# Discovery and characterization of five new eclipsing AM CVn systems

J. van Roestel,1 T. Kupfer,2 M. J. Green,3 S. Wong,4 L. Bildsten,4,5 K.
Burdge,1 T. Prince,1 T. R. Marsh,6 P. Szkody,7 C. Fremling,1 M. J. Graham,1 V.
S. Dhillon,8 S. P. Littlefair,8 E. C. Bellm9, M. Coughlin,10 D. A. Duev1, D.
A. Goldstein11 R. R. Laher,12 B. Rusholme,12 R. Riddle,13 R. Dekany,1 S. R.
Kulkarni1
1Division of Physics, Mathematics, and Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA
2Department of Physics & Astronomy, Texas Tech University, Box 41051, Lubbock
TX 79409-1051, USA
3Department of Astrophysics, Faculty of Exact Sciences, Tel-Aviv University,
Ramat Aviv, Tel-Aviv 6139001, Israel
4Department of Physics, University of California, Santa Barbara, CA 93106, USA
5Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106, USA
6Department of Physics, University of Warwick, Coventry CV4 7AL, UK
7Department of Astronomy, University of Washington, Seattle, WA 98195, USA
8Department of Physics and Astronomy, University of Sheffield, Sheffield S3
7RH, UK
9DIRAC Institute, Department of Astronomy, University of Washington, 3910 15th
Avenue NE, Seattle, WA 98195, USA
10School of Physics and Astronomy, University of Minnesota, Minneapolis,
Minnesota 55455, USA
11Weights & Biases, Inc., 255 Kansas St, San Francisco, CA 94103, USA
12IPAC, California Institute of Technology, 1200 E. California Blvd, Pasadena,
CA 91125, USA
13Caltech Optical Observatories, California Institute of Technology, Pasadena,
CA 91125, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

AM CVn systems are ultra-compact, hydrogen-depleted and helium-rich, accreting
binaries with degenerate or semi-degenerate donors. We report the discovery of
five new eclipsing AM CVn systems with orbital periods of 61.5, 55.5, 53.3,
37.4, and 35.4 minutes. These systems were discovered by searching for deep
eclipses in the Zwicky Transient Facility (ZTF) lightcurves of white dwarfs
selected using Gaia parallaxes. We obtained phase-resolved spectroscopy to
confirm that all systems are AM CVn binaries, and we obtained high-speed
photometry to confirm the eclipse and characterize the systems. The spectra of
two long-period systems (61.5 and 53.3 minutes) show many emission and
absorption lines, indicating the presence of N, O, Na, Mg, Si, and Ca, and
also the K and Zn, elements which have never been detected in AM CVn systems
before. By modelling the high-speed photometry, we measured the mass and
radius of the donor star, potentially constraining the evolutionary channel
that formed these AM CVn systems. We determined that the average mass of the
accreting white dwarf is $\approx 0.8$ $\mathrm{M_{\sun}}$, and that the white
dwarfs in long-period systems are hotter than predicted by recently updated
theoretical models. The donors have a high entropy and are a factor of
$\approx$ 2 more massive compared to zero-entropy donors at the same orbital
period. The large donor radius is most consistent with He-star progenitors,
although the observed spectral features seem to contradict this. The discovery
of 5 new eclipsing AM CVn systems is consistent with the known observed AM CVn
space density and estimated ZTF recovery efficiency. Based on this estimate,
we expect to find another 1–4 eclipsing AM CVn systems as ZTF continues to
obtain data. This will further increase our understanding of the population,
but will require high precision data to better characterize these 5 systems
and any new discoveries.

###### keywords:

keyword1 – keyword2 – keyword3

††pubyear: 2021††pagerange: Discovery and characterization of five new
eclipsing AM CVn systems–C

##  1 Introduction

AM CVn systems are ultra-compact accreting binaries with degenerate or semi-
degenerate, hydrogen-depleted and helium-rich donors. They are part of the
family of cataclysmic variables; a primary white dwarf that is accreting mass
from a donor via Roche lobe overflow (Warner, 1995). For AM CVn binaries, the
donors are degenerate and the binary is very compact with orbital periods
ranging from 65 to as short as 5 minutes (see Solheim (2010) for a review).
Thousands of AM CVn systems are expected to be present in our Galaxy, but
their intrinsic faintness limits the known population to $\approx$ 60 AM CVn
systems (see Ramsay et al. 2018 for a recent compilation).

Because of their compactness and short orbital periods, AM CVn stars are an
excellent tool to study accretion physics under extreme conditions (e.g. Kotko
et al., 2012; Coleman et al., 2018; Cannizzo & Ramsay, 2019; Oyang et al.,
2021). Their short orbital periods also means that their angular momentum
losses are dominated by gravitational wave radiation. Several hundred AM CVn
systems will be detectable by the Laser Interferometer Space Antenna (LISA,
Amaro-Seoane et al., 2017) satellite and are one of the most abundant types of
persistent LISA sources (Kremer et al., 2017; Breivik et al., 2018; Kupfer et
al., 2018). LISA will continuously observe these systems and will show the
orbital period evolution of short period AM CVn systems. AM CVn systems are
also potential progenitors of rare transient events. Bildsten et al. (2006)
discuss that, as a layer of He builds up on the accreting white dwarf,
recurring He-shell flashes can occur which would look like He-Novae. The mass
of the He-shell becomes larger and the time between flashes longer as the
systems evolve to longer orbital periods. This can result in a very energetic
‘final-flash’ which can be dynamical and eject material from the white dwarf,
dubbed a ‘.Ia’ transient (Shen et al., 2010). The most important open question
regarding AM CVn systems is their formation channel, and how AM CVn systems
fit into the overall picture of compact binary evolution (see Section 2.1 and
also Nelemans et al. 2001; Toloza et al. 2019).

In this paper, we present the search of Zwicky Transient Facility (ZTF)
lightcurves to find new eclipsing white dwarf binaries and the discovery of
five new AM CVn systems and their characterization. In Section 2 we briefly
summarize the current understanding of the formation channels and
observational properties of AM CVn systems, and discuss the currently known
eclipsing systems. Section 3 presents the method and search of the ZTF
lightcurves used to find the five new eclipsing AM CVn systems. Section 4
presents all the follow-up observations, high-speed photometry and phase-
resolved spectroscopy, and archival data used to characterize the systems.
Section 5 presents the methods we used to characterize the systems, and
Section 6 presents the results of this analysis. In Section 7 we discuss the
implications of the measurements, specifically what the results imply for the
formation channels of AM CVn systems. We summarize this paper in Section 8 and
end with a short discussion on future work in Section 9.

##  2 Background

###  2.1 Formation channels

There are three proposed AM CVn formation channels, but their relative
importance is uncertain. In the white dwarf channel, a short period double
white dwarf binary is formed after going through two common envelope phases
(Ivanova et al., 2013). The resulting binary is a typical CO white dwarf with
a low mass helium white dwarf companion (Paczyński, 1967; Tutukov & Fedorova,
1989; Deloye et al., 2007). They evolve closer together as a result of
gravitational wave radiation, and start stable mass transfer at orbital
periods of $\approx$2-3 m and evolve to longer orbital periods. Simulations by
Nelemans et al. (2001) predicted the white dwarf channel to be the dominant
formation channel. However, Shen (2015) suggest that all double white dwarfs
merge (because of friction with ejected material in a nova eruption), and
double white dwarfs do not form AM CVn systems.

The second formation channel, the He-star channel, is similar to the white
dwarf channel, but instead of a low mass white dwarf, the donor-progenitor is
a non-degenerate He-burning star. These will also evolve to shorter periods,
but start mass-transfer at slightly longer periods of 10 minutes (Savonije et
al., 1986; Iben & Tutukov, 1987; Tutukov & Fedorova, 1989; Yungelson, 2008;
Brooks et al., 2016).

The third formation channel is the evolved-CV channel. In this scenario, the
donor must evolve off the main sequence at the same time as the start of mass
transfer. The system goes through a phase as a hydrogen-dominated CV (H-CV),
and will eventually change into a helium-dominated CV once all the hydrogen
has been stripped from the donor (e.g. Podsiadlowski et al., 2003). Models
indicate that this channel is rare and does not contribute significantly to
the AM CVn population. This is due to the finely tuned starting parameters and
long timescales required to remove all visible hydrogen from these systems
(Nelson et al., 2018). However, there are known helium CVs which are potential
progenitors of the channel (Thorstensen et al., 2002; Breedt et al., 2012;
Carter et al., 2013b; Green et al., 2020).

###  2.2 Observational properties

Although AM CVn systems are all accreting DB white dwarfs with a degenerate or
semi-degenerate donor, their observational characteristics vary significantly.
Their appearance (both photometric and spectroscopic) is very different
because of differences in the accretion rate, which is strongly correlated
with the orbital period. The accretion rate determines the behaviour of the
accretion disc (Kato et al., 2001; Nelemans et al., 2004; Ramsay et al., 2012;
Kupfer et al., 2013; Coleman et al., 2018) and also sets the accreting white
dwarf temperature (e.g. Bildsten et al., 2006).

Very short period ($P\lesssim\\!10$ m) AM CVn systems have high accretion
rates and, depending on the masses, can be ‘direct impact’ accretors. In these
systems, there is no accretion disk and the accretion stream directly impacts
the white dwarf. These systems emit X-rays, for example, HM Cnc and V407 Vul
(Marsh et al., 2004). Systems with periods $\gtrsim\\!10$ m form accretion
discs. System with periods of $\approx\\!10$–$22$ m of the high accretion
rate, the systems are in a constant ‘high state’ (Kotko et al., 2012).
Intermediate period systems ($\approx\\!22$–$45$ m) with lower accretion rates
show dwarf nova outbursts and feature large amounts of flickering in their
lightcurves (see Duffy et al., 2021). As the orbital period increases, the
outburst recurrence time increases exponentially (Levitan et al., 2015), and
the luminosity of the disk decreases as the orbital period decreases. In long-
period systems ($\gtrsim\\!45$ m), the accretion rate is low, outbursts are
very rare (recurrence times of $>\\!100$ years), and the accreting white dwarf
dominates the luminosity (Bildsten et al., 2006).

