produced. This is the final phase of CEE. The properties of post-CE systems
are also very important because they could be affected by the specific
circumstances of the CEE. For example, any remaining circumstellar matter,
likely in the form of a circumbinary disk, may change the orbit and
eccentricity of a surviving binary. In addition, the remnant cores might
further overfill its Roche lobe during the thermal evolution, driving mass
transfer again.

It is unclear whether the self-regulating spiral-in phase occurs or not if
non-local energy dissipation is common in the entire spiral-in phase. Either
the CE ejection or the merger of the binary may occur rapidly following the
plunge-in phase. On the other hand, as proposed by [295], a self-regulated
spiral-in could also be followed by another dynamical plunge if the mechanism
maintaining self-regulation somehow ends, and this second plunge could in turn
be followed by another self-regulated phase in some cases.

II. Role of the recombination energy and loss of orbital energy

The energy budget during the CEE is the most important aspect to determine the
outcomes of the CEE. In Section 3.2.1, we have discussed the terms of energy
included in the standard energy mechanism. Numerical simulations have shown
evidence that the recombination energy (one term of internal energy) could
play a crucial role in the final ejection of the CE because it gives the final
push to make a loose envelope unbound [280, 296]. Four types of ejection
processes from the simulations were identified in the simulation of [280],
that is, the initial ejection (pre-plunge-in), the plunge-in outflow, the
recombination runaway outflow, and the shell-triggered ejection. The first two
processes take place in all the simulations, including those ending up with a
merger. The amount of the pre-plunge-in ejection can be estimated roughly by
using the energy formalism on the material outside the orbit. The prompt
plunge-in ejection carries away substantially more mass, but it is hard to be
estimated properly. The shell-triggered ejection is caused by the hollow shell
fallback when the envelope is decoupling with the core-companion binary,
providing another prompt ejection process as a part of the CE. The
recombination runaway outflow starts during the slow spiral-in when the
expanded envelope is cooled down to start hydrogen recombination. With
detailed energy analysis, the authors further gave a description for the final
ejection phase as follows [296]: At the end of the plunge-in, the outer
envelope hosts most of the unbound mass and has some kinetic energy obtained
from the previous plunge-in, leading to an expansion of the envelope on the
dynamical time-scale. In this process, material in the outer envelope has a
parabolic (bound) trajectory with respect to the binary and will gain enough
energy to become unbound once the material expands and cools down enough to
start recombination. The recombination radius at which the released
recombination energy can remove material out of the potential well has been
derived in their simulations. They found that the recombination takes place at
large optical depths which are at least 10 in 3D models and could be above 100
in 1D models [297].

Hydrodynamical simulations of CEE showed that a significant fraction of the
released orbital energy is taken away by the ejected material. The appearance
of large-scale waves with tidal arms could trail the binary orbit. Spiral
shocks transfer angular momentum to the matter in the envelope and some of the
energy in the shocks will be dissipated as heat [298]. Obviously, this kind of
energy dissipation is far away from the secondary, which is different from
that normally expected, i.e., the orbital energy is thermalized locally by
viscous dissipation in the region of the in-spiraling companion. Even if the
energy thermalization is local, a significant fraction of energy would be
transferred to the surface and radiated away during the following self-
regulating phase. Furthermore, it is found that about 25 percent of the CE
could be flung out by the spiral waves, taking away a significant part of
orbital energy [298]. The simulations of 25 3D hydrodynamical interactions
between a low-mass RG and a WD companion by [299] have obtained the total
energy taken away by the ejecta, $E_{\rm tot,unb}^{\infty}$. The fraction of
energy loss related to the total energy can calculated via a fitting formula
as:

$\alpha_{\rm unb}^{\infty}=-\frac{E_{\rm tot,unb}^{\infty}}{\Delta E_{\rm
orb}}=-0.16-0.30M_{\rm d}+0.49M_{\rm 2}+2.27M_{\rm c}$ (3.13)

where $M_{\rm d}$, $M_{\rm c}$ and $M_{\rm 2}$ are the donor mass, the core
mass of the donor and the companion mass in solar mass, respectively. The
value of $\alpha_{\rm unb}^{\infty}$ ranges from about 50 percent to 20
percent when the donor increases in mass from 1.2 to 1.8 $M_{\odot}$. The
energy formalism is therefore preferred to be revised as

$\Delta E_{\rm orb}(1-\alpha_{\rm unb}^{\infty})+E_{\rm bind}+\eta(M_{\rm
d}-M_{\rm c})=0,$ (3.14)

where the parameter $\eta$ is used to assess the recombination energy stored
in the envelope, which is proportional to the mass of the envelope. The exact
value of $\eta$ varies with chemical composition of the envelope, and which is
$\eta=1.5\times 10^{\rm 13}$ erg ${\rm g}^{\rm-1}$ for fully ionized gas that
consists of 30% He and 70% H.

In order to make the results being used conveniently by BPS studies, a fitted
formula for $\alpha_{\rm CE}\lambda$ value, as a function of the $M_{\rm d}$,
$M_{\rm c}$ and $M_{\rm 2}$ is given [296] as below:

$\alpha_{\rm CE}\lambda=0.92+0.55M_{\rm d}-0.79M_{\rm 2}-1.19M_{\rm c}.$
(3.15)

They obtained that the maximum value of $\alpha_{\rm CE}\lambda$ is less than
1.03 in their studies. We should always keep in mind that this formula is
based on limited models and thus may be only valid in narrow parameter space.
Note that low-mass RG stars in these studies have an initial mass of 1.2, 1.4,
1.6, or $1.8\,M_{\odot}$ and a He core mass of $0.32$ or $0.36\,M_{\odot}$,
and the WD companions have a mass of 0.32, 0.36 or $0.40M_{\odot}$.

Despite the challenges of spatial scales, CE simulations for massive donors
have emerged gradually in recent two years [300, 301]. These simulations also
demonstrated the importance of internal energy, particularly the recombination
energy in the final CE ejection. The study of [301] presented the values of CE
parameters for the standard energy formulae. They performed 3D
magnetohydrodynamic (MHD) simulations of the CE evolution of a binary system
composed of an initially $10M_{\odot}$ red supergiant primary star and either
a BH ($5M_{\odot}$) or NS ($1.4M_{\odot}$) companion. The simulations showed
that about half of the CE in these systems is dynamically ejected, and the
ejected mass is still increasing at the end of the simulations, which leaves
some uncertainties on the true final ejecta mass. The authors suggested that
if all the remaining ionization energy is released by the expanding envelope
and can be used for unbinding the material, it is possible that almost the
full envelope will be eventually ejected. Assuming that the full envelope is
ejected, they found a value of $\alpha_{\rm CE}\lambda=1.17$ and 1.32 in the
cases with an NS and a BH companion, respectively. The inclusion of the
thermal and ionization energies (i.e. $\alpha_{\rm th}=1.0$) gives
$\lambda=1.91$ from the relaxed dynamical model, resulting in an ejection
efficiency of $\alpha_{\rm CE}=0.61$ and 0.69 in the NS and BH companion case,
respectively555However, the energy formalism with these values can only
reproduce the outcomes of their simulations if the core–envelope boundary is
adjusted to their setup and one does not apply the usual hydrogen mass
fraction of 0.1 or maximum compression point..

III. Additional energy sources

Most detailed hydrodynamical models of CEE encounter difficulty in ejecting
most of the envelope following the inspiral, showing that most material of the
CE remains bound to the binary though the CE expands to larger separations. In
addition to the internal energy, more possibilities for extra energy sources
have been considered in numerical simulations to reduce the difficulty in
ejecting the CE, including nuclear energy, accretion and jets, long
pulsations, dust-driven winds, and radiation energy for massive donors [25].

It is possible that nuclear energy come from either the burning at the base of
the CE or the ignition on the surface of the accretor. The former could be
explosive if H-rich material penetrates deep into the core of the giant star
and reaches the He-burning shell during the final self-regulating spiral-in
phase (e.g. [302]). The energy released in the explosion could exceed the
binding energy of the He shell and lead to a successful CE ejection
immediately. This CE ejection mechanism may help a less massive companion to
survive the CE and thus has an advantage for the formation of low-mass black-
hole X-ray binaries.

Mass transfer during the CEE will change gravitational potential energy into
thermal energy. Also, the accretion luminosity may be another energy source
for the CE ejection. In most cases of CEE, the Bondi-Hoyle-Lyttleton
(BHL)666The Bondi-Hoyle-Lyttleton (BHL) model was first developed by [303] for
spherical accretion onto a point mass traveling through a gas cloud (or the
interstellar medium) that is assumed to be free of self-gravity, and to be
uniform at infinity. prescription predicts a super-Eddington accretion rate.
However, hydrodynamical simulation of [298] showed that the true accretion
rate given by BHL prescription has been significantly overestimated and that
the contribution of the accretion luminosity could be negligible. Furthermore,
some authors argued that the accretion during the CEE is commonly not
significant at least for non-degenerate companions. This is because the CE has
typically a much higher specific entropy than that of the surface of the
accretor, which would cause a temperature inversion or roughly isothermal
layer between the accretor and the CEE and thus prevent further accretion. In
the cases with degenerate companions, the ignition of nuclear burning on the
surface of the accretor might be inhibited by the high-entropy accreted
material which is buoyant and hard to be compressed to the ignition
conditions. The jets driven by the accretion of compact companions (WDs or
NSs) during spiraling-in could be an effective way for assisting the CE
ejection. This has been observed by the simulations of [304] for binaries with
a $0.88\,M_{\odot}$ RG star with a radius of $83\,R_{\odot}$ and a
$0.3\,M_{\odot}$ compact companion. The simulations showed that the inclusion
of jets could unbind higher envelope mass approximately three times that of
identical simulations without jets. Meanwhile, the authors suggested that the
jets would play a crucial role in shaping the final outcomes of CEE by
generating high-velocity outflows in the polar directions, increasing the
final core-companion orbital separation, and causing a kick velocity of the
core-companion binary. However, small-scale simulations for the same binary by
[305] showed that the broken-out of the RG envelope in large-scale simulations
is choked and that the accreted angular momentum onto the companion is not
high enough to form a disk. The mass accretion rate onto the MS star is about
1-10% of the BHL rate.

