11institutetext: Natalie Webb 22institutetext: Institute de Recherche en
Astrophysique et Planétologie, 9 avenue du Colonel Roche, 31028 Toulouse,
France 22email<EMAIL_ADDRESS>
# Accreting white dwarfs

Natalie Webb

###### Abstract

White dwarfs are the most common endpoints of stellar evolution. They are
often found in close binary systems in which the white dwarf is accreting
matter from a companion star, either via an accretion disc or channelled along
the white dwarf magnetic field lines. The nature of this binary depends on the
masses and the separation of the two stellar components, as well as other
parameters such as the white dwarf magnetic field and the nature of the stars.
This chapter looks at the formation of white dwarfs and intrinsic properties,
before looking at the different populations of accreting white dwarf binary
systems that exist. The chapter covers the characteristics of the various sub-
populations and looks at how they evolve. The means to discover and study
various sub-classes of accreting white dwarfs in the optical and other bands
is discussed, and the role of these systems in the broader astrophysical
context is considered. Future missions that will find new systems and new
populations are also reviewed. Finally some of the current open questions
regarding accreting white dwarfs are presented.

##  1 Keywords

Close binaries – Novae – Cataclysmic variables – White dwarfs – Accretion
discs – Surveys – Globular clusters – Supernovae Ia

##  2 Introduction

White dwarfs are formed at the end of the lives of the least massive stars and
are thus the most common endpoints of stellar evolution. White dwarfs were the
first type of compact object to be discovered. William Herschel made the first
detection of a white dwarf, 40 Eridani B in 1783 Herschel (1785). However, it
wasn’t until 1915 when Adams (1915) demonstrated that Sirius’ companion had a
mass of around a solar mass. However, it had a low luminosity similar to that
of 40 Eridani B Adams (1914), revealing that these stars were different from
the general population of stars observed. The approximately flat optical
spectrum ($\sim$4000-7000Å), along with the low luminosity, which indicates a
small radius, caused Luyten (1922) to refer to these stars as white dwarfs. By
1924 it was clear that white dwarfs were very different to other stars known
Eddington (1924), but it was only in 1931 that Chandrasekhar (1931) understood
the nature of the dense degenerate matter constituting these objects. Today,
around a hundred thousand white dwarfs are known. Many are found in the Solar
neighbourhood (e.g. Jiménez-Esteban et al, 2018) and some are young enough and
therefore bright enough to be visible through a pair of binoculars (e.g. 40
Eridani B, magnitude $\sim$ 9.5).

White dwarfs are characterised by small radii, typically similar to the radius
of the Earth, but with masses reaching 1.4 M⊙ Chandrasekhar (1931). This
implies high densities and therefore strong gravitational fields, which become
stronger as the mass increases (and the radius decreases, see Section 3.5), so
that it becomes necessary to use general relativity to understand their nature
and their environment. If a white dwarf is in a close binary system, matter
from the companion star may be lost through the inner Lagrangian point of the
binary (see Chapter Accreting white dwarfs) and become trapped in the
gravitational field of the white dwarf, allowing it to fall onto the white
dwarf, rendering it hotter and brighter than an isolated white dwarf of a
similar age. Studying the evolution of these systems can give an insight into
the white dwarf nature and help to understand the accretion phenomenon which
is found throughout the Universe, in stellar and planetary formation, as well
as around supermassive black holes in the hearts of galaxies.

This chapter covers white dwarfs and the nature of the different binary
systems containing these compact objects, as well as the different populations
of these systems, where and how they are found and how they can be used to
understand different phenomena in the Universe.

##  3 What is a white dwarf?

In main-sequence stars, i.e. stars burning hydrogen in their core, the
hydrostatic equilibrium is achieved thanks to the gas pressure opposing the
pressure due to the mass of the star. However, white dwarfs are no longer
undergoing fusion. For white dwarfs to find a stable equilibrium, a different
pressure is required and this is through electron degeneracy. The physics of
the electron degeneracy dictate the white dwarf characteristics and
properties, including the maximum mass allowed and how the white dwarf
responds to accreted matter. The electron degeneracy and its implications for
the nature of the white dwarf are described in the following subsections.

###  3.1 Electron degeneracy

The pressure, $P_{e}$, that electrons ($e^{-}$) can exert is a function of
their density ($\rho_{e}$) i.e. $P_{e}$ = $f(\rho_{e}$). To determine this
pressure, the number of free-electron states in a volume, $V$, are calculated.
For simplicity, a cube with the length of the side, $L$, is considered. This
means $V=L^{3}$. Duplicating this cube to fill all the space available and
allowing the wave vectors of the free-electron quantum states to take only
discrete values, the electron wave function is given by $\psi\propto
e^{(ik\textperiodcentered x)}$, when considering the three dimensions,
$x,y,z$, $k$ = ($k_{x}$,$k_{y}$,$k_{z}$), so that the replication of the cube
leads to:

$k_{x}=n_{x}\frac{2\pi}{L}\quad{\rm where}\quad n_{x}=1,2,\ldots$ (1)

The allowed states are separated by $2\pi/L$, so the density of the states in
$k$ space ($N$) is

$dN=g\frac{L^{3}}{(2\pi)^{3}}d^{3}k\hskip 56.9055pt(d^{3}k\equiv
dk_{x}dk_{y}dk_{z})$ (2)

where $g$ is the degeneracy factor for spin (spin can be either up
($\uparrow$) or down ($\downarrow$)).

Taking the de Broglie relationship for the electron’s momentum, $p$,
associated with its wave vector: $p=\hslash\ k$, where $\hslash$ is the Hslash
constant divided by 2$\pi$, the density of the states in $k$ space is
converted to momentum space using:

$dN=g\frac{L^{3}}{(2\pi\hslash)^{3}}d^{3}p$ (3)

The number density of particles per unit volume ($n$) with momentum states in
$d^{3}p$ is then

$dn=g\frac{1}{(2\pi\hslash)^{3}}f(p)d^{3}p$ (4)

where $f(p)$ is the mode occupation number, i.e. the number of particles in
the cube with a particular wave function. For bosons (e.g. photons), $f(p)$ is
not restricted, but fermions (i.e. electrons with spin angular momentum
$\hslash$/2) must obey the Pauli exclusion principle. This states that

$f(p)\leq 1$ (5)

The electron gas is thus different to a classical gas and the Pauli exclusion
principle limits the density of the electron gas.

###  3.2 The equation of state of electron-degenerate matter

Treating the particle momenta as a Maxwellian distribution, where each
velocity ($v$) component has a Gaussian distribution with a standard
deviation, gives the following equation

$\Psi(v)=\frac{1}{(2\pi\sigma^{2})^{3/2}}e^{(-v^{2}/2\sigma^{2})}d^{3}v$ (6)

The velocity dispersion ($\sigma$) is a function of the temperature, $T$,
through the equipartition of energy: $\sigma^{2}=k_{B}T/m$ (where $k_{B}$ is
the Boltzmann constant and $m$ the particle mass). Writing Eq. 6 in terms of
the number density of particles in a given momentum range and then multiplying
by the total number density of particles and using the momentum equation
$p=mv$ gives,

$dn=\frac{n}{(2\pi mk_{B}T)^{3/2}}e^{(-p^{2}/2mk_{B}T)}d^{3}p$ (7)

Comparing Eqs. 6 and 4 implies a critical density ($n_{crit}$) where the
classical description would violate the Pauli exclusion principle (Eq. 5) for
$p=0$.

$n_{crit}=\frac{g}{(2\pi)^{3/2}}\frac{(mk_{B}T)^{3/2}}{\hslash^{3}}$ (8)

The gas will then be in the classical regime at high temperatures and fixed
density, but quantum effects become important as $T\rightarrow$ 0\.
Integrating Eq. 4 over all momentum states leads to the total number density
of particles:

$n=\frac{g}{(2\pi\hslash)^{3}}\int{f(p)d^{3}p}$ (9)

Then as $T\rightarrow$ 0, the states are occupied only up to the Fermi
momentum ($p_{F}$). Hence:

$n=\frac{g}{(2\pi\hslash)^{3}}\int_{0}^{p_{F}}{d^{3}p=\frac{g}{(2\pi\hslash)^{3}}\frac{4\pi}{3}p^{3}_{F}}$
(10)

The Fermi momentum is then correlated with the particle density as shown in
Eq. 11,

$p_{F}=2\pi\hslash\left(\frac{3}{4\pi g}\right)^{1/3}n^{1/3}$ (11)

If the gas density increases, the Fermi momentum will also increase. The
additional particles are forced to fill the higher momentum states because the
lower momentum states have already been occupied.

The pressure from the electrons can be determined by considering the pressure
as a momentum flux and therefore the flux of electrons in the $x$-direction is
just the number of electrons crossing a unit area per unit time, or
$n_{e}v_{x}$. The pressure is then approximated by $P\sim p_{x}n_{e}v_{x}$.
Using Eq. 4 the pressure in the $x$-direction (which by isotropy must be equal
to the pressure in any direction) is

$P=P_{x}=\frac{g}{(2\pi\hslash)^{3}}\int p_{x}v_{x}f(p)d^{3}p$ (12)

Using spherical polar coordinates in momentum space

$\int{p_{x}v_{x}dp_{x}dp_{y}dp_{z}}=\frac{1}{3}\int{(p_{x}v_{x}+p_{y}v_{y}+p_{z}v_{z})dp_{x}dp_{y}dp_{z}}=\frac{1}{3}\int{p\textperiodcentered
v4\pi p^{2}dp}$ (13)

so

$P=\frac{g}{3}\frac{1}{(2\pi\hslash)^{3}}\int_{0}^{\infty}pvf(p)4\pi p^{2}dp$
(14)

As $T\rightarrow$ 0 (Eq. 10), electron speeds do not achieve relativistic
values so that $pv=p^{2}/m_{e}$, and

$P_{e}=\frac{4\pi
g}{3(2\pi\hslash)^{3}}\int_{0}^{p_{F}}\frac{p^{2}}{m_{e}}p^{2}dp=\frac{g}{30\pi^{2}\hslash^{3}m_{e}}p_{F}^{5}$
(15)

Rewriting the pressure as a function of density $\rho_{e}=m_{e}n_{e}$ using
Eq. 11

$p_{F}=\left(\frac{6\pi\hslash\rho_{e}}{gm_{e}}\right)^{1/3}$ (16)

and substituting into Eq. 15, the pressure for non-relativistic electrons can
be determined as follows:

$P_{e}=\frac{g}{30\pi^{2}\hslash^{3}}\left(\frac{6\pi^{2}\hslash^{3}}{g}\right)^{5/3}\rho^{5/3}m_{e}^{-8/3}=K_{1}\rho_{e}^{5/3}\hskip
8.5359pt$ (17) ${\rm where}\ \
K_{1}=\frac{\pi^{2}\hslash^{2}}{5m_{e}^{8/3}}\left(\frac{6}{g\pi}\right)^{2/3}$
(18)

As before, as the densities increase (Eq. 11), the Fermi momentum increases
until it reaches relativistic values. Some particles may be compelled into
momentum states with velocities approaching the speed of light, $c$. Replacing
$v$ with $c$ gives the relativistic expression

$P_{e}=\frac{4\pi g}{3(2\pi\hslash)^{3}}\int_{0}^{p_{F}}pcp^{2}dp=\frac{g\
c}{24\pi^{2}\hslash^{3}}p_{F}^{4}$ (19)

or

$P_{e}=\frac{gc}{24\pi^{2}\hslash^{3}}\left(\frac{6\pi\hslash^{3}\rho_{e}}{gm_{e}}\right)^{4/3}=K_{2}\rho_{e}^{4/3}\hskip
8.5359pt$ (20) ${\rm where}\ \ K_{2}=\frac{\pi\hslash
c}{4m_{e}^{4/3}}\left(\frac{6}{g\pi}\right)^{1/3}$ (21)

Equations 17 and 20 reveal that in a degenerate gas, the pressure depends on
the density alone and is thus independent of temperature. In a partially
degenerate gas a residual temperature dependence remains.

