11institutetext: Institut für Astronomie und Astrophysik, Eberhard-Karls
Universität Tübingen, Sand 1, 72076 Tübingen, Germany
11email<EMAIL_ADDRESS>22institutetext: INAF -
Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo,
Italy 33institutetext: INAF - Osservatorio Astronomico di Capodimonte, Via
Moiariello 16, 80131 Naples, Italy 44institutetext: Leibniz-Institut für
Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany

# New X-ray detections of magnetic period-bounce cataclysmic variables from
XMM-Newton and SRG/eROSITA

Daniela Muñoz-Giraldo 11 Beate Stelzer, 1122 Domitilla de Martino 33 Axel
Schwope 44

(Received XX; accepted XX, 2023)

###### Abstract

Context. A great portion of the cataclysmic variable population, between
40$\%$ and 70$\%$, is predicted to be made up of period-bouncers, systems with
degenerate donors that have evolved past the period minimum. However, due to
their intrinsic faintness, only a few of these systems have been observed and
confidently identified so far.

Aims. We have searched for X-ray emission as a proof of accretion in order to
confirm period-bounce cataclysmic variables.

Methods. In this study we use data from XMM-Newton and eROSITA on a pilot
sample of three period-bounce candidates with a magnetic white dwarf, which
are expected to exhibit stronger X-ray emission than non-magnetic systems due
to more efficient conversion of matter accretion onto the white dwarf.

Results.  In a dedicated XMM-Newton observation of the period-bounce candidate
SDSS J151415.65+074446.5 we discovered X-ray modulation at the binary orbital
period confirming it as an accreting system. The X-ray light curve and the
X-ray spectrum display characteristics of magnetic Polar-type systems allowing
for the first time the determination of the X-ray luminosity and mass
accretion rate for this system. Catalog data from eROSITA on the Spektrum-
Roentgen-Gamma satellite for V379 Vir and SDSS J125044.42+154957.4 allowed a
first look into the X-ray behavior of period-bounce candidates with this new
all-sky instrument. From the eROSITA measurements the X-ray luminosity and
mass accretion rate were determined for the first time for SDSS
J125044.42+154957.4, and the earlier result for V379 Vir from XMM-Newton was
confirmed.

Conclusions. All three cataclysmic variables with a magnetic white dwarf and
very low-mass donor studied in this work present evidence for X-ray emission
at a similar level of $L_{\rm x}\,{\rm[erg/s]}\approx 10^{29}$, which,
together with the detection of X-ray orbital modulation in two of them, V379
Vir and SDSS J151415.65+074446.5, unambiguously proves the presence of
accretion in these systems. The detection of these period-bouncers at faint
X-ray luminosity levels with the all-sky X-ray survey eROSITA offers new
prospects for the identification of additional period-bouncers, providing
impetus for theoretical studies of binary evolution.

###### Key Words.:

X-rays: binaries, cataclysmic variables – stars: individual: SDSS
J151415.65+074446.5 – stars: individual: V379 Vir – stars: individual: SDSS
J125044.42+154957.4

##  1 Introduction

Cataclysmic variables (CVs) are interacting compact binaries where a white
dwarf (WD) accretes matter from a Roche-lobe filling, late-type donor (Warner,
1995). Two main types of CVs are distinguished according to the accretion
geometry of the system. Usually, accretion onto the WD occurs via a disc, but
if the WD magnetic field is strong enough ($\gtrsim$ 10MG), the formation of
an accretion disc is prevented and accretion flow proceeds directly from the
companion towards the magnetic polar regions. In these high field systems,
called Polars, the WD rotation is phase-locked at the orbital period (Cropper,
1990). Polars are known to switch between high and low mass accretion rate
states spending roughly half their time in low-states, seen in the case of the
prototype AM Her (Hessman et al., 2000), making their identification and
characterization challenging (Pretorius et al., 2013).

In terms of evolution, all CVs follow a track from longer orbital periods
towards shorter ones driven by angular momentum loss which causes the orbital
separation of the system to decrease (Knigge et al., 2011). In the course of
this evolution the system passes through a period gap as a detached binary,
between an orbital period ($P_{\rm orb}$) of $2-3$ h, and re-emerges, when the
donor is again filling its Roche lobe, as an active CV (Howell et al., 2001).
Through angular momentum loss due to gravitational radiation the evolution
continues towards even shorter periods until the system reaches a period
minimum located at an orbital period of $P_{\rm min}\approx 80$ min (see e.g.
Howell et al., 2001; Patterson, 2011; Pretorius et al., 2013; Goliasch &
Nelson, 2015; McAllister et al., 2019; Pala et al., 2020, 2022). At the period
minimum the donor is out of thermal equilibrium because its mass-loss
timescale is much shorter than its thermal timescale, causing the donor to
stop shrinking in response to mass-loss (King, 1988). This results in the
increase of the system’s orbital separation and consequently the CV bouncing
back to longer orbital periods. These systems are dubbed ”period-bouncers”
(Patterson, 1998). The actual contribution from period-bouncers to the total
CV population is highly disputed, with estimations ranging between 40$\%$ and
70$\%$, depending heavily on the formation and evolution model used as well as
the assumptions made about the systems parameters (see e.g. Schreiber et al.,
2015; Belloni et al., 2020; Pala et al., 2020). However, as a matter of fact,
only a small number of period-bouncers and candidates have been observed and
identified, mainly from the Sloan Digital Sky Survey (SDSS) (Gänsicke et al.,
2009; Littlefair et al., 2008), with a few of them harbouring a magnetic WD
possibly accreting at low rates (see e.g. Schmidt et al., 2005; Breedt et al.,
2012).

