As $\epsilon_{L}$ decreases, less energy is
injected into the model, therefore the TŻO must provide more of the energy
itself. As the mass of the NS varies little in our models this is also
equivalent to changing the opacity of the material at the base of the
convection zone. The material below the knee then moves to higher temperatures
and densities to provide the necessary energy via nuclear burning and mass
accretion to support the star. The region below the knee does not evolve at a
constant temperature, as previously found in Thorne & Zytkow (1975, 1977) and
C92. Instead as the temperature increases, the material avoids the pair-
instability region by evolving around the instability region. The T–$\rho$
tracks converge for $\epsilon_{L}\leq 0.1$, until
$\log(\rho/\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3})\approx 6.0$.

At $\log(\rho/\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3})\approx 6.0$ the core
begins evolving to cooler temperatures. At this temperature both hydrogen and
helium are depleted in the material below the knee. With a very low mass
fraction of carbon and oxygen, due to the low initial metallicity, and our use
of the approx21.net nuclear network, the nuclear energy generation rate goes
to 0 as there are no available nuclear reactions. The inner regions then begin
to cool due to an increase in thermal neutrino losses as the density
increases.

Finally, the increase in temperature at
$\log(\rho/\mathrm{g}\,{\mathrm{c}\mathrm{m}}^{-3})\approx 8.0$ is due to a
new convection zone setting in near the inner boundary of the model. This is
due to the (small) amount of energy that is still being injected into the
model. Models with less injected energy cool further and reach higher
densities before showing this uptick.

##  7 Final fate

Once the large amplitude surface pulsations begin the computational timescale
decreases and we begin evolving the models on the pulsation timescale. This
leads to timesteps of order $10^{2}$–$10^{3}$ seconds, which is much smaller
than the $10^{8}$ seconds we take during the normal phase of a TŻOs evolution
and thus is infeasible to evolve over longer time frames. Numerical
convergence issues also occur as shocks form in the outer envelope, preventing
the models from being evolved for more than a few tens of years at this point.

Thus we can only speculate what happens next. The TŻOs are undergoing large
pulsations, and in other $\epsilon$-mechanism pulsators this is expected to
lead to pulsation-driven mass loss (Baraffe et al., 2001; Nakauchi et al.,
2020) or, as in normal RSGs, pulsations may lead to pulsational driven
superwinds (Yoon & Cantiello, 2010). This could rapidly decrease the envelope
mass over $\approx 100$ years, leaving behind a bare NS. Or perhaps the TŻO
will undergo a neutrino runaway when nuclear burning ceases and energy losses
from neutrinos cause the TŻO to collapse (Podsiadlowski et al., 1995), leaving
a BH behind, though this seems less likely as our models avoid the pair-
instability region.

If the envelope was entirely ejected from the NS, it seems unlikely that the
NS mass will have increased significantly enough to be visible as a higher
mass NS. The accretion rate onto the NS is $\approx
10^{-9}$–$10^{-8}\,\rm{M}_{\odot}\,\rm{yr}^{-1}$. In our models, this leads to
the NS gaining at most $\approx 0.002\,\,\rm{M}_{\odot}$ during its evolution.
The NS may also be spun down by braking between the NS’s magnetic field and
the envelope (Liu et al., 2015). In this scenario the NS will become a slow
spinning NS inside a slow moving CSM that appears like a supernovae remnant,
that has been proposed for RCW 103 (Liu et al., 2015).

The other option is that the TŻO runs out of CNO material, which provides the
bulk of the nuclear energy generated (Biehle, 1994). In this case, the
envelope may collapse onto the NS forming a BH, possibly leading to a
transient event (Moriya, 2018; Moriya & Blinnikov, 2021). In a
5$\,\rm{M}_{\odot}$ TŻO model the total mass of CNO elements decreases at a
rate of $\approx 10^{-10}\,\rm{M}_{\odot}\,\rm{yr}^{-1}$. With an initial mass
of $\approx 10^{-5}\,\,\rm{M}_{\odot}$ of CNO elements (for
$\rm{Z_{init}}=10^{-5}$), this gives an upper limit of $\approx 10^{6}$ years,
assuming the burn rate continues at the same rate. This is comparable to the
lifetime given assuming a steady state wind mass loss.

