# Galactic Model Parameters and Space Density of Cataclysmic Variables in Gaia
Era: New Constraints to Population Models

Remziye Canbay Istanbul University, Institute of Graduate Studies in Science,
Programme of Astronomy and Space Sciences, 34116, Beyazıt, Istanbul, Turkey
Selçuk Bilir Istanbul University, Faculty of Science, Department of Astronomy
and Space Sciences, 34119, Beyazıt, Istanbul, Turkey  Aykut Özdönmez Atatürk
University, Faculty of Science, Department of Astronomy and Space Sciences,
25240, Erzurum, Turkey  Tansel Ak Istanbul University, Faculty of Science,
Department of Astronomy and Space Sciences, 34119, Beyazıt, Istanbul, Turkey

(Received January 05, 2023; Revised February 16, 2023; Accepted February 21,
2023)

###### Abstract

The spatial distribution, Galactic model parameters and luminosity function of
cataclysmic variables (CVs) are established using re-estimated trigonometric
parallaxes of Gaia DR3. The data sample of 1,587 CVs in this study is claimed
to be suitable for Galactic model parameter estimation as the distances are
based on trigonometric parallaxes and the Gaia DR3 photometric completeness
limits were taken into account when the sample was created. According to the
analysis, the scale height of All CVs increases from 248$\pm$2 to 430$\pm$4 pc
towards shorter periods near the lower limit of the period gap and suddenly
drops to 300$\pm$2 pc for the shortest orbital period CVs. The exponential
scale heights of All CVs and magnetic systems are found to be 375$\pm$2 and
281$\pm$3 pc, respectively, considerably larger than those suggested in
previous observational studies. The local space density of All CVs and
magnetic systems in the sample are $6.8^{+1.3}_{-1.1}\times$10-6 and
$2.1^{+0.5}_{-0.4}\times 10^{-6}$ pc-3, respectively. Our measurements
strengthen the 1-2 order of magnitude discrepancy between CV space densities
predicted by population synthesis models and observations. It is likely that
this discrepancy is due to objects undetected by CV surveys, such as the
systems with very low $\dot{M}$ and the ones in the period gap. The
comparisons of the luminosity function of white dwarfs with the luminosity
function of All CVs in this study show that 500 times the luminosity function
of CVs fits very well to the luminosity function of white dwarfs. We conclude
that the estimations and data sample in this study can be confidently used in
further analysis of CVs.

Cataclysmic Variables – solar neighbourhood

††journal: AJ

##  1 Introduction

Cataclysmic variables (CVs) are short-period semi-detached binary stars. A
cataclysmic variable’s primary component is a white dwarf which is accreting
matter from a Roche-lobe filling low-mass main-sequence star, the secondary
component, via a gas stream. Since the matter stream has a high angular
momentum and the primary star is small, an accretion disc surrounding the
white dwarf is created. A bright spot is also formed where the matter stream
impacts the disc. Magnetised white dwarfs in CVs have accretion columns
instead of discs to transfer matter from the secondary component (Warner,
1995; Hellier, 2001; Knigge, 2011; Knigge, Baraffe & Patterson, 2011).

The standard formation and evolution scenario developed for CVs is
concentrated on the explanation of the features seen in the orbital period
distribution of these systems, since the most precisely determined parameter
of a CV is its orbital period. The sharp cut-off at about 80 min (Willems et
al., 2005; Gänsicke et al., 2009), period minimum, and the period gap between
roughly 2 and 3 h (King, 1988; Knigge, Baraffe & Patterson, 2011) are the most
striking features of the CV period distribution. The standard theory
successfully explains these features as its main predictions are supported by
observations. However, there are still some observational properties to be
explained, for example (1) Predicted and observed fractions of CVs above and
below the period gap (de Kool, 1992; Kolb, 1993; Howell, Nelson & Rappaport,
2001; Gänsicke et al., 2009; McAllister et al., 2019) are not in agreement.
Standard CV population studies predict that more than 90% of CVs must be
located below the period gap, while observations imply almost equal numbers of
CVs below and above the gap. Although sky surveys like the Sloan Digital Sky
Survey (SDSS; Szkody et al., 2002, 2003, 2004, 2005, 2006, 2007, 2011)
revealed a larger population of short-period CVs, the disagreement still
remains. (2) The fraction of observed post-period minimum CVs, period
bouncers, is much smaller than that predicted by the population models based
on the standard theory (Patterson et al., 2005; Unda-Sanzana et al., 2008;
Littlefair et al., 2008; Patterson, 2011; Kato et al., 2015, 2016; McAllister
et al., 2017; Neustroev et al., 2017; Pala et al., 2018). McAllister et al.
(2019) found that 30% of donor stars in a sample of 225 CVs are likely to be
brown dwarfs in period bouncers. However, only 5% of volume-limited sample of
CVs in Pala et al. (2020) included period bouncers. (3) Although the standard
theory predicts an orbital period minimum of about 65-70 min (see Kalomeni et
al., 2016, and references therein), observed values are about 76-82 min
(Knigge, Baraffe & Patterson, 2011; McAllister et al., 2019). (4) The observed
white dwarf masses in CVs have been significantly larger than those of single
white dwarfs (see Zorotovic & Schreiber, 2020, and references therein). In
addition, white dwarf mass in CVs does not change with orbital period
(McAllister et al., 2019). (5) Population studies based on the standard
evolutionary model predict space densities 1-2 orders of magnitude larger than
observed values (Zorotovic & Schreiber, 2020). Although additional angular
momentum loss mechanisms and models (Patterson, 1998; Knigge, Baraffe &
Patterson, 2011; Schreiber, Zorotovic & Wijnen, 2016; Pala et al., 2017;
Zorotovic & Schreiber, 2017, 2020; Belloni et al., 2018; Liu & Li, 2019;
Metzger et al., 2021; Sarkar & Tout, 2022) were suggested to solve the
disagreements, the only published, self-consistent simulations of CV evolution
were performed by Hillman et al. (2020), whose multi-Gyr models of novae take
into account every nova eruption’s thermonuclear runaway, mass and angular
momentum loses, feedback due to irradiation and variable mass transfer rate
($\dot{M}$), and orbital size and period changes. Hillman et al. (2020)
reproduced the observed range of mass transfer rates at a given orbital
period, with large and cyclic Kyr-Myr timescale changes. It should be noted
that the magnetic systems may have different evolutionary scenarios from non-
magnetic CVs (see references in Belloni et al., 2020).