Only the electrons are considered in determining the pressure. In Eqs. 17 and
20 it can be seen that the particle mass is in the denominator. As the proton
is 1836 times more massive than the electron, its contribution to the pressure
is therefore negligible.

###  3.3 The Chandrasekhar mass

As the white dwarf is supported by electron degeneracy, there must be a limit
to the pressure that can be provided, simply because there is a limit to the
phase space density of the electrons. Therefore, it is intuitive that there is
a maximum mass for the white dwarf. This mass can be deduced by estimating the
energy density, $U$, of the degenerate gas in a similar way to that which was
done for the pressure. Integrating over momentum space and incorporating a
term, $\epsilon(p)$, to convey the energy per mode, the energy density is
given by

$U=\frac{g}{(2\pi\hslash)^{3}}\int_{0}^{\infty}\epsilon(p)f(p)4\pi p^{2}dp$
(22)

As $T\rightarrow$ 0, $f(p)=1$, but is zero otherwise. In the relativistic case
$\epsilon(p)=pc$. Using this in Eq. 22 and integrating gives rise to the
electron energy density ($U_{e}$)

$U_{e}=\frac{g}{(2\pi\hslash)^{3}}\frac{1}{4}4\pi cp_{F}^{4}$ (23)

Using Eq. 11 to express $U_{e}$ as a function of the electron number density
gives

$U_{e}=\frac{3}{4}\left(\frac{6\pi^{2}}{g}\right)^{1/3}\hslash cn_{e}^{4/3}$
(24)

The non-relativistic case for $\epsilon(p)=p^{2}/2m_{e}$ then gives

$U_{e}=\frac{3\hslash^{2}}{10m_{e}}\left(\frac{6\pi^{2}}{g}\right)^{2/3}n_{e}^{5/3}.$
(25)

The total kinetic energy ($E_{K}$) of the degenerate electrons is proportional
to their energy density multiplied by the volume. In the relativistic case,
$E_{K}\propto U_{e}V\propto n_{e}^{4/3}V\propto M^{4/3}/R$, where $M$ is the
mass and $R$ is the radius of the white dwarf. The gravitational energy
($E_{p}$) is proportional to $M^{2}/R$. The total energy ($E_{tot}$) is then
the sum of both terms : $E_{tot}=(AM^{4/3}-BM^{2})/R$, where $A$ and $B$ are
constants. For a set of masses, the total energy must be positive. The total
energy is reduced as the radius increases (the star expands) at which point
the electrons become only mildly relativistic. The star then exists as a
stable white dwarf. If however the mass increases, the total energy becomes
negative and the radius decreases until gravitational collapse is inevitable,
see Section 4.5 for more discussion on this point.

To determine the mass at which the electron degeneracy pressure is no longer
sufficient to support the white dwarf, $A$ and $B$ must be determined
Chandrasekhar (1931). These values depend on the molecular weight per electron
of the material inside the white dwarf and the density profile. The critical
mass at which the electron pressure is overcome was first determined by
Subrahmanyan Chandrasekhar Chandrasekhar (1931), and is named after him, the
Chandrasekhar mass. This was subsequently determined to be $\sim$1.4 M⊙
Ostriker et al (1966).

###  3.4 White dwarf formation

Stars with masses up to $\sim$8 - 10.6 M⊙, depending on the initial stellar
metallicity Maeder and Meynet (1989); Meynet (1991); Ibeling and Heger (2013);
Doherty et al (2015); Woosley and Heger (2015), leave behind white dwarfs at
the end of their lives. There are different types of white dwarf (see Section
3.5), and their nature is often related to the initial mass of the progenitor.
Stars on the main sequence of the Hertzsprung Russell diagram undergo hydrogen
fusion in their core, providing a force that counteracts the gravitational
force due to the mass of the star, so that the star remains in hydrostatic
equilibrium. Hydrogen fusion produces helium. When insufficient hydrogen
remains in the core to continue the fusion process, the star leaves the main
sequence with a helium-flash. For the lowest mass stars, there is insufficient
pressure and the temperature is too low for the helium to undergo fusion.
These stars leave behind remnants that are constituted mainly of hydrogen or
helium. Stars with a mass $\gtrsim$1.85 M⊙ Meynet (1991) can continue to burn
helium in a stable fashion and produce carbon-oxygen cores. Once insufficient
matter remains for the fusion process to continue, the gravitational force due
to the mass of the star dominates. The outer parts of the star are blown away
during the planetary nebula phase and the core will then collapse due to
gravity.

###  3.5 White dwarf characteristics

#### Rotation rates

In non-degenerate, i.e. normal matter, the volume increases with the mass. For
degenerate matter, the opposite occurs, so as the mass of the white dwarf
increases, the radius decreases (see also Sec. 3.3). This was described by
Nauenberg (1972),

$R=7.8\times 10^{8}{\rm
cm}\left[\left(\frac{1.44}{M}\right)^{2/3}-\left(\frac{M}{1.44}\right)^{2/3}\right]^{1/2}$
(26)

The radius of a solar mass white dwarf is approximately the same as the radius
of the Earth ($\sim$6000 km). Less massive white dwarfs have larger radii, for
example a 0.7 M⊙ white dwarf would have a radius of $\sim$7500 km.
Chandrasekhar mass white dwarfs have smaller radii, $\sim$1200 km. The compact
nature of white dwarfs implies that the gravitational force responsible for
holding it together ($GMm/R^{2}$) where $m$ is the mass of a particle on the
surface of the white dwarf, is extremely high. This means that white dwarfs
can rotate extremely rapidly without flying apart due to the centripetal force
($mv^{2}/R$). The velocity ($v$) at the surface of the star is $(2\pi r)/P$,
where $P$ is the period of the rotation of the star and therefore the
theoretical minimum rotational period of a white dwarf can be calculated

$\frac{GMm}{R^{2}}>\frac{m4\pi^{2}R}{P^{2}}\hskip 14.22636pt\Rightarrow\hskip
14.22636ptP>2\pi\sqrt{\frac{R^{3}}{GM}}$ (27)

From Eq. 27, the minimum period is for a minimum radius, which corresponds to
a white dwarf with the Chandrasekhar mass. This implies a minimum rotation
period of $\sim$0.5 s. Without energy injected into the system to maintain
this high rotation rate, the white dwarf will rapidly spin down. Indeed, no
white dwarf with such a short period has been detected to date, where the
shortest rotation period of a white dwarf is 24.93 s for the white dwarf in
the cataclysmic variable LAMOST J024048.51+195226.9 Pelisoli et al (2021), as
mass transfer in such a binary can transfer angular momentum to the white
dwarf, spinning it up, see Section 4.4.

#### Magnetic field

As angular momentum should be conserved as the star evolves to a white dwarf,
so the magnetic flux may also be conserved. Some main-sequence stars can show
magnetic fields as high as 3 $\times$ 102 \- 3 $\times$ 104 G1111 G (Gauss) is
the CGS unit for magnetic fields and corresponds to $10^{-4}$ T (Tesla).
Wickramasinghe and Ferrario (2010). The magnetic flux, $B$, is proportional to
the inverse of the radius squared and therefore magnetic fields of the order
108 G can be generated when the main sequence star collapses to the white
dwarf Wickramasinghe and Ferrario (2010). White dwarfs with lower magnetic
fields of 103-104 G may have had their fields generated by convective dynamos
during stellar evolution to the white dwarf state or for fields of 105-106 G,
through contemporary dynamos in cooling white dwarfs that develop extensive
convective envelopes.

#### Temperature and cooling

White dwarfs are born with high temperatures. The initial temperature can be
estimated using the contraction of the thermally unsupported stellar core down
to the radius at which degeneracy pressure will halt the contraction. The
virial theorem states that just before reaching the final point of
equilibrium, the thermal energy ($E_{th}=\frac{3}{2}Nk_{B}T$) will equal half
of the potential energy ($GM^{2}/R$). For a pure helium composition, the
number of nuclei in the core is $M/4m_{p}$, and the number of electrons is
$M/2m_{p}$, so

$E_{th}=\frac{3}{2}\frac{M}{m_{p}}\left(\frac{1}{2}+\frac{1}{4}\right)k_{B}T=\frac{9}{8}\frac{M}{m_{p}}k_{B}T$
(28)

Using the virial theorem,

$k_{B}T\sim\frac{4}{9}\frac{GMm_{p}}{R}$ (29)

for a solar mass white dwarf, its temperature at birth is $\sim 1\times
10^{9}$ K. This hot temperature implies that the emission peaks in the X-ray
range. Once the core becomes an exposed white dwarf, its radiation ionises the
gas blown off during the Asymptotic Giant Branch (AGB) phase, giving rise to
the planetary nebula discussed above. The white dwarf will start to cool.
Neglecting the envelope around the white dwarf and assuming a uniform
temperature throughout, the maximum cooling rate can be determined. The
radiative energy loss is obtained by equating the luminosity, $L$, of the
blackbody, given by the Stefan-Boltzmann law, to the rate of change of thermal
energy (Eq. 28)

$L=4\pi R^{2}\sigma
T^{4}\sim\frac{dE_{th}}{dt}=\frac{3}{8}\frac{Mk_{B}}{m_{p}}\frac{dT}{dt}$ (30)

where $\sigma$ is the Stefan-Boltzmann constant. Separating the variables $T$
and $t$ and integrating, the cooling time to reach a temperature $T$ is

$\tau_{cool}\sim\frac{3}{8}\frac{Mk_{B}}{m_{p}}\frac{1}{4\pi
R^{2}\sigma}\frac{1}{3}\frac{1}{T^{3}}\sim 3\times 10^{9}{\rm
yr}\left(\frac{T}{10^{3}K}\right)^{-3}$ (31)

In reality this is too rapid. The outer, non-degenerate envelope acts as an
insulator and the inner regions crystallise, modifying the cooling with time
(e.g. Wood, 1992). One of the interests in understanding exactly how white
dwarfs cool is that they have cooling times comparable to the age of the
Universe. It may therefore be possible to use these objects to determine the
history of star formation in our Galaxy by modelling their luminosity
function. Studying these systems in binary systems in which accretion onto the
surface of the white dwarf occurs (see Section 4), can help understand how
white dwarfs cool as the white dwarf is constantly being heated and cooled due
to increasing and decreasing accretion.

#### Composition

Most white dwarfs are composed of carbon and oxygen, but the atmosphere is
usually observed to be predominantly hydrogen ($\sim$83% of spectroscopically
confirmed white dwarfs). White dwarfs with hydrogen atmospheres are called DA
white dwarfs. 7% of white dwarfs (e.g. Ourique et al, 2019) show helium
spectra and are known as DO white dwarfs if they are hot enough to show He II
lines (from singly ionised helium) and DB if they only show He I lines (from
neutral helium). DC white dwarfs are too cool to show absorption lines in
their optical spectra ($\sim$5% of white dwarfs) and are believed to have
helium atmospheres which explains why hydrogen lines are not seen (e.g.
Ourique et al, 2019). DQ white dwarfs, showing carbon lines or bands, count
for only 1% of white dwarfs. White dwarfs with other types of metal lines in
their optical spectra are known as DZ white dwarfs and make up about 4% of the
population (e.g. Ourique et al, 2019). The atmosphere is always dominated by
the lightest element present, as the high surface gravity leads to a
stratification of the atmosphere.

