believed to happen for LMXBs and UCXBs, which correspond to the brightest
X-ray sources found in Milky Way globular clusters, as there is continuously
growing strong evidence for such a correlation (Pooley & Hut, 2006; Ivanova et
al., 2008). There are currently 21 bright X-ray sources in 15 Milky Way
globular clusters, being 8 persistent and 13 transients (van den Berg, 2020),
and they are still consistent with being dynamically formed, especially
considering their mass density, which is much higher than that of systems in
the Milky Way disk.

Given the problem with reproducing the properties of BH-LMXBs in binary models
(Wiktorowicz et al., 2014) and the above-mentioned over-abundance of LMXBs in
the Milky Way globular cluster population in comparison with the Milky Way
disk population, a natural solution to the problem could be a dynamical origin
for the field population (i.e. those LMXBs not belonging to stellar
aggregates). Recent Monte Carlo numerical simulations of evolving globular
clusters (Kremer et al., 2018) show that progenitors of LMXBs are ejected from
their host clusters over a wide range of ejection times, contributing to
${\sim 300}$ NS-LMXBs and ${\sim 180}$ BH-LMXBs to the Milky Way field
population, not all of them being indeed observable due to observational
selection effects. More than 20 galaxies within ${\sim 25}$ Mpc have been
deeply investigated with Chandra and Hubble Space Telescope (Lehmer et al.,
2020). These observations have revealed that, in order to properly describe
the X-ray luminosity function of the field population of LMXB, a component
that scales with the globular cluster specific frequency is needed. In
addition, it has a shape consistent with that found for the globular cluster
population of LMXBs. This then indicates that most, if not all, LMXBs in the
fields of these galaxies are formed similarly to those in globular clusters
and in turn could have origin in globular clusters. These binaries could be
ejected from the globular clusters due to energetic dynamical interaction
leading to their formation, or the cluster itself could dissolve due to strong
tidal interaction. However, for galaxies with low globular cluster specific
frequencies, the X-ray luminosity function has a component that scales with
stellar mass, suggesting then that a substantial population of LMXBs in these
galaxies are formed through binary evolution, without invoking dynamics.

These findings may alleviate quite a bit the problems of reproducing the
proprieties of BH-LMXBs in the Milky Way field. However, it is not clear
whether enough systems are predicted to form through this channel, nor whether
their positions in the Milky Way as well as their kinematic properties, nor
their numbers could be explained as ejected from globular clusters, or from
dissolving star clusters. For instance, while the observed systems are almost
entirely concentrated in the Milky Way disk (Corral-Santana et al., 2016),
those predicted to form in globular clusters and latter ejected to the field
are spherically distributed around the Milky Way disk, reflecting the observed
distribution of globular clusters in the Milky Way. In addition, the number of
systems predicted by this channel seems too low to account for the large
number of BH-LMXBs inferred from observations (${\sim 10^{4}-10^{8}}$,
Tetarenko et al., 2016). Therefore, the current theoretical and observational
evidence indicates that a dynamical origin in star cluster is unlikely to be
the dominant formation channel for these systems. A more promising alternative
to bring into agreement predicted and observed properties seems to be the
standard CE evolution, coupled with a revision of the strength of orbital AML
during their evolution, as we will see later.

Regarding the fainter X-ray sources such as CVs, the situation is much more
difficult to be addressed, due to the several biases and selection effects
(Belloni & Rivera, 2021), e.g. the photometric incompleteness in the more
crowded central parts of the clusters is most likely playing a key role in the
characterization of these sources (e.g. Cohn et al., 2021). Therefore, any
conclusions drawn from a comparison between the numerical simulations and
observations of faint X-ray sources should be taken with a grain of salt.
Despite that, it is still possible to infer the impact of dynamics on this
population, if observational selection effects are somehow addressed. Recent
numerical simulations suggest that fewer CVs should be expected in dense
globular clusters relative to the Milky Way field, due to the fact that
dynamical destruction of CV progenitors is more important in globular clusters
than dynamical formation of CVs (Belloni et al., 2019). This is consistent
with observations (Cheng et al., 2018), because the faint X-ray populations,
primarily composed of CVs and chromospherically active binaries, are under-
abundant in globular clusters with respect to the Solar neighbourhood and
Local Group dwarf elliptical galaxies. Regarding the dynamical formation of
CVs in globular clusters, these simulations show that the detectable CV
population is predominantly composed of CVs formed via standard CE evolution
(${\gtrsim 70}$%). This happens because the main dynamical scenarios have a
very low probability of occurring, which resulted in (or very weak, if at all)
correlation between the predicted number of detectable CVs and the predicted
globular cluster stellar encounter rate. Despite the fact that these results
are consistent with recent observations (Cheng et al., 2018), we shall mention
that, even though it is not expected to be dominant, dynamics should play a
sufficiently important role to explain the relatively larger fraction of
bright CVs (i.e. those close to the MS in the colour-magnitude diagram)
observed in denser clusters.

Regarding SySts and SyXBs, despite a few reasonable candidates have been
proposed over the years, none so far has been confirmed in the entire Milky
Way globular cluster population. This issue has been partially addressed in
Monte Carlo numerical simulations of evolving globular clusters (Belloni et
al., 2020b), in which SySts with WDs formed without Roche lobe overflow have
been investigated. These simulations show that their orbital periods are
typically sufficiently long (${\gtrsim 10^{3}}$ days) so that dynamical
disruption of their progenitors is virtually unavoidable in very dense
clusters. On the other hand, in less dense clusters, some of these SySts are
still predicted to exist, although their identification should be rather
difficult in observational campaigns, mainly due to their locations within the
clusters and rareness. Despite these simulations aimed at explaining the lack
of SySts in globular clusters, they do not necessarily correspond to definite
answers to the issue, since most known SySts have orbital period between
${\sim 200}$ and ${\sim 10^{3}}$ days, which means that most SySts were not
addressed in the simulations. Further simulations are still missing, in which
the formation of SySts through dynamically stable non-conservative mass
transfer is consistently taken into account and the impact of dynamics on
their progenitors is evaluated.

So far, we have discussed the population of accreting compact objects in old
star clusters. These star clusters cannot harbour HMXBs, since at their
present-day ages, the entire population of HMXBs have already become single
NSs/BHs or binaries hosting either NSs and/or BHs. However, in very young
starburst galaxies (only a few Myr old), HMXBs are expected to be the dominant
source of X-rays. Given that high-mass stars, and potentially all stars
(Kroupa, 1995; Lada & Lada, 2003), are most likely formed in binaries (e.g.
Sana et al., 2012) in embedded clusters (e.g. Marks & Kroupa, 2012), it is not
surprising that observations suggest that the HMXBs in starburst galaxies are
consistent with being formed in star clusters (e.g. Kaaret et al., 2004). Even
though young star clusters are most likely HMXB factories, these HMXBs are not
necessarily formed dynamically. The analysis of the X-ray sources in the
merging Antennae galaxies and in the dwarf starburst NGC 4449 suggests that
high stellar density and in turn dynamical interactions are not a primary
driver of HMXB formation (Johns Mulia et al., 2019), which is consistent with
star cluster direct $N$-body simulations (Garofali et al., 2012). Therefore,
despite being rather frequent in starburst galaxies and likely formed in
compact star clusters, there is evidence supporting that HMXBs form directly
from primordial high-mass binaries, without any significant influence of
dynamics.

##  6 Secular Evolution

The mass transfer in accreting compact objects is dynamically stable and can
be separated into three main modes: (i) Roche lobe overflow (if ${R_{\rm
d}\sim R_{\rm RL}}$), (ii) atmospheric Roche lobe overflow (if ${0.9\,R_{\rm
RL}\lesssim R_{\rm d}\lesssim\,R_{\rm RL}}$), and (iii) wind accretion (if
${R_{\rm d}\lesssim 0.9\,R_{\rm RL}}$). In CVs, LMXBs, AM CVns, and UCXBs, the
donor is filling its Roche lobe. Therefore, mass transfer in these systems can
be driven by orbital AML or by nuclear expansion of the donor, with or without
an initial phase of thermally unstable mass transfer. On the other hand,
despite the fact that in most SySts, SyXBs, sg-HMXBs, and WR-HMXBs, the donor
is under-filling its Roche lobe, mass transfer still occurs, mainly when the
extended atmosphere of the donor fills its Roche lobe, or due to accretion of
a significant part of the winds from the donor.

Among the observational properties, the donor mass and the mass transfer rate
deserve special attention, as these two quantities are somewhat tied to the
secular evolution of accreting compact objects. We show these properties of
accreting compact objects in Figures 7 and 8, against their orbital periods,
together with evolutionary tracks computed by us with the mesa code and other
tracks from previous binary models. In what follows, we will describe the
dynamically stable evolution of accreting compact objects as well as the up-
to-date observational information that can be used to constrain evolutionary
models, which includes their orbital periods, their mass transfer rates, and
their donor and compact object masses, but without being exclusively limited
to them.

Figure 7: Comparison between observed and predicted properties of accreting
compact objects in the plane donor mass ($M_{\rm d}$) versus orbital period
($P_{\rm orb}$). The observational data correspond to BH-LMXBs (Wiktorowicz et
al., 2014; Shao & Li, 2020), BH-HMXBs (Crowther et al., 2010; Orosz et al.,
2009, 2007; Casares et al., 2014; Zdziarski et al., 2013; Silverman &
Filippenko, 2008; Liu et al., 2013; Miller-Jones et al., 2021), NS-LMXBs
(Bassa et al., 2009; Van et al., 2019), UCXBs (Van et al., 2019), eclipsing
sg-HMXBs (Clark et al., 2002; Quaintrell et al., 2003; van der Meer et al.,
2007; Mason et al., 2010, 2011; Mason et al., 2012; Bhalerao et al., 2012;
Falanga et al., 2015; Pearlman et al., 2019), eclipsing AM CVns (Copperwheat
et al., 2011; Green et al., 2018; van Roestel et al., 2021), eclipsing CVs
(McAllister et al., 2019), S-type SySts (most of which are eclipsing,
Mikołajewska, 2003; Stanishev et al., 2004; Brandi et al., 2009; Mikołajewska
et al., 2021; Hinkle et al., 2009), SyXBs (Hinkle et al., 2006, 2019), and
SSXBs (Kalomeni et al., 2016). In particular, we labeled the only known Be-
HMXB hosting a BH (MWC 656, Casares et al., 2014), a WR-HMXB with an extremely
short orbital period (Cyg X–3, Zdziarski et al., 2013), and the two SySts T
CrB (Stanishev et al., 2004) and RS Oph (Brandi et al., 2009), which are among
the few SySts in which the red giant is nearly filling its Roche lobe, and in
addition, given their high carbon–oxygen WD masses, they will most likely
evolve towards Type Ia supernovae (Liu et al., 2019). The theoretical
evolutionary tracks correspond to the lines, of which all coloured tracks were
computed by us using the mesa code, adopting solar metallicity, assuming the
RVJ prescription (Rappaport et al., 1983) (Equation 19, with $\gamma_{\rm
MB}=3$), for the MB, and considering tidal interaction (Hurley et al., 2002)
and stellar wind mass loss, with the Reimers parameter equals to 0.5 for FGB
stars and the Blöcker parameters equals to 0.02 for AGB stars. In all these
coloured tracks, we chose several different initial orbital periods and
initial masses for the point-mass compact object and its zero-age MS star
companion. The black tracks correspond to previous binary evolution models, in
which the donor is initially a helium WD (Wong & Bildsten, 2021), a helium
star (Heinke et al., 2013; Wang et al., 2021), or a MS star (Kalomeni et al.,
2016; Chen et al., 2021). Given that the evolutionary tracks pass through most
observational points, we could naively conclude that our current knowledge on
how accreting compact objects evolve is not bad. However, major problems arise
when we take into account the entire populations of different types of
accreting compact objects, and confront results from population synthesis with
observations. These problems are most likely connected with our poor
understanding of the mechanisms driving the evolution of accreting compact
objects. Despite that, there has been important progress in the last couple of
years, which can definitely guide us towards a solution to these problems.
Figure 8: Comparison between observed and predicted properties of accreting
compact objects in the plane mass transfer rate ($\dot{M}_{\rm d}$) versus
orbital period ($P_{\rm orb}$). We only considered in the plot those systems
in which the donor is a Roche lobe filling low-mass star, i.e. BH-LMXBs
(Coriat et al., 2012), NS-LMXBs (Van et al., 2019), UCXBs (Van et al., 2019),
AM CVns (Ramsay et al., 2018), and CVs (Pala et al., 2022). Some of the
coloured evolutionary tracks in Figure 7 are shown, which were computed by us
with the mesa code. The black lines correspond to evolutionary tracks from
previous binary models, in which the donor in initially either a WD (Wong &
Bildsten, 2021), or a helium star (Heinke et al., 2013; Wang et al., 2021), or
a MS star (Goliasch & Nelson, 2015; Chen et al., 2021), the latter having an
initial phase of thermally unstable mass transfer. We also included in the
figure, as grey thick lines, the critical mass transfer rates (Coriat et al.,
2012), above which the accretion disks are stable and the systems are
persistent sources, and transient otherwise, considering irradiated accretion
disks or not. By comparing the observational data with these critical mass
transfer rates, we can see that the disk instability model reasonably well
explains persistent and transient sources, except perhaps in the case of Cyg
X–2 (but see Coriat et al., 2012) and some AM CVns. There are two
representative tracks for CVs, the solid red line computed by us and the long-
dashed black line (Goliasch & Nelson, 2015), which were both computed with the
RVJ prescription (Rappaport et al., 1983) (Equation 19, with $\gamma_{\rm
MB}=3$). We can clearly see that this MB prescription is not adequate to
explain the persistent and transient sources with orbital periods longer than
${\sim 3}$ h. In addition, CVs hosting fully convective M-type MS stars, at
orbital periods shorter than ${\sim 2}$ h, exhibit a spread in the mass
transfer rates that cannot be explained by only gravitational wave radiation.
With respect to NS-LMXBs, it is also clear that the RVJ prescription (dashed
red lines) is not able to account for the persistent sources, which has been a
long-standing problem of LMXB evolution. However, when a MB law (Van et al.,
2019) similar to the recently developed CARB prescription (Van & Ivanova,
2019) is adopted (dot-dashed black lines, Chen et al., 2021), mass transfer
rates are great enough so that they can be explained. The persistent UCXBs
with long orbital periods (${\gtrsim 40}$ min) can apparently only be
explained if the donor is initially a helium star, which is consistent with
observations (e.g. Heinke et al., 2013). There are two particular systems, one
AM CVn (SDSS J1908$+$3940) and one CV (SDSS J1538$+$5123), that are very
interesting, since they apparently cannot be explained by any of the tracks we
included in the figure. While it has been proposed that this CV might be
either a newly born system or a post-nova system, it is not clear what might
be happening with this AM CVn.

###  6.1 Cataclysmic Variables and Low-Mass X-ray Binaries

As mentioned earlier, CVs and LMXBs are formed through an episode of CE
evolution. More specifically, after CE evolution, a post-CE binary is formed
in a detached binary hosting a compact object and a non-degenerate star.
Because of orbital AML and/or nuclear evolution of the future donor, the
orbital separation decreases up to the formation of the accreting compact
binary, when the non-degenerate companion starts filling its Roche lobe and
becomes a donor. The donors in CVs and LMXBs are usually MS stars, subgiants
or unevolved red giants.