Depending on the completeness of the samples, reliable observational
constraints can be obtained from the stellar statistics (Ak et al., 2008;
Özdönmez, Ak & Bilir, 2015), and a proposed evolutionary scheme must also be
in agreement with the data obtained from stellar statistics (Duerbeck, 1984).
In this respect, Galactic model parameters and the space density of a group of
objects are key parameters to constrain and to test population models based on
evolutionary schemes. A wide range of observational results is remarkable,
while the predicted space densities are systematically 1-2 order of magnitude
larger than those derived from observations. For example, previous CV
population synthesis models predicted space densities $10^{-5}$-$10^{-4}$ pc-3
(Ritter & Burkert, 1986; de Kool, 1992; Kolb, 1993; Politano, 1996; Willems et
al., 2005, 2007; Goliasch & Nelson, 2015; Belloni et al., 2018), while
observations indicated $10^{-7}$-$10^{-4}$ pc-3 (Warner, 1974; Patterson,
1984, 1998; Thomas & Beuermann, 1998; Ringwald, 1993; Schwope et al., 2002;
Araujo-Betancor et al., 2005; Pretorius et al., 2007a, b; Ak et al., 2008;
Revnivtsev et al., 2008; Pretorius & Knigge, 2012; Pretorius, Knigge &
Schwope, 2013; Schwope, 2018). In a recent study, Pala et al. (2020) measured
very precise space densities of $4.8^{+0.6}_{-0.9}\times 10^{-6}$ and
$1.2^{+0.4}_{-0.5}\times 10^{-6}$ pc-3 for All CVs and magnetic CVs (mCVs),
respectively, using the European Space Agency’s (ESA) Gaia Data Release 2
(Gaia DR2; Gaia Collaboration, 2018). They assumed a scale height of 280 pc
for their analysis and the sample was composed of only 42 objects within 150
pc from the Sun. They assumed that this restriction reduces the uncertainties
in the derived space densities related to the unknown age and scale height of
the CV population and the uncertainties from astrometric solutions. Belloni et
al. (2020) claimed that these are the most reliable observational space
density estimates ever found. They also concluded that the space densities
given in Pala et al. (2020) are in very good agreement with their predicted
values if potential period bouncers are excluded from the space density
estimation. However, it should be noted that Belloni & Schreiber (2020)
performed binary population models using an up-to-date version of the BSE code
(Hurley, Tout & Pols, 2002) and found that their model fails to explain some
observational properties of magnetic CVs. Thus, the agreement between the
space density measurements of Pala et al. (2020) and predictions by Belloni et
al. (2020) does not mean that the previous population synthesis models based
on the standard evolution and formation theory are wrong. It is likely that
the space densities of $10^{-5}$-$10^{-4}$ pc-3 proposed by previous CV
population synthesis models are correct and the current observational
measurements suffer from incomplete sky surveys as the surveys are probably
missing the most of very low $\dot{M}$ systems, CVs in the orbital period gap,
whose lifetime is predicted to be longer (Hillman et al., 2020), and the most
of period bouncers.

Besides these studies, observational Galactic model parameters, luminosity
functions and space densities of CVs, i.e. intrinsic properties of Galactic CV
population, should still be determined from suitable data by using the methods
of observational Galactic structure studies (Karaali, Bilir & Hamzaoǧlu, 2004;
Bilir et al., 2006a, b, c, 2008; Karaali et al., 2007; Cabrera-Lavers et al.,
2007), as the number of systems with reliable distance estimates is high
enough to use these methods. Sky surveys such as Gaia (Gaia Collaboration,
2016, 2018, 2021a, 2022) and SDSS (Szkody et al., 2002, 2003, 2004, 2005,
2006, 2007, 2011) determined faint systems and presented reliable distances of
CVs. Using these data, it is possible to decrease the selection effects that
may be strong for faint systems. These data and methods can allow us to
determine observational constraints for evolutionary models of CVs and mCVs.

In this study, we use ESA’s Gaia Data Release 3 (Gaia DR3; Gaia Collaboration,
2022) and Bailer-Jones et al. (2021) to obtain reliable distances for a CV
sample. We analyse the spatial distribution of CVs and discuss the
completeness of the CV sample. Then, we use this sample to estimate the
Galactic model parameters and space densities of All CVs and mCVs with scale-
heights obtained from exponential and ${\rm sech}^{2}$ functions fitted to
$z$-histograms.

##  2 Data

In order to construct our CV sample, we collected CVs from AAVSO’s
International Variable Star Index111www.aavso.org/vsx/ database. Their
equatorial and Galactic coordinates were taken from
SIMBAD222https://simbad.u-strasbg.fr/simbad/sim-fid database. We also included
CV’s found in the previous studies (e.g. Hofmann et al., 2018; Halpern et al.,
2018; Szkody et al., 2018; Bernardini et al., 2019; Kato, 2019; Yu et al.,
2019; Belloni et al., 2020; Kato et al., 2020; Schwope et al., 2020) in this
sample. Orbital periods were mainly taken from Ritter & Kolb (2003) and Downes
et al. (2001). Superhump periods of objects, whose orbital periods are
unknown, were assumed to be their orbital periods (Kato et al., 2020, and
references therein). The number of systems is 10,852 in this very rough
sample. We ignored the objects classified as non-mCVs and removed objects for
which trigonometric parallax measurements are not present in Gaia DR3. All the
objects in the preliminary sample were checked with respect to equatorial
coordinates in order to avoid from duplication. In order to prevent the
misidentification of CVs due to adjacent objects, we also checked each
object’s Gaia position individually using
Aladin333https://aladin.u-strasbg.fr/AladinLite/ and removed misidentified
objects from the database. To ensure a robust match, we crossmatched each
object in the catalogue to a $15^{{}^{\prime\prime}}$ radius in Gaia DR3,
propagated the subsets of Gaia DR3 $15^{{}^{\prime\prime}}$ crossmatches to
the J2000 epoch using proper motion, and made a final crossmatch at J2000 with
a radius of $2^{{}^{\prime\prime}}$, given the precision of the catalogue. The
132 CVs in the directions of the globular clusters were excluded from the
statistics to avoid mismatching with the data in the Gaia catalogue. As a
result of this selection process, Gaia photometric and astrometric data of
5,621 CVs were obtained. We showed the matching procedure for nine objects as
an example. The results for matching these objects at different $G$ apparent
magnitudes from the Gaia DR3 catalogue with panSTARR $g$ images on
Aladin444http://aladin.cds.unistra.fr/aladin.gml are shown in Figure 1.

Figure 1: Location of nine CVs with different $G$ apparent magnitudes selected
from the Gaia DR3 catalogue are shown on panSTARR $g$ images using Aladin.
Target objects are located in the centres of the images.

As we need a precise data sample to extract Galactic model parameters of CVs,
we also made strict cuts on quality flags, even though these cuts remove
numerous systems from the sample. We retain matches with
phot_g_mean_flux_over_error, $f_{\rm G}/{\delta f_{\rm G}},>8$, more than
eight “good” astrometric observations, as characterized by Gaia DR3,
astrometric_excess_noise<2, and $\varpi>0.1$ mas. The remaining sample
includes 4,149 CVs. In the sample, magnetic CVs classified as DQ Her
(Intermediate polars) or AM Her type (Polars) objects are indicated as mCV,
remaining systems as CV. Number of magnetic CVs is only 205 in this sample.
Although the sample includes 4,149 objects classified as CV, we know orbital
periods only for 1,187 of them.

We used Gaia DR3 data (Gaia Collaboration, 2022) to obtain the distances of
CVs in our catalogue. In order to do this, we matched our catalogue with Gaia
DR3 catalogue and found Gaia ID for each CV. It is possible to estimate the
distances of CVs by simply inverting their trigonometric parallaxes taken from
Gaia DR3 catalogue. However, Bailer-Jones et al. (2018, 2021) indicated that
the nonlinearity of the transformation and the asymmetry of the resulting
probability distribution must be taken into account, and they re-estimated the
Gaia EDR3 (Gaia Collaboration, 2021a) parallaxes. In such a re-estimation, we
do not expect too different distance estimates from Gaia DR3 data and Bailer-
Jones et al. (2021)’s approximation for a certain system. Thus, we matched our
catalogue with the catalogue of Bailer-Jones et al. (2021) using Gaia DR3 IDs
to obtain precise distances and distance errors of CVs and compared distances
of CVs estimated from Gaia DR3 and Bailer-Jones et al. (2021) in Figure 2,
where different $G$ apparent magnitude intervals are represented by coloured
symbols. This comparison shows that there is a considerable scatter especially
for fainter systems, while the scatter is much less for CVs with $G\leq 18.5$
mag. Thus, we limited our CV sample to the systems for which $G\leq 18.5$ mag,
which is set as the faint limit of the sample. We set the bright limit as
$G=9$ mag, since there is no brighter object in our sample. The final sample
comprises of CVs with $9\leq G\leq 18.5$ mag and includes 1,714 CVs, 767 of
them with known orbital periods. The relative distance error of the sample is
less than 1.66 and the median value is 0.06.

Figure 2: Comparison of CV distances obtained from Gaia DR3 and Bailer-Jones
et al. (2021) ($d_{\rm BJ}$). Different $G$ apparent magnitude intervals are
shown in panels (a), (b), (c) and (d). The red line represents the one-to-one
line and blue dashed lines 500 and 1000 pc distances from the red line.

