# A catalogue of cataclysmic variables from 20 years of the Sloan Digital Sky
Survey with new classifications, periods, trends and oddities.

Keith Inight,1 Boris T. Gänsicke,1 Elmé Breedt,2 Henry T. Israel,6 Stuart P.
Littlefair,6 Christopher J. Manser,1,4 Tom R. Marsh,1 Tim Mulvany,1 Anna
Francesca Pala,3 John R. Thorstensen5
1Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
2Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, United Kingdom
3European Space Agency, European Space Astronomy Centre, Camino Bajo del
Castillo s/n, Villanueva de la Cañada, E-28692 Madrid, Spain
4Astrophysics Group, Department of Physics, Imperial College London, Prince
Consort Rd, London, SW7 2AZ, UK
5Department of Physics and Astronomy, Dartmouth College, Hanover NH 03755, USA
6Department of Physics and Astronomy, University of Sheffield, Sheffield S3
7RH, UK E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a catalogue of 507 cataclysmic variables (CVs) observed in SDSS I
to IV including 70 new classifications collated from multiple archival data
sets. This represents the largest sample of CVs with high-quality and
homogeneous optical spectroscopy. We have used this sample to derive unbiased
space densities and period distributions for the major sub-types of CVs. We
also report on some peculiar CVs, period bouncers and also CVs exhibiting
large changes in accretion rates. We report 70 new CVs, 59 new periods, 178
unpublished spectra and 262 new or updated classifications. From the SDSS
spectroscopy, we also identified 18 systems incorrectly identified as CVs in
the literature. We discuss the observed properties of 13 peculiar CVS, and we
identify a small set of eight CVs that defy the standard classification
scheme. We use this sample to investigate the distribution of different CV
sub-types, and we estimate their individual space densities, as well as that
of the entire CV population. The SDSS I to IV sample includes 14 period bounce
CVs or candidates. We discuss the variability of CVs across the Hertzsprung-
Russell diagram, highlighting selection biases of variability-based CV
detection. Finally, we searched for, and found eight tertiary companions to
the SDSS CVs. We anticipate that this catalogue and the extensive material
included in the Supplementary Data will be useful for a range of observational
population studies of CVs.

###### keywords:

white dwarfs, dwarf novae, cataclysmic variables

††pubyear: 2020††pagerange: A catalogue of cataclysmic variables from 20 years
of the Sloan Digital Sky Survey with new classifications, periods, trends and
oddities. –LABEL:lastpage

##  1 Introduction

Cataclysmic variables (CVs, see Warner 1995 for an exhaustive overview) are an
important class of white dwarf binaries within multiple astrophysical
contexts, including accretion disc physics (González Martínez-País et al.,
2014) and potential gravitational wave sources for LISA (Danzmann & LISA
Study, 1996). These compact binaries will become increasingly important as the
inventory becomes more complete and the emerging population statistics can be
used to test evolutionary models. Although $\sim 10\,000$ CV candidates are
known (see for example Watson et al. 2017) that sample contains a fair amount
of contaminants, including flaring M-dwarfs, young stellar objects, variable
AGN and unexplained transients (see discussion in section 2 of Pala et al.
2020). To be useful for evolutionary modelling a sample of CVs needs not only
to be reasonably complete over a given volume but also possess astrometric
data for each CV and, crucially, an identification spectrum.

The sample of 507 CVs spectroscopically observed by the Sloan Digital Sky
Survey (SDSS) (Gunn et al., 1998; York et al., 2000) presented here is a
magnitude-limited sample covering over 25.6 per cent of the sky (section
LABEL:sec:SDSSfootprint).

The four phases of SDSS (hereafter SDSS I to IV) were executed between 2000
and 2020 (Abazajian et al., 2003; Adelman-McCarthy et al., 2008; Aihara et
al., 2011; Aguado et al., 2019) and addressed many science goals of which the
search for quasars (Richards et al., 2002) and latterly luminous red galaxies
and quasars (Dawson et al., 2013) in the Baryon Oscillation Spectroscopic
Survey (BOSS and eBOSS) are of particular relevance here. Between 2000 and
2009 SDSS obtained photometry, using the five $ugriz$ filter passbands, of
over 14 000 square degrees, predominantly of the northern sky. The imaging
camera (Gunn et al., 1998) was installed on a purpose-built 2.5 m telescope at
Apache Point, New Mexico. This photometry was used to select targets for
subsequent spectroscopic surveys using a multi-object, fibre-fed spectrograph
capable of taking 640 (SDSS I and II) and 1000 (SDSS III and IV) simultaneous
spectra spanning $\simeq 3800-9200$ Å (SDSS I and II) and $\simeq
3600-10\,000$ Å (SDSS III and IV, Smee et al. 2013).

Whereas CVs were targeted within SDSS I via a multi-dimensional colour
selection (Szkody et al., 2002), the vast majority of CVs were serendipitously
identified from the spectroscopy of quasar candidates. SDSS I (York et al.,
2000) and II (Dilday et al., 2005; Yanny et al., 2009) have proven to be a
major source of optical spectroscopy of CVs, both of previously known systems
and new discoveries (Szkody et al., 2002, 2003, 2004, 2005, 2006, 2007, 2009,
2011).

SDSS III and IV were two subsequent phases spanning from July 2008 until June
2014 (Eisenstein et al., 2011; Dawson et al., 2013) and from July 2014 until
August 2020 (Blanton et al., 2017; Dawson et al., 2016) respectively.

SDSS I and II yielded spectra of 285 CVs (Szkody et al., 2011) and have been
particularly successful in identifying CVs with no history of outbursts
(Gänsicke et al., 2009). Specifically they provided a rich source of short
period ($80-86$ min) CVs that provided the crucial observational evidence for
the existence of a “period minimum spike”. Other groundbreaking achievements
include the identification of low accretion rate, X-ray weak, polars (Schmidt
et al. 2008 and references therein) and CVs with pulsating white dwarfs (Woudt
et al., 2005; Gänsicke et al., 2006; Mukadam et al., 2007). Beyond these
individual discoveries the systematic detection of all types of CVs has
enabled statistical studies of population density and CV evolution (Gänsicke,
2005). These achievements from SDSS I and II provided the motivation to
identify CVs in SDSS III and IV. Further motivation came from the opportunity
to refine the attributes of the known CVs using new astrometric and
photometric survey data.

We report here the identification of 70 new CVs found in SDSS I to IV(see list
in Appendix LABEL:section:appendix1). We have also revisited the remaining 437
CVs observed during SDSS I to IV in the light of Gaia (Gaia Collaboration,
2020) astronomy and photometry from the Catalina Real-Time Transient Survey
(CRTS, Djorgovski et al. 2011), Zwicky Transient Facility (ZTF, Masci et al.
2018; Bellm et al. 2019) and the Transiting Exoplanet Survey Satellite (TESS,
Ricker et al. 2015) which was not previously available. This has resulted in
59 new orbital periods (including some unpublished orbital periods from other
sources, details are provided in the supplementary data) and 262 new/revised
classifications. SDSS objects are identified by their coordinates, e.g. SDSS
J152212.20+080340.9, which we abbreviate throughout the paper as J1522+0803.

The paper is laid out as follows. We discuss CV evolution in section 2 and in
section 3 we provide an overview describing how the different types evolve in
the overall CV life-cycle using examples from SDSS I to IV. In section 4 we
describe the methods used to identify new CVs and classify existing ones. Our
results are described in section LABEL:section:results and discussed in
section LABEL:section:discussion with an overall summary in section
LABEL:section:summary.

##  2 CV evolution

Whilst we do not seek to replicate the extensive literature on CVs (a
comprehensive coverage can be found in Warner 1995 and Hellier 2001) we do
explain the evolutionary relevance of the different types of CV found by SDSS.

CVs are born out of a common-envelope event (Paczynski, 1976; Ivanova et al.,
2013) in which the higher-mass star in a binary system evolves off the main
sequence and becomes a giant, engulfing its lower-mass companion. Friction in
the common envelope extracts angular momentum and energy, resulting in a
dramatic reduction in the orbital separation of the two stars. The result of
the common envelope is a close binary consisting of a white dwarf (the core of
the former red giant) plus a low-mass companion; this is known as a post
common envelope binary (PCEB). Most PCEBs exit the common envelope in a
detached configuration (and many are observed in this state, e.g. Rebassa-
Mansergas et al. 2012). However, the stellar wind from the companion causes
magnetic braking (Verbunt & Zwaan, 1981) and results in a loss of angular
momentum. Meanwhile tidal forces on the secondary will eventually result in
tidal locking to the orbital period (Fleming et al., 2019). Magnetic braking
will eventually shrink the orbital separation to the point where the secondary
star fills its Roche lobe, and starts mass transfer onto the white dwarf –
initiating the further evolution of the system as a CV.

Magnetic braking continues into the CV phase resulting in the system
continuing to lose angular momentum and as a consequence the system separation
and $P_{\mathrm{orb}}$ will decrease.

At around $\mbox{$P_{\mathrm{orb}}$}=3$h the donor stars in CVs become fully
convective, possibly resulting in a re-configuration of their magnetic field
(see observational evidence in Reiners & Basri 2009 and also fig. 15 in
Kochukhov 2021). In the standard scenario of CV evolution (Knigge et al.,
2011), magnetic braking is strongly reduced, and the donor star, slightly
bloated due to prior mass loss, shrinks to its thermal equilibrium radius,
detaching from the Roche-lobe and stopping the mass transfer. The
observational motivation for this “disrupted magnetic braking scenario”
(Rappaport et al., 1983) is a paucity of CVs within the range
$\mathrm{2.15\,\mathrm{h}\lesssim}\mbox{$P_{\mathrm{orb}}$}\lesssim 3.18$ h;
this is the so-called “period gap” (Whyte & Eggleton, 1980; Howell et al.,
2001; Knigge, 2006). Some of this under-representation appears to be due to
selection effects (see for example fig. 9 in Inight et al. 2021 and section
LABEL:section:perioddistribution). In the case of strongly magnetic white
dwarfs, magnetic braking could be reduced, and these systems may not
experience a detached phase in their evolution (Webbink & Wickramasinghe,
2002).

For periods $\lesssim 3$ h, gravitational radiation (Paczyński, 1967; King,
1988) supersedes magnetic braking as the primary cause of angular momentum
loss, which will eventually cause the separation and orbital period to
decrease sufficiently for the donor to fill its Roche lobe and the system to
resume accretion as a CV.

Throughout the evolution of a CV the mass of the donor, the rate of mass loss
from the donor ($\dot{M}$), and $P_{\mathrm{orb}}$ all continue to decrease
until the donor mass is too low to sustain hydrogen burning and the system
becomes a brown dwarf. This point is the “period minimum” of the system
(Paczynski & Sienkiewicz, 1983). Following further mass loss from the donor
the orbital separation (and hence period) increases and it becomes a “period
bouncer” (e.g. Pala et al. (2018)).

The previous discussion has assumed that the donor is still on the main
sequence when it commences accretion. However the donors in some CVs have
undergone substantial nuclear evolution. Because the orbital period at which a
donor fills its Roche lobe is related to its density $\rho_{\mathrm{d}}$ via
$\mbox{$P_{\mathrm{orb}}$}\propto\left(\rho_{\mathrm{d}}\right)^{-0.5}$
(equation 12 in Knigge et al. 2011) it follows that, for the same donor mass,
CVs with nuclear evolved donors have longer orbital periods than their un-
evolved counterparts (e.g. Rodríguez-Gil et al. 2009). At periods $\lessapprox
5$ h, evolved donors have a higher temperature and spectral type than main
sequence stars of the same mass. They can appear in the period gap (Rebassa-
Mansergas et al., 2014) and evolve to periods below the period minimum
(Thorstensen et al., 2002b).

##  3 CV sub-types

“Cataclysmic Variables” were named as such because of the large brightness
variations (eruptions and outbursts) observed among classical novae and dwarf
novae. The modern definition of a CV dates from the 1960’s (see for example
Kraft 1963) and consists of a semi-detached binary in which a white dwarf
accretes from a Roche-lobe filling donor star – and the current roster of CVs
includes many systems that have no recorded outbursts (Gänsicke et al., 2009).

Figure 1:  Diagram depicting the CV classification scheme used in this paper.
The physical key parameters that define the boundaries between different
classes are $\dot{M}$ and $P_{\mathrm{orb}}$ (which, together, establish
whether or not the accretion disc is subject to thermal instabilities) and $B$
(which determines whether or not a disc forms in the first place). Depending
on the exact physical setup, some individual CVs may straddle the boundaries
between the sub-types defined above. AM CVn systems, with three possible
formation channels (Section 3.1.4), follow the same distribution as non-
magnetic CVs.