Due to the high density of white dwarfs, the atmosphere also has a high
pressure. In the optical and ultra-violet spectra, broad line wings are
observed due to pressure broadening (or collisional broadening, Foley, 1946),
as collisions between atmospheric atoms reduce the effective lifetime of a
state, leading to broader lines. This is most notable in the Balmer lines seen
in the optical spectra of DA white dwarfs, but as white dwarfs are hot and
therefore also emit strongly at shorter wavelengths, the effect is also seen
in the Lyman lines in the ultra-violet, for example Barstow et al (2003).
Isolated white dwarfs can be detected in soft (low energy) X-rays, but the
spectra are thermal in nature and do not usually show any lines. X-ray
emission is stronger when the white dwarf is in a binary system accreting from
a companion star (see Section 4 and Chapter Accreting white dwarfs).

#### Observed masses and radii

The Montreal White Dwarf Database Dufour et al (2017) provides data for
$\sim$56000 spectroscopically confirmed white dwarfs. The average temperature
for the white dwarfs in this database is $\sim$7500 K, but with a large range
spanning $\sim$3000-180000 K. The mean mass of the sample is 0.57 M⊙, but with
the range spanning very low ($\sim$0.01 M⊙) masses up to masses in slight
excess of the Chandrasekhar mass, $\sim$1.49 M⊙. The average surface gravity
is log(g) $\sim$ 8, with the range spanning 3.1-10. The magnetic fields are in
the range 1 $\times$ 105 \- 9 $\times$ 108 G. These values are similar to the
almost complete sample of white dwarfs detected out to 100 pc with Gaia
Jiménez-Esteban et al (2018). This survey also gives an average white dwarf
radius of $\sim$ 0.012 R⊙ (8346 km), with a range 0.002-0.02 R⊙ (1391-13910
km), but with a possible bimodal distribution with maxima at 0.01 R⊙ (6955 km)
and at 0.0125 R⊙ (8694 km), which remains to be explained.

##  4 Accreting white dwarfs

The majority of stars are found in binary systems, where the two stars orbit
about their common centre of mass. The orbital period, or the time it takes
for the two stars to orbit each other can be thousands of years ($P_{orb}$).
These systems will evolve as described in the Chapter Accreting white dwarfs
and one or both of the stars can evolve to become a white dwarf, depending on
its initial mass, see Section 3.4. Due to the compact nature of the white
dwarf (see Section 3.1), the orbital period can become very short (see Section
4.5), as little as tens of minutes, so that the stars are very close, and they
will have a strong influence on one another. The different types of close
binary systems containing white dwarfs are discussed here.

###  4.1 Roche Lobe Overflow and accretion

A full account of accretion and ejection is beyond the scope of this Chapter
which are discussed in Chapters Accreting white dwarfs and Accreting white
dwarfs, however, several notions concerning these topics are necessary to
understand some of the aspects covered in this Chapter.

To understand a binary system and the mutual influence of the two stars, the
total potential (gravitational and centripetal forces), should be considered.
This is approximated by the Roche potential, $\Phi_{R}$ and assumes point like
stars in circular orbits. Considering Cartesian coordinates ($x,y,z$) that
rotate with the binary, the $x$-axis is described by the line traversing the
stellar centres (separated by a distance $a$ and of masses $M_{1}$ and
$M_{2}$), the $y$-axis is in the direction of orbital motion of the primary
and the $z$-axis is perpendicular to the orbital plane. The gravitational
potential $\Phi_{G}$ is then defined as

$\Phi_{G}=-\frac{GM_{1}}{(x^{2}+y^{2}+z^{2})^{0.5}}-\frac{GM_{2}}{[(x-a)^{2}+y^{2}+z^{2}]^{0.5}}$
(32)

and the centripetal potential $\Phi_{C}$ is written as

$\Phi_{C}=-\frac{1}{2}\Omega^{2}[(x-\mu a)^{2}+y^{2}]$ (33)

where $\mu$ is the mass of the second star ($M_{2}$) with respect to their
combined masses ($M_{1}+M_{2}$), also known as the mass function and $\Omega$
is $2\pi/P_{orb}$. The Roche potential can be written as

$\Phi_{R}=-\frac{GM_{1}}{(x^{2}+y^{2}+z^{2})^{0.5}}-\frac{GM_{2}}{[(x-a)^{2}+y^{2}+z^{2}]^{0.5}}-\frac{1}{2}\Omega^{2}[(x-\mu
a)^{2}+y^{2}]$ (34)

Close to the star, the potential is dominated by its gravitational potential,
so the stellar surfaces are almost spherical. Further from the stellar centre,
both effects become important: the tidal effect, which causes an elongation in
the direction of the companion star and flattening due to the centripetal
force. The surfaces are then distorted so that their major axis is along the
line connecting the stellar centres. The innermost equi-potential surface
encompassing both stars, defines the Roche lobe of each star. Matter inside a
Roche lobe is gravitationally bound to that star.

The topology of the surfaces can be determined by the Lagrange points, the
point at which the gradient of the potential becomes zero. There are five such
points where the effective gravity from the two stars and the centrifugal
force cancel. The inner Lagrangian point, $L_{1}$ (also called the ‘saddle
point’), forms a pass between the two Roche lobes. $L_{1}$, $L_{2}$ (situated
behind the Roche lobe corresponding to M2) and $L_{3}$ (behind the Roche lobe
corresponding to M1) are unstable, i.e. a small perturbation will displace a
body placed at these points. However, $L_{4}$ (situated above L1 in the
rotation plane) and $L_{5}$ (situated below L1 in the rotation plane) are
stable, i.e. resist gravitational perturbations.

If one star fills its Roche lobe exactly, e.g. M2, while the other, here the
white dwarf, is still smaller than its Roche lobe, the binary is said to be
semi-detached, or an interacting binary. If the star $M_{2}$ expands, it will
encounter a hole in its surface, near $L_{1}$. Here, hydrostatic equilibrium
is no longer possible and matter will flow through the nozzle around $L_{1}$
into the Roche lobe of the white dwarf. This is known as Roche-lobe overflow.
The matter that arrives in the white dwarf Roche lobe is gravitationally bound
to it. The matter has angular momentum, due to the rotation of the binary
system and so it will circle around the white dwarf, eventually forming a
ring. Due to friction in the ring, the material will spread into a disc shape,
known as an accretion disc.

The matter forms a thin disc around the white dwarf if the mass transfer rate
is low. In this case, the height of the disc is much smaller than the radius
(typically 0.1-3%). The matter loses angular momentum due to frictional forces
within the disc, allowing matter to fall onto the white dwarf, heating it up
and causing it to become brighter, either over the whole surface or at the
poles, depending on the magnetic field configuration, see Section 4.4. The
accretion disc is cool towards the exterior, mainly emitting in the infra-red,
but becomes hotter towards the centre where it achieves high enough
temperatures that result in X-ray emission. Each disc annulus emits a
blackbody spectrum that corresponds to the temperature of that annulus. The
accretion disc then emits the totality of the annuli, called a disc blackbody.
The lowest frequencies $\nu$ are the Rayleigh-Jeans tail emission from the
outer regions of the disc, where the intensity $I_{\nu}\propto\nu^{2}$. The
middle part of the spectrum results from the sum of the blackbodies
($I_{\nu}\propto\nu^{1/3}$) and the highest frequencies emanate from the inner
edge of the accretion disc with a temperature $T_{in}$, that has an
exponential cutoff, $I_{\nu}\propto e^{-h\nu/kT_{in}}$. Spectral lines may be
superposed on this emission.

Accretion is an efficient way of producing energy. As particles with mass $m$
fall from infinity onto the compact white dwarf, they lose their gravitational
potential energy and the change in energy $\Delta E_{acc}$ is given by

$\Delta E_{acc}=\frac{GMm}{R}$ (35)

where $M$ and $R$ are the mass and the radius of the white dwarf.

Outbursts occur as matter builds up in the disc, so that the density and
therefore the temperature increases. At a critical density, the matter,
essentially hydrogen if accreting from a hydrogen rich star, ionises, and thus
becomes more viscous, allowing matter to pass through the disc and fall onto
the white dwarf. This results in an important increase in the luminosity until
sufficient matter is evacuated from the disc, so that it becomes less dense
and cooler, thus ending the outburst. Theory predicts that the outbursts start
either in the outer regions of the disc and move inwards through the disc, and
are termed as outside-in or they start in the inner regions of the disc and
move outwards, and are called inside-out outbursts Smak (1984). It is unclear
prior to an outburst whether the outburst is going to be outside-in or inside-
out. It is likely that the type of outburst that takes place, is due to the
amount and location of material left in the disc after the previous outburst
Smak (1984). For a detailed treatment see Chapter Accreting white dwarfs.

###  4.2 Outflows and jets

Momentum can be extracted from the radiation field of the accretion disc
through line opacity Castor et al (1975), driving a wind (or outflow) from the
accretion disc surface. However, magnetic fields threading the accretion disc
can also play a role in launching the wind. Broad, blue-shifted, high
excitation absorption lines, along with P Cygni line profiles observed from
cataclysmic variables (novalikes and dwarf novae in outburst) seen face on are
evidence for winds from the accretion discs in these systems. These winds are
not thought to be completely radiatively driven, as wind line strengths are
not directly correlated with the strength of the photo-ionising continuum e.g.
Froning (2005).

Jets, collimated beams of high velocity material, often emitted perpendicular
to the accretion disc have been detected in some accreting white dwarf
systems. The energy to launch jets is thought to come either from the rotation
of the compact object Blandford and Znajek (1977) or from differential
rotation in the accretion disc Blandford and Payne (1982). Therefore, it is
logical to expect that jets may be launched from accreting white dwarfs if the
magnetic field is not so strong that it disrupts the central regions of the
disc. Indeed, jets have been detected from a wide range of accreting white
dwarfs, including supersoft sources, e.g. Tomov et al (1998) and in symbiotic
systems e.g. Taylor et al (1986). Alternative models also exist. Observational
evidence shows that the jets are often brightest in the radio domain, with
spectra indicating that the emission mechanism is synchrotron, supporting
these theories.

Evidence for jet emission from a cataclysmic variable, outside a nova
eruption, was first detected from SS Cyg Körding et al (2008). These jets
appear to be similar to jets detected in X-ray binaries (stellar mass black
holes or neutron stars accreting from a companion star), as radio flares best
explained by transient jets were detected as SS Cyg moved from quiescence to
outburst in the same way as X-ray binaries. The power associated with these
jets was analogous to X-ray binary jets. However, Hameury et al (2017) state
that the evolution throughout the outburst observed in cataclysmic variables
and that in X-ray binaries may be somewhat different. Indeed, although the
flaring/jet emission observed is generally similar to that observed in X-ray
binaries, there are a number of differences, notably the persistence of radio
emission in the plateau phase (peak of outburst) of dwarf novae outbursts.
Further, novalikes, which have persistently high accretion rates have also
been shown to have jet emission e.g. Coppejans and Knigge (2020).

