11institutetext:  Diogo Belloni 22institutetext: Departamento de Física,
Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso, Chile
22email<EMAIL_ADDRESS>33institutetext: Matthias R. Schreiber
44institutetext: Departamento de Física, Universidad Técnica Federico Santa
María, Av. España 1680, Valparaíso, Chile; Millennium Nucleus for Planet
Formation (NPF), Valparaíso, Chile 44email<EMAIL_ADDRESS>
# Formation and Evolution of Accreting Compact Objects

Diogo Belloni and Matthias R. Schreiber corresponding author

###### Abstract

Accreting compact objects are crucial to understand several important
astrophysical phenomena such as Type Ia supernovae, gravitational waves, or
X-ray and $\gamma$-ray bursts. In addition, they are natural laboratories to
infer fundamental properties of stars, to investigate high-energy phenomena
and accretion processes, to test theories of stellar and binary evolution, to
explore interactions between high-density plasma and very strong magnetic
fields, to examine the interplay between binary evolution and dynamical
interactions (in the case they belong to dense star clusters), and they can
even be used as a probe for the assembling process of galaxies over cosmic
time-scales. Despite the fundamental importance of accreting compact objects
for astrophysics and recent progress with the comprehension of these
fascinating objects, we still do not fully understand how they form and
evolve. In this chapter, we will review the current theoretical status of our
knowledge on these objects, and will discuss standing problems and potential
solutions to them.

## Keywords

Cataclysmic Variables, AM CVn Stars, Symbiotic Stars, Super-soft X-ray
Binaries, Low-mass X-ray Binaries, Ultra-compact X-ray Binaries, High-mass
X-ray Binaries

###### Contents

  1. 1 Introduction
  2. 2 The Accreting Compact Object Zoo
  3. 3 Modes of Mass Transfer
  4. 4 Stability of Mass Transfer Through Roche Lobe Filling
    1. 4.1 Dynamical Time-scale Mass Transfer
    2. 4.2 Thermal Time-scale Mass Transfer
    3. 4.3 Nuclear or Orbital Angular Momentum Loss Time-scale Mass Transfer
  5. 5 Formation Channels
    1. 5.1 Common-Envelope Evolution
    2. 5.2 Dynamically Stable Non-conservative Mass Transfer
    3. 5.3 Combination of Dynamically Stable Non-conservative Mass Transfer and Common-Envelope Evolution
    4. 5.4 Additional Channels Through Dynamical Interactions in High-Density Environments
  6. 6 Secular Evolution
    1. 6.1 Cataclysmic Variables and Low-Mass X-ray Binaries
    2. 6.2 AM CVns and Ultra-Compact X-ray Binaries
    3. 6.3 Symbiotic Stars and Symbiotic X-ray Binaries
    4. 6.4 Supergiant and Wolf–Rayet High-Mass X-ray Binaries
  7. 7 Conclusion

##  1 Introduction

Compact objects are remnants of stellar evolution and they are mainly divided
into three categories, which strongly depend on the mass and metallicity of
their main-sequence (MS) star progenitor. Low-mass stars (${\lesssim 2}$ M⊙)
and intermediate-mass stars (${\sim 2-8}$ M⊙) end their lives as white dwarfs
(WDs), while more massive stars evolve into either neutron stars (NSs) or
black holes (BHs). Understanding stellar remnants represents a crucial part of
understanding stellar evolution and stellar structure. Compact objects,
especially as part of a binary, offer the potential to study physical
phenomena in the most extreme environments and produce some of the most
fascinating events in the Universe.

Compact objects as members of binaries are frequently found to accrete
material from their companion stars. Accreting compact objects are then
binaries in which a compact object accretes matter from a companion, which can
be a star, another compact object or even a planet in some cases. They are
excellent laboratories for studying accretion disk physics and the
generation/evolution of extremely strong magnetic fields. These objects are
also among the most luminous high-energy sources in the Universe, emitting
detectable radiation from X-rays to $\gamma$-rays, thereby permitting detailed
studies of plasma physics and the related emission processes under extreme
conditions.

Not only the secular evolution of accreting compact objects is intrinsically
related to mass transfer, also their formation is in many cases associated
with at least one episode of mass transfer between the stars composing the
progenitor binary. Unlike single stars, the stars within a binary may interact
and in turn may be affected by one another depending on their nature and the
orbital separation. Mass transfer in a binary through the inner Lagrangian
point occurs when at least one star fills its Roche lobe, which is triggered
by either nuclear evolution of this star, for example the expansion of a giant
star, or by orbital shrinkage due to orbital angular momentum loss (AML). In
the former case, mass transfer can be maintained because the star expands at
least at the same rate as its Roche lobe radius. The latter occurs in all
binaries because two masses orbiting their common centre of mass always loose
angular momentum due to gravitational wave radiation. Depending on the nature
of the mass losing star, orbital AML mechanisms that are stronger than
gravitational wave radiation can be present. The loss of angular momentum
reduces the separation between the two stars, and therefore, the radius of the
companion can continuously fill its Roche lobe.

Regardless of what is causing mass transfer, the Roche lobe filling star has
to lose mass because it tries to keep itself inside its Roche lobe radius,
which implies further mass loss than that originally expected from single
stellar evolution. Mass transfer in binaries affects the evolution of the
stellar components. For example, the mass growth of the helium core of a giant
star can be halted before helium burning occurs if the giant star is losing
its entire envelope to the accreting companion and this is why most helium-
core WDs are found to be members of binaries. In other cases, the accreting
star can rejuvenate by becoming more massive than it was at the onset of mass
transfer, which alters its natural evolution. Such a mass gain could allow,
for instance, a star that would otherwise evolve into a WD (or NS) to evolve
into a NS (or BH), or more generally a more massive compact object than in the
case of single star evolution. Another example corresponds to the non-expected
chemical abundances in some stars such as barium stars, which can only be
explained if their companion were mass-losing stars that managed to synthesise
those elements.

In this chapter we will discuss the state-of-the-art of our understanding of
the formation and evolution of accreting compact objects. We start with a very
brief review of the different types of accreting compact objects and the main
processes that produce their X-ray emission. Their formation, the current
state, and the future evolution depend fundamentally on the mechanisms that
are driving mass transfer in binaries. We therefore subsequently review the
present understanding of the formation and evolution of these systems with a
discussion of the modes and the stability of mass transfer, before we discuss
their possible formation channels and their secular evolution. We will then
end this review with a discussion of the major current problems in our
understanding of the formation and evolution of accreting compact objects and
briefly describe potential avenues towards solving these issues.

##  2 The Accreting Compact Object Zoo

Cataclysmic variables (CVs) are close binaries composed of WDs accreting from
low-mass Roche lobe filling MS stars and are definitely the best-studied and
most numerous type of interacting compact binaries. While several thousand CVs
are known, the most important samples currently available are volume limited
(Pala et al., 2020; Inight et al., 2021). These representative samples provide
key observational constraints on CV formation and evolution theories, but at
the same time we have to admit that they are still rather small and to some
degree subject to low-number statistics. The majority of CVs are non-magnetic
(e.g. Pala et al., 2020) and most non-magnetic CVs are dwarf novae, which
exhibit repetitive outbursts due to thermal-viscous instabilities in their
accretion disks (e.g. Hameury et al., 1998; Lasota, 2001, 2016). All types of
CVs are X-ray sources. In the case of non-magnetic CVs, the X-ray emission is
produced in the transition region between the accretion disk and the WD, the
boundary layer, from where approximately one-half of the accretion energy is
expected to be released (Patterson & Raymond, 1985a, b). In magnetic CVs,
matter is accreted onto the WD in a field-channeled accretion flow, starting
at the region where the magnetic field captures the mass flow from the
secondary star, and extending down to the surface of the WD. This accretion
flow is supersonic when it reaches regions close to the surface of the white
dwarf and a shock is formed. The matter in the post-shock region, which is the
region between the shock and the WD surface, is compressed and heated to
temperatures up to a few tens of keVs. Bremsstrahlung from this post-shock
region is responsible for most of the X-ray emission of magnetic CVs (Mukai,
2017; Belloni et al., 2021). When a certain amount of hydrogen-rich matter is
accumulated on the surface of an accreting WD, hydrogen shell burning is
ignited. Non-magnetic and magnetic CVs experience unstable hydrogen burning
that produces nova eruptions (e.g. Chomiuk et al., 2020) and because of the
associated mass loss, the WD masses in CVs are supposed to remain
approximately constant over a nova cycle (e.g. Yaron et al., 2005).

While mass transfer in CVs is driven by orbital AML, binaries consisting of a
WD and a slightly more massive Roche lobe filling MS star experience thermal
time-scale mass transfer. In such systems, accretion rates exceeding values of
${\sim 10^{-7}-10^{-6}}$ M⊙ yr-1, depending on the WD mass, can be reached and
the hydrogen-rich material accreted by the WD can stably burn on the surface
of the WD (e.g. Nomoto et al., 2007). Nova eruptions and the corresponding
erosion of the WD mass are avoided in these systems, and therefore, they are
considered to be potential Type Ia supernova progenitors (Di Stefano, 2010a).
Hydrogen burning on the surface of a WD is producing effective temperatures
somewhat below $100$ eV, producing X-ray emission in the $0.01–2$ keV range.
Stably hydrogen-burning WDs in close binaries are therefore called super-soft
X-ray binaries (SSXBs). The first two SSXBs were discovered in the eighties
(Long et al., 1981) and about a decade later one additional system was found
(Trümper et al., 1991). Today the discovery of more than 100 systems has been
reported in external galaxies, the Magellanic clouds, and the Milky Way, but
this number and the total emission in X-rays seem to be too low to account for
the full rate of Type Ia supernovae (Gilfanov & Bogdán, 2010; Galiullin &
Gilfanov, 2021). The relatively low number of SSXBs and the corresponding low
soft X-ray emission are potentially related to the fact that circumstellar
matter can suppress the soft X-ray flux (Nielsen et al., 2013).

A binary where a compact object is accreting material at a relatively high
rate (greater than several $10^{-9}$ M⊙ yr-1) from a low-mass-evolved red
giant is called a symbiotic star (SySt), in case the compact object is a WD
(Kenyon, 1986), or a symbiotic X-ray binary (SyXB), in case the accretor is a
NS (Yungelson et al., 2019). In most cases (${\sim 80}$%), the donor is a
normal evolved giant, typically M-type red giant, but in some cases it can be
a G-/K-type yellow giant. Their near-infrared emission shows the presence of a
stellar photosphere, and for this reason these systems are called S-type SySt
(Mikołajewska, 2003, 2007; Mikolajewska, 2010; Mikołajewska, 2012). In the
remaining systems, the donor is a Mira-type variable and their near-infrared
emission is consistent with being originated from a combination of a reddened
Mira-type variable and a warm optically thick dust shell, which makes these
systems to be called D-type SySts (Gromadzki & Mikołajewska, 2009). In SySts,
the accretion rates are typically large enough to cause stable nuclear burning
on the surface of the WD, which leads to the emission of soft X-rays similar
to those of SSXBs. In most SySts and SyXBs, the mass transfer from the red
giant to the compact object is due to gravitationally focused wind accretion
and only in some cases due to atmospheric Roche lobe overflow. SySts are
important for understanding mass transfer in wide binaries and the formation
of astrophysical jets. Most importantly, as SSXBs, SySts are promising
candidates for being Type Ia supernova progenitors (Di Stefano, 2010b; Liu et
al., 2019; Iłkiewicz et al., 2019).

AM CVn stars are compact binaries with orbital periods in the range of ${\sim
5-65}$ min (Ramsay et al., 2018). These binaries are so close that a MS star
cannot be part of the system as it would just not fit. In AM CVns, a WD
accretes helium-dominated material from a Roche lobe filling companion,
usually a lower-mass WD or a semi-degenerate helium-rich star. Mass transfer
in AM CVns is driven by orbital AML through gravitational wave radiation which
is expected to be detectable by the Laser Interferometer Space Antenna (LISA)
mission (Nelemans, 2013; Korol et al., 2017). Observationally, AM CVns are
characterized by their hydrogen-deficient optical spectra and their blue
colour. More than 50 AM CVns are known by the time of writing (e.g. Ramsay et
al., 2018) and their number is constantly increasing thanks to dedicated
spectroscopic and photometric transit surveys (Carter et al., 2013; Breedt et
al., 2014). X-ray emission can be triggered in two different ways in AM CVns
and the details depend to some degree on the formation channel and the nature
of the donor. In case the donor is a WD, the onset of mass transfer occurs as
a direct impact during which a small region of the accreting WD is heated and
emits a fraction of the released accretion luminosity at high energy
wavelength (Postnov & Yungelson, 2014). In the presence of an accretion disk
around the accreting WD, X-ray emission is produced in the boundary layer in
full analogy to the X-ray emission of non-magnetic CVs.

Low-mass X-ray binaries (LMXBs) consist of a NS or a BH that is accreting from
a Roche lobe filling low-mass star (Tauris & van den Heuvel, 2006). As in CVs,
mass transfer in LMXBs is driven by orbital AML due to magnetic braking and/or
gravitational wave radiation. In analogy to dwarf novae, LMXBs also show
outbursts that are most likely triggered by thermal-viscous instabilities in
their accretion disks (Hameury, 2020). An important difference between LMXBs
and CVs is that in the former irradiation of the accretion disk is more
important and may fully suppress disk instabilities. In CVs, disk irradiation
is dominating most likely only in post-novae systems (Schreiber et al., 2000;
Schreiber & Gänsicke, 2001). LMXBs also show thermonuclear X-ray bursts
originating from unstable hydrogen and helium burning on the surface of the
accreting NS (Strohmayer & Bildsten, 2006), similarly to the nova eruptions
occurring in accreting WDs. A typical LMXB emits almost all of its radiation
in X-rays and these systems are therefore among the brightest X-ray sources.
Since the seventies, an over-abundance of LMXBs in globular cluster in
comparison to the Milky Way field has been observationally established, which
naturally suggests that dynamical interactions, which are very common in the
dense central parts of globular clusters, facilitate their formation (e.g. van
den Berg, 2020).

Ultra-compact X-ray binaries (UCXBs) are binaries with periods shorter than
about an hour. UCXBs are very similar to AM CVns with the main difference
being that the accreting compact object is either a NS or a BH. As in the case
of AM CVns, such short orbital periods exclude MS stars to form part of the
binary, and therefore, the donors are either WDs or semi-degenerate helium-
rich stars (Nelemans & Jonker, 2010). UCXBs are mainly observed at X-rays and
some of these objects show thermonuclear X-ray bursts (Koliopanos et al.,
2021). As in the case of LMXBs, there is currently a clear consensus that the
relative number of UCXBs is much larger in globular clusters than in the Milky
Way field (van den Berg, 2020), and dynamics therefore seem to provide an
important formation channel also for these systems.