The very low-mass donors (late-M and L dwarfs) of period-bouncers have
extremely weak or no coronal X-ray emission which is hardly detectable with
present-day instrumentation (Audard et al., 2007; De Luca et al., 2020).
Therefore, the detection of X-ray emission is a key diagnostic of ongoing mass
accretion in the system and hence it is the most promising path for the
identification of period-bouncers. Another method for detecting mass accretion
relies on the presence of H$\alpha$ emission. However, contrary to X-rays,
this is not a reliable diagnostic of mass accretion in binaries with M or L
dwarfs, since only upper limits to the mass accretion rate ($\dot{M}_{\rm
acc}$) can be obtained due to the possible contribution of the donor’s
chromospheric activity or its irradiation by the WD.

Candidate period-bouncers can also be identified through an excess of emission
in the infrared spectrum over the WD contribution. Such emission could be
produced by a very late-type donor, one of the key characteristics of these
systems. Since CVs with a magnetic WD typically show a higher ratio between
X-ray flux and optical flux and, thus, higher X-ray luminosity as they do not
present a disk through which accretion luminosity is dissipated, we have
narrowed our search for period-bounce candidates to binaries composed by a
magnetic WD with evidence for a very late-type companion. Of the few binaries
where the presence of a very late-type companion has been confirmed
spectroscopically, so far four have a magnetic WD, EF Eri and V379 Vir (also
known as SDSS J121209.31+013627.7), detected as an X-ray source by White
(1981) and Burleigh et al. (2006) respectively, and the two siblings of V379
Vir, SDSS J125044.42+154957.4 and SDSS J151415.65+074446.4 (henceforth SDSS
1250 and SDSS 1514), that have no X-ray emission reported in the literature.
The nature of EF Eri is, presently, still unclear. Its orbital period locates
EF Eri near the $P_{min}$ for CVs, but the presence of a degenerate donor in
the system is still debated. Schwope & Christensen (2010) favored a degenerate
donor based on its estimated $J$ band brightness and the semi-empirical donor
sequence of Knigge (2006). However, with Gaia-DR3 the distance to EF Eri was
revised to a much larger value, increasing the estimated absolute magnitude of
the donor above the substellar limit and, thus, weakening the arguments for a
degenerate donor in the system.

Stelzer et al. (2017) detected X-ray orbital modulation on V379 Vir from a
deep XMM-Newton observation, proving that the system is accreting, and thus
confirming it as a period-bounce system of the Polar type. Through modelling
of the X-ray light curve, Stelzer et al. (2017) derived an X-ray luminosity of
$\sim 3\times 10^{29}$ erg/s that, assuming a mass of 0.8$M_{\odot}$ and a
radius of $7\times 10^{8}\rm cm$ for the WD, yields a mass accretion rate in
the system of $3.2\times 10^{-14}\,{\rm M_{\odot}/yr}$. The estimated value
for the wind driven mass accretion rate in this system is orders of magnitude
too weak to explain the observed accretion rate. Additionally, donor stars in
CVs are typically oversized as they are driven out of thermal equilibrium due
to mass transfer (Knigge et al., 2011), making Roche-lobe overflow the most
likely accretion mechanism.

Yet, it still has to be demonstrated that V379 Vir is a typical magnetic
period-bouncer rather than a peculiar system. To this end, we study here the
X-ray emission of the two systems (SDSS 1250 and SDSS 1514) composed of
similar stellar components, namely a cool ($\sim$ 10000 K) DA magnetic WD with
a very low-mass companion (late-M or L estimated spectral type). Neither of
them has been previously detected in the X-rays. We use XMM-Newton with its
EPIC/pn detector, that offers the largest collecting area of existing X-ray
instruments.

Most of the few confirmed detections of period-bouncers (see e.g. Patterson et
al. (2005), McAllister et al. (2017), Neustroev et al. (2017), Pala et al.
(2018), Schwope et al. (2021), Amantayeva et al. (2021), Kawka et al. (2021),
Neustroev & Mäntynen (2022)) have occurred through detailed observations of
specific sources using different instruments in the X-ray, optical and/or
infrared bands. As a result, the sample of period-bouncers with complete and
reliable system parameters is quite small, and may not be representative of
the period-bouncers as a class. With the launch of the extended ROentgen
Survey with an Imaging Telescope Array (eROSITA; Predehl et al., 2021) onboard
the Spektrum-Roentgen-Gamma mission (SRG; Sunyaev et al., 2021) we are now
able to conduct new studies of the X-ray activity of period-bouncers. Even
though the high sensitivity of instruments like XMM-Newton is required to
identify accreting period-bounce systems, the enormous statistical samples of
faint X-ray sources that are being observed by eROSITA is expected to boost
the number of new detections and new identifications among period-bounce
candidates. In this article we carry out the analysis of the first detections
of X-ray emission from period-bouncers based on the eROSITA All-Sky surveys,
the last of which so far was completed in winter 2021.

The pilot sample of period-bounce candidates is introduced in more detail in
Sect. 2. For SDSS 1514 we present a dedicated XMM-Newton observation in Sect.
3. In Sect. 4 we discuss the detections of period-bounce candidates from this
sample in the source catalogs of eROSITA. We give our conclusions in Sect. 5.

##  2 Sample of period-bounce candidates

Our pilot sample of short-period systems that we selected for our search of
X-ray emission is drawn from Breedt et al. (2012): V379 Vir, SDSS 1250 and
SDSS 1514. These systems are composed by a magnetic WD and a presumably very
low-mass donor. The WD has been previously identified as magnetic from SDSS
spectra, specifically from the Zeeman splitting of the Balmer absorption lines
(Schmidt et al. 2005; Vanlandingham et al. 2005; Külebi et al. 2009). This
sample of candidates also shares some other characteristics, including large
radial velocity variations, variable H$\alpha$ emission, a $\sim$ 10000 K WD,
and light curves taken with the Catalina Real-time Transient Survey that show
no large-scale variability, eclipses or high states.