Emission corresponding to jets has also been detected from several other
cataclysmic variables, see Coppejans and Knigge (2020) for a review. However,
one system, VW Hyi, has not shown evidence for jets, even though it is very
nearby ($\sim$54 pc Pala (2019)) and the reason for this is unclear. It may be
that the jet emission, which is transient, was simply missed. Alternatively,
it may be due to the mass of the white dwarf in this system which is
significantly less massive than the white dwarfs in the other systems
exhibiting jet emission, where the white dwarf in VW Hyi is 0.67 M⊙ and those
in the other systems have masses $>$0.8 M⊙ with the white dwarf in U Gem
measuring 1.2 M⊙ Ritter and Kolb (2011). This lower mass and thus larger
radius (see Section 3.3) will influence the accretion rate, where a lower
accretion rate implies less accretion power, which could lead to weaker jets
or even no jets.

Magnetic cataclysmic variables have also been seen to show radio emission
reminiscent of jets, but it is not yet clear if this radio emission is coming
from a jet or from somewhere else within the system (secondary star,
interacting matter between the two stars, etc) Coppejans and Knigge (2020).

###  4.3 Binary components and the diversity in accreting white dwarfs

The diversity in the white dwarf masses and magnetic fields, as well as the
diversity in the companion star mass and nature (main sequence or evolved star
or compact object) along with the evolutionary stage of the binary, gives rise
to a very wide range of accreting white dwarf systems. This diversity is
covered in the following sections.

###  4.4 Cataclysmic Variables

Figure 1: Different classes of accreting white dwarfs and their sub-classes.

The term Cataclysmic Variable (CV) describes a wide variety of semi-detached
close binary systems, where the primary is a white dwarf, accreting matter
from a companion (secondary) star via Roche Lobe overflow (see Section 4.1 and
Chapter Accreting white dwarfs). A block diagram showing the different classes
and sub-classes of accreting white dwarfs is given in Figure 1. CVs are
considered non-magnetic when the field is not strong enough to alter the
accretion flow, often taken to mean that the accretion disc extends all the
way down to the surface of the white dwarf.

The sub-classes of CVs, were originally defined from the shape of the optical
photometric lightcurves. Nowadays, spectroscopic and polarimetric observations
are used to classify the various CVs into sub-classes. Many of the sub-classes
can be categorised simply by the mass transfer rate and the type (if any) of
outbursts that they exhibit. By plotting the mass transfer rate against the
orbital period, CVs can be shown to fall roughly into different categories
Osaki (1996) and shown in Fig. 2. The main features of each category are
further described in the following subsections.

Figure 2: Different classes of dwarf novae shown with respect to their
approximate mass transfer rate as a function of orbital period. The period gap
is indicated as a grey vertical bar and the critical mass transfer rate (see
Section 4.4) is indicated as a dotted line.

#### Classical novae

Classical novae have been known for hundreds (or maybe thousands) of years as
they can be so bright, they were initially mistaken for new (nova) stars
appearing in the Universe, which explains the origin of their name. Classical
novae are now known to be luminous transients, brightening in the optical by
up to 9 magnitudes in a few days. This occurs when degenerate shells of
accreted material burn explosively (thermonuclear runaway) on the surface of
white dwarfs. The speed of the decline phase gives the nova class. These are
usually expressed as either t2 or t3, indicating the time to decay by either
two or three magnitudes from the peak magnitude. The classification is then
given by $very\ fast<$10 days, $fast$ 11-25 days, all the way to $very\ slow$
151-250 days. Shorter decay times usually indicate brighter novae.

In general, nova progenitors are novalike binaries, exhibiting high rates of
mass transfer to their white dwarfs before and after an eruption. At the
outset of a Classical nova, there is an optically thick, high velocity wind
and the mass loss decreases with time. As the photosphere recedes, the degree
of ionisation increases giving rise to stronger and stronger, higher emission
lines, coming from successively deeper shells with lower and lower velocities.
Thus the emission lines narrow. Absorption lines come from two distinct shells
that collide. The ejecta are enriched in carbon, nitrogen, oxygen and neon
Cassatella et al (2004). The typical mass of the ejected material, the nova
shell, is estimated at 10${}^{-5}-10^{-4}$ M⊙ Yaron et al (2005). The outburst
is so powerful that the nova shell is often observed for a long time after the
eruption.

The nature of the white dwarf, either carbon-oxygen or oxygen-neon, plays a
role in the type of ejecta. Analysing the ejecta can give insight into the
nucleosynthesis occurring during the outburst. As the radius of pseudo-
photosphere shrinks, the peak of the emission shifts from the optical to soft
X-rays. For some novae the photosphere does not recede to the regions within
the outflow that are hot enough to produce X-rays and they move directly to
what is called a supersoft source phase (described in the following).
Alternatively, they can move to this phase after an X-ray bright phase.

Some classical novae are recurrent, outbursting every few decades or so. Some
novae can also stem from symbiotic systems.

Supersoft sources

Following the classical nova explosion, a fraction of the envelope is ejected
and the expanding shell becomes optically thin to X-rays, giving rise to a
bright source of supersoft X-rays in some sources. This is powered by residual
hydrogen burning continuously on the surface of the white dwarf. As the
envelope mass is depleted, the photospheric radius decreases at constant
bolometric luminosity (close to the Eddington value, the luminosity at which
the force due to radiation pressure equals the gravitational force from the
mass of the star, supposing spherical accretion, and thus stopping a higher
accretion rate and thus higher luminosity, $\sim$ 1038 erg s-1 for white
dwarfs) with an increasing effective temperature. This leads to a shift of the
spectral energy distribution from the optical to the soft X-rays. This is
known as a supersoft source (SSS) Greiner (1996) and it has a hot white dwarf
atmosphere spectrum. They were first discovered with the Einstein observatory,
in the Large Magellanic cloud Long et al (1981).

The duration of this supersoft source phase is related to the nuclear burning
timescale of the remaining hydrogen rich envelope and depends, among other
factors, on the white dwarf mass Sala and Hernanz (2005). The typical duration
for this phase is a few months Pietsch et al (2005). Low resolution X-ray
spectra are well fitted with blackbodies with temperatures of the order of 105
K (emission $\lesssim$ 1 keV), but higher resolution X-ray spectra show broad
absorption lines such as carbon, oxygen and nitrogen, see Chapter Accreting
white dwarfs.

#### Dwarf novae and novalikes

The term dwarf nova is an all encompassing term for cataclysmic variables that
show (semi-)regular outbursts. These outbursts are fainter than those observed
from classical and recurrent novae. There are many types of dwarf novae, which
are described below. The novalikes set themselves apart from the other dwarf
novae as they are almost always in a hot state, similar to the novae
outbursts. However, they can occasionally be seen to fade as described below.

U Gem stars

U Geminorum (U Gem) stars have orbital periods longer than the top end of the
period gap (see Section 4.5) and have low mass transfer rates (low
${\rm\dot{M}}$), see Fig. 2. The secondary star is usually a K- or early
M-dwarf Ritter and Kolb (2011). U Gem stars exhibit regular outbursts,
primarily detected in the optical domain, where they increase in brightness by
2-5 magnitudes, for example the outbursts of AL Com, V544 Her, V660 Her, V516
Cyg and DX And Spogli et al (1998) and see also Fig. 6. U Gem outbursts occur
every few weeks or months and last for days-weeks, depending on the size of
the system and the mass transfer rates, e.g. U Gem (Sion et al, 1997) and SS
Cyg Cannizzo and Mattei (1998). The majority of outbursts observed from U Gem
stars are outside-in (see Section 4.1), but some inside-out outbursts have
also been observed, e.g. Webb et al (1999). The quiescent lightcurves of
eclipsing systems, show a brightening prior to eclipse, as the bright spot
(the region where material from the secondary impacts the accretion disc)
rotates into view. The spectra of U Gem stars taken in quiescence, show Balmer
and helium emission lines which are common for dwarf novae in general. The
emission lines can show a double-peaked profile, consistent with an accretion
disc.

SU UMa stars

SU Ursae Majoris (SU UMa) stars have periods below the period gap (see Section
4.5), or $\lesssim$2 hours and have low $\dot{M}$ Osaki (1996), see Fig. 2.
The secondary is usually a late-ish M-dwarf Ritter and Kolb (2011). SU UMa
systems exhibit semi-regular outbursts, but also have occasional super-
outbursts Wagner et al (1998). In a super-outburst, the system brightens
slightly more than it would in an ordinary outburst (up to about a magnitude
brighter), but the outburst lasts about 5 times longer than an ordinary
outburst. SU UMa systems are theoretically predicted to show only ‘outside-in’
outbursts Smak (1984). Evidence of the bright spot in the quiescent
photometric lightcurve is observed, e.g. OY Car Wood et al (1989) and the
spectra show emission lines typical of dwarf novae. SU UMa stars can also be
recognised by ‘superhumps’ seen in their photometric lightcurves, during
super-outburst, e.g. in VY Aquarii Patterson et al (1993) and Fig. 3. Due to
tidal instabilities, where the gas in the disc experiences the gravitational
force of the secondary, a brightening (hump) in the lightcurve is observed.
This occurs in systems where the mass ratio is $0.03<q<0.33$, where $q$ is the
mass of the secondary ($M_{2}$) divided by the mass of the primary ($M_{1}$).
The period of the hump is greater than the orbital period by a few percent.
‘Negative’ superhumps have also been observed, where their period is less than
the orbital period of the system e.g. Pavlenko et al (2021).

Figure 3: The V-band lightcurve of the cataclysmic variable Z Cha, taken from
the AAVSO database. The data were taken on 14th February 2022. Three eclipses
of V$\sim$1.4 depth are visible in the 5 hours of data. Shortly after each
eclipse, a $\sim$0.2 magnitude brightening is visible. These are known as
superhumps.

Z Cam stars

Z Camelopardalis (Z Cam) stars have orbital periods $\gtrsim$3 hours (above
the period gap) and have a higher mass transfer rate than either the SU UMa or
the U Gem stars, e.g. TZ Per Ringwald (1995). The mass transfer rate is
approximately the critical mass transfer rate, above which the system is
stable and no outbursts occur (as with the novalikes). Below the critical mass
transfer rate the system is unstable and the system exhibits outbursts, as in
the U Gem stars. Therefore, they exhibit outbursts and are also included in
the dwarf nova group, but these in turn can exhibit standstills, e.g.
Honeycutt et al (1998). A standstill can occur on the decline from outburst.
As the brightness of the disc fades, it can sometimes be seen to stall and
remain at the same brightness ($\sim$0.7 magnitudes fainter than the peak of
outburst) for weeks to years, e.g. in AT Cnc Nogami et al (1999). This is
because the mass transfer rate has increased above a critical rate and the
system has entered a quasi-stable state. Until the mass transfer rate drops,
the system will remain in standstill. Photometry of the standstills reveal
small fluctuations in brightness and some flickering e.g. Nogami et al (1999).
During standstill, the spectra of the Z Cam stars show the Balmer jump and the
Balmer series in absorption e.g. Nogami et al (1999). He II (4686Å) can also
sometimes be seen in emission.

ER UMa stars

ER Ursae Majoris (ER UMa) stars share many of the same characteristics as the
SU UMa stars, as they have an orbital period below the period gap ($\lesssim$2
hours). They are indeed often thought to simply be a sub-class of the SU UMa
stars, e.g. Gao et al (1999). Like the Z Cam stars, they have a mass transfer
rate that lies near to the critical mass transfer rate, see Fig. 2, and so
they also show standstills in some of their outbursts Kato and Kunjaya (1995).
Superhumps, however, are not just observed in the photometric lightcurve
during superoutburst, but have also been detected during ‘normal’ outbursts,
e.g. ER UMa Gao et al (1999).

Permanent superhumpers

These systems have short orbital periods, but have a mass transfer rate above
the critical mass transfer rate e.g. Retter and Naylor (2000) and see Fig. 2.
They therefore do not exhibit outbursts. There is evidence for a permanent
superhump in their photometric lightcurves.