If a compact object is accreting from a high-mass star (${\gtrsim 10}$ M⊙),
the system is known as a high-mass X-ray binary (HMXB, e.g. Kretschmar et al.,
2019). As in the cases above, the potential energy of the accreted material in
HMXBs is converted into X-ray emission. Although HMXBs are therefore strong
X-ray sources, they are also bright at optical wavelengths given that their
optical emission is dominated by the massive companion. In some cases, the
donor is an O- or early B-type supergiant that is filling or close to filling
its Roche lobe and these systems are consequently called supergiant HMXBs (sg-
HMXBs, e.g. Martínez-Núñez et al., 2017). Despite hosting a supergiant, the
orbital periods of some sg-HMXBs are shorter than ${\sim 10}$ days, and they
are persistent X-ray sources, powered by the capture of matter from the strong
stellar wind of the supergiant or, in a few cases, through Roche lobe overflow
(Chaty, 2011). In contrast, supergiant fast X-ray transients (SFXTs) have
orbital periods typically longer than ${\sim 10}$ days and exhibit peculiar
characteristics compared to the remaining sg-HMXBs, such as bright short time-
scale flares. A second sub-type of HMXBs are B-emission HMXBs (Be-HMXBs),
which were discovered in the 1970s (Maraschi et al., 1976). In virtually all
cases these systems contain accreting NSs, but there is one confirmed Be-HMXB
hosting a BH (Casares et al., 2014), and a few Be-HMXBs have been proposed to
contain WDs (e.g. Kennea et al., 2021; Coe et al., 2020). The donors in these
systems are rapidly rotating B stars on or close to the MS, and unlike the
donors in sg-HMXBs, given their typical orbital period (${\sim 10-400}$ days),
the donors in Be-HMXBs are deep inside their Roche lobes (e.g. Reig, 2011).
Be-HMXBs can be in slightly eccentric orbits (Pfahl et al., 2002), or highly
eccentric orbits, and their X-ray emission is originated when the compact
object accretes from the Be star circumstellar decretion disk either during
periastron passages, resulting in orbital-phase-dependent X-ray outbursts
lasting several days, or due to enhanced activity of the Be star, which leads
to X-ray outbursts lasting several weeks not correlated with any particular
orbital phase (Ziolkowski, 2002). As a final sub-class, we mention Wolf–Rayet
HMXBs (WR-HMXBs), in which the donor is a massive helium star. Only few such
systems are known and they are likely descendants from sg-HMXBs and Be-HMXBs
(van den Heuvel et al., 2017). Among HMXBs, Be-HMXBs correspond to the most
numerous class (${\sim 60}$%, Liu et al., 2006), followed by sg-HMXBs (${\sim
32}$%, Liu et al., 2006). This is a direct consequence of selection effects
favouring the discovery of Be-HMXBs in comparison with sg-HMXBs. As previously
mentioned, Be-HMXBs are discovered during outbursts, which faint persistent
sg-HMXBs would not have. For this reason, it is very likely that the
contribution of sg-HMXBs to the intrinsic population of HMXBs is much higher.

##  3 Modes of Mass Transfer

The Roche lobe is the last equipotential surface in a binary that keeps all
the mass attached to one star. Any expansion of a star beyond its Roche lobe
or any orbital shrinkage of the Roche lobe due to orbital AML may cause a star
to fill its Roche lobe thereby forcing mass to flow over to the other star.
Determining the rate at which mass is transferred through the inner Lagrangian
point in binaries represents in general a difficult hydro-dynamical problem.
One of the most frequently used and reasonable prescriptions properly takes
into account the finite scale height of the stellar atmosphere of the Roche
lobe filling donor.

For stars with extended atmosphere Roche lobe overflow can occur through the
inner Lagrangian point even when the donor is slightly under-filling its Roche
lobe, i.e. ${R_{\rm d}<R_{\rm RL}}$, where $R_{\rm d}$ is the donor radius and
$R_{\rm RL}$ is the Roche lobe radius of the donor. Taking this into account,
the mass transfer rate $\dot{M}_{\rm d}$ can be estimated as (Ritter, 1988)

$\dot{M}_{\rm d}\ =\ -\,\dot{M}_{0}\,\exp\left(\,-\,\frac{R_{\rm
RL}\,-\,R_{\rm d}}{H_{\rm P}\,/\,\gamma(q)}\,\right)\ ,$ (1)

where $H_{\rm P}$ is the pressure scale height at the photosphere of the
donor.

The quantity $\gamma(q)$ is a function of the mass ratio $q=M_{\rm d}/M_{\rm
a}$, where $M_{\rm d}$ is the donor mass and $M_{\rm a}$ is the accretor mass,
given by

$\displaystyle\gamma(q)\ =\ \begin{cases}0.954\ +\ 0.025\,\log_{10}q\ -\
0.038\,\left(\log_{10}q\right)^{2},&\ {\rm if}\ \ 0.04\lesssim q\leq 1\\\
0.954\ +\ 0.039\,\log_{10}q\ +\ 0.114\,\left(\log_{10}q\right)^{2},&\ {\rm
if}\ \ 1\leq q\lesssim 20\end{cases}$ (2)

Finally, $\dot{M}_{0}$ is the mass transfer rate when the donor exactly fills
its Roche lobe given by

$\dot{M}_{0}\ =\ F(q)\,\rho_{\rm
ph}\,\left(\,\frac{2\pi}{\sqrt{2.71828}}\,\right)\,\left(\,\frac{R_{\rm
RL}^{3}}{G\,M_{\rm d}}\,\right)\,\left(\,\frac{k_{\rm B}\,T_{\rm eff}}{m_{\rm
p}\ \mu_{\rm ph}}\right)^{3/2}\ ,$ (3)

where $G$ is the gravitational constant, $k_{\rm B}$ is the Boltzmann
constant, $m_{\rm p}$ is the proton mass, $M_{\rm d}$ is the donor mass,
$T_{\mathrm{eff}}$ is the donor effective temperature, $\mu_{\rm ph}$ and
$\rho_{\rm ph}$ are the mean molecular weight and density at its photosphere,
respectively and $F(q)\,=\,1.23\,+\,0.5\log_{10}q$, valid for $0.5\lesssim
q\lesssim 10$. Outside the ranges of validity, $F(q)$ and $\gamma(q)$ are
usually evaluated using the value of $q$ at the edge of their respective
ranges (Paxton et al., 2015).

In case the donor is over-filling its Roche lobe, i.e. $R_{\rm d}>R_{\rm RL}$,
the mass transfer rate can be expressed as (Kolb & Ritter, 1990)

$\dot{M}_{\mathrm{d}}\ =\ -\dot{M}_{0}\,-\,2\,\pi\,F(q)\,\frac{R_{\rm
RL}^{3}}{G\,M_{\rm
d}}\,\int_{P_{\mathrm{ph}}}^{P_{\mathrm{RL}}}\Gamma_{1}^{1/2}\left(\frac{2}{\Gamma_{1}+1}\right)^{\frac{\Gamma_{1}+1}{2\Gamma_{1}-2}}\left(\,\frac{k_{\rm
B}\,T}{m_{\rm p}\,\mu}\right)^{1/2}dP\ ,$ (4)

where $T$ is the temperature, $\Gamma_{1}$ is the first adiabatic exponent,
and $P_{\mathrm{ph}}$ and $P_{\mathrm{RL}}$ are, respectively, the pressures
at the photosphere and at the radius corresponding to $R_{\rm RL}$.

We shall mention that, in case donor is strongly under-filling its Roche lobe,
the mass transfer rates provided by Equation 1 are negligible. However, mass
can still be transferred from the donor, if the stellar winds from the donor
are sufficiently strong and dense and a part of the this wind is accreted by
the companion. In what follows, we discuss in more detail these modes of mass
transfer, starting with the stability of mass transfer when the donor is
filling its Roche lobe, i.e. ${R_{\rm d}\sim R_{\rm RL}}$.

##  4 Stability of Mass Transfer Through Roche Lobe Filling

To understand the evolution of binaries containing a Roche lobe filling star,
it is of fundamental importance to understand the reaction of the donor and
the Roche geometry to mass loss. Despite the fact that the constraints on a
Roche lobe filling star are modified due to mass loss and the star
consequently can no longer be considered as evolving in isolation, under
certain conditions the star may adjust its internal structure in response to
mass loss because most of its mass is concentrated towards its centre. Whether
hydrostatic and/or thermal equilibrium can be maintained in a Roche lobe
filling star and whether mass transfer is stable or unstable depend entirely
on the response of the donor radius and its Roche lobe radius to mass loss on
the corresponding time-scales (Webbink, 1985).

###  4.1 Dynamical Time-scale Mass Transfer

Changes of the radius of the donor star as a response to variations of its
mass and through time can be expressed as

${\rm d}\ln R_{\rm d}\ =\ \frac{\partial\ln R_{\rm d}}{\partial t}\ +\
\left(\,\frac{\partial\ln R_{\rm d}}{\partial\ln M_{\rm d}}\,\right)\,{\rm
d}\ln M_{\rm d}\ ,$ (5)

where $t$ is the time. While the first term on the right side is connected
with stellar evolution and relaxation processes connected with thermal
readjustment, the second term corresponds to the adiabatic response of the
donor to mass loss, i.e. the hydrodynamic response of the donor radius to
variations in its mass.

The changes in the donor Roche lobe radius as a response to variations in
mass, time, and angular momentum can be written as

${\rm d}\ln R_{\rm RL}\ =\ \frac{\partial\ln R_{\rm RL}}{\partial t}\ +\
\left(\,\frac{\partial\ln R_{\rm RL}}{\partial\ln M_{\rm d}}\,\right)\,{\rm
d}\ln M_{\rm d}\ .\ $ (6)

The first term on the right side represents the changes in the Roche lobe
radius due to orbital AML in the absence of mass transfer, and the second term
is associated with the response of the Roche lobe radius to mass loss.

Whether mass transfer from a Roche lobe filling star is stable or not can be
evaluated by comparing two logarithmic derivatives associated with the donor.
These derivatives are that of the donor radius with respect to its mass, for a
fixed initial profile in composition and specific entropy, and that of its
Roche lobe radius with respect to its mass. The former is usually called
adiabatic radius–mass exponent $\zeta_{\rm ad}$, defined as

$\zeta_{\rm ad}\equiv\left.\frac{\partial\ln\,R_{\rm d}}{\partial\ln\,M_{\rm
d}}\right|_{\rm ad}\ ,$ (7)

while the latter is termed the Roche lobe radius–mass exponent $\zeta_{\rm
RL}$, defined as

$\zeta_{\rm RL}\equiv\frac{{\rm d}\ln\,R_{\rm RL}}{{\rm d}\ln\,M_{\rm d}}\ .$
(8)

Combining Equations 5 and 6, and using the definitions 7 and 8, we can express
the change in the donor radius relative to its Roche lobe radius as

${\rm d}\ln\left(\,\frac{R_{\rm d}}{R_{\rm RL}}\,\right)\ =\
\left(\,\frac{\partial\ln R_{\rm d}}{\partial t}\ -\ \frac{\partial\ln R_{\rm
RL}}{\partial t}\,\right)\,{\rm d}t\ +\ \Bigl{(}\,\zeta_{\rm ad}-\zeta_{\rm
RL}\,\Bigr{)}\,{\rm d}\ln M_{\rm d}\ .$ (9)

The condition for the donor to remain within its Roche lobe in hydrostatic
equilibrium during an episode of nearly stationary mass transfer is obviously
${\rm d}\ln(R_{\rm d}/R_{\rm L})\sim 0$. Given that mass transfer is
occurring, the difference in the time derivatives in Equation 9 must be
positive because the donor radius needs to change more strongly than its Roche
lobe radius to guarantee that the donor keeps filling its Roche lobe. In
addition, as the donor is losing mass, ${\rm d}\ln M_{\rm d}$ is necessarily
negative. Therefore, the difference between the adiabatic and Roche lobe
radius–mass exponents needs to be positive to keep the donor in hydrostatic
equilibrium. In other words, by comparing these two derivatives we are able to
determine whether the response to mass transfer of the donor inner layers can
occur adiabatically, i.e. whether the entropy profile in the stellar interior
remains unchanged despite the mass loss. If, on the other hand, $\zeta_{\rm
ad}<\zeta_{\rm RL}$, the mass transfer is unstable and occurs on the dynamical
time-scale, because the mass transfer rate is only limited by the sonic
expansion of the envelope through the inner Lagrangian point.

Another way of reaching the same conclusion is by analyzing perturbations of
the mass transfer rate (e.g. Ritter, 1988). During nearly steady mass
transfer, i.e. ${\rm d}\ln(R_{\rm d}/R_{\rm L})\sim 0$, the well-known mass
transfer rate equation as a function of $\zeta_{\rm ad}$ and $\zeta_{\rm RL}$
can be derived from Equation 9 as

$\dot{M}_{\rm d}\ =\ -\ \left(\frac{M_{\rm d}}{\zeta_{\rm ad}-\zeta_{\rm
RL}}\right)\,\left(\,\frac{\partial\ln R_{\rm d}}{\partial t}\ -\
\frac{\partial\ln R_{\rm RL}}{\partial t}\,\right)\ .$ (10)

Considering a system in an unperturbed phase of stationary mass transfer, i.e.
$\ddot{M}_{\rm d}\sim 0$, we can evaluate the dynamical stability of mass
transfer after a sudden non-null perturbation $\delta\dot{M}_{\rm d}$. After
the perturbation, $\dot{M}_{\rm d}=\dot{M}_{\rm d,0}+\delta\dot{M}_{\rm d}$,
where $\dot{M}_{\rm d,0}$ is the unperturbed mass transfer rate, and
$\ddot{M}_{\rm d}\propto\delta\dot{M}_{\rm d}\,\dot{M}_{\rm d}\,(\zeta_{\rm
ad}-\zeta_{\rm RL})$. Given that $\dot{M}_{\rm d}<0$, the mass transfer will
be dynamically stable only if the response to a sudden increase in
$|\dot{M}_{\rm d}|$ is a decrease in $|\dot{M}_{\rm d}|$. This happens only if
$\zeta_{\rm ad}-\zeta_{\rm RL}\,>\,0$. Otherwise, the sudden increase in
$|\dot{M}_{\rm d}|$ will result in further increase of $|\dot{M}_{\rm d}|$,
which leads then to dynamically unstable mass transfer.

Even though realistic and detailed binary calculations must be performed to
investigate the stability of mass transfer (e.g. Chen & Han, 2008; Woods et
al., 2012; Pavlovskii & Ivanova, 2015; Ge et al., 2020), polytropes can shed
some light on this process in a very simple way and provide a reasonable
first-order approach to the problem in some cases (e.g. Hjellming & Webbink,
1987; Soberman et al., 1997). More specifically, composite polytropes, which
are supposed to reproduce rather well the structure of fully convective low-
mass MS stars, provide $\zeta_{\rm ad}$ in very good agreement with those
computed from detailed stellar models (Ge et al., 2020). On the other hand,
condensed polytropes, which are approximations of red giants, are unable to
reproduce the non-ideal gas effect and inefficient convection occurring in
detailed stellar models, which implies that condensed polytropes likely
underestimated $\zeta_{\rm ad}$ of red giants (Ge et al., 2020). If simple
stellar models do not provide realistic estimates, the adiabatic radius–mass
exponent needs to be calculated by solving the differential equations for
stellar structure.