CVs are classified into a number of sub-types. Historically they have been
categorised based upon their observable characteristics with categories
typically being named after a prototype system, frequently the first
discovery. The number of sub-types, their definition, and relationship between
different sub-types remain in flux, making their use challenging for
scientists not regularly working in this research area – and some care has to
be taken with the temptation to define new sub-types for each additional
observational wrinkle in a particular CV. Nevertheless, dividing CVs into sub-
types is meaningful, as their observational characteristics depend on a small
number of physical parameters, which link to the evolutionary state and the
accretion geometry of an individual system (see Fig. 1).

Here we provide an overview of the main CV categories that have been
historically used with examples from SDSS I to IV and detail their state of
evolution and mode of accretion. The key physical parameters are the orbital
period, $P_{\mathrm{orb}}$, the rate of mass loss from the donor, $\dot{M}$,
and the white dwarf magnetic field strength, $B$.

The strength of the magnetic field of the white dwarf is of particular
importance in determining the accretion geometry (Norton et al., 2008). We
therefore discuss the classification and evolution of magnetic and non-
magnetic systems separately.

###  3.1 Non-magnetic CVs

####  3.1.1 Dwarf Novae

The dwarf nova thermal limit cycle starts with increasing mass being retained
in the disc which continues to grow until it becomes thermally unstable and a
dwarf nova occurs. The magnitude of a dwarf nova outburst is typically $\Delta
m=3-5$ (Meyer & Meyer-Hofmeister, 1983) and typically lasts for a few days
(although superoutbursts such as that of GW Lib in 2007 can reach $\Delta
m\approx 9$ and last several weeks) after which the disc cools and the cycle
repeats itself. Four sub-types are discussed below (SU UMa, WZ Sge, ER UMa and
Z Cam) with the remaining dwarf nova CVs being classified as U Gem. In
quiescence dwarf nova spectra (Fig. 2) exhibit hydrogen and also sometimes
helium emission lines whilst in outburst their spectra resemble those of
novalike variables (see below).

Figure 2:  J0813+4528 is an example of a dwarf nova. Top panel: The quiescent
SDSS spectrum (smoothed with a five-point box-car filter) shows strong
emission lines. The continuum reveals the spectroscopic signatures of the
donor star, indicative of a long orbital period (6.94 h); this is a U Gem type
dwarf nova above the period gap. Bottom left panel: ZTF ($g$-band) light curve
showing outbursts. Bottom right panel: Expanded light curve of the outburst
highlighted in red in the bottom left panel showing the decay over $\simeq 3$
d.

##### SU UMa

dwarf novae are characterised by occasional “superoutbursts” (Fig. 3)
typically lasting several weeks and recurring with a supercycle of a few
hundred days (Osaki, 1989; Buat-Ménard & Hameury, 2002; Patterson et al.,
2005b). These superoutbursts are significantly brighter than the regular dwarf
novae which occur in between the superoutbursts. Hirose & Osaki (1990) showed
that as the separation of CVs decreases tidal forces eventually cause the disc
to become eccentric and non-axisymmetric. The material in the outer edge of
the disc rotates more rapidly than the orbital period and will stabilise at a
resonant ratio of 3:1. The gradual accumulation of mass in the disc eventually
results in a superoutburst; Hirose & Osaki (1990) showed analytically that
superoutbursts only occur when $q=M_{\mathrm{d}}/M_{\mathrm{WD}}\lesssim 0.3$,
with $M_{\mathrm{WD}}$ and $M_{\mathrm{d}}$ the masses of the white dwarf and
of the donor star. Assuming a typical value of $M_{\mathrm{WD}}\simeq
0.8\,\mbox{$\mathrm{M}_{\sun}$}$ and using the relation $M_{\mathrm{d}}\approx
0.065P_{\mathrm{orb}}^{5/4}$ (Warner 1995 section 2.8) implies
$\mbox{$P_{\mathrm{orb}}$}\lesssim 2.8$ h. After reaching the peak luminosity,
the superoutburst light curves are modulated by periodic oscillations called
superhumps. The superhump period is a few per cent longer than
$P_{\mathrm{orb}}$ and the relation $P_{\mathrm{orb}}=0.9162\times
P_{\mathrm{sh}}+5.39/60$ h can be used to estimate $P_{\mathrm{orb}}$
(Gänsicke et al., 2009). In reverse, if $P_{\mathrm{sh}}$ and
$P_{\mathrm{orb}}$ are both measured, the superhump period excess
$\epsilon=(P_{\mathrm{sh}}-P_{\mathrm{orb}})/P_{\mathrm{orb}}$ can be used to
estimate $q$ (Patterson et al., 2005b; Kato & Osaki, 2013) (note that the
latter authors distinguish between the slightly different superhump periods
observed as the superoutbursts evolve).

Figure 3:  J0032+3145 is an example of an SU UMa CV. Top panel: Smoothed SDSS
spectrum in quiescence showing the typical, slightly double-peaked, Balmer
emission lines. The Helium emission lines are not always present in SU UMas.
Bottom left panel: ZTF ($g$-band) light curve showing outbursts and a
superoutburst in red. Bottom right panel: Expanded light curve showing the
decay of the superoutburst and characteristic superhumps.

##### ER UMa

Dwarf novae (Fig. 4) are a sub-type of SU UMa CVs characterised by an
unusually short supercycle ($20-50$ d) so that they spend over a third of the
time in superoutburst. The intervening time between superoutbursts is
punctuated by normal outbursts every few days (Robertson et al., 1995; Kato et
al., 2013). The cause for this behaviour is an $\dot{M}$ around four times
that of a typical SU UMa CV.

Figure 4:  J0912+5053 (DI UMa) is an example of an ER UMa CV. It has a high
$\dot{M}$ which is inconsistent with its short orbital period of 78 min at the
period minimum (Kato et al., 1996; Fried et al., 1999), resulting in extremely
frequent superoutbursts. The supercycle is 25 d whilst normal outbursts occur
every four days. Top panel: Smoothed SDSS spectrum in quiescence showing
Balmer emission lines. Bottom left panel: ZTF ($g$-band) light curve showing
frequent superoutbursts. Bottom right panel: Expanded light curve showing the
frequency of the superoutbursts.

##### WZ Sge

dwarf novae are also a sub-type (Patterson et al., 2002; Kato, 2015) of SU UMa
CVs. Their donors have lost most of their mass and they have low $\dot{M}$
with very infrequent, albeit bright, superoutbursts (Fig. 5). Most WZ Sge
systems do not exhibit normal outbursts in between their superoutbursts. The
low $\dot{M}$ and consequential low disc luminosity, often result in the white
dwarf being visible in the optical spectrum. In many cases this is the best
evidence for a WZ Sge classification as commonly used definitions (Kato et
al., 2001; Kato, 2015) are based upon observation of an outburst, together
with the characteristic early superhumps, which may not occur for decades.
Note that low $\dot{M}$ is not sufficient in itself for a classification;
there also needs to be evidence of the low mass of the donor.

Figure 5:  J1610$-$0102 (V386 Ser) is an example of a WZ Sge CV. Top panel:
Smoothed SDSS spectrum in quiescence showing double-peaked Balmer emission
lines from the accretion disc. The spectrum of the white dwarf is visible at
the blue end with characteristic wide Balmer absorption lines. Bottom panel:
ZTF ($g$-band) light curve showing a single characteristically large
superoutburst. Many systems believed to be WZ Sge stars, based on their
spectra and periods, have no such recorded outbursts and identification rests
on other evidence such as spectra.

##### Z Cam

dwarf novae are dwarf novae with the unusual characteristic (Fig. 6) of having
“standstills” of luminosity midway between the quiescent level and the
elevated level associated with a dwarf nova outburst (Simonsen et al., 2014).
This is believed to be due to having an $\dot{M}$ that fluctuates around the
critical level and is too high to sustain the thermal limit cycle associated
with dwarf novae. The IW And stars (Hameury & Lasota, 2014) form a sub-
category of Z Cam and show continual outbursts (not unlike dwarf novae) during
the “standstill”.

Figure 6:  J1028+2148 (MV Leo) is an example of a Z Cam CV. Top panel:
Smoothed SDSS spectrum in “standstill” showing Balmer emission lines. Bottom
left panel: ZTF (merged and scaled $r-$ and $g-$bands shown in blue) light
curve showing standstill periods as well as outbursts and the low, quiescent,
level. The outburst is highlighted in red. Bottom right panel: Expanded light
curve showing a period of “standstill” followed by an outburst and a decline
to quiescence.

####  3.1.2 Classical Novae

Classical novae (Bode & Evans, 2012) were first observed by ancient Chinese
astronomers as eruptions with $\Delta m=8-15$ that decayed over a period of a
few to several hundred days (Clark et al., 1977). They are caused by an
accumulation of hydrogen on the surface of the white dwarf which eventually
reaches a critical mass and pressure thereby initiating a runaway nuclear
reaction (Starrfield, 1971) that expels an expanding shell (Cohen & Rosenthal,
1983; Shara et al., 2012) and may also influence the evolution of the CV
(Nelemans et al., 2016; Shen & Quataert, 2022) by causing frictional angular
momentum loss (Schreiber et al., 2016). Classical novae can therefore occur in
all types of CVs including magnetic ones (Stockman et al., 1988). The
accumulation of hydrogen is influenced by $\dot{M}$ and therefore classical
novae are more common in high $\dot{M}$ systems typically with periods
$\mbox{$P_{\mathrm{orb}}$}\gtrsim 3$ h.

By definition an eruption of a classical nova has been recorded at some time.
The spectrum of a classical nova varies considerably depending upon the time
since this eruption. The rapidly expanding shell will initially dominate the
spectrum, with broad emission lines of hydrogen and helium. As it expands
further and becomes more diffuse it resembles a planetary nebula and exhibits
forbidden lines such as $[\text{S\,{ii}}]$ 6717 Å. The rapid expansion of the
shell also results in spectral lines exhibiting P Cygni profiles. Eventually
(years to decades) the spectrum becomes that of the pre-eruption CV. Most
classical novae revert to being novalike variables (Fig. 7), which is
unsurprising as the nova recurrence time is inversely related to the accretion
rate, and hence the nova population is dominated by novalike variables
(Townsley & Bildsten, 2005).

Figure 7:  J1545+1422 (CT Ser) is an example of an old nova that erupted in
1948, and that now appears as a novalike variable. Top panel: Smoothed SDSS
spectrum showing Balmer and He ii emission lines. Top inset panel: Expanded
spectrum showing the relative strength of the H$\beta$ and He ii lines.
(Munari et al., 2014) Bottom panel: ZTF ($g$-band) light curve illustrating
the relatively constant emission from the disc in its high state.

####  3.1.3 Novalike variables

Novalike variables (Dhillon, 1996) have bright spectra dominated by the hot
and stable disc (Fig. 8). The hydrogen in their discs is almost fully ionised
and their spectra therefore exhibit absorption lines, often with inset
emission lines. They often have strong disc winds which can appear as P Cygni
spectral lines (Inight et al., 2022). Novalikes sometimes exhibit changes
between high and low states and observational phenomena in their light curves
have given rise to a number of sub-types. The VY Scl sub-type shows relatively
deep ($\Delta m=2-5$) low states which can last a few weeks to about a year,
during which short brightenings occur (Bruch, 2019), the nature of which is
still discussed (Scaringi et al., 2017). This contrasts with the IW And class
(Section 3.1.1) in which characteristic outbursts occur during the (higher
state) standstills (Kato, 2019).

Figure 8:  J0758+1043 is an example of a novalike variable. Top panel:
Smoothed SDSS spectrum showing Balmer H$\alpha$ and H$\beta$ emission lines
together with H$\gamma$ and H$\delta$ absorption lines. Top inset panel:
Expanded spectrum showing the H$\beta$ line and absence of He ii lines. The
strength of He ii 4686 Å emission lines varies with orbital phase far more
than Balmer emission lines $-$ and often exceeding the strength of the
H$\beta$ line (Hoard et al., 2014). Bottom panel: ZTF ($g$-band) light curve
showing the relatively constant emission from the disc in its hot, steady
state.

####  3.1.4 AM CVn

AM CVn systems (Fig. 9) are identified by helium lines and an absence of
hydrogen lines in the optical spectrum (Ramsay et al., 2018). They are short-
period systems ($\mbox{$P_{\mathrm{orb}}$}\simeq 5-68$ min) but their
evolutionary origin is unclear with at least three potential formation
channels leading from their progenitors (Solheim, 2010). AM CVn systems are
rare, with only $\simeq 60$ systems known (Ramsay et al., 2018). They are good
verification binaries for the space-based LISA gravitational wave detector
(Amaro-Seoane et al., 2017) given their short periods and separations
(Nelemans et al., 2004; Liu et al., 2021).

Figure 9:  J0926+3624 is an example of an AM CVn CV. Top panel: The smoothed
SDSS spectrum shows He i and He ii emission lines but no hydrogen lines. The
absence of hydrogen lines is a characteristic of AM CVn CVs. Bottom panel: ZTF
(merged and scale $r-$ and $g-$bands) light curve showing outbursts.