Using the stellar evolution code MESA, 1D hydrodynamic simulations of a low-
mass red giant undergoing a synthetic CE event in the final slow spiral-in
phase has been carried out [306]. In these calculations, they investigated the
response of the giant’s envelope by including the heating of the envelope due
to frictional dissipation in multiple configurations, finding that the
envelopes of their models become dynamically unstable and develop large-
amplitude pulsations with periods in the range of 3–20 years. In some cases,
the shocks and associated rebounds that emerged with the pulsations are strong
enough to eject the matter up to 0.1 $M_{\odot}$ (which is about 10% of the
envelope mass) at a speed exceeding the escape velocity of the surface. Such
ejections repeat within a few decades, leading to an averaged mass-loss rate
of the order of $10^{\rm-3}\,M_{\odot}{\rm yr}^{\rm-1}$. They therefore
suggested that long-period, large-amplitude pulsations could be a candidate
mechanism for removing the entire envelope over the duration of the slow
spiral-in phase.

It has been proposed that the dust-driven winds can be produced following the
CEE [307]. In this case, such winds might evaporate the CE through similar
processes operating in the envelope ejection of AGB stars [307]. In AGB stars,
the pulsations drive the envelope expansion, leading to the material in the
envelope being cooled down to low temperatures to enable dust condensation.
The radiation pressure then accelerates the dust and drives winds when the
dust is coupled to the gas. As a consequence, the dust-driven winds eventually
erode the whole envelope of AGB stars. It has been shown that the inspiral
phase of CEE can effectively replace the role of stellar pulsations that
appeared in AGB stars and drive the CE expansion to the scales comparable with
those of AGB envelopes, giving rise to efficient mass-loss through dust-driven
winds [307]. By modelling the CEE on a moving mesh, it was found that the
model with setting the red giant into corotation with the orbital motion
ejects more mass from the system than the non-rotating model, resulting in a
larger final orbital separation [308]. Massive stars are expected to be
significantly different from low-mass stars because their envelopes have
significantly been supported by radiation pressure. Very recently, 3D
hydrodynamical simulations of a CE event involving a $12\,M_{\odot}$ massive
red supergiant donor have found that the inclusion of radiation energy leads
to that 60% more mass being ejected (which is up to two-thirds of the total
envelope mass becoming unbound, or more) and thus a final separation is 10%
larger than that in models with an ideal gas [300]. In addition, the inclusion
of recombination energy could eject almost the entire envelope (at least
three-quarters of the envelope), and lead to the post-plunge-in separation
increase of a further 20%. This seems to suggest that the He recombination
energy is the major source that contributes to the additional mass ejection.
More importantly, these simulations further demonstrated the roles played by
the internal energy, radiation transport, and convection in the ejection of
the CE for binaries with massive donors.

IV. Magnetic fields in the common envelope evolution

It is generally believed that magnetic WDs and CVs are related to the CE
evolution. The magnetic fields could be generated or amplified in the
differentially rotating envelope during the dynamical spiral-in (e.g. [309])
or in an accretion disc around the giant core formed from the tidally
disrupted companion in merging phases (e.g. [310]). The MHD simulations for
the CEE were preformed for the first time by [311]. In these simulations, a
$1\,M_{\odot}$ companion has been spiraled into the envelope of a
$2\,M_{\odot}$ RG star with a $0.4\,M_{\odot}$ He core. Various initial
surface magnetic field strengths of $10^{\rm-10}$, $10^{\rm-6}$, and
$10^{\rm-2}$ G were adopted in their simulations to investigate the influence
of magnetic fields. The simulations showed that magnetic fields are strongly
amplified in the accretion stream that appeared around the $1\,M_{\odot}$
companion on a dynamical timescale, leading to 10-100 kG field strengths
throughout the entire envelope after 120 d. The temporal and spatial scales
are compatible with the magnetorotational instability in the accretion flow
around the companion, thus providing a new way of generating magnetic fields
during the dynamical spiral-in of the CE phase. This short-timescale dynamical
amplification of magnetic fields, however, is difficult to anchor at the WD
surface as pointed out by [312] because the strengths of generated magnetic
fields during the CE phase are several orders of magnitude weaker than those
in WDs (up to $10^{\rm 9}$ G).

Similarly, other studies further followed the 3D MHD simulations for an AGB
star engulfing a companion by [313]. In their simualtions, the AGB star has a
mass of $0.970\,M_{\odot}$, with a core of $0.545\,M_{\odot}$ and a radius of
$173\,R_{\odot}$ at the onset of the CE evolution. The companion is unresolved
and represented by a point particle, with a mass of 0.243, 0.486, and
$0.729\,M_{\odot}$, respectively. The simulation starts from the plunge-in of
the companion into the envelope and covers hundreds of orbits of the binary
system until the envelope has been ejected completely. In this study, magnetic
fields are strongly amplified in two consecutive episodes, i.e., when the
companion spirals in the envelope, and when the companion forms a contact
binary with the core of the former giant star. A magnetically driven, high-
velocity gas outflow is launched self-consistently in the second episode. The
outflow is bipolar and is collimated by the ejected CE. They also fund that
initial parameters of the simulated system have significant impacts on the
morphology of the ejected CE material and therefore provide an explanation for
the variety of shapes observed in PNe. Their simulations could well reproduce
typical morphologies and velocities of young PNe, which leads to an
establishment of a direct link between observed bipolar PNe and the enigmatic
CE phase of binary evolution. As proposed by the authors, the magnetic driving
mechanism could be a universal consequence of CE evolution responsible for a
substantial fraction of observed PNe, and it is likely to also exist in CEE of
other binary stars that lead to the formation of SNe Ia, X-ray binaries, and
GW merger events.

#### 3.2.4 Evidence for the common envelope evolution

Although the CEE plays an important role in binary evolution, direct
observational evidence for this process has been lacking for several decades.
The existence of extremely short-orbital period binaries only means that they
are very likely to be produced from the CEE. There are two major reasons for
this situation. First, the timescale for the CEE is very short ($\sim
1000-10,000$ yr), leading to such events having a low possibility of being
discovered. Second, the binaries and the merge products (e.g. TZO) during the
CEE stage have similar observational properties as some normal stars.

At the termination of the CEE, substantial materials are expected to be
ejected, even when the CEE leads to a merger of the binary
eventually777Partial ejection is likely to happen for merger events because
the orbital energy deposited into the CE at an early stage during the merger
may exceed the binding energy of the outer layers and the upper layers may
also contain a substantial amount of the orbital angular momentum of the
companion (i.e. the merger timescales is too short to transport the angular
momentum across the whole envelope).. Motivated by the models of type IIP SNe
888In the model of type IIP SNe, the ejected stellar plasma expands and cools,
and the recombination occurs and changes the opacity, leading to the
propagation of a photosphere that moves inwards with respect to the mass
variable., observational signature of the CEE was proposed by [314] in which
the radiation from the ejected matter is assumed to be dominated by a
recombination front as the ejected matter cools. According to this
description, the recombination defines a photosphere as the radiation surface
which does not grow with the ejected matter expanding, therefore resulting in
a plateau shape in the light curves. The plateau luminosity $L_{\rm p}$ and
the timescale $t_{\rm p}$ were estimated by applying an analytic model for the
luminosity and duration of Type II SNe light curves from [315]. The results
are presented in Figure 1 of [314] and compared with a sample of luminous red
novae (LRNe; i.e. V1309 Sco, M85 OT, M31-RV, and V838 Mon), a subset of
intermediate luminosity red transients (ILRTs). The effective photospheric
temperature is expected to be $\sim 5,000$K for thick ejecta, leading to that
the ejection events (outburst) being red in color naturally. When the ejected
envelope has fully recombined, the material may become transparent suddenly.
Such expected characteristics, as well as the ejection velocities and event
rate, are consistent with those of LRNe.

After the pioneered work of [314], lots of attention has been paid to the LRNe
[28, 27, 316, 317, 318]. In particular, the properties of two LRNe in nearby
galaxies (M101 OT2015-1 and M31LRN 2015) which have pre-outburst sources being
observed were presented [27, 28]. From archival data spanning 15-8 years
before the outburst, it has found that an F-type yellow supergiant with a
luminosity of $8.7\times 10^{4}\,L_{\odot}$ and an effective temperature of
$\sim 7000$K is consistent with the observations of M101 OT2015-1 [27]. M101
OT2015-1 has an estimated mass of $18\pm 1\,M_{\odot}$ and is crossing the
Hertzsprung gap after H has been exhausted in the core. Based on the observed
properties, this object has been argued to be a binary system, in which the
more evolved system is overfilling the Roche lobe [27]. By analyzing the
optical photometry and spectroscopy along with the color and magnitude of the
pre-outburst source detected by HST for stellar-merger transient M31LRN 2015
in the Andromeda galaxy, it was found that its primary star is a
$3-5.5\,M_{\odot}$ subgiant-branch star with a growing radius of
$30-40\,R_{\odot}$, which is consistent with a picture where a subgiant
primary star engulfs its companion [28]. Also, it has been suggested that
transient M31LRN 2015 has lasted less than 10 orbits of the original binary,
which had a pre-merger period of $\sim 10$ days [28].

Very recently, a sample of 14 LNRe was presented, among which 6 have pre-
transient observations [319]. It has been found that most LRN progenitor stars
in this sample are moderately evolved, with radii between $\sim 1-10$ times
that of MS stars of their mass. It is a bit strange since giant donors are
more likely to experience unstable mass transfer and step into the CEE in
binary evolution. It may mean that the progenitors of the CEE are more likely
to be giant stars or supergiant stars rather than MS stars, which is, however,
inconsistent with the observations from these 14 LNRe. Two reasons might be
responsible for this discrepancy. First, the sample is still small and giant-
like donors are expected to be found in the future. Second, the mass ejection
events of the CEE from giant-like progenitors have properties different from
that of LRNe. However, a final answer for the reasons behind the discrepancy
can only be given by a better understanding of the observational properties of
ejection events of the CEE and LRNe. Considering the fact that the ejecta of
the CEE has lower velocities, higher density, and lower specific thermal
energy in comparison to that of type II SNe and that recombination could be an
important source of luminosity, a new model for synthetic light curves of LRNe
was developed [319]. In this new model, theoretical light curves of LRNe
typically present two peaks, which are consistent with observations. The early
phase light curves are powered by the initial thermal energy of the hot,
fastest ejecta layers, and the second peak (the plateau) is powered by H
recombination from the bulk of the ejecta. This new model was applied to the
sample of LRNe to infer the ejecta properties such as the mass, velocity, and
launching radius [319]. These properties are expected to be helpful for
comparison with the progenitor properties from pre-transient imaging in future
observations, which will give us a better understanding of mass ejection.