Once $\zeta_{\rm ad}$ has been obtained, it needs to be compared to
$\zeta_{\rm RL}$ in order to evaluate whether mass transfer is dynamically
stable or not. The Roche lobe radius–mass exponent $\zeta_{\rm RL}$ can be
expressed as

$\zeta_{\rm RL}\ =\ \frac{\partial\ln a}{\partial\ln M_{\mathrm{d}}}\ +\
\left(\,\frac{\partial\ln(R_{\mathrm{RL}}/a)}{\partial\ln
q}\,\right)\,\left(\,\frac{\partial\ln q}{\partial\ln
M_{\mathrm{d}}}\,\right),$ (11)

where $a$ the semimajor axis. The three terms in Equation 11 can be written as
(Eggleton, 1983; Soberman et al., 1997; Woods et al., 2012)

$\frac{\partial\ln a}{\partial\ln M_{\mathrm{d}}}\ =\ (\,1\,-\,\alpha_{\rm
ml}\,)\,\left[\,\frac{2\,q^{2}\,-2\,+\,q\,(1\,-\,\beta_{\rm
ml})}{q\,+\,1}\,\right]\ -\ \frac{q}{q\,+\,1}\ ,$ (12)

$\frac{\partial\ln(R_{\rm RL}/a)}{\partial\ln q}\ =\
\frac{2}{3}\,-\,\left(\frac{q^{1/3}}{3}\right)\left[\frac{1.2q^{1/3}\,+\,1/(1\,+\,q^{1/3})}{0.6q^{2/3}\,+\,\ln(1\,+\,q^{1/3})}\right]\
,$ (13)

and

$\frac{\partial\ln q}{\partial\ln M_{\rm d}}\ =\
1\,+\,\left(\,1\,-\,\alpha_{\rm ml}\,-\,\beta_{\rm ml}\,\right)\,q\ ,$ (14)

where $0\leq\alpha_{\rm ml}\leq 1$ corresponds to the fraction of mass lost
from the vicinity of the donor in the form of fast isotropic winds, and
$0\leq\beta_{\rm ml}\leq 1$ corresponds to the fraction of mass lost from the
vicinity of the accretor due to isotropic re-emission. Given that the fraction
of mass leaving the donor that is effectively accreted is $1\,-\,\alpha_{\rm
ml}\,-\,\beta_{\rm ml}$, the fully conservative case corresponds to
$\alpha_{\rm ml}=\beta_{\rm ml}=0$, while $\alpha_{\rm ml}+\beta_{\rm ml}=1$
represents the fully non-conservative case. In a more general situation,
$\zeta_{\rm RL}$ depends on several other processes, which are related to
orbital AML and mass loss from the system (Soberman et al., 1997), such as
emission of gravitational waves, magnetic wind braking, tidal interactions,
and torques exerted by circumbinary disks. The most important dependencies of
$\zeta_{\rm RL}$ with respect to the stability of mass transfer are those on
the mass ratio as well as on the fraction of mass lost from the binary.

The relation between $\zeta_{\rm ad}$ and $\zeta_{\rm RL}$ for a binary
composed of a red giant donor with initial mass $1.2$ M⊙ and a companion of
mass $1.1$ M⊙ is illustrated in the left-hand panel of Figure 1 (taken from a
binary model, Woods et al., 2012). As mentioned before, the limit for
dynamical stability strongly depends on $\beta_{\rm ml}$. In particular, the
greater $\beta_{\rm ml}$, the smaller $\zeta_{\rm RL}$. This means that the
greater $\beta_{\rm ml}$, the better the chances for the red giant donor to
remain in hydrostatic equilibrium. If wind mass loss is included in the
simulations, $\zeta_{\rm RL}$ decreases even further for more evolved red
giants (i.e. red giants with larges core masses), for a given $\beta_{\rm
ml}$. For instance, when mass loss through winds is not considered, fully
conservative mass transfer is dynamically stable only for core masses
${\gtrsim 0.63}$ M⊙, but this limit becomes ${\gtrsim 0.46}$ M⊙, if wind mass
loss is included.

Figure 1: Adiabatic ($\zeta_{\rm ad}$) and Roche lobe ($\zeta_{\rm RL}$)
mass–radius exponents for a red giant donor of initial mass $1.2$ M⊙ with a
companion of mass $1.1$ M⊙, as a function of the red giant core mass ($M_{\rm
d,c}$), taken from a binary model (left-hand panel, Woods et al., 2012), and
the critical mass ratio separating dynamically stable from unstable mass
transfer for a donor of mass $5$ M⊙, as function of its radius ($R_{\rm d}$)
during its evolution, taken from three different binary models (right-hand
panel, Chen & Han, 2008; Pavlovskii & Ivanova, 2015; Ge et al., 2020). In the
computation of $\zeta_{\rm RL}$ in the left-hand panel, for six different
fractions of mass lost due to isotropic re-emission from the vicinity of the
accretor ($\beta_{\rm ml}$), mass loss through stellar winds is considered
(red lines) or not (blue horizontal lines). In the right-hand panel, wind mass
loss is not considered in either binary model and fully conservative mass
transfer is assumed. It is clear from the left-hand panel that non-
conservative ($\beta_{\rm ml}>0$) mass transfer is more easily dynamically
stable in comparison with the fully conservative case ($\beta_{\rm ml}=0$),
regardless of whether wind mass loss is taken into account or not. This
happens because $\zeta_{\rm RL}$ becomes smaller as $\beta_{\rm ml}$
increases, which allows mass transfer to be dynamically stable for smaller
core mass, i.e. the more mass loss through isotropic re-emission is allowed,
the more extended the range of stability. The inclusion of wind mass loss
enhances this effect, by decreasing even further $\zeta_{\rm RL}$, for core
masses greater than ${\sim 0.3}$ M⊙, extending in turn even further the range
of stability. The right-hand panel, on the other hand, clearly illustrates how
the assumptions made in binary models can drastically change the boundary
separating dynamically stable from unstable mass transfer. For instance, the
solid critical mass ratio strongly increases during the AGB evolution, which
is not seen in the other two models. In addition, the dashed critical mass
ratio is on average much greater than the other two models.

Even though the fundamental principles related to dynamical stability of mass
transfer are well-understood, outcomes of binary models are strongly dependent
on the physical assumptions. As mentioned earlier, for a given binary at the
onset of mass transfer, $\zeta_{\rm RL}$ strongly depends on the mass ratio
and on the assumed mass loss and orbital AML prescriptions. In addition,
$\zeta_{\rm ad}$ depends entirely on the donor mass and its evolutionary
status. Thus, for a given binary, it is possible to derive a critical mass
ratio separating dynamically stable from unstable mass transfer, as a function
of the evolutionary status of the donor. This critical mass ratio is plotted
as a function of the donor radius on the right-hand side of Figure 1, for a
donor with mass $5$ M⊙, taken from three different binary models (Chen & Han,
2008; Pavlovskii & Ivanova, 2015; Ge et al., 2020). Even though all binary
models show the initial decrease in the critical mass ratio as the donor
evolves along the first giant branch (FGB), it is clear from the figure that
these models provide qualitative and quantitative different results. In
particular, one of them delivers on average a much higher critical mass ratio,
in comparison with the other two. In addition, while in two models the
critical mass ratio slowly decreases as the donor climbs the asymptotic giant
branch (AGB), the other model predicts a rapid increase of the critical mass
ratio. The main differences in the three works are connected with the mass
transfer rate equation (whether it is limited by the local sound speed or
not), the radius used to quantify $\zeta_{\rm ad}$ (surface or inner) and the
Lagrangian radius adopted to quantify $\zeta_{\rm RL}$ (inner or outer).

Therefore, despite significant progress during the last decade, the limiting
mass ratio for the onset of dynamically unstable mass transfer remains quite
uncertain. This has significant implications for our general understanding of
the formation of close binaries hosting compact objects, as dynamically
unstable mass transfer is considered a crucial ingredient in most formation
scenarios proposed for these objects. If mass transfer is dynamically
unstable, there are mainly two outcomes. If the donor is either an MS star or
a degenerate object, then most likely the binary will merge on the dynamical
time-scale of the donor, with a non-null fraction of the total mass being
eventually lost in such an energetic event. On the other hand, if the donor
has evolved beyond the MS, common-envelope (CE) evolution will be triggered,
resulting in a strong decrease of the orbital period. In fact, CE evolution
triggered by dynamically unstable mass transfer represents the standard
scenario for the formation of close binaries containing a compact object.
Without a proper understanding of the conditions for dynamically unstable mass
transfer, it will be impossible to reliably predict the relative numbers of
objects as important as SN Ia progenitors and binaries that are main
gravitational wave sources.

###  4.2 Thermal Time-scale Mass Transfer

If the Roche lobe filling donor in a binary can maintain hydrostatic
equilibrium, the natural follow-up question is whether it must sacrifice its
own thermal equilibrium or not to restrain its surface within its Roche lobe.
The thermal equilibrium of a star, i.e. the balance between the nuclear energy
production and the atmospheric radiation losses, is disturbed as a response to
mass loss. However, as in the case of hydrostatic equilibrium, the donor may
be able to re-establish thermal equilibrium, if mass transfer proceeds on a
time-scale slower than its thermal time-scale, i.e. if the donor interior has
enough time to relax to thermal equilibrium. Otherwise, mass transfer will be
thermally unstable, i.e. it will proceed on the thermal time-scale of the
donor. In order to verify whether this happens or not, we use the thermal
equilibrium radius–mass exponent $\zeta_{\rm eq}$, defined by

$\zeta_{\rm eq}\equiv\left.\frac{{\rm d}\ln\,R_{\rm d}}{{\rm d}\ln\,M_{\rm
d}}\right|_{\mathrm{th}}\ .$ (15)

In order to calculate $\zeta_{\rm eq}$, the mass-radius relation for stars in
thermal equilibrium must be known. If mass transfer is dynamically stable, it
will also be thermally stable if $\zeta_{\mathrm{eq}}>\zeta_{\mathrm{RL}}$. In
this case, mass transfer needs to be driven by an external process and
proceeds on a time-scale different from the dynamical and thermal time-scales.
On the other hand, if $\zeta_{\rm ad}>\zeta_{\rm RL}>\zeta_{\rm eq}$, mass
transfer will be driven by thermal readjustment and proceed on the donor
thermal time-scale.

###  4.3 Nuclear or Orbital Angular Momentum Loss Time-scale Mass Transfer

If mass transfer is dynamically and thermally stable, continuous mass transfer
can still be generated by either stellar or binary evolutionary processes. In
this case, mass transfer is not self-stimulated as in the previous cases,
since mass transfer does not proceed by the virtue of the inability of the
donor to remain in either dynamical or thermal equilibrium. Dynamically and
thermally stable mass transfer is driven by either nuclear expansion of the
Roche lobe filling star or orbital AML. In the former case, the donor keeps
filling its expanding Roche lobe, while in the latter case, the Roche lobe and
the donor shrink as a response to mass loss. In case mass transfer is driven
by nuclear expansion of the donor, mass transfer proceeds at a rate such that
$R_{\rm d}\sim R_{\rm RL}$. On the other hand, if mass transfer is driven by
orbital AML, then the mass transfer rate is entirely dependent on the strength
of orbital AML. The main orbital AML mechanisms driving the evolution of
binaries are gravitational wave radiation (GR, e.g. Paczyński, 1967) and
magnetic wind braking (MB, e.g. Huang, 1966; Mestel, 1968).

Using the weak-field approximation of general relativity, it is possible to
derive the orbital angular momentum changes due to emission of gravitational
waves from two point masses as (Hurley et al., 2002)

$\frac{{\dot{J}}_{\rm GR}}{J_{\rm orb}}\ =\ -8.315\times
10^{-10}\,\left[\frac{M_{1}\,M_{2}\,(\,M_{1}\,+\,M_{2}\,)}{a^{4}}\,\right]\,\left[\,\frac{1+\frac{7}{8}e^{2}}{\left(1-e^{2}\right)^{5/2}}\,\right]\
\ {\rm yr}^{-1}\ ,$ (16)

where $a$ is the semimajor axis in $R_{\odot}$, $M_{1}$ and $M_{2}$ are the
binary component masses in M⊙, $e$ is the eccentricity, and $J_{\rm orb}$ is
the orbital angular momentum given by

$J_{\rm orb}\ =\
a^{2}\,\left(\frac{M_{1}\,M_{2}}{M_{1}+M_{2}}\right)\,\left(\frac{2\,\pi}{P_{\rm
orb}}\right)\,\sqrt{1-e^{2}}\ ,$ (17)

where $P_{\rm orb}$ is the orbital period in years. Emission of gravitational
waves acts all the time in every binary. However, from the strong dependence
on the orbital separation, only for very close binaries it will have any
impact on the orbital angular momentum evolution. Of course, how close a given
binary must be so that GR is non-negligible depends on the component masses
and on the eccentricity.

While Equation 16 is well-established and gravitational wave radiation is
well-understood, the same can unfortunately not be said about MB. The main
physical processes that drive MB are quite easy to understand but as usual the
devil is in the details. It is clear that a highly ionized wind leaving the
star is forced to corotate with the magnetic field of the star out to the
Alfvén radius, thereby taking away more angular momentum than the specific
angular momentum of the star that consequently spins down. If the star is
tidally locked to the orbit, i.e. if the star is synchronized with the orbital
motion, the mass loss through winds removes significant amounts of orbital
angular momentum, even if the wind mass-loss rate is relatively small. The
torques produced by the magnetized winds depend on the strength of the
magnetic field, the Alfvén radius, the wind mass-loss rate, and the stiffness
of the magnetic field at large distances from the star. However, the details
of these dependencies are very uncertain and neither observationally nor
theoretically well-constrained. Consequently, in the last decades several
attempts to formulate empirical laws or recipes that are applicable to
particular cases have been developed and a large number of MB prescriptions is
available on the market (Knigge et al., 2011). Unfortunately, as we will see,
for a given fixed star mass, these MB prescriptions provide dramatically
different orbital AML rates spanning easily several orders of magnitude.

In what follows, we briefly mention some important MB prescriptions but
discuss later in more detail their impact on the evolution of accreting
compact objects. A MB prescription that is widely used for stars in open star
clusters and based on several simplifying assumptions for the winds, magnetic
field geometry, and flow acceleration profile was developed by Kawaler (1988)
as

$\dot{J}_{\rm MB,Kaw}\ =\ -\,K_{W}\,\left(\frac{\Omega}{{\rm
s^{-1}}}\right)^{1\,+\,4a_{K}n/3}\,\left(\frac{\dot{M}_{\rm
wind}}{-\,10^{-14}\,\,{\rm M}_{\odot}\,{\rm
yr}^{-1}}\right)^{1\,-\,2n/3}\,\left(\frac{R}{{\rm
R}_{\odot}}\right)^{2\,-\,n}\,\left(\frac{M}{{\rm M}_{\odot}}\right)^{-n/3}\ \
{\rm dyn~{}cm}\ ,$ (18)

where $a_{K}$ depends on the scaling between the secondary magnetic field and
spin, $n$ depends on the secondary magnetic field geometry, $\dot{M}_{\rm
wind}<0$ is the star mass-loss rate through winds, and $\Omega$, $R$, and $M$
are the star spin, radius, and mass, respectively. Even though the parameter
$K_{W}$ depends on the ratio of the wind speed to the escape speed at the
Alfvén radius and how the star magnetic field scales with the star radius and
spin, it is usually approximated by ${\sim 2.035\times
10^{33}(24.93^{n})(6.63\times 10^{5})^{4n/3}}$. With ${a_{K}=1}$ and
${n=1.5}$, Equation 18 describes the standard Skumanich law (Verbunt & Zwaan,
1981), as orbital AML due to MB becomes proportional to $\Omega^{3}$.