###  3.2 Magnetic CVs

Perhaps a third (Pala et al., 2020) of CVs have a white dwarf with a magnetic
field $B\gtrsim 1$ MG and these are termed magnetic. The origin of this strong
field is not well understood; it may be formed during the common-envelope
event or sometime later (Belloni & Schreiber, 2020). Schreiber et al. (2021)
propose a model whereby the magnetic CV is born without a strong field and the
white dwarf is spun up by accretion. The cooling white dwarf in the CV then
forms a dynamo as it crystallises. The strengthening magnetic field causes the
system to become detached and the only accretion is from the donor wind which
follows magnetic field lines. This pre-polar stage (Hakala et al., 2022)
continues until after the donor becomes fully convective and mass transfer
recommences at which point the CV is a polar or intermediate polar (Wilson et
al., 2021). This model is consistent with observational evidence (fig. 9 in
Ferrario et al. 2015) as intermediate polars (with $B\lesssim 10$ MG) are, in
general, observed to have orbital periods $\mbox{$P_{\mathrm{orb}}$}\gtrsim 3$
h whilst polars (with $B\gtrsim 10$ MG) are observed to have typical orbital
periods $\mbox{$P_{\mathrm{orb}}$}\lesssim 2$ h .

####  3.2.1 Polar

The magnetic field of the white dwarf in polars (Cropper, 1990; Wu, 2000),
typified by the prototype AM Her, is generally greater than 10 MG. In this
case the Alfvén radius is beyond the circularisation radius, preventing the
formation of an accretion disc, and the accretion flow follows field lines
from the inner Lagrange point to impact on the white dwarf. These shock-heated
impact zones ($\approx 10^{8}$ K) give rise to X-ray radiation and also
characteristic He ii emission lines (Fig. 10). The cyclotron radiation is
sometimes observed in spectra as “humps” (see for example Thomas et al. 2000).
These humps are harmonics of the cyclotron frequency $f=(eB)/(2\pi m_{e})$,
with $e$ and $m_{e}$ the charge and the mass of the electron, due to the
magnetic field $B$ and can appear in the infrared (e.g. AM Her, $B\simeq 15$
MG Bailey et al. 1991), optical (e.g. MR Ser, $B\simeq 25$ MG; Schwope et al.
1993) or ultraviolet (e.g. AR UMa, $B\simeq 200$ MG; Gänsicke et al. 2001)
according to field strength. Cyclotron emission is directional and results in
cyclotron beaming which manifests itself as periodic behaviour. Polars can
also be identified either by polarimetry, as cyclotron radiation is polarised
(Chanmugam & Dulk, 1981; Shahbaz, 2019), or by the Zeeman splitting of Balmer
lines from the white dwarf (Ferrario et al., 1995) that is sometimes visible
in spectra obtained during low states when accretion nearly ceases (Fig. 11)
and can be used to estimate the strength of the magnetic field. A consequence
of the strong magnetic field is that the spin period of the white dwarf
normally equals $P_{\mathrm{orb}}$. This is thought to be due to the
interaction of the strong white dwarf magnetic field with the donor magnetic
field generating a torque which over time causes the white dwarf to spin down
until it equals $P_{\mathrm{orb}}$; the equilibrium situation (Campbell,
1984). Asynchronous polars (Piirola et al., 1994; Campbell & Schwope, 1999)
have a white dwarf spin period slightly different from $P_{\mathrm{orb}}$
which can be detected in periodograms. Asynchronous polars are believed to
have been knocked out of exact synchronisation by an event such as a nova
(Pavlenko et al., 2018).

Figure 10:  J0921+2038 is an example of a polar. Top panel: The smoothed SDSS
spectrum (blue) shows Balmer, He i and characteristic He ii emission lines.
The four cyclotron humps (brown, magenta, green and red) provide clear
evidence that this is a polar. Schmidt et al. (2008) determined that these are
the fifth, sixth, seventh and eighth harmonics (shown in grey) due to a
magnetic field of $B=32$ MG. Bottom panel: ZTF ($g$-band) light curve showing
the variability due to cyclotron beaming. There are no outbursts as polars do
not have discs in which to store accreting material and hence they do not
undergo a thermal limit cycle.  Figure 11:  J1105+2506 (ST LMi) is an example
of a polar showing Zeeman splitting. The spectrum is shown in the top
panel.The lower panel shows a model of the H$\beta$ line evolving vertically
(grey lines) with increased magnetic field (Schimeczek & Wunner, 2014). The
spectrum is over-plotted in blue with flux scaled to roughly match the
amplitude of the grey lines. The split lines are identified by vertical
dashes; they coincide with the grey lines at $B\simeq 13$ MG.

####  3.2.2 Pre-polars

White dwarfs in pre-polars (Reimers et al., 1999; Schwope et al., 2002a;
Parsons et al., 2021) (also called low-accretion rate polars) accrete via a
stellar wind from the donor rather than through Roche-lobe overflow. These
detached systems therefore have very low accretion rates with $\dot{M}$ less
than one per cent of that of other CVs (Schmidt et al., 2005). Cyclotron humps
are the defining characteristic of pre-polars; typically the second and third
harmonics are visible in the optical $-$ although the humps can be masked by
the donor spectrum (Fig. 12). Their cyclotron emission does exhibit circular
polarisation (Hakala et al., 2022), but they have very low X-ray luminosities
(Vogel et al., 2011). In the evolution scenario of Schreiber et al. (2021)
pre-polars represent a natural phase in the evolution of CVs once the white
dwarf develops a strong magnetic field.

Figure 12:  J0049+2226 is an example of a pre-polar. Top panel: The smoothed
SDSS spectrum (blue) shows Balmer emission lines. The donor is visible at the
red end of the spectrum. The cyclotron hump visible between 4200 Å and 5000 Å
(brown) is clear evidence that this is a polar or pre-polar. Following Parsons
et al. (2021) we subtracted a model spectrum (Kesseli et al., 2017) of an M3
dwarf (result in grey) to reveal a further cyclotron hump (magenta) at $6840$
Å. Modelling of harmonic frequencies for varying magnetic fields yielded a
good fit for a magnetic field strength of $B\simeq 78$ MG with the humps
forming the second and third harmonics. The fact that the donor is visible
implies that the accretion rate is low and this implies that this is a pre-
polar accreting from the donor wind (Schreiber et al., 2021). Bottom panel:
ZTF ($g$-band) light curve showing the variability attributable to cyclotron
beaming which revealed the orbital period ($P_{\mathrm{orb}}=5.92$ h).

####  3.2.3 Intermediate polars

Intermediate polars (Patterson 1994) such as their prototype DQ Her, have
weaker white dwarf magnetic fields than polars. The magnetic field is too weak
to force synchronisation of the spin period of the white dwarf resulting in a
spin period significantly shorter than the orbital period (Norton et al.,
2004) $-$ the defining characteristic of an intermediate polar (Fig. 13). The
vast majority of intermediate polars have a truncated accretion disc (Hameury
& Lasota, 2017). In common with non-magnetic CVs material from the donor
passes through the inner Lagrange point and pursues a ballistic path until it
merges with the disc. The disc is truncated at the Alfvén radius and material
then follows field lines from the innermost part of the disc to one or both
magnetic poles. The presence of the disc can result in outbursts (Hameury &
Lasota, 2017). Magnetically funneled accretion onto the magnetic poles of the
white dwarf results in pulsations in X-ray light curves (Hellier et al.,
1991), which are the definitive observational proof of a CV being an
intermediate polar .

Figure 13:  J0946+1350 (HY Leo, HS 0943+1404) is an example of an intermediate
polar. Top panel: The smoothed SDSS spectrum shows Balmer, He i and He ii
emission lines. Middle panel: ZTF ($g$-band) light curve showing the
variability due to the spin of the white dwarf and the accretion stream. The
alternation between high and low states is also evident. Bottom panel:
Periodogram taken from archival Jacobus Kapteyn Telescope data showing the
orbital frequency ($\Omega$), spin period ($\omega$) and the beat period
together with numerous aliases. For a detailed period analysis, see Rodríguez-
Gil et al. (2005).

##  4 Methods

In compiling this CV catalogue, we followed a two-step process. In the first
step we identified a shortlist of potentially new CVs from the 5 789 200
spectra of SDSS III and IV. In the second step we then undertook a detailed
classification of these potential CVs, as well as revisited all the CVs
previously identified in SDSS I and II.

###  4.1 Identification

SDSS I to IV focused upon a number of surveys none of which explicitly
targeted CVs. Fortuitously the (BOSS and eBOSS) (Dawson et al., 2013, 2016)
targets included quasars which share a similar colour parameter space to CVs.
Since the publication of the first comprehensive orbital period distribution
of SDSS (Gänsicke et al., 2009), we progressively built up a list of candidate
CVs based upon visual inspection of the flux and wavelength calibrated SDSS
spectra (Weaver et al., 2015; Dawson et al., 2016). The spectra were reviewed
multiple times by a range of individuals using a number of pre-selections in
terms of colour and filtering on the redshifts determined by the SDSS project.

This visual process was augmented by a supervised machine learning process. A
data set of known CVs and proven non-CVs was used to train a Random Forest
(RF) classifier (Liaw & Wiener, 2001). The RF model used features extracted
from the equivalent widths of 12 spectral lines and the average gradient of
the spectrum measured over five intervals. The different identification
methods resulted in a substantial overlap in the sub-sets of CV candidates
that they identified, and whereas we cannot claim to have identified all the
CVs that were spectroscopically observed by SDSS, we are confident that our
search has picked up the majority of them. CVs that remain overlooked will
either have extremely low signal-to-noise ratio spectroscopy, or have colours
very close to those of single main-sequence stars, and weak emission lines.

###  4.2 Classification

No single method can reliably classify CVs – it is necessary to consider all
the available information (SDSS spectroscopy, time-series photometry, Gaia
astrometry, colours from broad-band photometry, imaging) and seek a
classification of CV sub-type that is consistent with all of the data by
making use of the diagnostics outlined in section 3. Where possible, we
attempted to classify the SDSS CVs down to the sub-types illustrated in Fig.
1. However, in $\simeq 13$ per cent of cases only high level classifications
such as dwarf nova or magnetic CV are possible due to the limited amount of
information that is currently available. In addition, in the $\simeq 20$ per
cent of cases where a classification is very likely but not certain a colon
(:) has been appended.

We began the classification of each object with a literature search using
Simbad (Wenger et al., 2000) and a search of the AAVSO VSX database (Watson et
al., 2006). Where an existing classification exists we checked that it is
consistent with the SDSS I to IV spectra together with data from Gaia, VizieR,
ZTF, CRTS and TESS. This process resulted in refining 262 classifications. It
also revealed 18 systems, previously categorised as CVs that are now known not
to be CVs (see section LABEL:section:misclassifications). For each CV, the
references for the initial discovery, orbital period measurement, and optical
spectroscopy are documented in the Supplementary Data.

Detailed consideration of the available data is discussed in the following
sections.

####  4.2.1 Spectral analysis

We first plotted the SDSS spectrum and then used the spectrum to calculate a
synthetic Gaia $G$-band magnitude using the  python pyphot
111https://github.com/mfouesneau/pyphot package. In addition to the SDSS
spectrum we also created a broad-band (ultraviolet to infrared) spectral
energy distribution (SED) diagram using data downloaded from VizieR (Allen et
al., 2014) and over-plotted the spectrum. These plots are available as PNG
files in the supplementary data.

We initially compared the SED with the SDSS spectrum. There can be a number of
possible explanations if the spectrum is not consistent with the SED. Firstly
the flux calibration of all or part of the spectrum may be faulty which we
establish by comparing the synthetic magnitude with archive observations.
Secondly the spectrum may have been taken at a time when the CV was in a
different state to that when the photometry was obtained. Thirdly the SED may
have been contaminated by flux from a nearby object – which is why we check
the PanSTARRS images (see Section 4.2.4). In some cases, the photometry from
VizieR traces distinct SEDs, suggesting that photometry of the system was
obtained in different accretion states. In these cases, the epochs of the SED
photometry can be compared with the available light curves (section 4.2.2).

We next tried to identify the donor and/or the white dwarf in the SDSS
spectrum. At the blue end of the spectrum broad Balmer absorption lines are
indicative of the white dwarf. Conversely if the Balmer jump is visible in
emission then the disc or accretion stream is much brighter than the white
dwarf, which is typical of SU UMa CVs, polars, and intermediate polars.
Absorption from the Mg triplet at $5167,5172\,\mathrm{and}\,5183$ Å (Fig. 2)
is indicative of an early M-type or late K-type donor (suggestive of long,
$\gtrsim 2$ h, orbital periods) whereas a rising continuum in the red part of
the spectrum (Fig. 5), often broken by TiO “sawtooth” molecular absorption
bands indicates a mid-to-late type M dwarf (suggestive of shorter orbital
periods). In addition, the detection of a blackbody-shaped bump in the
ultraviolet or infrared part of the photometric SED can identify a white dwarf
or M dwarf, respectively. Care was taken to distinguish the Balmer absorption
lines from a white dwarf from those originating in a disc. Disc absorption
lines tend to be narrower and flat-bottomed as they are formed by Doppler
broadening due to Keplerian velocities in the disc whilst the Balmer
absorption lines of the white dwarf are pressure-broadened due to the high
surface gravities in the atmosphere. A clear white dwarf signature in the
spectrum is indicative of a low level of accretion. If in addition the donor
is not visible it suggests that the system is a WZ Sge candidate.