#### Low-Mass Unevolved M-/K-Type Main-Sequence Star Donors

Let us start with the case in which the donor is initially a low-mass
unevolved MS star, corresponding to post-CE binaries hosting MS stars with
types M and K (${\lesssim 0.8}$ M⊙), which do not significantly evolve within
the Hubble time. This case is illustrated in Figures 7 and 8 by the solid red
track, for which the point-mass compact object and zero-age MS star have both
the same initial mass of $0.8$ M⊙, and the initial orbital period is $12$ h.
In this case, the MS donor has a radiative core, which implies that MB is the
main orbital AML mechanism driving the binary evolution. MB causes mass
transfer rates high enough so that the MS donor is only able to maintain
thermal equilibrium if it is slightly bloated, which means that MS donors are
somewhat oversized in comparison with isolated MS stars (Knigge et al., 2011).
When the MS donor becomes fully convective, MB is expected to cease or, at
least, become much less efficient (disrupted MB scenario), most likely as a
consequence of a rise in the secondary star magnetic complexity (Taam &
Spruit, 1989; Garraffo et al., 2018), rather than the originally proposed
demise of the dynamo (Rappaport et al., 1983; Spruit & Ritter, 1983). From
this point on, the main driving mechanism becomes GR. This leads to a
significant drop in the mass transfer rate and a slowdown of the evolution, as
AML through GR is in general orders of magnitude weaker than MB. Such a drop
in the mass transfer rate allows the MS donor to re-establish thermal
equilibrium at its normal size (i.e. decrease in size), and the system becomes
a detached binary since the companion of the compact object is no longer
filling its Roche lobe. This detachment occurs at the upper edge of the so-
called orbital period gap (a paucity of observed accreting systems in the
orbital period range of ${\sim 2-3}$ h).

Even though the system is now detached, it keeps losing angular momentum due
to GR and continues to evolve towards shorter orbital periods. When the
orbital period is sufficiently short, the MS donor fills its Roche lobe again,
and mass transfer restarts, i.e. the system is again a CV evolving now with
lower mass transfer rates towards shorter orbital periods, which happens at
the lower edge of the orbital period gap. When the donor mass becomes too low
to sustain hydrogen burning, it becomes a hydrogen-rich degenerate object,
i.e. a brown dwarf. From this point on, the donor expands in response to the
mass loss, leading to an increase in the orbital period. The so-called orbital
period minimum is then a natural consequence of the transition from a fully
convective MS star to a brown dwarf.

The edges of the orbital period gap, and consequently its width, as well as
the orbital period minimum depend on the metallicity (Stehle et al., 1997;
Kalomeni et al., 2016), the accreting compact object mass, and the orbital AML
rate (Knigge et al., 2011; Belloni et al., 2020a). For instance, regarding the
impact of the metallicity, the higher the central helium fraction, the shorter
the orbital period minimum, and the orbital period minimum can be as short as
${\sim 10-40}$ min, when the central helium fraction is nearly unity (Kalomeni
et al., 2016). The orbital period minimum is also affected by orbital AML in
the following way. For systems having fully convective donors at a given
orbital period, the stronger/weaker the orbital AML, the higher/lower the mass
transfer rate, and in turn the smaller/greater the donor mass, which allows
the donor to reach the mass limit for hydrogen burning at longer/shorter
orbital periods. Orbital AML is also expected to have a strong impact on the
orbital period gap, due to variations in the MB strength for systems with
donors having convective cores. The stronger/weaker MB, the more/less bloated
the donor is, which means that it becomes fully convective at a lower/higher
mass (see Figure 2), having in turn a smaller/greater radius. This implies
that system will become semi-detached again at a shorter/longer orbital
period, i.e. the lower edge of the orbital period gap is located at
shorter/longer orbital periods. For a sufficiently weak MB , such as the RVJ
prescription with $\gamma_{\rm MB}\geq 5$, the donor bloating is negligible,
especially near fully convective boundary, and the system evolves virtually
without detaching. On the other hand, for a very strong MB, such as that
provided by the CARB prescription or by the Kawaler prescription with
$n\gtrsim 1.5$, the donor becomes fully convective at a mass as small as
${\sim 0.18}$ M⊙, and the orbital period gap can start at an orbital period as
long as ${\sim 7-8}$ h. This happens because, as the donor approaches the
fully convective boundary, it is increasingly driven out of thermal
equilibrium, leading to an increase in its radius by a factor of ${\gtrsim 2}$
in comparison with isolated M-type MS stars with similar masses. Then, as a
response of the donor to mass loss, the orbit expands.

The above outlined evolution successfully describes several features of the
evolution of LMXBs and CVs. However, it is clearly not complete. For instance,
it does not include the impact of the magnetic field of the compact object on
the evolutionary picture for CV and LMXB, which can be significant as we will
explain in what follows. If the compact object magnetic moment, which is
proportional to its magnetic field strength and to the cube of its radius, is
sufficiently large (${\gtrsim 10^{33}}$ G cm3), then a coupling of the donor
magnetic field lines and the compact object magnetic field lines may lead to
an increase of the portion of the donor winds that remains trapped to the
binary (Li et al., 1994; Webbink & Wickramasinghe, 2002). This effect would
make winds to carry away less angular momentum, resulting in reduced orbital
AML via MB and, in turn, diminished mass transfer rates. While NS magnetic
fields seem to considerably decay during LMXB evolution (Zhang & Kojima,
2006), the magnetic field strength in most accreting WDs are expected to
change only negligibly, which is a consequence of mass loss due to nova
eruptions (Zhang et al., 2009). Given the small sizes of NSs (only ${\sim
10-20}$ kilometers) and their relatively weak magnetic fields (${\sim
10^{8}-10^{9}}$ G) due to accretion-induced field decay, their magnetic
moments are usually not strong enough to cause an important impact on LMXB
evolution.

The situation is different for WDs, as they are much bigger (${\sim
5000-10000}$ kilometers), which implies that, if they host sufficiently strong
magnetic fields (${\gtrsim 10^{7}}$ G), the CV evolution is most likely
different from the remaining CVs hosting WDs with weaker magnetic fields
(Belloni et al., 2020a). In case the WD magnetic moment is higher than ${\sim
10^{34}-10^{35}}$ G cm3, MB is completely suppressed and the CV evolution is
driven by only GR, i.e. at a much lower orbital AML rate, without detaching.
Recent binary models provide strong support to this magnetic evolutionary
picture for such CVs, as the predicted mass transfer rates are in agreement
with the observations and the orbital period gap is absent in this population,
which reasonably well agrees with observations (Belloni et al., 2020a). It
even seems that CV evolution in general cannot be understood ignoring the
origin of WD magnetic fields and its impact on orbital AML and exchange
between the stellar components and the orbit. Including magnetic field
generation due to a rotation- and crystallization-driven dynamo (Isern et al.,
2017; Schreiber et al., 2021) and the impact of the generated field on the
evolution of CVs can reproduce several observational facts that remain
otherwise inexplicable, such as the existence of AR Sco, which hosts a radio-
pulsing WD (Marsh et al., 2016), the relative numbers of magnetic and non-
magnetic CVs (Pala et al., 2020), and the absence of strong magnetic WDs among
young detached post-CE binaries (Liebert et al., 2005).

#### Subgiant or A-/F-/G-Type Main-Sequence Star Donors

Let us now consider the case in which the donor is a nuclear-evolved MS star
(i.e. near the end of the MS) or a subgiant at the onset of mass transfer to
the compact object. In other words, we are now looking at post-CE binaries
hosting MS stars of spectral type G or earlier (${\gtrsim 1}$ M⊙). If the
orbital period of the post-CE binary is sufficiently long, the future donor
star will have enough time to evolve on the MS, or even become a subgiant,
before filling its Roche lobe. For stars having convective envelopes, the
onset of mass transfer occurs due to a combination of orbital AML due to MB
and nuclear evolution of the future donor, while for stars having radiative
envelopes, it is triggered only by nuclear evolution of the future donor. The
evolution for these two types of donors is illustrated in Figures 7 and 8 by
the long-dashed and dot-dashed black tracks, which were taken from different
binary models (Goliasch & Nelson, 2015; Kalomeni et al., 2016; Chen et al.,
2021), by the two dashed red and green tracks with shortest initial orbital
periods ($1$ day), and by the two yellow tracks (initial orbital periods set
as $1$ and $3$ days). The difference between the three sets of coloured tracks
is the initial mass of the compact object, which is $0.8$ M⊙ in the red
tracks, $1.2$ M⊙ in the green tracks, $1.4$ M⊙ in the yellow tracks; and the
zero-age MS mass, which is $1.2$ M⊙ for the green and red tracks, and $3.0$ M⊙
in the yellow tracks.

Depending on the mass and structure of the donor as well as on the mass of the
compact object, the mass transfer can be thermally unstable at the very
beginning of the evolution. Thermally unstable mass transfer typically occurs
when the nuclear-evolved MS or subgiant donor is significantly more massive
than the compact object at the onset of Roche lobe overflow, as in the case of
the yellow tracks in Figures 7 and 8. More generally, for a given donor
evolutionary state, the higher the mass ratio at the onset of Roche lobe
overflow, the greater the chances that the mass transfer will be thermally
unstable. The mass transfer rates during this phase are typically larger than
those provided by thermally stable mass transfer and can exceed ${\sim
10^{-7}-10^{-6}}$ M⊙ yr-1, which is high enough to completely drive the donor
out of thermal equilibrium. In case the compact object is a WD, systems
observed at this phase appear typically as SSXBs. The orbital period during
this thermally unstable phase decreases as a consequence of the donor response
to thermal time-scale mass loss. However, when mass transfer stabilizes, the
donor re-establishes thermal equilibrium and the orbital period can further
decrease or reverse and increase.

Let us now discuss the evolution following the onset of thermally stable mass
transfer. For stars with radiative envelopes, orbital AML due to MB does not
take place, and the orbital period should increase as a response to mass
transfer. For stars with convective envelopes, there are two possibilities. In
case the nuclear time-scale of the donor is shorter than its MB-driven mass-
loss time-scale, then the binary evolution will be driven by the donor nuclear
evolution. Mass transfer then proceeds on the nuclear time-scale of the donor,
the orbital period increases, and the evolution is usually called divergent.
On the other hand, in case MB is sufficiently strong, such that the MB-driven
mass-loss time-scale becomes shorter than nuclear time-scale, the evolution
will be driven by orbital AML and the orbital period will decrease. In this
case, the evolution is called convergent, and the system will evolve to either
a CV or LMXB with a hydrogen-rich degenerate donor (in case the MS donor did
not have time to start developing a degenerate helium core) or an AM CVn or
UCXB with helium-rich degenerate donors (in case the donor is initially a
subgiant or in case the MS donor managed to develop a degenerate helium core).

The evolution towards a hydrogen-rich degenerate donor is similar to the case
in which the donor is an unevolved low-mass MS star (compare long-dashed black
and solid red tracks in Figure 8). However, the evolution towards AM CVns and
UCXBs is rather different from that of the previous cases, which can be seen
by comparing the long-dashed and dot-dashed black tracks with the solid red
track in Figure 7, and the dot-dashed black track with the long-dashed black
and solid red tracks in Figure 8. For instance, the orbital period minimum
becomes much shorter for donors that were more evolved at the onset of Roche
lobe overflow. This happens because the donors have helium cores that are
partially degenerate, with masses that are too low to trigger helium burning.
This allows their radii to considerably shrink as the binary evolves, so that
the binary can reach orbital periods as short as ${\sim 6}$ min, before the
donor becomes degenerate and the orbital period subsequently increases as a
response to mass transfer.

More generally, whether convergent or divergent evolution takes place depends
on the strength of orbital AML due to MB, on the fraction of mass loss from
the binary, on the metallicity, and on the binary properties at onset of Roche
lobe overflow, most importantly the orbital period, donor structure, and
compact object and donor masses (e.g. Nelson et al., 2004; Deng et al., 2021).
For instance, the impact of the compact object mass can be seen Figures 7 and
8 by comparing the dashed red and green tracks with the shortest initial
orbital period, in which the RVJ prescription with $\gamma_{\rm MB}=3$ was
adopted. We can see that the red track in which the point-mass compact object
mass is $0.8$ M⊙is convergent, while the green track in which the point-mass
compact object mass is $1.2$ M⊙is divergent. In addition, when the RVJ
prescription is compared with the CARB prescription, the bifurcation period
obtained with the latter is much longer (Istrate et al., 2014; Deng et al.,
2021; Chen et al., 2021; Soethe & Kepler, 2021). This is a natural consequence
of the orbital AML rates provided by the CARB prescription being much larger
than those obtained with the RVJ prescription.

#### Comparison with Observations

Let us start our comparison between theory and observations discussing how
good the disk instability model (e.g. Lasota, 2001) is to explain persistent
and transient systems. By inspecting Figure 8, we can see that in general this
model successfully explains why some CVs and LMXBs exhibit outbursts, while
others do not. Irradiation heating of the accretion disk is usually negligible
for the stability of disks in CVs (Dubus et al., 2018) while in the case of
LMXBs X-ray irradiation needs to be taken into account (Coriat et al., 2012).
The net effect of including irradiation is that a system at a given orbital
period can be persistent for lower mass transfer rates in comparison with non-
irradiated accretion disks which is in general the case (compare CVs and LMXBs
in Figure 8). However, the disk instability model apparently fails to explain
two systems in Figure 8, namely the CV SDSS J1538$+$5123 and the NS-LMXB Cyg
X–2.

The time-averaged mass transfer rates in CVs can be estimated with the disk
instability model itself (Dubus et al., 2018) or by analyzing the ultraviolet
emission from the WD during quiescence (Townsley & Gänsicke, 2009), the latter
being used for SDSS J1538$+$5123\. From the ultraviolet spectra, the WD
effective temperature can be estimated, and this is considered a good
indicator of the mass transfer rate, since the WD is compressionally heated
due to the accretion process, and the higher the mass transfer rate, the
higher the WD effective temperature (Townsley & Bildsten, 2003, 2004). SDSS
J1538$+$5123 is a dwarf nova (i.e. transient) located below the orbital period
gap and therefore should according to the disk instability model have mass
transfer rates much lower than the ${\sim 1.8\pm 0.7\times 10^{-9}}$ M⊙ yr-1
inferred from its high WD effective temperature. However, there is no problem
with the disk instability model here and this high WD effective temperature is
most likely related to a recent nova eruption (Pala et al., 2017), and
therefore, the temperature is not a good indicator of the time-averaged mass
transfer rate in this particular case.

The persistent system Cyg X–2 is the only NS-LMXB that does not agree with the
predictions of the disk instability model, which clearly indicates that this
system should be transient. One possibility for this disagreement would be
that Cyg X–2 is in fact a transient source undergoing a very long outburst
(Coriat et al., 2012), similarly to what happens with the well-known BH-LMXB
GRS 1915$+$105, which is a transient source and has the longest orbital period
($33.85$ days) of this population. Given that the longer the orbital period,
the larger the accretion disk, the disks in these wider systems can accumulate
enough mass to fuel an outburst for decades. The accretion disk in Cyg X–2 is
estimated to have a mass of ${\sim 2.5\times 10^{28}}$ g, which is massive
enough to sustain outbursts lasting up to ${\gtrsim 80}$ years (Coriat et al.,
2012).

Let us move now to compare expected secular evolution of CVs and LMXBs with
observations. When we compare the tracks with the observational data in these
figures, we can see that most CVs are consistent with having initially
unevolved low-mass MS donors (solid red track). However, some CVs clearly
reveal abnormal (non-solar) abundances in their spectra, indicating that the
material has undergone thermonuclear processing by the CNO cycle (see Sparks &
Sion, 2021, for list of systems). These CVs contribute to ${\sim 5-15}$% of
the entire CV population (Gänsicke et al., 2003; Pala et al., 2020), and given
that their donors are not expected to possess this material, two scenarios
have been proposed to explain them. As a first possibility, they are
descendants of SSXBs, i.e. they are systems hosting initially nuclear-evolved
MS stars that underwent a short phase of thermally unstable mass transfer
before reaching their present-day configurations (Schenker et al., 2002;
Podsiadlowski et al., 2003). In this way, the donor has its outer layers
stripped during this phase, revealing CNO-processed material. The other
possibility would be donor contamination by the accretion of CNO-processed
material from ejected by the WD during nova eruptions (Stehle & Ritter, 1999).