Although the number of CVs in the sample is the largest ever used in similar
analyses, it is still questionable if this sample is sufficiently large to be
representative of the entire CV population and subject to magnitude-related
selection effects. Since the standard theory predicts that most CVs should be
intrinsically faint objects, it seems that apparent magnitude limits of
surveys are one of the strongest selection effects. Therefore, the
completeness limits of the data must be taken into account in a study based on
stellar statistics.

In order to set completeness limits to the sample, we first obtained
interstellar absorption in $V$-band $A_{\rm V}$ for CVs in the sample by using
MWDUST555https://github.com/jobovy/mwdust code which produces two and three-
dimensional Galactic dust map (Bovy et al., 2016), based on Schlegel,
Finkbeiner & Davis (1998)’s dust maps as re-calibrated by Schlafly &
Finkbeiner (2011). MWDUST code can provide two and three-dimensional data
according to Galactic coordinates from the Sun to all directions until the
edges of our Galaxy contributed by the Galactic dust. Since the distances of
systems are well-known from trigonometric parallaxes, we preferred to use two-
dimensional data and showed the rough absorption value in $V$-band according
to Galactic latitude ($b$) and longitude ($l$) toward the direction of an
object by $A_{\infty}(V)$, practically means up to infinite, but actually up
to the edge of the Galaxy.

The total absorption in $V$-band for the distance $d$ to the star is
calculated as following (Bahcall and Soneira, 1980)

$A_{\rm d}(V)=A_{\infty}(V)\Biggl{[}1-\exp\Biggl{(}\frac{-\mid d\times\sin
b\mid}{H}\Biggr{)}\Biggr{]},$ (1)

here $H$ is the scaleheight for the interstellar dust which is adopted as 125
pc (Marshall et al., 2006). As the distance of the system ($d_{\rm BJ}$,
hereafter also denoted as $d$) $d$ is known from Bailer-Jones et al. (2021),
the total absorption for the system $A_{\rm d}(V)$ could be estimated. We
obtained the colour excess $E_{\rm d}(B-V)$ for each CV in the sample using
$E_{\rm d}(B-V)=A_{\rm d}(V)/3.1$. The total absorptions in $G$, $G_{\rm BP}$,
and $G_{\rm RP}$ bands were obtained by using relations given as follows:

$\displaystyle A(G)=0.83627\times 3.1E_{\rm d}(B-V),$ $\displaystyle A(G_{\rm
BP})=1.08337\times 3.1E_{\rm d}(B-V),$ (2) $\displaystyle A(G_{\rm
RP})=0.63439\times 3.1E_{\rm d}(B-V).$

The selective absorption coefficients in Equation (2) were taken from
Cardelli, Clayton, & Mathis (1989). Once we know the total absorption values,
we could calculate the de-reddened apparent magnitudes in $G$, $BP$ and
$RP$-bands, $G_{0}$, $G_{\rm BP}$ and $G_{\rm RP}$, respectively. The absolute
magnitudes $M_{\rm G}$ of CVs were calculated using the distance modulus
formula $G_{\rm 0}-M_{\rm G}=5\times\log(d_{\rm BJ})-5$, where $d_{\rm BJ}$ is
the distance obtained from Bailer-Jones et al. (2021). The absolute magnitudes
$M_{\rm G}$ of CVs in the preliminary sample against their distances are shown
in Figure 3.

Figure 3: The absolute magnitudes $M_{\rm G}$ of CVs in the preliminary sample
against their distances $d_{\rm BJ}$ obtained from Bailer-Jones et al. (2021).
Red dashed lines show distances estimated from the bright ($G=9$ mag) and
faint ($G=18.5$ mag) limiting apparent magnitudes for absolute magnitude
intervals of 1 mag. Blue solid lines correspond to the bright and faint
apparent magnitude limits.

Red dashed lines in Figure 3 show distances estimated from the bright and
faint brightness limits ($9\leq G\leq 18.5$ mag) for absolute magnitude
intervals of 1 mag. These boxes limited by red dashed lines define the
completeness limits of the data for certain absolute magnitude intervals. We
removed systems beyond (out of the boxes defined by red dashed lines in Figure
3) these limiting magnitudes from the CV sample in order to obtain a complete
catalogue in a certain volume with the Sun in its centre. We found that the
majority of systems beyond 4 kpc in Figure 3 are discovered by SDSS. The final
sample includes 1,587 CVs, 704 of them with known orbital periods. There are
only 124 mCVs in this sample, 117 of them with a known orbital period.
Analyses in this paper were performed using this final sample.

The final sample is given in Table 1 including equatorial coordinates
$(\alpha,\delta)_{\rm J2000}$, object groups (magnetic (mCVs) or non-magnetic
(non-mCVs)), orbital periods ($P_{\rm orb}$), Gaia DR3 trigonometric
parallaxes ($\overline{\omega}$) and relative parallax errors
($\sigma_{\overline{\omega}}/{\overline{\omega}}$), proper motions
($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$), $G$-band apparent magnitudes from
Gaia DR3. Distances $d_{\rm BJ}$ (also denoted as $d$) in Table 1 were taken
from Bailer-Jones et al. (2021). This sample is the largest ever used in
similar analyses of CVs. Besides, it includes the most reliable distance
information for these systems. Objects, that are not classified as magnetic
system in the literature, are denoted as CV in our catalogue.

Table 1: The data sample. The object name, equatorial coordinates
$(\alpha,\delta)_{\rm J2000}$, orbital periods $P_{\rm orb}$ and object groups
(magnetic (mCV) or non-magnetic (non-mCVs)). $G$ denotes $G$-band apparent
magnitudes, $\overline{\omega}$ trigonometric parallaxes,
$\sigma_{\overline{\omega}}/{\overline{\omega}}$ relative parallax errors,
$\mu_{\alpha}\cos\delta$ and $\mu_{\delta}$ proper motion components from Gaia
DR3. Distance $d_{\rm BJ}$ is calculated using the system parallax in Bailer-
Jones et al. (2021). First and the last two rows of Table 1 is given here. The
table can be obtained electronically.

ID | Star Name | $\alpha$ | $\delta$ | Object | $P_{\rm orb}$ | $\overline{\omega}$ | $\sigma_{\overline{\omega}}/{\overline{\omega}}$ | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | $G$ | $d_{\rm BJ}$ | Ref
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (hh:mm:ss) | (dd:mm:ss) | groups | (d) | (mas) |  |  (mas yr-1) |  (mas yr-1) | (mag) | (pc) |
0001 | 2XMM J000134.1+625008 | 00 01 34.18 | +62 50 07.40 | non-mCVs | …. | 0.2519 | 0.0968 |  2.268$\pm$0.091 |  0.486$\pm$0.0941 |  17.9224$\pm$0.0043 |  4133$\pm$1764 | ….
0002 | EF Tuc | 00 01 55.10 | -67 07 43.3 | non-mCVs | 0.145 | 0.7120 | 0.0203 |  13.348$\pm$0.020 |  -1.455$\pm$0.0230 |  14.8480$\pm$0.0196 |  1330$\pm$36 | (01)
…. | …. | …. | …. | …. | …. | …. | …. | …. | …. | …. | …. | ….
…. | …. | …. | …. | …. | …. | …. | …. | …. | …. | …. | …. | ….
…. | …. | …. | …. | …. | …. | …. | …. | …. | …. | …. | …. | ….
1713 | 2dFGRS TGS429Z114 | 23 50 32.90 | -31 25 48.5 | non-mCVs | - | 0.2672 | 0.0585 |  -2.237$\pm$0.043 |  -3.851$\pm$0.045 |  16.6399$\pm$0.0028 |  3338$\pm$715 | ….
1714 | BC Cas | 23 51 17.45 | +60 18 10.0 | non-mCVs | - | 0.5360 | 0.0588 |  -0.861$\pm$0.059 |  -2.52$\pm$0.06 |  17.2256$\pm$0.0095 |  1831$\pm$198 | ….