Next, we considered the emission lines detected. In addition to the Balmer and
He i lines, we probed for the presence of $[\text{S\,{ii}}]$ 6717 Å which is
indicative of contamination by a nebula and is not seen in CV spectra. Broad
and strong emission lines at arbitrary wavelengths are associated with highly
red-shifted quasars. Finally, He ii 4686 Å emission is indicative of a polar
or intermediate polar particularly if the ratio of its equivalent width to
that of H$\beta$ is greater than 0.4 (Silber, 1992)

We also considered the shape of emission lines. Balmer emission lines are
indicative of an accretion flow, either the disc in a non-magnetic CV, or a
stream in polars. In non-magnetic CVs, the width of the Balmer emission from
the disc will depend upon the inclination of the system (Horne & Marsh, 1986),
with broad lines and the detection of double-peaks indicating a high
inclination. Furthermore deep central absorption dips embedded in broad double
peaked Balmer lines are a good indication that a system is eclipsing.
Asymmetric emission lines are associated with the velocity dispersion of the
accretion stream of polars whilst very thin lines are typical of background
nebulae and detached systems.

Lastly we also superimposed upon the SDSS spectrum the five SDSS photometric
magnitudes obtained in the original observations. These can reveal
inconsistencies and in particular can reveal cyclotron humps (due to one
observation being taken at an orbital phase when the cyclotron emission
peaked).

####  4.2.2 Light curves

Archival light curves from the CRTS, ZTF and TESS were obtained from online
archives; Caltech, IRSA (Rebull et al., 2019) and Lightkurve (Lightkurve
Collaboration et al., 2018) respectively. We reviewed these looking for
outbursts, to provide confirmation of a dwarf nova classification, and also
for eclipses to obtain orbital periods. The long span of this data (over 20
years) also reveals many state changes that are indicative of certain CVs (see
Figs. 10 and 13) and novalike variables (Fig. 8). The light curves for all CVs
are shown in the supplementary data. We generated periodograms in an attempt
to determine orbital periods from the light curves. Prior to generating the
periodogram we cropped any outbursts from the light curve which was then de-
trended typically using a low-order polynomial fit. Two periodograms of the
light curve were then produced using a Fourier transform and also an
orthogonal multi-harmonic analysis of variance (Schwarzenberg-Czerny, 1996).
Potential periods were identified from peaks in the periodogram that exceeded
a false alarm probability of five per cent; the results were then validated by
phase-folding the original light curve on the potential period (see Fig. 14).
The shape of the phase-folded light curve not only confirms that the period is
valid but also helps to identify cases in which the orbital period may be
twice that of the strongest signal in the periodogram. The photometric orbital
modulation of CVs can have various potential causes. Eclipses may occur once
(of the white dwarf) or (rarely) twice (of the white dwarf and the donor, e.g.
Ashley et al. 2020) per orbital cycle whilst ellipsoidal modulation due to the
deformed shape of the donor star will yield two peaks per orbital cycle.
Cyclotron beaming is another case where the waveform may show multiple peaks
per orbital cycle, as does the emission from two poles in magnetic CVs.

Knowledge of the period assists the classification; SU UMa and ER UMa CVs
typically have $\mbox{$P_{\mathrm{orb}}$}\lesssim 3$ h (Ritter & Kolb, 1998),
WZ Sge have even shorter periods whilst U Gem and Z Cam CVs typically have
$\mbox{$P_{\mathrm{orb}}$}\gtrsim 3$ h.

Whilst some periodograms show an unambiguous period others leave room for
doubt due to aliases and harmonics, we have indicated such periods with a
colon (:).

Figure 14:  The Fourier transform of the ZTF light curve of J0158$+$2823, an
eclipsing CV, yields a particularly clear periodogram. Top left panel: The
aliases around the central frequency are separated by a frequency of once per
day $-$ reflecting the nightly sampling. Top right panel: The central peak
provides a good estimate of a period which can then be used to seed a sine fit
to the data yielding $f=22.51981(3)\,\mathrm{d}^{-1}$. Bottom panel: A plot of
the data folded at twice the the best-fit period with a sinusoidal fit
superimposed. In this case $P_{\mathrm{orb}}=2/f$ due to ellipsoidal
modulation although the corresponding frequency ($11.2599\,\mathrm{d}^{-1}$)
is barely visible in the periodogram. This can be seen in the bottom panel
with the deep eclipse of the white dwarf by the donor at $\Phi\simeq 0.5$ and
the two peaks at $0.25$ in phase either side. In this case there is no doubt
which is the correct alias but more generally the period uncertainty has to
take account of the possibility of an incorrect alias being selected.

####  4.2.3 Hertzsprung–Russell and Colour-Colour Diagrams

The position of a CV on the Hertzsprung–Russell (HR) diagram aids
classification as many CV types congregate in specific sections of the diagram
(see Fig. 15 in Abril et al. 2020 and fig. 8 in Inight et al. 2021). Absolute
magnitudes require knowledge of the distance of a CV which is normally derived
from Gaia parallaxes. Hence this diagnosis is not available for CVs without a
Gaia counterpart. Even when Gaia data exists the position of a CV can be
misleading if the parallax accuracy is poor or the CV is subject to
significant reddening. A further issue is that CVs are intrinsically variable
with outbursts, eclipses and orbital variations contributing to the observed
spread in their distribution in the HR diagram. Fig. 15 shows a subset of the
CVs from SDSS I to IV where systems with inaccurate parallaxes
($\varpi/\sigma_{\varpi}<10$) or inaccurate magnitudes
(phot_g_mean_flux_over_error<10,  phot_bp_mean_flux_over_error<10, or
phot_rp_mean_flux_over_error<10) have been removed in line with Lindegren et
al. (2018); Lindegren (2018). Inspection of Fig. 15 shows that the location of
a CV in the HR diagram can aid its sub-type classification. The HR diagrams,
where available, are included as PNG files in the supplementary data.

Where a Gaia counterpart does not exist, or does not have a parallax, a
colour-colour diagram based on SDSS photometry is an alternative (see Fig.
16). This is typically the case for faint ($m\gtrsim 20$) systems. These
approaches are somewhat complementary as SDSS photometry saturates at
$\mathrm{m\simeq 15}$ and is not available for bright CVs.

Figure 15:  HR diagram showing all the CVs from SDSS I to IV where the
astrometric and photometric errors are less than 10 per cent. The grey
background consists of those Gaia objects closer than 100 pc with GALEX
observations to increase the proportion of white dwarfs (which are UV-bright).
Note the concentration of AM CVn and WZ Sge CVs near the white dwarf cooling
sequence with SU UMa CVs spreading out towards the mains sequence. Novalike
variables and Z Cam CVs are concentrated around
$\mathrm{5<G_{\mathrm{abs}}<7}$ as their high $\dot{M}$ results in large
optical luminosity.  Figure 16:  Colour-colour diagram showing the CVs from
SDSS I to IV which have reliable SDSS photometry (where the magnitude of each
colour is greater than 15.5 to avoid saturation and the magnitude error is
less than 0.05). The grey background shows a collection of spectroscopically
observed SDSS stars to guide the eye. The dense lower area forms the main
sequence and the upper area is the white dwarf cooling sequence. Note the
concentration of SU UMa and WZ Sge CVs on and above the white dwarf cooling
sequence.

####  4.2.4 Imaging

Pan-STARRS (Chambers et al., 2016) cutout images for each CV were obtained
from the PS1 Image Cutout Server at STScI (Flewelling et al., 2020). We
superimposed circles for each Gaia object (with the radius being proportional
to the Gaia $G$-band magnitude) and for the SDSS fibre (2 arcsec diameter)
(Fig. 17). The resulting image shows whether there is potential contamination
of a spectrum from neighbouring objects, nebulae or background galaxies. In
addition we superimposed circles for any GALEX detections. CVs often have
GALEX observations but the accuracy of the coordinates of GALEX observations
can be incorrect by a few arcseconds and the image helps to assess whether a
GALEX observation could be associated with a different object. These images
are available as PNG files in the supplementary data.

Figure 17:  Pan-STARRS cutout of J0752+3628 showing the potential for the contamination of fibre spectroscopy. J0751+1444 has an apparent Gaia magnitude $m_{\mathrm{G}}=20.0$ whereas its neighbour, at a distance of 3.35 arcsec, has $m_{\mathrm{G}}=16.4$. The red circle denotes the target, the black dotted circle denotes the nominal coverage of an SDSS fibre (2 arcsec) and the blue circle is centred on the Gaia coordinates of the neighbour scaled according to the Gaia $G$ magnitude. The Gaia coordinates are based upon EDR3 with an epoch of 2016 whilst the Pan-STARRS epoch is around 2012 and so a lack of alignment is a prompt to check proper motions to see if the contaminant was closer when the spectrum was observed. The pink square shows a GALEX observation.  Table 1:  CVs from SDSS I to IV. We provide the geometric distances from Bailer-Jones et al. (2021) and CV sub-types according to the definitions in section 3. SDSS $g$-band magnitudes are from data release 7 (Abazajian et al., 2009). The references for the initial discovery (ID), spectrum (Sp) and orbital and superhump periods ($P$ – where known) are also shown. New discoveries, orbital periods and sub-type identifications are shown in blue. Sub-type identifications in black are from VSX (Watson et al., 2017). : Tentative values, * Superhump periods, †low accretion SU UMa, ‡CVs with unusual state changes, §period bouncers. This is a short extract of the table, the full data is included in the supplementary data.  SDSS | Name |  Gaia EDR3 source_id |  $P$ (h) |  SDSS $g$ (mag) |  Distance (pc) |  CV Sub-type | References
---|---|---|---|---|---|---|---
ID | Sp | P
J000014.74+255603.1 |  |  |  | 22.17 |  | CV: | 1 | 1 |
J000720.77+200721.6 | CSS 110921:000721+200722 |  |  | 21.48 |  | WZ Sge: | 15 | 1 |
J000844.32-014014.6 | Gaia15abi | 2544817841421994624 |  | 17.93 | $1236^{+220}_{-230}$ | NL:/Polar | 15 | 1 |
J001107.26+303235.9 | V0402 And | 2861506395998660096 | 0.063499(3)* | 20.4 | $793^{+434}_{-225}$ | SU UMa | 11 | 1 | 5
J001153.08-064739.2 | SDSS J001153.08-064739.1 | 2443219535337313280 | 0.10028081(8) | 17.83 | $477^{+29}_{-19}$ | U Gem | 6 | 6 | 6
J001158.28+315543.7 | CSS 101111:001158+315544 | 2861949606559861120 | 0.0554(1): | 21.85 |  | SU UMa | 15 | 1 | 1
J001231.54+280011.1 |  | 2859847683924640384 |  | 20.98 |  | WZ Sge: | 1 | 1 |
J001538.25+263656.7 | CSS 090918:001538+263657 | 2856493142666940544 | 0.10150(6) | 17.92 | $557^{+36}_{-34}$ | SU UMa | 15 | 7 | 7
J001856.92+345444.2 | V0479 And | 2876157251000218496 | 0.594093(4) | 17.46 | $2027^{+408}_{-259}$ | Polar: | 4 | 4 | 8
J002049.51+280020.2 | MASTER OT J002049.54+280020.5 | 2857063063352792320 | 0.05658(1) | 19.64 | $347^{+59}_{-45}$ | WZ Sge: | 13 | 1 | 1
J002243.55+061002.7 | MGAB-V295 | 2747622177049417344 | 0.079(1): | 19.88 | $2113^{+1596}_{-715}$ | SU UMa | 9 | 1 | 1
J002500.17+073349.2 | CSS 081123:002500+073350 | 2748173586426407424 |  | 19.68 | $1484^{+412}_{-306}$ | DN/IP: | 15 | 1 |
J002637.05+242915.6 | CSS 091026:002637+242916 | 2807191999055199232 | 0.08533(1) | 18.75 | $588^{+100}_{-87}$ | Polar | 15 | 1 | 1
J002728.00-010828.5 | EN Cet | 2541910801397761152 | 0.05933(5) | 20.78 | $1218^{+461}_{-547}$ | WZ Sge | 10 | 4 | 2
J002842.53+311819.4 |  | 2859593555004235520 | 1.217(1) | 18.59 | $3642^{+1152}_{-985}$ | NL | 3 | 3 | 1
J002910.96+324125.8 |  |  |  | 22.12 |  | DN | 1 | 1 |
J003151.76+202051.0 | MGAB-V534 | 2796463449923353856 | 0.080871025(2) | 18.06 | $368^{+19}_{-16}$ | SU UMa† | 9 | 1 | 14
J003153.46+084118.5 | ASASSN-17kg | 2749846901389046656 | 0.0585(3)* | 19.15 | $283^{+24}_{-17}$ | WZ Sge | 12 | 1 | 16
J003203.62+314510.4 | CSS 091220:003204+314510 | 2859621764350006784 |  | 19.1 | $960^{+335}_{-184}$ | SU UMa | 15 | 1 |
J003459.90+273619.0 | CSS 081031:003460+273620 | 2857545615813275136 | 0.07337(1) | 21.12 | $1349^{+828}_{-462}$ | SU UMa | 15 | 1 |

References: 1 This work, 2 Dillon et al. (2008), 3 Hou et al. (2020), 4 Szkody
et al. (2005), 5 Kato et al. (2012), 6 Rebassa-Mansergas et al. (2014), 7
Thorstensen et al. (2016), 8 González-Buitrago et al. (2013), 9 MGAB catalog,
10 Esamdin et al. (1998), 11 Antipin (1998), 12 Kepler et al. (2015), 13
Gentile Fusillo et al. (2019), 14 VSNET23583, 15 CRTS candidate CV, 16
VSNET21317

##  5 Results

In the interests of brevity the full observational data and analysis of the
over 500 SDSS I to IV CVs has been consigned to the supplementary data.
Illustrative examples are given here (Table LABEL:tab:cvtable and Fig.
LABEL:fig:spectraandLCexample) together with a brief discussion of a small set
of the most interesting systems. Spectra have been smoothed by using a box-car
median filter of width five points. A list of the 70 newly discovered CVs is
included in Table LABEL:tab:newCVs.