Using 9 LRNe with known masses of the quiescent progenitors, a linear fit
between the progenitor mass (in logarithmic, ${\rm log}M_{\rm pro}$) and
V-band second peak (or the plateau) magnitude ($M_{\rm V}$) has been given
[320], which can be written as

${\rm log}(M_{\rm pro}/M_{\odot})=(-0.162\pm 0.020)M_{\rm v}-(0.701\pm
0.048),$ (3.16)

or

$M_{\rm v}=(-5.56\pm 0.68){\rm log}(M/M_{\odot})-(4.91\pm 0.26).$ (3.17)

This linear fit could be used to infer the progenitor mass or predict the
plateau luminosity conveniently.

Some peculiar single stars (such as magnetic stars, blue stragglers, rapid
rotators, T CrB stars, etc) have been considered to be likely formed by the
merging of binaries through the CEE. An unusual, ring-shaped ultraviolet
nebula with the star at its center, TYC 2597-731-1, has been discovered [321].
It is likely an unobstructed stellar merger in an evolutionary stage between
its dynamic onset of the CE ejection and the theorized final equilibrium
state. Both the spectrum and the position suggest TYC 2597-731-1 as an old
star. But TYC 2597-731-1 has an abnormally low surface gravity and a
detectable long-term luminosity decay which is uncharacteristic for its
evolutionary stage. Meanwhile, TYC 2597-735-1 exhibits many features that can
be accounted for by stellar mergers, including ${\rm H_{\alpha}}$ emission,
radial velocity variations, enhanced ultraviolet radiation, and excess
infrared emission as well as signs of dusty circumstellar disks, stellar
activity, and accretion. The comparison between observations and stellar
evolution models suggests that TYC 2597-735-1 merged with a lower-mass
companion about several thousand years ago. Therefore, this object could be
used to infer how two stars merge into a single star directly.

On the other hand, a recent study claimed that they likely observed the
ejected CE around a short-period sdO+WD binary J1920-2001 [321]. The binary
system J1920-2001 has an orbital period of 3.5 hr and a mass ratio (the WD
mass over the sdO mass) of 0.738. In this binary, the sdO star has a mass of
$0.55M_{\odot}$ and it is overfilling its Roche lobe to likely transfer mass
to the WD via an accretion disk. Based on the binary model for sdB stars [214,
232], this binary was likely produced from a CE ejection channel, in which the
progenitor of the sdO star is either an RGB star or an early AGB star. In
addition, strong Ca H&K lines have been observed to blue-shifted by $\sim
200\,\rm km\,s^{-1}$ in J1920-2001. This outstanding feature likely originated
from the CE ejected about 10,000 years ago [321]. It seems not surprising that
the ejected CE has been detected around short-period sdB stars because such
binaries are mainly produced through the CE ejection. Applying a similar
method to the LAMOST survey is expected to discover much more candidates of
sdB binaries like J1920-2001, which will be very helpful for us to better
understand the CE process and thus examine the current binary evolution
theory.

The discovery of Thorne-Zytkow objects (TZOs) that contain an NS inside an RG
would be direct evidence of stellar mergers from the CEE. However, none of
such objects have been discovered yet even though growing survey projects are
being carrying out. Detecting the inner structures of RGs directly would
provide some clues on this.

### 3.3 Non-conservative mass transfer

The treatment of non-conservative mass transfer is one of the key
uncertainties in binary evolution. There are some important questions
associated with the non-conservative mass transfer that remain unclear: How
much mass can be lost from the binary system during the stable mass-transfer
phase? How much specific angular momentum can be taken away by the mass loss?
What is the dominant mechanism to drive the angular momentum loss? The non-
conservative mass transfer was first described in 1970s (e.g. [322, 323, 208])
and it was referred to as “liberal” [324] in very early studies to distinguish
binary evolution with conservative mass transfer. The non-conservative mass
transfer has been widely studied in detail by recent studies from both
theoretical and observational sides. On the one hand, theoretical studies have
shown that mass transfer in binary systems could be non-conservative, causing
the mass and angular momentum loss from the system (e.g. [323, 325, 326, 327,
328]). For instance, it has been suggested that the accretion disk around the
donor star could become geometrically thick at near the disk outer radius if
the mass transfer rate in a binary system is sufficiently high, which leads to
that a large fraction of the transferred mass can be lost from the system
through the outer Lagrangian (L2) point [328]. On the other hand, various
studies have concluded that non-conservative mass transfer seems to be
required to explain observational properties of classes of particular
binaries, including classical Algols [329, 330, 331, 332], sdB stars [214,
232], accreting millisecond X-ray pulsars [333], and ultra-compact X-ray
source [334], etc. For instance, some studies have concluded that about 60 per
cent of transferred masses from $\beta\,\mathrm{Per}$ loses from the system
and takes away about 30 per cent of the angular momentum during mass transfer
through RLOF [335]. In addition, in order to reproduce the short-period sdB
binaries with WD companions, the first phase of mass transfer seems to has to
be stable and non-conservative [214, 232]. Also, the total number of Algol
binaries predicted by the non-conservative mass transfer model has been found
to match the observation better [331]. Recently, it has been suggested that
the large orbital period derivative in the ultra-compact X-ray source XB
1916-053 is very likely due to a high non-conservative mass transfer [334].

Different treatments of the mass and angular momentum loss in non-conservative
mass transfer play an important role in determining the distributions of
binary properties and their fates. Three mechanisms are generally adopted to
deal with orbital angular momentum loss in a binary system: (i) _gravitational
radiation_ , (ii) _magnetic braking_ (MB), and (iii) _mass loss_.
Gravitational radiation has been well studied theoretically, which is thought
to be dominant in angular momentum loss of tight binary systems. In
particular, the gravitational wave signals have been confirmed by the
observations [165, 336]. The standard prescription from [337] is commonly used
for the calculation of gravitational radiation in binary evolution. The
prescription for MB is still uncertain. Earlier theoretical models generally
employed the standard “Skumanich MB” law that derived from observations of the
time-dependence of rotational braking of Sun-type stars [338, 339, 340]:

$\frac{{\rm d}J_{\rm MB,Sk}}{{\rm d}t}=-3.8\times
10^{-30}\;M_{2}\,R_{\odot}^{4}\,\left(\frac{R_{2}}{R_{\odot}}\right)^{\gamma_{\rm
mb}}\Omega_{2}^{3}\quad{\rm dyne\,cm}\;,$ (3.18)

where $\gamma_{\rm mb}$ ( which is from 0 to 4) is a dimensionless parameter;
$M_{2}$, $R_{2}$ and $\Omega_{2}$ are the mass, radius and angular velocity of
the donor star, respectively. Recently, it is found that the standard
“Skumanich MB” law gives mass transfer rates too weak to explain most observed
persistent low-mass X-ray binaries (LMXBs) [341]. The MB law was therefore
modified by considering a scaling of the magnetic field strength with the
convective turnover time, and scaling of MB with the wind mass-loss rate, i.e.

$\frac{{\rm d}J_{\rm MB}}{{\rm d}t}=\frac{{\rm d}J_{\rm MB,Sk}}{{\rm
d}t}\;\left(\frac{\Omega_{2}}{\Omega_{\odot}}\right)^{\beta}\left(\frac{\tau_{\rm
conv}}{\tau_{\rm\odot,conv}}\right)^{\xi}\left(\frac{\dot{M}_{\rm
2,wind}}{\dot{M}_{\rm\odot,wind}}\right)^{\alpha}\;,$ (3.19)

where $\Omega_{\odot}=3.0\times 10^{-6}\;{\rm s}^{-1}$, $\tau_{\rm conv}$ is
the turnover time of convective eddies and $\tau_{\rm\odot,conv}=2.8\times
10^{6}\;{\rm s}$, $\dot{M}_{\rm 2,wind}$ is the wind mass-loss rate of the
donor star and $\dot{M}_{\rm\odot,wind}=2.54\times 10^{-14}\;M_{\odot}\,{\rm
yr}^{-1}$ [342]. Four cases based on different combinations of values of
$\beta$, $\xi$, and $\alpha$ were defined (see also Table 1 of [343]):

$(\beta,\,\xi,\,\alpha)=\,\left\\{\begin{array}[]{ll}(0,\,0,\,0)&\mbox{\hskip
2.84544pt standard Skumanich MB}\\\ (0,\,2,\,0)&\mbox{\hskip 2.84544pt
convection-boosted MB}\\\ (0,\,2,\,1)&\mbox{\hskip 2.84544pt intermediate
MB}\\\ (2,\,4,\,1)&\mbox{\hskip 2.84544pt wind-boosted MB}\\\
\end{array}\right.$ (3.20)

The models with “intermediate MB” law (i.e. $\beta=0$, $\xi=2$ and $\alpha=1$)
has been found to reproduce the observed orbital characteristics of LMXBs.