V379 Vir is the only candidate with a near-IR spectrum where the donor’s
photospheric features of spectral type L8 were identified together with the
contribution of cyclotron emission from the magnetic accreting WD (Farihi et
al., 2008). In absence of spectroscopic evidence, IR photometry can be used to
estimate the donor spectral type of all three candidates: L5 or later for V379
Vir estimated by Schmidt et al. (2005) from comparing the absolute $J$ band
magnitude of the system to observed absolute $J$ band magnitudes of L and T
dwarfs; L3 or later for SDSS 1514 estimated by Breedt et al. (2012) from the
near-IR excess in the spectral energy distribution (SED); and M8 for SDSS 1250
estimated by Steele et al. (2011) from comparing the absolute $JH$ band
magnitude of the system to the observed absolute $JH$ band magnitudes of M, L
and T dwarfs. At this spectral type the secondary of SDSS 1250 would be
slightly above the substellar limit on the donor sequence of Knigge (2006).
However, the photometry used to estimate the donor spectral type was
considered uncertain by Steele et al. (2011), making a spectroscopic
confirmation mandatory for this system.

Some relevant system properties are summarized in Table 1. We identify the
candidates by their shortened SDSS name or variable star designation (col. 1).
Col. 2 refers to the distance obtained from the Gaia-DR3 parallax, col. 3 is
the orbital period derived by different authors using the H$\alpha$ emission
line, col. 4 refers to the estimated spectral type of the donor, as described
above, and col. 5 shows the estimated WD magnetic field strength.

Table 1: Relevant properties of our pilot sample of period-bounce candidates. Name | $d^{1}$ | $P_{\rm orb}$ | SpTcomp | $B_{\rm WD}$
---|---|---|---|---
| [pc] | [min] |  | [MG]
V379 Vir | $155\pm 4$ | $88.4^{2}$ | L5-L8 | $7^{2}$
SDSS 1514 | $181\pm 8$ | $88.7^{3}$ | L3 | $36^{4}$
SDSS 1250 | $132\pm 3$ | $86.3^{3}$ | M8 | $20^{4}$
1 distances from Gaia-DR3 parallax. 2 Farihi et al. (2008).
3 Breedt et al. (2012). 4 Külebi et al. (2009).

###  2.1 Determination of the white dwarf mass

To estimate the individual WD radii and masses we fitted WD atmosphere models
by Koester (2010) to the SDSS spectra111https://dr9.sdss.org/ of the period-
bounce candidates in our pilot sample considering values for $T_{\rm eff}$ and
$\log(g)$ previously reported in the literature. Because these systems are
magnetic WDs the Balmer lines cannot be used and our fits rely solely on the
continuum. Since we select the appropriate model ($T_{\rm eff}$, $\log(g)$)
beforehand, the fit has a single parameter, the ratio between the observed
flux from the SDSS spectrum and the model surface flux. This ratio represents
the dilution factor, $(d/R_{*})^{2}$ where $d$ is the distance and $R_{*}$ the
stellar radius. From the Gaia-DR3 distance we can, therefore, derive the WD
radius. From the radius and an appropriate mass-radius relation we obtained
the WD mass.

For SDSS 1514 and SDSS 1250 we used the $10000$ K DA WD spectrum with
$\log(g)=8.0$ corresponding to the parameters derived by Breedt et al. (2012)
for the two stars from the same SDSS spectra. With the Gaia-DR3 distances in
Table 1 we obtain a WD radius of $7.0\times 10^{8}$ cm for SDSS 1514 and of
$7.2\times 10^{8}$ cm for SDSS 1250, and from these values we derive the WD
masses using the mass-radius relation by Nauenberg (1972) of 0.80$M_{\odot}$
and 0.77$M_{\odot}$ respectively.

For V379 Vir we used a 11000 K DA WD spectrum with $\log(g)$=8.0 following the
WD temperature obtained by Burleigh et al. (2006) for the same SDSS spectra
and assuming a $\log(g)$ similar to the other candidates in our pilot sample.
With the Gaia-DR3 distance in Table 1 we obtain a WD radius of $8.3\times
10^{8}$ cm and from this value we derive a WD mass using the mass-radius
relation by Nauenberg (1972) of 0.64$M_{\odot}$.

We report here that Gentile Fusillo et al. (2021) derived a WD mass for all
three systems in our sample using Gaia photometry and astrometry to fit
stellar parameters. From this fitting process they obtain an effective
temperature of 10668 $\pm$ 444 K, 8057 $\pm$ 1234 K, 8211 $\pm$ 311 K and a
$\log(g)$ of 8.0 $\pm$ 0.1, 7.6 $\pm$ 0.5, 7.8 $\pm$ 0.1 for V379 Vir, SDSS
1514 and SDSS 1250 respectively. Using a pure-H atmosphere model Gentile
Fusillo et al. (2021) obtains a WD mass of 0.62 $\pm$ 0.07$M_{\odot}$, 0.41
$\pm$ 0.20$M_{\odot}$ and 0.48 $\pm$ 0.05$M_{\odot}$ for V379 Vir, SDSS 1514
and SDSS 1250 respectively. The values for V379 Vir are consistent with the
ones applied to the SDSS spectra. For the other two candidates, the values are
considerably lower than the ones we obtained from the SDSS spectra.

##  3 XMM-Newton observation of SDSS J1514

XMM-Newton observed SDSS 1514 on January 14 2020 for 41ksec (Obs-ID
0840380201; PI Stelzer) with all EPIC instruments (Strüder et al. 2001, Turner
et al. 2001) using the THIN filter and with the Optical Monitor (OM; Mason et
al., 2001) in FAST MODE using the V band filter.