Figure 18: Left panels: Spectra with Balmer lines shown in green and He lines
in pink. Right panels: Light curves with CRTS shown in orange, ZTF in red
($r$-band) and green ($g$-band) and TESS in black. This is a short extract of
the data: the complete set is included in the supplementary data.

###  5.1 Unusual systems

The following notes discuss a small selection of the systems in SDSS I to IV
which have unusual features and are worthy of further study.

####  5.1.1 J0028+3118

The spectrum displays single-peaked Balmer emission lines above a flat
continuum as well as a very strong He ii 4686 Å emission line. The system is
located close to the main sequence in the HR diagram. There is a ${\simeq}1$
yr state change of ${\simeq}2$ magnitudes at $\mathrm{MJD}\simeq 58750$
together with three shorter and smaller brightening events in the ZTF light
curve. The ZTF light curve (Fig. LABEL:fig:interestingJ0028) suggests that
$P_{\mathrm{orb}}=29.20(1)$h and also that the system is eclipsing. J0028+3118
is a high-accretion novalike system but the very long period is unusual. The
extraordinarily strong He ii 4686 Å suggests a hot and helium-rich accretion
stream $-$ possibly the result of a hydrogen depleted donor that is still
burning (Harrison, 2018).

####  5.1.2 J0223$-$0520

The spectrum shows double-peaked Balmer emission lines with the higher order
ones embedded in broad absorption lines from the white dwarf (Fig.
LABEL:fig:interestingJ0223). The blue slope of the continuum, and the shape of
the white dwarf absorption lines suggest a relatively high white dwarf
temperature, which is consistent with the SED being dominated by the white
dwarf. The system is brighter in the far-ultraviolet (GALEX
$m_{\mathrm{FUV}}=19.5$) than in the optical ($m_{\mathrm{g}}=19.9$). This is
not consistent with its position at the lower end of the white dwarf cooling
sequence in the HR diagram with $G_{\mathrm{BP}}-G_{\mathrm{RP}}=0.35$, which
would imply a temperature $T_{\mathrm{eff}}\simeq 8000$ K. The ZTF light curve
shows variability but does not reveal an orbital period. There is an XMM X-ray
detection which is unusual for a low-$\dot{M}$ system (Webb et al., 2020).
This is probably a WZ Sge. It is possible that the raised temperature of the
white dwarf is the consequence of a previous eruption.

####  5.1.3 J0932+4725 and J0947+4524

Double-peaked Balmer, He i, He ii and Ca emission lines are visible in the
SDSS spectra of these systems as well as the Balmer jump in emission (Fig.
LABEL:fig:interestingJ0947). There is deep central absorption in the emission
lines suggesting that they are eclipsing – which is confirmed by Homer et al.
(2006) for J0932+4725, and by the ZTF light curve for J0947+4524. Both are
closer to the white dwarf cooling sequence than the main sequence in the HR
diagram. In the CRTS and ZTF light curves J0932+4725 exhibits outbursts and
superoutbursts whilst J0947+4524 exhibits two small ($\Delta m\simeq 1$)
outbursts. J0932+4725 has a period of 1.588 h (Homer et al., 2006) whilst we
obtained a period of 1.34963(2) h from the ZTF light curve.

The large ratio of the helium to hydrogen line strengths suggests that the
donor is likely to be He-enriched. An HST observation of J0932+4725 by Pala et
al. (2017) shows a large N v/C iv line flux ratio. Gänsicke et al. (2003)
attribute this type of nitrogen excess to a system having passed through a
phase of thermal timescale mass transfer and now accreting CNO processed
material from a donor stripped of its external layers. J0932+4725 is therefore
most probably such an evolved system. Whilst there are no HST spectra of
J0947+4524 the optical spectrum is very similar to that of J0932+4725 and
suggests that this may also have an evolved donor. Systems with evolved donors
are interesting as they are potential precursors to AM CVn systems
(Podsiadlowski et al., 2003; Goliasch & Nelson, 2015)

The superoutbursts confirm that J0932+4725 is an SU UMa CV. J0947+4524 is a
dwarf nova but we cannot classify it more precisely.

####  5.1.4 J1146+6759

Double-peaked Balmer and He i emission lines are visible in the spectrum as
well as the Balmer jump in emission. The donor is not detected. J1146+6759 is
located midway between the main sequence and the white dwarf cooling sequence
in the HR diagram. The system was observed in a superoutburst on 2011 January
4 and a superhump period of 0.0615(8) d was reported (vsnet-alert
12567222http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/12567),
classifying this system as an SU UMa dwarf nova. The ZTF light curve shows
very unusual variability (Fig. LABEL:fig:interestingJ1146) with a steady
decline in amplitude of $\Delta m\simeq 1.5$ over three years. This is
followed by a rapid return to the original magnitude and then an outburst. It
demonstrates that some dwarf novae can undergo state changes (section
LABEL:section:state_changes)

####  5.1.5 J1150+4041

The spectrum shows strong Balmer, He i and He ii asymmetric emission lines,
suggesting that this is a magnetic CV. Neither the white dwarf nor the donor
are visible. The ZTF light curve shows continuous variability with an
amplitude of $\Delta m\simeq 1$, but no outbursts. The periodogram computed
from the ZTF data shows a clear and strong signal at an unusually long period
of 141.65(5) h, or 5.9 d (Fig. LABEL:fig:interestingJ1150). If this signal is
genuinely the orbital period of the system, it would exceed that of the
current record of 5.7 d (V1017 Sgr, Sekiguchi 1992). However, in contrast to
V1017 Sgr, the spectrum of J1150+4041 does not reveal any signature of a donor
star, normally detected at such a long orbital period. Moreover, the system is
located relatively close to the white dwarf cooling sequence in the Gaia HR
diagram, i.e. intrinsically faint. We have currently no consistent explanation
for the observed properties of J1150+4041, but speculate that the long period
might be the beat between the white dwarf spin and the orbital period in a
slightly asynchronous polar. Radial velocity and fast photometry follow-up is
required to unravel the true nature of this system.

####  5.1.6 J1224+1841

The spectrum shows a blue continuum with Balmer emission lines, which are
embedded in Balmer absorption lines for H$\beta$ and the higher Balmer lines.
J1224+1841 is located in the area of the HR diagram occupied by novalike
variables. The ZTF light curve shows multiple outbursts and also a drop of
$\Delta m\simeq 2$ to a low state lasting several hundred days (Fig.
LABEL:fig:interestingJ1224). There appear to be outbursts during the low state
although it is possible that they are very short reversions to the high state.
The CRTS light curve shows two similar low states. J1224+1841 belongs to the
growing family of CVs bordering between novalike variables and dwarf novae
(Kato, 2019; Kato et al., 2021; Inight et al., 2022). Radial velocity follow-
up should be carried out to determine the orbital period of this system.

####  5.1.7 J1244+5936

This system is also known as Gaia16adi (Wyrzykowski et al., 2012). A double-
peaked H$\alpha$ emission line is visible in the spectrum as well as other
Balmer, He i and He ii emission lines (Fig. LABEL:fig:interestingJ1244). The
deep central absorption features in the higher Balmer lines, dipping below the
continuum flux level, suggest that the system is eclipsing, which is confirmed
by the ZTF photometry. Absorption lines from the white dwarf are visible
together with TiO molecular bands from the donor. It is closer to the white
dwarf cooling sequence than the main sequence in the HR diagram. The ZTF and
CRTS light curves show $\simeq 10$ outbursts with at least one low amplitude
superoutburst accompanied by superhumps. We obtained a reliable period of
1.74954(1) h from the ZTF light curve and a consistent period from the CRTS
light curve. The folded light curve shows that it is eclipsing with a hot spot
just before the eclipse. J1244+5936 appears to be an SU UMa albeit with a very
low $\dot{M}$. It may have recently started accreting – possibly after passing
through the period gap; although theory (D’Antona et al., 1989) suggests that
this transitional period will be short ($\sim 10^{6}$ yr).

####  5.1.8 J1522+3913

The spectrum (Fig. LABEL:fig:interestingJ1522) shows Balmer emission lines
above a relatively red continuum. Narrow absorption lines reveal a
$\simeq\mathrm{G}$-type donor star. The SED of the system exhibits a clear
ultraviolet excess in the GALEX bands, most likely from the accretion disc.
The Gaia HR diagram shows that J1522+3913 is just evolving off the main
sequence into the giant branch. The ZTF light curve shows at least one
outburst. We obtained a reliable period of 10.7686 d (258.4464 h) from the ZTF
light curve confirming the period found by Drake et al. (2014) from the CRTS
data. The folded light curve shows a clear double hump, consistent with the
detection of ellipsoidal modulation. To our knowledge J1522+3913 is the
longest known orbital period for a CV $-$ twice that of V1017 Sgr
($\mbox{$P_{\mathrm{orb}}$}=5.714$ d, Sekiguchi 1992). We note that a small
number of symbiotic stars are located in this region of the HR diagram as
well, however, J1522+3913 does not show any of the defining features in its
spectrum (see the definition in section 1.1 of Merc et al. 2021). Instead, we
suggest that the very long orbital period, and the location in the HR diagram
are consistent with a CV in which the donor has evolved off the main sequence
(e.g. for example GK Per Álvarez-Hernández et al. 2021).

####  5.1.9 J1549+1739

The SDSS spectrum shows single-peaked Balmer emission lines and a very strong
He ii emission line on top of a strong blue continuum (Fig.
LABEL:fig:interestingJ1549). There is no sign of the white dwarf or the donor.
J1549+1739 is located in the novalike variable area of the HR diagram. Whilst
there are no outbursts in the CRTS and ZTF light curves, the system meanders
on time scales of about a year between $m\simeq 17.5-19$. This is probably a
novalike variable but the strong He ii and its light curve are unusual.
Spectroscopic follow up is encouraged to explore the spectral changes between
the bright and faint states, and to determine the orbital period of the
system.

####  5.1.10 J1712+3218

The spectrum (Fig. LABEL:fig:interestingJ1712) shows Balmer and Helium
emission lines, including strong He ii, and the Bowen blend above a blue
continuum. Narrow absorption lines of Mg, Na and Ca are detected, which may be
from the donor star. J1712+3218 is located in the novalike variable area of
the HR diagram. The ZTF light curve shows variability with clear eclipsing
behaviour. Drake et al. (2014) report an orbital period of 17.998 h and
concluded that this is a Beta Persei system. We determine
$\mbox{$P_{\mathrm{orb}}$}=8.9599(1)$ h from the eclipses in the ZTF light
curve. We also have unpublished MDM time series spectroscopy confirming the
orbital period. We classify J1712+3218 as a novalike variable. This is an
unusually long period for a novalike variable although it would be consistent
with an early donor that is visible even though the disc is in a high state.

####  5.1.11 J1726+5432

This system has two SDSS spectra obtained at MJDs 51813 and 51997. Both
spectra show single-peaked Balmer, He i and He ii emission lines as well as
the Balmer jump in emission (Fig. LABEL:fig:interestingJ1726). The SED shows
that it has historically been brighter. There is no Gaia parallax. It is above
the white dwarf cooling sequence in the SDSS $u-g,g-r$ diagram, although the
SDSS photometry was obtained during a brighter state. The ZTF light curve
shows no outbursts but low-level variability. We find an unusually long period
of 15.2783(3) h from the ZTF light curve which, together with the helium-rich
spectrum may indicate an evolved donor. In the absence of outbursts we cannot
assign a sub-type and have classified this generically as a CV.