Ramsay et al. (2018) give an overview of the currently known sample of AM CVn
systems. This sample has been built up using various methods. Many AM CVn
systems (including AM CVn itself) were identified by their blue colour and
identification spectra. This method was used most recently by Carter et al.
(2013a) who used SDSS colour information. The second main method of finding AM
CVn systems is by their outbursts. Breedt et al. (2012); Breedt et al. (2014)
obtained spectra of Catalina Real-time Transient Survey (CRTS) dwarf novae and
found three new AM CVn systems. Levitan et al. (2015) used the Palomar
Transient Factory (PTF) to identify cataclysmic variables and identified AM
CVn systems with follow-up spectroscopy. van Roestel et al. (2021a) combined
the colour selection and outburst selection methods and identified 9 new AM
CVn systems. Isogai et al. (2019) also focused on outbursting CVs but instead
used high-cadence photometry to find the period and identify a system as an AM
CVn system. Searching for short period variability was also used to identify a
new AM CVn system using Kepler data (Fontaine et al., 2011; Kupfer et al.,
2015; Green et al., 2018b). Finally, in parallel with this work, Burdge et al.
(2020) searched for compact binaries by searching for short-period variability
in Zwicky Transient Facility data.

###  2.3 Eclipsing AM CVn systems

Among the sample of known AM CVn systems, only a few have been confirmed as
eclipsing. Eclipsing AM CVn systems where the primary white dwarf is eclipsed
by the donor are extremely valuable since they allow the binary parameters to
be measured. Hardy et al. (2017) provides examples of modelling eclipsing
accreting cataclysmic variable lightcurves with various accretion rates.

Three eclipsing AM CVns have been studied in detail so far. The first is
PTF1J1919+4815, in which just the edge of the disk and bright spot are
eclipsed (Levitan et al., 2014). The second is YZ LMi where the white dwarf is
partially eclipsed (Anderson et al., 2005; Copperwheat et al., 2011).
Gaia14aae (ASASSN-14cn), is the only published AM CVn in which the white dwarf
is confirmed to be fully eclipsed by the donor (Campbell et al., 2015; Green
et al., 2018a, 2019a). Besides these well-studied cases, there are also a
number of promising candidates. Burdge et al. (2020) presented the discovery
of a candidate 17.20 orbital period eclipsing AM CVn system also using ZTF
data. In addition, follow-up of ES Cet by Bakowska et al. (2020) indicates
that this system is also eclipsing, although it seems that in this case only
the disk is eclipsed and not the accreting white dwarf.

Because for YZ LMi and Gaia14aae the white dwarf is eclipsed, the donor mass
and radius have been measured to a few per cent accuracy by modelling the
lightcurve. These precise measurements and their comparison with models show
an interesting result (Copperwheat et al., 2011; Green et al., 2018a). The
donor for YZ LMi is relatively large and is consistent with either the helium
star channel or the white dwarf channel. The donor in Gaia14aae is large,
which is consistent only with mass-radius models describing the evolved CV
channel. This is a surprising result since the evolved CV channel is expected
to be the most uncommon channel of the three. These studies demonstrate that
eclipsing AM CVn systems allow us to precisely characterize them and better
understand their evolutionary history. This motivated us to perform a
dedicated search for eclipsing AM CVn systems.

##  3 Target selection

Table 1: Overview of the five new eclipsing AM CVn stars discovered by their eclipses using ZTF lightcurves. The parallax is taken from Gaia eDR3 Brown et al. (2020a). The distance is calculated using a prior based on the white dwarf population (Kupfer et al., 2018). The dust-extinction is taken from Green et al. (2019b).  Name | RA | Dec | $P_{\mathrm{orb}}$ | ZTF-$g$ | ZTF-$r$ | $G$ | $BP-RP$ | parallax | distance | E(g-r)
---|---|---|---|---|---|---|---|---|---|---
|  |  | minutes | AB-mag | AB-mag | Vega-mag | Vega-mag | mas | pc |
ZTFJ1637+49 | 16h37m43.6s | 49∘17${}^{{}^{\prime}}$40.9${}^{{}^{\prime\prime}}$ | 61.5 | 19.34 | 19.49 | 19.40 | 0.27 | 4.88 $\pm$ 0.27 | $207^{+8}_{-8}$ | $<0.01$
ZTFJ0003+14 | 00h03m22.4s | 14∘04${}^{{}^{\prime}}$59.0${}^{{}^{\prime\prime}}$ | 55.5 | 20.19 | 20.14 | 20.20 | 0.35 | 4.03 $\pm$ 0.54 | $263^{+29}_{-40}$ | $0.10^{+0.02}_{-0.02}$
ZTFJ0220+21 | 02h20m08.6s | 21∘41${}^{{}^{\prime}}$55.8${}^{{}^{\prime\prime}}$ | 53.3 | 19.72 | 19.82 | 19.76 | 0.09 | 2.94 $\pm$ 0.50 | $350^{+39}_{-58}$ | $0.14^{+0.02}_{-0.02}$
ZTFJ2252−05 | 22h52m37.1s | −05∘19${}^{{}^{\prime}}$17.4${}^{{}^{\prime\prime}}$ | 37.4 | 19.05 | 19.16 | 19.09 | 0.04 | 1.95 $\pm$ 0.31 | $536^{+82}_{-93}$ | $0.01^{+0.01}_{-0.02}$
ZTFJ0407−00 | 04h07m49.3s | −00∘07${}^{{}^{\prime}}$16.7${}^{{}^{\prime\prime}}$ | 35.4 | 19.23 | 19.41 | 19.42 | 0.07 | 1.33 $\pm$ 0.37 | $810^{+190}_{-230}$ | $0.14^{+0.01}_{-0.02}$
Figure 1: The ZTF lightcurves of the five new AM CVn stars folded on their
orbital periods. ZTF $g$, $r$, and $i$-bands are displayed in green, red, and
gold. Dots show ZTF PSF-photometry, crosses show ZTF alert photometry. The
narrow eclipse in each of the lightcurves was used to discovery these objects.
The two short period systems also show irregular variability in their
lightcurves (see Section 6.4 and Figure 9).

###  3.1 Zwicky Transient Facility lightcurves

As part of the Zwicky Transient Facility (ZTF), the Palomar 48-inch (P48)
telescope images the sky every night (Graham et al., 2019; Bellm et al.,
2019b; Dekany et al., 2020). Several sub-surveys are carried out by ZTF (Bellm
et al., 2019a), including an all-sky survey (publicly available), but also
higher cadence surveys of smaller areas (e.g. van Roestel et al., 2019; Kupfer
et al., 2021). Most surveys use $g$ and $r$ bands, but a small fraction of the
observations are done in $i$ band. The exposure times are predominantly 30
seconds for $g$ and $r$ and 60 seconds for $i$ exposures. The median limiting
magnitude, averaged over the lunar cycle, is $r\approx 20.5$ in all three
bands. ZTF images are automatically processed and two main data products are
generated. The first are the ‘alerts’ which are based on difference imaging
and are mainly designed to identify transients. The second main data product
is PSF-photometry of persistent sources in the science images (for a full
description see Masci et al., 2019).

###  3.2 Search method

To find deep eclipsing white dwarfs, we combined the PSF photometry data from
the science images with the alert data generated from the difference images.
We do this because the PSF-photometry is only reported for sources that are
detected with a significance of 5-standard deviations in the science image. If
the target shows a deep eclipse, no PSF-photometry is reported. Alerts are
only generated for 5-sigma detection on the difference images, both positive
or negative. A non-detection of a source thus generates a negative alert. The
negative alerts (like the positive alerts) are vetted by a ‘real-bogus’
system, which is based on deep-learning (Duev et al., 2019).

To identify periodic eclipses we used the Boxed-Least-Square (BLS)
algorithm111https://github.com/johnh2o2/cuvarbase (Kovács et al., 2002). We
normalised the $g$, $r$, and $i$, data and combined them into a single
lightcurve, rejecting any points which were flagged as potentially bad
($5-10$%, mostly due to clouds). In addition, we removed any points which were
more than 5 standard deviations above the median level of lightcurve in order
to reject outbursts. We searched a linear frequency grid and used a frequency
step inversely proportional to the baseline and the minimum eclipse duration,
and use an additional oversampling factor of 10. This typically results in a
grid of a few million frequencies to be searched. To identify the best period,
we simply used the period with the highest power. In the case of multiple best
periods, we used the longest period.

As a post-processing step, we fit the phase-folded lightcurve with a simple
trapezoid plus two sinusoids to model an eclipse, reflection effect, and
ellipsoidal modulation. The eclipse width is shared between colour bands, but
the depth of the eclipse and amplitude of any sinusoidal component is
independently determined for each band. Periods, the period power, the
goodness-of-fit of the model, and other statistics were saved for later
selection of lightcurves for visual inspection.

###  3.3 Application to the ZTF lightcurves

We analysed the ZTF-lightcurves DR3 and private data (available on 2019
December 11th) of all the sources in the Gaia DR2 white dwarf catalogue from
Gentile Fusillo et al. (2019), in our search for eclipsing white dwarfs. We
processed all lightcurves with more than 80 epochs in $g$ and $r$ combined
(241 775 objects), and searched a period range from 0.02 d (28.8 m) to 5 d on
a linear scale in frequency. The lower limit was chosen to reduce the
computational time and because Burdge et al. (2020) has already searched for
shorter period systems. We visually inspected the folded lightcurves for all
objects in the white dwarf catalogue which had at least one alert associated
with it (6 412 objects). For the rest, we only inspected lightcurves for which
the model fit showed an improvement of $\Delta\chi^{2}>50$ compared to fitting
a straight line (27 116 objects). In this work, we present the AM CVn systems
found using this method, a full catalog of eclipsing white dwarf and other
interesting discoveries (e.g. van Roestel et al., 2021b) will be presented in
a future publications.