Non-magnetic novalikes

There are many sub-classes of novalikes, but in general, they have orbital
periods longer than $\sim$3 hours and have a higher mass transfer rate than
even the Z Cam stars. They are therefore in a stable state and have never been
observed to go into outburst. Little or no evidence of a bright spot is seen
in the photometric or spectroscopic lightcurves. The optical/UV spectra show a
larger number of high-excitation emission lines than the other cataclysmic
variable, e.g. UU Aquarii Baptista et al (2000). At these high accretion
rates, the novalikes (like dwarf novae in outburst) have an optically thick
boundary layer. In this region close to the surface of the white dwarf, the
gas moving at Keplerian velocities is decelerated to the same speed as the
surface velocity of the white dwarf. Some of the gas arrives at the surface of
the compact object and spins it up through viscous shear stress Stoeger
(1980), but the rest, is simply dissipated in the boundary layer. As the
boundary layer is just a small thin region, the radiation emitted from this
region is high energy. Depending on the nature of the boundary layer, whether
it is optically thin or optically thick, and the rotation rate of the compact
object, the radiation emitted can be from the far ultra-violet to hard X-rays.
A radiation-driven accretion disc wind is also a common phenomenon among
novalike CVs.

For orbital periods between three and four hours, a class of CVs exist called
SW Sex stars. These systems show single Balmer emission lines as well as a
prominent HeII 4686 Å emission line. They show transient absorption at some
phases, and are often discovered as eclipsing systems. A few SW Sex stars show
low circular polarisation, indicating a relatively high magnetic field for the
white dwarf. Some novalikes, including the SW Sex stars, occasionally fade by
several magnitudes, for weeks to years. During these low states, the mass
transfer from the secondary star is interrupted, and the accretion disk
becomes very faint, or vanishes altogether. Consequently, the white dwarf and
the secondary star can be observed directly. The white dwarfs in these systems
appear to be very hot, which suggests that the high accretion rates observed
during the high states could persist for centuries Szkody and Gaensicke
(2012).

AM CVn binaries

AM Canum Venaticorum (AM CVn) stars are a class of compact binaries in which
the white dwarf accretes hydrogen-poor and helium-rich material from a
degenerate (or semi-degenerate) companion. These objects are characterised in
the optical/UV by their strong helium spectral lines; their lack of (or very
weak) hydrogen features and by their short orbital periods ($<$70 min), see
Fig. 5.

Many AM CVns show superoutbursts aswell as regular outbursts. These
superoutbursts are generally similar to those seen in the short period SU UMa
stars. During the superoutbursts, these systems also show superhumps. Thanks
to their strongly variable nature, recent sensitive large sky surveys have
revealed a significant population of these systems, e.g. Pichardo Marcano et
al (2021).

These systems are expected to be persistent gravitational wave sources that
will constitute a large proportion of the gravitational wave emission detected
by the future observatory LISA (Laser Interferometer Space Antenna), see also
Section 5.3. They will therefore be useful calibrators for the mission and
locating them and determining their periods, component masses etc will be
essential for fully exploiting this mission. A larger population will also be
discovered with LISA, with 2200-5200 expected in our galaxy Liu et al (2021).

#### Other non-magnetic CVs

Symbiotic stars

Symbiotic stars are wide, interacting binaries in which a compact object
accretes from a red giant companion. This section will focus on the systems in
which the compact object is a white dwarf. Due to the larger size of the
companion, the binary separation is larger than in the systems discussed
above, spanning a few to hundreds of astronomical units, thus the orbital
periods range from years to decades. They were first discovered in 1932,
thanks to optical spectroscopy revealing the TiO bands seen in the red giant
spectra, along with emission lines from the binary interaction Merrill and
Humason (1932), though at that time, their nature remained unexplained.

Matter is lost from the red giant to the white dwarf via the inner Lagrangian
point, either via a wind or through Roche lobe overflow. In some symbiotic
stars, the emission is generated by the release of gravitational potential
energy as matter falls onto the white dwarf. For the others, the emission
arises primarily from the quasi-steady burning of a hydrogen-rich shell on the
white dwarf surface e.g. Sokoloski et al (2017). The systems are highly
variable over the orbital period in all wavelengths. Ellipsoidal modulation in
the near infra-red is seen if the red giant fills its Roche lobe. In this
case, more emission is seen when the star is viewed side on, presenting a tear
drop shape, than when seen from behind, when it appears spherical. Modelling
this modulation can provide the inclination of the system, as a face on system
(inclination, $i$=0∘) shows no modulation and a side on system ($i$=90∘) shows
maximum modulation. Strong irradiation of the matter between the companion and
the white dwarf gives rise to strong Balmer emission lines, which vary
significantly as they rotate in and out of eclipse. In X-ray, they can show
similar low resolution spectra to the supersoft sources or harder sources
where the boundary layer is detected. Some show evidence for colliding winds
between the two components.

The systems also show longer term variability, showing outbursts as in the
cataclysmic variables and some have shown classical nova eruptions. In
symbiotic stars, the equivalent of polars should not exist due to the much
larger binary separation, but there is no obvious reason why IP-like systems
should not exist.

These systems are particularly interesting to study as they are wide and
therefore the timescales are much longer than in the systems discussed above,
making them easier to study. These systems are also potential progenitors of
type 1a supernovae.

Be star-white dwarf systems

In these systems, the white dwarf accretes from a B star showing emission
lines due to a circumstellar disc. Accretion usually occurs as the white
dwarf, which is in an elliptical orbit, passes through the disc at periastron.
Few such systems are known to date, although they are predicted to exist in
large number Raguzova (2001). These systems are particularly interesting from
an evolutionary point of view as more massive stars evolve more rapidly than
lower mass stars. As seen above, the maximum mass of a star that ends its life
as a white dwarf is 10.6 M⊙. The companion star should therefore be less
massive than this mass, ruling out a large number of the B stars.

Be-white dwarf binaries often show transient (as the white dwarf passes
through/ near the disc), soft X-ray spectra, similar to the supersoft sources.
However, the $\gamma$-Cas systems, also thought to be Be star binaries but
with very hard $\gamma$-ray spectra, may also be a part of this family, e.g.
Motch et al (2015).

#### Magnetic CVs

Polars and Intermediate Polars

These magnetic CVs used to be classified together with the novalikes. However,
when AM Her was shown to be linearly and circularly polarised Tapia (1977),
where previously only magnetised white dwarfs had been observed to be
polarised, a new sub-class of magnetic cataclysmic variables was realised.
Since then, many other CVs with magnetised white dwarfs have been discovered.
When the magnetic field of the white dwarf is strong enough to completely
disrupt the accretion disc, with fields typically of B$\gtrsim$107 G, they are
known as the AM Her stars or polars. Polars accrete without an intervening
accretion disk. The matter lost from the inner Lagrangian point of the
secondary follows a ballistic trajectory, and is subsequently captured by the
magnetic field of the white dwarf. It then follows the field lines onto the
poles of the white dwarf. This material heats up through strong adiabatic
shocks and cools subsequently through plasma emission in the X-ray domain and
optical cyclotron radiation. The field magnetically locks the white dwarf spin
period to the orbital period, so all variations are seen on the orbital
timescale.

Asynchronous polars can be found, but the difference between the orbital and
spin period are generally $<$1%. The other properties are common with those of
polars. For white dwarf magnetic fields of 1$\lesssim$B$\lesssim$10 MG, the
acretion disc is only partially disrupted, in the central regions, so material
that is accreted onto the white dwarf, does so via magnetic field lines that
connect the accretion disc to the white dwarf. These systems are known as
intermediate polars (IPs) or DQ Her type systems Patterson (1994). The white
dwarf magnetic moment in this case is insufficient to tidally lock the two
stars and the white dwarf rotates at higher frequencies than the orbital
frequency of the binary system (e.g. Warner, 1995). The spin period of the
primary is significantly shorter than the orbital period, so these systems are
often identified by pulsations on the spin period.

The secondary stars of the polars and intermediate polars (IPs) are late-type
stars Ritter and Kolb (2011). The photometric lightcurves show that these
magnetic CVs tend to hover about a maximum brightness, and occasionally, after
years, fall to a lower brightness, e.g. AM Her itself Watson et al (1980).
Some of the IPs, however, have not yet been observed to show a low state.
There is also flickering, some flaring and quasi-periodical oscillations
evident in the lightcurve, e.g. AM Her Beuermann et al (1991). The optical/UV
spectrum shows many emission lines Patterson (1994). In the X-ray, the
lightcurve shows either one or two bright pulses per orbital period, depending
on whether one or both poles are visible, when the heated polar column becomes
visible due to the viewing angle, see for example Fig. 4 and Chapter Accreting
white dwarfs. A single wide sinusoidal “pulse” can be visible if one pole is
always visible. A central, short duration, dip due to the eclipse of the white
dwarf is seen if the inclination is $\gtrsim$75∘. The short duration is due to
the small size of the white dwarf. With regards to the X-ray spectra, these
vary strongly from polar to polar, but with sufficient counts and resolution,
multi-temperature shock models with complex absorbers fit the data at higher
energies and some polars show very strong soft excesses. Many polars also show
an iron emission line at 6.4 keV, with equivalent widths of $\sim$150 eV Mukai
(2017). One way in which IPs are often identified is through the modulation of
the emission on the spin period of the white dwarf. A third period, in
addition to the orbital period (Porb) and the spin period (Pspin) is sometimes
observed. This is the beat period between the two periods, such that 1/P =
1/Pspin \- 1/Porb.

Magnetic CVs are interesting to study as the mass, radius and composition of
the white dwarf can be constrained Mukai et al (1990), and the accretion
processes can be probed Ramsay et al (2004). Further, short orbital period CVs
are often faint in the optical domain, but can be bright in X-rays if they are
magnetic. Using the high energy domain can then be helpful for identifying CVs
with the shortest orbital periods close to the period bounce, to help
understand how such objects evolve.

Figure 4: The 0.2–12.0 keV pn lightcurve of the polar 3XMM J183333.1+225136,
folded on the optical period of 7715.8 s and shown with bins of 128.6 s. Two
periods are shown for clarity. Only one pole is visible in this system,
causing the bright phase between from phases $\sim$0.8-1.2. An eclipse of the
white dwarf is seen at phase 0.05. Courtesy of Webb et al (2018).

###  4.5 Accreting white dwarfs in the broader astrophysical context

#### The origin and evolution of accreting white dwarfs

Binary systems containing accreting white dwarfs often evolve from a
primordial binary systems composed of stars of $\lesssim$8 M⊙ masses, see
Section 3.4 and Iben and Tutukov (1985). The most massive star will generally
evolve first to become a white dwarf, passing through the red giant phase,
that if the two stars are close enough, will lead to a common envelope, which
will permit angular momentum to be removed from the binary and allow the
binary to harden (the two stars will become closer), see Chapter Accreting
white dwarfs. This may allow the two stars to become semi-detached, see
Section 4.1, allowing them to interact as a Cataclysmic variable, see Section
4.4. Depending on the relative masses of the component stars, the binary
separation, the white dwarf magnetic field strength, the local environment, as
well as other characteristics, the different types of CVs, see Section 4.4,
will be formed. Later, however, the companion star can evolve through the red
giant phase, causing higher rates of accretion as te star overfills its Roche
lobe, or the system to become a symbiotic, see Section 4.4 and possibly a
second common envelope phase, removing further angular momentum from the
system. Eventually, the binary will become a double white dwarf system, see
Section 4.4, and again depending on the relative masses of the component
stars, the binary separation, the white dwarf magnetic field strength and the
local environment, accretion may continue, see Section 4.5.