Even though the first scenario over predicts the number of such CVs by a
factor of ${2-6}$ (Schenker et al., 2002), it has been for a long time
accepted as most likely. This is because it is apparently rather difficulty
for the donor to capture enough nova-processed material (Sparks & Sion, 2021),
e.g. because its geometrical cross-section is small and the high-velocity
material that the secondary geometrically intercepts is not highly nuclear-
processed. However, it has been recently suggested that these problems can be
overcome if a significant part of the ejected material leaves the WD with
velocities smaller than the escape velocity (Sparks & Sion, 2021). This is
possible because the thermonuclear runaway on the WD causes a strong
propagating shock wave, which leaves behind a large velocity gradient in the
expanding shell. This implies that part of this shell will reach escape
velocity but the remaining material will be gravitationally bound to the
binary, leading to a non-rotating CE-like structure surrounding the CV. This
nova-induced CE-like structure will then contain a large amount of non-solar
metallicity material, which can be efficiently accreted by the donor, creating
a significant non-solar convective region. However, given that virtually all
CVs are supposed to undergo nova cycles and that only ${\sim 10-15}$% of them
exhibit peculiar abundances, it remains unclear under which conditions this
sort of pollution can efficiently work, and whether it can explain why most CV
donors fail in to efficiently accrete. These questions could and should be
addressed in future detailed binary models, since their answers are
potentially related to major problems connected with CV secular evolution, as
we will see later.

Before moving to LMXBs, we just mention a few interesting long period (${\sim
14-48}$ h) CVs that host subgiant donors, i.e. BV Cen (Gilliland, 1982) and GK
Per (Álvarez-Hernández et al., 2021) in the Milky Way field, and AKO 9 in the
globular cluster 47 Tuc (Knigge et al., 2003). Interestingly, the masses of
the WDs in BV Cen (${\sim 1.02-1.44}$ M⊙, Watson et al., 2007) and GK Per
(${\sim 0.92-1.19}$ M⊙, Álvarez-Hernández et al., 2021) are significantly
higher than the average among CVs (${\sim 0.8\pm 0.2}$ M⊙, Zorotovic et al.,
2011; McAllister et al., 2019; Pala et al., 2022), which is consistent with
the evolutionary picture in which the WD mass has grown due to the high mass
transfer rates expected when the donor is initially an evolved MS star or a
subgiant. This makes this type of CVs very good candidates for being Type Ia
supernova progenitors, under certain conditions, although they correspond to a
negligible fraction of the entire CV population.

When we compare the distribution of LMXBs and CVs in Figure 7, together with
the evolutionary tracks, it is very likely that the contribution of systems
with initially evolved donors is much larger for LMXBs in comparison to CVs,
as they have on average significantly longer orbital periods for the same
range of donor masses. While the evolutionary sequence given by solid red line
in this figure seems to nicely explain at least around half of NS-LMXBs, a lot
of them can only be explained if the donor is initially an evolved MS star or
a subgiant. The situation is even more dramatic for BH-LMXBs, since only two
systems host M-type MS donors, and the majority of systems with orbital
periods shorter than ${\sim 1}$ day have K-type MS donors (Shao & Li, 2020).
In addition, it seems that a significant fraction of LMXBs start their
evolution having initially intermediate-mass donors (Podsiadlowski &
Rappaport, 2000), especially in the case of BH-LMXBs. Similarly to CVs, the
LMXBs with longest orbital periods (${\gtrsim 1}$ days) most likely have
subgiant or red giant donors, such as Cyg X–2 (Casares et al., 1998), GRO
J1744–28 (Doroshenko et al., 2020), GX 13$+$1 (Bandyopadhyay et al., 1999),
XTE J1550–564 (Orosz et al., 2011), GX 339–4 (Muñoz-Darias et al., 2008), GS
1354–64 (Casares et al., 2009), GRO J1655–40 (Israelian et al., 1999), V4641
Sgr (MacDonald et al., 2014), GS 2023$+$338 (Khargharia et al., 2010), and GRS
1915$+$105 (Harlaftis & Greiner, 2004). These systems are then evolving
towards longer orbital periods, as mass transfer is driven by the nuclear
expansion of the donor.

Even though the evolutionary picture that we have discussed reasonably well
explains the main observational features of individual CVs and LMXBs, as
illustrated in Figures 7 and 8, there are still major problems when we take
into account the entire populations and compare observed properties with those
predicted by population synthesis that are most likely intrinsically connected
with the orbital AML mechanisms driving the binary evolution. In particular,
the standard RVJ prescription (Equation 19) for MB usually assumed for models
of CVs and LMXBs is probably the origin of most of the problems we are
currently facing. For instance, the predicted fraction of CVs hosting brown
dwarfs is much higher than observed (Belloni et al., 2020a; Pala et al.,
2020), which suggests that the overall evolutionary time-scale should be much
longer than predicted by models using the RVJ prescription. In addition, the
predicted mass transfer rates of CVs above the orbital period gap are
typically much higher than those inferred from observations, except for
systems near the upper edge of the orbital period gap, for which predicted
rates are much smaller than those inferred from observations (Belloni et al.,
2020a; Pala et al., 2022). This suggests that the dependence of MB with the
orbital period should be the opposite of what is predicted by the RVJ
prescription. While this recipe leads to a decreasing orbital AML rate as the
CV evolves (see Figure 2), observations suggest that the overall rate should
increase as the CV evolves. Moreover, the predicted mass transfer rates for
NS-LMXB are so low that they cannot explain the persistent sources
(Podsiadlowski et al., 2002), which suggests that MB should be much stronger
in these systems than predicted by the RVJ prescription. Furthermore, the
predicted donor mass distribution of BH-LMXBs peaks at a much higher value
than the observed distribution (Wiktorowicz et al., 2014), which suggests that
MB should also be stronger in these systems. Finally, when the RVJ
prescription is adopted, a well-known fine-tuning problem arises when trying
to reproduce millisecond pulsars in close detached binaries (orbital periods
in the range ${\sim 2-9}$ h) with helium WD companions of mass ${\lesssim
0.2}$ M⊙, (Istrate et al., 2014; Chen et al., 2021), which represents a big
problem in the formation of these binaries.

Regarding LMXBs and their descendants, recent binary models (Van et al., 2019;
Van & Ivanova, 2019), in which the CARB prescription has been proposed,
indicate that the problem with the persistent NS-LMXBs can be solved with this
recipe that provides much higher orbital AML rates (see Figure 2) and
therefore higher mass transfer rates, especially for persistent systems with
shorter orbital periods. For this reason, the CARB prescription might also
solve the problem with BH-LMXBs, since it could shift the predicted donor mass
distribution towards lower values and hopefully reproduce the observed
distribution. In addition, this new prescription might also be consistent with
the fast orbital decay observed in some BH-LMXBs, such as XTE J1118$+$480
(González Hernández et al., 2012), A0620–00 (González Hernández et al., 2014),
and GRS 1124–68 (González Hernández et al., 2017), which cannot be explained
with the RVJ prescription. When applied to close detached millisecond pulsar
binaries, the CARB prescription also provides a solution to the fine-tuning
problem (Soethe & Kepler, 2021). In addition, not only compact binaries are
well-reproduced by this prescription but also wide-orbit binaries can be
explained, and in general, the relation between the helium WD mass and the
orbital period predicted by this prescription is in agreement with
observations (Soethe & Kepler, 2021).

Regarding CVs, it has been proposed that not only the problem with the mass
transfer rates but also the problem with the fraction of post-orbital period
minimum could be solved with another MB prescription (Belloni et al., 2020a;
Pala et al., 2020). A reasonable candidate to solve both problems is the
Kawaler prescription with $n\sim 1.25$, which is shown in Figure 2. This
prescription predicts an overall evolutionary time-scale substantially longer
than that obtained with the RVJ prescription, which may perhaps contribute
significantly to reduce the predicted number of CVs that manage to reach the
orbital period minimum. Moreover, given that the orbital AML rates provided by
this prescription are around one order of magnitude higher than GR, the
constraints provided by detached WD–MS post-CE binaries (Schreiber et al.,
2010) are not expected to be violated, which strengthens even more the
potential of solving CV problems with this prescription.

In addition to the problems most likely connected with the standard RVJ
prescription for MB, there are other problems that are probably associated
with another orbital AML mechanism, i.e. consequential AML, which is not
included in the standard evolutionary picture. For instance, the observed CV
WD mass distribution has a peak at ${\sim 0.8}$ M⊙ (Zorotovic et al., 2011;
McAllister et al., 2019; Pala et al., 2022), while most predicted CV WD masses
are on average smaller than that (Zorotovic et al., 2011; McAllister et al.,
2019; Pala et al., 2022; Zorotovic & Schreiber, 2020). In addition, for CVs
hosting fully convective MS donors, there is a clear spread in their mass
transfer rates inferred from observation that is definitely not predicted when
GR is taken as the only driver of their evolution (Pala et al., 2022).
Moreover, still when only GR is considered, the predicted mass transfer rates
among post-orbital-period-minimum CVs are much lower than those inferred from
observations (Pala et al., 2022), and the predicted orbital period minimum is
significantly shorter than observed (Howell et al., 2001; Gänsicke et al.,
2009; Knigge et al., 2011). These problems might be solved if the total AML
for these systems is stronger than predicted by GR (Knigge et al., 2011;
Schreiber et al., 2016; Nelemans et al., 2016). In case there is another
orbital AML mechanism driving the evolution together with GR for CVs with
fully convective MS donors and brown dwarf donors, then the predicted mass
transfer rates might be significantly enhanced, which would cause the CVs to
evolve faster so that their donors become degenerate at longer orbital
periods. The problem with the WD masses can be solved if the strength of this
additional orbital AML is inversely proportional to the WD mass, such that
most CVs with low-mass WDs would become dynamically unstable at the onset of
mass transfer (Schreiber et al., 2016). Similarly, the spread in the mass
transfer rates for CVs with fully convective MS donors might also be solved
with this dependence on the WD mass, if perhaps the additional orbital AML
also depends on the age of the CV.

The most natural choice of additional orbital AML to enhance the mass transfer
rates would be consequential AML, and there are very good candidates for the
above-highlighted dependence on the WD mass, and eventually on the time since
the onset of mass transfer(Schenker et al., 1998; Schreiber et al., 2016;
Nelemans et al., 2016). The frictional orbital AML produced by novae depends
strongly on the expansion velocity of the shell (Schenker et al., 1998). For
low-mass WDs (${\lesssim 0.55}$ M⊙), the expansion velocity is small (Yaron et
al., 2005), and this may lead to strong orbital AML by friction that makes
most CVs with low-mass WDs dynamically unstable. In particular, given the
velocity gradient in the nova shell we mentioned earlier, it is reasonable to
expect that a nova-induced CE-like structure surrounding the binary will be
formed after the eruption. This idea has been tested in binary models
(Nelemans et al., 2016), which confirms that such a configuration drastically
affects the stability of mass transfer in CVs with low-mass WDs. As we
mentioned earlier, this form of consequential AML could also explain the
existence of CVs with non-solar abundance, as an alternative to the post-SSXB
scenario, which predicts more such CVs (a factor of ${2-6}$) than observed.
Other promising mechanisms are circumbinary disks (Spruit & Taam, 2001;
Willems et al., 2005), or a combination of both, e.g. if the circumbinary disk
is fed by a fraction of the accreted matter that is re-emitted by WD due to
the nova eruptions.

###  6.2 AM CVns and Ultra-Compact X-ray Binaries

As we mentioned earlier, AM CVns and UCXBs can be formed in three ways, either
through two CE evolution, or through dynamically stable non-conservative mass
transfer followed by CE evolution, or through CE evolution followed by
dynamically stable non-conservative mass transfer. Since the later channel
from CVs/LMXBs has just been addressed, we focus on the other two, which have
always a CE evolution for the second episode of mass transfer. Similarly to
CVs and LMXBs, after CE evolution, a detached post-CE binary is formed and
evolves towards shorter orbital periods due to orbital AML, leading to the
onset of mass transfer, when the donor fills its Roche lobe. However, unlike
these systems, the main mechanism driving the evolution of AM CVns and UCXBs
is always GR, since typically their donors are initially helium WDs or helium
stars.

#### Helium White Dwarf or Helium Star Donors

Let us start with the case in which the donor is initially a helium WD. This
case is illustrated in Figures 7 and 8 by the dotted black track in each
figure, which were taken from a binary model (Wong & Bildsten, 2021). Even
though the orbital period corresponding to the onset of mass transfer depends
on the mass ratio and on helium WD properties, mainly its radius, and in turn
its mass, it is of order of a few minutes. After that, the further binary
evolution depends on the initial entropy and the subsequent thermal evolution
of the donor, and is usually separated into three phases (e.g. Deloye et al.,
2007; Solheim, 2010; Kaplan et al., 2012; Wong & Bildsten, 2021).

In the very beginning, the donor contracts in response to mass loss as the
outermost radiative layer is stripped off, leading to an increase in the mass
transfer rate. The mass transfer rate then grows from zero to its maximum,
which can be up to ${\sim 10^{-8}-10^{-6}}$ M⊙ yr-1, depending the donor
properties, while the donor radius decreases to a minimum value. During this
phase, the mass-loss time-scale becomes much shorter than the donor thermal
time-scale, and the orbital period decreases as a response to mass transfer.
After this quick phase, which lasts ${\lesssim 10^{6}}$ years, the donor
radius evolution reverses and the donor expands in response to mass loss. This
happens because, during the first phase, the entropy profile eventually
becomes sufficiently shallow that the expansion of the underlying layers
starts to dominate. The binary then evolves towards longer orbital periods
driven by GR, with decreasing mass transfer rates. During this second phase,
which can last ${\sim 1}$ Gyr, the mass-loss time-scale is much shorter than
the donor time-scale and the donor responds adiabatically to the mass loss.
The third phase starts when the mass loss time-scale becomes comparable to or
shorter than the donor thermal time-scale in the underlying layers. This
allows the donor to cool down and contract, although the contraction ends when
the donor has shed sufficient entropy to reach its fully degenerate
configuration. This phase starts at an orbital period of ${\sim 40}$ min.

Let us now address the case in which the companion of the compact object was
very close to the tip of the FGB just before CE evolution, resulting in turn
in a post-CE binary hosting the compact object and a helium star. In case the
post-CE binary orbital period is sufficiently short, the compact object
companion will still be a helium star, i.e. still burning helium, at the onset
of mass transfer. The binary evolution for this case is illustrated in Figures
7 and 8 by the solid and short-dashed black tracks taken from two binary
models (Heinke et al., 2013; Wang et al., 2021). Depending on the mass, the
lifetime of a helium star is typically ${\sim 10-10^{3}}$ Myr, more massive
stars having shorter lifetimes, and its radius can increase up to ${\sim 30}$%
during its life (Yungelson, 2008), and how large it can become strongly
depends on the amount of hydrogen left after CE evolution (Bauer & Kupfer,
2021). Thus, unlike the previous case, the onset of Roche lobe filling is
driven by a combination of orbital AML due to GR and nuclear expansion of the
donor. Therefore, the orbital period at which the onset of mass transfer
occurs depends on the masses of the compact object and helium star as well as
on the evolutionary status of the helium star. In particular, the longer the
post-CE binary orbital period, the more time the helium star has to evolve,
becoming larger, and in turn the longer the orbital period at the onset of
mass transfer. Given the orbital AML rates provided by GR and the helium star
lifetimes, to have a helium star at onset of mass transfer, the post-CE binary
orbital must be typically shorter than ${\sim 2}$ h (Yungelson, 2008). After
mass transfer starts the further evolution for systems hosting helium star
donors is different from those having helium WDs (Savonije et al., 1986;
Yungelson, 2008; Solheim, 2010; Wang et al., 2021).