(01) Downes et al. (2001), (02) Hardy et al. (2017), (03) Ritter & Kolb
(2003), (04) Szkody et al. (2013), (05) González-Buitrago et al. (2013), (06)
Steiner et al. (2007), (07) Paterson et al. (2019), (08) Kozhevnikov (2019),
(09) Thorstensen et al. (2017), (10) Watson, Henden, & Price (2006), (11) Wils
et al. (2011), (12) Thorstensen, Alper, & Weil (2016), (13) Rude & Ringwald
(2012), (14) Nesci, Tuvikene, & Gualandi (2019), (15) Patterson et al. (2003),
(16) Hamilton-Drager et al. (2018), (17) Sterken et al. (2007), (18)
Thorstensen (2020), (19) Coppejans et al. (2016), (20) Plavchan et al. (2008),
(21) Copperwheat et al. (2011), (22) Shafter & Misselt (2006), (23) Bruch
(2019), (24) Masci et al. (2019), (25) Hümmerich, Bernhard, & Srdoc (2014),
(26) McAllister et al. (2015), (27) Thorstensen et al. (2020), (28) Kato et
al. (2015), (29) Kozhevnikov (2014), (30) Thorstensen & Fenton (2003), (31)
Drake et al. (2014), (32) Thorstensen, Peters, & Skinner (2010), (33) Schaefer
(2021), (34) Ruiz-Carmona, Groot, & Steeghs (2020), (35) Kato et al. (2009),
(36) Kozhevnikov (2018), (37) Pala et al. (2020), (38) Halpern & Thorstensen
(2015), (39) Alfonso-Garzón et al. (2012), (40) Rodríguez-Gil et al. (2012),
(41) Kato et al. (2017), (42) Kozhevnikov (2003), (43) Halpern et al. (2018),
(44) Worpel et al. (2018), (45) Kato (2020), (46) Rodríguez-Gil & Torres
(2005), (47) Thorstensen & Skinner (2012), (48) Joshi et al. (2020), (49) Dai
et al. (2020), (50) Bernardini et al. (2019), (51) Bruch & Diaz (2017), (52)
Belloni et al. (2020), (53) McAllister et al. (2019), (54) Mason et al.
(2013), (55) Uemura et al. (2010), (56) Beuermann et al. (2021), (57) Samus’
et al. (2017), (58) Kato et al. (2020), (59) Pal et al. (2020), (60) Tappert
et al. (2013), (61) Vogt et al. (2018), (62) Rutkowski et al. (2011), (63)
Avilés et al. (2020), (64) Chen et al. (2020), (65) Koen & O’Donoghue (1995),
(66) Gabdeev et al. (2019), (67) Ringwald & Velasco (2012), (68) Han et al.
(2016), (69) Mukai, Zietsman, & Still (2009), (70) Hambsch (2014), (71) Kato
et al. (2013), (72) Bond & Miszalski (2018), (73) Zubareva et al. (2011), (74)
Yu et al. (2019), (75) Sheets et al. (2007), (76) Breus, Petrík, & Zola
(2019), (77) Gasque et al. (2019), (78) Myers et al. (2017), (79) Kozhevnikov
(2017), (80) Rude & Ringwald (2012a), (81) Altan et al. (2019), (82) Weil,
Thorstensen, & Haberl (2018)

In Figure 4a, we show the distance histogram of CVs in the final sample which
is limited using bright and faint limiting magnitudes in $G$-band, 9 and 18.5
mag, respectively. The cumulative distribution of CV distances is presented in
Figure 4b. Relative distance errors $\sigma_{d_{\rm BJ}}/{d_{\rm BJ}}$
obtained from Bailer-Jones et al. (2021) are also shown in Figure 4c. As
expected, the relative errors are increasing with the distance. Relative
distance errors are good indicators of the precision of the distance
measurements, which is very important in our study. While relative errors for
66% of the systems in the final sample is $\sigma_{d_{\rm BJ}}/{d_{\rm BJ}}$
$\leq$ 0.10, 88% of them have relative distance errors of $\sigma_{d_{\rm
BJ}}/{d_{\rm BJ}}$ $\leq$ 0.25. These values show that reliable constraints to
population models of CVs can be obtained from the sample in this study.

Figure 4: Distance histogram of CVs in the sample which is limited using
bright and faint limiting magnitudes in $G$-band. Relative distance errors
$\sigma_{d_{\rm BJ}}/{d_{\rm BJ}}$ are also shown in the lowest panel. Figure
5: Gaia DR3 HR diagram of CVs for which orbital periods $P_{\rm orb}$ are
known in the final sample. The separating point of the orbital period gap is
accepted as 2.6 h (Ak et al., 2010). White dwarf, main-sequence and giant star
regions are shaded in blue and grey, respectively (Abril et al., 2020).
Figure 6: The distribution of CVs according to an equatorial ($\alpha$,
$\delta$) and Galactic ($l$, $b$) coordinates. Distributions of magnetic and
non-magnetic systems are shown with different colours, blue and black,
respectively. Red dashed lines represent the Galactic plane (upper panel) and
the celestial equator (lower panel). The green star in the upper panel
indicates the location of the Galactic centre on the plot.

Precision of distance estimations can be also seen in the HR diagram of CVs,
for which the orbital periods are known. Figure 5 shows HR diagram of CVs
below and above the orbital period gap using Gaia photometry. The separating
point of the orbital period gap is accepted as 2.6 h (Ak et al., 2010). CVs
above the gap are generally brighter than those located below, as expected.
This discrimination is very clear now due to precise distance measurements. It
is clear that colour range for systems with $P_{\rm orb}\leq 2.6$ h is
narrower compared to that of systems with $P_{\rm orb}>2.6$ h. Figure 5 shows
that there are short-period systems brighter than $M_{\rm G}\approx 5$ mag.
These are probably SU UMa type dwarf novae, which were in the superoutburst
phase when they were observed. Blue and grey shaded parts in the HR diagram of
CVs represent the regions where the white dwarf, main sequence and giant stars
are located, respectively. The data for these shaded regions were taken from
Abril et al. (2020).

##  3 The Analysis and comparisons

###  3.1 Spatial distribution

The distribution of CVs in the final sample according to equatorial and
Galactic coordinates are plotted in Figure 6, upper and lower panels,
respectively. The panels in Figure 6 indicate that the CVs are symmetrically
distributed about the Galactic plane in general. The densest regions in both
panels correspond to the Solar vicinity. In order to inspect the Galactic
distribution of CVs in the Solar neighbourhood, we also calculated the Sun-
centred rectangular Galactic coordinates of CVs ($X$ towards Galactic centre,
$Y$ Galactic rotation, $Z$ north Galactic pole) in the sample and displayed
their projected positions on the Galactic plane ($X-Y$ plane) and on a plane
perpendicular to it ($X-Z$ plane) in Figure 7. Median heliocentric rectangular
Galactic coordinates ($X$, $Y$, $Z$) are 80, 93 and -18 pc, respectively, for
all systems in the sample and -51, -4 and 24 pc for magnetic systems,
respectively. These values are summarized in Table 2.