Alternatively they can form binaries through tidal capture of a star in a
dense stellar environment, such as globular clusters (see Section 4.5) or
galactic centres. Binaries can subsequently encounter neighbouring stars. If
the binding energy of the binary is larger than the kinetic energy of the
star, angular momentum will be removed from the binary and it will harden.
Binaries can undergo a number of such encounters, which will have a strong
impact on their evolution. The third body leaves the encounter with the extra
energy extracted from the binary e.g. (Hut et al, 1992). This significantly
modifies the evolutionary path of the binary, see for example Ivanova et al
(2006), creating different population sizes of the same types of white dwarf
binaries created through primordial binary evolution, described above.

#### Observed orbital period distribution

White dwarf binaries have a very wide range of periods, dictated by the mass
of the two components and the separation that allows them to be semi-detached.
In reality, the Roche lobe filling star’s mean density, $\rho_{\odot}$, is
almost wholly responsible for determining the orbital period (in hours)
$P_{h}$, where $P_{h}=\kappa/\rho_{\odot}^{0.5}$ (where $\kappa\simeq$ 8.85 is
only a weak function of the mass ratio) e.g. Kolb and Baraffe (1999). As can
be seen in the Fig. 5, the orbital periods can be as long as 5 days and as
short as 5.4 minutes. For the shortest periods, these are the AM CVn systems,
see Fig. 5 (the blue bars). The secondary star in AM CVns is either a helium
rich white dwarf or a helium rich semi-degenerate star and therefore has a
small radius, allowing such short orbital periods. However, for CVs with main
sequence secondary stars, the periods are longer. Using the above equation, a
minimum period of $\sim$70 minutes is calculated. In the distribution of
periods shown in Fig. 5, the lower bound to the bulk of the CVs (which are not
AM CVn type systems) is $\sim$72 minutes, commensurate with this value.

Figure 5: Distribution of the orbital periods of white dwarf binaries taken
from Ritter and Kolb (2011). The period gap is clearly visible between
$\sim$2.3 and $\sim$2.8 hours, as is the period spike at around 1.2 hours.
Different types of systems are indicated, when the nature is known. The polar
category also includes intermediate polars.

The period spike

The donor star density on the main sequence increases as it tends towards
lower masses, as it loses matter to the white dwarf. Nearing the hydrogen-
burning minimum mass, the increasing electron degeneracy brings around
structural changes so that continued mass loss reduces the density. Therefore
as the donor star transitions from a main-sequence star to a brown dwarf, the
mean orbital period derivative changes from negative to positive (passing
through zero at period minimum, i.e. Ṗ${}_{orb}=0$), so the period begins to
increase instead of steadily decreasing e.g. Kolb and Baraffe (1999). This
means that there should be a large number of CVs around the period minimum,
known as the period spike. This is indeed seen in Fig. 5. However, X-ray
luminosity functions imply a larger, as yet unseen population of these faint,
short period CVs, e.g. Pretorius and Knigge (2012). The shortest period CVs
correspond to CVs with the lowest mass white dwarfs Ritter and Kolb (2011).

The period gap

There is a dearth of CVs, including IPs, with periods between approximately
2.3-2.8 hours, and this is called the period gap. It is thought to arise due
to disrupted magnetic braking. Magnetic braking (see Chapter Accreting white
dwarfs) is responsible for driving the evolution of CVs with periods greater
than approximately 3 hours. As the rotation of the secondary is slowed,
angular momentum is lost from the binary and the companion expands slightly to
maintain thermal equilibrium. At masses of 0.2-0.3M⊙, stars lose their
radiative cores and become fully convective Grossman et al (1974). The
magnetic field in solar-type, low-mass stars is expected to be anchored at the
interface between the radiative core and the convective envelope (the
tachocline, Charbonneau and MacGregor (1997); MacGregor and Charbonneau
(1997)). Since fully convective stars do not possess such an interface, the
magnetic field and thus the magnetic braking, are assumed to vanish at this
point Knigge et al (2011). It is then the gravitational radiation that drives
the evolution, which at these periods is less efficient, so the angular
momentum lost from the system and the mass transfer rate drop. The secondary
will shrink back to its original size and will thus be inside the Roche lobe,
so the binary becomes non-detached. Through the emission of gravitational
radiation removing energy from the binary system, the separation between the
two stars could decrease Rappaport et al (1983) and the secondary could
eventually come into contact again with its Roche lobe. Mass transfer could
then recommence Rappaport et al (1983). This would happen at an orbital period
of about 2.3 hours King (1988) and hence CVs are generally observed above and
below the period gap and not within it.

This would then imply that the companion stars just above and below the gap
should have identical masses, but different radii. Indeed Patterson et al
(2005) showed that there is a clear discontinuity in donor radii at $\simeq$
0.2M⊙ at the interface between the longer-period systems from the shorter-
period systems. Indeed, donors in systems just below the period gap have radii
consistent with regular main sequence stars of this mass, while the donors
just above the gap have radii that are inflated by $\simeq$ 30%. This gives
excellent support to the disrupted magnetic braking model.

Some magnetic systems have periods that fall within the period gap. As the
white dwarf in a polar rotates synchronously with the orbital period, so the
secondary star field lines that are normally open in other binary systems,
become connected with the field lines of the magnetic white dwarf. This
material, that would normally be forced to co-rotate with the secondary out to
large radii and hence cause magnetic braking, becomes trapped in what is known
as a ‘dead zone’ Wickramasinghe and Wu (1994). In such a way, magnetic braking
cannot be an efficient loss of angular momentum in polars. The orbital
evolution of polars should then be driven mainly by angular momentum loss due
to gravitational radiation Wickramasinghe and Wu (1994). This scenario
explains the higher proportion of polars in the period gap when compared with
other CVs, as there is no sudden cut-off in magnetic braking.

#### Exceeding the Chandrasekhar mass

As seen in the Section 3.3, the maximum mass for a white dwarf is the
Chandrasekhar mass. If the white dwarf accretes matter such that the mass
exceeds this limit, the electron degeneracy is not sufficient to support the
gravitational force due to this mass. Exceeding this mass implies that the
core will shrink further and the density will increase. At high densities, of
the order of 107 g cm-3, the highest energy electrons have energies greater
than the difference between the masses of the neutrons (mn) and the protons
(mp), mnc2 \- mpc2 = 1.294 MeV. At this time it becomes energetically
favourable for the protons to capture electrons via the weak-interaction i.e.
for inverse $\beta$-decay to occur (Eq. 36)

$e^{-}+p\rightarrow\ n+\nu_{e}$ (36)

where $p$ is the proton, $n$ is the neutron and $\nu_{e}$ is the electron-
neutrino. The inverse $\beta$-decay generates more neutrons than electrons and
protons due to the fact that the electrons still in the star are degenerate
and have filled all the available states, stopping new electrons from being
created via neutron $\beta$-decay. Thus a neutron star is created. This is
known as accretion induced collapse. At birth, the nucleons have free fall
energies, $\sim$100 MeV per nucleon ($\sim 10^{12}$ K). The neutron star cools
quickly through neutrino emission, so that after just a few seconds it is
$\sim 10^{11}$ K Miller (2013). The core continues to cool in such a way. This
is known as the Urca (name originating from the place at which the process was
elaborated) process (or Durca for direct Urca):

$n\rightarrow\ p+e^{-}+\bar{\nu_{e}}$ (37)

so the neutron star will have a temperature of the order of 109 K within
years. Over approximately a century, the surface temperature stabilises at
about 100 times the temperature of the core.

Before this can occur, carbon burning can take place. This process creates a
large instability, which at some point can turn into a runaway. The burning
then proceeds very rapidly through the star and can disrupt it completely
within seconds. Depending on how the burning proceeds, a luminous supernova,
known as type Ia supernova, can be produced, e.g. Goobar and Leibundgut
(2011). This process can be modelled and gives rise to a standard luminosity
for the supernova. Thanks to their high energy ($\sim$1046 J), they can be
detected out to cosmological distances. As the luminosity (L) is standard for
these events, the distance (d) to the supernova can be determined from the
measured flux, and knowing that the flux is L/(4$\pi$d2). The distances can
then be used to understand the local expansion rate of the Universe as well as
its structure. It is therefore important to understand how they occur, whether
it is simply through the accretion onto the white dwarf or through the
coalescence of two white dwarfs e.g. Iben and Tutukov (1985), how the burning
proceeds Goobar and Leibundgut (2011), etc.

Type 1a supernovae are detected at all wavelengths and the emission stems from
the radioactive decay of newly synthesised material, as well as Compton
scattering.

Despite this theory, there has been much debate in the literature as to
whether a white dwarf can indeed accrete so much mass as to reach the
Chandrasekhar mass (or close to this value). This is because if mass transfer
is too slow, novae occur, which appear to remove much of the accreted mass. If
the mass transfer is higher, hydrogen burns stably, but only a small part
avoids expansion and mass loss e.g. van Kerkwijk et al (2010) and references
therein. However, the average white dwarf mass in accreting white dwarf
binaries listed in Ritter and Kolb (2011) is 0.82 M⊙ with a dispersion of
0.23, with the lowest mass given of 0.16 M⊙ and the highest 1.95$\pm$0.3 M⊙
(CH UMa) which is massive even for a neutron star (M${}_{NS}>1.4$ M⊙) and
therefore maybe misidentified. The second highest is 1.55$\pm$0.24 M⊙,
commensurate with the maximum mass of a white dwarf, the Chandrasekhar mass
(see Section 3.3). All the other white dwarfs have masses between 0.4$\pm$0.1
- 1.3$\pm$0.3 M⊙, with a distribution close to a Gaussian distribution. These
masses are significantly higher than those of single white dwarfs, see Section
3.5.

The high masses observed do not appear to be seen in CV progenitors and they
can not be explained through selection effects, where more massive accreting
white dwarfs are brighter and thus easier to detect. This suggests that there
is indeed a mass increase in CVs. However, this is difficult to explain, as
our current understanding of CV evolution and thermal timescale mass transfer
suggests that such a mass increase is difficult to produce Wijnen et al
(2015). However, if additional angular momentum can be lost through the mass
transfer process, lower mass CVs may merge, leaving preferentially CVs with
higher masses Pala et al (2020). Such a model may also explain the low space
density of CVs observed (compared to current theory), as well as the number of
CVs observed above and below the period gap, which again is at odds with the
theory and the number of period bouncers observed, also compared to
theoretical predictions.

Some studies suggest that some type Ia supernovae originate from the merging
of two carbon-oxygen white dwarfs e.g. Liu et al (2016). Further, mergers of
white dwarfs with helium sub-giants may also produce type 1a supernovae Liu et
al (2016). The future low frequency gravitational wave mission LISA will make
huge advances in our understanding of the evolution of double white dwarf
systems and the Galactic merger rate, how the AM CVn systems form and evolve,
the effect of tidal coupling in white dwarfs and its role in stabilising mass
transfer, as well as the discovery of many new eclipsing white dwarfs Marsh
(2011), see also Section 5.3.

#### White dwarfs in globular clusters

Globular clusters are dense spherical systems of $\sim$104-106 old stars (e.g.
Hénon, 1961). Their old age implies that they should also contain many compact
objects. Through dynamical evolution, globular cluster cores should become
increasingly more concentrated (Hénon, 1961) and eventually collapse. This
core collapse should occur within $\sim$108-1010 years unless further energy
is supplied to the central regions of the cluster. Therefore, many Galactic
globular clusters should have already undergone core collapse. However,
observations reveal that $\sim$80% have not yet done so, see e.g. Harris
(1996).