During the Roche lobe overflow, the mass-loss time-scale is typically shorter
than the thermal time-scale of the donor, which makes the donor slightly
bloated as a response to mass loss. This causes a rapid increase in the mass
transfer rate, reaching values of a few $10^{-8}$ M⊙ yr-1, after the residual
hydrogen is entirely consumed. In a more realistic situation, a non-negligible
residual outer hydrogen envelope is left in the helium star after CE
evolution, which plays a significant role during the early evolution (Bauer &
Kupfer, 2021). The donors can spend tens of Myr transferring this small amount
of hydrogen-rich material at low rates (${\sim 10^{-10}-\sim 10^{-9}}$ M⊙
yr-1), before finally transitioning to a phase where helium is transferred at
much higher rates (${\gtrsim 10^{-8}}$ M⊙ yr-1). Since the donor should be
modestly evolving and expanding, in the absence of orbital AML, the orbital
period should increase in response to mass loss. However, since GR is
sufficiently strong, the donor nuclear evolution time-scale is longer than the
GR-driven mass-loss time-scale, which leads to convergent evolution, and the
orbital period decreases. A significant fraction of the energy released
through nuclear burning is absorbed in the donor envelope, which leads to a
decrease in its luminosity and an increase of its thermal time-scale. As the
binary evolves and the donor mass drops, nuclear burning quickly becomes less
important, which makes the convective core to be replaced by an outer
convection zone that penetrates inwards. At the same time, as the orbital
period decreases, the mass-loss time-scale also decreases, causing the mass
transfer rate to slightly increase. Towards the end of this initial phase, the
donor thermal time-scale becomes much longer than the mass-loss time-scale,
and the thermal structure of the donor is severely perturbed. The mass
transfer rate then increases faster, as the donor tries to remain within its
Roche lobe. At some point, helium burning in the centre of the donor virtually
vanishes and the donor is driven far out of thermal equilibrium. The mass
transfer rate then reaches its maximum and the orbital period its minimum.
This typically occurs when the donor mass is ${\sim 0.20-0.26}$ M⊙, at an
orbital period of ${\sim 8-11}$ min. Since the star becomes more degenerate,
its radius starts to increase and the further evolution resembles that of
systems hosting helium WD donors, in which the donor is evolving adiabatically
in response to mass loss. The mass transfer rate drops while the binary
evolves towards longer orbital periods.

In case the compact object is a WD, some interesting features are expected in
the first phases of AM CVn evolution hosting initially helium star donors. It
has been shown by binary models (Fink et al., 2007; Brooks et al., 2015) that,
if the initial mass of the helium star is ${\gtrsim 0.4}$ M⊙ and the accreting
WD mass is initially ${\gtrsim 0.8}$ M⊙, enough mass is accumulated onto the
accreting WD, leading to a first thermonuclear flash that is likely vigorous
enough to trigger a detonation in the helium layer. This thermonuclear runaway
originated from the detonation in the helium layer of the accreting WD may
create converging shock waves that reach the core. If the carbon in the core
is also detonated, the thermonuclear runaway is seen as a Type Ia supernova.
Those that survive the first flash and eject mass will have a temporary
increase in orbital separation, but orbital AML through GR drives the donor
back into contact, resuming mass transfer and triggering several subsequent
weaker flashes.

Despite the fact that the three evolutionary channels leading to AM CVns and
UCXBs are in principle viable, there are theoretical arguments against two of
them. It has been argued that most, if not all, WDs accreting initially from
helium WDs merge at the beginning of the evolution (Shen, 2015). This would
happen due to nova eruptions during the initial phases of accretion that would
trigger dynamical friction within the expanding nova shell, shrinking the
orbit and causing the mass transfer rate to become dynamically unstable. If
that is true, most, if not all, AM CVns would come from WDs initially
accreting from helium stars or from CVs with subgiant or nuclear-evolved MS
donors. It has been argued, based on the RVJ prescription for orbital AML due
to MB, that only a negligible fraction of AM CVns could have originated from
CVs (Goliasch & Nelson, 2015; Liu et al., 2021). The reason for that is the
narrow range of CV properties from which AM CVns originate. However, the
situation here is very similar to the fine-tuning problem related to the
formation of close detached millisecond pulsar binaries. As the CARB
prescription, which predicts higher orbital AML rates than the standard RVJ
prescription, can solve the millisecond pulsar fine-tune problem, it would not
be surprising if it can also increase the probability of forming AM CVns from
CVs.

#### Comparison with Observations

After describing the main features of AM CVn and UCXB evolution, we can
confront prediction with observations. Let us start with a brief discussion of
the persistent and transient sources with respect to what the disk instability
model predicts. By inspecting Figure 8, we can see that this model provides
quite convincing results, similarly to the case of CVs and LMXBs, especially
for UCXBs (Heinke et al., 2013). However, some AM CVns deserve attention, like
ASASSN-14cc and ASASSN-14mv. The disk instability model predicts that these
two systems should be persistent, but they have undergone outbursts (Ramsay et
al., 2018). In addition, SDSS J1525$+$3600 and GP Com have never shown
outbursts (Ramsay et al., 2018), although the model predicts that they should
be transient. Finally, despite their very low mass transfer rates (${\lesssim
10^{-11}}$ M⊙ yr-1), SDSS J1208$+$3550, SDSS J1137$+$4054, V396 Hya and SDSS
J1319$+$5915 are marked in the figure as persistent, since they have never
been observed in outburst. Even though it is very likely that the lack of
outbursts related to the last four systems is due to the fact that their
accretion disks are stable and cold, it is not clear what might be happening
with the other systems.

When we compare the evolutionary tracks for the three channels leading to AM
CVns and UCXBs in Figures 7 and 8, we can see that they can reasonably well
explain observations, and some particular systems can be better described by
particular channels. For instance, as noted in previous studies (Heinke et
al., 2013), UCXBs can be separated into three groups: (i) transients with
orbital periods longer than ${\sim 40}$ min, (ii) persistent with orbital
periods shorter than ${\sim 25}$ min, and (iii) transient with orbital period
longer than that. Since the mass transfer rate drops as an UCXB evolves with a
WD donor, sources in the groups (i) and (ii) could in principle be explained
by any scenario. However, those in the group (iii), i.e. the long-period
persistent UCXBs 4U 1626–67, 4U 1916–053, and 4U 0614$+$091, can apparently
only be explained if the donor is initially a helium star or perhaps if they
are descendants of LMXBs. This is because if the donor is initially a helium
WD, mass transfer rates in this orbital period range are expected to be
similar to those in group (i), i.e. orders of magnitude lower than those
inferred for the group (iii).

This could have important implications when comparing the population of UCXBs
in the Milky Way field with those belonging to globular clusters. As already
noted before (Zurek et al., 2009; Heinke et al., 2013), the persistent UCXBs
in globular clusters have orbital periods much shorter than those of 4U
1626–67, 4U 1916–053, and 4U 0614$+$091 (the longest orbital period is $22.58$
min for M15 X–2, Dieball et al., 2005). According to Heinke et al. (2013),
this fact could be naturally explained if long-period persistent UCXBs have
initially helium star donors that ignite helium under non-degenerate
conditions in the centre. This is because UCXBs in globular clusters cannot
form in such a way, and the reason for that is twofold. First, the typical
evolutionary time-scale for an UCXB to be seen as persistent is only a few
tens of Myr (Wang et al., 2021). Second, for low metallicities, only MS stars
with masses ${\gtrsim 1.85}$ M⊙ will evolve without developing a degenerate
helium core on the FGB (Han et al., 2002), which is a much higher mass than
that of MS stars close to the turn-off point in globular clusters (${\sim
0.8-0.9}$ M⊙, e.g. Cohn et al., 2010; Lugger et al., 2017; Rivera Sandoval et
al., 2018; Cohn et al., 2021). This would then explain the lack of long-period
persistent UCXBs in globular clusters. However, helium stars can also
originate from less massive stars, which undergo the helium core flash to
ignite helium off-centre (e.g. Han et al., 2002; Bauer & Kupfer, 2021), and it
is not clear why such helium stars should behave in a different way than those
formed under non-degenerate conditions. In addition, there is an over-
abundance of bright X-ray sources in globular clusters in comparison to the
Milky Way field, which indicates that dynamics are playing a significant role
in their formation. Furthermore, it has been shown that strong MB can lead to
mass transfer rates high enough to explain long-period persistent UCXBs (Van
et al., 2019; Van & Ivanova, 2019). Therefore, the difference in the orbital
period distribution of persistent UCXBs in the two environments most likely
arises from a more complex interplay between dynamics and binary evolution
than previously thought by Heinke et al. (2013).

There is one AM CVn (SDSS J1908$+$3940) that seems challenging to be explained
by any evolutionary channel. This system has an orbital period of ${\sim 18}$
min and an inferred mass transfer rate of ${\sim 4\times 10^{-7}-10^{-6}}$ M⊙
yr-1 (Ramsay et al., 2018). For its orbital period, this mass transfer rate is
orders of magnitude higher than expected from any type of initial donor, which
is ${\lesssim 10^{-8}}$ M⊙ yr-1. Future observational efforts and binary
models with focus on this system could help to understand whether any of these
formation channels could explain its properties. If not, perhaps something
atypical is occurring with this system that pushed its inferred mass transfer
rate to such a high value.

Unfortunately, except for a few particular systems, we still do not know which
of these three formation channels is the dominant among AM CVns and UCXBs, if
any is dominating at all, which compromises studies of their intrinsic
populations. It seems clear as well that, for the overwhelming majority of
systems, having only the orbital period, donor mass and mass transfer rate are
not enough to distinguish between the different evolutionary scenarios, and
more information is needed. There are a few ways to better evaluate the
contribution of each channel, by gathering more information about these
systems.

For instance, by identifying and studying the properties of potential
progenitors, such as the system LAMOST J0140355$+$392651 (ElBadry et al.,
2021). This system has an orbital period of $3.81$ h and hosts a bloated,
relatively cool, low-mass (${\sim 0.15}$ M⊙) proto WD and a massive (${\sim
0.95}$ M⊙) WD companion. Its properties can be well-explained as a CV entering
the detached phase, when the donor envelope has been entirely consumed. In
this case, the proto WD was a nuclear-evolved MS star before the onset of mass
transfer. The system will evolve through this detached phase due to GR and
mass transfer will eventually resume, after a few Gyr. This sort of
investigation provides key information to understand binary evolution leading
to AM CVns and UCXBs. However, the number of these systems is far too low to
provide conclusive evidences for a particular channel being dominant or not.

Another possibility is by constraining the properties of their donors with
detailed analyses of their chemical composition, which allows us to
investigate in more detail their evolutionary history (Nelemans et al., 2010).
Those formed through CV/LMXB evolution are expected to retain a greater
fraction of their hydrogen in comparison with systems formed in the other two
channels. This would imply that the presence of hydrogen in their spectra is
needed to rule out the other channels as well as systems not exhibiting
hydrogen in their spectra are most likely not formed in such a way. In both
helium WD and helium star channels, the transferred material is expected to be
mainly composed of helium with CNO-processed material, mainly nitrogen, and
abundance ratios of these elements are believed to be an effective
discriminant between these two channels. A third possibility is by means of
the donor mass-radius measurements obtained from precise analyses of eclipsing
systems, since each evolutionary channel predicts on average different radius
for the same mass (Deloye et al., 2007; Yungelson, 2008; Goliasch & Nelson,
2015), although some overlap is also expected depending on the initial
properties of the donor as well as how the WD donor cooling is treated. In
general, systems having initially helium WD donors evolve with smaller radii
than systems having initially helium star donors, and these evolve with
smaller radii than those originated from CVs/LMXBs.

The last option we discuss is applicable to UCXBs and consists of
investigating the existence/lack of thermonuclear explosions that occur on the
surface of the NS as well as their features. It has been proposed that there
is a strong correlation between the iron K$\alpha$ line strength in their
X-ray spectra and the abundance of carbon–oxygen in the accretion disc
(Koliopanos et al., 2013). This line occurs at ${\sim 6.4-6.9}$ keV and is a
result of the reflection of the radiation by the accretion disk and the
surface of the WD facing the NS, and for this reason it can provide clues on
the disk chemical composition. This correlation coupled with the fact that
UCXBs with prominent and persistent iron K$\alpha$ emission also show bursting
activity, it has been proposed that UCXBs with persistent iron emission have
helium-rich donors, while those that do not, likely have carbon–oxygen or
oxygen–neon–magnesium-rich donors (Koliopanos et al., 2021). This possibility
can be further explored, but more elaborated X-ray reflection models will be
eventually needed.

Having in mind these possibilities, we illustrate with eclipsing AM CVns how
they may help to disentangle the formation channels. Thus far, there are seven
eclipsing AM CVns with accurate measurements of the masses and radii of their
donors (Copperwheat et al., 2011; Green et al., 2018; van Roestel et al.,
2021). When considering only the radii and masses, YZ LMi can be easily
explained by either the helium WD or the helium star channels, ZTF J0407–0007
by the helium WD channel, and ZTF J2252–0519 by the helium star channel. The
remaining (Gaia14aae, ZTF J1637$+$4917, ZTF J0003$+$1404 and ZTF J0220$+$2141)
have very large radii, which could be only explained by the CV channel.
However, when the chemical properties of these seven systems are taken into
account, the situation becomes much more complicated. Regarding those systems
apparently formed through the CV channel, since no hydrogen was detected in
the spectra of any of them, it seems more likely that they have formed in a
different way. Therefore, the situation for these systems apparently seems
rather challenging. A possibility to bring into agreement predicted and
observed WD donor radii and chemical properties is by avoiding the phase in
which the donor can contract and cool down. This phase is predicted to exist
because, at some point, the donor thermal time-scale becomes comparable or
shorter than the mass-loss time-scale, since the orbital AML rate is
continuously dropping as the AM CVn evolves. If there is an additional source
of orbital AML, e.g. consequential AML, such that the donor cooling is
significantly delayed, then the donors could keep their large sizes, even when
the orbital period is longer than ${\sim 40}$ min. However, this possibility
at the moment is very speculative and future detailed binary models could
verify whether consequential AML is able to do that, and if so, for which
mechanisms and under which conditions.

After describing compact objects accreting from Roche lobe filling donors, we
can turn to a discussion of other modes of transfer, in which the donor is not
filling the Roche lobe and is either a low-/intermediate-mass-evolved red
giant, or a supergiant, or a Wolf–Rayet star.

###  6.3 Symbiotic Stars and Symbiotic X-ray Binaries

We have argued before that, in order to explain their orbital periods (${\sim
200-10^{3}}$ days), most S-type SySts should form through an episode of
dynamically stable non-conservative mass transfer. After this event, the
resulting post-stable-mass-transfer binary, with an orbital period longer than
${\sim 100}$ days, hosts a WD orbiting a low-mass MS star or perhaps a
subgiant or red giant, depending on the mass ratio of the zero-age MS–MS
binary. On the other hand, those S-type SySts hosting massive WDs as well as
SyXBs are most likely formed through CE evolution, leading to long-period
post-CE binaries, since dynamically stable non-conservative mass transfer
cannot explain their properties. Finally, given the very long orbital periods
of D-type SySts (${\gtrsim 40}$ years), their WD progenitors are not expected
to form during an episode of Roche lobe filling, i.e. their WDs are formed
similarly to single WDs. In the three situations, given the relatively long
orbital period of the binary, the WD companion has enough room to evolve and
expand, and the binary will become a SySt or a SyXB under certain conditions.
Even though in what follows we discuss the evolution of these systems with
focus on SySts, the case of SyXBs is very similar, except from the fact that
the compact object is a NS.

The donors in SySts are either located near the tip of the FGB or on the AGB
(e.g. Gromadzki et al., 2013; Mikołajewska, 2007). The typical luminosity of
accreting WDs in SySts is ${\sim 10^{2}-10^{4}}$ L⊙ (Mikolajewska, 2010), and
in most cases these are powered by nuclear burning due to the very high
accretion rates, which are typically ${\sim 10^{-8}-10^{-7}}$ M⊙ yr-1. In
addition, in most SySts, the red giant is under-filling its Roche lobe filling
(Mikołajewska, 2012; Boffin et al., 2014), but in some cases it is nearly
filling its Roche lobe, as evidenced by the clear signature of ellipsoidal
variations in their light curve (Gromadzki et al., 2013). Therefore, in what
follows, we discuss two promising modes of mass transfer for these systems,
which are (i) atmospheric Roche lobe overflow (Ritter, 1988), if ${0.9\,R_{\rm
RL}\lesssim R_{\rm d}\lesssim R_{\rm RL}}$, or (ii) 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 ${R_{\rm d}\lesssim 0.9\,R_{\rm RL}}$.