Figure 1 shows the folded ZTF lightcurves of the five eclipsing AM CVn systems
we discovered in the sample of eclipsing white dwarfs. ZTFJ1637+49 (also
discovered by Keller et al. 2021) is located in a region of the sky that is
observed at a higher cadence by ZTF as part of a ZTF-partnership survey. For
this reason, many more epochs are available compared to the rest of the sky.
This allowed us to discover this object first, despite the eclipse being the
most shallow and narrow of the five objects, and the fact that no alerts were
generated for this object. ZTFJ0220+21 and ZTFJ0407−00 both show negative
‘alerts’ in both $g$ and $r$ band. A visual inspection of their lightcurves
confirmed their orbital period. ZTFJ0407−00 also showed one outburst, which
combined with its period, immediately confirmed its nature as an AM CVn
system.

ZTFJ2252−05 was identified later in a general search of persistent sources
which have a Gaia parallax (Bailer-Jones et al., 2018) and are located close
to the white dwarf track, and have negative ZTF-alerts. This sample contained
a total of 11 766 sources ZTF sources. ZTFJ2252−05 triggered three negative
alerts, and a BLS-period search of the lightcurve identified a potentially
short period, which was confirmed using CRTS and PTF data. ZTFJ2252−05 was not
found in the initial search because its Gaia parallax was very uncertain, and
not included in the catalogue by Gentile Fusillo et al. (2019).

ZTFJ0003+14 was identified as a candidate in the initial search but was
initially not followed up because of the sparse ZTF-lightcurve and red $BP-RP$
colour. With more epochs available, the eclipse became more prominent and
follow-up data confirmed its nature as an AM CVn system.

##  4 Observations

For each system, we obtained high-speed photometry and phase-resolved
spectroscopy. Table 6 presents an overview of all follow-up observations. In
addition, we used multiple surveys to study the long timescale evolution and
spectral energy distribution.

###  4.1 LRIS spectroscopy

Phase-resolved spectroscopy of all systems was obtained using the Keck I
Telescope (HI, USA) and the Low Resolution Imaging Spectrometer (LRIS; Oke et
al. 1995; McCarthy et al. 1998). We used a $1^{\prime\prime}$ slit, with the
R600 grism for the blue arm, and R600 grating for the red arm. With this
setup, spectral resolution is approximately $R\approx 1100$ in the blue arm
and $R\approx 1400$ in the red arm. For one spectrum we used the R300 grating,
which has a resolution of $R\approx 800$ .

On 2019 May 7th, we obtained 8 spectra of ZTFJ1637+49 with a 600 s exposure
times each. During this night, the weather was good and seeing $\approx 0.8$
arcsec. Due to technical problems, no standard star was observed this night so
a standard star from another night was used. Additional spectra with 300 s
exposure times were obtained on 2019 June 5th. A sequence of 4 hours of
spectra were taken, 47 in the blue arm and 41 in the red arm (the difference
is due to a longer readout time). Weather conditions were bad at the beginning
of the night, but during the course of the observations the weather cleared.

Spectra of ZTFJ0220+21 and ZTFJ0407−00 were obtained on 2020 January 24th. The
weather was good and the seeing ranged between 0.4 and 0.8 arcsec. We used
exposure times of approximately 10% of the orbit: 300 seconds for ZTFJ0220+21
and 200 seconds for ZTFJ0407−00. A standard star spectrum for flux calibration
was obtained in the middle of the night.

Spectra of ZTFJ2252−05 were obtained on 2020 September 16th. The weather was
good and seeing 0.4 arcsec during the observation. We again used exposure
times of approximately 10% of the orbit, 220 seconds. The readout time for the
red arm is much longer in the red arm than in the blue arm due to a technical
problem and we, therefore, obtained 11 spectra in blue and 8 in red. A
standard star spectrum for flux calibration was obtained in the middle of the
night.

We obtained an identification spectrum of ZTFJ0003+14 on 2020 December 10th.
The exposure time is 300 seconds and conditions were good with a seeing of
0.4″. We used the lower resolution R300R grating in the red band for this
spectrum.

A standard long-slit data reduction procedure was performed with the Lpipe
pipeline222http://www.astro.caltech.edu/~dperley/programs/lpipe.html (Perley,
2019). The pipeline reduced LRIS spectral data to spectra using the standard
procedure, including calibration with a standard star. Wavelength calibration
was done using lamp-spectra obtained at the beginning and end of each
sequence. Fine-tuning of the wavelength calibration was done using sky-lines.

In order to identify spectral lines, we interpolated the spectra to a common
wavelength grid, averaged them and normalised the result using a 7th order
polynomial fitted by minimizing the absolute error. The absolute error is less
sensitive to emission and absorption lines compared to the squared error
commonly used.

###  4.2 CHIMERA fast cadence photometry

CHIMERA (Harding et al., 2016) is a dual-channel photometer that uses frame-
transfer, electron-multiplying CCDs mounted on the Hale 200-inch (5.1 m)
Telescope at Palomar Observatory (CA, USA). The pixelscale is 0.28
arcsec/pixel (unbinned). We used the conventional amplifier and used 2x2
binning on most nights (except when the seeing was excellent) to reduce the
readout noise and readout time. Each of the images were bias subtracted and
divided by twilight flat fields333https://github.com/caltech-
chimera/PyChimera.

We used the ULTRACAM pipeline to do aperture photometry (Dhillon et al.,
2007). We used an optimal extraction method with a variable aperture of 1.5
times the FWHM of the seeing (as measured from the reference star). A
differential lightcurve was created by simply dividing the counts of the
target by the counts from the reference star. Timestamps of the images were
determined using a GPS receiver.

We obtained CHIMERA data for each target in the $g$ and $r$ or $i$ filters
during dark or grey time (see Table 6). The seeing for the CHIMERA
observations was typically between 0.8 and 1.5 arcsec.

###  4.3 HiPERCAM fast cadence photometry

HiPERCAM is a high-speed camera for the study of rapid variability (Dhillon et
al., 2016) and is able to obtain images in $ugriz$ simultaneously. HiPERCAM
was mounted on the 10.4m Gran Telescopio Canarias on La Palma (Spain). With
this telescope, HiPERCAM has a $0.081^{\prime\prime}$/pixel scale and a
$3.1^{\prime}$ field-of-view across the diagonal.

We obtained 76 minutes of data in $ugriz$ of ZTFJ1637+49. The observation was
timed such that the lightcurves covers two eclipses. The data were reduced
using the dedicated HiPERCAM pipeline444https://github.com/HiPERCAM/,
including debiasing and flat-fielding. Differential lightcurves were
constructed using a single reference star. Timestamps were obtained from a GPS
receiver.

###  4.4 ’Alopeke fast cadence photometry

’Alopeke is a low noise, dual-channel imager mounted on the Differential
Speckle Survey Instrument (DSSI, Scott, 2019) of the 8m Gemini North Telescope
on Mauna Kea (Hawaii, USA). In the blue arm we used the $g$ filter and in the
red arm we used the $i$ filter. The pixelscale of the wide-field mode is 0.07
arcsec/pixel.

Alopeke was used to obtain 5 lightcurves of the eclipse of ZTFJ0220+21. Images
were bias subtracted and flatfielded using the standard procedure. We used
aperture photometry to extract lightcurves using the ULTRACAM pipeline. No
bright stars were close enough to use differential photometry. Instead, we
used a large aperture of 2.5 times the average FWHM, and used the raw
lightcurve. No GPS receiver was available for Alopeke, so we used computer
timestamps instead.

###  4.5 Archival photometry

To study the long timescale photometric variability and the spectral energy
distribution, we obtained single epoch or averaged photometry data from
multiple other survey telescopes: Galex (Bianchi et al., 2017), Gaia eDR3
(Brown et al., 2020b), and WISE (Marocco et al., 2020). We also obtained
multi-epoch photometry from SDSS (Alam et al., 2015), Pan-STARRS (Chambers et
al., 2016), CRTS (Drake et al., 2009), ASAS-SN (Shappee et al., 2014; Kochanek
et al., 2017), PTF (Law et al., 2009; Rau et al., 2009) and ATLAS (Tonry et
al., 2018; Smith et al., 2020).

All systems except ZTFJ0003+14 have been detected by Galex. ZTFJ1637+49 and
ZTFJ0220+21 only have a NUV detection, while ZTFJ2252−05 and ZTFJ0407−00 have
been detected in both the FUV and NUV bands. The area of ZTFJ0003+14 has been
observed by Galex, but it is too faint to be detected.

Gaia data is available for all 5 systems. This includes parallax and proper
motion measurements.

All WISE data were recently reanalysed by co-adding images by Marocco et al.
(2020). ZTFJ1637+49, ZTFJ0220+21, ZTFJ2252−05, and ZTFJ0407−00 have been
detected in the W1 and W2 bands.

For ZTFJ1637+49, ZTFJ0003+14, ZTFJ0220+21, and ZTFJ2252−05 a single epoch of
SDSS $ugriz$ data is available. The $r$-band observation of ZTFJ1637+49 is
$\approx 0.5$ mag fainter, and was obtained when the targets was in-eclipse.
ZTFJ0407−00 is located in the Stripe 82 survey area of SDSS, and has been
observed on 12 separate nights by the SDSS survey. Two observations were
obtained in-eclipse.

Pan-STARRS observed the five targets approximately 50 times each combined over
the 5 Pan-STARRS filters ($grizy$). A few observations were obtained during
eclipse.