###  3.1 X-ray data

SDSS 1514 is detected in all EPIC instruments, giving a net count rate of
0.0042 $\pm$ 0.0004 cts/s in the MOS1 and of 0.0121 $\pm$ 0.0011 cts/s in the
MOS2 cameras, respectively. Considering how faint the source is in the MOS
instrument, we limit the analysis to EPIC/pn, which provides the highest
sensitivity of the EPIC detectors, as can be seen in the net source count
rates presented in Table 2. The data analysis was carried out with XMM-
Newton’s Standard Science Analysis System (SAS) version 19.1.0. The
observation is slightly affected by flaring particle background, therefore we
retained only the events for which the count rate measured over the full
detector area fulfills RATE $\leq$ 0.6 cts/s, leaving an exposure time of 37
ks for the analysis. We filtered the data for pixel patterns (PATTERN $\leq$
4), quality flag (FLAG = 0) and events channels (200 $\leq$ PI $\leq$ 12000).
Source detection was performed in three energy bands: 0.2 - 1.0 keV (S), 1.0 -
2.0 keV (M) and 2.0 - 12.0 keV (H) using a customized procedure based on the
steps implemented in the SAS task EDETECT_CHAIN.

Table 2: X-ray count rate and pulsed fraction (PF) from a sine fit for SDSS 1514 EPIC/pn data in different energy bands. Energy | Band | net source rate | PFsine
---|---|---|---
[keV] | label | [cts/s] |
0.2-12.0 | B | 0.0335 $\pm$ 0.0013 | 1.0
0.2-1.0 | S | 0.0181 $\pm$ 0.0009 | 1.0
1.0-2.0 | M | 0.0096 $\pm$ 0.0006 | 1.0
2.0-12.0 | H | 0.0058 $\pm$ 0.0006 | 0.96 $\pm$ 0.02

For the spectral and temporal analysis we defined a circular photon extraction
region with radius of $30^{\prime\prime}$ centered on the EPIC/pn source
position. To ensure a homogeneous signal, the background was taken as the
average of three adjacent circular regions on the same CCD chip, each with a
radius of $30^{\prime\prime}$. The background substraction of the light curve
was carried out with SAS task EPICLCCORR, which also corrects for instrumental
effects, on an event list that was previously barycenter corrected using the
SAS tool BARYCEN.

The X-ray light curve shows a clear periodic modulation in all energy bands
with larger amplitude for softer emission, as can be observed in Fig. 1. The
modulation displays an on-off behaviour that is typical for Polars (Cropper,
1990). In the minimum, the count rate drops to approximately zero suggesting
that the area of accretion, presumably the magnetic pole cap of the WD, is
completely occulted. A Lomb-Scargle periodogram analysis of the broad band
light curve yields a period of $P_{\rm orb}=87.93\pm 0.30$ min. This value and
its 1$\sigma$ error were derived with a bootstrap approach from 5000 simulated
broad band light curves that were generated by drawing the count rates of
individual bins randomly from the range defined by the count rate errors. This
period is in good agreement with the published period derived from the
H$\alpha$ emission (88.7 min; Breedt et al., 2012).

Figure 1: EPIC/pn X-ray light curve of SDSS 1514 in four energy bands,
labelled in the upper left corner, and OM $V$-band light curve. The bin size
is 500 s in all panels. X-ray light curves represent the background-
substracted source signal (black) and for comparison the background signal
(red).

To first approximation, we fitted a sinusoid to the X-ray light curve from
which we determined the pulsed fraction (PF) taking into account the
uncertainties on y-offset and amplitude of the sine curve. The values obtained
for the individual energy bands (see Table 2) are consistent with the 100$\%$
PF typically observed in Polars (Norton & Watson, 1989). The phase-folded
light curve shown in Fig. 2 was obtained using the HEASARC tool EFOLD which
uses an input of 11 phase bins to calculate a new bin time of 479.61 s
corresponding to the duration of a phase bin in the folded light curve. The
folded light curve displays more clearly the already mentioned on-off
behaviour with a faint phase that is slightly longer than the bright phase.

Radial accretion at the magnetic pole can be described by the duration of the
faint phase ($\gamma$) in fractions of the orbital period, where the WD is
viewed at an angle $i$ to the rotation axis (the inclination), such that the
magnetic pole is located at a colatitude $\beta$ (angle between the rotation
and the magnetic axis) given by Cropper (1990) as

$\beta=\arctan\bigg{(}\frac{\cot{i}}{\cos{\pi\gamma}}\bigg{)}\,.$ (1)

We do not see evidence of eclipses in the $V$ band lightcurve (Fig. 1), such
that we can set a rough upper limit to the inclination of $i<75^{\circ}$.
Considering that we observed occultation in the X-ray light curve, a lower
limit was assumed at $i>10^{\circ}$ such that the hot-spot comes in and out of
view due to the rotation of the WD. Together with the observed duration of the
faint phase ($\gamma=0.46$) determined from the folded light curve in Fig. 2
we find for the colatitude of the magnetic pole a range of
$65^{\circ}<\beta<89^{\circ}$ using Eq. (1). This result gives us information
about the geometry of the system. For $\beta<90^{\circ}$ the accreting pole is
on the same side of the orbital plane as our line-of-sight. The high value for
$\beta$ suggests a large misalignment between the rotation axis of the WD and
its magnetic field. From Fig. 2 the shape of the folded X-ray light curve
indicates that the rise and decline to the maximum are approximately of the
same duration. This suggests that the emission region is symmetrical however
with the available data we cannot discern if the extent is vertical or
lateral.