On the other hand, in the context of accreting compact objects driven by
orbital AML, usually the so-called RVJ prescription (Rappaport et al., 1983)
is adopted, which is given by

$\dot{J}_{\rm MB,RVJ}\ =\ -\,3.8\times 10^{-30}\,\left(\,\frac{M}{\rm
g}\,\right)\,\left(\,\frac{R_{\odot}}{{\rm
cm}}\,\right)^{4}\,\left(\,\frac{\Omega}{\rm
s^{-1}}\,\right)^{3}\,\left(\,\frac{R}{R_{\odot}}\,\right)^{\gamma_{\rm MB}}\
\ {\rm dyn~{}cm}\ ,$ (19)

where $0\leq\gamma_{\rm MB}\leq 4$ is a free parameter that largely affects
how $\dot{J}_{\rm MB}$ varies during the binary evolution. With $\gamma_{\rm
MB}=4$, Equation 19 describes the standard Skumanich law.

Despite the fact that the RVJ prescription can explain several observations of
accreting objects, it has serious problems in different contexts, mainly
connected with the mass transfer rates it provides and the corresponding
evolutionary time-scales during CV and LMXB evolution (e.g. Podsiadlowski et
al., 2002; González Hernández et al., 2017; Belloni et al., 2020a; Pala et
al., 2022). In order to overcome part of these difficulties, recently a new
convection and rotation boosted (CARB) prescription that enhances orbital AML
due to MB was proposed (Van et al., 2019; Van & Ivanova, 2019). The CARB
prescription can be expressed as (Van & Ivanova, 2019)

$\dot{J}_{\rm MB,CARB}\ =\ -\,\frac{2}{3}\,\left(-\dot{M}_{\rm
wind}\right)^{-1/3}\,R^{14/3}\,\Omega_{\odot}\,B_{\odot}^{8/3}\,\left(\,\frac{\Omega}{\Omega_{\odot}}\right)^{11/3}\,\left(\frac{\tau_{\rm
conv}}{\tau_{\odot,\rm conv}}\right)^{8/3}\,\left(\,v_{\rm
esc}^{2}\,+\,\frac{2\,\Omega^{2}\,R^{2}}{K_{2}^{2}}\,\right)^{-2/3}\ ,$ (20)

where $v_{\rm esc}$ is the surface escape velocity, $\tau_{\rm conv}$ is the
convective turnover time, $B$ is the surface magnetic field strength, and
$K_{2}=0.07$. The main difference between the RVJ and the CARB prescription is
that the former is the result of an empirical fitting scheme, while the latter
was obtained in a self-consistent physical way, considering wind mass loss,
the dependence of the magnetic field strength on the outer convective zone and
the dependence of the Alfvén radius on the donor spin.

We illustrate in Figure 2 the orbital AML rates provided by the three MB
prescriptions mentioned above. The evolutionary sequences have been computed
by us using the Modules for Experiments in Stellar Astrophysics (mesa) code
(Paxton et al., 2011; Paxton et al., 2013, 2015, 2018, 2019), in which we
evolved a Roche lobe filling star with initial mass of $1$ M⊙, orbiting a
point-mass companion of mass $1.4$ M⊙. In the simulations, we take into
account stellar wind mass loss, adopting the Reimers (1975) prescription with
the scaling factor equal to $1$, and assume ${\beta_{\rm ml}=0.8}$ and
${\alpha_{\rm ml}=0}$. For MS masses ${\lesssim 0.4}$ M⊙, the CARB
prescription clearly provides higher orbital AML rates than the RVJ and the
Kawaler prescriptions. For MS masses greater than that, the CARB, RVJ and
Kawaler prescriptions provide comparable orbital AML rates if $n_{K}=1.5$ and
$\gamma_{\rm MB}=0$. Finally, GR is typically much weaker than any of these
prescriptions, expect for the Kawaler recipe with $n\lesssim 1$.

Figure 2: Comparison of the orbital AML rates ($-\dot{J}$) due to GR and due
to some MB prescriptions, during the evolution of a low-mass Roche lobe
filling star, as a function of its mass ($M$). All tracks have been computed
by us with the mesa code, assuming solar metallicity, wind mass loss (Reimers
parameter equals to 1), $\beta_{\rm ml}=0.8$, $\alpha_{\rm ml}=0$, a point-
mass accretor with initial mass of $1.4$ M⊙, and the initial mass of the donor
is $1.0$ M⊙. Each track represents the evolution from the onset of Roche lobe
filling to the moment when the star becomes fully convective (when MB is
supposed to cease or at least significantly decrease). The dot-dashed red
lines correspond to the CARB prescription (Van & Ivanova, 2019), for which we
chose two different initial orbital periods, namely $12$ h to illustrate the
case of an unevolved MS star (helium-core mass equals to zero), and $4$ days
to illustrate the subgiant case (helium-core mass ${\sim 0.1}$ M⊙). The solid
blue lines show the RVJ prescription (Rappaport et al., 1983), for three
different values of $\gamma_{\rm MB}$, and in all these cases we chose an
initial orbital period of $12$ h, corresponding in turn to unevolved MS stars.
The Kawaler (1988) prescription is marked by the long-dashed green lines, for
three different values of $n$, assuming in all three sequences $a_{K}=1$, for
which we also chose an initial orbital period of $12$ h, corresponding to the
unevolved MS star case. Finally, the short-dashed black lines correspond to
the strength of GR in each of the evolutionary sequences, which can be easily
associated with them by comparing the mass at which the MB becomes
inefficient, i.e. the lowest mass in each track. It is clear that the CARB
recipe provides on average higher orbital AML rates than the RVJ and the
Kawaler prescriptions, the latter for $n\lesssim 1.5$, especially for masses
${\lesssim 0.4}$ M⊙. On the other hand, the CARB, RVJ and Kawaler
prescriptions provide comparable orbital AML rates for larger masses, if
$n_{K}=1.5$ and $\gamma_{\rm MB}=0$. In addition, the Kawaler prescription is
much more sensitive to the chosen parameters, in comparison with the RVJ
prescription, as it provides much more different orbital AML rates, for small
variations of $n_{K}$. Moreover, except for the Kawaler prescription with
$n\lesssim 1$, GR in all evolutionary sequences contributes only negligibly to
the total orbital AML. Finally, regarding the type of the star in the CARB
tracks, we can see that the orbital AML rate is initially higher for a more
evolved star but decreases as the star mass decreases, resulting in a weaker
MB at masses ${\lesssim 0.4}$ M⊙, in comparison with an unevolved MS star.

In addition to GR and MB, there are other sources of orbital AML acting on
interacting binaries that arise as a consequence of mass transfer. For this
reason this type of orbital AML is called consequential AML. Among the several
mechanisms proposed for consequential AML, the most important are a fast
isotropic wind from the vicinity of the donor, isotropic re-emission from the
vicinity of the accretor (King & Kolb, 1995), a circumbinary disk (Spruit &
Taam, 2001) and winds from the accretion disk (Cannizzo & Pudritz, 1988).
Consequential AML is by definition an orbital AML mechanism that is mass-
transfer-dependent, i.e. it is null in the absence of mass transfer. It is
usually written in a simple parameterized form as

$\frac{\dot{J}_{\rm CAML}}{J_{\rm orb}}\ =\ \nu\,\frac{\dot{M_{d}}}{M_{d}}\ ,$
(21)

where $\nu$ depends on the binary properties, including perhaps the mass
transfer rate itself, and is the parameter characterizing the strength of
consequential AML. Its form depends entirely on the mechanism driving orbital
AML.

In the case of a circumbinary disk, $\nu$ is given by (Spruit & Taam, 2001)

$\nu_{\rm CB}\ =\ \gamma\,\delta_{\rm ml}\,(\,1\,+\,q\,)\ ,$ (22)

where $\delta_{\rm ml}$ is the fraction of the mass lost from the binary
components that feeds the circumbinary disk, and the parameter $\gamma$ takes
into account the characteristics of the circumbinary disk as a function of
time and depends on the inner edge of the circumbinary disk, the binary
semimajor axis, the viscous time-scale at the inner edge of the circumbinary
disk and the mass of the circumbinary disk.

In the case of mass loss from the binary, $\nu$ may correspond to the specific
angular momentum of the accretor (isotropic re-emission) or the donor (fast
isotropic wind), given by (King & Kolb, 1995)

$\nu_{\rm ml,a}\ =\ \beta_{\rm ml}\,\left(\,\frac{q^{2}}{1\,+\,q\,}\,\right)\
,\hskip 42.67912pt\nu_{\rm ml,d}\ =\ \alpha_{\rm
ml}\,\left(\,\frac{1}{1\,+\,q\,}\,\right)\ .\hskip 42.67912pt$ (23)

Fast isotropic winds and isotropic re-emission may also generate additional
orbital AML due to friction arising from the interaction of the ejected
material and the accretor/donor. This frictional AML then strongly depends on
the velocity of the ejected material and on the orbital velocity of the
accretor/donor as well as on the geometrical cross-section of the
accretor/donor (Shara et al., 1986; Schenker et al., 1998; Schreiber et al.,
2016; Sparks & Sion, 2021).

We have already illustrated in Figure 2 that MB typically provides orbital AML
rates several orders of magnitude higher than those expected from GR. The
rates predicted by consequential AML may be higher or lower in comparison with
MB, depending on the form of the two types of orbital AML. For instance,
frictional AML due to isotropic winds and/or isotropic re-emission is
typically negligible, unless the expansion velocity of the ejected material is
very small (Liu & Li, 2019). The same also occurs for consequential AML due to
mass loss (isotropic winds or isotropic re-emission), which typically provides
rates smaller than those related to either MB or GR alone. On the other hand,
consequential AML due to a circumbinary disk can be the dominant source of
orbital AML depending on the parameters of the disk and the rate at which it
is fed (Willems et al., 2005; Ma & Li, 2009).

So far we have briefly described the current understanding of the different
processes that can drive mass transfer in binaries. This included the
stability of a given binary against mass transfer on the dynamical and thermal
time-scales of the donor as well as mass transfer driven by the nuclear
evolution of the donor and orbital AML. The largest uncertainties in current
models are probably related to the conditions separating dynamically stable
from dynamically unstable mass transfer, the strength and parameter
dependencies of orbital AML through MB, and the processes that drive
consequential AML. On the basis of the above-described knowledge of mass
transfer in binaries, formation channels for the different types of accreting
compact binaries have been developed in the last decades which we will review
in the following section.

##  5 Formation Channels

Most accreting compact objects form from the evolution of zero-age MS–MS
binaries. Some may form from stellar mergers in hierarchical systems, i.e.
from triples, quadruples and, eventually, higher-order systems. Dynamical
interactions, which are common in high-density environments such as the
central parts of dense star clusters, could also lead to the formation of a
sufficiently important fraction of accreting compact objects, especially for
those originating from high-mass stars. While for some specific systems, and
some particular types of accreting compact objects, the triple and/or the
dynamical scenario is required to explain their properties, in most cases the
contribution from these two scenarios is expected to be much smaller than that
of binary evolution. This is simply because of the lower frequency of triples
and higher-order hierarchical systems in comparison to binaries (e.g. Moe & Di
Stefano, 2017) and because strong dynamical interactions only occur frequently
in dense star clusters. In what follows, we therefore mainly focus on
accreting compact objects formed through binary evolution but add a few
comments on additional formation channels through dynamical interactions at
the end of this section.

The current small orbital separations of close binaries in which a compact
object accretes from a non-degenerate low-mass companion, i.e. CVs and LMXBs,
imply that these systems are currently much smaller than the progenitor of the
compact object was as a giant. Consequently, the orbit of the binary must have
been significantly reduced during the formation of the compact object, most
likely through CE evolution generated by dynamically unstable mass transfer
(Webbink, 1975; Paczyński, 1976).

The current orbital periods of most wide binaries in which a WD accretes from
the winds of an evolved low-mass red giant, i.e. most S-type SySts, are
significantly longer than those of post-CE binaries but short enough that mass
transfer must have occurred when the WD was formed. Therefore, in these cases,
the most likely scenario is that the orbit expanded during the formation of
the WD, which happens when the mass transfer is dynamically stable and most
likely non-conservative (Webbink, 1988).

SyXBs are similar to SySts, in the sense that they are also wide-orbit
binaries in which the compact object accretes from the wind of an evolved low-
mass red giant. However, their compact objects are NSs, which implies that
stable non-conservative mass transfer cannot be the formation channel, because
the required mass ratio of the zero-age MS–MS binary implies that the non-
degenerate companions should be high-mass stars, which is not the case. A
possible formation scenario for these systems could be the formation of a
long-period post-CE binary, hosting a helium star paired with a low-mass star,
followed by either electron-capture or core-collapse supernova leading to the
NS (Lü et al., 2012; Yungelson et al., 2019).

Regarding compact objects accreting from high-mass non-degenerate stars, i.e.
HMXBs, those in wide orbits (e.g. Be-HMXBs) are most likely formed through an
episode of dynamically stable non-conservative mass transfer, while those
having orbital periods of only a few days (e.g. sg-HMXBs) seem to have formed
through CE evolution (Tauris et al., 2017). On the other hand, NSs/BHs
accreting from the strong and dense winds of Wolf–Rayet stars, i.e. WR-HMXBs,
require two episodes of mass transfer, one to form the accreting compact
object and the other one to form the Wolf–Rayet star. Even though the first
episode of mass transfer is most likely dynamically stable non-conservative
mass transfer (Qin et al., 2019), there is evidence that in some WR-HMXBs the
second episode of mass transfer was CE evolution, while others can be more
easily explained when the second episode was dynamically stable non-
conservative mass transfer (van den Heuvel et al., 2017).

Finally, in those very close binaries in which the compact object is accreting
from a Roche lobe filling WD or semi-degenerate helium-rich star, i.e. AM CVns
and UCXBs, two episodes of mass transfer are needed, and at least one of these
episodes must have been CE evolution. More specifically, these systems may
form either through two CE phases, or through dynamically stable non-
conservative mass transfer followed by CE evolution, or finally through CE
evolution followed by dynamically stable non-conservative mass transfer
(Solheim, 2010; Nelemans et al., 2010).

In what follows, we will in more detail address the main formation channels of
accreting compact objects, which are illustrated in Figure 3.