CRTS has detected each of the objects 200-300 times each, except for
ZTFJ0003+14 which is too faint. Non-detections are not reported by CRTS, and
only one in-eclipse can be seen which is in the lightcurve of ZTFJ2252−05. One
outburst has been seen in CRTS data in the lightcurve of ZTFJ2252−05.

ASAS-SN observed each of the sky locations of each source approximately 2000
times. Although ASAS-SN is not able to detect the objects in quiescence, ASAS-
SN observed outbursts of ZTFJ2252−05 and ZTFJ0407−00. ZTFJ0003+14 is too close
to a bright star to be detected by ASAS-SN.

PTF data was available for ZTFJ2252−05 and ZTFJ0407−00, with 43 and 112 epochs
all in $r$-band. For ZTFJ0407−00, two of the epochs are non-detections, and
one epoch is a clear faint outlier. For ZTFJ2252−05, there are 4 in-eclipse
points and one non-detection. There are no positive outliers in either
lightcurve which would indicate an outburst. ZTFJ1637+49 and ZTFJ0220+21 are
not in the footprint of PTF, and ZTFJ0003+14 is too faint and too close to a
bright star to be detected by PTF.

##  5 Methods

###  5.1 Timing

To determine the ephemerii of the mid-eclipse times, we use both the ZTF
lightcurve, archival data (SDSS, PS and CRTS) and the follow-up photometry. We
fit the normalised and combined ZTF $g$, $r$, and $i$ lightcurves with the
best lcurve model (see section 5.4) as determined from the high-cadence
photometry. The only free parameters are the period and mid-eclipse time. As a
prior, we use the measured mid-eclipse times from the CHIMERA, HiPERCAM, and
Alopeke lightcurves. In addition, we also add the epochs of any individual in-
eclipse points from SDSS, Pan-STARRS and PTF, and use half their exposure time
as the uncertainty. The uncertainties (standard deviations) are determined by
using emcee (Foreman-Mackey et al., 2013).

###  5.2 Doppler tomography

To better understand the geometry of the system, we use Doppler tomography
(Marsh, 2001). Doppler tomography converts a series of 2D spectra obtained at
different phases of the orbit into a velocity map (also called a Doppler map).
We used the He-I lines at 6678.15, 7065.17, and 7281.35 Å for ZTFJ1637+49,
ZTFJ0220+21, and ZTFJ0407−00, because these are not blended with other lines.
For ZTFJ0003+14 and ZTFJ2252−05, more blue spectra are available and in this
case we use He-I lines in the blue spectra: 3613.64, 3634.23, 3888.64,
4026.19, and 4921.93Å. To make the Doppler maps from the LRIS spectra, we used
the python package doppler555https://github.com/trmrsh/trm-doppler. We used a
blurring scale of 40 $\mathrm{kms^{-1}}$, and manually set the target reduced
$\chi^{2}$ value to avoid over- or underfitting the map.

###  5.3 Modelling of the spectral energy distribution

To estimate the white dwarf temperature, we fit the spectral energy
distribution of the systems by simply modelling them as a blackbody. The model
parameters are the temperature ($T_{\mathrm{eff}}$), the radius of the white
dwarf ($R_{\mathrm{WD}}$), the distance ($d$), and the amount of reddening
using the Fitzpatrick (1999) reddening law with the parameter $E_{B-V}$.

We constrain the distance by putting a Gaussian prior directly on the parallax
using Gaia eDR3 values. We also use a Gaussian prior on the radius of the
white dwarf using the values obtained using the lightcurve modelling. Finally,
we constrain the value of $E_{B-V}$ using the Pan-STARRS dustmap (Bayestar19,
Green et al., 2019b), with the conversion $E_{B-V}=0.884E_{g-r}$.

We converted the magnitudes to flux using the zeropoints for each
filter666http://svo2.cab.inta-csic.es/theory/fps/ (Rodrigo et al., 2012;
Rodrigo & Solano, 2020). We use the effective wavelength for each filter to
calculate the model flux.

The observed flux is the combined flux from the white dwarf, disk, brightspot,
and the donor. Using the $gri$ lightcurves of the fully eclipsing systems, we
can disentangle the contributions from the different components by modelling
the lightcurves in those three bands (see Figures 3–6). For each system, we
use the model to determine how much the disk and brightspot contribute. We use
this to correct the $gri$ magnitudes from PS, SDSS, and ZTF magnitudes.

To find the optimal solution and uncertainties we use emcee. We use a
likelihood function assuming Gaussian uncertainties. In order to account for
any systematic errors, we include an additional error term in the model which
is optimized together with the other parameters. The measurements in other
bands are used as upper limits (implemented as one-sided Gaussians). We
exclude the two WISE bands from the fit as these are likely dominated by the
disk and/or donor.

For ZTFJ1637+49 and ZTFJ0003+14 we deviated from this general approach.
ZTFJ1637+49 only shows partial eclipses, and we cannot use the eclipse depth
and shape to disentangle the contribution to the overall luminosity by the
different components. We use the uncorrected magnitudes to fit the white dwarf
model and assume the other components can be neglected, which is justified
because this is a very long period system with a very low accretion rate and
cold donor. We exclude the SDSS $r$ band measurement which was obtained during
an eclipse and is 0.5 mag fainter than other $r$-band observations.

For ZTFJ0003+14, the SDSS data was significantly brighter than the PS and Gaia
data, which suggests that the system was in outburst when SDSS observed it.
We, therefore, use the SDSS $ugriz$ data as upper limits only.

###  5.4 Lightcurve modelling

We use Lcurve777https://github.com/trmrsh/cpp-lcurve to model the lightcurves
and infer the binary parameters. Lcurve uses 3D grids to simulate two stars
(spherical or with a Roche geometry). In addition, a disk and bright-spot can
be added to the model. A detailed description of Lcurve is given by
Copperwheat et al. (2011). In this section, we discuss the model setup for
each of the systems. For a detailed discussion on the subtleties of modelling
eclipsing AM CVn lightcurves, we refer the reader to Green et al. (2018a).

The lightcurves show a range in contributions from the disk and brightspot,
and therefore we use different model configurations and free parameters for
each of the AM CVn systems. The basic model is a spherical white dwarf with a
dark, Roche-lobe filling donor. The free parameters for this model are the
inclination ($i$), mass-ratio ($q$, which sets the radius of the donor,
$r_{2}$), scaled radius of the accreting white dwarf ($r_{1}$), the mid-
eclipse time ($t_{0}$), and the velocity scale ($[K_{1}+K_{2}]/\sin i$), with
$K_{1,2}$ the observable radial velocity amplitude. We fixed the temperature
of the white dwarf to the values obtained from the fit to the SED (Section
6.3). We set the donor temperature to $2000$ K, which in practice means that
the donor does not contribute to the model lightcurve. We fixed the orbital
period to the period derived from the ZTF data (see Section 5.1). Limb-
darkening of the white dwarf is approximated by using the 4-parameter Claret
law (Claret & Bloemen, 2011) with the parameters taken from the DB models by
Claret et al. (2020) for the closest temperature value and $\log(g)=8.5$. In
addition, we imposed two restrictions on the white dwarf radius using the
approximation of the mass–radius relation of Eggleton from Rappaport et al.
(1989). The first is that it cannot be smaller than a zero-temperature white
dwarf. The second is a Gaussian prior on the white dwarf radius relative to
the white dwarf M-R relation with an uncertainty of 5%.

To find the best parameter values and uncertainties, we use a Markov chain
Monte Carlo (MCMC) method as implemented in emcee. To make sure the data and
prior are fitted properly, we rescaled the uncertainties in the lightcurves.
For the final fit of each lightcurve, we use 512 walkers and at least 2000
generations.

Since the quality and quantify of the data differ, and the systems are also
intrinsically different, we use a different setup in each case, which are
described below.

####  5.4.1 ZTFJ1637+49

At very long orbital periods, both the disk and bright spot components do not
contribute significantly to the lightcurve, as is the case for ZTFJ1637+49.
This is supported by the fact that the eclipse depths are identical in each of
the 5 HiPERCAM bands. We therefore use just the basic model without a disk or
brightspot to model the lightcurve.

For the $g$ and $r$ band, multiple nights of data are available. To combine
the lightcurves, we first fit a model to each lightcurve individually. We then
remove any trend in the lightcurve using a polynomial and rescale the
uncertainties. We then combine the lightcurves per band (without any binning
or averaging) and fit the combined lightcurves. For the $u$, $i$, and $z$
bands, only the HiPERCAM lightcurve is available, and we fit these directly.

####  5.4.2 ZTFJ0003+14

We have obtained high-speed photometry for ZTFJ0003+14, three orbits in the
$g$ and $r$, and a single eclipse in $g$ and $i$. Conditions were bad, with
seeing ranging between 2-4″. The lightcurves of ZTFJ0003+14 do not show any
obvious variability due to a disk and no obvious eclipse of the brightspot or
any other variability besides the eclipse. The white dwarf eclipse seems
consistent with no light. We do however include both a disk and brightspot in
the model to account for any small amount of in-eclipse light.

For the disk, we use the same setup as in Green et al. (2018a) and keep most
parameters fixed. The only two free parameters of the disk are the temperature
($T_{disc}$) and the outer radius ($r_{disc}$). For the brightspot, we keep
most parameters fixed to their default values. The only two free parameters of
the brightspot are the temperature ($T_{spot}$) and spot length ($l_{spot}$).
We fix the spot position to the outer radius of the disk.