#### Atmospheric Roche Lobe Overflow

As the red giant evolves, its radius approaches its Roche lobe, and at some
point the mass transfer rate provided by Equation 1 will become relevant. If
the red giant mass is not much higher than the compact object mass,
$\zeta_{\rm ad}>\zeta_{\rm RL}$, and mass transfer is dynamically stable. We
show in Figure 7, as dashed green lines, several evolutionary tracks in which
the initial point-mass compact object mass is $1.2$ M⊙, and the zero-age MS
mass is $1.25$. For those with initial orbital periods longer than ${\sim 5}$
days but shorter than ${\sim 100}$ days, the onset of mass transfer occurs
when the donor is an unevolved red giant on the FGB. The mass transfer in this
case is similar to what is depicted in Figure 6, and it is characterized by an
initial phase of thermally unstable mass transfer, until the donor mass is
sufficiently low such that the mass transfer stabilizes. After that, mass
transfer proceeds on the nuclear time-scale of the donor. The mass transfer
rates nicely correlate with the evolutionary status of the donor. First, the
longer the initial orbital period, the more evolved the donor at the onset of
mass transfer. Consequently, the more evolved the donor, the higher the mass
transfer rates. For instance, during thermally unstable mass transfer, the
rates for the binaries with shortest and longest initial orbital period are
${\sim 3\times 10^{-8}}$ and ${\sim 6\times 10^{-6}}$ M⊙ yr-1, respectively.
During nuclear time-scale mass transfer, the rates for the same binaries are
${\sim 5\times 10^{-9}}$ and ${\sim 7\times 10^{-8}}$ M⊙ yr-1, respectively.

There are at least three SySts in which their red giant donors are very close
to filling their Roche lobes and their WDs are massive enough so that mass
transfer can be regarded as dynamically stable. T CrB is a well-studied SySt
with a relatively short orbital period ($227$ days), hosting a massive
carbon–oxygen WD ($1.37\pm 0.13$ M⊙) and a red giant with mass $1.12\pm 0.23$
M⊙ (Stanishev et al., 2004). RS Oph is another well-known SySt, which also
hosts a massive carbon–oxygen WD ($1.2-1.4$ M⊙). However, its orbital period
($453$ days) is longer than that of T CrB and it hosts a less massive red
giant donor (${0.68-0.80}$ M⊙, Brandi et al., 2009). This likely indicates
that both SySts could share similar evolutionary sequences, in which the
orbital period is expanding as the red giant is consumed. The last SySt we
know in which the red giant donor is nearly filling its Roche lobe is V3890
Sgr (Mikołajewska et al., 2021). Similarly to the other two systems, this SySt
also hosts a massive carbon–oxygen WD ($1.35\pm 0.13$ M⊙). However, its longer
orbital period ($747$ days) combined with the mass of its red giant donor
($1.05\pm 0.11$ M⊙) suggests that the evolutionary pathway of V3890 Sgr is
slightly different from those of T CrB and RS Oph, starting at a longer
orbital period.

The properties of T CrB and RS Oph can be nicely reproduced if the post-CE
binary hosts a WD with mass ${\sim 1.15}$ M⊙, and a MS star companion with
mass ${\sim 1.4}$ M⊙, as illustrated in Figure 7 by the solid green track
passing through these two systems. As mentioned earlier, T CrB and RS Oph
could indeed be explained with the same evolutionary sequence, if the WD mass
in the former is smaller, which would imply that T CrB may evolve to something
similar to RS Oph. In order to reproduce the properties of these two systems
with the same evolutionary sequence, the post-CE binary orbital period needs
to be as long as ${\sim 200}$ days, and the fraction of mass lost from the
vicinity of the WD due to isotropic re-emission needs to be $\beta_{\rm
ml}\sim 0.6$. The predicted masses of the WD and red giant during the T CrB
stage are ${\sim 1.25}$ and ${\sim 1.07}$ M⊙, and during the RS Oph stage,
they are ${\sim 1.38}$ and ${\sim 0.67}$ M⊙. The WD will exceed the
Chandrasekhar limit and explode in a Type Ia supernova when the orbital period
is ${\sim 500-800}$ days, depending on the assumed Chandrasekhar limit. After
the onset of mass transfer, the binary reaches the T CrB stage after ${\sim
0.2}$ Myr. The transition from the T CrB stage to the RS Oph stage takes
${\sim 1}$ Myr, and from RS Oph stage to the Type Ia supernova ${\lesssim
1-4}$ Myr, depending on the assumed Chandrasekhar limit.

The properties of V3890 Sgr can also be reproduced with similar WD and
companion masses after CE evolution, as well as the same mass-loss fraction
during SySt evolution. However, the post-CE binary orbital period must be much
longer (${\sim 600}$ days) than in the previous case. An example of
evolutionary sequence (rightmost solid green track) is shown in Figure 7. The
predicted masses of the WD and red giant at the orbital period of V3890 Sgr
are ${\sim 1.25}$ and ${\sim 0.96}$ M⊙. Similarly to the previous case, the
evolution will terminate when the WD exceeds the Chandrasekhar limit.

As mentioned earlier, mass transfer will be dynamically stable only if the red
giant mass is comparable to or only slightly more massive than the compact
object mass. This condition is usually not satisfied in SySts (Mikołajewska,
2003), since their WD masses are typically ${\lesssim 0.6}$ M⊙, while their
red giant donors usually have masses of ${\sim 1.2-2.2}$ M⊙, implying mass
ratios ${\gtrsim 2.5}$. Such mass ratios are likely high enough to cause mass
transfer to be dynamically unstable. In this case, as the red giant radius
approaches the Roche lobe radius, the mass transfer rates provided by Equation
1 always increase, leading to CE evolution. This behaviour is illustrated in
Figure 9 by the solid lines. The evolutionary tracks in this figure correspond
to some of the blue tracks in Figure 7, and they all start with a zero-age MS
star with mass $1.6$ M⊙ orbiting a point-mass WD with mass $0.53$ M⊙, but each
one of them starts at a different orbital period. Before the onset of CE
evolution, the atmospheric Roche lobe overflow model provides accretion rates
that can only explain SySts that are close filling their Roche lobe, i.e.
${R_{\rm d}\gtrsim 0.90\,R_{\rm RL}}$.

Moreover, the SySt lifetime predicted by this model is ${\sim 0.01-0.1}$ Myr.
This is orders of magnitude shorter than those typically assumed in studies
that try to estimate the expected number of SySts in the Milky Way (e.g.
Kenyon et al., 1993). In general, regardless of the assumptions, these
investigations predict much more Milky Way SySts than observed. This problem
could be partially/completely overcome with the atmospheric Roche lobe
overflow model, if the intrinsic population of S-type SySts in the Milky Way
hosts red giants close to Roche lobe filling. However, if only a small
fraction in the intrinsic population shows evidence for ellipsoidal
variability, like currently observed (Gromadzki et al., 2013), then this model
most likely does not correspond to the dominant mode of mass transfer for
these systems. In what follows, we discuss the case in which the red giant is
strongly under-filling its Roche lobe.

Figure 9: Predicted accretion rate ($\dot{M}_{\rm a}$) onto the WD during pre-
SySt and SySt evolution, as a function of the orbital period $P_{\rm orb}$,
according to the atmospheric Roche lobe overflow model (Equation 1), the
Bondi-Hoyle-Lyttleton model (Equations 27 and 28, with $\alpha_{\rm
wind}=1.5$), the wind Roche lobe overflow model (Equations 27, 29, and 30,
with ${T_{\rm dust}=1\,500}$ K and ${p=1}$), and the wind compression model
(Equation 31, assuming ${5\leq\rho^{\rm c}/\rho^{\rm sph}\leq 15}$). The grey
thick horizontal line marks the minimum accretion rate such that the system
can be considered a SySt, which is ${\sim 10^{-8}}$ M⊙ yr-1. The evolutionary
tracks were computed by us using the mesa code and correspond to some of those
shown in Figure 7. For these selected tracks, we start the simulations with a
zero-age MS star of mass $1.6$ M⊙ orbiting a point-mass WD with mass $0.53$
M⊙, which are consistent with the average values for the red giant and WD in
SySts (Mikołajewska, 2003). We chose nine different initial orbital periods,
from left to right: $300$, $450$, $600$, $800$, $1000$, $1700$, $2400$,
$3100$, and $3800$ days, and evolve each one of them until the onset of CE
evolution. In all tracks, the orbital period increases/decreases as the SySt
evolves mainly as a response to red giant donor spin evolution, since it
quickly becomes synchronized with the orbit while ascending the FGB. For the
three binaries with shortest initial orbital period, Roche lobe filling occurs
when the donor is on the FGB, while for the remaining binaries it takes place
when the donor is on the AGB. In case the orbital period during the SySt phase
is shorter than $10^{3}$ days, we computed the accretion rates provided by the
atmospheric Roche lobe overflow model (solid lines) and those obtained with
gravitationally focused wind accretion, for which we assumed either the Bondi-
Hoyle-Lyttleton model, if the donor is on the FGB (dashed blue lines), or the
wind Roche lobe overflow model, if the donor is on the AGB (dashed red lines),
or the wind compression model (green areas). Furthermore, we only show the
accretion rates provided by the wind Roche lobe overflow model when the
orbital periods are longer than $10^{3}$ days. The average Bondi-Hoyle-
Lyttleton and wind Roche lobe overflow accretion rate efficiencies ($\eta_{\rm
wind,BHL}$ and $\eta_{\rm wind,wRLOF}$) are indicated in the figure, while
those for the compression wind model are given by $(5-15)\times\eta_{\rm
wind,BHL}$. Also indicated is the typical SySt lifetime ($\tau$), based on the
atmospheric Roche lobe overflow and wind compression model, for orbital
periods shorter than $10^{3}$ days, and based on the wind Roche lobe overflow
model, for orbital periods longer than that. SySts with orbital periods
shorter than $10^{3}$ days cannot be easily explained by the Bondi-Hoyle-
Lyttleton model, nor the wind Roche lobe overflow model, as these mass
transfer modes do not provide accretion rates high enough to trigger the
symbiotic phenomenon. However, the required rates in this period range can be
obtained with the wind compression model, if the accretion efficiency is
${\gtrsim 5}$ times higher than that from the Bondi-Hoyle-Lyttleton model. On
the other hand, the wind Roche lobe overflow model is clearly able to explain
SySts with AGB donors at orbital periods longer than $10^{3}$ days. Finally,
the atmospheric Roche lobe overflow model always provides sufficiently high
accretion rates, when $R_{\rm d}\gtrsim 0.85-0.9\,R_{\rm RL}$.

#### Gravitationally Focused Wind Accretion

In the case of low-mass red giants, the mass transfer rates provided by
Equation 1 in the atmospheric Roche lobe overflow model are only relevant if
${R_{\rm d}\gtrsim 0.85-0.9\,R_{\rm RL}}$. If this condition is not satisfied,
then the accretion rates onto the WD are negligible, and in turn this model
cannot explain the observed WD luminosities in SySts. This implies that mass
transfer must be associated with accretion of part of the stellar winds from
the red giant, i.e.

$\dot{M}_{\rm a}\ =\ -\,\eta_{\rm wind}\,\dot{M}_{\rm wind}\ ,$ (27)

where $\eta_{\rm wind}$ is the orbit-averaged wind accretion rate efficiency.
Observations suggest that the wind mass-loss rate from a typical red giant
donor in SySts is ${\gtrsim 10^{-7}}$ M⊙ yr-1 (e.g. Skopal, 2005), implying
that ${\eta_{\rm wind}\gtrsim 0.1}$ to reach the accretion rates of ${\gtrsim
10^{-8}-10^{-7}}$ M⊙ yr-1 needed to explain the hydrogen nuclear burning.

The most simple case of wind accretion occurs when we assume that stellar
winds are supersonic and spherically symmetric, which is the so-called Bondi-
Hoyle-Lyttleton model (Hoyle & Lyttleton, 1939; Bondi & Hoyle, 1944; Edgar,
2004). In this case, the WD can accrete part of the stellar wind as it orbit
through it and the accretion rate efficiency, i.e. the fraction of the stellar
winds that is effectively accreted by the WD, can be estimated as

$\eta_{\rm wind,BHL}\ =\ \frac{\alpha_{\rm
wind}}{2\sqrt{1\,-\,e^{2}}}\,\left(\frac{G\,M_{\rm a}}{a\,v_{\rm
wind}^{2}}\,\right)^{2}\,\left[1\,+\,\left(\frac{v_{\rm orb}}{v_{\rm
wind}}\right)^{2}\,\right]^{-3/2}\ \leq 1\ ,$ (28)

where $v_{\rm wind}$ and $v_{\rm orb}$ are the wind and orbital velocities,
respectively, and $\alpha_{\rm wind}$ is a parameter to be adjusted based on
observations.

Depending on the properties of the red giant and the orbital separation, the
winds can be gravitationally focused towards the orbital plane and more easily
accreted by the WD. Even though there are a few models available to estimate
gravitationally focused wind accretion rates (e.g. Abate et al., 2013; Skopal
& Cariková, 2015; Saladino et al., 2019), we will only consider two models to
illustrate how this mechanism can lead to accretion efficiencies much higher
than those obtained with the Bondi-Hoyle-Lyttleton model. These two models are
(i) the wind Roche lobe overflow model (Mohamed & Podsiadlowski, 2007, 2012;
Abate et al., 2013) and (ii) the wind compression model (Bjorkman &
Cassinelli, 1993; Skopal & Cariková, 2015), and they look very promising in
the context of SySts.

Let us start with the wind Roche lobe overflow model. During the AGB phase,
winds are most likely driven by a combination of pulsation-induced shock waves
and radiation pressure on dust grains (Höfner & Olofsson, 2018). Stellar
pulsation and convection induce strong shock waves in the extended atmosphere,
pushing the gas outwards, which reaches cooler regions. Since most of the gas
in a cool AGB star is in molecular form, some species may condense into dust
grains, giving rise to a dusty shell around the star. These dust grains are
then accelerated outwards by radiation pressure and the gas is dragged along.
In the presence of a companion, the gas can remain within the AGB Roche lobe,
if the wind acceleration radius is larger than the AGB Roche lobe radius
(Abate et al., 2013). When this happens, instead of having the red giant
effectively filling its Roche lobe due to nuclear evolutionary processes, the
slow winds fill its Roche lobe, enhancing in turn the wind accretion onto the
WD (Mohamed & Podsiadlowski, 2007, 2012; de Val-Borro et al., 2009; de Val-
Borro et al., 2017; Saladino et al., 2019). This implies that the accretion
rates predicted by the wind Roche lobe overflow model are expected to be
higher than those provided by the Bondi-Hoyle-Lyttleton model, but smaller
than those obtained with the atmospheric Roche lobe overflow model.

The accretion rate efficiency in the wind Roche lobe overflow model can be
expressed as (Abate et al., 2013)

$\eta_{\rm wind,wRLOF}\ =\
q^{2}\,\left(\,\frac{25}{9}\,\right)\,\left[\,-\,0.284\,\left(\,\frac{R_{\rm
dust}}{R_{\rm RL}}\,\right)^{2}\,+\,0.918\,\left(\,\frac{R_{\rm dust}}{R_{\rm
RL}}\,\right)\,-\,0.234\,\right]\ \leq 0.5\ ,$ (29)

where the maximum value $0.5$ comes from the results of hydro-dynamical
simulations (Mohamed & Podsiadlowski, 2007, 2012), and $R_{\rm dust}$ is the
dust condensation radius, corresponding the radius of the wind acceleration
zone, given by (Höfner, 2007)

$R_{\rm dust}\ =\ \left(\,\frac{R_{\rm d}}{2}\,\right)\,\left(\,\frac{T_{\rm
dust}}{T_{\rm eff}}\,\right)^{-(4+p)/2}$ (30)

where $T_{\rm dust}$ is the dust condensation temperature and $p$ is a
parameter characterizing wavelength dependence of the dust opacity.