Figure 3: Main formation channels leading to accreting compact objects.
Starting from a zero-age MS–MS binary, when the more massive MS star evolves
and fills its Roche lobe, mass transfer stars. Mass transfer is dynamically
stable if the adiabatic radius–mass exponent ($\zeta_{\rm ad}$) is greater
than the Roche lobe radius–mass exponent ($\zeta_{\rm ad}$), and unstable
otherwise, leading to CE evolution. During dynamically stable mass transfer,
the orbital period increases or decreases as a response to mass transfer, and
the orbital period at the end of mass transfer depends on the binary
properties at the onset of Roche lobe filling and on the fraction of the
transferred mass that is lost ($\alpha_{\rm ml}+\beta_{\rm ml}+\delta_{\rm
ml}$) during the event. On the other hand, if CE evolution happens, drag
forces remove orbital energy, which is used with a certain efficiency
($\alpha_{\rm CE}$) to eject the envelope, potentially together with a
fraction of other sources of energy, such as recombination energy
($\alpha_{\rm rec}$) and/or thermal energy ($\alpha_{\rm th}$). The outcome of
CE evolution is a binary with a much shorter orbital period. Compact objects
in close orbits accreting from a Roche lobe filling non-degenerate low-mass
companion (i.e. CVs and LMXBs) are believed to form during one episode of CE
evolution. NSs accreting from the winds of low-mass red giants in wide orbits
(i.e. SyXBs) are also expected to form in an episode of CE evolution. On the
other hand, WDs in wide orbits accreting from the winds of low-mass red giants
(i.e. SySt) are expected to form through an episode of dynamically stable non-
conservative mass transfer. Regarding NSs or BHs accreting from high-mass
companions (i.e. HMXBs), they can be formed through dynamically stable non-
conservative mass transfer or CE evolution. Finally, compact objects accreting
from Roche lobe filling WDs or semi-degenerate helium-rich stars (i.e. AM CVns
and UCXBs) are formed through two episodes of mass transfer, at least one of
them being CE evolution.

###  5.1 Common-Envelope Evolution

When the more massive star of a zero-age MS–MS binary evolves off the MS and
fills its Roche lobe, it will give rise to dynamically unstable mass transfer,
if $\zeta_{\rm ad}<\zeta_{\rm RL}$. In this case, the mass-loss time-scale
becomes so short that the donor cannot remain within its Roche lobe, which
leads to the formation of a CE around the dense giant core and the MS star.
Even though a CE event in many situations is triggered by dynamically unstable
mass transfer, there are also other mechanisms able to do that, such as the
Darwin instability (e.g. Hut, 1980), which occurs if the future donor angular
momentum at synchronism exceeds one-third of that of the orbit. The Darwin
instability corresponds to a tidal instability. It occurs when not enough
orbital angular momentum is extracted from the orbit to maintain the star
synchronized as it evolves. Tidal forces will spin up the star by removing
orbital angular momentum, resulting in a binary with smaller orbital
separation and in turn even less orbital angular momentum. This implies that
the star will need even more orbital angular momentum to stay synchronized
with the orbital motion. This leads to a runaway process of orbital decay and
ultimately a CE event.

Because of drag forces during CE evolution, the MS star and the core spiral in
towards their common centre of mass and the CE is expelled. In this process, a
large fraction of the initial orbital energy and orbital angular momentum is
lost with the envelope which leads to a post-CE binary with an orbital period
orders of magnitude shorter than that of the initial MS–MS binary. In case
dynamically unstable mass transfer was generated from a massive star to its
companion, the post-CE binary hosts a helium star, which is the progenitor of
the compact object, and the binary separation may increase due to a natal kick
during the collapse of the helium star.

Despite a lot of effort and (computational) resources that are spent towards a
better understanding of CE evolution, we still struggle to identify and
describe some of the main physical processes involved in this crucial
evolutionary phase. The main reason for this failure lies in the fact that
modelling CE evolution is very demanding from a computational point of view,
since it involves a complex mix of physical processes operating over a huge
range of scales (Ivanova et al., 2013). In the absence of reliable simulations
that cover the full complexity of CE evolution, a very simplistic approach is
frequently used to overcome this problem and to make predictions for
populations of post-CE binary characteristics. These simple equations based on
energy and angular momentum conservation provide useful constraints on the
outcome of CE evolution.

#### The Energy Budget of Common-Envelope Evolution

In the so-called energy formalism, the outcome of CE evolution is usually
approximated by the balance between the change in the orbital energy and the
envelope binding energy. This energy conservation equation is parameterized
with a parameter $0\leq\alpha_{\rm CE}\leq 1$, which corresponds to the
efficiency with which orbital energy is used to eject the envelope. In other
words, $\alpha_{\rm CE}$ represents the fraction of the difference in orbital
energy (before and after the CE phase) that unbinds the envelope, i.e.

$E_{\rm bind}\ =\ \alpha_{\rm CE}\ \Delta E_{\rm orb}\ =\ -\ \alpha_{\rm CE}\
\left(\,\frac{G\,M_{\rm d,c}\,M_{\rm a}}{2\,a_{f}}\ -\ \frac{G\,M_{\rm
d}\,M_{\rm a}}{2\,a_{i}}\,\right)\ ,$ (24)

where $E_{\rm bind}$ is the envelope binding energy, $E_{\rm orb}$ is the
orbital energy, $M_{\rm d,c}$ is the core mass of the donor, $a_{i}$ is the
semimajor axis at onset of CE evolution, and $a_{f}$ is the semimajor axis
after CE ejection.

The binding energy is usually approximated by

$E_{\rm bind}\ =\ -\ \frac{G\,M_{\rm d}\,(M_{\rm d}\,-\,M_{\rm
d,c})}{\lambda\,R_{\rm d}}\ ,$ (25)

where $\lambda$ is the binding energy parameter, which depends on the
structure of the donor (Dewi & Tauris, 2000; Claeys et al., 2014).

In case thermal and/or recombination energies stored inside the envelope are
assumed to help in ejecting the envelope, the binding energy equation is
usually written as

$E_{\rm bind}\ =\ -\int_{M_{\rm d,c}}^{M_{\rm d}}\frac{G\;m}{r(m)}\;{\rm d}m\
+\ \alpha_{\rm th}\int_{M_{\rm d,c}}^{M_{\rm d}}\varepsilon_{\rm th}(m)\;{\rm
d}m\ +\ \alpha_{\rm rec}\int_{M_{\rm d,c}}^{M_{\rm d}}\varepsilon_{\rm
rec}(m)\;{\rm d}m$ (26)

where $\varepsilon_{\rm th}$ and $\varepsilon_{\rm rec}$ are the specific
thermal and recombination energies and $\alpha_{\rm th}$ and $\alpha_{\rm
rec}$ are the fractions of the thermal and recombination energies that are
assumed to contribute to unbinding the envelope. Thermal energy is the
thermodynamic internal energy of the CE, which is constrained by the virial
theorem, and can be released and partially used to help in the CE ejection.
Recombination energy is stored in ionized helium or hydrogen within the CE.
These ions can recombine in the expanding layers of the gaseous CE, once it
has sufficiently cooled and expanded, thereby releasing energy (Lucy, 1967)
that can contribute to unbinding the CE. The binding energy parameter
$\lambda$ can be computed by equating Equations 25 and 26. When $\alpha_{\rm
th}>0$ and/or $\alpha_{\rm rec}>0$, additional sources of energy contribute to
the ejection of the CE, which implies that less orbital energy is needed to
unbind the envelope.

Deriving constraints on the values of $\alpha_{\rm CE}$, $\alpha_{\rm th}$,
and $\alpha_{\rm rec}$ has been a focus of close compact binary research in
the last decades. Unfortunately, several slightly different CE prescriptions
are available, which needs to be taken into account when comparing the results
of different studies. In what follows, we briefly describe the constraints
obtained so far for binaries hosting WDs, for which the progenitors are
intermediate-mass MS stars, and for those hosting NSs or BHs, which originate
from the evolution of high-mass MS stars.

Regarding low-/intermediate-mass stars, the best way for constraining the CE
evolution is by using samples of detached WD–MS post-CE binaries because of
the large number of known systems and because it is relatively easy to
characterize them (Nebot Gómez-Morán et al., 2011). Reconstruction of the
evolution of observationally characterized post-CE binaries consisting of WDs
with M-type companions suggests that CE ejection is rather inefficient, with
$\alpha_{\rm CE}\sim 0.2-0.3$ (Zorotovic et al., 2010). Comparisons between
predictions from binary population synthesis with observations reached similar
conclusions for this type of post-CE binary (Toonen & Nelemans, 2013; Camacho
et al., 2014; Cojocaru et al., 2017). Recently, observational surveys have
become capable of identifying close WD binaries with MS companions more
massive than M-type MS stars (spectral types A, F, G, and K), and early
results indicate the existence of a population of post-CE binaries that can be
reproduced with the same small CE efficiency (Parsons et al., 2015; Hernandez
et al., 2021). However, for some of these systems, a small fraction of
recombination energy seems to be required to contribute to CE ejection
(Zorotovic et al., 2014a).

Close binaries hosting hot subdwarf B stars (i.e. helium-core-burning stars
with very thin hydrogen-rich envelopes), mostly paired with WDs but some also
with M-type MS stars, are also reasonably good targets for estimating the
efficiency of CE ejection, as CE evolution is certainly involved in their
formation. Contrary to WD–MS post-CE binaries, results from binary population
synthesis of hot subdwarf B stars indicate that the observations are better
reproduced with $\alpha_{\rm CE}\sim 0.75$ and a large fraction (${\sim
0.75}$) of the internal energy (thermal plus recombination) contributing to
expelling the envelope (Han et al., 2002, 2003) which indicates that both CE
ejection and the conversion of thermal and recombination energies into kinetic
energy are efficient processes. However, given that the sample of WD–MS
binaries is cleaner in the sense that only one phase of mass transfer is
involved, the observational constraints indicating a small value for the CE
efficiency should be considered as being stronger. In addition, we have to be
aware that the energy prescription for CE evolution is too simplistic and
several unknown physical processes are not properly taken into account which
might affect the constraints derived from observations of different systems on
the CE efficiency.

With respect to progenitors of accreting compact objects hosting NSs and BHs,
the situation is more complicated because we know less about CE evolution in
this mass range (Ivanova et al., 2013) and the situation might be quite
different from the previous case. The main reason for these expected
differences is that CE evolution leading to WD–MS post-CE binaries is
generated by dynamically unstable mass transfer from red giant donors with a
convective envelope (Taam & Sandquist, 2000), while for high-mass stars with
solar metallicity the onset of Roche lobe filling usually happens when the
donor is in the Hertzsprung gap and has a radiative envelope. In addition,
metallicity has a strong impact on the evolution of high-mass stars prior to
the onset of Roche lobe overflow, which is independent of the effect of
metallicity-dependent winds, thereby altering the evolutionary stage at which
Roche lobe filling occurs (Klencki et al., 2020). In most cases, this causes
$\lambda$ to be far lower for high-mass stars than for intermediate-mass stars
(Kruckow et al., 2016), implying that in several situations, there would be
apparently not enough energy available to eject the CE.

For a long time, this has been believed to be the case for BH-LMXBs, which
would require an abnormally high efficiency for the envelope ejection in
binary models (e.g. Yungelson & Lasota, 2008), since it is very difficult for
the binary hosting a Roche lobe filling high-mass star paired with a much less
massive companion to eject the CE at the expense of only orbital energy.
However, more recent calculations have shown that this is not an actual issue
in the formation of BH-LMXBs, since they could be produced without invoking
unrealistically high CE efficiencies (Wiktorowicz et al., 2014). The real
problem arises when trying to reproduce the donor mass distribution in this
population. Despite the fact that it is possible to reproduce individual BH-
LMXBs in binary models, we cannot yet reproduce the properties of the entire
population with just one model (Wiktorowicz et al., 2014). We will come back
to this issue later.

Regarding SyXBs, in the absence of other sources of energies, in population
synthesis aiming at explaining their observational properties, $\alpha_{\rm
CE}$ needs to be as high as $4$ (Yungelson et al., 2019), which is
unrealistically high. This result could suggest that there are other sources
of energy that are required to help to expel the envelope or that these X-ray
binaries do not form through standard CE evolution . For instance, they could
instead be preferentially formed from accretion-induced collapses of a massive
oxygen–neon–magnesium WD (e.g. Hinkle et al., 2006). Another possibility,
perhaps more realistic, is that the uncertainties and simplifications involved
in stellar and binary evolution calculations, for example mixing in stellar
interiors, core-collapse/electron-capture supernova and CE evolution, are the
main reason for the problems in reproducing these systems, as well as BH-LMXBs
(e.g. Belczynski et al., 2021).

#### Common-Envelope Evolution from Hydro-dynamical simulations

Despite the above-described attempts to constrain $\alpha_{\rm CE}$,
$\alpha_{\rm th}$ and $\alpha_{\rm rec}$, it seems unlikely that there is a
universal recipe within the simplistic energy formalism able to account for
all types of accreting compact objects. In the case of low-/intermediate-mass
stars, evidence is growing evidence that the efficiency of expelling common
envelopes is rather low, and overall the energy formalism delivers reasonable
results. In contrast, it is very difficult to explain the properties of
accreting compact objects coming from high-mass stars based on the energy
balance of CE evolution, regardless of the assumed values for the CE evolution
parameters. This difficulty must to some degree arise from our inability to
understand the role played by other sources of energies acting together with
orbital energy, as well as correctly computing the CE binding energy in an
unbiased way.

CE evolution can be divided into several subsequent phases, which helps to
better understand why this event is so complicated to be modelled. Most of the
problems arise from the fact that each of these phases occurs at a particular
time-scale, involving a particular physical process (e.g. Podsiadlowski,
2001). The first phase is the loss of co-rotation, which can last hundreds of
years, and corresponds to very small changes in the orbital separation in
comparison with the orbital period. During this phase, the accretor orbits
either outside the future CE or inside the CE outer layers (e.g. Ivanova &
Nandez, 2016). The second phase is called plunge-in, which corresponds to the
rapid spiral-in with huge changes in the orbital separation. During this
phase, the accretor plunges inside the CE, and the orbital energy is
transferred to the CE, driving its expansion, in case there is enough energy
available for that. At the end of this phase, most of the CE mass is outside
the orbit of accretor and the core of the donor (e.g. Ivanova & Nandez, 2016).
During the third phase, when the CE is well-extended, a self-regulating
spiral-in occurs on the thermal time-scale of the CE, during which energy
released due to the spiral-in is transported to the CE surface and
subsequently radiated away. This phase corresponds to small changes in the
orbital energy (e.g. Ivanova & Nandez, 2016). CE evolution terminates at the
end of the self-regulating spiral-in phase with either the ejection of the CE
or the accretor or the core of the donor filling its Roche lobe. The latter
most likely result is a merger but could also provide another pathway for CE
ejection (e.g. Podsiadlowski et al., 2010).