Figure 2: X-ray light curve (EPIC/pn 0.2 - 12 keV) of SDSS 1514 folded using
the period determined from the X-ray signal. The bin size is 479.61s.

To constrain the properties of the X-ray emitting plasma, we fitted the
EPIC/pn spectrum of SDSS 1514 using the X-Ray Spectral Fitting Package (XSPEC;
Arnaud, 1996). We performed two separate fits considering, in both cases, a
simple absorber (tbabs), the first with a single-component thermal model
(apec) and the second with a thermal Bremsstrahlung model (brems). The apec
fit was carried out using the abundances from Asplund et al. (2009).

The brems and the apec models provide consistent values for the X-ray
temperature of the system (see Table 3). For the apec model we obtain a sub-
solar abundance that is however ill-constrained, and its uncertainty also
allows for zero metallicity. This might be explained by the lack of strong
emission lines at the plasma temperature of about $4$ keV as well as the low
S/N ratio of the spectrum, thus justifying the use of a pure Bremsstrahlung
model. The neutral hydrogen column density also appears to be unconstrained
and compatible with no absorption of the source. The range obtained from our
spectral fits for the gas column density falls within the upper limit for the
galactic absorption222https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl
in the direction of SDSS 1514, estimated at $2.2\times 10^{20}$ $\rm cm^{-2}$
(Bekhti et al., 2016). The brems and the apec models yield similar $\chi_{\rm
red}^{2}$, and we can not distinguish between these two models at the
statistics of our observation. We show in Fig. 3 the observed EPIC/pn spectrum
together with the apec model and residuals.

Table 3: Best-fit parameters for the XMM-Newton EPIC/pn spectrum of SDSS 1514 and values corresponding to upper and lower 90% confidence ranges. The flux is given for the XMM-Newton broad band (0.2 -12 keV) Model |  $\chi_{red}^{2}$ (dof) | $N_{\rm H}$ |  kT | Z | Flux | Emission measure
---|---|---|---|---|---|---
|  | $[\rm cm^{-2}]$ | [keV] | [$Z_{\odot}$] | $[\rm erg\,\rm cm^{-2}\rm s^{-1}]$ | $[\rm cm^{-3}]$
TBABS*APEC | 1.03 (27) | $1.9_{0.0}^{6.1}\times 10^{20}$ | $3.75_{2.55}^{5.92}$ | $0.04_{0.0}^{0.71}$ | $7.20_{6.22}^{8.41}\times 10^{-14}$ | $2.16_{1.72}^{2.46}\times 10^{51}$
TBABS*BREMSS | 1.00 (28) | $1.6_{0.0}^{6.6}\times 10^{20}$ | $3.84_{2.38}^{6.36}$ | – | $7.21_{6.18}^{8.37}\times 10^{-14}$ | $2.16_{1.87}^{2.66}\times 10^{51}$
Figure 3: Time-averaged EPIC/pn X-ray spectrum of SDSS 1514 with the apec
model and residuals.

We have also performed several apec fits with abundances fixed at different
values and found that this has a negligible influence on the X-ray flux.
Therefore, we adopt an unabsorbed flux of $f_{\rm x}=7.3\times 10^{-14}$
erg/$\mbox{cm}^{2}$/s in the $0.2-12$ keV band, taken as the average flux from
all the different apec fits we performed. For the $0.1-2.4$ keV ROSAT band the
average X-ray flux is $4.5\times 10^{-14}$ erg/$\mbox{cm}^{2}$/s. This latter
value is in accordance with the upper limit placed on the X-ray flux by the
ROSAT non-detection at $\leq 6\times 10^{-14}$ erg/$\mbox{cm}^{2}$/s for this
energy band (Breedt et al., 2012). Using the best-fit apec parameters and a
dummy response which covers a broad energy range we calculated the bolometric
X-ray flux to obtain a bolometric correction factor of 1.10.

With its Gaia-DR3 distance we determine the bolometric X-ray luminosity to
$\log({L_{\rm x}})$ [erg/s] $=29.5$, and from this value we derive a mass
accretion rate of ${\dot{M}}_{acc}=3.3\times 10^{-14}\,M_{\odot}$ / yr
obtained adopting the WD mass and radius derived in Sect. 2.1 for the SDSS
spectra of SDSS 1514. A mass accretion rate of ${\dot{M}}_{acc}=9.4\times
10^{-14}\,M_{\odot}$ / yr was obtained adopting the WD mass and radius SDSS
1514 from Gentile Fusillo et al. (2021). Similar to the conclusion reached by
Stelzer et al. (2017), the estimated lower limit for the accretion rate in
this system is orders of magnitude too high to be accounted for by a wind
driven mass accretion rate mechanism, making Roche-lobe overflow the most
likely accretion mechanism for SDSS 1514.

Because we are working with magnetic targets a contribution of cyclotron
emission to the overall flux (and therefore luminosity) of the systems might
be present. Cyclotron flux depends on several parameters like the electron
temperature $kT_{e}$ and density $N_{e}$, the magnetic field strength and the
angle between the line of sight and the magnetic field vector (see Schwope et
al. (1990)). Hence even a rough estimate of cyclotron flux needs assumptions
on several unknown parameters. In addition at low accretion rates the plasma
is likely in a non-hydrodynamic regime. Thus such quantification is out of
scope of the present work. We, instead, prefer to report accretion rates for
the sources obtained from X-ray data that should be taken henceforth as lower
limits.