Globular clusters should be home to both primordial binary systems and
binaries formed due to encounters in the dense environment. The interaction of
the binary with cluster stars (see Section 4.5) can inject extra energy into
the cluster and therefore be important in helping to delay globular cluster
core collapse.

The formation mechanisms of CVs in globular clusters is currently unclear. In
the field, collisions are rare and therefore CVs usually evolve from their
primordial binaries (see Chapter Accreting white dwarfs). However, in a
globular cluster, primordial binaries may have a wide separation and as they
interact with a cluster star and/or another binary, the primordial binary can
break up, especially if it is located in a dense cluster region, such as the
core. Alternatively, if initial binary is harder, successive encounters can
cause it to harden further. Instead the binary could undergo an exchange
encounter Davies (1997), where the encountered star exchanges into the binary
if it is more massive than one of the binary stars. A final alternative is
that a single white dwarf can encounter a cluster star. This encounter can
raise tides on the cluster star and transfer energy and angular momentum into
the envelope. If sufficient energy is transferred, the two objects will either
merge or form a binary system, which may become a CV Fabian et al (1975).

Due to mass segregation, binaries sink to the centre of the cluster Davies
(1997). Indeed the distribution of the binaries depends essentially on the
central density of the cluster. A higher core density will result in a larger
number of encounters, increasing the number of CVs formed through encounters,
whilst decreasing the primordial CV population. Primordial CVs are therefore
expected to be found outside the core and CVs formed through encounters should
be found inside the core. Belloni and Rivera (2021) show that there is a
bright and a faint population of CVs associated with core collapse globular
clusters, and they are correlated with position in the cluster. This may
provide some support to this idea.

One of the most noticeable differences between globular cluster CVs and field
CVs is the lack of outbursts observed in the cluster CVs. Many types of field
CVs (U Gem, SU UMa, Z Cam etc) show outbursts every few weeks to months, see
above. However, the evidence for outbursts from globular cluster CVs is poor,
with few outbursts detected. Amongst these are two outbursts of the CV V2 in
the globular cluster 47 Tuc Paresce and de Marchi (1994); Shara et al (1996).
Through imaging of the globular cluster M 5, Shara et al (1987) detected a
probable outburst and decline to quiescence of the CV V101. Naylor et al
(1989) confirmed the CV identification through optical observations in
outburst and quiescence. Ciardullo et al (1990) searched for nova outbursts,
as indicators for the presence of CVs in 54 of M 31’s globular clusters. With
an effective survey time of about two years, no cluster exhibited evidence for
a nova explosion.

It is unclear why there should be so few outbursts from globular cluster CVs,
as there is no reason physical reason for why CV activity should be suppressed
in these systems. Shara et al (1996) suggested that tidal capture may lead to
runaway mass transfer in almost all cases, which could explain the lack of CV
oubursts observed. Alternatively, this may result in low accretion rates, so
the time between outbursts could be long. Following the detection of two CVs
in an open cluster, where one of which underwent an outburst, Kaluzny et al
(1997) noted that it would be difficult to identify a CV undergoing a similar
type of outburst in a globular cluster if either photometric variability or
searching for strong Balmer emission objects was used to find CVs in globular
cluster centres. This could help to explain why few CVs in outburst have been
detected using optical observations. It has been suggested that the globular
cluster CVs detected using X-rays may be dominated by magnetic CVs, e.g. the 5
magnetic CVs discussed in Grindlay (1999) that were detected initially by
Rosat. Magnetic CVs have high Lx /Lopt ratios meaning that they are
preferentially detected when observing in X-rays, and they show few outbursts
because their accretion discs are disrupted by the magnetic fields Patterson
(1994). Indeed, the CVs observed with XMM-Newton for which there are X-ray
spectra all have fits that are consistent with the high temperature
bremsstrahlung models, typical of magnetic CVs e.g. Webb et al (2006). This
could help explain why the globular cluster CVs detected through X-ray
observations have not been seen to outburst.

Globular clusters may also give us insight into the evolution of cataclysmic
variables, as they may give rise to many accretion induced collapse systems
Ivanova et al (2008). Indeed, the slow pulsar 3XMM J004301.4+413017 in the M
31 globular cluster B091D is a good example of such a system Zolotukhin et al
(2017). Studying globular clusters may therefore help in understanding type 1a
supernovae.

##  5 Discovering accreting white dwarfs

Accreting white dwarfs are often found through their strong variability e.g.
Pala et al (2020) in almost all electromagnetic wavebands, either when they go
into outburst or through their variability over their orbital period or
sometimes due to the modulation on the spin period. Due to the accretion onto
the white dwarf, it can become very hot and radiate at high energies, notably
in the X-ray. Extreme events such as novae can also be detected in the
$\gamma$-ray domain Abdo et al (2010). Therefore, searching for high energy
(UV/X-ray/$\gamma$-ray) sources can be a good way to reveal these objects, as
few objects emit at such high energies. Dedicated surveys can also reveal new
accreting white dwarfs. The rest of this section discusses some of the surveys
that have been very successful in identifying new accreting white dwarfs and
how these systems have helped to understand the nature of these objects and
their effect on their environment.

###  5.1 Accreting white dwarfs found in optical surveys

#### Accreting white dwarfs found in the SDSS survey

The Sloan Digital Sky Survey (SDSS) York et al (2000), carried out with the
2.5-meter telescope at Apache Point Observatory, New Mexico, continues to
survey much of the northern hemisphere and has found almost 300 CVs in its
Legacy survey (2000-2008) Szkody et al (2011). This survey made a map in five
bands u, g, r, i and z of more than 8400 square degrees of the North Galactic
Cap and three stripes in the South Galactic Cap as well as more than 1.3
million medium resolution ($\sim$ 3 Å) spectra in the band 3800-9200Å. Small
numbers of additional CVs continue to be discovered through continuing
observations. The CVs were primarily identified through spectroscopic coverage
of selected objects from the photometric survey and the survey has provided an
unprecedented number of CV spectra. A wide variety of systems were therefore
discovered including polars, intermediate polars, novalikes, dwarf novae and
objects with pulsating white dwarfs Szkody et al (2011). Thanks to the
spectroscopic observations, periods of many of these sources have also been
determined. Given the depth of the survey and the fact that optical luminosity
functions become independent of X-ray luminosity functions at (2-3) $\times$
1030 erg s-1, all of the non-magnetic accreting white dwarfs should have been
identified in the SDSS footprint out to 400-600 pc Revnivtsev et al (2014).

The SDSS has also been responsible for finding many of the 18 known pulsating
white dwarfs in cataclysmic variables. The pulsations have been found in the
photometric lightcurves. Alternatively, the newly discovered CVs were shown
later to pulsate Szkody (2021). Through accretion, the white dwarfs heat up
and then cool again after the outburst. This allows them to transit the
instability region, between 11000 and 12500 K for a white dwarf of 0.6 M⊙ e.g.
Szkody (2021), when global pulsations associated with their normal modes
occur. Through asteroseismology, it is then possible to gain an insight into
the interiors of these stars

#### Accreting white dwarfs found in the Gaia survey

The Gaia mission Gaia Collaboration et al (2016) aimed to detect more than a
billion stars with excellent positional accuracy. Using two identical, three-
mirror anastigmatic telescopes, with apertures of 1.45 m $\times$ 0.50 m,
around 1.7 billion objects have been detected since 2013, with the majority of
them in our own galaxy Brown et al (2018). Amongst these, there is a
significant population of CVs, including new CVs, notably discovered through
their variability combined with colour and luminosity information. This has
revealed 3253 CV candidates, where most are below the period gap Schwab and
Bloom (2021). Gaia has also been extremely important for determining the
distance to many of these objects which enables the luminosity to be
determined, aiding the search for CVs and has allowed the space density of the
various classes of cataclysmic variables to be determined and compared to
theory, to hone our understanding of the evolution of these objects e.g.
Schwope (2018).

Abril et al (2020) have been able to identify different types of CV by
studying their position on Hertzsprung-Russell diagrams (luminosity versus
temperature). Amongst the results, a number of period bouncers and 9 detached
CVs that are currently in the period gap and have ceased to accrete (see
Section 4.5) have been discovered.

#### Accreting white dwarfs found in other optical surveys

The American Association of Variable Star Observers (AAVSO) is a non-profit
worldwide scientific and educational organisation of amateur and professional
astronomers who are interested in variable stars. It was founded in 1911 and
the members follow thousands of variable stars. This network has allowed many
accreting white dwarfs to be followed over the long term, see for example Fig.
6, as well as signalling the start of outbursts which have subsequently been
followed up by members and professional astronomers alike e.g. Webb et al
(1999). The AAVSO have also allowed simultaneous multi-band photometry of an
outburst to be taken, something that can be difficult to organise on
professional facilities alone e.g. Webb et al (1999). Some of the major
results of the AAVSO work on accreting white dwarfs have been presented in
Szkody and Gaensicke (2012).

Figure 6: The V-band lightcurve of the cataclysmic variable U Gem, taken from
the AAVSO database. The data span May 1994 to November 2021 (27.5 years) and
are made up of 57409 observations of the 204252 observations for this source
in the AAVSO database. The $\sim$1.5 magnitude eclipses are clearly visible,
as well as the outbursts in which U Gem brightens by as much as 5 magnitudes
in the V-band.

Several fully-automated, wide-field surveys have operated at the Palomar
observatory aimed at a systematic exploration of the optical transient sky.
This started with the Palomar Transient Factory (PTF) from March 2009 Rau et
al (2009), using an 8.1 square degree camera installed on the 48 inch Samuel
Oschin telescope with colours and light curves for detected transients
obtained with the automated Palomar 60 inch telescope. The intermediate
Palomar Transient Factory (iPTF) used an upgraded large field camera
subtending 7.8 square degrees on the sky. More recently, the Zwicky Transient
Facility (ZTF, Smith et al (2014)) took over with a three year, wide (field of
view 47∘2), three band photometric (g, r, i) survey, carried out with the
Palomar 48” telescope. Searching for blue (g-r $\leq$ 0.6) and variable
(magnitude change $\geq$ 2 magnitudes) within two days, sources, ZTF have so
far found 93 known and 279 new CV candidates since March 2018, at distances
between 108 pc and 2.096 kpc. Most of the new CVs are at the faint end of the
luminosity distribution and are therefore the lowest mass-transfer systems
Szkody et al (2021).

Unprecedented lightcurves of a few tens of CVs were taken over the nine years
of Kepler/K2 observations (2009-2018) e.g. Koch et al (2010), which was
designed to search for transits by exoplanets using optical photometry taken
with the 1.4 m space based telescope. Cadences as short as 1 minute provided
extremely detailed lightcurves of known and new CVs Molnár et al (2016). New
CVs were found through orbital variability as well as by detecting outburst
from systems with short outburst intervals. As of 2018, the Transiting
Exoplanet Survey Satellite (TESS) observatory Ricker et al (2016) was launched
and carries out a photometric (600-1000 nm) survey of a rectangular field of
24 deg $\times$ 90 degrees for 27.4 days at a time at a cadence of 30/10/2
minutes Pichardo Marcano et al (2021). TESS has observed many CVs with high
quality lightcurves. This has enabled a detailed study of the long-term
lightcurves of 9 AM CVns, showing that normal outbursts are more common than
previously thought, where it was believed that superoutbursts were more common
in these systems Pichardo Marcano et al (2021) and that the duration of the
latter are in fact shorter than that inferred from incomplete data. They have
also allowed quasi-periodic oscillations to be discovered in the light curve
of TX Col, a disc-less intermediate polar, that appear to be due to episodes
of enhanced accretion, forming a torus of diamagnetic blobs near the binary’s
circularisation radius. The QPOs would then be interpreted as beats between
the white dwarf’s spin frequency and unstable blob orbits within the white
dwarf’s magnetosphere. These blobs may then lead to the formation of a disc
Littlefield et al (2021). Many other new, as well as known CVs, symbiotics and
other accreting white dwarfs have now been studied in detail, with many more
results expected.