####  5.4.3 ZTFJ0220+21

For ZTFJ0220+21, one night of $g$ and $r$ band data are available from Chimera
and multiple sequences of $g$ and $i$ band data from ’Alopeke. We first fit a
model to the individual lightcurves. We divide the data by the model, and fit
the residual with a 3rd order polynomial function and remove this trend from
the data. After dividing out the polynomial fit, the ’Alopeke lightcurves
still showed correlated ‘flickering’ behaviour, likely due to small changes in
atmospheric conditions. To remove these, we fitted the residuals using
Gaussian process regression (McAllister et al., 2017, similar as in [). The
kernel we used is a combination of a white kernel and a Matern 3/2 kernel with
the timescale constrained to 3-12 seconds. Finally, we divide the data by the
polynomial fit and GPR model, and rescale the errorbars to reflect the
remaining noise in the data. We then combined all detrended lightcurves per
band using the ZTF period.

ZTFJ0220+21 is similar to ZTFJ0003+14; it does not show any obvious
variability due to a disk and no obvious eclipse of the brightspot or any
other variability besides the eclipse. However, in this case, a small amount
of light is detected in eclipse. Although no brightspot eclipse can be seen in
the data, we do include both a disk and brightspot in the model to account for
any in-eclipse light. To model the lightcurves, we use the same setup and free
parameters as for ZTFJ0003+14 ($T_{\mathrm{disc}}$, $r_{\mathrm{disc}}$,
$T_{\mathrm{spot}}$, $l_{\mathrm{spot}}$).

####  5.4.4 ZTFJ2252−05

We obtained CHIMERA data of this system on two nights. The lightcurves show
clear variability on long timescales. We first fit a basic lcurve model and
subtract this from the lightcurves. We then smoothed the residuals with a
Gaussian kernel of 2.5 minutes, and subtract this smoothed curve from the
initial lightcurve, which removed any non-periodic variability.

The folded lightcurves show the eclipse by the white dwarfs and a clear
eclipse by the disk. There appears to be some asymmetry in the eclipse due to
a brightspot. We therefore model the disk with two free parameters
($T_{\mathrm{disc}}$, $r_{\mathrm{disc}}$), and use length, exponent, angle,
yaw, temperature, and cfrac (again, see Green et al. 2018a for a description
of each parameter). We again fixed the brightspot location to the outer edge
of the disk.

####  5.4.5 ZTFJ0407−00

The first two nights of Chimera data were obtained just after a superoutburst
(see Section 6.4). These lightcurves show a strong contribution by the disk
and vary significantly. We attempted to model these lightcurves, but the
strong variability between eclipses makes this very challenging. We therefore
obtained more data seven months later. These lightcurves show almost no
flickering or other variability which is not periodic at the orbital period.
In addition, the lightcurves show a clear brightspot eclipse at a level of
10%. For the estimation of model parameters, we excluded the data that were
obtained during the first two nights.

Since this lightcurve shows an obvious brightspot eclipse, we use more free
parameters to model this system than for ZTFJ2252−05. The free parameters for
the disk are the temperature ($T_{disc}$) and the outer radius ($r_{disc}$),
we also allow the temperature gradient to vary. To model the brightspot (which
causes the observed asymmetry in the eclipse), we use the same free parameters
as for ZTFJ2252−05.

Figure 2: The lightcurves of ZTFJ1637+49 with the best-fit models overplotted
in the $ugriz$ bands (blue, green, red, gold, and purple; from top to bottom).
The eclipse depth is identical is each of the 5 bands, which suggests that
there is no contribution from the disk or brightspot. Therefore, the
lightcurve model (black line) does not include these components, and the model
only includes a white dwarf that is eclipsed by a cold and dark, Roche-lobe
filling donor.

Figure 3: The combined lightcurves of ZTFJ0003+14 in the $gri$ bands, similar
to 3. The full model is shown by the black line, the contribution from the
disk is shown by the dotted line, and the contribution by the brightspot is
shown by the dashed-dotted line. The model shows that the white dwarf
dominates the luminosity, and disk and/or brightspot only contribute $\lesssim
10\%$ to each lightcurve (see also Table 4).

Figure 4: The combined lightcurves of ZTFJ0220+21 in $gri$-bands (top to
bottom, see also Figure 3). Similar to ZTFJ0003+14, the white dwarf dominates
the luminosity and disk and/or brightspot only contribute $\lesssim 10\%$.

Figure 5: The combined lightcurves for ZTFJ2252−05in the $gri$-bands (top to
bottom). The full model is indicated by the black line, while the contribution
from the disk and brightspot are indicated with a dotted and dashed-dotted
line. The lightcurves have been detrended, but flickering remains visible. The
disk contributes significantly more compared to the longer period systems
(27–51%), with larger contributions for longer wavelengths. In each of the
lightcurves, a few percent (3–7%) contribution from the brightspot is also
visible, shown in the inset. Figure 6: The combined lightcurves of of
ZTFJ0407−00 in $gri$-bands (top to bottom, similar to Figure 3). Compared to
ZTFJ2252−05 the disk contributes less for this system (13–22%). The overall
contribution by the brightspot is similar as for ZTFJ2252−05 but the eclipse
is clearly visible in the $g$ and $r$ lightcurves.

##  6 Results

###  6.1 Ephemerii

The mid-eclipse ephemerii are given in Table 2. The combination of archival
data that spans a long-baseline and accurate eclipse time measurements means
that the periods can be constrained to $\approx 10^{-8}$ days or better, and
the zeropoint to a few seconds. We did not detect any significant deviations
from the eclipse arrival time, except for system for ZTFJ0407−00. More data is
needed verify these eclipse arrival time variations and determine their
nature, which we will be the topic of future work.

Name |  $t_{0}$ ($\mathrm{BJD_{TDB}}$) | period (d)
---|---|---
ZTFJ1637+49 | 2458370.73498(4) | 0.042 707 771(8)
ZTFJ0003+14 | 2458323.02453(1) | 0.038 564 45(8)
ZTFJ0220+21 | 2458428.74931(8) | 0.037 041 443(2)
ZTFJ2252−05 | 2458469.64845(7) | 0.025 968 133 6(9)
ZTFJ0407−00 | 2458386.99699(2) | 0.024 588 123(3)
Table 2: The orbital period and mid-eclipse times as measured from the ZTF and
archival photometry.

###  6.2 Dopplermaps

Figure 7: Dopplermaps of the five eclipsing AM CVn systems. For ZTFJ1637+49,
ZTFJ0220+21, and ZTFJ0407−00 we used spectra from the LRIS red arm, and for
ZTFJ0003+14 and ZTFJ2252−05 we used spectra from the LRIS blue-arm. Red dashed
lines show the Roche lobes of the binary system, with the white dwarf located
in the center of the image and the donor (with the small Roche-lobe) on top.
In this coordinate system, the accretion flow originates at the top of the
donor. ZTFJ1637+49shows emission from the white dwarf (the ’central-spike’).
The primary brightspots are visible at the top or top-left of the figure for
all systems. ZTFJ1637+49 and ZTFJ0220+21show possible secondary spots at the
bottom-left, and ZTFJ2252−05 shows a possible secondary spot at the bottom
right. The structure in the map of ZTFJ0003+14 is caused by the sparse
sampling.

The Dopplermaps are shown in Figure 7. All systems show a clear disk in the
Dopplermaps. The size of the disk increases slightly as the orbital decreases
and the velocities of the components increase. For ZTFJ1637+49 and
ZTFJ0220+21, (with the best data) show a brightspot at the top of the figure,
and possibly a secondary brightspot at the lower right of the map, also seen
in Gaia14aae (Green et al., 2020). ‘Central spike’ emission is clearly visible
for ZTFJ1637+49 which originates from the white dwarf (Kupfer et al., 2016;
Green et al., 2020). The other Dopplermaps are less clear, but all seem to
show brighter parts in the disk that are consistent with bright-spots.

###  6.3 Spectral energy distribution and white dwarf temperature

Figure 8: The SED of each system, showing Galex, SDSS, PS, median ZTF, UKIDSS,
and WISE data. Triangles show observed flux measurements which were used as
upperlimits in the fit. Filled dots show the estimated flux by the white dwarf
by removing the contributions from the disk and brightspot (Table 4).
Errorbars are omitted if they are smaller than the symbol size. We fitted the
data with a blackbody in order to measure the white dwarf temperature (black
line). Grey lines show the uncertainty in the fit. The dashed line shows the
expected contribution from the donor star if we assume an equilibrium
temperature that is set by irradiation from the white dwarf. For ZTFJ1637+49
the correction of the observed flux is assumed to be 0.

The spectral energy distributions are shown in Figure 8. Also shown are the
best-fit white dwarf models (see Section 5.3), and the white dwarf
temperatures as reported in Table 3. In general, the models agree well with
the data, but do show some inconsistencies at extreme wavelengths. For
shortest period systems, a FUV measurement is available and the model
underestimates the FUV flux in both cases. This is possibly due to emission
from the accretion disk, but could also be due to the blackbody assumption.

The flux in the WISE bands, which were not used to fit the model, are also
underestimated in all cases. To check if this would be consistent with
emission from the donor, we calculated the donor equilibrium temperature based
on irradiation by the white dwarf. This suggests that the excess in the far IR
for ZTFJ1637+49 could be by the donor (see also Green et al. 2020). For the
other objects, the estimated contribution by the donor is insufficient to
explain the excess light, and the disc is the more likely source of the IR-
excess.

###  6.4 Long timescale variability

Figure 9: The lightcurves of the two shortest period systems ZTFJ2252−05 (top)
and ZTFJ0407−00 (bottom). The consequative panels show an increasingly zoomed
in range, indicated by black boxes in the previous panels. The figures show
photometry from SDSS (squares), CRTS (dots), PTF (hexagons), ASAS-SN
(triangles, points as upperlimits) and ZTF data (PSF photometry as dots and
alerts as crosses). The $ugriz$ filters are shown in blue, red, green, yellow
and purple, $V$-band observations are shown in black. "C" and "L" indicate
when Chimera and LRIS data were obtained. ZTFJ2252−05: Two outbursts were
detected by CRTS and ASAS-SN, three years apart. No other outbursts can be
seen in the rest of the data, including well-sampled ATLAS and ZTF data of the
last 5 years. ZTFJ0407−00: No outbursts or long-timescale variability can be
identified in the SDSS, CRTS and PTF data. The ATLAS cyan and orange filter
data are shown in cyan and orange dots, with the data binned to one day bins.
The middle panel shows ZTF and ASAS-SN data from 2017 to 2020 which shows two
large-amplitude, multi-day long outbursts. The bottom panels shows the most
recent data, in which 4 short duration ($\lesssim$1 day) outbursts which occur
very shortly after a superoutburst. These echo outbursts have also been seen
in other AM CVn systems (e.g. Green et al., 2020; Duffy et al., 2021; Pichardo
Marcano et al., 2021).