The EPIC/pn X-ray count rate in the minimum of the light curve (0.64
$\leq\,\phi\,\leq$ 1.09) is 0.0083 cts/s with a standard deviation of 0.0091
cts/s, that is the system is undetected. We produced an EPIC/pn sensitivity
map for the combined time intervals corresponding to the range of phases
$\phi$ that define the minimum. Then we multiplied the value of the count rate
of this map at the source position with the rate-to-flux conversion factor,
$CF$, derived from the time-averaged X-ray spectrum. This way we obtain the
upper limit to the X-ray emission at the orbital minimum of $\log{L_{\rm
x,min}}$ [erg/s] $<28.3$. The coronal emission of the only two L dwarfs
detected in X-rays so far is $\log{L_{\rm x,min}}$ [erg/s] $<27.0$ (Audard et
al., 2007; De Luca et al., 2020), that is we do not obtain any constraints on
the L dwarf’s X-ray emission from this observation.

###  3.2 Optical data

$V$ band photometry acquired with the OM in fast mode, simultaneously with the
X-ray observation, was extracted with the SAS task OMFCHAIN, and the time
series was barycentre corrected.

A Lomb-Scargle periodogram analysis of the $V$ band did not yield a
significant periodicity, and when folded on the X-ray period, no phase-related
variability was observed. SDSS 1514 is optically faint, and at the time of the
observation, the system was at $V_{\rm OM}$ = 18.92 $\pm$ 0.32 mag, about one
magnitude above the sensitivity limit of the OM. The variability for the $V$
band was obtained by converting the OM count rates into fluxes using the
conversion factor given in the XMM-Newton SAS User
Guide333https://heasarc.gsfc.nasa.gov/docs/xmm/sas/USG/ommag.html. We used the
values for the average flux (${flux}_{\rm avg}$) and for the maximum flux
(${flux}_{\rm max}$) in the light curve to estimate the magnitude change as
$\Delta mag=-2.5\log_{10}({flux}_{\rm max}/{flux}_{\rm avg})$. This gives a
variability amplitude of $\Delta mag=0.47$ mag or around $2.5\,\%$. This
percentage is consistent with the variability estimate made by Burleigh et al.
(2006) for the $V$ band light curve of V379 Vir of about 3$\%$. However, as
can be seen in Fig. 1 the variation in the OM light curve of SDSS 1514 is not
significant, due to the low statistics of the OM data.

##  4 eROSITA detections

eROSITA has carried out four full sky surveys, called eRASS 1 to eRASS 4.
Source catalogs from eRASS data are produced at Max Planck Institut für
extraterrestrische Physik (MPE) in Garching with the eROSITA software eSASS
described by Brunner et al. (2022). These catalogs comprise all eRASS sources
in the western half of the sky in terms of Galactic coordinates, that is
galactic longitude $l\geq 180^{\circ}$, which is the sky area with german data
rights.

To obtain the highest sensitivity for detecting the presumably faint period-
bounce candidates we used the merged catalog eRASS:3, that was generated from
summing data from the first three all-sky surveys. The latest version of the
eRASS:3 catalog available to us in December 2022 was produced with the data
processing version 020444The source catalog used in our work is
all_s3_SourceCat1B_221007_poscorr_mpe_clean.fits (for eRASS:3).. Source
detection was performed in this catalog for a single eROSITA energy band,
$0.2-2.3$ keV.

In order to search for the period-bouncers from our pilot sample in eRASS data
we performed an initial match between our targets and the Gaia-DR3 catalog.
Using the corresponding Gaia-DR3 proper motions, we corrected the coordinates
of the objects to the mean observing date of eRASS:3. Then we matched them
with the eRASS:3 catalog allowing for a maximum separation of
$30^{\prime\prime}$. After having established the closest matches for our
pilot sample, we carried out a visual inspection in a $30^{\prime\prime}$
radius region around the X-ray source to assure that there where no other
potential optical counterparts. In Table 4 we report the X-ray parameters
obtained from the merged eRASS:3 catalog for the matches of the two period-
bouncers with eROSITA detections, V379 Vir and SDSS 1250. SDSS 1514 is located
in the sky area with exclusive Russian eROSITA data rights and, therefore, not
included in the eRASS:3 catalog. As can be seen from Table 4, the sources are
faint in the eRASS surveys, with only tens of counts in the merged data from
the first three eROSITA surveys, yet significantly above the detection
likelihood limit of $5$ set for the eRASS:3 merged catalog.

Table 4: X-ray parameters from the eROSITA merged catalog eRASS:3 for V379 Vir and SDSS 1250 given for the eROSITA single band (0.2-2.3 keV). Name | Detection | Separation | Counts | Count rate | Flux (apec) | X-ray luminosity∗ | Mass accretion rate∗
---|---|---|---|---|---|---|---
| likelihood | [′′] | [cts] | [cts/s] | [erg $cm^{-2}$ $s^{-1}$] | [erg$s^{-1}$] | [$M_{\odot}\rm yr^{-1}$ ]
V379 Vir | 44.5 | 2.2 | 30.1 $\pm$ 6.3 | 0.08 $\pm$ 0.02 | (0.75 $\pm$ 0.13) $\times 10^{-13}$ | log($\rm L_{bol}$)= 29.5 | $5.3\times 10^{-14}$
SDSS 1250 | 19.9 | 2.9 | 16.5 $\pm$ 4.9 | 0.05 $\pm$ 0.01 | (0.46 $\pm$ 0.12) $\times 10^{-13}$ | log($\rm L_{bol}$)= 29.2 | $1.7\times 10^{-14}$
∗ Values do not consider potential contributions from cyclotron emission to
the X-ray luminosity .

In the eROSITA eRASS:3 catalog the single band flux has been calculated for a
power law model with an index $\Gamma=2.0$ and a galactic absorption of
$N_{\rm H}=3\times 10^{20}\rm cm^{-2}$ (Brunner et al., 2022). Since this
spectral model is not appropriate for CVs, we computed fluxes for an apec
model with help of a simulation that provided us with the eROSITA conversion
factor, $CF_{\rm eROSITA,APEC}$, from count rate to thermal flux.