####  5.1.12 J1831+4202

He i absorption lines are detected in the spectrum superimposed on a blue
continuum (Fig. LABEL:fig:interestingJ1831). There is no sign of the white
dwarf or the donor. J1831+4202 has previously been classified as a CV (Girven
et al., 2011) and a DB white dwarf (Genest-Beaulieu & Bergeron, 2019; Geier et
al., 2019a) however the DB classification is inconsistent with the variability
in the Gaia and ZTF light curves which indicates that this is an interacting
binary. It is close to AM CVn in the HR diagram and we hence classify this
system as an accretion-disc dominated AM CVn. The drop-outs in the Gaia and
ZTF light curves are unusual for this type of system.

####  5.1.13 J2119+0332

Double-peaked Balmer and He i lines are visible in the spectrum as well as the
Balmer jump in emission (Fig. LABEL:fig:interestingJ2119). There are also Na
absorption lines at $\simeq 8190$ Å and at $\simeq 5890$ Å together with broad
Mg at $\simeq 5170$ Å suggesting a K-type donor. J2119+0332 is closer to the
main sequence than the white dwarf cooling sequence in the HR diagram. There
are no outbursts in either the CRTS or the ZTF light curves. We found
ellipsoidal modulation and a reliable orbital period of 2.37223(3) h from the
ZTF light curve, i.e. within the period gap. The period and relatively low
luminosity are inconsistent with a main-sequence K-type donor and reminiscent
of SDSS J001153.08$-$064739.2 (Rebassa-Mansergas et al., 2014) and QZ Ser
(Thorstensen et al., 2002a) where the donor has undergone nuclear evolution on
the main sequence before mass transfer to the white dwarf commences. In these
cases a period of rapid mass transfer has resulted in a stripped donor and a
short period. The strong Na absorption lines are consistent with CNO burning.

###  5.2 Non-CVs

The classification of a number of systems that were previously classified as
CVs have now, with the benefit of new data such as from Gaia and ZTF, been
revised. Most of the systems previously classified as CVs turned out to be
either single white dwarfs or white dwarf plus main sequence binaries, in
which emission lines from a nebular background or from the chromosphere of the
companion were mistaken for signatures of accretion flows. The new
classifications of these non-CV systems are listed in Table LABEL:tab:notacv.

##  6 Discussion

###  6.1 Distribution of periods

We have assembled periods for over half of the CVs in SDSS I to IV, either
from the literature or from our analysis of the ZTF data. The main reasons why
the remaining systems have no period measurements are either a low inclination
(which will reduce the amplitude of any orbital modulation), faintness (i.e.
low signal-to-noise ratio of the time-series photometry or spectroscopy), or
sparse sampling of the available time-series data. With the possible exception
of low $\dot{M}$ short-period CVs (which do not have much orbital modulation
from the hot spot and are inherently fainter) these factors are not likely to
correlate with the period. The distribution of the SDSS CV periods is
therefore assumed to be relatively unbiased, which we compare in Fig.
LABEL:fig:PeriodDist with that of version 7.24 (December 2015) of the Ritter &
Kolb (2003) catalogue of CVs (hereafter R&K). This final version of the R&K
catalogue lists 1429 CVs (including some from SDSS I to IV); however it
focuses on well-studied CVs, many of which were discovered by virtue of their
outbursts.

Interestingly the period gap is clearly evident in the period distribution of
CVs from the R&K catalogue but is not so evident in the SDSS I to IV sample.
The proportion of SDSS CVs with periods below three hours is also greater than
in the R&K catalogue, presumably due to spectroscopic identification of CVs
with infrequent outbursts as reported by Gänsicke et al. (2009), and the fact
that the SDSS CV sample reaches, on average, fainter apparent magnitudes than
the R&K sample (19.7 and 18.1 respectively). This is consistent with the
results obtained by Pala et al. (2020) and Inight et al. (2021). Fig.
LABEL:fig:PeriodDist highlights that selection effects have to be taken
carefully into account when constructing observed CV samples to develop,
support, or disprove theories of CV evolution, such as the canonical
“disrupted magnetic braking framework” (Rappaport et al., 1983; Knigge et al.,
2011) as well as suggestions that the structure of the donor star may have no
strong dependence upon the efficiency of magnetic braking (Andronov et al.,
2003; El-Badry et al., 2022).

The orbital period distribution broken down by CV-sub-type is shown in Fig.
LABEL:fig:PeriodDistbytype. From the top panel it can be seen that U Gem CVs
typically have periods above the period gap whilst most SU UMa CVs are located
below the gap. Later in the evolution process low accretion WZ Sge CVs
approach the period minimum and subsequently evolve back towards longer
periods as “period bouncers”. The CVs below the period minimum have helium-
rich donors which were partially evolved before accretion commenced or formed
directly from a detached white dwarf/brown dwarf binary (Kolb & Baraffe, 1999;
Littlefair et al., 2007; Podsiadlowski et al., 2003). Polars tend to have
shorter periods whilst those of intermediate polars and pre-polars tend to be
longer. The “period gap” is not so apparent in magnetic CVs, (see for example
Mason et al. 2019) and may be a result of less efficient magnetic braking due
to the white dwarf magnetic field (Webbink & Wickramasinghe, 2002). Novalike
variables congregate above the period gap alongside classical novae. Many
classical novae become novalike variables following their eruptions although
there are exceptions where they become dwarf novae in the period between
eruptions (Honeycutt et al., 1995; Mróz et al., 2016). Short-period novalike
variables are very rare; BK Lyn is the most famous example: Patterson et al.
(2013) speculate that this is due to being the result of a relatively recent
($\sim\,2000$ yr ago) nova, and it has now morphed into an ER UMa dwarf nova.

The evolution of orbital periods of CVs across the HR diagram is shown in Fig.
LABEL:fig:HRDiagram. CVs originate in or near the main sequence, with their
luminosity dominated either by a luminous accretion disc or donor star, and
progressively migrate towards the white dwarf cooling sequence as their donor
mass decreases and periods shorten. In most WZ Sge the contribution of the
accretion disc to their optical flux is small, and that of their donor stars
entirely negligible.

###  6.2 Distribution of CV sub-types

The distribution of the SDSS I to IV CV sub-types is shown in Fig.
LABEL:fig:Hist_of_types, where we compare it with that of the 150 pc volume-
limited sample (Pala et al., 2020) and also with that of the R&K catalogue.

The proportion of magnetic CVs (polars, pre-polars and intermediate polars) in
our sample is 20 per cent which is consistent with the 18 per cent found in
Ritter & Kolb (2003) but is much lower than the 34 per cent found in Pala et
al. (2020). This may be due to the small sample size (42) in Pala et al.
(2020) which has a statistical $2\sigma$ uncertainty of $\pm 19$ per cent. By
comparison our sample (507) has an uncertainty of $\pm 0.03$ per cent.
Alternatively the difference may be because the absolute luminosity of polars
is on average lower than non-magnetic systems causing a selection effect in
our magnitude-limited sample.

The proportion of novae and novalikes is lower in Pala et al. (2020) than our
sample. This is due to their intrinsic brightness rendering them visible at
far greater distances than other CVs and hence there will be a higher
proportion in our magnitude-limited sample. The proportion is greater still in
Ritter & Kolb (2003) due to historic novae identified by their eruptions.

We found a smaller proportion of WZ Sge CVs than Pala et al. (2020) most
probably because they are inherently faint and only observed at relatively
short distances.

###  6.3 Variability of CVs

The photometric variability of CVs is widely used as a selection criterion
when searching for CV candidates. However, different CV sub-types display very
different types of variability, both in terms of amplitude and time-scales,
and hence specific variability selection methods may introduce strong biases
towards (or against) individual sub-types.

To illustrate this point, we computed three variability criteria for the SDSS
I to IV CVs: (1) The difference between the maximum and minimum magnitude over
the full set of ZTF observations which will be sensitive to outbursts and
eclipses, where we only included systems with at least 150 observations. (2)
The reduced $\chi^{2}$ of the ZTF photometry with respect to its median. (3)
the Gaia $G$-band variability defined as (Guidry et al., 2021)

$G_{\mathrm{var}}=\frac{\texttt{phot\\_g\\_mean\\_flux\\_error
}\times\sqrt{\texttt{phot\\_g\\_n\\_obs}}}{\texttt{phot\\_g\\_mean\\_flux}}$
(1)

The distribution of these three types of variability are shown in the Gaia HR
diagram in Fig. LABEL:fig:HRvariability).

The maximum amplitude of variability in the ZTF light curves (top panel) shows
a concentration half-way between the white dwarf cooling track and the main
sequence. This reflects the predominant population of SU UMa type CVs in this
area of the HR diagram (see Fig. 15), which have frequent dwarf nova outbursts
with amplitudes of $m\simeq 3-5$, i.e. it is very likely that ZTF, despite the
sparse sampling, will have caught them during an outburst. In contrast, the
regions occupied predominantly by novalike variables
($G_{\mathrm{BP}}-G_{\mathrm{RP}}\simeq 0.25$, $G_{\mathrm{abs}}\simeq 6$) and
WZ Sge systems ($G_{\mathrm{BP}}-G_{\mathrm{RP}}\simeq 0.25$,
$G_{\mathrm{abs}}\simeq 12$) show only relatively small variations between the
minimum and maximum brightness detected by ZTF. The is reflected by their
stable, hot (novalike variables) or cool (WZ Sge CVs) accretion discs.
However, a few systems in the region occupied by WZ Sge CVs show very large
amplitude, $\Delta m\simeq 8$, variability, implying that ZTF observed them
during one of their rare super-outbursts. The ZTF reduced $\chi^{2}$ diagram
shows overall a similar trend, however, the area occupied by novalike
variables exhibits a higher level of variability than in the maximum amplitude
diagram, most likely because many of these systems have persistent short-term
variability (flickering, Bruch 1992), and on average higher quality ZTF light
curves as they are intrinsically bright. Finally, the Gaia variability diagram
shows again a lack of variability in the area occupied by novalike variables
and WZ Sge stars, and overall a lower number of strongly variable systems.
This is most likely due to the fact that the Gaia data spans a shorter
baseline ($\simeq 2.8$ yr), and has on average less data points than the ZTF
photometry. Future Gaia releases will extend the baseline to at least 5 yr,
possibly up to 10 yr, increasing the diagnostic power of the Gaia variability
index.

###  6.4 Space Density

Measuring the space density of the various CV sub-types is fundamentally
important for validating evolutionary models.

Assuming that the Galaxy is axially symmetrical the space density of a class
of objects is a function of the height above the Galactic disc and the radial
distance from the Galactic centre. Our analysis is sufficiently local that
radial variations will be small and we henceforth ignore this factor. We
further assume an exponential vertical distribution with scale height $h$
where $h$ is assumed to be dependent upon the typical age of the CV sub-type.
The space density at a height $z$ above the plane is therefore:

$\rho\left(z\right)=\rho_{0}\times\exp\left(-\frac{\left|z\right|}{h}\right)$
(2)

Our objective is to estimate the space density of the CV sub-types. Hence if
the estimate of the number of a sub-type in a volume $V$ is $N_{\mathrm{obs}}$
the space density is given by:

$\rho_{0}=\frac{N_{\mathrm{obs}}}{\displaystyle\int_{V}{\exp\left(-\frac{\displaystyle\left|z\right|}{\displaystyle
h}\right)}dv}$ (3)

which for a suitably large sample volume would become a constant dependent
only on the choice of $h$. This poses four challenges which are addressed in
the following sections. (1) some CVs within the SDSS footprint would have been
too bright or too faint to be observed by SDSS; (2) an appropriate value of
$h$ for each CV sub-type needs to be chosen; (3) SDSS was not complete and did
not target a proportion of CVs within its footprint; (4) the SDSS footprint
only achieved partial sky coverage and a method of extrapolating the results
derived from the SDSS observations is needed.