If the NS inside the TŻO did collapse into a BH and form a transient event,
this may look like a type IIn SN. The pre-SN pulsations likely removed
1–10$\,\rm{M}_{\odot}$ of material in the 100–1000 years before the final
collapse. At collapse it is then likely that there is only a few solar masses,
at most, of H-rich envelope left. Our default 5$\,\rm{M}_{\odot}$ TŻO has only
a $\approx 2.5\,\rm{M}_{\odot}$ envelope left before the pulsations begin,
while at 20$\,\rm{M}_{\odot}$ has $\approx 5.5\,\rm{M}_{\odot}$ envelope
before the pulsations begin. We would not expect any significant
$\mathrm{{}^{56}Ni}$ production in the event, and as the TŻO was fully mixed
there would not be any observed change in the composition at late times. To
power a SN transient event would require it either to be entirely powered by
CSM interaction, energy released as the NS collapses into a BH, or if the BH
generates a jet from additional mass accretion (Fryer et al., 1996; Qin et
al., 1998; Fryer et al., 2014).

##  8 Discussion

In this work, we have been agnostic as to how a TŻO formed, whether it was a
direct impact, common envelope merger, or dynamical merger. Each of the
formation pathways may impact the resulting TŻO. Common-envelope mergers
require that the companion star is evolved, with a helium-rich core and an
envelope that is increasing in radius. While direct impacts and dynamical
mergers are less sensitive to the companion star’s evolutionary state, at
least regarding when a merger can occur. Merging at different points in the
star’s pre-TŻO life will lead to a different initial metal fraction than our
assumed solar-scaled metal distribution, and limits what values of
$\rm{Y_{init}}$ are possible. There could be differences in whether or not a
merger results in a TŻO or instead causes enhanced mass loss, leaving behind a
tight binary with an NS, thus failed TŻO’s are progenitors for double NS
systems (DNS). Detections of TŻOs could place constraints on the uncertain
merger rate of DNS systems as currently being probed by gravitational wave
detections.

We assumed that the starting point for a TŻO can be approximated as a normal
star, which then becomes fully convective. However, it is possible that the
companion was the accretor in a binary system. The material it gains will be
enriched by the nuclear burning in the donor (Farmer et al., 2021, 2023) and
the internal structure of the accretor will be altered by the mass accretion
(Renzo & Götberg, 2021; Renzo et al., 2023). This may lead to possible
different outcomes during the merger, which may include additional mass loss,
changes in the probability of a successful TŻO formation, and to initially
enrich the TŻO in additional metals.

It was argued in C93 that it was necessary to use a two-stream convection
model to follow the nucleosynthesis. Here the composition is tracked for
material that is convectivly moving towards the knee and away from the knee.
As material moves towards the NS it undergoes proton captures, before mixing
away from the knee, where it can then undergo beta decays, before being mixed
back towards the knee. As the burning timescale is comparable to the mixing
timescale at the base of the envelope the chemical composition of material
flowing up/down may be significantly different. MESA takes assumes a diffusion
equation for chemical transport. The effect of having a diffusion model is
that we likely over predict the production of heavy nuclei. In the MESA models
some heavy nuclei will be able stay near the base of the envelope, where the
temperatures are highest, for much longer than they would if they where being
mixed outwards on a timescale similar to their burning timescale. Thus they
can undergo additional proton captures and produce even heavier nuclei. This
may suggest we are even less likely to see a detectable nucleosynthetic signal
in TŻOs in the local Universe.