Figure 7: The spatial distribution of CVs in the final sample with respect to the Sun. $X$, $Y$ and $Z$ are the Sun centred rectangular Galactic coordinates. Magnetic systems (mCVs) are shown with blue symbols. Number histograms created according to coordinates are shown to the right and top of the figure.  Table 2: The median distances ($d$) and heliocentric rectangular Galactic coordinates ($X$, $Y$, $Z$) of CVs in the sample. Values are separately listed for All CVs and magnetic (mCVs) systems. $N$ denotes the number of objects. Group | $N$ | $\tilde{d}$ | $\tilde{X}$ | $\tilde{Y}$ | $\tilde{Z}$
---|---|---|---|---|---
|  | (pc) | (pc) | (pc) | (pc)
All CVs | 1587 | 989 | 80 | 93 | -18
mCVs |  124 | 559 | -51 | -4 |  24

Figures 6 and 7 show that there is no considerable bias according to the
spatial distribution of CVs in our study. It should be noted that only
magnetic CVs near the Sun can be detected as they are faint objects, in
general. The median distances of CVs in the final sample are 989 and 559 pc
for All CVs and magnetic systems, respectively. The median distances of All
CVs and magnetic systems in the CV sample of Özdönmez et al. (2015) were found
to be 423 and 385 pc, respectively. Comparisons of median distances in both
studies and Figure 5 in this study with Figure 6 of Özdönmez et al. (2015)
reveals that much farther objects are included in our sample, thanks to Gaia
mission. In addition, the number of CVs with distance estimates in this study
is higher, and the distances are more accurate than those in Özdönmez et al.
(2015).

###  3.2 Galactic model parameters

Using the number of objects per unit volume, it is possible to obtain
information on their Galactic population. Galactic locations of objects must
be known in order to estimate their Galactic model parameters. This
information allows us to estimate scale length and scale height of the objects
in question. Based on deep sky surveys, it is known that the scale length of
the thin disc stars is expected to be larger than 2.6 kpc (Bilir et al.,
2006a; Juric et al., 2008). Besides All CVs being members of the thin-disc
population of the Galaxy according to their Galactic kinematics (Ak et al.,
2015), most of the systems in our sample are located in distances less than 2
kpc, as shown in Figure 7, and the median distance of All CVs in the final
sample is 989 pc, which is much less than 2.6 kpc. Thus, our sample was not
inspected according to population types and the scale length estimation was
not performed.

In order to find the Galactic model parameters of CVs, $z$-histograms that
demonstrate the vertical distribution of objects in the Galaxy must be
studied. Although the Galactic model parameters are derived using the
exponential functions, Bilir et al. (2006a) showed that the observed vertical
distribution in the Galaxy is smoother in the Solar neighbourhood, and is
well-approximated by a secans hiperbolicus square function (${\rm sech}^{2}$).
Thus, the number of stars at a distance $z$ from the Galactic plane is
described in our study by using both exponential and secans hiperbolicus
square functions

$n(z)=n_{0}\exp\Biggl{(}-\frac{\mid z\mid}{H}\Biggr{)}$ (3)

and

$n(z)=n_{0}~{}{\rm sech}^{2}\Biggl{(}-\frac{\mid z\mid}{H_{z}}\Biggr{)},$ (4)

respectively. $n(z)$ based on ${\rm sech}^{2}$ can be also expressed as

$n(z)=n_{0}~{}\Biggl{(}\frac{4}{\exp{(-2z/H_{z})}+\exp{(2z/H_{z})+2}}\Biggr{)}$
(5)

(Bilir et al., 2006a). Here, $z$ is the distance of objects from the Galactic
plane and $n_{0}$ is the number of stars for $z=0$ pc. $H$ and $H_{z}$ are the
exponential and ${\rm sech}^{2}$ scale heights, respectively. $z$ is described
as $z=z_{0}+d\sin(b)$, with $b$ being the Galactic latitude of the star, $d$
distance of the object and $z_{0}$ distance of the Sun from the Galactic plane
(24 pc; Juric et al., 2008). The relation between the exponential scale height
$H$ and the ${\rm sech}^{2}$ scale height $H_{z}$ is $H=1.08504\times H_{z}$
(Bilir et al., 2006a). To sample the posteriors, we used the Markov Chain
Monte Carlo (MCMC) emcee package of the affine-invariant ensemble sampler
(Goodman & Weare, 2010), kindly provided by Foreman-Mackey et al. (2013). To
obtain initial model parameters, we used a nonlinear least-squares algorithm
(Python library LMFIT). Using these priors, we ran the MCMC to sample the
posteriors for 128 initial conditions and tested these walkers in 15,000 steps
of chains. Thus, we obtained the most plausible model parameters and their
errors through minimising chi-square ($\chi_{\rm min}^{2}$).

The best fits to the $z$-histograms of All CVs and magnetic CVs in the sample
are shown in Figure 8. We also demonstrate the 2-D posterior probability
distributions of the model parameters sampled by MCMC in Figure 8. The scale
height $H$ and the number of stars in the Solar neighbourhood $n_{0}$ obtained
from the analyses are listed in Table 3.

Figure 8: The $z$-histograms for CVs. The upper panel shows the $z$-histogram for All CVs in the sample and the lower panel magnetic systems (mCVs). The blue dashed line represents the ${\rm sech}^{2}$ function and the red solid line the exponential function. The 2-D posterior probability distributions of the model parameters sampled by MCMC are demonstrated to the right of the $z$-histograms. Table 3: The Galactic model parameters for All CVs and mCVs in the sample. Model functions are given in the third column. Here, exp denotes for exponential function and ${\rm sech}^{2}$ for secans hiperbolicus square function. $N$ is the number of systems in the object group, $n_{0}$ is the number of stars in the Solar neighbourhood, $H$ the scale height for the model function. Group | $N$ | Function | $n_{0}$ |  $H$(pc)
---|---|---|---|---
All CVs | 1587 | exp | 229$\pm$1 | 375$\pm$2
|  | ${\rm sech}^{2}$ | 196$\pm$1 | 370$\pm$1
mCVs | 124 | exp | 41$\pm$1 | 281$\pm$3
|  | ${\rm sech}^{2}$ | 35$\pm$1 | 279$\pm$3
Figure 9: The number of stars for $z=0$ pc ($n_{0}$) versus scale height
($H$), estimated with 15,000 trials for the Monte Carlo simulations. The
simulations were performed for All CVs in the sample, keeping magnetic systems
in the sample as they are and assuming 22$\%$ of the remaining sample to be
magnetic by arbitrary selection. The left and right panels are plotted for
exponential and ${\rm sech}^{2}$ scale heights, respectively. Red straight
lines show the most probable values. 1$\sigma$ values are indicated with blue
straight lines.

It is remarkable that exponential and ${\rm sech}^{2}$ functions give very
similar scale heights for the two groups in Table 3. Nevertheless, the
exponential functions well represent the $z$-histograms in Figure 8
constructed for All CVs and magnetic CVs in the sample. From Table 3, we found
that there is a considerable difference between the scale heights obtained for
All CVs and magnetic systems, 375$\pm$2 and 281$\pm$3 pc, respectively. This
can be expected as the median distances of these two groups of systems are 989
and 559 pc, respectively. Thus, we conclude that magnetic CVs are different
than All CVs in the sample with respect to the scale height. In addition, as a
consequence of reduced magnetic braking with the strong magnetic fields (Li et
al., 1994), magnetic CVs evolve slower than non-magnetic systems (Araujo-
Betancor et al., 2005). Thus, we expect that their Galactic parameters should
be different. Ak et al. (2013) estimated the contribution of thick-disc CVs in
the Solar neighbourhood to the Galactic model parameters from the Monte Carlo
simulations and found that only about 6 per cent of CVs in the Solar
neighbourhood are members of the thick-disc population of the Galaxy.
Therefore, the effect of thick disc systems on the scale heights in Table 3
must be negligible.

Another interesting finding is that, if the $z$ histogram of CVs is accepted
to be exponential, it must be about 25 missing CVs within the sphere with a
radius of about 100 pc with the Sun at the centre, which reminds that the
number of period bouncers discovered in sky surveys is less than expected from
the population models based on the standard theory. McAllister et al. (2019)
found that 30% of donor stars in their sample are likely to be brown dwarfs in
period bouncers, while only 5% of CVs are located within 150 pc from the Sun
in Pala et al. (2020) included period bouncers.