By considering the dependence of the magnetic field strength and the Alfvèn
radius on the convective turnover time and the donor’s rotation, another new
MB prescription, the so-called “Convection And Rotation Boosted (CARB)”, was
further presented in the same year [341]. It is found that the CARB MB
prescription can well reproduce the observed mass transfer rates of all
observed Galactic persistent NS LMXBs. There are also more MB prescriptions
have also been presented by other studies [344, 345, 346]. The strength and
weaknesses of different MB prescriptions presented by various studies in
explaining the observational characteristics of LMXBs have been discussed in
detail by [347]. They showed that convection-boosted MB prescription by [343]
seems to provide the best match with the observations. With using the newly
suggested MB prescriptions, different studies [343, 348] have investigated
their effects on the formation and evolution of LMXBs, radio binary
millisecond pulsars (BMSPs), extremely low-mass helium WDs and ultra-compact
X-ray binaries (UCXBs). Interestingly, it has been found that the initial
orbital period range of LMXBs which can evolve via detached NS+WD binaries
into UCXBs becomes significantly wider if the “intermediate MB prescription”
of [343] is adopted. This therefore helps to relieve the long-standing fine-
tuning problem in producing UCXBs presented by earlier studies [349, 350,
351]. But it was found that the MB prescription of [343] fails to explain the
wide-BMSPs [348]. Further study showed that the CARB MB prescription can
reproduce the LMXB phase as well as being able to form UCXBs and wide BMSPs
systems [352].

There are also other ways that could affect mass transfer in a binary system
such as the spin of the accretor. For example, by calculating the maximum
accreted mass of an NS in the recycling process in a binary system with
considering the effect of NS spin evolution, it has been found that massive
NSs can accrete more material than low-mass NSs with other initial parameters
fixed. In addition, the spin evolution of an NS can affect the maximum NS
accreted masses due to the propeller effects [353]. In summary, despite
substantial efforts on the theoretical and observational sides, the
constraints on the mass and angular momentum loss in the non-conservative mass
transfer of a binary system are still quite weak. Future observations and
multi-dimensional hydrodynamical simulations of mass transfer in a binary
system are still needed to give a better understanding of non-conservative
mass transfer.

### 3.4 Extended binary evolution channels

#### 3.4.1 The chemically homogeneous evolution of massive binaries

Figure 3.7: Schematic overview of the chemically homogeneous evolution (bottom
tracks) in a close binary system. For a comparison, the ‘normal’ evolution
(top tracks) of a binary system is also presented. The figure is reprinted
from [354] (see also [355]).

The first direct detection of gravitational waves GW150914 was announced by
the LIGO and Virgo scientific collaboration on 2015 September 14, which has
been confirmed to come from the merger of binary black holes (BBHs) with
masses of about $30\,\mathrm{M_{\odot}}$ [165, 356, 164]. This reveals the
existence of BHs with such heavy masses in the form of binaries. To form such
heavy BHs, it requires the star to form in sub-solar metallicity environments
because a higher metallicity can cause the strong stellar winds to lead to
that the remnant mass of the star being below $30\,\mathrm{M_{\odot}}$ [357,
358]. In the literature, several possible formation scenarios have been
proposed to produce such heavy BBHs (see [165], and references therein),
including (i) the dynamical formation in dense stellar clusters [359, 360,
361], (ii) the classical isolated binary evolution channel involves non-
conservative mass transfer or common envelope ejection [362, 363], (iii) the
chemically homogeneous evolution in tidally locked close binaries [364, 365,
366], and (iv) the coalescence of primordial black holes [367]. In this
section, we will only focus on the discussion of the chemically homogeneous
evolution scenario.

The so-called “chemically homogeneous evolution” was originally described for
_rotating single stars_ by Maeder in 1987 [368]. If the stars rotate rapidly,
they may experience efficient mixing throughout core H burning, which allows
nuclear-burning products produced in the center of the stars to transport to
their envelopes significantly [368, 369, 370, 371]. This leads to that the
build-up of a chemical gradient between core and envelope is prevented and the
star evolves chemically homogeneously to avoid the strong post-MS expansion.
As a consequence, the star stays compact and their envelopes become He-rich
gradually during their MS evolution. Meanwhile, the stars become hotter and
more luminous and slowly evolve towards the He MS. In Fig. 3.7, we show a
comparison between chemically homogeneous evolution in a close binary system
and its nomral evolution. Different studies on the chemically homogeneous
evolution of the stars have shown that this channel is favoured at low
metallicity environments, in which a strong angular-momentum loss is avoided
and the stars remain fully mixed during their MS evolution (e.g. [372, 373,
374, 375, 376]). The chemically homogeneous evolution of rapidly rotating
single stars have been suggested as progenitors of long-duration gamma-ray
bursts [377, 372] because that this channel can produce rapidly spinning iron
cores.

Interestingly, it has been suggested that the tidal interaction of both stars
in massive tight binary systems can force the stars to spin rapidly, which is
sufficient to reach the conditions for chemically homogeneous evolution
([378]; see also [379, 364]). By considering the mixing induced by rotation
and angular momentum transport by magnetic fields, detailed studies have
provided the initial binary parameter space that allows for chemically
homogeneous evolution in massive close binaries (e.g. [378, 379, 364]). The
chemically homogeneous evolution could force two stars in a massive close
binary to shrink and remain inside their Roche lobes to avoid the CE phase.
This leads to the two stars gradually converting nearly all their H into He.
As a consequence, two massive He stars are expected to be formed in this
evolutionary scenario. These two massive He stars may eventually collapse to
form two massive BHs that can explain BH masses inferred from GW150914 [378,
364]. Merging binary black holes formed through chemically homogeneous
evolution in massive tight binary systems have been investigated with both
detailed binary stellar evolution calculations (e.g. [364, 379]) and Monte
Carlo simulations (e.g. [365, 378]). It has been suggested that about 500
merging BHs events per year could be detected with advanced ground-based
detectors operating at full sensitivity in the chemically homogeneous
evolution channel, and the typical total masses of two BHs in this channel are
$50$-$110\,\mathrm{M_{\odot}}$. Also, this channel seems to favour producing
binary BHs with comparable masses with mass ratios $>0.6$ [365, 378].
Furthermore, it has been found that the chemically homogeneous evolution
channel can fully explain the inferred parameters and merger rates of the
GW150914 [365, 378].

However, whether all of the detected GW events were produced through a single
formation scenario is still unclear [380, 381]. In addition, as for other
proposed formation channels, the chemically homogeneous evolution channel also
has some uncertainties regarding the predictions of the formation of BBHs such
as the simplified treatment of the mixing processes. Also, solid observational
evidence for the existence of chemically homogeneously evolving stars is still
missing [378, 366, 365], although a few observed binary systems have been
suggested to undergo chemically homogeneous evolution such as HD 5980 [382]
and VFTS 352 [383]. Finally, it seems that the chemically homogeneous
evolution channel has difficulties in explaining relatively low-mass BBHs such
as GW151226 with a total mass of about $21.8\,\mathrm{M_{\odot}}$.

#### 3.4.2 Wind Roche-lobe overflow mass transfer

Figure 3.8: Schematic illustration of the wind Roche-lobe overflow (WRLOF) in
a binary system (reproduced from the talk of Ph. Podsiadlowski, see also
[384]). The WRLOF occurs when the radius of the wind acceleration zone (red
circle) of the donor star is larger than (or comparable to) the Roche-lobe
radius (blue curve) of the mass-losing star. The accreting star is given in a
black dot.

The wind mass transfer plays an important role in the formation of peculiar
stars formed in the binary scenario, in particular for the formation of
Carbon-enhanced metal-poor (CEMP) stars . Early studies on the formation of
CEMP stars generally adopted the canonical Bondi-Hoyle-Lyttleton (BHL) model
[303] for the treatment of wind mass transfer from an AGB companion. The BHL
model assumes that the wind speed is much faster than the orbital speed of the
accreting star. However, AGB stars in wide binary systems are expected to have
a slow wind speed of $5-35\,\mathrm{km\,s^{-1}}$ that is comparable with the
orbital velocity their orbital velocities (e.g. [385, 386, 387]). A number of
recent hydrodynamical simulations of mass transfer in a binary system with an
AGB donor have shown that the accretion efficiency of AGB winds and the amount
of angular-momentum loss can be significantly different from those described
by the BHL model (e.g. [388, 389, 384, 390, 391, 392, 393, 394, 395], and
references therein).

In particular, there are some studies have presented a new mass transfer mode
in binary systems with an evolved giant donor - the so-called “wind Roche-lobe
overflow (i.e. wind-RLOF, or WRLOF)” [384, 396] \- in which wind material
fills the donor’s Roche-lobe and transfers to the accretor through the inner
Lagrangian point, as for the standard RLOF. As shown in Fig. 3.8, the WRLOF
occurs when the radius of the wind acceleration region of the donor star is
larger than the Roche-lobe radius of the donor star. The accretion rate in
WRLOF model could be 100 times higher than that given by the canonical BHL
model [384, 396, 397]. This new mass transfer mode (i.e. WRLOF) between RLOF
and wind mass-transfer is expected to have important implications for stellar
and galactic chemical evolution such as the formation of SNe Ia [398, 399],
CEMP stars [397] and X-ray binaries [400, 401], etc. Very interestingly, by
investigating the effect of WRLOF model on the formation of CEMP stars, it has
been found that the number of CEMP stars predicted by the WRLOF model can
increase by a factor of $1.2$-$1.8$ compared to that given by the classical
BHL model [397].

To comprehensively investigate how the WRLOF affects the evolution and fates
of binary systems, one need to provide a realistic prescription of the WRLOF
that can be implemented into 1D stellar evolution and population-synthesis
codes. However, how mass and angular momentum accretion (or loss) efficiencies
depend on binary properties (e.g. binary orbital period, mass ratio, mass-loss
rate and wind velocity, and so on) is still quite uncertain, which requires
future improvements of two- or three-dimensional hydrodynamial simulations of
WRLOF to cover a wider range of binary parameters, and to better consider the
detailed wind acceleration mechanism due to star pulsation and radiation
pressure on dust [392, 384].

### 3.5 Binary population synthesis

The traditional stellar evolution theory generally focuses on the evolution of
a single object (a star and/or binary) at a time. This leads to that it is not
able to present the statistical properties of stellar and/or binary
populations. However, a galaxy consists of thousands of millions of stars, and
more than half of stars are in binaries. Understanding the properties of
galaxies and different types of stars requires a statistical investigation of
stellar populations. In addition, growing large surveys such as the Sloan
Digital Sky Survey (SDSS) ([75]), Large Area Multi-Object Fiber Spectroscopic
Telescope (LAMOST) survey ([79]), Gaia ([53]), _Kepler_ ([49]) and TESS
([50]), have revealed the statistical properties of stars and galaxies from
the observational side, which makes significant progress in astrophysics.
Therefore, to provide insight into the formation and evolution properties of
different stars and binaries rely on not only individual studies of each
object but also systematical investigations of stellar and/or binary
populations from the theoretical side, and then comparing theoretical
predictions with the observations. This leads to the proposal of “binary
population synthesis (BPS)” approach.