##  9 Conclusions

In this work, we have computed the first set of MESA TŻOs. We did this by
adjusting the inner boundary of the model to approximate the NS at the centre
of each TŻO. We have then followed the evolution of the TŻOs. We have also
explored in detail the pulsation periods with the GYRE stellar oscillation
instrument, and computed detailed nucleosynthetic signatures with a large
fully-coupled 399 isotope nuclear network. Our results can be summarised as
follows:

  * •

We find that TŻOs evolve to lower luminosities and lower temperatures during
their lifetime. We have also expanded the range of possible locations for TŻOs
to be between $\rm{\log\left(T_{eff}/K\right)}\approx 3.47$–$3.6$ and
$\rm{\log\left(L/L_{\odot}\right)}\approx 5.0$–$5.5$.

  * •

We do not find a gap in the parameter space where models can not exist. This
is because our models are denser than previously predicted, which prevents
models from evolving into the pair-instability region.

  * •

We have computed the pulsation periods of our TŻOs and find the longest
periods to be $\approx 250,500$ and 1000-2000 days.

  * •

If HV 2112 is a TŻO, we predict there should be a currently undetected
1500–3000 day pulsation period. If detected this will also imply
$\rm{\alpha_{mlt}}\approx 3$ in the envelopes of TŻOs.

  * •

If VX Sgr were a TŻO and its $\sim 28,000$ day pulsation period were real,
this would imply it is a very helium enriched TŻO.

  * •

Based on the measured pulsation periods of HV 11417 we would infer it to be a
massive, but very young TŻO. However this is inconsistent with the measured
luminosity of HV 11417. Thus we rule out HV 11417 as a TŻO.

  * •

Our results and the predicted lifetimes depend strongly on the mass-loss rates
due to RSG pulsations. The lifetime of the TŻO will depend on the total mass-
loss rate experienced by the TŻO. If RSG pulsations remove significant amounts
of material then the lifetime of a TŻO may only be 100–1000 years.

  * •

Assuming the RSGs do not experience significantly higher mass loss than we
assume, then we estimate a lifetime in the range of $10^{4}$–$10^{5}$ years.
Contrary to non-TŻO stars, the higher the initial mass of the TŻO the longer
it lives.

  * •

We have computed several of our models with a large 399 isotope fully-coupled
nuclear network. We reconfirm the previous findings that Rb, Sr, Y, Zr can be
enhanced due to the irp-burning. However, the level of enhancement is
sensitive to the initial composition of the TŻOs.

  * •

At higher initial metallicities TŻOs do not show any metal enrichment due to a
lower knee temperature, caused by the increasing opacity at the knee as the
metallicity increases. Thus in the local Universe TŻOs may not be
distinguishable from non-TŻOs based on their nucleosynthesis alone.

  * •

We propose a new observational signal, that of molecules containing
$\mathrm{{}^{44}Ti}$. Due to the high-temperature burning and fully convective
envelope of the TŻO, $\mathrm{{}^{44}Ti}$ can be mixed to the surface before
it decays. There in the cooler envelope, it can form $\mathrm{{}^{44}Ti}$O2
and $\mathrm{{}^{44}Ti}$O, which could be detectable due to the shift in their
molecular lines compared to stable Ti-containing molecules.

  * •

TŻOs represent a class of stars that are exceptional tests of the numerical
capabilities of a stellar evolution code. This work has led to many
improvements and code fixes in the MESA stellar evolution code that have
applications far outside that of TŻOs.

## Acknowledgements

We acknowledge helpful discussions with T. Maccarone, F. Timmes, B. Paxton, R.
Smolec, J. Schwab, A. Jermyn, R. Townsend. This work has been supported by the
following grants at some point in time; NASA under TCAN grant NNX14AB53G (PI
F. Timmes), NSF under SI2 grant 1339600 (PI F. Timmes), the Netherlands
Organization for Scientific Research (NWO) through a top module 2 grant with
project number 614.001.501 (PI S.E. de Mink). Support for this work was
provided by NASA through the NASA Hubble Fellowship Program grant #HST-
HF2-51457.001-A awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS5-26555. This work was also supported by the Cost
Action Program ChETEC CA16117. This research was supported by the Munich
Institute for Astro-, Particle and BioPhysics (MIAPbP) which is funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy – EXC-2094 – 390783311. This research was
supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958. This research has made use of NASA’s Astrophysics Data System.