Although we selected magnetic systems (AM Her or DQ Her type systems)
according to their classification in the literature, in fact, it can be
unclassified magnetic systems in the sample. In a sense, the sample could be
contaminated by them. As a result the scale heights estimated for All CVs in
Table 3 could not be reliable. In order to find the effect of magnetic system
contamination on the scale height of All CVs, we decided to perform Monte
Carlo simulations. An inspection shows that about 30$\%$ of CVs in the
catalogue of Ritter & Kolb (2003) are classified as either AM Her or DQ Her
system. Thus, we assumed that 30$\%$ of the systems in our sample is magnetic.
As we know that 124 of CVs in the sample are classified as magnetic, which is
about 8$\%$ of objects in this study. Keeping magnetic systems in the sample
as they are, 22$\%$ of the remaining sample were assumed to be magnetic by
arbitrary selection. We calculated $n_{0}$ and $H$ for each run of 15,000
trials for the Monte Carlo simulations. Figure 9 shows $n_{0}$ versus $H$ for
the logarithmic and ${\rm sech}^{2}$ functions. After 15,000 trials for the
Monte Carlo simulations performed on All CVs in the sample, the most probable
values of the scale heights for the logarithmic and ${\rm sech}^{2}$ functions
were found to be 398$\pm$31 and 406$\pm$18 pc, respectively. Error values are
1$\sigma$ errors. A comparison with the values in Table 3 shows that these
values are in agreement, within errors, with those given for All CVs in the
sample and the effect of the magnetic system’ contamination on the scale
height of all systems can be negligible for our sample. Note that the most
probable $n_{0}$ values obtained from the Monte Carlo simulations are
158$\pm$7 and 132$\pm$5 for the logarithmic and ${\rm sech}^{2}$ functions,
respectively. Differences of these values from those in Table 3 are expected
as the system numbers in different $z$ distances were changed during the
simulations.

A comparison of the scale heights found in this study with the ones in
Özdönmez et al. (2015) reveals that their results are very different from
those listed in Table 3. They found the exponential scale heights to be
213${}^{+11}_{-10}$ and 173${}^{+18}_{-15}$ pc for All CVs and magnetic
systems in their sample, respectively. It is clear that these values are very
different than 375$\pm$2 and 281$\pm$3 pc in Table 3. The scale height of
about 375 pc found for All CVs in this study is also considerably larger than
those suggested by Patterson (1984, 190$\pm$30 pc), van Paradijs, Augusteijn &
Stehle (1996, 160-230 pc) and Ak et al. (2008, 158$\pm$14 pc). Note that Pala
et al. (2020) assumed a scale height of 280 pc for their analysis.

The scale height differences between this study and the previous estimates
should be resulted from the number of systems in the analysis and the accuracy
of distance estimates. As our data sample is based on reliable distances based
on precise trigonometric parallax measurements and it includes the highest
number of systems in a similar analysis in the literature, we believe that the
results in Table 3 can be confidently used in population studies and further
analysis of CVs.

We also derived the scale height of CVs in terms of the orbital period to find
if this Galactic model parameter changes according to the orbital period. In
order to compare our results with those in Özdönmez et al. (2015), the CVs in
our sample were divided into four-period intervals as defined in their study.
Thus, there are 203 systems in the period interval $1.37\leq P_{\rm orb}$(h)
$<2.25$, 157 systems in $2.25\leq P_{\rm orb}$(h) $<3.7$, 105 systems in
$3.7\leq P_{\rm orb}$(h) $<4.6$ and 171 in $4.6\leq P_{\rm orb}$(h) $<12$. The
best fits to the $z$-histograms of CVs grouped according to these period
intervals are shown in Figure 10. The scale height $H$ and the number of stars
in the Solar neighbourhood $n_{0}$ obtained from the minimum $\chi^{2}$
analysis are listed in Table 4 for the period intervals given above. Since the
number of systems is too small for magnetic CVs in these period ranges, we
list the Galactic model parameters only for All CVs in the sample.

The $z$-histograms in Figure 10 are well represented by exponential functions,
in general. A comparison of scale heights in Table 4 with those given in
Özdönmez et al. (2015) shows the same trend, although the values in this study
are more reliable. The scale height increases monotonously from 248$\pm$2 to
430$\pm$4 pc while the orbital period decreases from 12 to 2.25 h. However, it
drops to 300$\pm$2 pc for the shortest orbital period CVs with $P_{\rm
orb}<2.25$ h. We found a similar trend also for the ${\rm sech}^{2}$ function.
The scale height for the systems in the interval $2.25\leq P_{\rm
orb}{\rm(h)}<3.70$ is the highest for all CVs. Note that only $\sim 14\%$ of
the systems in this interval are classified as mCV in our sample.

Figure 10: The $z$-histograms for All CVs in the sample in terms of the orbital period. Orbital period ($P_{\rm orb}(\rm h)$) intervals are shown in brackets. The best fits to the $z$-histograms are also presented. The blue dashed line represents the ${\rm sech}^{2}$ function and the red solid line the exponential function. Table 4: The Galactic model parameters for All CVs in the sample in terms of the orbital period $P_{\rm orb}(\rm h)$. $n_{\rm 0}$ and $H$ are as defined in Table 3. $P_{\rm orb}(\rm h)$ | $N$ | Function | $n_{0}$ |  $H$ (pc)
---|---|---|---|---
[1.37 , 2.25) | 203 | $\exp$ | 86$\pm$1 | 300$\pm$2
|  | ${\rm sech}^{2}$ | 74$\pm$1 | 286$\pm$1
[2.25 , 3.70) | 157 | $\exp$ | 43$\pm$1 | 430$\pm$4
|  | ${\rm sech}^{2}$ | 37$\pm$1 | 424$\pm$3
[3.70 , 4.60) | 105 | $\exp$ | 41$\pm$1 | 269$\pm$4
|  | ${\rm sech}^{2}$ | 34$\pm$1 | 278$\pm$3
[4.60 , 12.00) | 171 | $\exp$ | 69$\pm$1 | 248$\pm$2
|  | ${\rm sech}^{2}$ | 57$\pm$1 | 258$\pm$2

###  3.3 Space density

Space density is an important parameter for population synthesis studies based
on theoretical evolutionary models of a selected object type. The space
density of a group of stars is derived by dividing the number of stars in
consecutive distances from the Sun to the corresponding partial spherical
volumes: $D=N/\Delta V_{i,i+1}$ (Bilir et al., 2006a, b, c). Here, $D$ is the
space density, $N$ denotes the number of stars in the partial spherical volume
$\Delta V_{i,i+1}$ which is defined by consecutive distances $d_{i}$ and
$d_{i+1}$ from the Sun. The logarithmic space density is preferred to compare
the results in the literature, which is defined as $D^{*}=\log D+10$. The
logarithmic density functions of All CVs and magnetic systems in the Solar
neighbourhood are shown in Figure 11, where $r^{*}$ denotes the centroid
distance of the partial spherical volume which is defined as
$r^{*}=[(d^{3}_{i}+d^{3}_{i+1})/2]^{1/3}$. The local space density $D_{0}$ is
the space density estimated for $r^{*}=$ 0 pc from the exponential fits shown
in Figure 11. The logarithmic and local space densities of CV groups in the
Solar neighbourhood are listed in Table 5.

Table 5: The logarithmic and local space densities of CVs. Symbols for subgroups are as in Table 3. $N$ denotes the number of stars in the subgroup, $D_{0}$ is the local space density and $D^{*}$ logarithmic space density. Group | $N$ | $D^{*}$ | $D_{0}$
---|---|---|---
|  |  | ($\times$10-6 pc-3)
All CVs | 1587 | 4.83$\pm$0.07 | $6.8^{+1.3}_{-1.1}$
mCVs | 124 | 4.33$\pm$0.09 | $2.1^{+0.5}_{-0.4}$
Figure 11: The logarithmic density functions of All CVs (panel a) and mCVs
(panel b) in the sample. Dashed lines represent exponential fits applied to
the data.