BPS approach was developed by astronomers in the 1990s to rapidly evolve a
large sample of binary stars over the course of their lifetimes to study their
evolution and properties of various types of close binary populations with a
simplified treatment for some physical processes such as the CEE, the
stability and rate of mass transfer (see [17], for a recent review). BPS has
been very successfully used for studies of many aspects of astrophysics. For
instance, it has been widely used to investigate the formation and evolution
of exotic objects, including double black holes (BBHs; [402, 380, 403]),
double neutron stars (e.g. [404, 405, 406]), double white dwarfs (e.g. [274,
407, 408]), Type Ia supernovae (e.g. [409, 277, 236, 410, 411, 412, 413, 227,
414]), gamma ray bursts (e.g. [415, 416]), X-ray binaries (e.g. [348]),
pulsars (e.g. [417, 418]), novae (e.g. [419, 420, 421]), cataclysmic variables
(e.g. [422, 423]), hot subdwarfs stars (e.g. [214, 232, 217, 424]), barium
stars (e.g. [268, 425]), blue stragglers (e.g. [211, 216]), etc. In addition,
it is a robust approach to studying the spectral energy distribution and the
chemical evolution of galaxies (e.g. [426, 427]). Many BPS codes have been
developed, e.g. Scenario Machine ([428, 429]), SeBa ([430, 277, 422, 413]),
Yunnan Model ([407, 214, 232, 431, 426, 432]), BSE ([433]), StarTrack ([404]),
BiSEPS ([417]), BPASS ([434]), SYCLIST ([435]), COMPAS ([380]), MOBSE ([436]),
Combine ([402]), dart-board ([437]), COSMIC ([403]), etc.

Figure 3.9: Major steps in BPS studies. The figure is adopted from [232], see
also Figure 2 of [17].

As shown in Fig. 3.9, BPS studies generally involve several major steps as
follows (see also Fig. 2 of [17]): (i) Generating a large sample of binaries
by setting initial distributions of some parameters (e.g. masses, periods,
mass ratios); (ii) Following the evolution of both components of every binary
system and their interactions over the course of the binary lifetime; (iii)
Deriving the diverse properties and observables of binary populations such as
the frequency of an important astrophysical event; (iv) Comparing the results
of BPS calculations with the corresponding observations.

The details of the BPS approach have been reviewed by Han et al [17],
including the role of BPS, its general picture and main ingredients, and the
various components that comprise it. Here, only skim the surface of it. First,
to generate sample binaries, we need to set some basic input parameters and
assumptions, which are described below.

  1. (1)

_The star formation rate (SFR)_ — One can simply assume a constant SFR over
the last $15\,\mathrm{Gyr}$ in most cases.

  2. (2)

_The initial mass function (IMF)_ — The IMFs of [438] and [439] are usually
adopted for analytic studies and BPS studies, respectively (see also [440]). A
canonical discussion on the IMF has been given by [441]. Generally, the
primary mass is generated with the formula of [206],

$M_{1}={0.19X\over(1-X)^{0.75}+0.032(1-X)^{0.25}},$ (3.21)

where $X$ is a random number uniformly distributed between 0 and 1, and the
adopted ranges of primary masses are 0.8 to $126.0\,M_{\odot}$.

  3. (3)

_The mass-ratio distribution_ — BPS studies mainly take a constant mass-ratio
distribution (e.g. [442, 4]),

$n(1/q)=1,\qquad 0\leq 1/q\leq 1,$ (3.22)

where $q=M_{1}/M_{2}$. An alternative distribution of mass-ratio is the case
where the masses of both binary components are chosen randomly and
independently from the same IMF.

  4. (4)

_The orbital period distribution_ — We simply assume that all stars are
members of binary systems and that the distribution of separations is constant
in $\log a$, where $a$ is the separation, for wide binary separations, and
falls off smoothly at close separations (for a discussion, see [443, 4]),

$an(a)=\begin{cases}\alpha_{\rm sep}({a\over a_{0}})^{m},\&a\leq
a_{0};\cr\alpha_{\rm sep},\&a_{0}<a<a_{1}\cr\end{cases}$ (3.23)

where $\alpha_{\rm sep}\approx 0.070$, $a_{0}=10\,R_{\odot}$,
$a_{1}=5.75\times 10^{6}\,R_{\odot}=0.13\,{\rm pc}$, and $m\approx 1.2$. This
distribution means that there is an equal number of wide binary systems per
logarithmic interval and that approximately 50% of stellar systems are
binaries with orbital periods less than 100 yr.

  5. (5)

_The orbital eccentricity distribution_ — Under an assumption of that all
binaries are circularized, the eccentricity can set to be $e=0$.

Figure 3.10: Comparison between merger rate distributions inferred from the
observations and theoretical predictions from the polytropic (_solid curves_)
and Ge’s (_dotted curves_) models. Different coloured points with error bars
present the observational rates of DWD mergers that produce
$0.8-1.32\,M_{\odot}$ WDs (_red points_ ; [444]), SNe Ia (_green points_ ;
[445]), ELM WD mergers (_blue point_ ; [446]), and all DWD mergers (_gray
point_ ; [447]), respectively. The figure is reproduced from [408] with
permission © ESO.

Once a large sample of initial binary systems is generated, we need to follow
the evolution of both components of every binary system until the studied
important objects are formed. The studies of binary populations in principle
need to follow the evolution of every binary system in detail. However, it is
not feasible to evolve a population of binary stars with a detailed stellar
evolution code. Therefore, the evolution of each component in a binary system
in BPS studies is commonly done by using either the rapid fit evolutionary
calculations or interpolations in available stellar evolution model grids.
This simplifying treatment for the evolution of the single star might
introduce some uncertainties in the results although these are typically small
compared to other uncertainties of BPS studies.

Moreover, there are different physical processes that need to deal with for
the interactions between two components of a binary system, including mass-
transfer mechanisms such as stellar wind and RLOF, mass transfer rate,
prescriptions for the stability of mass transfer, tidal evolution, CEE, and
mass and angular momentum loss from the binary system, etc. These physical
processes are generally treated with some simplifying assumptions in BPS
studies. For instance, prescriptions for the stability of mass transfer in BPS
studies are commonly treated as the parametrization (e.g. [433, 17]). This
also introduces more uncertainties in BPS results. However, most of these
physical processes remain quite uncertain and cannot be modelled in detail
with a stellar evolution code.

In summary, BPS is a robust approach to deriving the observables of binary
populations in the studies of many aspects of astrophysics. However, one
should keep in mind that theoretical predictions from BPS studies still have
uncertainties due to weak constraints on the initial inputs and assumptions,
and some physical processes in binary evolution. For instance, a series of BPS
calculations were performed by adopting two different descriptions for mass
transfer stability in a binary system (i.e. the adiabatic mass loss model [22]
— Ge’s model and the composite polytropic model) to investigate the influence
of a mass transfer stability criterion on double white dwarf (DWD)
populations[408]. The study suggested that mass transfer stability plays an
important role in the formation and properties of DWD populations as well as
in the progenitors of SNe Ia and detectable GW sources. In addition, they
found that Ge’s model seems to provide a better match with observational
properties of DWD populations such as the merger rate distribution per Galaxy
and the space density in the Galaxy (Fig. 3.10; [408]).

## 4 Stellar Nucleosynthesis

Figure 4.1: Schematic overview of how the main nuclear processes in the star
depend on its masses at solar metallicity. The figure is reproduced base on
Figure 1 of [448].

Nucleosynthesis first occurred within a few minutes of the Big Bang, which
produced light elements (hydrogen, helium, and lithium). Elements heavier than
helium (which are known in astronomy as metals) are thought to be produced in
the interiors of stars during their evolution and in the final explosion. The
evolution of nucleosynthesis of a single star primarily depends on its initial
mass and metallicity (which is denoted by the symbol $Z$). The first-
generation stars formed after the Big Bang had abundances of about 76%
hydrogen, 24% helium, and a tiny fraction of heavier elements (i.e. an
extremely low Z).

Figure 4.1 presents the evolution of nucleosynthesis in stars. It starts from
the hydrogen core burning with the $pp$ chains or the CNO cycle, then to
helium-core burning with triple-$\alpha$ reactions, and to carbon and oxygen
core burning, and so on. Finally, an onion-like structure has been formed with
the iron core in the center. For low and intermediate-mass stars, the carbon-
and oxygen-core burning and the following series of reactions may not happen
at all. The envelope would be ejected to space via superwind, leaving white
dwarfs as remnants. An important process for those stars is the slow neutron
capture process (s-process) when the stars are in the AGB stage, leading to
the production of several elements heavier than Fe. An intermediate neutron
process (known as i-process) has been proposed to occur in AGB stars to
account for the abundant pattern observed in metal-poor carbon-enhanced
stars(see Section 4.4). Those elements may be dredged up to the surface to be
observed by us. In fact, the productions in the core or burning shell have
several chances to be taken onto the surface when the stars climb upwards the
giant branches, which are known as the first/second/third dredge-up due to the
development of the surface convection zone. Massive stars end their lives via
supernovae or direct collapse to black holes. Abundant nucleosynthesis will
take on in the supernovae. The rapid neutron capture process (r-process) is
believed to occur in core-collapse supernovae, producing some other heavier
elements. Fig. 4.2 summarizes the element origin from stellar evolution.

Figure 4.2: Schematic overview of the nuclear processes in the Universe on the
chart of nuclides. Here, r-, s-, i-, rp-, and $\rm\gamma-$process respectively
represent the rapid neutron capture, slow neutron capture, intermediate
neutron capture, rapid proton capture, and photodisintegration process.