Another way to enhance wind accretion is by means of asymmetric radiation-
driven winds from rotating red giants. Based on the wind compression disk
model (Bjorkman & Cassinelli, 1993), which explains the equatorial
circumstellar disks in rapidly rotating Be stars, it has been proposed that
gravitational focusing in SySts can be achieved if the radiation-driven wind
from the red giant donor is not spherically symmetric and is partially
confined to the equatorial plane around the donor (Skopal & Cariková, 2015).
Compared to Be stars, normal red giants are slow rotators. Despite that, red
giants in SySts rotate faster than isolated stars with comparable spectral
type (Zamanov et al., 2007; Zamanov et al., 2008), and this mechanism can
still work, leading to the formation of a circumstellar gaseous disk-like
component around the red giant donor. Therefore, part of the wind leaving the
red giant can be compressed to its equatorial plane, which increases the mass
loss around the equatorial plane, proving then accretion rate efficiencies
that are enhanced in comparison with the spherically symmetric case, i.e. the
Bondi-Hoyle-Lyttleton model.

The required assumption that winds are asymmetric seems to be consistent with
observations. For instance, based on the velocity profile of the wind from the
giant near the orbital plane of the eclipsing SySts EG And and SY Mus, it was
possible to conclude that their winds are not spherically symmetric (Shagatova
et al., 2016). In addition, the properties of the nebular [O iii]
$\lambda$5007 line in EG And provides further support for the wind being
focusing towards the orbital plane (Shagatova et al., 2021).

The accretion rate efficiency in the wind compression model can be expressed
as (Skopal & Cariková, 2015)

$\eta_{\rm wind,WC}\ =\ \eta_{\rm wind,BHL}\,\left(\,\frac{\rho^{\rm
c}(r)}{\rho^{\rm sph}(r)}\,\right)\ ,$ (31)

where $r$ is the distance between the center of the red giant and the WD,
$\rho^{\rm sph}(r)$ is the density of the spherically symmetric wind at a
distance $r$, and $\rho^{\rm c}(r)$ is the local density of the compressed
wind at a distance $r$ in the equatorial plane. The quantity $\rho^{\rm
c}(r)/\rho^{\rm sph}(r)$ also depends on the radius and rotational velocity of
the red giant, on the initial and terminal wind velocities, and on how the
wind is accelerated. For typical rotational velocities of red giants in SySts
(${\sim 6-10}$ km s-1, Zamanov et al., 2008) and wind terminal velocities
(${\sim 20-50}$ km s-1), the accretion rate efficiencies obtained with the
wind compressional model, in comparison with those from the Bondi-Hoyle-
Lyttleton model, can be enhanced by a factor of ${\sim 5-15}$ (Skopal &
Cariková, 2015).

We compare in Figure 9 the accretion rates onto the WDs in SySts predicted by
the atmospheric Roche lobe overflow model, the Bondi-Hoyle-Lyttleton model,
the wind Roche lobe overflow model, and the wind compression model. It is
clear from the figure that the Bondi-Hoyle-Lyttleton model cannot account for
the accretion rates in SySts. On the other hand, the wind Roche lobe overflow
model can reasonably well explain the SySts with orbital periods longer than a
few thousand days. However, for those systems with orbital periods shorter
than ${\sim 10^{3}}$ days, this model is most likely unable to provide the
required accretion rates, since in this orbital period range it provides
accretion rate efficiencies comparable to the Bondi-Hoyle-Lyttleton model
(Abate et al., 2013). The wind compression model is a good candidate to
explain SySts in this range, as it provides accretion rates ${\gtrsim
10^{-8}}$ M⊙ yr-1. As final comment, the SySt lifetimes predicted by these
gravitationally focused wind accretion models (${\sim 0.1-1}$ Myr) may also
help to bring into agreement predicted and observed numbers of Milky Way
SySts, although further and more detailed calculations are needed to better
address this issue.

###  6.4 Supergiant and Wolf–Rayet High-Mass X-ray Binaries

We have just described the modes of mass transfer in SySts and SyXBs. Given
that sg-HMXBs and WR-HMXBs are the massive counterparts of these systems, the
same modes could be in principle directly applicable to sg-HMXBs and WR-HMXBs.

Let us start with sg-HMXBs hosting NSs in which the supergiants are nearly
filling their Roche lobes, such as Cen X–3, SMC X–1, LMC X–4 (van der Meer et
al., 2007), and XMMU J013236.7$+$303228 (Bhalerao et al., 2012). The mass
transfer in these systems can be described by the atmospheric Roche lobe
overflow mode in which the extended photosphere of the supergiant is in
contact with the Roche radius. In this case, matter flows to the compact
object through the inner Lagrangian point and an accretion disk around the NS
is formed. Given that the donor masses in these systems are several times
higher than the masses of their NS companions, mass transfer will quickly
become dynamically unstable mass transfer, triggering CE evolution. The most
likely outcome of this situation is a complete coalescence of the system,
leading to the formation of a Thorne–Żytkow object (Thorne & Zytkow, 1975),
which would be a very cool red supergiant with a degenerate neutron core,
although these objects remain thus far hypothetical.

In other sg-HMXBs and WR-HMXBs, their donors are under-filling their Roche
lobes, and the most likely mode of mass transfer for them is gravitationally
focused wind accretion. An important implication of this mass transfer mode is
that it can naturally explain wind-fed ultraluminous X-ray sources (ULXs,
Kaaret et al., 2017), such as the WR-HMXB M101 ULX–1 (El Mellah et al.,
2019a). In addition, it also offers an explanation for the existence of
accretion disks in the WR-HMXBs IC 10 X–1 and NGC 300 X–1, since this model is
consistent with the formation of a wind-captured disk (El Mellah et al.,
2019b). Moreover, results from recent binary models indicate that the
contribution of wind-fed ULXs to the overall ULX population is up to ${\sim
75-96}$% for young (${<100}$ Myr) star-forming environments and up to ${\sim
49-87}$% for environments like the Milky Way disk (Wiktorowicz et al., 2021).
All this implies that Roche lobe filling is not the only mode of mass transfer
able to provide large accretion rates onto the NS/BH in ULXs.

As a final comment, there is potential important application of
gravitationally focused wind accretion in the context of HMXBs, which is
related to the formation of Be-HMXBs hosting BHs. Having the uncertainties in
the stability criterion for dynamical mass transfer in mind (Olejak et al.,
2021; Belczynski et al., 2021), binary models suggest that these systems
should form through an episode of CE evolution (Grudzinska et al., 2015). In
case there is no efficient mass and angular momentum transfer from the BH
progenitor to the B star, the post-CE binary will host a Wolf–Rayet star
orbiting a slowly rotating B star, which will not exhibit the typical
characteristics of Be-HMXBs. However, this problem may be overcome in case of
efficient wind accretion prior the CE evolution, such that the B star would
enter this event already as a rapidly rotating star. Further binary models,
with consistent spin-up prescription, are still needed to better evaluate this
possibility.

##  7 Conclusion

Multi-wavelength observations of several types of binaries as well as large
systematic surveys, such as the Sloan Digital Sky Survey, have provided new
observational constraints on theories of the formation and evolution of
accreting compact objects. Together with improved numerical modelling, these
constraints have led to substantial progress in our understanding of the
formation of these fascinating objects that have been outlined in the previews
sections of this review. Despite these improvements, there are still important
problems to be solved. In what follows, we will list the most critical open
questions and potential avenues towards obtaining answers.

For the formation of accreting compact objects, mass transfer from a giant
star to its companion star is fundamental. We clearly need to better
understand the conditions that lead to dynamically stable and unstable mass
transfer. This will only be possible if we find ways to test which assumptions
for the mass transfer equation, the radius of the donor and the Lagrangian
radius provide more realistic results. Comparison of detailed model
predictions with observed samples of stars that evolve through dynamically
stable mass transfer or common-envelope evolution may perhaps allow us to
reach that goal. Post-common-envelope binaries consisting of WDs and MS star
companions may be numerous enough and easy enough to characterize to provide
the required observational constraints. The increasing number of merging
binary black holes/neutron stars detected by the LIGO and Virgo
interferometers as well as combined gravitational wave and electromagnetic
observations of compact binary mergers will also provide useful and strong
multi-messenger constraints in the high-mass regime.

Very much related to the point above, we need to better understand common-
envelope evolution. If we ever want to understand the relative numbers of
accreting compact objects and different binaries hosting compact objects, we
need to be able to reliably predict the outcome of common-envelope evolution
for a given system, which apparently is currently not the case. To reach this
goal, on one hand we need to aim at developing modelling tools that cover the
different physical processes and time-scales involved in common-envelope
evolution. On the other hand, observational surveys that provide clean and
ideally volume limited samples of close binaries can provide key information
for theoretically unconstrained parameters. First steps in this direction have
been performed for white dwarf binaries with low-mass star companions, but
large, complete, and unbiased samples are still not available.

Dynamically stable non-conservative mass transfer is potentially the dominant
(but most likely not the only) formation mechanism for systems hosting white
dwarfs with orbital periods between ${\sim 100}$ and ${\sim 1000}$ days. These
systems are often found in eccentric binaries, which should not be the case if
our understanding of tidal evolution is correct. Despite the several proposed
mechanisms to enhance the eccentricity during dynamically stable non-
conservative mass transfer, they all fail to some extent. Either the theory of
tidal interaction and/or the eccentricity pumping mechanisms need serious
revision, or another more promising, still unknown, mechanism is needed. On
the top of that, we have virtually no idea how much mass and angular momentum
are lost during dynamically stable non-conservative mass transfer, which makes
predictions rather uncertain. Given that mass loss is a huge problem already
in single star evolution, it is unlikely that we will soon find the answer by
performing theoretical simulations. Decisive progress therefore again depends
to a significant degree on observational constraints from samples of post-
mass-transfer systems.

A problem equally serious as those listed above is that we do not properly
understand orbital angular momentum loss in close binaries with both accreting
and non-accreting compact objects. Problems related very likely to our
ignorance of magnetic braking and consequential angular momentum loss are
numerous when we confront standard prescriptions with observations, e.g. the
average white dwarf mass in cataclysmic variables is much larger than
expected, mass transfer rates in cataclysmic variables and low-mass X-ray
binaries inferred from observations are very different from those predicted by
our models, the predicted donor masses in black hole low-mass X-ray binaries
are significantly higher than observed, the observed orbital period minimum in
cataclysmic variables is significantly longer than predicted, the predicted
close millisecond pulsar binaries come from a range of the parameter space so
narrow that their formation becomes compromised.

Promising revisions in our understanding of the orbital angular momentum loss
due to magnetic braking have been proposed in the past years, such as the
convection and rotation boosted prescription, which provides stronger orbital
angular momentum loss rates and can potentially solve the major problems
related to the mass transfer rates in low-mass X-ray binaries and consequently
the formation of their millisecond pulsar binary descendants. In addition,
given that a dynamical origin of black hole low-mass X-ray binaries appears
unlikely, this new prescription perhaps in combination with consequential
angular momentum loss, e.g. due to circumbinary disks, seems a fruitful path
to explore in binary models aiming at reproducing their characteristics.
However, this prescription cannot be the solution for cataclysmic variables,
since observations suggest lower orbital angular momentum loss rates than
those predicted by the standard prescription, which implies a weaker magnetic
braking.

Any important question arising when all these systems are taken into account
is whether there is a universal magnetic braking prescription able to
simultaneously explain all of them. Given the intrinsic different donor types
in cataclysmic variables and low-mass X-ray binaries, a possible universal
prescription could be achieved if the donor structure is also taken into
account in the formulation of magnetic braking. The overwhelming majority of
donors in cataclysmic variables are M-type MS stars, while most donors in low-
mass X-ray binaries seem to be MS stars of earlier spectral types or
subgiants. If a magnetic braking prescription derived from first principles
could provide weaker orbital angular momentum loss rates for MS stars with
initial masses smaller than ${\sim 0.8}$ M⊙ and stronger for MS stars
initially more massive than that, then such a recipe would be a reasonable
candidate for a universal magnetic braking prescription.

All that said, despite the significant progress achieved during decades of
investigation, formation and evolution of accreting compact objects is a
remarkably interesting wide open field. Future observational and theoretical
efforts will hopefully improve our physical understanding of these systems.