We inspected the ZTF and archival data to study the long-timescale behaviour
of the AM CVn systems. As expected, the long period systems, ZTFJ1637+49,
ZTFJ0003+14, and ZTFJ0220+21, do not show any outburst or other long timescale
variability in the ZTF data or the other archival data that spans a decade or
more. For ZTFJ0003+14, only ZTF data are available that spans three years, and
no outbursts or trends are visible. We do note that the source is $\approx$0.6
mag brighter in SDSS compared to ZTF and Pan-STARRS data. ZTFJ0003+14 likely
showed an outburst, and SDSS observed it while it was on its way back to
quiescence.

Both ZTFJ2252−05 and ZTFJ0407−00 do show variability on longer timescales,
shown in Figure 9. ZTFJ2252−05 shows low-level variability in the lightcurve
(besides the eclipse), but no clear overall trends. Only two outbursts have
been detected, one by CRTS and one by ASAS-SN, both reaching magnitude 16. The
lightcurves are not sampled well enough to make any statement on the duration
of the outbursts.

ZTFJ0407−00 also shows variability on longer timescales. No outbursts have
been detected by SDSS, CRTS, Pan-STARRS, or PTF. The ZTF and ASAS-SN data
obtained in the last three years show two (possibly three) high amplitude
outbursts that last multiple days. As can be seen in the middle panel of
Figure 9 for ZTFJ0407−00, the quiescence luminosity increased after the last
superoutburst by about 0.6 magnitudes and is slowly decreasing. This system
shows at least 4 short duration (<1 day) outbursts.

###  6.5 Binary parameters

| $P$ | $q$ | $i$ | $T_{1}$ | $M_{1}$ | $M_{2}$ | $R_{1}$ | $R_{2}$
---|---|---|---|---|---|---|---|---
| min |  | $\deg$ | kK | $\mathrm{M_{\sun}}$ | $\mathrm{M_{\sun}}$ | $\mathrm{R_{\sun}}$ | $\mathrm{R_{\sun}}$
ZTFJ1637+49 | 61.5 | $0.026\pm 0.010$ | $82.7\pm 0.09$ | $11.2\pm 0.3$ | $0.90\pm 0.05$ | $0.023\pm 0.008$ | $0.009\pm 0.001$ | $0.068\pm 0.007$
ZTFJ0003+14 | 55.5 | $0.0214\pm 0.010$ | $86.5\pm 2.0$ | $11.6\pm 0.8$ | $0.79\pm 0.11$ | $0.017\pm 0.011$ | $0.0106\pm 0.001$ | $0.057\pm 0.012$
ZTFJ0220+21 | 53.5 | $0.0174\pm 0.005$ | $85.3\pm 2.1$ | $14.2\pm 1.0$ | $0.83\pm 0.07$ | $0.014\pm 0.006$ | $0.010\pm 0.001$ | $0.054\pm 0.007$
Gaia14aae | 49.7 | $0.0290\pm 0.0006$ | $86.27\pm 0.10$ | $17.0\pm 1.0$ | $0.872\pm 0.007$ | $0.0253\pm 0.0007$ | $0.00924\pm 0.00009$ | $0.0603\pm 0.0003$
ZTFJ2252−05 | 37.4 | $0.034\pm 0.006$ | $87.0\pm 1.0$ | $15.2\pm 0.9$ | $0.76\pm 0.05$ | $0.026\pm 0.008$ | $0.010\pm 0.001$ | $0.049\pm 0.004$
ZTFJ0407−00 | 35.4 | $0.024\pm 0.004$ | $86.5\pm 0.7$ | $17.4\pm 1.2$ | $0.79\pm 0.06$ | $0.019\pm 0.003$ | $0.010\pm 0.001$ | $0.044\pm 0.002$
YZ LMi | 28.3 | $0.041\pm 0.002$ | $82.6\pm 0.3$ | $17.0\pm 1.0$ | $0.85\pm 0.04$ | $0.035\pm 0.003$ | - | $0.047\pm 0.001$
Table 3: Binary parameters of all well-characterized eclipsing AM CVn systems. The 5 ZTF systems are from this work, parameters of Gaia14aae are from Green et al. (2018a), and parameter of YZ LMi are from Copperwheat et al. (2011). In each case, we have used the results based on modelling of the $g$-band lightcurve, which is the most accurate in all cases. | WD | disc | brightspot
---|---|---|---
| $g$ | $r$ | $i$ | $g$ | $r$ | $i$ | $g$ | $r$ | $i$
ZTFJ1637+49 | 100% | 100% | 100% | - | - | - | - | - | -
ZTFJ0003+14 | 97% | 93% | 89% | 3% | 0% | 11% | 0% | 7% | 0%
ZTFJ0220+21 | 92% | 93% | 89% | 8% | 0% | 10% | 0% | 7% | 0%
ZTFJ2252−05 | 65% | 58% | 46% | 27% | 36% | 51% | 7% | 5% | 3%
ZTFJ0407−00 | 82% | 75% | 70% | 13% | 16% | 22% | 5% | 9% | 8%
Table 4: The contribution to the overall luminosity of white dwarf, disk, and
brightspot in the $gri$ filter for the best-fitting models. The white dwarf is
typically dominant for long period systems and at shorter wavelengths, while
the disk contributes more at short orbital periods and longer wavelengths.
Interesting to note is that the disk is more prominent in ZTFJ2252−05 than in
ZTFJ0407−00 despite having a slightly longer orbital period. At long periods
and with low-SNR data, there can be some degeneracy between the disc and
brightspot. For the $r$-band model of ZTFJ0003+14 and ZTFJ0220+21 no light
from the disk is modelled but instead the brightspot contributed 7% of the
light. We suspect that this is due to minor systemics in the data and that the
7% contribution of light is actually from the disk.

The lightcurves and their best fit models are shown in Figures 3–6, and the
best-fit parameters and uncertainties can be found in the Appendix (Tables
7–11). An overview of the binary parameters is given in Table 3 and Table 4
lists the contribution of each component to the overall luminosity. We first
discuss the overall results, and discuss the result for each object
individually at the end of this section.

As expected and can be seen in Table 4, the relative contribution by the disk
and brightspot increases with decreasing orbital periods. In general, for the
long period systems (ZTFJ1637+49, ZTFJ0003+14, and ZTFJ0220+21) the brightspot
is not detected. However, the $r$-band models for ZTFJ0003+14 and ZTFJ0220+21
do include a significant contribution by the brightspot. Since this feature is
not seen in the other bands, we expect this is due some asymetry in the
lightcurve caused by residual flickering, and the additional light is actually
from the disk. For the two short-period systems (ZTFJ2252−05 and ZTFJ0407−00)
there seems to be a detection of the brightspot eclipse. However, this is a
subtle feature in the lightcurve. The brightspot eclipse for ZTFJ0407−00, but
only marginally for ZTFJ2252−05. We note that the contribution from the disk
is largest for ZTFJ2252−05 despite having a longer orbital period than
ZTFJ0407−00. We discus this further in Section 7.2.

Table 3 shows the binary parameters as derived from the lightcurve modelling.
The main sources of uncertainty is the degeneracy is between $q$ and $i$
(Green et al., 2018b). For a lightcurve with where only the primary eclipse is
detected, $q$ and $i$ are fully degenerate. This degeneracy can be lifted with
additional features in the lightcurve; either by measuring the brightspot
eclipse, or the eclipse shape which is set by a combination of the limb-
darkening of the white dwarf and the white dwarf mass-radius relation used.
888A radial velocity semi-amplitude measurement or surface gravity measurement
would also work. As we have discussed above, only ZTFJ0407−00 shows a clear
detection of the brightspot eclipse. Therefore, the uncertainty on the mass-
ratio is high for the other systems, up to $\approx 50$% is the worst case
(ZTFJ0003+14). Correspondingly, the uncertainties on the inclinations are
0.5–2 degrees. We do note that of the 5 systems, ZTFJ1637+49 has be best
constrain on the inclination despite only showing a grazing eclipse. This can
be explained by the lack of any disk or brightspot and the high quality
$g$-band data.

In general, the mass of the accreting white dwarf is high, $0.76$–$0.90$
$\mathrm{M_{\sun}}$, discussed further in Section 7.4. The uncertainty on the
white dwarf mass ranges between 0.05–0.11 $\mathrm{M_{\sun}}$ (6–14%). The
uncertainty is mainly determined by the uncertainty on $r_{1}$ and the
assumption of the M-R relation, and only has a weak dependence on $q$ and $i$.

The donor mass and radius The uncertainty is the mass-ratio is also the
largest source of uncertainty in the mass and radius of the donor, which

####  6.5.1 ZTFJ1637+49

The models for ZTFJ1637+49 agree well with the data for each of the colour
bands and do not show any systematic deviations. This justifies the use of a
model with only the accretor and donor, without any contribution from a disk
or brightspot.

The best-fit parameters consistently indicate that the inclination is $82.8\pm
1.1$ degrees, which is expected for a grazing eclipse. Inspection of the
posteriors shows that $q$ and $i$ are strongly correlated (as expected). The
mass-ratio ($q$) is $0.025\pm 0.001$. The results are consistent between
bands.