Figure 4: Change in the orbital separation (left-hand panel, Ivanova & Nandez,
2016) and envelope mass (right-hand panel, Sand et al., 2020) through time
during CE evolution, taken from two different CE evolution models. In the
left-hand panel, the onset of the plunge-in phase (when
$(\,|\,\dot{a}\,|\,P_{\rm orb}\,)\,/\,a$ becomes greater than 0.1), the end of
the plunge-in phase (when $(\,|\,\dot{a}\,|\,P_{\rm orb}\,)\,/\,a$ becomes
smaller than 0.1), and the onset of the self-regulating spiral-in (when
$|\,(\,\dot{E}_{\rm orb}\,P_{\rm orb}\,)\,/\,E_{\rm orb}\,|$ becomes smaller
than 0.01) are indicated with crosses. The transition between the end of the
plunge-in phase and the onset of the self-regulating spiral-in is not sudden
in these simulations, although it lasts for only a few days. In the model with
companion mass of $0.1$ M⊙, there is not enough orbital energy available to
expel the CE, and the binary is expected to merge during the self-regulating
spiral-in phase. However, in the model with companion mass of $0.2$ M⊙, even
though orbital energy is transferred to the CE, it is not enough to entirely
eject the CE during the self-regulating spiral-in phase. Despite the fact that
a strong reduction in the orbital separation occurs, like in the $0.2$ M⊙
model, in most numerical simulations, the CE is not entirely ejected, which
indicates that other sources should contribute to the CE ejection together
with orbital energy. In the right-hand panel, there is a comparison between
the case without internal energy and the case with internal energy (thermal
plus recombination), which has been widely proposed as the key source of
energy usually missing in the simulations. It is clear in this simulation that
the CE can be ejected with the help from internal energy. However, there are
strong physical arguments suggesting that this energy would be simply radiated
way, not actually contributing to the CE ejection. If confirmed beyond doubts,
other sources of energy or transfer mechanisms must be playing a role and
should be further explored as alternatives to recombination energy.

We illustrate CE evolution in the left-hand panel of Figure 4, where we show
the evolution of the orbital separation from just before the plunge-in phase
until just after the self-regulating spiral-in, for two models taken from a
study based on a three-dimensional smoothed particle hydrodynamics code
(Ivanova & Nandez, 2016). In these simulations, the authors chose a low-mass
red giant donor with a mass of $1.8~{}M_{\odot}$, a core mass of
$0.318~{}M_{\odot}$ and radius of $16.3~{}R\odot$, and several companion
masses. The figure clearly illustrates that there is a huge orbital shrinkage
during the plunge-in phase until the onset of the self-regulating spiral-in
phase, which is precisely what the energy formalism tries to reproduce.
However, in those cases that avoid merger, the CE is not always effectively
ejected, which represents a long-standing problem in numerical simulations of
CE evolution (e.g. Ohlmann et al., 2016; Soker, 2017).

Given the above-mentioned inefficient ejection of the CE predicted in
simulations, there should exist at least one energy source, perhaps more, in
addition to orbital energy, that would be able to expel the remaining bound
matter (e.g. Soker, 2017). It has been argued for a long time that thermal
energy and mainly hydrogen and helium recombination energy could solve this
problem (e.g. Webbink, 2008; Nandez et al., 2015; Ivanova, 2018). The impact
of including thermal and recombination energies is illustrated in the right-
hand panel of Figure 4, which shows the evolution of the unbound mass fraction
through time, taken from a numerical simulation with a moving-mesh
hydrodynamics code (Sand et al., 2020). In this simulation, the donor is on
the AGB undergoing helium-shell burning and no thermal pulses, with a mass of
$0.97$ M⊙ and a core mass of $0.545$ M⊙. Almost the entire CE mass is unbound
when the stored internal (thermal plus recombination) energy is used, most of
which is only released at rather late times, when the expanding CE
sufficiently cools down. This is not the case when only the potential energy
contributes to the binding energy, since ${\sim 10}$% of the CE mass is still
inside the orbit of the donor core and the companion, after the plunge-in
phase. No further dynamical spiral-in is expected, because of the low density
of the remaining envelope and the co-rotation of the material, avoiding in
turn any drag force.

The example above, together with other simulations, provides support to the
claim that internal (mainly recombination) energy might be a key ingredient in
CE evolution. On the other hand, most of the available recombination energy
could be simply radiated away (e.g. Soker et al., 2018; Grichener et al.,
2018), since the optical depth is expected to be low when recombination
occurs, allowing radiation to more easily escape. In addition, despite the
fact that the amount of available recombination energy depends on the core
definition, it is expected to be smaller in high-mass stars than in
intermediate-mass stars. Therefore, some simulations suggest that
recombination energy is unlikely to help in the removal of the CE in this mass
range (Ricker et al., 2019).

In any event, it is widely accepted by now that only dynamical spiral-in is
most likely not enough to completely remove the CE. In case recombination
energy is not significantly helping, other sources of energy would be needed
(e.g. Soker, 2017). These additional contributions could come from nuclear
energy (Podsiadlowski et al., 2010), accretion energy (Voss & Tauris, 2003),
dust-driven winds (Glanz & Perets, 2018), interaction of the binary (core of
the donor + companion) with a circumbinary disk (Kashi & Soker, 2011), large-
amplitude pulsations (Clayton et al., 2017), or jets launched by the companion
as it accretes mass from the circumbinary envelope when it is about to exit
the CE from inside (Soker, 2017; Moreno Méndez et al., 2017; López-Cámara et
al., 2019, 2020; López-Cámara et al., 2022). Even though there are several
mechanisms available, they have been poorly explored in hydro-dynamical
simulations, and it is still not clear which one is dominant, if any at all,
and under which conditions. This means that CE evolution is a subject far from
being well-understood, and, hopefully, in the next years, further hydro-
dynamical simulations will shed more light on this extremely important phase
for the formation of accreting compact objects.

###  5.2 Dynamically Stable Non-conservative Mass Transfer

Even though the energy formalism is successful to some extent in avoiding the
messy physics involved in the several phases of CE evolution, it fails to
reproduce some observed systems, in particular double-helium WD binaries.
Explaining these systems with two consecutive CE events requires that energy
is generated during CE (which implies that the energy conservation law is
violated), or at best, that an unknown source of energy, much greater than the
orbital energy, is contributing to expelling the envelope (which can be
considered unrealistic).

To solve this problem, an alternative but similarly simplistic formalism was
invented based on conservation of angular momentum, which is usually called
angular momentum (or $\gamma$) formalism (Nelemans et al., 2000; Nelemans &
Tout, 2005). Although this new approach could somehow work mathematically, it
does not provide a physical explanation for the required CE phases without
spiral-in (e.g. Webbink, 2008; Ivanova et al., 2013).

We believe that most systems for which no reasonable solutions in the
framework of CE energy balance exist are actually not formed through CE
evolution, but rather dynamically stable non-conservative mass transfer.
Indeed, there is strong theoretical evidence that an increase in the orbital
separation can be reached during an episode of dynamically stable non-
conservative mass transfer on a thermal and/or nuclear time-scale, which can
potentially reproduce otherwise inexplicable systems. In what follows, we
discuss in more detail systems that may form through this dynamically stable
non-conservative mass transfer channel.

#### Low-/Intermediate-Mass Stars

There are several types of binaries hosting (proto) WDs with orbital periods
longer than ${\sim 100}$ days. In some of them, it is clear that some sort of
interaction in their history must have occurred, because of observed chemical
anomalies. For instance, barium stars, which are paired with WDs, are
F-/G-type MS stars or G-/K-type red giants exhibiting strong absorption lines
of ionized barium in their spectrum as well as of other s-process elements
(e.g. Jorissen et al., 2019; Escorza et al., 2019), which could not have been
synthesized by them. In a similar way, carbon-enhanced metal-poor stars show
abundances of s-process elements (e.g. Jorissen et al., 2016; Hansen et al.,
2016) that cannot be explained in the context of single star evolution.
S-process elements are produced during the thermally-pulsing AGB phase (e.g.
Käppeler et al., 2011), and it is therefore widely accepted that these
elements were transferred to the barium stars and carbon-enhanced metal-poor
stars via stellar winds or dynamically stable Roche lobe overflow, when the WD
progenitor was at this evolutionary stage.

Other types of binaries having comparable orbital periods include post-AGB
binaries, with orbital periods in the range of ${\sim 100-2500}$ days (Oomen
et al., 2018). In addition, some blue stragglers, which are typically found in
old star clusters and correspond to MS stars bluer, brighter, and more massive
than those around the cluster turnoff point, are in binaries with orbital
periods between ${\sim 100}$ and ${\sim 3000}$ days. While some hot subdwarf B
stars are members of close binaries, a certain fraction of these stars are
members of binaries with rather wide orbits. Hot subdwarf B stars are post-FGB
binaries that must have formed when their progenitors filled their Roche lobe
just before reaching the tip of the FGB. The typical orbital periods for wide
binaries containing hot subdwarf B stars range from ${\sim 700}$ to ${\sim
1300}$ days. Moreover, most S-type SySts have orbital periods between ${\sim
200}$ and ${\sim 1500}$ days. In some cases, there is evidence for negligible
mass accretion during these episodes of mass transfer, such as for hot
subdwarf B stars (Vos et al., 2018), while in other cases, such as for barium
stars and carbon-enhanced metal-poor stars, only a non-negligible amount of
mass accretion can explain their chemical properties (e.g. Miszalski et al.,
2013; Abate et al., 2015a, b).

Figure 5: Diagram depicting the orbital period ($P_{\rm orb}$) versus
eccentricity for several types of low-/intermediate-mass post-mass-transfer
binaries with orbital period longer than ${\sim 100}$ day, namely barium star
binaries (Van der Swaelmen et al., 2017), blue straggler binaries (Carney et
al., 2001; Mathieu & Geller, 2009), post-AGB binaries (Oomen et al., 2018),
s-process-polluted carbon-enhanced metal-poor star (CEMP-s) binaries (Jorissen
et al., 2016; Hansen et al., 2016), wide-orbit hot subdwarf B binaries (Vos et
al., 2015; Otani et al., 2018), S-type SySts (Mikołajewska, 2003; Brandi et
al., 2009; Fekel et al., 2007, 2008; Hinkle et al., 2009; Fekel et al., 2010),
and R Aqr, which is the only D-type SySt with radial velocity data suitable
for orbital determination (Gromadzki & Mikołajewska, 2009). The hatched
regions correspond to the location of post-CE binaries ($P_{\rm orb}\lesssim
100$ days) and of systems formed without Roche lobe filling ($P_{\rm
orb}\gtrsim 2000$ days), taken from a binary model (Belloni et al., 2020b). It
is clear from the diagram that there is a significant fraction of systems in
eccentric binaries with orbital periods between ${\sim 100}$ and ${\sim 2000}$
days, which is precisely between the region occupied by post-CE binaries and
the region occupied by systems formed without Roche lobe filling. Therefore,
most of these systems apparently cannot be explained by CE evolution, nor by
the lack of Roche lobe filling. Indeed, the overwhelming majority of these
systems are most likely formed during dynamically stable non-conservative mass
transfer, accompanied by an eccentricity pumping mechanism that would
efficiently work against tidal circularization for some system, but not for
all of them, as it must also account for those systems in circular orbits.

All these binaries, in addition to having orbital periods usually longer than
${\sim 100}$ days, have in many cases non-null eccentricities. We show the
orbital period versus eccentricity diagram for them in Figure 5, where we also
added a hatched region indicating the location of the post-FGB and post-AGB
binaries formed without Roche lobe overflow (taken from a binary model,
Belloni et al., 2020b). We can see from the figure that those systems with
orbital periods longer than ${\sim 2000}$ days fit reasonably well within the
expected region for WD binaries where the WD progenitor never filled its Roche
lobe. The same certainly holds for most D-type SySts, which have long orbital
periods (${\gtrsim 40}$ years), although only for one system the orbital
period has been accurately measured. The mass exchange that is required to
explain the formation of barium stars and s-process-polluted carbon-enhanced
metal-poor stars with periods exceeding ${\sim 2000}$ days has likely been
driven by efficient wind accretion, instead of Roche lobe overflow.

In contrast, systems with orbital periods shorter than ${\sim 2000}$ days must
have experienced Roche lobe overflow. Given their current orbital period, the
progenitors of the WD (or hot subdwarf B stars) in these systems must have
filled their Roche lobes at some point along their evolution. It is difficult
to explain the periods of these systems as a result of CE evolution, since
orbital periods of post-CE binaries are usually shorter than ${\sim 100}$ days
(Nie et al., 2012; Zorotovic et al., 2014b) and often just a few h (Nebot
Gómez-Morán et al., 2011). In addition, even though some of these binaries are
on circular orbits, many others have eccentric orbits, which is not expected
for post-CE binaries, since tidal forces during CE evolution would clearly
have managed to circularize their orbits. Consequently, Figure 5 indicates
that a different mode of mass transfer has to be considered to understand
these systems. For orbital periods longer than the typical outcome of CE
evolution, but shorter than those that might avoid Roche lobe overflow, i.e.
${\gtrsim 100}$ and ${\lesssim 2000}$ days, dynamically stable and non-
conservative mass transfer represents a reasonable alternative.

Indeed, several theoretical studies show that dynamically stable non-
conservative mass transfer can explain these systems (e.g. Webbink, 1988; Chen
& Han, 2008; Chen et al., 2013; Gosnell et al., 2019; Vos et al., 2020).
Depending on the combination of initial masses, metallicity, and orbital
period of the low-/intermediate-mass zero-age MS–MS binaries, the more massive
star could fill its Roche lobe while on the FGB (having or not a degenerate
core) or on the AGB. The properties of post-stable-mass-transfer binaries then
depend on the above-mentioned combination of initial masses, initial orbital
period, and metallicity as well as on the fraction of mass and orbital angular
momentum lost by the binary during the process.

We show evolutionary tracks for dynamically stable non-conservative mass
transfer in Figure 6. It is clear that orbital periods between a few hundreds
and a few thousands days can be achieved during dynamically stable non-
conservative mass transfer. Extrapolating the results shown in the figure
towards smaller initial periods one can easily imagine that stable mass
transfer may also lead to orbital periods ${\lesssim 100}$ days.