Two components in a binary system could interact with each other. As a result,
the surface material of one star (i.e., the donor star) can be transferred to
the other star (i.e., the accretor) through either RLOF, wind, or WRLOF. This
leads to the formation of different important objects such as novae,
kilonovae, type Ia supernovae, carbon-enhanced metal poor stars, barium stars,
and so on. Therefore, one can expect that nucleosynthesis in binary stars are
significantly different from that in single star as discussed in the above
section. Here, we briefly discuss the nucleosynthesis of some peculiar objects
that are widely thought to be formed in binary systems.

### 4.1 Novae

It is commonly understood that a nova outburst results from a thermonuclear
runaway (TNR) on the surface of a WD (usually CO- or ONe-WD) accreting
hydrogen- or helium-rich matter from a low-mass companion in a close binary
system [449, 450, 451]. The TNR is triggered by the accumulation of a
hydrogen-rich envelope on the WD continues until a critical pressure is
achieved at the base of the accreted envelope, which requires a mass-arrection
rate of $\lesssim 10^{-9}\,\rm M_{\odot}\,yr^{-1}$ onto the WD. Once the TNR
starts, the temperature rises rapidly to 10-100 million K and the entire
envelope is burnt within a few minutes, ejecting mass and elements. Nova
outbursts play an important role in the chemical evolution of Galaxy. Although
it is generally thought that novae can provide about 0.3 per cent of the
interstellar medium in the Galaxy, growing evidence both in observational and
theoretical sides have suggested that novae may be the most important
contributors of $\rm{}^{7}Li$, $\rm{}^{13}C$, $\rm{}^{15}N$, $\rm{}^{17}O$, as
well as the radioactive isotopes like $\rm{}^{7}Be$, $\rm{}^{22}Na$ and
$\rm{}^{26}Al$ in the Galaxy [452, 451]. However, more observations are still
needed to confirm such processes in novae.

The nucleosynthesis in novae is thought to operate dominantly by both the pp
chains as well as the CNO cycle. Meanwhile, the NeNa and MgAl cycles are also
thought to operate in nova nucleosynthesis to alter the isotopic compositions
of Ne and Mg. However, the details of nucleosynthesis in novae are very hard
to model because of short timescale of the event, determination of density and
temperature of the burning region. In addition, convection could play a
critical role in the nova explosion. However, it is unclear that how the
accreted shell is mixed with the outer layers of the underlying WD, which also
poses challenges to modelling nova nucleosynthesis [451, 453]. The
observations have detected the enrichments of intermediate-mass elements
(e.g., C, N, O, Ne, Na, Mg, and Al) in nova ejecta [449, 452], which is
thought to be strong evidence of mixing between the material of WD and the
accreted matter. Nonetheless, the exact mixing fraction in novae remain
unknown. In addition, the large uncertainties of some challenging reactions
such as $\rm{}^{18}F(p,\alpha)\rm^{15}O$, $\rm{}^{25}Al(p,\gamma)\rm^{26}Si$
and $\rm{}^{30}P(p,\gamma)\rm^{31}S$ also have a strong impact on nova
nucleosynthesis [454, 455], which still requires improvements in these related
reaction rates by future experimental investigations.

### 4.2 Merge of double neutron stars and kilonovae

Theoretical studies on the inspiral and merger of double neutron stars have
been carried out since the 1970s (e.g. [456]), showing that neutron-rich
matter that undergoes rapid neutron capture (i.e. r-process) nucleosynthesis
is ejected in the violent merger of two neutron stars (e.g. [457, 458, 459,
460, 461, 462, 463, 464, 465, 466]). Therefore, the mergers of double neutron
stars have long been proposed to be one of the main astrophysical sites for
the occurrence of the r-process which can assemble into rare heavy elements
such as gold and platinum. The radioactive decay of unstable nuclei
synthesized during the merger of double neutron stars can power a rapidly
evolving electromagnetic transient called a “kilonova” (e.g. [467, 468, 469,
470, 471]). Detecting the electromagnetic spectrum of a kilonova is therefore
expected to provide the direct evidence for the r-process nucleosynthesis in
the merger of double neutron stars. However, there has been a lack of direct
observational evidence for this hypothesis.

The first gravitational wave (GW) signal of the merger of double neutron stars
was detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO)
in August of 2017, i.e. GW170817 [336], which opens the new era of
multimessenger astronomy (e.g. [472, 473, 465, 474, 475]). More importantly,
the discovery of an electromagnetic counterpart (i.e. a kilonova AT2017gfo,
[476, 477, 478]) to GW170817 for the fist time provides the important and
direct opportunity to probe the nucleosynthesis in merger of double neutron
stars [464, 465]. Theoretical studies have shown that the ejected matter of
neutron star mergers contains two distinct components: (i) The neutron-rich
ejecta composed of heavy r-process elements ($\mathrm{A\geq 140}$), which
radiate red/infrared light, producing a “red” kilonova; (ii) The less neutron-
rich ejecta composed of light r-process elements ($\mathrm{A\leq 140}$), which
radiate blue/optical light, producing a “blue” kilonova (e.g. [464, 479,
480]). The behaviour of the AT2017gfo has been found to be in good agreement
with theoretical predictions for kilonovae. Furthermore, the ejecta mass for
the red and blue components of the kilonova light curve of AT2017gfo has been
roughly estimated to be about $5\times 10^{-2}\,\mathrm{M_{\odot}}$ and
$2\times 10^{-2}\,\mathrm{M_{\odot}}$, respectively [464, 481]. Given that a
substantial ejecta mass has been inferred from AT2017gfo, the question is
whether or not the binary neutron-star mergers is the dominant producer of
heavy r-process elements in the Universe. Early studies have suggested that a
mass production rate of r-process nuclei of $\sim
10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$ is required to explain the r-process
abundances in the Milky Way [482, 464]. Under an assumption of that all
neutron star mergers have the same productions of r-process elements as for
AT2017gfo and by taking a Galactic binary neutron-star merger rate of $8\times
10^{-5}\,\mathrm{yr^{-1}}$ (which is however still uncertain; [483, 484, 356,
485]), the mass production rates of heavy and light r-process elements from
neutron star mergers are expected to be $\sim 4\times
10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$ and $\sim 2\times
10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$, respectively. This is approximately
consistent with the Galactic r-process production rates. As a consequence,
Kasen et al (2017) [464] suggested that the binary neutron-star mergers may be
the dominant contributors to the production of heavy r-process elements in the
Galaxy. However, there are still some uncertainties on the estimates of binary
neutron-star rate and ejecta mass derived from the observation of a kilonova
such as the radioactive heating rate and viewing angle effects (for a detailed
discussion on these uncertainties, see [464]). To give a stronger conclusion
on the question of whether the binary neutron-star mergers is the dominant
producer of heavy r-process elements in the Universe, future GW detections of
more neutron star mergers and follow-up multi-messenger observations of their
electromagnetic counterparts are still needed.

### 4.3 Type Ia supernovae

SNe Ia are widely accepted to be thermonuclear explosions of WDs. SNe Ia have
been used as the standard candles to measure the accelerating expansion of the
Universe and constrain the properties of Dark Energy [486, 487, 488]. SNe Ia
are the important elements factories in the universe [489, 490, 491, 492].
They are responsible for the production of iron-group elements (IGEs; e.g. Ti,
Cr, Mn, Fe, Co, Ni, Cu and Zn) and intermediate-mass elements (IMEs; e.g. Mg,
Si, S, Ar and Ca). The nucleosynthetic outcomes of SNe Ia strongly depend on
the local fuel density of the exploding WD. This means that different
explosion mechanisms (which are still mysterious) of SNe Ia may lead to
various nucleosynthetic outcomes [493, 492, 494]. For instance, stable IGEs
can only be synthesized when the densities are higher a few
$10^{8}\,\mathrm{g\,cm^{-3}}$, which requires a CO WD above
$1.2\,\mathrm{M_{\odot}}$. Depending on whether SNe Ia are triggered when the
WDs approach the near Chandrasekhar mass or the sub-Chandrasekhar mass, a
number of explosion models which involve deflagrations and/or detonations have
been proposed in the literature, such as near Chandrasekhar-mass
deflagrations, near Chandrasekhar-mass detonations, near Chandrasekhar-mass
delayed detonations, gravitationally-confined detonations, sub-Chandrasekhar-
mass double detonations, and violent mergers, etc (for a review of SN Ia
explosion models, see [493, 495]).

Because a detonation with supersonic speed does not allow the WD to expand to
lower densities, the near Chandrasekhar-mass detonation model produces too
much ${}^{56}\mathrm{Ni}$ and IGEs but too little IMEs (e.g. Si, S and Ti) to
match the observations of normal SNe Ia [489]. As a consequence, deflagrations
of near Chandrasekhar-mass WDs have been proposed for SNe Ia. A subsonic
deflagration allows the WD to expand to lower densities, which reduces the
production of ${}^{56}\mathrm{Ni}$ and gives an efficient IME yields. One
widely discussed near Chandrasekhar-mass deflagration mode is the 1D ‘W7
model’ which was originally proposed by Nomoto et al [490]. The
nucleosynthetic yields of the W7 model such as ${}^{56}\mathrm{Ni}$ mass and
IMEs are consistent with those of normal SNe Ia. However, the W7 model uses a
parametrized description for the thermonuclear burning process. 3D near
Chandrasekhar-mass deflagration model has difficulties in producing enough
${}^{56}\mathrm{Ni}$ mass for normal SNe Ia ($\sim 0.5\,\mathrm{M_{\odot}}$),
and it leaves too much-unburned material such as C, O and Mg. However, weak
deflagrations of near Chandrasekhar-mass WDs have recently suggested to well
reproducing the observational features of a sub-luminous subclass of SNe Ia,
the so-called SNe Iax (e.g. [496, 497]). The near Chandrasekhar-mass delayed
detonation model has been developed to overcome the shortcoming of
nucleosynthetic yields of pure deflagration or detonation model [498, 499]. In
this model, an initial subsonic deflagration allows the WD to expand at an
early phase, leading to the later-on detonation triggered at lower densities.
On the one hand, the delayed detonation model enhances the yields of IMEs but
reduces the production of ${}^{56}\mathrm{Ni}$ mass compared with those in the
pure detonation model [500, 493]. On the other hand, it produces more
${}^{56}\mathrm{Ni}$ mass and IGEs than those in the pure deflagration model.