The derived binary parameters (masses and radii) are all consistent between
bands. The $g$-band results are the most accurate and we use these in the rest
of the paper. The mass of the white dwarf is $M_{1}=0.90\pm 0.05$
$\mathrm{M_{\sun}}$. The donor has a mass of $M_{2}=0.023$ $\mathrm{M_{\sun}}$
and radius of $R_{2}=0.068\pm 0.007$ $\mathrm{R_{\sun}}$.

####  6.5.2 ZTFJ0003+14

The fit to the lightcurves of ZTFJ0003+14 are good and no discrepancies can be
seen. The best-fit models indicate that almost no light is visible when the
system is in eclipse. The parameter estimates for the $g$ and $i$ band are
consistent with each-other, but the $r$-band parameters differ. This is
possibly because the $r$-band model has a 7% contribution from the brightspot,
while the $g$ and $i$ models attribute any additional light to the disk. This
is likely due some residual systematics in the $r$-band data. Taking this and
the data quality into account, we conclude that the $g$-band solution is the
most accurate and precise.

As is the case with ZTFJ1637+49, with just a white dwarf eclipse and no
additional information, there is a degeneracy between $i$ and $q$, which are
both poorly constrained; $i=86.5\pm 2.0$ and $q=0.0214^{+0.016}_{-0.007}$,
which results in relatively poor constrains on the mass and radius. The mass
of the white dwarf is $M_{1}=0.79^{+0.13}_{-0.10}$ $\mathrm{M_{\sun}}$. The
donor has a mass of $M_{2}=0.0165^{+0.018}_{-0.006}$ $\mathrm{M_{\sun}}$ and
radius of $R_{2}=0.068\pm 0.007$ $\mathrm{R_{\sun}}$.

####  6.5.3 ZTFJ0220+21

Models for ZTFJ0220+21 agree well with the data, and do not show any
variability that the models do not account for. We note that in all
lightcurves, the eclipse is not consistent with zero, and up to 10% of the
light is emitted by the disk and/or spot. Similar as for ZTFJ0003+14 the
$r$-band model does model some contribution by the brightspot, but the $g$ and
$i$ models does not include any contribution from the brightspot. Again, this
is likely due to systematics in the data.

The parameter estimates between bands deviate significantly. The $g$ and
$r$-bands roughly agree, but the $i$-band solution deviates significantly. It
is skewed towards very high inclination solutions and very low-mass ratios. A
comparison between the models shows a different eclipse shape in the $i$-band
compared to the other two models. The difference is small, but does seem to
have an effect on the parameter estimates. The difference is possibly a result
of the detrending process. Given that the quality of the $g$-data is best, we
use this result in the rest of the paper.

As is the case for ZTFJ1637+49 and ZTFJ0003+14 the solution for ZTFJ0220+21 is
also affected by a degeneracy between $i$ and $q$. However, because of the
abundance of $g$-band data, the parameters are beter constrained compared to
ZTFJ0003+14 $i=85.3^{+0.8}_{-0.9}$ and $q=0.018^{+0.007}_{-0.004}$. The mass
of the white dwarf is $M_{1}=0.83^{+0.07}_{-0.08}$ $\mathrm{M_{\sun}}$. The
donor has a mass of $M_{2}=0.014^{+0.007}_{-0.005}$ $\mathrm{M_{\sun}}$ and
radius of $R_{2}=0.054^{+0.008}_{-0.006}$ $\mathrm{R_{\sun}}$.

####  6.5.4 ZTFJ2252−05

This shorter period system shows significant variability out of eclipse. While
we did detrend the data and combined a number of orbits, some correlated
variability remains in the lightcurve. Besides this, the model is in good
agreement with the data. In this system, the disk contribution is clearly
visible as the slopes before and after the main eclipse by the white dwarf.
There also seems to be some asymmetry present in the eclipse and it seems to
be fainter at egress, which is likely the eclipse of the brightspot. This is
however at the same level as the residual flickering, and might not be
significant.

Estimates of the system parameters show roughly consistent between bands. For
sake of consistency, we adopt the $g$-band results in the rest of the paper.
Because there is no significant detection of the brightspot eclipse, there is
still a strong degeneracy between $q$ and $i$. The inclination and mass-ratio
are $i=87.0^{+0.5}_{-1.5}$ and $q=0.030^{+0.002}_{-0.003}$. The mass of the
white dwarf is $M_{1}=0.76^{+0.05}_{-0.04}$ $\mathrm{M_{\sun}}$. The donor has
a mass of $M_{2}=0.026^{+0.010}_{-0.002}$ $\mathrm{M_{\sun}}$ and radius of
$R_{2}=0.049^{+0.006}_{-0.002}$ $\mathrm{R_{\sun}}$.

####  6.5.5 ZTFJ0407−00

This system has an orbital period very similar to ZTFJ2252−05, but does not
show as much flickering and has a smaller contribution to the luminosity by
the disk (Table 4). The contribution by the brightspot is more significant and
can be seen by the broad ’hump’ which peaks at phase $\approx 0.8$. In
addition, there is also a possible contribution by a ‘superhump’ that causes a
peak in the lightcurve at phase 0.2 (also seen in YZ Lmi, see Copperwheat et
al. 2011). In this system, the brightspot eclipse is clearly visible (at phase
0.02–0.08) as shown by the inset.

The parameter estimates from the $g$ and $i$ band both agree, but the $r$-band
result indicate a lower inclination and higher mass-ratio. There is no obvious
difference between lightcurves. Comparing the models, there is a subtle
difference between the duration of the eclipse between bands. This could be
the result of the flickering and/or the detrending process. We again adopt the
$g$-band solution for the rest of the paper, as it is deemed the most reliable
given the higest SNR and largest amount of data available.

The inclination and mass-ratio are $i=86.5^{+0.8}_{-0.5}$ and
$q=0.024^{+0.003}_{-0.004}$. The mass of the white dwarf is
$M_{1}=0.79^{+0.04}_{-0.05}$ $\mathrm{M_{\sun}}$. The donor has a mass of
$M_{2}=0.019^{+0.003}_{-0.004}$ $\mathrm{M_{\sun}}$ and radius of
$R_{2}=0.044^{+0.002}_{-0.003}$ $\mathrm{R_{\sun}}$.

###  6.6 The average spectra and spectral lines

Figure 10: The averaged and normalised LRIS spectra of ZTFJ1637+49, ZTFJ0003+14, ZTFJ0220+21, ZTFJ2252−05, and ZTFJ0407−00 from top to bottom, labelled by their orbital periods. The spectra have only been shifted and have not been re-scaled. The different absorption and emission lines are indicated by colored vertical lines, and telluric lines are indicated with black ticks at the top and bottom. |  ZTFJ1637+49 |  ZTFJ0003+14 |  ZTFJ0220+21 |  ZTFJ2252−05 |  ZTFJ0407−00
---|---|---|---|---|---
He-I | e, de | e | de | de | de
He-II | e | e | e | de | de
N-I | e, a |  | a | a? | a?
O-I |  |  | a |  |
Na-I | a, e? |  | a | a | a?
Mg-I | a | a | a | a | a?
Mg-II |  |  | a |  | a?
Si-I |  |  | a | de | de
K-I | a |  |  |  |
Ca-II |  |  | a | a | a
Zn-I | a |  |  |  |
Table 5: List of elements identified from emission and absorbtion lines in the
averaged LRIS spectra (see Figure 10). ‘e’ indicates emission lines, ‘de’
indicates double-peaked emission lines, and ‘a’ for absorption lines.

Spectra of AM CVn systems are a combination of light from the white dwarf, the
disk, the brightspot, and sometimes a boundary layer on the interface between
the disk and white dwarf (e.g. Kupfer et al., 2015). The donor is cold and not
visible. For systems in quiescence, the overall shape of the spectral energy
distribution is set by the continuum of the white dwarf. The brightspot is
small but hot and is sometimes seen as excess UV emission.

In quiescence, the disk is optically thin and appears as emission lines. If
the system is viewed near edge-on, the lines are double-peaked. In GP Com
(Smak, 1975; Morales-Rueda et al., 2003; Kupfer et al., 2016) and other
systems, a single-peaked emission line is also visible in the centre of many
double-peaked He lines, dubbed the ‘central spike’. Radial velocity
measurements show that these lines originate from the white dwarf (Marsh,
1999; Roelofs et al., 2005; Kupfer et al., 2015) and is likely boundary layer
emission from the surface of the white dwarf (e.g. Green et al., 2019a; Gehron
et al., 2014).

In some cases, absorption lines of metals are present in the spectra of AM CVn
systems (Groot et al., 2001; Anderson et al., 2008; Kupfer et al., 2016).
Radial velocity shifts of absorption lines are consistent with that of the
white dwarf (Kupfer et al., 2016; Green et al., 2020), which confirms that
these lines are formed in the atmosphere of the white dwarf.

The average spectra are shown in Figure 10, and the spectral lines listed in
Table 5. To identify the lines, we systematically searched the spectra element
by element and checked for the presence of spectral lines. First, we confirmed
that no Hydrogen lines are present. We continued with He-I lines, which are
found both in emission (as double peaks) and absorption in all spectra.
ZTFJ1637+49 also shows a strong ‘central spike’ component. He-II is also
present in all spectra as a small but significant emission line, which is also
likely a ‘central spike’. There also seems to be some double-peaked emission
from He-II, but other lines close to He-II-4686Å makes this hard to determine
with certainty. ZTFJ0003+14 is an exception and shows a strong emission line
at He-II-4686Å.