The Catalina Real-Time Transient Survey Djorgovski et al (2011) discovers
dwarf novae through their optical outbursts in the long time span of
observations. Observations are taken during the darkest 21 nights per 28 days,
in sets of four images, separated by 10 min. Exposures typically last for 30
seconds. Almost 2000 CVs have been detected since 2004 using the three
telescopes carrying out this survey, MLS a 1.5 m telescope has detected 1214
CVs/SN, CSS a 0.7 m telescope has detected 601/SN and SSS a 0.5 m telescope
has detected 109 CVs/SN. Many of these are new CVs and are detected out to
$\sim$ 4 kpc Drake et al (2014). Many of these systems are among the shortest
period CVs, demonstrating that deeper limiting magnitudes probe a different
population of systems compared to the known bright CVs. Drake et al (2014).

The All-Sky Automated Survey for Supernovae (ASAS-SN) network currently
consists of 24 telescopes, across the world and therefore surveys the whole
visible sky. Aside from supernovae, it discovers many other transients
Jayasinghe et al (2018), including accreting white dwarfs, where almost 3000
CVs and symbiotic systems have been detected, where machine learning methods
are used intensively to classify objects Jayasinghe et al (2019). Work on
individual subgroups of CVs has also been prolific, such as the analysis of
superhump maxima for 56 SU UMa-type dwarf novae Kato et al (2014), where these
data support the idea that disk tilt is responsible for modulating the
outburst pattern in SU UMa-type dwarf novae.

###  5.2 Accreting white dwarfs found in X-ray surveys

X-ray surveys can reveal different types of accreting white dwarfs compared to
the optical surveys. One advantage of the X-ray surveys is that many fewer
objects emit at such high energies, making it easy to pick out the accreting
white dwarfs. However, the X-ray/$\gamma$-ray data can be ambiguous and
follow-up observations are often important to investigate the nature of the
X-ray source.

As noted in Section 4.4, magnetic CVs are particularly bright in X-rays and
many of them have been discovered in this domain e.g. Mukai (2017); Webb et al
(2018) and references therein. X-ray observations with Rosat (0.1-2.4 keV,
1990-1999) Truemper (1982) and later with XMM-Newton (0.2-12.0 keV, since
1999) Jansen et al (2001) and Chandra (0.2-10.0 keV, since 1999) Weisskopf
(1999) of globular clusters have identified many CV candidates, notably a
significant number of magnetic systems e.g. Webb and Barret (2005) and
references therein and see Section 4.5. INTEGRAL/IBIS has now identified 929
objects of which 79 are CVs Krivonos et al (2021). Identifying the hard X-ray
sources detected with INTEGRAL/IBIS (15 keV-10 MeV, since 2003) Winkler et al
(2003) and the Swift/BAT (15-150 keV) Gehrels and Swift Team (2004), de
Martino et al (2020) have shown that about a quarter of the hard Galactic
X-ray sources are accreting white dwarfs, including dwarf novae, novae,
novalikes, symbiotic systems. They are dominated by intermediate polars (64%
of the 59 INTEGRAL/IBIS accreting white dwarfs and 55% of the 88 Swift/BAT
accreting white dwarfs) and polars (10% of the INTEGRAL/IBIS accreting white
dwarfs and 15% of the Swift/BAT accreting white dwarfs). They also show that
the intermediate polars reach luminosities of 1034 erg s-1, with a peak at 5
$\times$ 1034 erg s-1 in the 14-159 keV band. These systems also show a soft
blackbody (softer for the polars than the intermediate polars), thought to be
emanating from the heated polar region.

Thanks to these recent X-ray observations, there is strong evidence that the
Galactic ridge X-ray emission may actually be the result of a large population
of accreting white dwarfs, and not the result of X-ray illumination of gas
clouds, Revnivtsev et al (2009). This is in contrast to the Galactic centre
X-ray emission, which may be largely attributed to accreting magnetic white
dwarfs Hailey et al (2016).

The S-CUBED survey, is a $\sim$weekly 60 s survey of 6${}^{\circ}\times
6^{\circ}$ centred on the Small Magellanic Cloud (SMC) carried out with the
Swift X-ray telescope (0.3-10 keV). It has been ongoing since 2016 Kennea et
al (2018), and during this time a number of rare Be star-white dwarf binaries
(see Section 4.4) have been discovered, e.g. Coe et al (2020). The binaries
were found as they brightened during an outburst, in which they reach a few
percent of the Eddington luminosity. Only 4 such objects are known, even
though a large population is predicted Raguzova (2001), whereas more than 100
X-ray binaries have been found. Using such a survey may reveal many more of
these rare objects Kennea et al (2021).

Repeated surveys of large areas with soft X-ray telescopes such as XMM-Newton,
Chandra or Swift/XRT (0.3-10.0 keV, since 2004) are the ideal way to discover
Super soft sources, see Section 4.4. Less than 20 such sources are known, with
many in the Large Magellanic Cloud (LMC) discovered with the Einstein
observatory Long et al (1981) (0.15-4.5 keV, 1978-1981) or Rosat. They can be
difficult to disentangle from magnetic CVs in our own galaxies, and indeed in
a recent survey of the LMC a further four were discovered, although two maybe
magnetic CVs in the Milky Way Maitra and Haberl (2021). The large number of
supersoft systems in the Magellanic clouds may simply be thanks to the large
area surveys and the low column density which allows these sources with very
soft spectra to be detected. In a study of supersoft sources in 4 spiral
galaxies, 2 lenticular galaxies and 3 ellipticals Galiullin and Gilfanov
(2021) showed that there is about a factor 8 less SSS in early-type galaxies
compared to later type galaxies, in alignment with results from population
synthesis codes Galiullin and Gilfanov (2021).

###  5.3 Future surveys that will detect accreting white dwarfs

Amongst the future missions that will be interesting for studying accreting
white dwarfs, four are highlighted in this section. Although the X-ray
detector eRosita (0.2-10.0 keV) on board the observatory Spektrum-RG was
launched in 2019 Predehl et al (2021), the data will not be released for a few
years, so the results are yet to come. It will survey the whole sky several
times in its lifetime. An early release of a portion of the sky was made
recently, giving insight into the full data release that will occur in a few
years time. A number of interesting accreting white dwarfs have already been
published, e.g. SRGt 062340.2$-$265715, a new novalike CV at a distance of 495
pc, possibly an X-ray underluminous magnetic CV, an intermediate polar, or an
overluminous nonmagnetic CV Schwope et al (2021), but many more accreting
white dwarfs are expected. Schwope (2012) stated that eRosita should reach a
flux limit of about 10-14 erg cm-2 s-1 (0.5$-$2 keV) so it may be possible to
discover all CVs brighter than log LX = 29, 30, 31, 32 erg s-1 out to 0.3,
0.9, 3, and 9 kpc, respectively. Schwope (2012) estimated that as many as 9000
CVs will be detected in half of the sky or 18000 in the whole sky.

The Vera C. Rubin observatory Ivezić et al (2019) will start to observe the
sky systematically and repeatedly from early 2024. Using the bands $u,g,r,i,z$
and $y$, it will reach magnitudes over the 10 years as deep as 27.5 for the
$r$ band and as deep as 25th magnitude in $g$ for a single visit. The high
cadence observations will allow thousands of new accreting white dwarfs to be
discovered across the galaxy and the frequent observations will allow orbital
periods to be determined and outbursts to be studied. 2000 double degenerates
are also expected to be discovered through the identification of periodicities
Korol et al (2017) and studying their evolution will provide an insight into
the type Ia supernovae.

In the 2030s, two major missions from the European Space Agency will be
launched, the Laser Interferometer Space Antenna (LISA) Amaro-Seoane et al
(2017) that will detect gravitational waves in the frequency range from tens
of micro-hertz to 1 Hz and the Advanced Telescope for High Energy Astrophysics
(Athena) Nandra et al (2013) that will observe the sky with a sensitive
Silicon Pore Optics telescope in the 0.2-12 keV range. The most prevalent
sources detected with LISA will be Galactic double white dwarfs that are about
a hundred thousand years before merger e.g. Lamberts et al (2019). Using
cosmological simulations of our galaxy along with binary population synthesis
models Lamberts et al (2019) suggest that as many as 12000 double white dwarfs
could be detected with LISA. These include Helium-Helium systems, Helium-
Carbon/Oxygen systems and Carbon/oxygen-Carbon/oxygen systems. Athena will be
able to provide unique plasma diagnostics of the disk-white dwarf boundary
layer in CVs, and the magnetically-channelled accretion flows in
(intermediate) polars using the X-ray Integral Field Unit (X-IFU). Athena will
be sensitive enough to routinely measure the gravitational red-shifted iron Kα
line from the surface of the white dwarf which will give a constraint on the
mass of the white dwarfNandra et al (2013). Athena observations will measure
the temperature, mass, and chemical composition of novae and supersoft sources
to get insight into SN Ia progenitors Motch et al (2013).

##  6 Open questions regarding accreting white dwarfs

A number of different questions have been addressed in this Chapter and whilst
some of the questions posed have been partially answered, progress is required
to fully understand these problems. One of the most intriguing questions is
why are the white dwarf masses in accreting white dwarfs on average more
massive than those in the field? To determine this, the evolution of these
systems should be understood, notably the quantity of angular momentum lost.
It will also be necessary to understand the physics behind the mass increase,
be it through accretion, mergers or something else. This could also help
elucidate the low space density of CVs observed, as well as the number of CVs
observed above and below the period gap and the number of period bouncers
observed, compared to theoretical predictions. It may also give insight into
the mechanism for type 1a supernovae, which still needs to be understood.

Another area which requires a deeper understanding is the radio emission
associated with jet emission from these systems. Are the CV jets equivalent to
jets observed in other accreting systems? What is the physical process behind
the jet emission and why do some accreting white dwarfs not show jets?

A number of questions remain around the CVs in globular clusters, namely what
is the origin of these systems, are the centrally located systems really the
products of encounters and those located in the periphery evolved from
primordial binaries? Also, why are so few outbursts observed from these CVs in
globular clusters compared to those in the field?

Future observations will give further insight into the accretion processes in
accreting white dwarfs. Whilst this has not been discussed in detail in this
Chapter, it is still a subject of intense research as it will also help in
understanding the evolution of these systems and other accreting systems such
as supermassive black holes in the centres of galaxies. Upcoming surveys, as
described in the previous sections, will also provide a wealth of new
accreting white dwarfs and insight into these systems which will allow many of
the questions discussed here to be resolved in the next two decades.

##  7 Cross-References

Chapter Accreting white dwarfs : Overall accretion disk theory, Shane Davis
Chapter Accreting white dwarfs : Overall binary evolution theory, Diogo
Belloni
Chapter Accreting white dwarfs : White dwarfs, Kim Page
Chapter Accreting white dwarfs : Accretion process, Thomas J. Maccarone
Chapter Accreting white dwarfs : The extreme of SMBH accretion/ejection,
William Alston

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