Figure 4.3: Nucleosynthesis yields of a variety of SN Ia explosion models,
including one C-ignited violent merger (VM), two sub-Chandrasekhar pure
detonations (PD081, PD115), three double detonations ($\rm M2a_{\odot}$,
CSDD-S, CSDD-L), one helium detonation (HeD-S), two Chandrasekhar-mass pure
deflagrations (R60, W7), and a Chandrasekhar-mass delayed detonation
(N100ddt). Here, it presents elemental ratios to Fe (with radioacitve isotopes
decayed to $\rm 2\times 10^{9}\,yr$) compared to their solar ratios. The
figure is reproduced from Figure 1 of [494] with permission © ESO.

The density structure of a sub-Chandrasekhar mass WD is different from that of
a near Chandrasekhar mass WD (see Fig. 1 of [492]). This is expected to give
different nucleosynthetic yields. The sub-Chandrasekhar mass WDs can also
trigger SN Ia explosion through a double-detonation mechanism (i.e. an initial
detonation of the He shell and a subsequent core carbon detonation), in which
a He shell is accumulated and ignited on the surface of the WD (e.g. [501,
502, 503, 504, 505, 506, 507]). Comparing to explosive C-O burning, the
initial detonation in a thick He shell ($\sim 0.1$-$0.2\,\mathrm{M_{\odot}}$)
typically leads to the over-production of $\alpha$-isotopes such as
${}^{36}\mathrm{Ar}$, ${}^{40}\mathrm{Ca}$, ${}^{44}\mathrm{Ti}$,
${}^{48}\mathrm{Cr}$ and/or ${}^{52}\mathrm{Fe}$, which seems to be
inconsistent with the observations of normal SNe Ia (e.g. [508, 373, 509]).
However, the nucleosynthetic products of the outer layer on the top of the WD
strongly depend on the mass, and chemical condition of the He shell. Recent
studies have suggested that a thin He shell ($<0.1\,\mathrm{M_{\odot}}$) with
pollution of CO seems to hold promise for explaining SNe Ia (e.g. [510, 511,
506, 507]). The double detonation mechanism could also occur during the
violent merger of two WDs, which is known as “ _He-ignited violent merger
model_ ” or the “ _dynamically driven double-degenerate double-detonation
(D6)_ ” model (e.g. [185, 510, 512]). In D6 model, because the He shell on the
top of primary WD is generally quite thin and the He detonation happens in the
dynamic stage rather than in the hydrostatic equilibrium state, the influence
of He detonation on the final nucleosynthesis products of the D6 model is not
significant. This brings the D6 model to have the potential to explain the
bulk of normal SNe Ia [510, 507]. The violent merger of two WDs can also
directly trigger a C-detonation in primary WD to cause an SN Ia explosion,
which is called as “ _C-ignited violent merger model_ ” (e.g. [513, 186,
514]). Generally, the violent merger model enhances the production of neutron-
rich electron capture nuclei such as ${}^{54}\mathrm{Fe}$ and
${}^{58}\mathrm{Ni}$ [492]. There are other explosion models have been
suggested for SNe Ia such as the “WD-WD collisions model”. The nucleosynthesis
yields of a variety of SN Ia explosion models999A detailed review of
nucleosynthetic yields of different explosion models of SNe Ia have been given
by [492]. have been presented in Fig. 4.3. Very recently, by analyzing the
nucleosynthesis yields of various SN Ia explosion models, it has concluded
that both the Chandrasekhar-mass explosion models and the models including a
He detonation can significantly contribute to the production of Mn, leading to
super-solar values of [Mn/Fe] [494]. In addition, they found that double-
detonation models are able to produce Zn and Cu in super-solar ratios with
respect to Fe [494].

### 4.4 Neutron-capture process in carbon-enhanced metal-poor stars

Carbon-enhanced metal-poor stars (which are usually referred to as CEMP stars)
are a subclass of chemically peculiar stars which show enhancement of carbon
($\mathrm{[C/Fe]}\gtrsim+1.0$; [515]). CEMP stars contribute to a substantial
fraction (about $20\%$; [516, 517, 518, 519]) of the observed very metal-poor
stars with $\mathrm{[Fe/H]}\lesssim-2.0$ [520], and the fraction of CEMP stars
increases as the decrease of metallicity (e.g. [518, 519, 521]; and references
therein). However, their exact origins are still poorly understood. CEMP stars
are generally divided into four subclasses according to the abundance of
barium (Ba) and europium (Eu) which are produced by slow ($s-$process) and
rapid ($r-$process) neutron-capture processes (but the exact definitions for
various subclasses of CEMP stars vary for different studies; [520, 522, 397]):
(i) _CEMP-s stars_ with $\mathrm{[C/Fe]}>1.0$, $\mathrm{[Ba/Fe]}>1.0$ and
$\mathrm{[Ba/Eu]}>0.5$. This subtype accounts for at least $80\%$ of all CEMP
stars [515]. (ii) _CEMP-r stars_ with $\mathrm{[C/Fe]}>1.0$ and
$\mathrm{[Eu/Fe]}>1.0$. (iii) _CEMP-r/s stars_ with $\mathrm{[C/Fe]}>1.0$,
$\mathrm{[Ba/Fe]}>1.0$, $\mathrm{[Eu/Fe]}>1.0$ and $\mathrm{[Ba/Eu]}>0$. (iv)
_CEMP-no stars_ with $\mathrm{[C/Fe]}>1.0$, $\mathrm{[Ba/Fe]}\leq 1.0$ and
$\mathrm{[Eu/Fe]}\leq 1.0$. One of the key questions of CEMP stars is that how
different subclasses of CEMP stars are formed, and whether they share the same
formation scenario. A variety of scenarios are invoked to explain the observed
abundance patterns of CEMP stars in the literature (e.g. [523, 515, 524]).
However, no single published scenario can consistently explain the
observational signatures of CEMP stars. It has been suggested that CEMP-$r$
and CEMP-$no$ stars may result from pre-enhancement of the interstellar medium
when these stars formed (e.g. [525, 526, 527]), whereas CEMP-$r$ stars were
also claimed to originate from the binary evolution (e.g. [528]). CEMP-$s$
stars are generally thought to result from mass transfer from an AGB companion
in a binary system because this formation scenario can explain the
simultaneous enrichments of C, Ba, and other $s$-elements in CEMP-$s$ stars.
In addition, this is supported by the fact of that most CEMP-$s$ stars show
radial-velocity variations (e.g. [516, 515, 529, 530, 531]; and references
therein). However, the formation scenario of CEMP-$r/s$ stars is puzzling.
Especially, the origin of enrichment of $r$-elements in CEMP-$r/s$ is strongly
debated by different studies in the literature (e.g. [523, 524, 532, 533,
534], and references therein). The current AGB models do not reach
sufficiently high neutron density to produce enough $r$-elements that match
CEMP-$r/s$ stars. The majority of CEMP-$r/s$ stars are found in binary systems
[535, 531]. It is therefore thought that CEMP-$r/s$ stars may be formed due to
both pre-enrichment with $r$-process material and pollution of $s$-process
matter from an AGB companion [523, 524, 536, 533]. However, this model seems
to have difficulties to explain the observed number of CEMP-$r/s$ stars [533].
Interestingly, the nucleosynthesis of the _intermediate neutron-capture
process_ (i.e. $i$-process; see Section 4.4) in AGB models can fit the
observed abundances of CEMP-$s/r$ stars very well [537, 533, 534, 538].
However, whether or not the $i$-process can be active in AGB stars remains
uncertain. By performing population synthesis calculations, different studies
have investigated the formation of CEMP stars and attempted to reproduce the
observed fraction ($9\%$-$25\%$; [539, 540, 516]) and distributions of
different elements (e.g. C, Ba and other heavy elements) of CEMP stars (e.g.
[535, 528, 425, 397, 532, 541]). However, only weak constraints have been
placed on the origins of CEMP stars due to the uncertainties on the
nucleosynthesis in AGB stars, initial mass function (IMF) at low metallicity,
binary fraction, wind mass transfer process etc.

CEMP stars often show clear enhancements in C and neutron-capture elements
such as Ba and Eu [515]. Three neutron-capture processes are involved in the
formation of different subclasses of CEMP stars: (i) slow ($s-$process)
neutron-capture process; (ii) rapid ($r-$process) neutron-capture process;
(iii) intermediate ($i-$process) neutron-capture process. The $r-$, $s-$ and
$i-$process require completely different physical conditions (e.g. [542,
543]). The $s$ process requires low neutron densities of
$10^{6}-10^{10}\,\mathrm{cm^{-3}}$, and the AGB stars and He-burning core of
massive stars are commonly thought to be the predominant producer of
$s$-process elements (e.g. [544, 545, 546, 547, 542, 543]). Most CEMP-$s$
stars are found in the binary systems, the pollution of $s$-process elements
from an AGB donor is therefore suggested to be the most likely origin of
CEMP-$s$ stars (see Section 3.4.2). High neutron densities of $\gtrsim
10^{20}\,\mathrm{cm^{-3}}$ is required to trigger $r$-process. The $r$-process
is generally thought to occur during supernova explosions, accretion-induced
collapses and merger of double neutron stars (e.g. [542, 548, 543], and
references therein). Because such extreme conditions are unlikely to reach in
AGB stars, the enhancement of $r$-process elements in CEMP-$r$ and/or
CEMP-$r/s$ stars are suggested to result from the pre-enhancement of
$r$-elements material when these stars formed [525, 526]. Despite substantial
efforts in the study of the effect of $s$\- and $r$-processes on the formation
of CEMP-$r/s$ stars, it is still difficult to explain the observed $s/r$
abundance pattern and other properties (e.g. the number, the observed
correlation between the enrichment in the $s$\- and $r$-process elements, the
ratios of heavy $s$-process elements to light $s$-process elements) of
CEMP-$r/s$ stars with different formation scenarios that involve $s$\- and
$r$-processes (e.g. [523, 522, 541, 533]). The $i$-process is expected to
happen when neutron densities are in the order of $\sim
10^{15}\,\mathrm{cm^{-3}}$. There are possible physical sites in stellar
evolution that may allow $i$-process to occur such as when hydrogen is mixed
into a convective helium-burning zoo during the evolution of low-mass AGB
stars (e.g. [549, 550, 551]). The $i$-process is able to produce sufficient
$r$-process elements and a high ratio of heavy $s$-process elements to light
$s$-process elements, it therefore has been used to successfully explain the
observed abundance patterns of CEMP-$r/s$ stars in the context of wind mass
transfer from an AGB or TPAGB donor in binary systems [537, 533, 534, 552,
538]. This is supported by most CEMP-$r/s$ belong to binary systems [553,
538]. However, how physical sites for the $i$-process can be identified in
stellar evolution is still quite uncertain. More future studies are still
needed to carry out the detailed calculations of the nucleosynthesis of the
$i$-process and the evolution of low-mass AGB stars, which will provide
insight into the importance of $i$-process in the formation of CEMP stars and
the related nucleosynthesis.