## Data Availability

All input files and all output data is made available at
https://doi.org/10.5281/zenodo.4534425.

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##  Appendix A Sensitivity to other parameter choices

(a) Wind efficiency

(b) Mixing Length alpha

(c) Initial Mass NS

(d) NS accretion rate

Figure 14: HRDs for variations in our TŻO modelling assumptions. Grey lines
and grey boxes have the same meaning as in Figure 1. All models are taken as
5$\,\rm{M}_{\odot}$ with a $M_{\rm NS}=1.4\,\,\rm{M}_{\odot}$, and use our
default parameters except for those being varied in each panel. Panel a:
variations in the wind mass loss efficiency, panel b: variations in the mixing
length alpha, panel c: variations in the mass of the NS, and panel d:
variations in the accretion rate onto the NS.  Figure 15: HRDs for variations
in the size of the nuclear network. Grey lines and grey boxes have the same
meaning as in Figure 1. All models are taken as 5$\,\rm{M}_{\odot}$ with a
$M_{\rm NS}=1.4\,\,\rm{M}_{\odot}$, and use our default parameters. The red
lines show $\rm{Y_{init}}$=0.28, the orange line shows $\rm{Y_{init}}$=0.4,
and the blue lines show $\rm{Y_{init}}$=0.6 and are evolved with
$\rm{Z_{init}}=10^{-4}$. Solid lines are models that are evolved with the 399
isotope network, while dotted lines show the 21 isotope network, with the same
$\rm{Y_{init}}$. All models were stopped after 25,000 years.

Figure 14 shows the HRDs for the parameter variations we have tested here,
while Figure 15 shows the HRD for a hydrogen-rich and helium-rich TŻO evolved
with our 399 isotope network.

###  A.1 Wind mass loss

Figure 14(a) shows variations in the wind mass loss efficiency $\eta_{\rm{\rm
VL}}$, which scales the wind mass loss rates of van Loon et al. (2005). All
models start with a similar evolution, however as the mass loss rate increases
TŻOs become less luminous for a given temperature. As $\eta_{\rm{\rm VL}}$
increases models stop their evolution at higher surface temperatures, live for
less time overall, and enrich the surface with fewer metals (at a given age).
The effect of wind mass loss on a stellar model is more complex than a single
scaling factor (Renzo et al., 2017).

###  A.2 Mixing length alpha

Figure 14(b) shows variations in the mixing length $\rm{\alpha_{mlt}}$. As
$\rm{\alpha_{mlt}}$ increases the tracks are more luminous and hotter, acting
like a more massive TŻO (See Figure 1). This increase in luminosity is due to
the knee temperature increasing with increasing $\rm{\alpha_{mlt}}$. Models
with $\rm{\alpha_{mlt}}<1.5$ act similar to the high metallicity models in
Section 3.1. That is, they have a high opacity at the knee, which leads to a
low $L_{\rm{knee}}$, and thus a low knee temperature. As $\rm{\alpha_{mlt}}$
increases, TŻOs produce more metals (at a given age). The age of the TŻO is
approximately independent of the assumed $\rm{\alpha_{mlt}}$.

###  A.3 Core mass

Figure 14(c) shows variations in the neutron star mass $M_{\rm NS}$. As
$M_{\rm NS}$ increases stars are more luminous, though at a similar surface
temperature. As the TŻO evolves, low mass NSs have both a larger change in
luminosity and a drop to a lower final luminosity. This is due to the
reduction in the lower $L_{\rm Edd}$ needed to support the envelope. This
behaviour is also related to the ratio of the NS mass to the total mass of the
TŻO. Higher initial mass TŻOs show a similar trend as the NS mass (and thus
the ratio) is varied. As the mass of the NS increases, the lifetime of the TŻO
decreases and the surface metal enrichment decreases.