Table 5 shows that the local space density of All CVs in the sample is
$6.8^{+1.3}_{-1.1}\times 10^{-6}$ pc-3. The space density of magnetic CVs are
about three times smaller than that found for all systems,
$2.1^{+0.5}_{-0.4}\times 10^{-6}$ pc-3. The local space density estimation in
this study takes into account the CVs located even further than 6 kpc, which
corresponds to a large Galactic volume. Note that the median distance of the
objects in the sample is 989 pc and the local space density estimation is
based on the objects within the completeness limits. Therefore, we believe
that the space density values obtained from the CV sample in this study are
reliable.

CV population synthesis models based on the standard formation and evolution
scenario predicts space densities $10^{-5}$-$10^{-4}$ pc-3 (Ritter & Burkert,
1986; de Kool, 1992; Kolb, 1993; Politano, 1996; Willems et al., 2005, 2007;
Goliasch & Nelson, 2015; Belloni et al., 2018) while the space densities found
in observational studies are on the order of $10^{-7}$-$10^{-4}$ pc-3 (Warner,
1974; Patterson, 1984, 1998; Thomas & Beuermann, 1998; Ringwald, 1993; Schwope
et al., 2002; Araujo-Betancor et al., 2005; Pretorius et al., 2007a, b; Ak et
al., 2008; Revnivtsev et al., 2008; Pretorius & Knigge, 2012; Pretorius et
al., 2013; Schwope, 2018) In a recent study, Pala et al. (2020) measured the
space density of $4.8^{+0.6}_{-0.9}\times 10^{-6}$ and
$1.2^{+0.4}_{-0.5}\times 10^{-6}$ pc-3 for All CVs and mCVs, respectively,
from Gaia DR2 (Gaia Collaboration, 2018). They assumed a scale height of 280
pc for their analysis and their data sample included 42 objects within 150 pc
from the Sun. Note that we found a scale height of $H=375$ pc for All CVs (see
Table 3).

The local space density estimated for All CVs in our sample
($6.8^{+1.3}_{-1.1}\times 10^{-6}$ pc-3) is 2-20 times smaller than those
predicted by population synthesis studies based on the standard evolution
scenario. However, it is very similar to the observational space density found
by Pala et al. (2020) ($4.8^{+0.6}_{-0.9}\times 10^{-6}$ pc-3) from Gaia DR2
(Gaia Collaboration, 2018), within the errors. The local space densities
obtained in this study are only about 1.5 times more than those they found.
Note that the fraction of mCVs in the whole sample used in this study is about
8$\%$.

###  3.4 Luminosity function

The luminosity function is defined as the space density of objects in a
certain absolute magnitude interval (Karaali et al., 2003, 2004, 2009; Ak et
al., 2007). We estimated the logarithmic luminosity functions $\phi$ for All
CVs and magnetic systems and presented them in Table 6, where $\Delta
V_{i,i+1}$ is the partial spherical volume which includes the objects located
between the distances $d_{i}$ and $d_{i+1}$. These distance limits correspond
to the bright and faint limiting apparent magnitudes in $G$-band, 9 $\leq
G\leq$ 18.5 mag, for the absolute magnitude $M_{\rm G}$ interval preferred
here. The logarithmic luminosity functions of all systems and magnetic systems
are plotted in the lower panel of Figure 12. As can be expected, the
luminosity function of All CVs is considerably different than the one
estimated for magnetic CVs.

Table 6: The logarithmic luminosity functions $\phi$ of CVs in the sample with 9 $\leq G\leq$ 18.5 mag. $N$ is the number of stars in the $M_{\rm G}$ absolute magnitude interval given in the first column. $\Delta V_{i,i+1}$ is the partial spherical volume that includes the objects between the distances $d_{i}$ and $d_{i+1}$ corresponding to the bright and faint limits in $G$-band for the absolute magnitude $M_{\rm G}$ interval. $\rho$ denotes the density and $r^{*}$ the centroid distance of the partial spherical volume. |  |  |  |  | All CVs | mCVs
---|---|---|---|---|---|---
$M_{{\rm G}}$ | $d_{1}$ | $d_{2}$ | $\Delta V_{1,2}$ | $r^{*}$ | $N$ | $\rho$ | $\phi$ | $N$ | $\rho$ | $\phi$
(mag) | (pc) | (pc) |  ($\rm pc^{3}$) | (kpc) |  |  ($\rm pc^{-3}$) |  |  |  ($\rm pc^{-3}$) |
$[$-3,-2) | 2512 | 125893 |  8.4$\times 10^{15}$ | 99.92 | 11 |  1.3$\times 10^{-15}$ | -14.9 | 1 |  1.2$\times 10^{-16}$ | -15.9
$[$-2,-1) | 1585 | 79433 |  2.1$\times 10^{15}$ | 63.05 | 8 |  3.8$\times 10^{-15}$ | -14.4 | 1 |  4.8$\times 10^{-16}$ | -15.3
$[$-1, 0) | 1000 | 50119 |  5.3$\times 10^{14}$ | 39.78 | 25 |  4.7$\times 10^{-14}$ | -13.3 | 1 |  1.9$\times 10^{-15}$ | -14.7
$[$ 0, 1) | 631 | 31623 |  1.3$\times 10^{14}$ | 25.10 | 22 |  1.7$\times 10^{-13}$ | -12.8 | 5 |  3.8$\times 10^{-14}$ | -13.4
$[$ 1, 2) | 398 | 19953 |  3.3$\times 10^{13}$ | 15.84 | 28 |  8.4$\times 10^{-13}$ | -12.1 | 3 |  9.0$\times 10^{-14}$ | -13.0
$[$ 2, 3) | 251 | 12589 |  8.4$\times 10^{12}$ | 9.99 | 87 |  1.0$\times 10^{-11}$ | -11.0 | 5 |  6.0$\times 10^{-13}$ | -12.2
$[$ 3, 4) | 158 | 7943 |  2.1$\times 10^{12}$ | 6.30 | 164 |  7.8$\times 10^{-11}$ | -10.1 | 10 |  4.8$\times 10^{-12}$ | -11.3
$[$ 4, 5) | 100 | 5012 |  5.3$\times 10^{11}$ | 3.98 | 229 |  4.3$\times 10^{-10}$ | -9.4 | 14 |  2.7$\times 10^{-11}$ | -10.6
$[$ 5, 6) | 63 | 3162 |  1.3$\times 10^{11}$ | 2.51 | 275 |  2.1$\times 10^{-09}$ | -8.7 | 12 |  9.1$\times 10^{-11}$ | -10.0
$[$ 6, 7) | 40 | 1995 |  3.3$\times 10^{10}$ | 1.58 | 210 |  6.3$\times 10^{-09}$ | -8.2 | 8 |  2.4$\times 10^{-10}$ | -9.6
$[$ 7, 8) | 25 | 1259 |  8.4$\times 10^{09}$ | 1.00 | 159 |  1.9$\times 10^{-08}$ | -7.7 | 13 |  1.6$\times 10^{-09}$ | -8.8
$[$ 8, 9) | 16 | 794 |  2.1$\times 10^{09}$ | 0.63 | 159 |  7.6$\times 10^{-08}$ | -7.1 | 24 |  1.1$\times 10^{-08}$ | -7.9
$[$ 9,10) | 10 | 501 |  5.3$\times 10^{08}$ | 0.40 | 99 |  1.9$\times 10^{-07}$ | -6.7 | 10 |  1.9$\times 10^{-08}$ | -7.7
$[$10,11) | 6 | 316 |  1.3$\times 10^{08}$ | 0.25 | 65 |  4.9$\times 10^{-07}$ | -6.3 | 10 |  7.6$\times 10^{-08}$ | -7.1
$[$11,12) | 4 | 200 |  3.4$\times 10^{07}$ | 0.16 | 28 |  8.4$\times 10^{-07}$ | -6.1 | 6 |  1.8$\times 10^{-07}$ | -6.7
$[$12,13) | 3 | 126 |  8.4$\times 10^{06}$ | 0.10 | 7 |  8.4$\times 10^{-07}$ | -6.1 | 1 |  1.2$\times 10^{-07}$ | -6.9
Figure 12: Logarithmic luminosity functions of CVs. Lower panel shows
luminosity functions for All CVs and mCVs. Upper panel demonstrates the
logarithmic luminosity function for All CVs in the sample. Red line represent
the 500 times the luminosity function of All CVs and the blue line logarithmic
luminosity function of white dwarfs taken from Gaia Collaboration (2021b).