To this end, we generated a ”fake” spectrum using the FAKEIT command available
in XSPEC together with the eROSITA response files. We chose an apec model with
$kT=2.62\,$keV, $N_{\rm H}=2.3\times 10^{20}\,\rm cm^{-2}$ and an abundance of
$0.11\,Z_{\odot}$, the values found from the XMM-Newton spectrum for one of
our targets (see Stelzer et al., 2017). We then produced another simulated
eROSITA spectrum for the power law model used in the eRASS catalogs. From both
synthetic spectra we then retrieved the flux and the count rate in the single
band, using these values to calculate a $CF_{\rm fake,APEC}$ and a $CF_{\rm
fake,powerlaw}$. The results are shown in Table 5, from where it can be seen
that these two $CF$ differ by only $3.36$ %. This result allows us to derive
$CF_{\rm eROSITA,APEC}$ from $CF_{\rm eROSITA,powerlaw}$ used in the eRASS
catalogs. The final values for the apec fluxes for the eROSITA single band
(0.2-2.3 keV), given in Table 4, were calculated using the catalog count rate
and the apec eROSITA conversion factor as, $Flux_{\rm APEC}$ = Count
Rate/$CF_{\rm eROSITA,APEC}$. Analogous to the analysis of the XMM-Newton
data, the apec parameters and a dummy response which covers a broad energy
range were used to calculate the bolometric X-ray flux. We obtain a bolometric
correction factor of 1.60. With the Gaia-DR3 distances we obtained the
bolometric X-ray luminosity and mass accretion rate of each of our targets
(see Table 4), where we used the WD mass and radius obtained for each object
in Sect. 2.1. Using, instead, the values for the WD masses obtained by Gentile
Fusillo et al. (2021) from Gaia photometry, combined with a WD radius from the
Nauenberg (1972) mass-radius relation, we obtain a mass accretion rate of
${\dot{M}}_{acc}=5.5\times 10^{-14}\,M_{\odot}$ / yr for V379 Vir and of
${\dot{M}}_{acc}=3.7\times 10^{-14}\,M_{\odot}$ / yr for SDSS 1250. In the
latter case, the mass accretion rate is more than twice as high than the value
we calculated based on the WD mass from the SDSS spectra.

Table 5: Rate-to-flux conversion factor, $CF$, in the eROSITA single band (0.2-2.3 keV) for the simulated spectra obtained using an apec model with $kT=2.62\,$keV and a power law model with $\Gamma=2.0$.  $CF_{fake,APEC}$ | $CF_{fake,powerlaw}$
---|---
[cts $\rm cm^{2}$ $\rm erg^{-1}$] | [cts $\rm cm^{2}$ $\rm erg^{-1}$]
$4.23\times 10^{11}$ | $4.09\times 10^{11}$

##  5 Discussion and conclusions

We have detected X-ray emission from all three magnetic period-bouncer
candidates, all emitting at a similar level of $L_{\rm x}\approx 10^{29}$
[erg/s], and hence displaying similar mass accretion rates under the
assumption that the WD mass is similar in all systems.

In Stelzer et al. (2017) an apec flux value was reported for V379 Vir in the
XMM-Newton broad band (0.2 - 12.0 keV) of 1.3 $\times 10^{-13}$ erg/$\rm
cm^{2}$/s. Since the X-ray spectra of V379 Vir and SDSS 1514 are very similar
we can use the XMM-Newton spectrum of SDSS 1514 analysed in this paper to
estimate the flux of V379 Vir in the eROSITA band at its XMM-Newton epoch. In
XSPEC we calculated for SDSS 1514 a flux ratio of $1.76$ between the XMM-
Newton broad band and the 0.2-2.3 keV band used in the eRASS:3 catalog. From
this we infer the flux of V379 Vir in the eRASS:3 energy band at the XMM-
Newton epoch to be $7.1\times 10^{-14}$ erg/$\rm cm^{2}$/s. A comparison with
the observed eROSITA flux presented in Table 4 for V379 Vir shows that there
is only a $5.7\,\%$ difference between the two values and, considering the
error bars, the two brightness measurements are consistent with each other,
that is the system’s X-ray luminosity (averaged over the orbital cycle) did
not change on timescales of few years.

For two of the systems,V379 Vir and SDSS 1514, we have found X-ray orbital
modulation from our long sensitive XMM-Newton observations, presented by
Stelzer et al. (2017) and in this paper respectively. The fluxes discussed in
the previous paragraph, therefore, represent time-averages over several
orbital cycles. In the case of eROSITA, the flux is the average over three
surveys separated from each other by six months. Within each survey V379 Vir
and SDSS 1250 were observed between 6 and 11 times for around 20 s to 40 s
with a gap of $\approx 4$ h between these individual exposures, defined by the
scanning law of eROSITA (see e.g. Predehl et al., 2021). Given the faintness
of the sources we refrain here from a search of variability in eRASS data.

The X-ray detection of our targets confidently proves that accretion is taking
place. We have thus demonstrated that X-ray emission is an efficient way of
identifying CVs with very low mass donors. While the periodic X-ray
variability testifying the accretion geometry requires the high sensitivity
that can be achieved only with dedicated X-ray pointings we have shown that
the flux limit of the merged data from the eROSITA surveys is sufficient to
provide a weak detection for systems at distances out to $\sim$ 200 pc. A
rough projection of the prospects for a systematic search for period-bouncers
in the eRASS:3 catalog can be done using the X-ray flux limit of the survey
and an estimate of the space-density of CVs in an exponential disk.