Figure 6: Evolution towards a post-FGB binary with the donor mass ($M_{\rm
d}$) of the orbital period ($P_{\rm orb}$), in the left-hand panel, and the
mass transfer rate ($\dot{M}_{\rm d}$), in the right-hand panel, during
dynamically stable non-conservative mass transfer using the mesa code. We
chose two different initial orbital periods ($100$ and $300$ days) and seven
different values for the transferred mass-loss fraction from the vicinity of
the accretor due to isotropic re-emission ($\beta_{\rm ml}$), from $0.1$ to
$0.7$ in steps of $0.1$, which are indicated in the figure (the upper curve in
each set of tracks in the left-hand panel corresponds to $\beta_{\rm
ml}=0.1$). The simulations start with a zero-age MS–MS circular binary, with
stars of masses $1.50$ and $1.43$ M⊙, with negligible rotation. In addition,
we considered solar metallicity, changes in the rotation of the stars, tidal
synchronization as well as wind mass loss (Reimers parameter equals to 0.5).
Furthermore, except for stellar winds, we assumed that any mass loss in the
form of fast isotropic wind from the donor is negligible ($\alpha_{\rm
ml}=0$), and ignored mass loss and orbital AML due to a circumbinary disk
($\delta_{\rm ml}=0$). We can see some nice correlations in the left-hand
panel of this figure, in which the colours indicate the evolution of the
accretor mass ($M_{\rm a}$) along each track. In particular, the smaller
$\beta_{\rm ml}$, the longer $P_{\rm orb}$ at the end of mass transfer, which
occurs because the increase in the orbital period due to mass transfer is much
stronger than the increase due to mass loss from the binary. In addition, as
expected, the smaller $\beta_{\rm ml}$, the larger the fraction of mass
effectively accreted by accretor, and in turn, the greater $M_{\rm a}$ at the
end of the mass transfer episode. Finally, the longer the initial $P_{\rm
orb}$, the longer $P_{\rm orb}$ at the end of mass transfer. The right-hand
panel of the figure illustrates the typical behaviour in such episodes of
dynamically stable mass transfer (e.g. Woods et al., 2012), namely the
transition from a thermally unstable to a thermally stable phase of mass
transfer, the latter being driven then by the nuclear evolution. The mass
transfer starts thermally unstable because the equilibrium radius–mass
exponent ($\zeta_{\rm eq}$) is smaller than the Roche lobe radius–mass
exponent ($\zeta_{\rm RL}$), which means that it is driven by the thermal
readjustment of the donor, and proceeds initially on the donor thermal time-
scale. However, as the donor loses mass and changes its structure, $\zeta_{\rm
eq}$ eventually becomes larger than $\zeta_{\rm RL}$, and mass transfer
proceeds on the donor nuclear time-scale. We can also see a nice correlation
in this panel, the smaller $\beta_{\rm ml}$, the greater the peak of
$\dot{M}_{\rm d}$ during thermally unstable mass transfer. This happens
because the donor is increasingly driven out of thermal equilibrium, as mass
transfer tends to be more conservative. On the other hand, the smaller
$\beta_{\rm ml}$, the lower the mass transfer rate during thermally stable
mass transfer, i.e. the slower the mass transfer, which has an impact on the
post-FGB mass. In particular, the smaller $\beta_{\rm ml}$, the larger $M_{\rm
d}$ at the end of mass transfer, which can also be seen in the left-hand panel
and happens because the overall evolution is slower when mass transfer tends
to be more conservative, allowing in turn the core mass to grow more as
$\beta_{\rm ml}$ decreases.

According to Figures 5 and 6, formation through dynamically stable but non-
conservative mass transfer therefore represents a reasonable scenario for most
blue stragglers, post-AGB binaries, long-period binaries hosting subdwarf B
stars, and a significant fraction of barium stars and s-process-polluted
carbon-enhanced metal-poor stars. Also most S-type SySts, which correspond to
${\sim 80}$% of known SySts (Mikołajewska, 2003, 2007; Mikolajewska, 2010;
Mikołajewska, 2012), should be formed through an episode of dynamically stable
non-conservative mass transfer. Their WD and red giant masses are mostly in
the range ${\sim 0.4-0.6}$ M⊙ and ${\sim 1.2-2.2}$ M⊙ (Mikołajewska, 2003),
which does not seem difficult to obtain with dynamically stable non-
conservative mass transfer, especially when taking into account a potential WD
mass growth during SySt evolution. Following the episode of mass transfer, the
WD companion will eventually evolve beyond the MS and become an evolved red
giant, either close to the tip of the FGB, or on the AGB. In most cases, the
mass transfer from the red giant to the WD is enhanced due to gravitationally
focused wind accretion (Mohamed & Podsiadlowski, 2007, 2012; Abate et al.,
2013; de Val-Borro et al., 2009; Skopal & Cariková, 2015; de Val-Borro et al.,
2017). If the accretion rate onto the WD is sufficiently high, the binary will
become a SySt.

Despite the general agreement between predictions and observed systems, there
are three S-type SySts that have very different WD and red giant masses, and
their red giants are nearly filling their Roche lobe, namely T CrB, (Stanishev
et al., 2004), RS Oph (Brandi et al., 2009; Mikołajewska et al., 2017), and
V3890 Sgr (Mikołajewska et al., 2021). In these three systems, the compact
objects are massive carbon–oxygen WDs with masses in the range ${\sim
1.2-1.4}$ M⊙, and their Roche lobe filling red giants have relatively low
masses (${\sim 0.7-1.2}$ M⊙). These characteristics exclude that these systems
formed through stable mass transfer as the mass of the donor star (i.e. WD
progenitor) in the progenitor system was far more massive than the accretor
(i.e. red giant progenitor). It is therefore very likely that these three
SySts have formed through CE evolution (Liu et al., 2019), and their WD/red
giant masses have substantially increased/decreased during their evolution.
Why CE evolution resulted in exceptionally long orbital periods in these cases
(${\sim 200-600}$ days) remains unclear. However, if additional energy (e.g.
recombination energy) is efficiently contributing to expelling the envelope,
such long post-CE periods seem possible (Rebassa-Mansergas et al., 2012;
Zorotovic et al., 2014b). Following the CE evolution, the WD companion becomes
an evolved red giant, its extended atmosphere eventually fills its Roche lobe,
and mass transfer proceeds similarly to what is illustrated in Figure 6. We
will discuss in more detail the secular evolution of the different types of
SySts later.

Even though dynamically stable non-conservative mass transfer is a very good
candidate to explain the orbital periods of a large fraction of systems with
orbital periods ranging from ${\sim 100}$ to ${\sim 2000}$ days, the observed
eccentricities remain to be explained. In order to have orbital periods
consistent with observations at the end of mass transfer, the initial orbital
periods cannot be too short, nor too long. This would imply then that, even
for an initially eccentric orbit, tidal interaction should be strong enough to
circularize the orbit before the onset of mass transfer. However, it is clear
from Figure 5 that many systems have eccentric orbits at the end of mass
transfer. This leads then to three possibilities, either we still do not fully
understand the tidal interaction phenomenon, specially the circularization
time-scales, or there is an eccentricity pumping mechanism operating during
mass transfer, or both effects are acting together.

Concerning the first possibility, i.e. tidal forces, recent observational
efforts on ellipsoidal red giant binaries in the Large Magellanic Cloud
suggest that tidal interaction should be much weaker than expected from
theoretical considerations (Nie et al., 2017). Among the 81 binaries that were
investigated, $20$% have eccentric orbits, which should not exist according to
our understanding on tidal dissipation in convective stars. In order to
explain such systems, the circularization efficiency needs to be two orders of
magnitude smaller than predicted.

Regarding the possibility that the large eccentricities are generated by some
kind of eccentricity pumping, we note that there are currently several
mechanisms proposed that could at least help to explain the non-circular
orbits of the post-stable-mass-transfer binaries. First, the eccentricity can
increase if the mass-loss rate from the donor is higher near periastron than
near apastron. The required enhanced mass loss occurs during periastron
passages can be caused by either wind accretion or Roche lobe overflow or even
a combination of both, in which there is a smooth transition between wind
accretion and Roche lobe filling (Bonačić Marinović et al., 2008; Soker, 2000;
Vos et al., 2015).

Second, in the circumbinary disk scenario, eccentricity pumping occurs due to
the resonances in the interaction of the binary with a circumbinary disk
(Dermine et al., 2013; Vos et al., 2015). Since circumbinary disks are most
likely created only from mass lost from the outer Lagrangian point, the
eccentricity pumping due to circumbinary disks should actually work together
with the phase-dependent Roche lobe overflow scenario, by enhancing even
further the eccentricity. A third possibility is that the proto WD receives a
natal kick after the mass transfer episode, similarly to natal kicks in the
formation of NSs, although not that strong.

The final possibility is the triple scenario, in which dynamical and stellar
evolution with a tertiary companion would be responsible for creating
eccentric binaries hosting compact objects, via chaotic orbital evolution of
the stars, which can trigger close encounters, collisions, and exchanges
between the stellar components (e.g. Perets & Kratter, 2012). Even though all
these scenarios are promising and provide reasonable results for particular
systems, they all fail to some extent to explain the overall properties of the
systems, and further investigations are needed (e.g. Vos et al., 2015;
Rafikov, 2016; Oomen et al., 2020; Escorza et al., 2020).

There is an important question to be answered regarding the picture we
described above. If the accretion efficiency during dynamically stable non-
conservative mass transfer is high, what happens with the accretor spin? This
issue has been addressed in binary models (Matrozis et al., 2017), which
suggest that only ${\sim 0.05}$ M⊙ of accreted material is usually enough to
drive the accretor to critical rotation. This is a major problem to explain
barium stars and s-process-polluted carbon-enhanced metal-poor stars unless
there exists a mechanism allowing the accretor to transfer its spin angular
momentum back to the orbit with the help of tidal torques from the donor
(Matrozis et al., 2017). However, it is not yet clear what kind of mechanism
could do this job.

#### High-Mass Stars

After discussing dynamically stable non-conservative mass transfer among
low-/intermediate-mass stars, we now turn to the implication of this mass
transfer mode for the formation of HMXBs. We start by discussing the eclipsing
sg-HMXB M33 X–7, which hosts a rapidly spinning, $15.65$ M⊙ BH orbiting a
close to Roche lobe filling underluminous, ${\sim 70}$ M⊙ O-type supergiant in
a slightly eccentric orbit with a short orbital period of $3.45$ days (Orosz
et al., 2007). The existence of this HMXB can neither be explained by CE
evolution nor rotational mixing, but dynamically stable non-conservative mass
transfer offers a solution (Valsecchi et al., 2010). Starting with a zero-age
MS–MS binary with stellar masses of ${\sim 90}$ M⊙ and ${\sim 30}$ M⊙and an
orbital period of ${\sim 3}$ days, the more massive MS star fills its Roche
lobe and dynamically stable mass transfer occurs. A fraction of the mass
leaving the donor escapes the binary as strong isotropic winds and part is
effectively accreted by the accretor. After mass transfer finishes, due to the
strong winds when the initially more massive star becomes a Wolf–Rayet star,
the binary hosts a Wolf–Rayet star synchronized with the orbital motion, due
to tidal interaction, and a massive O-type MS star. The Wolf–Rayet star with
an iron–nickel core later collapses to a BH, leading to rapidly spinning BH in
a slightly eccentric short-period orbit with a high-mass O-type MS star. When
the O-type MS star becomes an O-type supergiant almost filling its Roche lobe,
the binary becomes a BH-HMXB and wind accretion explains its X-ray luminosity.
This formation picture is also consistent with other sg-HMXBs hosting rapidly
rotating BHs and O-type supergiants almost filling their Roche lobes, such as
Cyg X–1 (Miller-Jones et al., 2021) and LMC X–1 (Orosz et al., 2009), in which
dynamically stable non-conservative mass transfer can not only explain their
masses and orbital periods, but also their BH spins (Qin et al., 2019).

Even though the initial orbital periods in such models are very short (only a
few days), they are neither odd nor rare, when compared with observations. The
orbital period distribution of binaries hosting O-type MS stars, the so-called
Sana distribution (Sana et al., 2012), although extending until ${\sim
10^{6}}$ days, is dominated by systems with orbital periods shorter than
${\sim 10}$ days. According to the described formation channel, the orbital
period does not significantly increase after the onset of mass transfer, which
explain why sg-HMXBs can be formed through dynamically stable non-conservative
mass transfer and still have current orbital periods shorter than ${\sim 10}$
days.

In most known HMXBs, a NS accretes from the circumstellar decretion disk of a
rapidly rotating Be star (e.g. Reig, 2011), and their orbital periods are
predominantly in range from ${\sim 10}$ to ${\sim 400}$ days (e.g. Karino,
2021). According to the wind compression disk model (Bjorkman & Cassinelli,
1993), an equatorial disk is formed around a rapidly rotating star due to ram
pressure confinement by the stellar wind. The disk is formed because the
supersonic wind leaving the star surface at high latitudes travels along paths
that move it down to the equatorial plane, where the material passes through a
standing oblique shock on the top of the disk. The ram pressure of the polar
wind thus confines and compresses the disk. The emission lines typically
observed in an active Be star are attributed to this equatorial disk (Porter &
Rivinius, 2003). As mass from this outflowing disk is transferred to the
compact object via the accretion gate mechanism (Ziolkowski, 2002),
preferentially during periastron passages or enhanced activity of the Be star,
the system becomes bright in X-rays and observable as a Be-HMXB.

The typical age of a Be-HMXB is ${\sim 40-50}$ Myr (e.g. Williams et al.,
2013; Garofali et al., 2018), which is believed to be a direct consequence of
the NS formation and B star mass-loss activity time-scales. Given their
properties, specially their relatively long orbital periods and their rapidly
rotating Be stars, they have most likely formed through an episode of stable
non-conservative mass transfer (e.g. Rappaport & van den Heuvel, 1982; Tauris
et al., 2017), during which the B star spins up. Despite the fact that there
is only one known Be-HMXB in which the compact object is a BH (MWC 656,
Casares et al., 2014), it seems more likely that Be-HMXBs hosting BHs are
formed during CE evolution (Grudzinska et al., 2015), although given the
uncertainties in the stability criterion (Olejak et al., 2021; Belczynski et
al., 2021), dynamically stable non-conservative mass transfer is not
necessarily ruled out. The main problem with having Be-HMXBs hosting BHs
formed through CE evolution is that the existence of the Be star cannot be
easily explained, since a mechanism different from Roche lobe overflow would
be needed to spin up the B star. Gravitationally focused wind accretion prior
CE evolution seems to be a good candidate for that (Grudzinska et al., 2015;
El Mellah et al., 2019a).

In the Corbet’s diagram (Corbet, 1986), i.e. the orbital period versus NS spin
period diagram, Be-HMXBs exhibit a clear correlation, which is explained in
terms of the equilibrium period, defined as the period at which the rotation
velocity at the magnetospheric radius equals the Keplerian velocity (e.g.
Waters & van Kerkwijk, 1989). If the NS spin period is shorter than the
equilibrium period, then matter is efficiently ejected from the vicinity of
the NS due to the propeller mechanism (Illarionov & Sunyaev, 1975), and the NS
spin decreases. On the other hand, if the NS spin period is longer than the
equilibrium period, then mass and angular momentum is transferred onto the NS
thereby increasing its rotational velocity. The correlation comes from the
fact that the equilibrium period depends on the accretion rate, and in turn,
on the orbital period. In addition, Be-HMXBs can be separated into two groups
in the orbital period versus eccentricity diagram, one being those in low-
eccentricity orbits (${\lesssim 0.2}$) and the other one composed of high-
eccentricity orbits (${\gtrsim 0.2}$) (Pfahl et al., 2002; Townsend et al.,
2011). This could then indicate two distinct formation channels, with (core-
collapse supernova) and without (electron-capture supernova) strong natal
kicks during the helium star collapse into the NS.

###  5.3 Combination of Dynamically Stable Non-conservative Mass Transfer and
Common-Envelope Evolution

The orbits of AM CVns and UCXBs are so close that two episodes of mass
transfer must occur to explain their orbital periods, at least one of them
being CE evolution. Similarly, WR-HMXBs, which are compact objects accreting
part of the winds from massive helium stars, are also expected to be formed
through two episodes of mass transfer, given their short orbital periods.
However, unlike the previous case, these systems can be formed in two episodes
of dynamically stable non-conservative mass transfer. There are then three
possible combinations for AM CVns and UCXBs: (i) double CE evolution, (ii) CE
evolution followed by dynamically stable non-conservative mass transfer, and
(iii) dynamically stable non-conservative mass transfer followed by CE
evolution; and there is another possible combination for WR-HMXBs, in addition
to stable non-conservative mass transfer followed by CE evolution, which is
(iv) double dynamically stable non-conservative mass transfer. In what
follows, we will discuss in more detail these formation channels, based on
results from binary models.