####  6.4.1 CV brightness

SDSS I to IV spectroscopy did not target any objects with $g\lesssim 15$,
which has been taken into account in section LABEL:sec:completeness below. At
the faint end it will be apparent from the HR diagram (Fig.
LABEL:fig:HRDiagram) that some sub-types have a larger intrinsic luminosity
than others and will therefore be observable at greater distances by SDSS. We
therefore sought to establish for each sub-type a limiting distance
($R_{\mathrm{lim}}$) within which they would have reliably been detected by
SDSS. To derive this we first assume that CVs with $g<21$ will be reliably
detected from inspection of their SDSS spectrum if targeted; this is a
conservative assumption as for example Pâris et al. (2017) used $g<22$. In the
following, we decided to use the Gaia $G$-band magnitude rather than SDSS-$g$,
as the multiple observations of Gaia more adequately sample the average
brightness of an individual system. The two passbands yield very similar
apparent magnitudes for a given brightness. We calculated the mean ($\mu$) and
standard deviation ($\sigma$) of the absolute Gaia $G$ magnitude of each sub-
type from the SDSS I to IV CVs (ignoring objects where the error in the
Bailer-Jones et al. (2018) distance was greater than $20$ per cent). By
definition for any CV of a given sub-type with apparent magnitude $m$,
absolute magnitude $M$ and distance $d$ pc there is a 99.7 per cent
probability that $M<\mu+3\sigma$ hence

$\displaystyle m-5\log_{10}(d)+5<$ $\displaystyle\mu+3\sigma$ (4)
$\displaystyle m<$ $\displaystyle\mu+3\sigma+5\log_{10}(d)-5$ (5)

Now $d<R_{\mathrm{lim}}$ is the largest value of $d$ corresponding to $m=21$:

$\displaystyle 21=$ $\displaystyle\mu+3\sigma+5\log_{10}(R_{\mathrm{lim}})-5$
(6) $\displaystyle 5\log_{10}(R_{\mathrm{lim}})=$ $\displaystyle
5+21-\mu-3\sigma$ (7) $\displaystyle R_{\mathrm{lim}}=$ $\displaystyle
10^{1+0.2(21-\mu-3\sigma)}$ (8)

The values of $R_{\mathrm{lim}}$ for each sub-type are shown in Table
LABEL:tab:limitingdistance.

####  6.4.2 Scale heights of the different CV sub-types

As we shall see the choice of scale height in the model has a significant
effect upon the final estimates of the space density. Patterson (1984) used a
single scale height of 150 pc for all sub-types in their analysis as did
Duerbeck (1984) (who used $125\pm 22$ pc). Pretorius et al. (2007a) assumed
scale heights of 120, 260 and 450 pc (hereafter the P2007 model) for
long-$P_{\mathrm{orb}}$ systems ($\mbox{$P_{\mathrm{orb}}$}>3$ h), normal
short-$P_{\mathrm{orb}}$ systems ($\mbox{$P_{\mathrm{orb}}$}<3$ h) and period
bouncers respectively. This was based on the assumption that these systems had
increasing ages and therefore, as a result of scattering with other stars,
increased space velocities resulting in higher average heights above the
Galactic plane (Kolb & Stehle, 1996). We have used the assumptions of P2007 in
our work. Where systems do not have an observed period we assign them to one
of the three scale heights in P2007 based on their sub-type. U Gem CVs,
novalike variables and intermediate polars are assumed to be long-period CVs,
short-period systems include SU UMa CVs, polars and WZ Sge CVs with the third
category reserved for period bouncers. We attempted to analyse the SDSS I to
IV data in order to support the scale height assumptions in P2007 but in
practice our samples were not complete to a sufficient Galactic height.

####  6.4.3 Completeness

SDSS I to IV used complex targeting rules focusing largely on quasars and not
CVs (e.g. Richards et al. 2002; Ross et al. 2012; Dwelly et al. 2017).
Fortunately CVs have similar colours to quasars and a significant proportion
of quasar candidates were observed by SDSS. However, given that CV sub-types
occupy different regions of the Gaia HR diagram (and in the colour-colour
diagrams used by SDSS for their target selection) the proportion of each sub-
type observed by SDSS is not the same. Whilst in principle an analysis could
estimate the degree of completeness for each sub-type, such an endeavour would
be very complex. We have adopted an empirical approach by analysing version
7.24 of the R&K CV catalogue which contains 1429 CVs. We removed the 117 CVs
in the R&K catalogue that were originally discovered by SDSS leaving 1312
systems. The apparent magnitude (“mag1”) of these CVs in their normal state
was processed and systems that were too bright ($m<15$) or too faint ($m>21$)
were also removed333Ideally, this analysis would have been done by sub-type
but we discovered that the “type1” and “type2” fields in the R&K catalogue
were unreliable; in particular many WZ Sge had been classified as SU UMa. From
this we established that 208 CVs fell within the footprint of SDSS. We then
established that 97 of these (46.6 per cent) were re-discovered by SDSS and we
assumed that this proportion applies to all CVs and sub-types within the
footprint. We then defined a correction factor $C_{\mathrm{orr}}=2.14$ as the
reciprocal of this proportion.

Pala et al. (2020) used a different approach by quantifying the number of Gaia
objects within the SDSS footprint which fulfilled the quasar colour criteria
for SDSS I and II and did not include any cut in the red colours. This
resulted in 5300 objects and Pala et al. 2020 then found the proportion of
these objects with an SDSS spectrum (53 per cent) of which $\approx 6$ were
predicted to be in the sample. Only four were found (three had a spectrum)
from which they concluded that the completeness was 71 per cent (i.e.
$C_{\mathrm{orr}}=1.4$). It is unsurprising that the completeness of the full
SDSS I to IV sample is lower than that of the nearby, bright 150 pc sample.

####  6.4.4 SDSS Footprint

The coordinates of the centres of the plates used in the “BOSS”, “eBOSS” and
“SDSS” surveys, together with their radius (1.49 degrees), define the
coverage. However there are two problems to address; firstly the plates
overlap and secondly CVs are not uniformly distributed in space but
concentrated near the Galactic plane.

For convenience we define from the denominator of (LABEL:eq:scale2) an
effective volume for a sphere radius $R$:

$V_{\textrm{eff}}\left(h,R\right)=\int_{V(R)}{\exp\left(-\frac{\left|z\right|}{h}\right)}dv$
(9)

To address the overlap problem we model the celestial sphere using $2^{8}$
astropy HEALPix pixels (Górski et al., 2005). Each HEALPix pixel has the same
surface area ($\Omega$ steradians) on the unit sphere and can be represented
by a cone (Fig. LABEL:fig:diagram1). The effective volume of a sphere radius
$R$ is therefore the sum of the volume of the cones:

$V_{\textrm{eff}}(h,R)=\sum_{\mathrm{All\,pixels}}\int_{0}^{R}\exp-\left(\frac{\left|r\sin(b)\right|}{h}\right)(r^{2}\Omega
dr)$ (10)

The following substitutions are used:

$A=\frac{\left|R\sin(b))\right|}{h}$ (11)

and

$x=\left(\frac{rA}{R}\right)$ (12)

We then compared the coordinates (ra,dec) of each HealPix pixel with the
coordinates of the centre of each plate to identify whether that pixel was
covered by that plate. We then defined $\mathrm{SDSS_{pix}}$ as the de-
duplicated set of HEALPix pixels that lay within the coverage of one or more
plates. The effective volume of the SDSS sample is therefore:

$\begin{split}V_{\textrm{eff}}(h,R)&=\sum_{\mathrm{SDSS_{pix}}}\int_{0}^{A}\exp\left(-x\right)\left(\left(\frac{Rx}{A}\right)^{2}\Omega\left(\frac{Rdx}{A}\right)\right)\\\
&=\Omega
R^{3}\sum_{\mathrm{SDSS_{pix}}}\left(\frac{1}{A^{3}}\right)\int_{0}^{A}x^{2}\exp\left(-x\right)dx\end{split}$
(13)

Solving the integral analytically yields

$V_{\textrm{eff}}(h,R)=\Omega
R^{3}\sum_{\mathrm{SDSS_{pix}}}\left(\frac{1}{A^{3}}\right)\left(2-\left(A^{2}+2A+2\right)\exp\left(-A\right)\right)$
(14)

which can be computed numerically for a given $h$ and $R$.

####  6.4.5 Estimates of space density

Bringing the preceding sections together the three estimates of the space
density (for $h=120,260,450$ pc) are calculated from

$\rho_{0}=\frac{N_{\mathrm{obs}}\,\times
C_{\mathrm{orr}}}{V_{\mathrm{eff}}(h,R_{\mathrm{lim}})}$ (15)

and the results are listed in Table LABEL:tab:spacedensity.

Next, we need to discuss the accuracy of these results. From equation
(LABEL:equ:spaced) the primary sources of error in $\rho_{0}$ are in the
correction factor and the scale height. The true scale height in each case
will very likely lie between the two extreme values that we adopted, i.e.
$h=120$ pc and $h=450$ pc (Table LABEL:tab:spacedensity) and so a reasonable
estimate for $\rho_{0}$ (excepting possibly old WZ Sge systems) is
$\rho_{0}(260\,\mathrm{pc})$ with the extreme values forming a notional
$3\sigma$ error estimate. We have avoided quoting any statistical
uncertainties on our space density estimates as the true uncertainties are
likely to be dominated by the assumptions on the scale height.

It is interesting to compare these estimates with the those based on different
approaches used in previous works (Fig. LABEL:fig:space_densities) noting that
the assumed scale heights have to be comparable.

Section 9.5 in Warner (1995) proposed $\rho_{0}(103\,\mathrm{pc})=0.4\times
10^{-6}\,\mathrm{pc}^{-3}$ for U Gem type systems and
$\rho_{0}(176\,\mathrm{pc})=0.25\times 10^{-6}\,\mathrm{pc}^{-3}$ for SU UMa
systems. Both values are lower than contemporary estimates presumably as
earlier CV discoveries were primarily the result of following up transients
and limited to brighter targets.

Pretorius et al. (2007b) estimated
$\rho_{0}(120\,\mathrm{pc})=(15^{+25}_{-8})\times 10^{-6}\,\mathrm{pc}^{-3}$
for all non-magnetic CVs based on a very small sample of four systems. Our
results are towards the lower end of this range.

Pretorius & Knigge (2012) estimated a density of
$\rho_{0}(260\,\mathrm{pc})=(3.8^{+5.3}_{-1.9})\times
10^{-6}\,\mathrm{pc}^{-3}$ for non-magnetic CVs which compares with our value
of $\rho_{0}(260\,\mathrm{pc})=(4.29)\times 10^{-6}\,\mathrm{pc}^{-3}$ (by
adding the individual estimates from Table LABEL:tab:spacedensity).

Pretorius et al. (2013) estimated $\rho_{0}=(1.3^{+0.6}_{-0.4})\times
10^{-6}\,\mathrm{pc}^{-3}$ for magnetic CVs, based upon a scale height of
$120$ pc for long period systems and $260$ pc for short period systems, which
is consistent with our estimates. They also estimated values for polars and
intermediate polars which are consistent with this work.

Schwope (2018) estimated $\rho_{0}(120\,\mathrm{pc})=0.13\times
10^{-6}\,\mathrm{pc}^{-3}$ for intermediate polars. This is significantly
lower than our estimate.

Pala et al. (2020) estimated a composite
$\rho_{0}(280\,\mathrm{pc})=(4.8^{+0.6}_{-0.8})\times
10^{-6}\,\mathrm{pc}^{-3}$ for all CVs and our
$\rho_{0}(260\,\mathrm{pc})=5.78\times 10^{-6}\mathrm{pc}^{-3}$ is consistent
with this noting that we use a larger correction factor ($2.1$ compared with
$1.4$ in Pala et al. (2020)). Pala et al. (2020) also provided incomplete
(their footnote 9 states that they were unable to estimate the completeness of
individual sub-types) estimates for $\rho_{0}(260\,\mathrm{pc})$ for five sub-
types. These values are lower than our estimates – partly due to
(in)completeness but also because we use a larger correction factor ($2.1$
compared with $1.4$ in Pala et al. (2020)). Our larger sample size may also
have had an effect brought about by the effectiveness of SDSS in discovering
short-period systems.

Inight et al. 2021 estimated $\rho_{0}=(2.23^{+1.04}_{-0.6})\times
10^{-6}\,\mathrm{pc}^{-3}$ for their Gold Sample of 201 systems. The Gold
Sample is a reliable subset of the known CVs (at least 305) within 300 pc and
their estimate is therefore a lower limit for the true space density.

In principle, the fractions of sub-types from our space density estimates
should reflect the true fractions in the underlying population without
selection effects – as opposed to the number ratios in the magnitude limited
SDSS I to IV sample. We therefore compared these fractions with those from the
SDSS I to IV sample (Fig. LABEL:fig:comparison), where the main difference is
the larger proportion of WZ Sge stars and a lower fraction of SU UMa CVs when
looking at the space density results. This is in line with the expectation
that low-luminosity WZ Sge stars should be under-represented in a magnitude
limited sample. Whilst demonstrating that the SDSS I to IV sample is subject
to some selection effects, we also caution not to over-interpret the space
density ratios because of the significant effect of scale height assumptions.

In summary, the long-standing discussion of the CV space density seems to
converge to a value of $\simeq 5\times 10^{-6}\mathrm{pc}^{-3}$. However, the
uncertainties in scale heights continue to limit the accuracy of any estimates
of space density. We have calculated space densities for six sub-types and
Fig. LABEL:fig:space_densities demonstrates that these estimates are
consistent with other work.