## 5 Summary and outlook

In this review article, we have tried to highlight recent advances in our
theoretical and observational understanding of binary stars. Substantial
progress has been made in binary star research due to both the increasing
number of binary stars from different large surveys like SDSS, SEGUE, Gaia,
GALAH, LAMOST, APOGGE and _Kepler_ , and because of the rapid development of
numerical approaches and theoretical models. Nevertheless, there remain many
uncertainties. Future progress in binary stars, both observational and
theoretical, will enable us to have a better understanding of the fundamental
problems of stellar and binary astrophysics. Here, we point to some directions
that may lead to future progress.

  1. (1)

Further simulations of binary processes. A complete understanding of the
evolution of binary stars firmly relies on whether the related fundamental
physical processes are clearly determined. Binary mass transfer and common
envelope evolution (CEE) are the two most important physical processes in
binary evolution. Despite our attempt to model these two processes with a
number of multi-dimensional simulations over the years, it is still difficult
to reach a consensus due to the inherent multi-physics and multi-scale
challenges. As a consequence, there are still many open questions. For
example, what is the exact outcome of CEE? How much mass and angular momentum
are lost from the binary system in non-conservative mass transfer? What are
the exact mass-accretion efficiency and angular-momentum loss through wind
mass transfer (or wind Roche-lobe overflow) in binary systems? Given the rapid
development of numerical techniques and present-day computational power, we
expect that simulations in near future will have sufficient accuracy to firmly
constrain and/or solve these remaining fundamental problems of binary
evolution. This will enable us to give answers to the above questions.

  2. (2)

Further observations of exotic stars. In this article, we have reviewed the
recent discoveries of different types of binary stars and exotic objects
related to various evolutionary processes in binary evolution. Clearly, the
discoveries of these stars have significantly advanced our understanding of
binary star evolution. For example, the observed luminous red novae (LRNe) are
thought to be very likely candidates of CEE events. The discovery of samples
of post-CE binaries have placed constraints on the CE ejection efficiency,
although its exact value is still unknown. In addition, reproducing the
observational properties of short-period sdB binaries with WD companions have
required non-conservative mass transfer. In general, the strict constraints on
the fundamental evolutionary processes of binary stars is firmly based on the
discoveries of different new types of binary systems and exotic objects that
predicted from binary evolution. Relying on a number of present-day and future
large surveys, in the nearest future we expect the discovery of a much bigger
sample of such exotic events, including post-CE binaries, LRNe, sdB binaries,
chemical peculiar stars, extremely low-mass WDs, runaway and hypervelocity
stars, double compact objects, and so on. In particular, as pointed out above,
the discovery of Thorne-Zytkow objects (TZOs) that contain an NS inside a RG
is thought to be the direct evidence of stellar mergers from the CEE. However,
none of such objects have been discovered yet.

  3. (3)

Statistical properties of binary populations. The statistical properties of
binary populations, including binary fraction, orbital-period distribution,
mass-ratio distribution and orbital eccentricity distribution, as well as
their dependence on stellar types, environment and/or metallicities, are
essential for our understanding of different aspects of star formation to
galactic evolution. Moreover, the statistics of binaries are basic inputs of
binary population synthesis studies, which will thus have a major effect on
the formation and evolution of binary-related objects such as CVs, X-ray
binaries, SNe Ia, millisecond pulsars, compact binary systems that emit
detectable gravitational waves. Over the last few years, increasingly large
survey projects have led to the discovery of a huge number of binary stars,
which allows us to provide statistical properties of binary populations.
Indeed, significant progress in statistics of binaries such as binary
fraction, mass ratio distributions have been made. In particular, growing
observational evidence has suggested the non-universality of the initial mass
function (IMF) and the existence of correlations between different binary
properties. Nevertheless, the statistical properties of binaries are still
controversial, and their dependence on stellar types, environments or
metallicities is only poorly constrained. Different ongoing surveys introduced
in this review and upcoming massive survey projects like the _James Webb Space
Telescope (JWST)_ [554], the Vera C. Rubin Observatory _Legacy Survey of Space
and Time (LSST)[555]_, the _Chinese Survey Space Telescope (CSST)_ [556], and
the _Euclid space telescope_ [557] are expected to discover orders of
magnitude more binary stars in the future, from wide non-interacting pairs to
very close compact binaries. This will advance the studies of observational
properties of binary populations a significant step further.

  4. (4)

Development of binary population synthesis. Binary population synthesis (BPS)
provides the bridge between the theoretical properties of binary evolution
models and observational properties of different binary-related events such as
their rates, mass and orbital period distributions. However, the results given
by BPS studies still have uncertainties. The BPS method is developed to
rapidly evolve large samples of zero-age MS binaries initially generated with
given IMF, binary fraction, and distributions of orbital period, mass ratio
and eccentricity. On the one hand, uncertainties of these initial parameter
distributions have a significant influence on the overall population
properties of present-day binary populations. On the other hand, the evolution
of binary systems needs to deal with different fundamental physical processes
such as the CEE and mass transfer. As mentioned before, however, these binary
processes remain unsolved, and they are treated in a simplified way in BPS
studies, generally parameterized. This brings certain uncertainties on the
formation and evolution of binary-related events, and therefore leads to that
the comparison between BPS predictions and observations cannot give an
undisputed conclusion. For instance, studies have shown that different choices
of initial parameter distributions and binary evolution models could lead to
the predicted rates of double compact objects changed by more than an order of
magnitude. Future numerical simulations, as well as the discoveries of more
exotic objects predicted by different evolutionary phases of binary systems,
will give much stronger constraints on the fundamental processes of binary
evolution. Meanwhile, statistical properties of binary populations are
expected to be well-determined by much larger samples of binary stars from
ongoing and upcoming massive survey projects. Therefore, there is hope that an
accurate and universal BPS model may be developed in near future, which will
be very helpful for examining and testing the stellar and binary evolution
theory.

  5. (5)

Nucleosynthesis in binary stars. Stars are the main cosmic factories that
produce elements heavier than He. Most stars spend their lives in binary
systems or multiple systems, the nucleosynthesis related to binary stars
therefore plays an important role in the production of different elements. We
have reviewed the importance of the nucleosynthesis of binary-related objects
in the past decade, both in observation and theory. For example, GW170817 has
provided a great impact on the studies of r-process nucleosynthesis. Recent
observations have revealed that novae should be the promising sites of
$\rm{{}^{7}Li}$ production. In addition, the accretion of $i-$process elements
from an AGB star onto its companion in a binary system has been found to
provide a good explanation for the observed abundance patterns of CEMP-$r/s$
stars. On the observational side, we expect that future observations will
discover a big sample of chemically peculiar stars such as CEMP stars, Ba
stars and extremely metal-poor stars (Population III stars) enriched with
r/s-process elements. The statistical analysis of the chemical abundance
properties of these peculiar stars gives essential insights into r-process,
s-process, and big-bang nucleosynthesis, as well as binary evolution models.
Moreover, with more GW detection from mergers of double compact objects and
follow-up observations of their electromagnetic counterparts such as the
spectroscopic observations of r-process elements in kilonovae, we expect to
greatly deepen our understanding of nucleosynthesis of heavy nuclei and EoS of
neutron stars. On the theoretical side, future powerful computing resources
will allow us to remarkably advance the theoretical modelling of binary
processes to make binary models more realistic, which will provide more
reliable calculations of nucleosynthesis of different binary-related objects.
In addition, future computational power will enable us to consistently model
SN explosion, dynamics of mergers of double compact objects, explosive
nucleosynthesis, and synthetic outcome, and therefore eventually reveal the
origin of elements.

  6. (6)

Impacts on the cosmology. SNe Ia have been used as the standard candles to
successfully measure the cosmic distance and to place constraints on the
properties of Dark Energy. It is widely accepted that progenitors of SNe Ia
are in binary systems. However, the nature of SN Ia progenitors and their
explosion mechanism remains a mystery, which significantly affects the
precision of SN Ia cosmology. Future theoretical and observational progress on
the binary stars that have been mentioned above will allow us to narrow down
the constraints on SN Ia progenitor systems and explosion mechanisms. This
will help to reduce the uncertainty of SN Ia cosmology and therefore deepen
our understanding of Dark Energy.

## Acknowledgements

We would like to thank the anonymous referee for useful comments that helped
to improve this article. The authors would like to thank Hongwei Ge for
fruitful discussions and for his contributions to some figures of this
article. This work is supported by the National Natural Science Foundation of
China (NSFC, No. 12125303, 12288102, 12090040/3), the National Key R&D Program
of China (Nos. 2021YFA1600403/1 and 2021YFA1600400), the Chinese Academy of
Sciences (CAS) and the Natural Science Foundation of Yunnan Province (Nos.
202201BC070003, 202001AW070007), the International Centre of Supernovae,
Yunnan Key Laboratory (No. 202302AN360001) and the “Yunnan Revitalization
Talent Support Program”—Science & Technology Champion Project (NO.
202305AB350003).

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