The luminosity function of All CVs in the data sample is plotted in the upper
panel of Figure 12, where we also presented the white dwarf luminosity
function that demonstrate the collective evolution of white dwarfs (Gaia
Collaboration, 2021b). As can be seen from Table 6 and Figure 12, besides
magnetic systems have a luminosity function smaller than that estimated for
All CVs, they span a narrower absolute magnitude interval compared to the
absolute magnitude interval of all systems. The comparison of the luminosity
functions of white dwarfs and All CVs in our sample reveals that the
tendencies of both luminosity functions is almost the same and that the 500
times the luminosity function of All CVs corresponds to the continuation of
the luminosity function of white dwarfs towards the brighter absolute
magnitudes. A similar comparison was demonstrated by Özdönmez et al. (2015)
for CVs in their data sample and white dwarfs in the Anglo Australian
Telescope survey (Boyle, 1989) and the Palomar Green survey (Fleming, Liebert
& Green, 1986).

The single white dwarf masses are significantly smaller than those of the
white dwarfs in CVs (see Zorotovic & Schreiber, 2020, and references therein)
and it is claimed that the white dwarf mass does not depend on the orbital
period (McAllister et al., 2019; Pala et al., 2022). In addition, the mass of
CV white dwarfs does not show a monotonous increase during the evolution of
the system and the contribution of the primary component to the total
radiation of a CV is dominant -or much higher- in UV rather than that in
optical (Gänsicke, 2000). Simulations performed by Hillman et al. (2020)
predicts that the white dwarf masses in CVs decrease monotonically, by only a
few per cent throughout the evolution of cataclysmic variable. Therefore, it
is unlikely that the trend of CV luminosity function in the upper panel of
Figure 12 reflects the evolution of the white dwarf companion of these
systems. In any case, this comparison shows that we find one CV for about 500
white dwarfs in the Solar neighbourhood.

##  4 Conclusions

The spatial distribution, Galactic model parameters and luminosity function of
cataclysmic variables were precisely derived using distances from Bailer-Jones
et al. (2021), who re-estimated trigonometric parallaxes of ESA’s Gaia DR3
(Gaia Collaboration, 2021a). We compared distances obtained from Bailer-Jones
et al. (2021) and Gaia DR3 data (Gaia Collaboration, 2022) found that the
scatter is too much for the systems $G\geq 18.5$ mag. Thus, the data sample in
this study includes CVs with $9\leq G\leq 18.5$ mag. Number of CVs in the
sample decreased from 10,852 to 1,587, with 124 of them are magnetic systems,
by preventing the misidentification of CVs due to adjacent objects, checking
for duplication and by also taking into account the quality flags of parallax
measurements and completeness limits of the data.

Projected positions of CVs on the Galactic plane ($X-Y$ plane) and on a plane
perpendicular to it ($X-Z$ plane) demonstrate that systems in the sample are
symmetrically distributed about the Galactic plane, in general. So, we
conclude that there is no considerable bias according to the spatial
distribution of CVs in our study. The median distances of objects in the
sample are 989 and 559 for All CVs and magnetic systems, respectively.

The exponential scale heights were found to be 375$\pm$2 and 281$\pm$3 for All
CVs and mCVs in the sample, respectively. Thus, we conclude that a scale
height of 375 pc can be used in CV studies, in general. This value is
considerably larger than those previously suggested in observational studies
(Patterson, 1984; van Paradijs et al., 1996; Ak et al., 2008). It is also
significantly larger than that estimated by Özdönmez et al. (2015). Monte
Carlo simulations showed that the magnetic system’ effect on the scale height
of all systems can be negligible for the sample.

It seems that it must be 25 missing CVs within the sphere with a radius of
about 100 pc with the Sun at the centre. This reminds that the number of
period bouncers discovered in sky surveys is less than expected from the
population models based on the standard theory. Note that McAllister et al.
(2019) found that 30$\%$ of donor stars in their sample are likely to be brown
dwarfs in period bouncers.

The exponential scale heights of All CVs derived in terms of the orbital
period shows that the scale height increases from 248$\pm$2 to 430$\pm$4 pc
with the orbital period decreases from 12 to 2.25 h, and it almost suddenly
drops to 300$\pm$2 pc for the shortest orbital period CVs with $P_{\rm
orb}<2.25$ h. A similar trend was also found in Özdönmez et al. (2015). Note
that Pretorius et al. (2007b) modelled the Galactic population of CVs adopting
120, 260 and 450 pc for long, normal short orbital period systems and period
bouncers, respectively.

The local space density of All CVs and magnetic systems in the sample was
estimated to be $6.8^{+1.3}_{-1.1}\times$10-6 and $2.1^{+0.5}_{-0.4}\times
10^{-6}$ pc-3, respectively. The space densities estimated here for All CVs
and magnetic systems are in agreement with those calculated by Pala et al.
(2020), who used 42 CVs within 150 pc from the Sun with data obtained from
Gaia DR2 (Gaia Collaboration, 2018), within errors. They measured the space
densities of $4.8^{+0.6}_{-0.9}\times 10^{-6}$ and $1.2^{+0.4}_{-0.5}\times
10^{-6}$ pc-3 for all CVs and magnetic CVs, respectively. We claim that the
space density values in our study are the most reliable estimates ever found.
The measurements in this study strengthen the discrepancy between CV space
densities obtained from observations and those predicted by population
synthesis models based on the standard formation and evolution theory of these
systems. If the population synthesis models are correct, this disagreement
between the theory and observations means that the current CV surveys are
incomplete as they missed almost all very low $\dot{M}$ systems and CVs in the
period gap, through which the life time of a CV is predicted to be longer
(Hillman et al., 2020), and period bouncers.

The logarithmic luminosity functions derived for CVs in the sample are in
agreement with those shown in Özdönmez et al. (2015). The trend of the
logarithmic luminosity functions of CVs and white dwarfs are very similar.
Although 500 times the luminosity function of CVs looks like the extension of
the white dwarf luminosity function towards the brighter absolute magnitudes,
it is not likely that this similarity indicates the evolution of white dwarf
companion of CVs as the mass of CV white dwarfs does not show a monotonous
increase during the evolution of the system and contribution of the primary
component to the total radiation of a CV is dominant in UV (Gänsicke, 2000;
Pala et al., 2022).

To conclude, the results in this study can be used in population studies and
analysis of cataclysmic variables. We believe that the further $Gaia$
observations of cataclysmic variables and surveys focused on low $\dot{M}$ CVs
and fainter systems will lead to not only larger datasets but also more
precise distance measurements for these systems. Such observational results
will allow us to obtain more detailed and confident observational Galactic
model parameters to test population synthesis models.

##  5 Acknowledgments

We would like to thank Michael Shara, the referee, for his useful and
constructive comments concerning the manuscript. This work has been supported
in part by the Scientific and Technological Research Council of Turkey
(TÜBİTAK) 119F072. This work has been supported in part by Istanbul
University: Project number NAP-33768. This study is a part of the PhD thesis
of Remziye Canbay. 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 DPAC has
been provided by national institutions, in particular, the institutions
participating in the _Gaia_ Multilateral Agreement. This research has made use
of NASA’s (National Aeronautics and Space Administration) Astrophysics Data
System Bibliographic Services and the SIMBAD Astronomical Database, operated
at CDS, Strasbourg, France and NASA/IPAC Infrared Science Archive and
Extragalactic Database (NED), which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.

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