###  6.5 Low accretion rate systems

Period bouncers are CVs where the mass of the donor has shrunk below the level
needed to sustain hydrogen fusion in the core ($\simeq
0.061\mbox{$\mathrm{M}_{\sun}$}$) and the donor has evolved to a brown dwarf.
A consequence of this is that the orbital separation of the CV increases and
$P_{\mathrm{orb}}$ evolves back from the period minimum (often taken as $\sim
76.2$ min, Knigge 2006) towards longer periods. Patterson (2011) identified a
total of 22 candidate period bouncers, eight of which are in our sample (Table
LABEL:tab:bouncers). Together with one polar, the 13 non-magnetic period-
bouncers in Table LABEL:tab:bouncers represent 14 per cent of the WZ Sge CVs
and 2.6 per cent of our complete sample. The two new candidates from this work
(J0904+4402 and J2131$-$0039) were identified because of their spectra (white
dwarf visible and donor not apparent) and an orbital period significantly
larger than the period minimum. We estimated the space density of period
bouncers using 14 per cent of the space density of WZ Sge from Table
LABEL:tab:spacedensity to be
$\rho_{0}(450)=0.27\times\mathrm{10^{-6}pc^{-3}}$. Comparing this with
$\rho_{0}(P2007)=7.92\times\mathrm{10^{-6}pc^{-3}}$ implies that only $3.4$
per cent of CVs are period bouncers which is a long way from the 70 per cent
of all CVs predicted by Kolb (1993). Goliasch & Nelson (2015) modelled CV
evolution to predict the intrinsic proportion of period bouncers as being
$38-60$ per cent of all CVs and the observed proportion (taking account of
selection effects) as $2.5-11.2$ per cent. Although period bouncers are
intrinsically faint any observational bias is unlikely to fully account for
this difference. As long as period bouncers maintain a minimum amount of mass
transfer, and associated Balmer emission lines, SDSS spectroscopy is extremely
efficient at identifying WZ Sge CVs, independent of their orbital period and
their outburst frequency. The lack of observed period bouncers remains
puzzling, and could be due to the donors in period bouncers either becoming
detached or merging with the white dwarfs.

###  6.6 Unusual state changes

In contrast to dwarf novae, which undergo quasi-regular outbursts because of
thermal instabilities in their accretion discs, novalike variables are known
to exhibit state changes due to sustained changes in $\dot{M}$(Shafter et al.,
1985; Leach et al., 1999; Rodríguez-Gil et al., 2007). Polars are also known
to undergo state changes on a timescale of months to years (Kafka & Honeycutt,
2005; Ramsay et al., 2004; Šimon, 2016), and low states have been observed
occasionally in a small number of dwarf novae (Wood et al., 1995; Schreiber et
al., 2002). The cause of the high/low state transitions is still debated, but
a likely contender is star spots near the inner Lagrangian point (Livio &
Pringle, 1994; King & Cannizzo, 1998; Hessman et al., 2000)

Serendipitously whilst reviewing the ZTF light curves of the spectroscopically
identified CVs we have identified eight CVs (Figure. LABEL:fig:unusual) within
the SDSS I to IV sample that display erratic light curve morphologies, neither
displaying disc outbursts, nor clearly identifiable state changes – and which
have spectra that are inconsistent with being either magnetic CVs or novalike
variables. These systems defy the standard CV classification scheme outlined
in section 3. These eight examples of systems exhibit significant brightness
changes on timescales that indicate the cause is a change to $\dot{M}$ rather
than instabilities of their discs. Below, we discuss the properties of these
systems.

J0838+4910 is below the period gap with $\mbox{$P_{\mathrm{orb}}$}=1.69$ h.
Thorstensen et al. (2015) rightly concluded that this was an SU UMa based on
the measured orbital period and the CRTS light curve. The SDSS spectrum is
consistent with an SU UMa classification. However the ZTF data is
contradictory – showing no outbursts and two long-lasting faint states.

J0853+4848 has a spectrum that shows Balmer, He i and weak He ii emission
lines. The white dwarf is not visible but the M-type donor is detected in the
red part of the spectrum. The system has been brighter since the SDSS spectrum
was obtained. There is no CRTS data and the ZTF data shows one small outburst
but also state changes of up to two magnitudes. The period is 2.36 h which is
in the period gap.

J0922+3103 (AR Cnc) and J1544+2553 have periods of $5.15$ h and $6.03$ h
respectively and spectra which resemble U Gem CVs with double-peaked Balmer
and helium lines, and clear signatures of their M-dwarf donors. However their
light curves more closely resemble those of polars and they exhibit no
outbursts. This ambiguity prevents any further classification beyond that of a
CV. Palaversa et al. (2013) classified J1544+2553 as a novalike using an
automated process but this is not consistent with the strong lines in the
spectrum, the detection of the donor and the position in the HR diagram.

J1239+2108 (IR Com) has previously been identified as a CV with erratic
brightness changes (Richter et al., 1997), and SDSS obtained a spectrum when
accretion had virtually ceased for an extended period (Manser & Gansicke,
2014), revealing both the white dwarf and the donor star. It is on the lower
edge of the period gap with $\mbox{$P_{\mathrm{orb}}$}=2.09$ h.

J1244+3004 (CSS 080427:124418+300401) was found in a bright state within the
CRTS light curve for about ten years, albeit exhibiting outbursts (Thorstensen
et al., 2015). The ZTF light curve (Fig. LABEL:fig:unusual) shows a prolonged
drop of $m\simeq 2$. It’s period of $\mbox{$P_{\mathrm{orb}}$}=1.86$ h places
it below the period gap.

J2309+2135 (V405 Peg) was identified by Schwope et al. (2002b) as the
counterpart of a ROSAT X-ray source, and based on the detection of the donor
star, and the apparent magnitude, the authors speculated that V405 Peg may be
as close as $\simeq 30$ pc. Thorstensen et al. (2009) determined the period to
be 4.26 h, and measured a parallax of $\simeq 150\,$ pc – superseded by the
Gaia parallax of $\simeq 173$ pc. Schwope et al. (2014) obtained deep X-ray
observations, revealing a relatively low X-ray luminosity, and the authors
noted that the system displays characteristics of both magnetic and non-
magnetic CVs. The SDSS spectrum very clearly reveals the M-type donor star,
with little contribution from either an accretion disc or the white dwarf.

J2330+3033 has a spectrum showing both Balmer and He i emission lines, the
donor is clearly visible. It is eclipsing and Hardy et al. (2017) found
$\mbox{$P_{\mathrm{orb}}$}=3.74$ h. The CRTS and ZTF light curves show
variability but no outbursts.

We note that the periods of these systems cluster both near the lower edge of
the period gap, and in the $\simeq 4-6$ h range, indicating widely varying
evolutionary stages. The lack of outbursts is indicative of a low $\dot{M}$. A
low $\dot{M}$ is corroborated by the detection of the donor stars in the
short-period systems, as non-magnetic CVs within this period range are
typically SU UMa systems with accretion disc dominated spectra. The long-
period systems coincide with the period range where dwarf novae are usually
found (we encourage the reader to inspect section 7.2 of Knigge et al. 2011
for a discussion of the inconsistency between the CV sub-type distribution in
the $3-6$ h period range and the theoretical predictions of $\dot{M}$ and the
critical accretion rate for disc instabilities to (not) occur). We speculate
that these eight systems may either contain weakly magnetic white dwarfs, with
fields that are sufficiently low so that they do not develop the hallmarks of
polars or intermediate polars, but strong enough to affect the structure of
the disc. However, we also note that a number of intermediate polars exhibit
dwarf nova outbursts – so the tentative suggestion of weak fields does not
explain the absence of outbursts. An alternative idea is that these systems
are undergoing some type of state change in their donor stars, as possibly
suggested by their location near the lower edge of the period gap, and in the
range where dwarf novae and novalike overlap above the period gap.

###  6.7 Tertiary systems

Raghavan et al. (2010) suggest that around eight per cent of close binaries
are part of a triple system and have a distant companion. They are significant
as the distant star can influence the evolution of the inner binary through
Kozai cycles (Fabrycky & Tremaine, 2007). We searched Gaia EDR3 for such
systems by identifying candidate stars within 30 arcsec of each of the CVs in
our sample. Potential companions can be expected to have similar proper
motions. However proper motions, and hence differences between them, will
scale inversely with distance. We therefore adopted the following metric where
the denominator is a proxy for distance:

$\sqrt{\frac{\left(\mathrm{pmra_{cv}}-\mathrm{pmra_{cand}}\right)^{2}+\left(\mathrm{pmdec_{cv}}-\mathrm{pmdec_{cand}}\right)^{2}}{\mathrm{pmra_{cv}}^{2}+\mathrm{pmdec_{cv}}^{2}}}<0.05$
(16)

resulting in the eight systems in Table LABEL:tab:tertiary.

The projected separations, $a$, were calculated by multiplying the angular
separation by the Bailer-Jones et al. (2018) distance; the errors on the
Bailer-Jones et al. (2018) distances are too large to estimate the radial
separation. Estimating the binding energy is a useful test of whether these
systems are gravitationally bound to the CV or not:

$-E_{\mathrm{bind}}=G\frac{(M_{\mathrm{WD}}+M_{\mathrm{d}})M_{3}}{a}$ (17)

A rough estimate is sufficient for this purpose and so we assume typical
values of $M_{\mathrm{WD}}=0.83\,\mathrm{M}_{\odot}$ and a donor of mass
$M_{\mathrm{d}}=0.14\,\mathrm{M}_{\odot}$. We estimate $M_{3}$ from the
position of the star in the Gaia HR diagram; and find that the CV companion in
seven of the tertiaries is on the main sequence; the companion to J1206+5100
is a white dwarf. The calculated values shown in Table LABEL:tab:tertiary,
when compared with fig 6b of Burgasser et al. (2007) show clearly that these
eight CVs are tertiary systems. These eight are only 1.6 per cent of our
sample and should be considered a lower limit on the fraction of CVs in triple
systems. Our search is likely to be incomplete at larger distances where Gaia
may not resolve close tertiaries or not detect them because they are too
faint; in particular tertiaries with very low-mass or cool white dwarf
companions would be hard to identify.

##  7 Summary

We have reviewed the spectra of CVs observed by SDSS I to IV and publish 178
for the first time along with 59 new periods. We discovered 70 completely new
CVs and classified or updated 262. These 70 new CVs from SDSS III and IV
augment the 282 previously identified in SDSS I and II. This total is
remarkably consistent with the early prediction of 400 from Szkody et al.
(2002). We have analysed the results and position the different CV sub-types
in evolutionary terms together with revised estimates of space density. The
period bouncers and tertiary systems within our sample have been identified
and the abundances compared with previous estimates.

## Acknowledgements

We thank the anonymous referee for their helpful comments.

We acknowledge with thanks the variable star observations from the AAVSO
International Database contributed by observers worldwide and used in this
research.

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Science Foundation, and the U.S.
Department of Energy Office of Science. The SDSS-III web site is
http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven National
Laboratory, Carnegie Mellon University, University of Florida, the French
Participation Group, the German Participation Group, Harvard University, the
Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA
Participation Group, Johns Hopkins University, Lawrence Berkeley National
Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for
Extraterrestrial Physics, New Mexico State University, New York University,
Ohio State University, Pennsylvania State University, University of
Portsmouth, Princeton University, the Spanish Participation Group, University
of Tokyo, University of Utah, Vanderbilt University, University of Virginia,
University of Washington, and Yale University.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS acknowledges support and resources from the
Center for High-Performance Computing at the University of Utah. The SDSS web
site is www.sdss.org.

SDSS IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian (CfA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

Based on observations obtained with the Samuel Oschin Telescope 48-inch and
the 60-inch Telescope at the Palomar Observatory as part of the Zwicky
Transient Facility project. ZTF is supported by the National Science
Foundation under Grants No. AST-1440341 and AST-2034437 and a collaboration
including current partners Caltech, IPAC, the Weizmann Institute for Science,
the Oskar Klein Center at Stockholm University, the University of Maryland,
Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium
of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin,
Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr
University Bochum, Northwestern University and former partners the University
of Washington, Los Alamos National Laboratories, and Lawrence Berkeley
National Laboratories. Operations are conducted by COO, IPAC, and UW.

Data from the Catalina Real-Time Transient Survey has been used (Drake et al.,
2009)

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

This paper includes data collected with the TESS mission, obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the TESS mission is provided by the NASA Explorer Program. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555.

This work makes use of observations obtained at the MDM Observatory, operated
by Dartmouth College, Columbia University, Ohio State University, Ohio
University, and the University of Michigan.

Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic
Telescope LAMOST) is a National Major Scientific Project built by the Chinese
Academy of Sciences. Funding for the project has been provided by the National
Development and Reform Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences.

Some of the results in this paper have been derived using the healpy and
HEALPix package.

BTG was supported by grant ST/T000406/1 from the Science and Technology
Facilities Council (STFC). This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant agreement No. 101020057). This research was
supported in part by the National Science Foundation under Grant No.
PHY-1748958.

## Data Availability

All data used in this article are available from the sources referenced in the
text. The Supplementary Data including a machine-readable version of the
catalogue is included in the online version of this paper.

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