#### Evolution Through Two Episodes of Common-Envelope Evolution

Starting with a zero-age MS–MS binary, if the initial orbital period is not
too short nor too long, the initially more massive star will fill its Roche
lobe after evolving beyond the MS. If the initially more massive star is much
more massive than its companion, then mass transfer will be dynamically
unstable and CE evolution commences. The result of such an interaction will be
a helium star or a WD, depending on the mass and core mass of the donor at the
onset of CE evolution, orbiting the initially less massive MS star with an
orbital period orders of magnitude shorter than initially. In case of a helium
star, depending on its mass and composition, it will quickly collapse into a
WD, NS or BH, and in the latter cases, this collapse will produce a core-
collapse or electron-capture supernova.

If the orbital period of the post-CE binary is not too short, the initially
less massive MS will have room to grow and evolve off the MS before filling
its Roche lobe. If the compact object is much less massive than its companion,
then another CE will be triggered and the orbital period is shortened even
further. In case the companion of the compact object is initially a
low-/intermediate-mass MS star and if it fills its Roche lobe on the FGB, then
it will become a helium WD after CE evolution, if its mass is initially
smaller than ${\sim 2}$ M⊙, or a helium star if its mass is initially larger
than that. On the other hand, if the low or intermediate-mass star fills its
Roche lobe on the AGB, then the compact object companion will likely become a
carbon–oxygen WD. After the second CE evolution, the post-CE binary will be
brought into contact due to emission of gravitational waves, becoming in turn
an AM CVn or an UCXB, depending on the nature of the compact object, accreting
from either a helium/carbon–oxygen WD or a helium star, or even a hybrid WD,
in case the helium star becomes a WD before filling its Roche lobe.

This is a standard formation channel for AM CVns and UCXBs (e.g. Solheim,
2010) in which the accreting compact object is formed first during the binary
evolution. However, even if this scenario ends up having a non-negligible
contribution to these populations, it cannot account for the existence of some
particular types of binaries related to AM CVns and UCXBs, such as close
detached double-helium WDs in which the younger WD is more massive than the
older WD (e.g. Nelemans et al., 2000), or millisecond pulsars in close
detached binaries with helium WDs (e.g. Tauris et al., 2011). Therefore,
alternative scenarios involving dynamically stable non-conservative mass
transfer are needed.

#### Dynamically Stable Non-conservative Mass Transfer Followed by Common-
Envelope Evolution

Starting from a zero-age MS–MS binary, if the orbital period is sufficiently
short, the initially more massive star will fill its Roche lobe on the
Hertzsprung gap, or when it is an unevolved giant. If the initial masses of
the stars are comparable, or at least the initially more massive star is not
much more massive than its companion, then mass transfer will be dynamically
stable. The orbital period in this case may increase, as illustrated in Figure
6, leading then to a binary with an orbital period longer than the orbital
period the zero-age binary had initially, hosting a (proto) WD and a companion
that is more massive than the zero-age MS stars. This companion can still be
on the MS or already slightly evolved, since it could have started its
evolution off the MS, in case the initial masses are comparable. The companion
of the (proto) WD then keeps evolving until it fills its Roche lobe, leading
to dynamically unstable mass transfer, since it is much more massive than the
WD. The post-CE binary will have an orbital period orders of magnitude shorter
than at the onset of CE evolution, hosting a WD paired with a WD or helium
star, which will later collapse into a WD, NS or BH, depending on its mass. If
the post-CE binary orbital period is sufficiently short, the (proto) WD will
fill its Roche lobe, giving rise to an AM CVn or UCXBs, depending on the
nature of the accreting compact object.

Unlike the double CE evolution scenario, in the scenario in which dynamically
stable non-conservative mass transfer is followed by CE evolution, the
accreting compact object is formed second. This scenario has been invoked to
explain the existence of double-helium WDs in which the younger WD is more
massive than the older WD (Woods et al., 2012), which is a natural consequence
of this scenario. Most importantly, this scenario can solve the energy budget
problem we mentioned earlier (e.g. Webbink, 2008; Ivanova et al., 2013),
suggesting that the importance of dynamically stable non-conservative mass
transfer for binary evolution has been underestimated in several contexts.
However, the existence of millisecond pulsars in close detached binaries with
helium WDs cannot be explained within this formation channel, which means that
another channel most likely plays a key role in formation of UCXBs.

Regarding WR-HMXBs, they are believed to be the direct progenitors of merging
binary BHs, which means that these merger events can also be used to further
constrain WR-HMXBs (e.g. Kalogera et al., 2007). We have already discussed
that the properties of sg-HMXBs hosting BHs can be reasonably well explained
by an episode of dynamically stable non-conservative mass transfer. This is
also the most common outcome for the first episode of mass transfer in merging
BH binary models (e.g. Belczynski et al., 2016; Belczynski et al., 2020). In
these binaries, the BH is accreting from the winds of an O-type supergiant
that is almost filling its Roche lobe. When it effectively fills its Roche
lobe, depending on the binary properties, mass transfer can be dynamically
unstable or not. A CE evolution is the most common scenario for the formation
of merging binary BHs, specially those mergers with massive stellar-mass BHs,
such as GW190521 (e.g. Belczynski, 2020).

The sg-HMXB M33 X–7 will apparently undergo CE evolution, given that its
O-type supergiant is much more massive than its BH companion (Orosz et al.,
2007). On the other hand, in the sg-HMXB Cyg X–1, the O-type supergiant is
more massive than its BH companion only by a factor of two (Miller-Jones et
al., 2021), which implies that mass transfer will be most likely dynamically
stable when the O-type supergiant fills its Roche lobe. Finally, the mass of
the O-type supergiant in sg-HMXB LMC X–1 is around three times its BH
companion mass (Orosz et al., 2009), and it is much less clear whether mass
transfer could be dynamically stable or unstable in this case. Now regarding
the current population of confirmed WR-HMXBs, there are two systems that have
very short orbital periods, namely Cyg X–3, in the Milky Way ($4.8$ h,
Zdziarski et al., 2013), and CG X–1, in the Circinus galaxy ($7.8$ h, Qiu et
al., 2019), which mostly implies that the BHs orbit within the Wolf–Rayet
photospheres. These extremely short orbital periods suggest that there was a
strong orbital shrinkage, mostly likely only possible during CE evolution. For
the remaining confirmed WR-HMXBs, the orbital periods are longer than a day
and these systems could be explained by a weak spiral-in during dynamically
stable non-conservative mass transfer (van den Heuvel et al., 2017), as we
will discuss later.

#### Common-Envelope Evolution Followed by Dynamically Stable Non-conservative
Mass Transfer

The situation here is initially similar to the double CE evolution scenario,
i.e. from a zero-age MS–MS binary, the initially more massive star fills its
Roche lobe when it evolves off the MS, leading to dynamically unstable mass
transfer, because it is much more massive than its companion. However, unlike
in the double CE evolution scenario, the post-CE binary orbital period is
shorter, and the compact object companion is not allowed to significantly grow
before filling its Roche lobe. Therefore, in this case, the zero-age MS–MS
binary orbital period must be shorter than that in double CE scenario.

When the companion of the compact object fills its Roche lobe, depending on
its mass and the mass of the compact object, mass transfer can be either
dynamically unstable or dynamically stable. If the donor is much more massive
than the compact object, mass transfer will be dynamically unstable, which
implies that the compact object and its companion will most likely merge on
the dynamical time-scale of the donor. In case the donor is an MS star, this
is the natural outcome of dynamically unstable mass transfer. On the other
hand, if the donor is a Hertzsprung gap star, or an unevolved FGB star, then a
merger is also expected, since there would not be enough orbital energy
available to eject the CE, due to the short orbital period of the binary.

If the donor is a low-mass star, then mass transfer will most likely be
dynamically stable and non-conservative. The beginning of the evolution can be
thermally stable or not, depending on the masses of the accreting compact
object and the donor, and on the properties of the donor. In any event, the
binary will become a CV or a LMXB, depending on the nature of the compact
object. If the donor is an M-/K-type MS star, i.e. an MS star with mass
${\lesssim 0.8}$ M⊙, then it will eventually become a hydrogen-rich degenerate
star, i.e. a brown dwarf, when enough mass is lost. However, if the donor is
an evolved MS star of an earlier type or a Hertzsprung gap star, then it has
already or will develop a growing degenerate helium core. In these cases, when
the donor hydrogen envelope is entirely consumed, it becomes a semi-degenerate
helium-rich star. At this moment, the CV/LMXB becomes an AM CVn/UCXB.

The donor transition from a non-generate to a semi-degenerate configuration
may be preceded by a short detached phase, depending on the post-CE binary
properties (e.g. Chen et al., 2021; Soethe & Kepler, 2021). Alternatively, if
the post-CE binary properties are such that this detached phase happens at
longer orbital periods (${\gtrsim 1}$ day), then emission of gravitational
waves will not be strong enough to bring the binary close enough so that the
compact object companion could fill again its Roche lobe within a Hubble time.
The existence of millisecond pulsars in close detached binaries is essentially
explained in such a way, since the NS becomes a millisecond pulsar during the
LMXB evolution (the so-called recycling scenario, e.g. Bhattacharya & van den
Heuvel, 1991; Di Salvo & Sanna, 2020), and the close detached population are
those systems observed in the transition from LMXB to UCXB. Those systems that
directly become UCXBs, or effectively manage to become UCXBs after the
detached phase, are then UCXBs hosting accreting millisecond pulsars.

#### Evolution Through Two Episodes of Dynamically Stable Non-conservative
Mass Transfer

We have argued before that WR-HMXBs with extremely short orbital periods can
most likely only be explained with CE evolution. However, the remaining
confirmed WR-HMXBs have orbital periods longer than ${\sim 1}$ day, namely X–1
in the IC 10 galaxy ($34.9$ h, Silverman & Filippenko, 2008), X–1 in the NGC
300 galaxy ($32.8$ h, Crowther et al., 2010), and ULX–1 in the galaxy M101
($196.8$ h, Liu et al., 2013). Despite their short orbital periods, they could
have formed through dynamically stable non-conservative mass transfer, since
it is possible in such cases to avoid CE evolution during the formation of the
Wolf–Rayet stars, due to the high masses of the Wolf–Rayet progenitors.

A slow spiral-in during dynamically stable non-conservative mass transfer is
possible if the Wolf–Rayet progenitor has a radiative envelope and if it is at
most ${\sim 3-4}$ times the BH mass (van den Heuvel et al., 2017; Neijssel et
al., 2019), although the exact critical mass ratio depends on the several
model assumptions (e.g. Olejak et al., 2021; Belczynski et al., 2021). This
seems to be the fate of several WR-HMXB progenitors. Based on an analysis of
the small sample of Wolf–Rayet $+$ O-type MS binaries, with orbital periods
clustering at around one week (van der Hucht, 2001), it has been recently
suggested that, after the Wolf–Rayet star becomes a BH and its O-type
companion fills its Roche lobe, they are expected to become WR-HMXBs with
orbital periods as short as ${\sim 1}$ day (van den Heuvel et al., 2017).
These WR-HMXBs will then evolve to binary BHs, when the Wolf–Rayet donors
collapse to BHs. This then suggests that not only the formation of WR-HMXBs
can avoid CE, but also the formation of merging BHs (Olejak et al., 2021;
Belczynski et al., 2021).

#### Further Considerations on the Formation of Ultra-Compact X-ray Binaries

The first three scenarios highlighted above correspond to the most traditional
scenarios for the formation of AM CVns and UCXBs. Regarding UCXBs, there is an
additional channel, via accretion-induced collapse of an oxygen–neon–magnesium
WD (e.g. Belczynski & Taam, 2004). If such a WD triggers electron-capture
reactions at its centre upon reaching the Chandrasekhar mass limit, then the
burning that propagates outward makes central temperatures and pressures high
enough to cause a collapse of the WD into a NS (e.g. Nomoto & Kondo, 1991).
Binary models, in which different types of donors have been considered, mainly
MS stars and helium star donors, have shown that this is indeed possible (e.g.
Tauris et al., 2013), leading to the formation of millisecond pulsar binaries.

In the case of MS stars, after the accretion-induced collapse, the binary
temporarily detaches (for $10^{3}-10^{5}$ years), due to the small kicks, but,
due to orbital AML, it soon becomes an LMXB. And the subsequent formation of
the UCXB is similar to that already discussed. The UCXB progenitors in this
case are SSXBs with MS donors with masses of ${\sim 2.0-2.6}$ M⊙, for a
metallicity of 0.02, and of ${\sim 1.4-2.2}$ M⊙, for a metallicity of 0.001.
It is worth mentioning that this formation channel may also help to explain
the existence of millisecond pulsars in eccentric detached binaries (e.g.
Freire & Tauris, 2014). In the case of helium stars, the binary also
temporarily detaches and soon becomes an UCXB, due to orbital AML through
emission of gravitational waves. Unlike the previous case, the helium stars
quickly (${\lesssim 1}$ Myr) become carbon–oxygen WDs with masses of ${\sim
0.6-0.9}$, causing the binary to detach. Further orbital AML could eventually
make the binary semi-detached again, and the UCXB would host a millisecond
pulsar accreting from a carbon–oxygen WD donor. This happens for AM CVns with
helium star donors with masses of ${\sim 1.1-1.5}$ M⊙

Even though possible, the contribution from this channel is most likely small.
This is because of the narrow range in the parameter space of the accreting WD
that allows it to eventually exceed the Chandrasekhar mass limit. In
particular, during the accreting WD evolution, there is a very narrow range of
mass transfer rates such that the WD mass can effectively increase (e.g.
Nomoto et al., 2007; Shen & Bildsten, 2007; Wolf et al., 2013). In addition,
given that a typical mass of an oxygen–neon–magnesium WD is ${\sim 1.2}$ M⊙,
the donor cannot be much more massive than that, since otherwise mass transfer
becomes dynamically unstable, which restricts even more the parameter space of
the UCXB progenitors.

###  5.4 Additional Channels Through Dynamical Interactions in High-Density
Environments

In addition to the formation channels highlighted above, dynamical
interactions in crowded regions of star clusters, specially globular clusters,
can create conditions to form accreting compact objects from systems that
would otherwise not evolve into them. A way of addressing whether a type of
X-ray sources is predominantly dynamically formed in globular clusters is by
means of the cluster stellar encounter rate, which depends on the central
density, the core radius and the central velocity dispersion (Verbunt & Hut,
1987; Pooley et al., 2003). If dynamical interactions are supposed to play any
significant role in the formation of X-ray sources, a correlation between the
number of sources of a particular type within a cluster and its stellar
encounter rate could indicate that dynamical interactions played a significant
role in their formation, despite the several potential problems with this
interpretation (Belloni & Rivera, 2021). For a long time, this has been