If HV 11417 was a TŻO then it is likely to need a very low mass NS $M_{\rm
NS}\lesssim 1.2\,\,\rm{M}_{\odot}$. At higher core masses (for constant total
mass) the knee temperature is lower. This causes a slower metal enrichment in
the models and leads to lower production of very heavy elements. Thus if both
HV 2112 and HV 11417 were TŻOs, we would expect different nucleosynthetic
signals.

###  A.4 Core accretion rate

Figure 14(d) shows variations in the accretion efficiency onto the NS
$\epsilon_{\dot{M}}$. Between $\epsilon_{\dot{M}}=10^{-4}$–$10^{2}$, there is
no appreciable difference in evolution. When $\epsilon_{\dot{M}}=1.0$ typical
accretion rates are $\approx 10^{-9}$–$10^{-8}\,\rm{M}_{\odot}\,\rm{yr}^{-1}$.
The $\rm{M_{init}}=5\,\rm{M}_{\odot}$ model has a lifetime of $\approx 50,000$
years thus the NS can only gain $\approx 10^{-4}\,\,\rm{M}_{\odot}$. Only with
accretion rates $\epsilon_{\dot{M}}>10^{3}$ does the evolution change
significantly, due to the increase in the NS mass, and follows more closely
the evolution seen in C92. Accretion rates in C92 are $\approx
10^{-8}$–$10^{-7}\,\rm{M}_{\odot}\,\rm{yr}^{-1}$. These are 10-100 times
larger than our accretion rates but still smaller than what we find is needed
to force the TŻO to evolve to higher luminosities.

###  A.5 Nuclear network composition

Figure 15 shows the HRD for our models with both our 21 and 399 isotope
nuclear network, for $\rm{M_{init}}=5\,\rm{M}_{\odot}$. We can see that the
evolution is broadly similar, with the large nuclear network models being
slightly more luminous at the start, but showing a faster decline in their
luminosity with time. The largest difference is in the $\rm{Y_{init}}=0.6$
models where the large network model evolves in the HRD much less than the
approx21.net for the same amount of evolutionary time. This shows that using
an approx21.net is a reasonable assumption to make when considering the bulk
properties of a TŻO, especially when considering the significant computational
savings for not needing to evolve with a large nuclear network. But this
limits the nucleosynthesis signals that an be inferred from the models,
without further post-processing.

##  Appendix B Network choices

Figure 16: The surface composition relative to solar of a
5$\,\rm{M}_{\odot}$TŻO our default model assumptions (blue star) and a
5$\,\rm{M}_{\odot}$ model with a Cannon-like network (red square). This
snapshot was taken at $\approx 10,000$ years post TŻO formation. In black
dots, the Model A of C93. Vertical lines mark elements that may be useful for
detecting TŻOs.

We have also run an additional network that is the one shown in figure 17 of
C93, except we truncate at Cd, leaving 390 isotopes. Figure 16 shows the
surface mass fractions of our 5$\,\rm{M}_{\odot}$ default model and a
5$\,\rm{M}_{\odot}$ model using the Cannon-like nuclear network. There is very
little difference in the overall mass fractions except at V, Cr, Mn, and Nb.

By extending the network to Cd we see a sharp decline in the relative mass
fraction of elements heavier than Pd in model A of C93. This is similar
behaviour to what is seen in our models, albeit at a higher atomic number.
Thus it appears that our models differ mostly by having the element with the
highest production factor (Kr) at lower Z, compared to C93. This turn over to
lower production factors then also occurs at lower atomic numbers.

##  Appendix C MESA physics options

Radiative opacities are primarily from OPAL (Iglesias & Rogers, 1993, 1996),
with low-temperature data from Ferguson et al. (2005) and the high-
temperature, Compton-scattering-dominated regime by Poutanen (2017). Electron
conduction opacities are from Cassisi et al. (2007).

Nuclear reaction rates are a combination of rates from NACRE (Angulo et al.,
1999), JINA REACLIB (Cyburt et al., 2010), plus additional tabulated weak
reaction rates (Fuller et al., 1985; Oda et al., 1994; Langanke & Martínez-
Pinedo, 2000). Screening is included via the prescription of Chugunov et al.
(2007). Thermal neutrino loss rates are from Itoh et al. (1996).