## References

  * Abate et al. (2013) Abate C., Pols O. R., Izzard R. G., Mohamed S. S., de Mink S. E., 2013, A&A , 552, A26
  * Abate et al. (2015a) Abate C., Pols O. R., Karakas A. I., Izzard R. G., 2015a, A&A , 576, A118
  * Abate et al. (2015b) Abate C., Pols O. R., Izzard R. G., Karakas A. I., 2015b, A&A , 581, A22
  * Álvarez-Hernández et al. (2021) Álvarez-Hernández A., et al., 2021, MNRAS, 507, 5805
  * Bandyopadhyay et al. (1999) Bandyopadhyay R. M., Shahbaz T., Charles P. A., Naylor T., 1999, MNRAS, 306, 417
  * Bassa et al. (2009) Bassa C. G., Jonker P. G., Steeghs D., Torres M. A. P., 2009, MNRAS, 399, 2055
  * Bauer & Kupfer (2021) Bauer E. B., Kupfer T., 2021, ApJ, 922, 245
  * Belczynski (2020) Belczynski K., 2020, ApJL, 905, L15
  * Belczynski & Taam (2004) Belczynski K., Taam R. E., 2004, ApJ, 603, 690
  * Belczynski et al. (2016) Belczynski K., Holz D. E., Bulik T., O’Shaughnessy R., 2016, Nat., 534, 512
  * Belczynski et al. (2020) Belczynski K., et al., 2020, A&A , 636, A104
  * Belczynski et al. (2021) Belczynski K., et al., 2021, arXiv e-prints, p. arXiv:2108.10885
  * Belloni & Rivera (2021) Belloni D., Rivera L., 2021, in The Golden Age of Cataclysmic Variables and Related Objects V. p. 13 (arXiv:2008.12772)
  * Belloni et al. (2019) Belloni D., Giersz M., Rivera Sandoval L. E., Askar A., Ciecielag P., 2019, MNRAS, 483, 315
  * Belloni et al. (2020a) Belloni D., Schreiber M. R., Pala A. F., Gänsicke B. T., Zorotovic M., Rodrigues C. V., 2020a, MNRAS, 491, 5717
  * Belloni et al. (2020b) Belloni D., Mikołajewska J., Iłkiewicz K., Schreiber M. R., Giersz M., Rivera Sandoval L. E., Rodrigues C. V., 2020b, MNRAS, 496, 3436
  * Belloni et al. (2021) Belloni D., et al., 2021, ApJS, 256, 45
  * Bhalerao et al. (2012) Bhalerao V. B., van Kerkwijk M. H., Harrison F. A., 2012, ApJ, 757, 10
  * Bhattacharya & van den Heuvel (1991) Bhattacharya D., van den Heuvel E. P. J., 1991, Phys. Rep., 203, 1
  * Bjorkman & Cassinelli (1993) Bjorkman J. E., Cassinelli J. P., 1993, ApJ, 409, 429
  * Boffin et al. (2014) Boffin H. M. J., Hillen M., Berger J. P., Jorissen A., Blind N., Le Bouquin J. B., Mikołajewska J., Lazareff B., 2014, A&A , 564, A1
  * Bonačić Marinović et al. (2008) Bonačić Marinović A. A., Glebbeek E., Pols O. R., 2008, A&A , 480, 797
  * Bondi & Hoyle (1944) Bondi H., Hoyle F., 1944, MNRAS, 104, 273
  * Brandi et al. (2009) Brandi E., Quiroga C., Mikołajewska J., Ferrer O. E., García L. G., 2009, A&A , 497, 815
  * Breedt et al. (2014) Breedt E., et al., 2014, MNRAS, 443, 3174
  * Brooks et al. (2015) Brooks J., Bildsten L., Marchant P., Paxton B., 2015, ApJ, 807, 74
  * Camacho et al. (2014) Camacho J., Torres S., García-Berro E., Zorotovic M., Schreiber M. R., Rebassa-Mansergas A., Nebot Gómez-Morán A., Gänsicke B. T., 2014, A&A , 566, A86
  * Cannizzo & Pudritz (1988) Cannizzo J. K., Pudritz R. E., 1988, The Astrophysical Journal, 327, 840
  * Carney et al. (2001) Carney B. W., Latham D. W., Laird J. B., Grant C. E., Morse J. A., 2001, AJ, 122, 3419
  * Carter et al. (2013) Carter P. J., et al., 2013, MNRAS, 429, 2143
  * Casares et al. (1998) Casares J., Charles P., Kuulkers E., 1998, ApJL, 493, L39
  * Casares et al. (2009) Casares J., et al., 2009, ApJS, 181, 238
  * Casares et al. (2014) Casares J., Negueruela I., Ribó M., Ribas I., Paredes J. M., Herrero A., Simón-Díaz S., 2014, Nat., 505, 378
  * Chaty (2011) Chaty S., 2011, in Schmidtobreick L., Schreiber M. R., Tappert C., eds, Astronomical Society of the Pacific Conference Series Vol. 447, Evolution of Compact Binaries. p. 29 (arXiv:1107.0231)
  * Chen & Han (2008) Chen X., Han Z., 2008, MNRAS, 387, 1416
  * Chen et al. (2013) Chen X., Han Z., Deca J., Podsiadlowski P., 2013, MNRAS, 434, 186
  * Chen et al. (2021) Chen H.-L., Tauris T. M., Han Z., Chen X., 2021, MNRAS, 503, 3540
  * Cheng et al. (2018) Cheng Z., Li Z., Xu X., Li X., 2018, ApJ, 858, 33
  * Chomiuk et al. (2020) Chomiuk L., Metzger B. D., Shen K. J., 2020, arXiv e-prints, p. arXiv:2011.08751
  * Claeys et al. (2014) Claeys J. S. W., Pols O. R., Izzard R. G., Vink J., Verbunt F. W. M., 2014, A&A , 563, A83
  * Clark et al. (2002) Clark J. S., Goodwin S. P., Crowther P. A., Kaper L., Fairbairn M., Langer N., Brocksopp C., 2002, A&A , 392, 909
  * Clayton et al. (2017) Clayton M., Podsiadlowski P., Ivanova N., Justham S., 2017, MNRAS, 470, 1788
  * Coe et al. (2020) Coe M. J., Kennea J. A., Evans P. A., Udalski A., 2020, MNRAS, 497, L50
  * Cohn et al. (2010) Cohn H. N., et al., 2010, ApJ, 722, 20
  * Cohn et al. (2021) Cohn H. N., et al., 2021, MNRAS, 508, 2823
  * Cojocaru et al. (2017) Cojocaru R., Rebassa-Mansergas A., Torres S., García-Berro E., 2017, MNRAS, 470, 1442
  * Copperwheat et al. (2011) Copperwheat C. M., et al., 2011, MNRAS, 410, 1113
  * Corbet (1986) Corbet R. H. D., 1986, MNRAS, 220, 1047
  * Coriat et al. (2012) Coriat M., Fender R. P., Dubus G., 2012, MNRAS, 424, 1991
  * Corral-Santana et al. (2016) Corral-Santana J. M., Casares J., Muñoz-Darias T., Bauer F. E., Martínez-Pais I. G., Russell D. M., 2016, A&A , 587, A61
  * Crowther et al. (2010) Crowther P. A., Barnard R., Carpano S., Clark J. S., Dhillon V. S., Pollock A. M. T., 2010, MNRAS, 403, L41
  * Deloye et al. (2007) Deloye C. J., Taam R. E., Winisdoerffer C., Chabrier G., 2007, MNRAS, 381, 525
  * Deng et al. (2021) Deng Z.-L., Li X.-D., Gao Z.-F., Shao Y., 2021, ApJ, 909, 174
  * Dermine et al. (2013) Dermine T., Izzard R. G., Jorissen A., Van Winckel H., 2013, A&A , 551, A50
  * Dewi & Tauris (2000) Dewi J. D. M., Tauris T. M., 2000, A&A , 360, 1043
  * Di Salvo & Sanna (2020) Di Salvo T., Sanna A., 2020, arXiv e-prints, p. arXiv:2010.09005
  * Di Stefano (2010a) Di Stefano R., 2010a, ApJ, 712, 728
  * Di Stefano (2010b) Di Stefano R., 2010b, ApJ, 719, 474
  * Dieball et al. (2005) Dieball A., Knigge C., Zurek D. R., Shara M. M., Long K. S., Charles P. A., Hannikainen D. C., van Zyl L., 2005, ApJL, 634, L105
  * Doroshenko et al. (2020) Doroshenko V., Suleimanov V., Tsygankov S., Mönkkönen J., Ji L., Santangelo A., 2020, A&A , 643, A62
  * Dubus et al. (2018) Dubus G., Otulakowska-Hypka M., Lasota J.-P., 2018, A&A , 617, A26
  * Edgar (2004) Edgar R., 2004, New Astron. Rev., 48, 843
  * Eggleton (1983) Eggleton P. P., 1983, ApJ, 268, 368
  * El Mellah et al. (2019a) El Mellah I., Sundqvist J. O., Keppens R., 2019a, A&A , 622, L3
  * El Mellah et al. (2019b) El Mellah I., Sander A. A. C., Sundqvist J. O., Keppens R., 2019b, A&A , 622, A189
  * ElBadry et al. (2021) ElBadry K., et al., 2021, MNRAS, 505, 2051
  * Escorza et al. (2019) Escorza A., et al., 2019, A&A , 626, A128
  * Escorza et al. (2020) Escorza A., Siess L., Van Winckel H., Jorissen A., 2020, A&A , 639, A24
  * Falanga et al. (2015) Falanga M., Bozzo E., Lutovinov A., Bonnet-Bidaud J. M., Fetisova Y., Puls J., 2015, A&A , 577, A130
  * Fekel et al. (2007) Fekel F. C., Hinkle K. H., Joyce R. R., Wood P. R., Lebzelter T., 2007, AJ, 133, 17
  * Fekel et al. (2008) Fekel F. C., Hinkle K. H., Joyce R. R., Wood P. R., Howarth I. D., 2008, AJ, 136, 146
  * Fekel et al. (2010) Fekel F. C., Hinkle K. H., Joyce R. R., Wood P. R., 2010, AJ, 139, 1315
  * Fink et al. (2007) Fink M., Hillebrandt W., Röpke F. K., 2007, A&A , 476, 1133
  * Freire & Tauris (2014) Freire P. C. C., Tauris T. M., 2014, MNRAS, 438, L86
  * Galiullin & Gilfanov (2021) Galiullin I., Gilfanov M., 2021, A&A , 646, A85
  * Gänsicke et al. (2003) Gänsicke B. T., et al., 2003, ApJ, 594, 443
  * Gänsicke et al. (2009) Gänsicke B. T., et al., 2009, MNRAS, 397, 2170
  * Garofali et al. (2012) Garofali K., Converse J. M., Chandar R., Rangelov B., 2012, ApJ, 755, 49
  * Garofali et al. (2018) Garofali K., Williams B. F., Hillis T., Gilbert K. M., Dolphin A. E., Eracleous M., Binder B., 2018, MNRAS, 479, 3526
  * Garraffo et al. (2018) Garraffo C., Drake J. J., Alvarado-Gomez J. D., Moschou S. P., Cohen O., 2018, ApJ, 868, 60
  * Ge et al. (2020) Ge H., Webbink R. F., Chen X., Han Z., 2020, ApJ, 899, 132
  * Gilfanov & Bogdán (2010) Gilfanov M., Bogdán Á., 2010, Nat., 463, 924
  * Gilliland (1982) Gilliland R. L., 1982, ApJ, 263, 302
  * Glanz & Perets (2018) Glanz H., Perets H. B., 2018, MNRAS, 478, L12
  * Goliasch & Nelson (2015) Goliasch J., Nelson L., 2015, ApJ, 809, 80
  * González Hernández et al. (2012) González Hernández J. I., Rebolo R., Casares J., 2012, ApJL, 744, L25
  * González Hernández et al. (2014) González Hernández J. I., Rebolo R., Casares J., 2014, MNRAS, 438, L21
  * González Hernández et al. (2017) González Hernández J. I., Suárez-Andrés L., Rebolo R., Casares J., 2017, MNRAS, 465, L15
  * Gosnell et al. (2019) Gosnell N. M., Leiner E. M., Mathieu R. D., Geller A. M., Knigge C., Sills A., Leigh N. W. C., 2019, ApJ, 885, 45
  * Green et al. (2018) Green M. J., et al., 2018, MNRAS, 476, 1663
  * Grichener et al. (2018) Grichener A., Sabach E., Soker N., 2018, MNRAS, 478, 1818
  * Gromadzki & Mikołajewska (2009) Gromadzki M., Mikołajewska J., 2009, A&A , 495, 931
  * Gromadzki et al. (2013) Gromadzki M., Mikołajewska J., Soszyński I., 2013, Acta Astron., 63, 405
  * Grudzinska et al. (2015) Grudzinska M., et al., 2015, MNRAS, 452, 2773
  * Hameury (2020) Hameury J. M., 2020, Advances in Space Research, 66, 1004
  * Hameury et al. (1998) Hameury J.-M., Menou K., Dubus G., Lasota J.-P., Hure J.-M., 1998, MNRAS, 298, 1048
  * Han et al. (2002) Han Z., Podsiadlowski P., Maxted P. F. L., Marsh T. R., Ivanova N., 2002, MNRAS, 336, 449
  * Han et al. (2003) Han Z., Podsiadlowski P., Maxted P. F. L., Marsh T. R., 2003, MNRAS, 341, 669
  * Hansen et al. (2016) Hansen T. T., Andersen J., Nordström B., Beers T. C., Placco V. M., Yoon J., Buchhave L. A., 2016, A&A , 588, A3
  * Harlaftis & Greiner (2004) Harlaftis E. T., Greiner J., 2004, A&A , 414, L13
  * Heinke et al. (2013) Heinke C. O., Ivanova N., Engel M. C., Pavlovskii K., Sivakoff G. R., Cartwright T. F., Gladstone J. C., 2013, ApJ, 768, 184
  * Hernandez et al. (2021) Hernandez M. S., et al., 2021, MNRAS, 501, 1677
  * Hinkle et al. (2006) Hinkle K. H., Fekel F. C., Joyce R. R., Wood P. R., Smith V. V., Lebzelter T., 2006, ApJ, 641, 479
  * Hinkle et al. (2009) Hinkle K. H., Fekel F. C., Joyce R. R., 2009, ApJ, 692, 1360
  * Hinkle et al. (2019) Hinkle K. H., Fekel F. C., Joyce R. R., Mikołajewska J., Gałan C., Lebzelter T., 2019, ApJ, 872, 43
  * Hjellming & Webbink (1987) Hjellming M. S., Webbink R. F., 1987, ApJ, 318, 794
  * Höfner (2007) Höfner S., 2007, in Kerschbaum F., Charbonnel C., Wing R. F., eds, Astronomical Society of the Pacific Conference Series Vol. 378, Why Galaxies Care About AGB Stars: Their Importance as Actors and Probes. p. 145 (arXiv:astro-ph/0702444)
  * Höfner & Olofsson (2018) Höfner S., Olofsson H., 2018, A&AR, 26, 1
  * Howell et al. (2001) Howell S. B., Nelson L. A., Rappaport S., 2001, ApJ, 550, 897
  * Hoyle & Lyttleton (1939) Hoyle F., Lyttleton R. A., 1939, Proceedings of the Cambridge Philosophical Society, 35, 405
  * Huang (1966) Huang S.-S., 1966, Annales d’Astrophysique, 29, 331
  * Hurley et al. (2002) Hurley J. R., Tout C. A., Pols O. R., 2002, MNRAS, 329, 897
  * Hut (1980) Hut P., 1980, A&A , 92, 167
  * Iłkiewicz et al. (2019) Iłkiewicz K., Mikołajewska J., Belczyński K., Wiktorowicz G., Karczmarek P., 2019, MNRAS, 485, 5468
  * Illarionov & Sunyaev (1975) Illarionov A. F., Sunyaev R. A., 1975, A&A , 39, 185
  * Inight et al. (2021) Inight K., Gänsicke B. T., Breedt E., Marsh T. R., Pala A. F., Raddi R., 2021, MNRAS, 504, 2420
  * Isern et al. (2017) Isern J., García-Berro E., Külebi B., Lorén-Aguilar P., 2017, ApJL, 836, L28
  * Israelian et al. (1999) Israelian G., Rebolo R., Basri G., Casares J., Martín E. L., 1999, Nat., 401, 142
  * Istrate et al. (2014) Istrate A. G., Tauris T. M., Langer N., 2014, A&A , 571, A45
  * Ivanova (2018) Ivanova N., 2018, ApJL, 858, L24
  * Ivanova & Nandez (2016) Ivanova N., Nandez J. L. A., 2016, MNRAS, 462, 362
  * Ivanova et al. (2008) Ivanova N., Heinke C. O., Rasio F. A., Belczynski K., Fregeau J. M., 2008, MNRAS, 386, 553
  * Ivanova et al. (2013) Ivanova N., et al., 2013, A&AR, 21, 59
  * Johns Mulia et al. (2019) Johns Mulia P., Chandar R., Rangelov B., 2019, ApJ, 871, 122
  * Jorissen et al. (2016) Jorissen A., et al., 2016, A&A , 586, A158
  * Jorissen et al. (2019) Jorissen A., Boffin H. M. J., Karinkuzhi D., Van Eck S., Escorza A., Shetye S., Van Winckel H., 2019, A&A , 626, A127
  * Kaaret et al. (2004) Kaaret P., Alonso-Herrero A., Gallagher J. S., Fabbiano G., Zezas A., Rieke M. J., 2004, MNRAS, 348, L28
  * Kaaret et al. (2017) Kaaret P., Feng H., Roberts T. P., 2017, ARA&A, 55, 303
  * Kalogera et al. (2007) Kalogera V., Belczynski K., Kim C., O’Shaughnessy R., Willems B., 2007, Phys. Rep., 442, 75