To this end, we use the fainter eRASS detection among our pilot sample, SDSS
1250, as a reference point to estimate the distance limit for period-bouncers
of eRASS:3. For a detection likelihood det_ml $>20$, corresponding to the
value for SDSS 1250 (see Table 4), $95$ % of the sources in the eRASS:3
catalog have a flux higher than $2\times 10^{-14}\,{\rm erg/cm^{2}/s}$.
Combined with the ”typical” X-ray luminosity of our pilot targets
($10^{29}\,{\rm erg/s}$) this yields an approximate distance limit of $\sim$
200 pc for eROSITA discoveries of accreting WDs with very-low mass donors in
the merged eRASS 1 to eRASS 3 data base. We caution that this estimate may
apply only to magnetic systems which tend to be X-ray brighter than non-
magnetic CVs (Cropper, 1990). On the other hand, our detection likelihood
threshold is very conservative. Values of det_ml = 11 have been shown to
define samples with only modest contamination by spurious sources (Wolf et
al., 2021).

A possible starting point in the search of new period-bounce candidates is the
Gaia catalogue of WDs by Gentile Fusillo et al. (2021) which presents a list
of $\sim$ 360000 genuine WDs constructed through a comprehensive selection
based on Gaia magnitudes and colours with consideration for diverse quality
flags. In Fig. 4 we display as the green sample the ”high fidelity” WDs with
distances within the eRASS:3 limit estimated above.

Figure 4: Gaia color-magnitude diagram showing the position of the ”high
fidelity” WDs from Gentile Fusillo et al. (2021) with a distance limit of
$200$ pc (green). The three period-bouncers from our pilot sample that are
comprised in the Gentile Fusillo et al. (2021) catalog are shown in red. The
Gaia-DR3 sources with Bailer-Jones et al. (2021) distances are shown in grey
as a reference.

The ”high fidelity” WDs are objects with probability of being a WD $P_{\rm
WD}\geq 0.75$ as derived by Gentile Fusillo et al. (2019) from the position of
the objects in the Gaia color-magnitude diagram compared to the WD density map
obtained from confirmed SDSS WDs. The sample shown in Fig. 4 in green
comprises $91089$ objects. Overlaid are the three period-bouncers from our
pilot X-ray survey, all of which (marked in red) are present in the Gentile
Fusillo et al. (2021) catalog. All three are located in the same area as the
Gaia WD subsample, indicating that the Gaia WD catalog by Gentile Fusillo et
al. (2021) might include further period-bouncers.

In Fig. 4 we present in grey a large sample of Gaia-DR3 sources to serve as
reference. This sample was selected considering only the sources with a
distance obtained by Bailer-Jones et al. (2021) and a parallax error less than
1$\%$ of the parallax value.

The expected estimated number of period-bouncers within the eRASS:3 distance
limit is determined by the space density of CVs, which is highly uncertain,
ranging from $2\times 10^{-4}\,\rm pc^{-3}$ (de Kool, 1992) to $4.8\times
10^{-6}\,\rm pc^{-3}$ (Pala et al., 2020). Considering this, for the space
density of period-bouncers in the galactic mid-plane we assume $\rho_{\rm
0}\sim 10^{-5}\,{\rm pc^{-3}}$, twice the value estimated for ”normal” CVs
from their observed X-ray luminosity function (Pretorius & Knigge, 2012). An
exponential vertical density profile with scale height of $260$ pc for short
period systems (Pretorius & Knigge, 2012) then yields $\approx 133$ period-
bouncers within $200$ pc. Since the eRASS:3 catalog comprises only half of the
sky, the actual expected number of PBs hidden in this catalog is $\approx 67$.
The number will be lower if the majority of period-bouncers has lower X-ray
luminosity than the benchmark systems we studied in this work. However, the
theoretically predicted mass accretion rates (on the order of $\dot{M}_{\rm
acc}$ $\sim 10^{-11}$ ${\rm M_{\odot}/yr}$; Goliasch & Nelson, 2015) are much
higher than expected from the observed $L_{\rm x}$ values and this suggests
that the number of X-ray detections might be substantially higher than the
estimate above.

Considering the all-sky nature of eROSITA’s surveys this instrument is ideal
for population studies. As we have shown here, with the detections of V379 Vir
and SDSS 1250, the eRASS data is suitable for the search for new period-bounce
candidates. Once accretion has been proven in a system through an X-ray
detection, an infrared (IR) spectrum should be obtained to confirm the late
spectral type of the donor and, thus, to confidently verify the system as a
period-bouncer. The discovery of more period-bouncers and the observational
characterization of this object class is expected to provide new impetus for
theoretical studies of binary evolution.

###### Acknowledgements.

We thank an anonymous referee for reviewing the original manuscript and giving
helpful comments and useful advice. Daniela Muñoz-Giraldo acknowledges
financial support from Deutsche Forschungsgemeinschaft (DFG) under grant
number STE 1068/6-1. Domitilla de Martino acknowledges financial support from
ASI and INAF. This work is based on observations obtained with XMM-Newton, an
ESA science mission with instruments and contributions directly funded by ESA
Member States and NASA. This work is based on data from eROSITA, the primary
instrument aboard SRG, a joint Russian-German science mission supported by the
Russian Space Agency (Roskosmos), in the interests of the Russian Academy of
Sciences represented by its Space Research Institute (IKI), and the Deutsches
Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by
Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL
with support from the Max Planck Institute for Extraterrestrial Physics (MPE).
The development and construction of the eROSITA X-ray instrument was led by
MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg and ECAP
(FAU Erlangen-Nürnberg), the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität München also participated in
the science preparation for ero. The eROSITA data shown here were processed
using the eSASS/NRTA software system developed by the German eROSITA
consortium. 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.

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