##  Appendix D Surface mass fractions

Table 3 shows the surface chemical mass fractions from Figure 9(a). These are
for a 5$\,\rm{M}_{\odot}$ TŻO, with a $1.4\,\,\rm{M}_{\odot}$ NS, and varying
initial helium mass fraction. Values are quoted at 10,000 years after
formation. The time evolution of the isotopic data is available in the Zenodo
online material.

Table 3: Surface elemental mass fractions for the 399 isotope models with different $\rm{Y_{init}}$ values. mass fractions measured at 10,000 years post TŻO formation.  Element | $\rm{Y_{init}}=0.28$ | $\rm{Y_{init}}=0.40$ | $\rm{Y_{init}}=0.60$
---|---|---|---
H | 7.194E-01 | 5.993E-01 | 3.991E-01
He | 2.804E-01 | 4.005E-01 | 6.008E-01
Li | 6.087E-11 | 6.105E-11 | 6.127E-11
Be | 1.007E-12 | 1.010E-12 | 1.013E-12
B | 2.886E-11 | 2.894E-11 | 2.903E-11
C | 1.694E-05 | 1.701E-05 | 1.714E-05
N | 5.115E-06 | 5.220E-06 | 5.422E-06
O | 4.605E-05 | 4.617E-05 | 4.629E-05
F | 2.443E-09 | 2.449E-09 | 2.456E-09
Ne | 1.024E-05 | 1.026E-05 | 1.029E-05
Na | 2.045E-07 | 2.050E-07 | 2.056E-07
Mg | 3.883E-06 | 3.893E-06 | 3.904E-06
Al | 3.552E-07 | 3.561E-07 | 3.571E-07
Si | 4.328E-06 | 4.338E-06 | 4.350E-06
P | 4.793E-08 | 4.805E-08 | 4.818E-08
S | 2.160E-06 | 2.165E-06 | 2.171E-06
Cl | 2.869E-08 | 2.875E-08 | 2.888E-08
Ar | 3.850E-07 | 3.859E-07 | 3.871E-07
K | 2.283E-08 | 2.288E-08 | 2.317E-08
Ca | 3.934E-07 | 4.092E-07 | 4.990E-07
Sc | 9.772E-10 | 2.419E-09 | 1.074E-08
Ti | 3.252E-08 | 6.066E-08 | 2.027E-07
V | 1.423E-08 | 3.899E-08 | 1.462E-07
Cr | 1.467E-06 | 1.330E-06 | 7.577E-07
Mn | 2.691E-07 | 4.975E-07 | 1.459E-06
Fe | 8.010E-06 | 8.932E-06 | 1.181E-05
Co | 2.666E-08 | 4.457E-08 | 1.266E-07
Ni | 1.244E-06 | 3.187E-06 | 9.547E-06
Cu | 5.966E-07 | 1.795E-06 | 3.881E-06
Zn | 2.584E-06 | 7.746E-06 | 7.375E-06
Ga | 2.307E-06 | 5.217E-06 | 1.428E-06
Ge | 5.077E-06 | 1.017E-05 | 1.689E-06
As | 2.688E-06 | 4.337E-06 | 3.025E-08
Se | 6.139E-06 | 1.054E-05 | 1.111E-07
Br | 9.040E-06 | 6.373E-06 | 3.187E-09
Kr | 3.322E-05 | 7.479E-06 | 1.132E-09
Rb | 2.707E-06 | 9.555E-09 | 9.490E-11
Sr | 9.650E-06 | 2.832E-08 | 3.171E-10
Y | 3.129E-07 | 7.296E-11 | 6.551E-11
Zr | 1.542E-07 | 1.595E-10 | 1.589E-10
Nb | 4.436E-10 | 1.014E-11 | 1.012E-11
Mo | 3.195E-09 | 3.964E-11 | 3.963E-11
Tc | 1.056E-14 | 1.274E-15 | 6.608E-17
Ru | 2.988E-11 | 2.988E-11 | 2.988E-11