  * Kalomeni et al. (2016) Kalomeni B., Nelson L., Rappaport S., Molnar M., Quintin J., Yakut K., 2016, ApJ, 833, 83
  * Kaplan et al. (2012) Kaplan D. L., Bildsten L., Steinfadt J. D. R., 2012, ApJ, 758, 64
  * Käppeler et al. (2011) Käppeler F., Gallino R., Bisterzo S., Aoki W., 2011, Reviews of Modern Physics, 83, 157
  * Karino (2021) Karino S., 2021, MNRAS, 507, 1002
  * Kashi & Soker (2011) Kashi A., Soker N., 2011, MNRAS, 417, 1466
  * Kawaler (1988) Kawaler S. D., 1988, ApJ, 333, 236
  * Kennea et al. (2021) Kennea J. A., Coe M. J., Evans P. A., Townsend L. J., Campbell Z. A., Udalski A., 2021, MNRAS, 508, 781
  * Kenyon (1986) Kenyon S. J., 1986, The Symbiotic Stars. Cambridge Astrophysics, Cambridge University Press, doi:10.1017/CBO9780511586071
  * Kenyon et al. (1993) Kenyon S. J., Livio M., Mikolajewska J., Tout C. A., 1993, ApJL, 407, L81
  * Khargharia et al. (2010) Khargharia J., Froning C. S., Robinson E. L., 2010, ApJ, 716, 1105
  * King & Kolb (1995) King A. R., Kolb U., 1995, ApJ, 439, 330
  * Klencki et al. (2020) Klencki J., Nelemans G., Istrate A. G., Pols O., 2020, A&A , 638, A55
  * Knigge et al. (2003) Knigge C., Zurek D. R., Shara M. M., Long K. S., Gilliland R. L., 2003, ApJ, 599, 1320
  * Knigge et al. (2011) Knigge C., Baraffe I., Patterson J., 2011, ApJS, 194, 28
  * Kolb & Ritter (1990) Kolb U., Ritter H., 1990, A&A , 236, 385
  * Koliopanos et al. (2013) Koliopanos F., Gilfanov M., Bildsten L., 2013, MNRAS, 432, 1264
  * Koliopanos et al. (2021) Koliopanos F., Péault M., Vasilopoulos G., Webb N., 2021, MNRAS, 501, 548
  * Korol et al. (2017) Korol V., Rossi E. M., Groot P. J., Nelemans G., Toonen S., Brown A. G. A., 2017, MNRAS, 470, 1894
  * Kremer et al. (2018) Kremer K., Chatterjee S., Rodriguez C. L., Rasio F. A., 2018, arXiv e-prints, p. arXiv:1802.04895
  * Kretschmar et al. (2019) Kretschmar P., et al., 2019, New Astron. Rev., 86, 101546
  * Kroupa (1995) Kroupa P., 1995, MNRAS, 277, 1491
  * Kruckow et al. (2016) Kruckow M. U., Tauris T. M., Langer N., Szécsi D., Marchant P., Podsiadlowski P., 2016, A&A , 596, A58
  * Lada & Lada (2003) Lada C. J., Lada E. A., 2003, ARA&A, 41, 57
  * Lasota (2001) Lasota J.-P., 2001, New Astron. Rev., 45, 449
  * Lasota (2016) Lasota J.-P., 2016, in Bambi C., ed., Astrophysics and Space Science Library Vol. 440, Astrophysics of Black Holes: From Fundamental Aspects to Latest Developments. p. 1 (arXiv:1505.02172), doi:10.1007/978-3-662-52859-4˙1
  * Lehmer et al. (2020) Lehmer B. D., et al., 2020, ApJS, 248, 31
  * Li et al. (1994) Li J. K., Wu K. W., Wickramasinghe D. T., 1994, MNRAS, 268, 61
  * Liebert et al. (2005) Liebert J., et al., 2005, AJ, 129, 2376
  * Liu & Li (2019) Liu W.-M., Li X.-D., 2019, ApJ, 870, 22
  * Liu et al. (2006) Liu Q. Z., van Paradijs J., van den Heuvel E. P. J., 2006, A&A , 455, 1165
  * Liu et al. (2013) Liu J.-F., Bregman J. N., Bai Y., Justham S., Crowther P., 2013, Nat., 503, 500
  * Liu et al. (2019) Liu D., Wang B., Ge H., Chen X., Han Z., 2019, A&A , 622, A35
  * Liu et al. (2021) Liu W.-M., Jiang L., Chen W.-C., 2021, ApJ, 910, 22
  * Long et al. (1981) Long K. S., Helfand D. J., Grabelsky D. A., 1981, ApJ, 248, 925
  * López-Cámara et al. (2019) López-Cámara D., De Colle F., Moreno Méndez E., 2019, MNRAS, 482, 3646
  * López-Cámara et al. (2020) López-Cámara D., Moreno Méndez E., De Colle F., 2020, MNRAS, 497, 2057
  * López-Cámara et al. (2022) López-Cámara D., De Colle F., Moreno Méndez E., Shiber S., Iaconi R., 2022, MNRAS, 513, 3634
  * Lü et al. (2012) Lü G. L., Zhu C. H., Postnov K. A., Yungelson L. R., Kuranov A. G., Wang N., 2012, MNRAS, 424, 2265
  * Lucy (1967) Lucy L. B., 1967, AJ, 72, 813
  * Lugger et al. (2017) Lugger P. M., Cohn H. N., Cool A. M., Heinke C. O., Anderson J., 2017, ApJ, 841, 53
  * Ma & Li (2009) Ma B., Li X.-D., 2009, ApJ, 698, 1907
  * MacDonald et al. (2014) MacDonald R. K. D., et al., 2014, ApJ, 784, 2
  * Maraschi et al. (1976) Maraschi L., Treves A., van den Heuvel E. P. J., 1976, Nat., 259, 292
  * Marks & Kroupa (2012) Marks M., Kroupa P., 2012, A&A , 543, A8
  * Marsh et al. (2016) Marsh T. R., et al., 2016, Nat., 537, 374
  * Martínez-Núñez et al. (2017) Martínez-Núñez S., et al., 2017, Space Sci. Rev., 212, 59
  * Mason et al. (2010) Mason A. B., Norton A. J., Clark J. S., Negueruela I., Roche P., 2010, A&A , 509, A79
  * Mason et al. (2011) Mason A. B., Norton A. J., Clark J. S., Negueruela I., Roche P., 2011, A&A , 532, A124
  * Mason et al. (2012) Mason A. B., Clark J. S., Norton A. J., Crowther P. A., Tauris T. M., Langer N., Negueruela I., Roche P., 2012, MNRAS, 422, 199
  * Mathieu & Geller (2009) Mathieu R. D., Geller A. M., 2009, Nat., 462, 1032
  * Matrozis et al. (2017) Matrozis E., Abate C., Stancliffe R. J., 2017, A&A , 606, A137
  * McAllister et al. (2019) McAllister M., et al., 2019, MNRAS, 486, 5535
  * Mestel (1968) Mestel L., 1968, MNRAS, 138, 359
  * Mikołajewska (2003) Mikołajewska J., 2003, in Corradi R. L. M., Mikolajewska J., Mahoney T. J., eds, Astronomical Society of the Pacific Conference Series Vol. 303, Symbiotic Stars Probing Stellar Evolution. p. 9 (arXiv:astro-ph/0210489)
  * Mikołajewska (2007) Mikołajewska J., 2007, Baltic Astronomy, 16, 1
  * Mikolajewska (2010) Mikolajewska J., 2010, arXiv e-prints, p. arXiv:1011.5657
  * Mikołajewska (2012) Mikołajewska J., 2012, Baltic Astronomy, 21, 5
  * Mikołajewska et al. (2017) Mikołajewska J., Shara M. M., Caldwell N., Iłkiewicz K., Zurek D., 2017, MNRAS, 465, 1699
  * Mikołajewska et al. (2021) Mikołajewska J., Iłkiewicz K., Gałan C., Monard B., Otulakowska-Hypka M., Shara M. M., Udalski A., 2021, MNRAS, 504, 2122
  * Miller-Jones et al. (2021) Miller-Jones J. C. A., et al., 2021, Science, 371, 1046
  * Miszalski et al. (2013) Miszalski B., et al., 2013, MNRAS, 436, 3068
  * Moe & Di Stefano (2017) Moe M., Di Stefano R., 2017, ApJS, 230, 15
  * Mohamed & Podsiadlowski (2007) Mohamed S., Podsiadlowski P., 2007, in Napiwotzki R., Burleigh M. R., eds, Astronomical Society of the Pacific Conference Series Vol. 372, 15th European Workshop on White Dwarfs. p. 397
  * Mohamed & Podsiadlowski (2012) Mohamed S., Podsiadlowski P., 2012, Baltic Astronomy, 21, 88
  * Moreno Méndez et al. (2017) Moreno Méndez E., López-Cámara D., De Colle F., 2017, MNRAS, 470, 2929
  * Muñoz-Darias et al. (2008) Muñoz-Darias T., Casares J., Martínez-Pais I. G., 2008, MNRAS, 385, 2205
  * Mukai (2017) Mukai K., 2017, PASP, 129, 062001
  * Nandez et al. (2015) Nandez J. L. A., Ivanova N., Lombardi J. C. J., 2015, MNRAS, 450, L39
  * Nebot Gómez-Morán et al. (2011) Nebot Gómez-Morán A., et al., 2011, A&A , 536, A43
  * Neijssel et al. (2019) Neijssel C. J., et al., 2019, MNRAS, 490, 3740
  * Nelemans (2013) Nelemans G., 2013, in Auger G., Binétruy P., Plagnol E., eds, Astronomical Society of the Pacific Conference Series Vol. 467, 9th LISA Symposium. p. 27 (arXiv:1302.0138)
  * Nelemans & Jonker (2010) Nelemans G., Jonker P. G., 2010, New Astron. Rev., 54, 87
  * Nelemans & Tout (2005) Nelemans G., Tout C. A., 2005, MNRAS, 356, 753
  * Nelemans et al. (2000) Nelemans G., Verbunt F., Yungelson L. R., Portegies Zwart S. F., 2000, A&A , 360, 1011
  * Nelemans et al. (2010) Nelemans G., Yungelson L. R., van der Sluys M. V., Tout C. A., 2010, MNRAS, 401, 1347
  * Nelemans et al. (2016) Nelemans G., Siess L., Repetto S., Toonen S., Phinney E. S., 2016, ApJ, 817, 69
  * Nelson et al. (2004) Nelson L. A., Dubeau E., MacCannell K. A., 2004, ApJ, 616, 1124
  * Nie et al. (2012) Nie J. D., Wood P. R., Nicholls C. P., 2012, MNRAS, 423, 2764
  * Nie et al. (2017) Nie J. D., Wood P. R., Nicholls C. P., 2017, ApJ, 835, 209
  * Nielsen et al. (2013) Nielsen M. T. B., Dominik C., Nelemans G., Voss R., 2013, A&A , 549, A32
  * Nomoto & Kondo (1991) Nomoto K., Kondo Y., 1991, ApJL, 367, L19
  * Nomoto et al. (2007) Nomoto K., Saio H., Kato M., Hachisu I., 2007, ApJ, 663, 1269
  * Ohlmann et al. (2016) Ohlmann S. T., Röpke F. K., Pakmor R., Springel V., 2016, ApJL, 816, L9
  * Olejak et al. (2021) Olejak A., Belczynski K., Ivanova N., 2021, A&A , 651, A100
  * Oomen et al. (2018) Oomen G.-M., Van Winckel H., Pols O., Nelemans G., Escorza A., Manick R., Kamath D., Waelkens C., 2018, A&A , 620, A85
  * Oomen et al. (2020) Oomen G.-M., Pols O., Van Winckel H., Nelemans G., 2020, A&A , 642, A234
  * Orosz et al. (2007) Orosz J. A., et al., 2007, Nat., 449, 872
  * Orosz et al. (2009) Orosz J. A., et al., 2009, ApJ, 697, 573
  * Orosz et al. (2011) Orosz J. A., Steiner J. F., McClintock J. E., Torres M. A. P., Remillard R. A., Bailyn C. D., Miller J. M., 2011, ApJ, 730, 75
  * Otani et al. (2018) Otani T., Oswalt T. D., Lynas-Gray A. E., Kilkenny D., Koen C., Amaral M., Jordan R., 2018, ApJ, 859, 145
  * Paczyński (1967) Paczyński B., 1967, Acta Astron., 17, 287
  * Paczyński (1976) Paczyński B., 1976, in Eggleton P., Mitton S., Whelan J., eds, IAU Symposium Vol. 73, Structure and Evolution of Close Binary Systems. p. 75
  * Pala et al. (2017) Pala A. F., et al., 2017, MNRAS, 466, 2855
  * Pala et al. (2020) Pala A. F., et al., 2020, MNRAS, 494, 3799
  * Pala et al. (2022) Pala A. F., et al., 2022, MNRAS, 510, 6110
  * Parsons et al. (2015) Parsons S. G., et al., 2015, MNRAS, 452, 1754
  * Patterson & Raymond (1985a) Patterson J., Raymond J. C., 1985a, ApJ, 292, 535
  * Patterson & Raymond (1985b) Patterson J., Raymond J. C., 1985b, ApJ, 292, 550
  * Pavlovskii & Ivanova (2015) Pavlovskii K., Ivanova N., 2015, MNRAS, 449, 4415
  * Paxton et al. (2011) Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton B., et al., 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton B., et al., 2015, ApJS, 220, 15
  * Paxton et al. (2018) Paxton B., et al., 2018, ApJS, 234, 34
  * Paxton et al. (2019) Paxton B., et al., 2019, ApJS, 243, 10
  * Pearlman et al. (2019) Pearlman A. B., Coley J. B., Corbet R. H. D., Pottschmidt K., 2019, ApJ, 873, 86
  * Perets & Kratter (2012) Perets H. B., Kratter K. M., 2012, ApJ, 760, 99
  * Pfahl et al. (2002) Pfahl E., Rappaport S., Podsiadlowski P., Spruit H., 2002, ApJ, 574, 364
  * Podsiadlowski (2001) Podsiadlowski P., 2001, in Podsiadlowski P., Rappaport S., King A. R., D’Antona F., Burderi L., eds, Astronomical Society of the Pacific Conference Series Vol. 229, Evolution of Binary and Multiple Star Systems. p. 239
  * Podsiadlowski & Rappaport (2000) Podsiadlowski P., Rappaport S., 2000, ApJ, 529, 946
  * Podsiadlowski et al. (2002) Podsiadlowski P., Rappaport S., Pfahl E. D., 2002, ApJ, 565, 1107
  * Podsiadlowski et al. (2003) Podsiadlowski P., Han Z., Rappaport S., 2003, MNRAS, 340, 1214
  * Podsiadlowski et al. (2010) Podsiadlowski P., Ivanova N., Justham S., Rappaport S., 2010, MNRAS, 406, 840
  * Pooley & Hut (2006) Pooley D., Hut P., 2006, ApJL, 646, L143
  * Pooley et al. (2003) Pooley D., et al., 2003, The Astrophysical Journal Letters, 591, L131
  * Porter & Rivinius (2003) Porter J. M., Rivinius T., 2003, PASP, 115, 1153
  * Postnov & Yungelson (2014) Postnov K. A., Yungelson L. R., 2014, Living Reviews in Relativity, 17, 3
  * Qin et al. (2019) Qin Y., Marchant P., Fragos T., Meynet G., Kalogera V., 2019, ApJL, 870, L18
  * Qiu et al. (2019) Qiu Y., et al., 2019, ApJ, 877, 57
  * Quaintrell et al. (2003) Quaintrell H., Norton A. J., Ash T. D. C., Roche P., Willems B., Bedding T. R., Baldry I. K., Fender R. P., 2003, A&A , 401, 313
  * Rafikov (2016) Rafikov R. R., 2016, ApJ, 830, 8
  * Ramsay et al. (2018) Ramsay G., et al., 2018, A&A , 620, A141
  * Rappaport & van den Heuvel (1982) Rappaport S., van den Heuvel E. P. J., 1982, in Jaschek M., Groth H. G., eds, IAU Symposium Vol. 98, Be Stars. pp 327–344
  * Rappaport et al. (1983) Rappaport S., Verbunt F., Joss P. C., 1983, ApJ, 275, 713
  * Rebassa-Mansergas et al. (2012) Rebassa-Mansergas A., et al., 2012, MNRAS, 423, 320
  * Reig (2011) Reig P., 2011, Ap&SS, 332, 1
  * Reimers (1975) Reimers D., 1975, Memoires of the Societe Royale des Sciences de Liege, 8, 369
  * Ricker et al. (2019) Ricker P. M., Timmes F. X., Taam R. E., Webbink R. F., 2019, IAU Symposium, 346, 449
  * Ritter (1988) Ritter H., 1988, A&A , 202, 93
  * Rivera Sandoval et al. (2018) Rivera Sandoval L. E., et al., 2018, MNRAS, 475, 4841
  * Saladino et al. (2019) Saladino M. I., Pols O. R., Abate C., 2019, A&A , 626, A68
  * Sana et al. (2012) Sana H., et al., 2012, Science, 337, 444
  * Sand et al. (2020) Sand C., Ohlmann S. T., Schneider F. R. N., Pakmor R., Röpke F. K., 2020, A&A , 644, A60
  * Savonije et al. (1986) Savonije G. J., de Kool M., van den Heuvel E. P. J., 1986, A&A , 155, 51
  * Schenker et al. (1998) Schenker K., Kolb U., Ritter H., 1998, MNRAS, 297, 633