We continued the search with the CNO elements. N-I lines are in emission in
the spectra of ZTFJ1637+49 and absorption for ZTFJ0220+21. The spectrum of
ZTFJ0407−00 seems to show a broad emission feature consistent with emission
from the N-I multiplet at 8223Å (for an overview of N lines see Kupfer et al.,
2017). There are also some hints of emission lines by N-I around 8680Å. There
is no clear detection of N-I in the spectra of ZTFJ0003+14 and ZTFJ2252−05,
possibly because the SNR of these spectra a lower. Oxygen is clearly detected
in ZTFJ0220+21: there is a clear detection of an absorption feature at
7772/4/5 Å. This feature is not detected in the other spectra. We found no
carbon lines in any of the spectra, which has implications on the nature of
the donor star, see Section 7.3.

Next, we searched for the presence of metal lines in the spectra. Both Na-I
and Mg-I/II are detected as absorption lines. Mg-I can be seen in the triplets
at 3832Å and 5172Å. They are strongest for the long period system and decrease
in strength for the short-period systems. Mg-II lines are weaker but can be
seen at 4481Å and possibly at 4384/90. Na-I lines are especially strong in
ZTFJ1637+49, but barely detectable in the other spectra (3302/3, 5682/8,
6154/60, and 5889/95Å). Ca-II lines (3933/68Å and also 3706/36Å) are seen for
ZTFJ0003+14, ZTFJ0220+21, ZTFJ2252−05, and ZTFJ0407−00, but not for
ZTFJ1637+49. Finally, ZTFJ0220+21, ZTFJ2252−05, and ZTFJ0407−00 all show Si-II
lines in their spectra; as double-peaked emission lines (6347/71Å) for the two
short-period systems, and as absorption lines for ZTFJ0220+21. There are also
lines visible at 3856/62Å for ZTFJ0220+21 and ZTFJ2252−05. No Si-II is visible
for ZTFJ1637+49, and ZTFJ0003+14.

There are a few rare elements present. In ZTFJ1637+49, two Zn-I absorption
lines can be seen (3345Å and 4810Å). In addition, there are two lines present
(7665/95Å) in ZTFJ1637+49 which are consistent with the strongest two K-I
lines. We do note that these are located close to telluric absorption lines.

##  7 Discussion

###  7.1 Search for eclipsing AM CVn binaries

####  7.1.1 Recovery efficiency and space density

Gaia has measured a parallax for all 7 eclipsing AM CVn systems, and the
distance is known with a precision of 5% to 30%. With a sample size of 5 (and
2 previously known eclipsing AM CVn systems), only an order of magnitude
estimate of the local space density is possible. However, there is a two order
of magnitude difference between the predicted AM CVn space density (Nelemans
et al., 2001; Kremer et al., 2017) and the measured space density by Carter et
al. (2013a), and it is therefore useful to check if the yield is consistent
with the measured space density to exclude the possibility of a previously
missed population. In addition, an estimate of the recovery efficiency is
useful to estimate how many systems ZTF and other surveys are set to discover
in the near future.

We have done a very thorough search of objects in the white dwarfs catalogue
by Gentile Fusillo et al. (2019), and limit the discussion to this sample. Of
the 7 known eclipsing AM CVn systems, only ZTFJ2252−05 is not in this
catalogue, and we therefore exclude it in this estimate. Both already known
eclipsing AM CVn systems, Gaia14aae and YZ Lmi, are part of this catalogue. We
recovered Gaia14aae in our search, but we initially did not recover YZ LMi
because its orbital period is just short of the initial cutoff of 28.8 minutes
(0.02 days). Therefore, we searched all lightcurves with one or more alerts
again, but now down to periods of 14.4 minutes (0.01 days) days and did
recover YZ LMi with the correct period, but no other systems were found.

A total of 241 775 objects in this sample have ZTF lightcurves with more than
80 epochs (49.6% of the total catalogue). To estimate the recovery efficiency
for each object, we assume that if there are 5–7 or more in-eclipse points in
the lightcurve, we are able to identify the source as an eclipsing AM CVn
system. This can be easily calculated as it is just a function of the total
number of epochs in the lightcurve and the eclipse duty cycle (see Appendix
A).

We estimate the eclipse duty-cycle for each system and calculate the average
probability to recover them from the 241 775 ZTF lightcurves. The average
recovery efficiencies are 33% for ZTFJ1637+49, and 60–75% for the other
systems (including Gaia14aae and YZ Lmi). This suggests that there are
$\approx$ 1–4 eclipsing systems we have not detected which do have ZTF
lightcurves and another 6–10 outside of the ZTF footprint.

In order to estimate the space density, we use the basic $\mathrm{1/V_{max}}$
method (Schmidt, 1968), with distances from Bailer-Jones et al. (2021) (see
Table 1). Since the AM CVn systems are very close, we do not correct for
expected changes in the space density and assume a uniform distribution. We
use the Gaia parallax and a magnitude limit of 20.5 to estimate the total
volume each of the systems could have been recovered. We correct this estimate
by the recovery efficiency we estimated earlier, a correction for the fraction
of systems not in the ZTF footprint, and a correction factor of 8 (the average
correction factor to account for eclipsing Roche-lobe filling binaries with
mass-ratios between $0.01<q<0.05$). This results in an estimated space density
of $\rho=6^{+6}_{-2}\times 10^{-7}$ $\mathrm{pc^{-3}}$. This consistent with
the measured space density by Carter et al. (2013a), $\rho=5\pm 3\times
10^{-7}$ $\mathrm{pc^{-3}}$ and again confirms the discrepancy with the
population synthesis predictions (e.g. Nelemans et al., 2001). With this
search, we exclude the possibility that this discrepancy is due to a large
hidden population of faint, low-accretion rate AM CVn systems.

####  7.1.2 Observational properties compared with the known population of AM
CVn systems

The currently known sample of AM CVn binaries has been found by various
searches based on static colours, outbursts, short periods variability, or by
large spectroscopic surveys. These search methods have different inherent
biases. Here, we briefly discuss the biases of each method, and compare them
to the search for AM CVn system using eclipses.

Searches for outbursting AM CVn (e.g. Levitan et al., 2015) will predominantly
find systems with periods between 22 and $\mathord{\sim}50$ minutes. A search
based on SDSS colours by Carter et al. (2013a) has mostly found AM CVn systems
with intermediate periods as well. These systems stand out because their
colour is a combination of a DB white dwarf and some contribution by a disk.
Shorter period systems (direct impact accretion systems) are easy to find
because of their high X-ray luminosity but these are intrinsically rare due to
the fast evolution. Large scale spectroscopic surveys allow for the discovery
of systems at all periods (e.g. Anderson et al., 2005). Therefore, most of the
long period ($\gtrsim 50$ min) systems have been discovered using SDSS
spectra.

A search for AM CVn systems by their eclipses has very different biases. It is
biased towards high inclination systems. Second, systems with deep eclipses
are easier to detect than shallow eclipses (assuming an efficient pipeline
that can handle ‘dropout’). Eclipses are more shallow for short period systems
where the disk or brightspot dominate the luminosity. However, short period
systems can show strong superhump periods which are close to the orbital
period. If outbursts and flickering dominate the lightcurve, the period is
more difficult to determine, which is an additional complicating factor.
Eclipsing long period systems are easy to find; the white dwarf dominates the
luminosity which means the eclipses are deep. In conclusion, using eclipses to
find AM CVn systems is biased to low accretion rate and therefore long orbital
periods. The eclipse probability does not decrease significantly with the
orbital period since the donor size increases with the orbital period.

(a) The Gaia observational H-R diagram (eDR3) with as background stars within
200 pc. (b) The SDSS $u-g$ versus $g-r$, colour-colour diagram of the white
dwarf catalog (Gentile Fusillo et al., 2019). The red line shows the colour-
selection used by Carter et al. (2013a).

Figure 11: The new eclipsing systems compared to known AM CVn systems in the
HR-diagram and colour-colour diagram. The newly discovered eclipsing systems
are shown as triangles and a diamond and the known AM CVn systems as dots. The
marker colour indicates the orbital period of the system. Note that we
corrected the SDSS measurements for ZTFJ1637+49 and ZTFJ0003+14 that were
obtained in-eclipse. In Section 7.1.2 we discuss why these systems were not
identified as AM CVn systems earlier and the implications for future searches
based on colours and parallaxes.

To determine why none of these eclipsing systems have been identified earlier
as AM CVn systems, we plot the detected eclipsing AM CVn systems and the known
sample in an HR diagram and a $u$-$g$–$g$-$r$ colour diagram (Figure 11).

ZTFJ1637+49 and ZTFJ0220+21 are located on the white dwarf track in the HR
diagram, and close to the DB track in colour-space. Both of these systems are
dominated by the DB white dwarf, and light emitted from the disk or brightspot
is almost negligible. Both these systems appear as typical DB white dwarfs.
ZTFJ0003+14 is very red and an outlier in both figures. This could be due to
reddening ($E_{g-r}=0.14$). In the HR-diagram, it is located slightly above
the white dwarf locus, which is expected given the small contribution from a
disk and the low temperature. ZTFJ2252−05 and ZTFJ0407−00 are located above
the white dwarf track in the HR diagram which suggests that the contribution
by the disk and the brightspot was small but not negligible when they were
observed by Gaia.

In conclusion, finding short period, non-eclipsing systems like ZTFJ2252−05
and ZTFJ0407−00 is easier because they stand out from the white dwarfs in the
HR diagram. In addition, they also show outbursts and flickering. However,
systems like ZTFJ1637+49, ZTFJ0003+14, ZTFJ0220+21 do not stand out in either
diagram and also do not show any outburst. If they are not eclipsing, the only
solution is to obtain spectra of all white dwarfs and identify them by the He
emission lines. Upcoming large spectroscopic surveys like SDSS-V (Kollmeier et
al., 2017), 4MOST (de Jong et al., 2012), and WEAVE (Dalton et al., 2012,
2014; Dalton, 2016), have programs to observe white dwarfs and will be able to
find long period AM CVn systems that are non-eclipsing.

We note that all 5 systems are in the SDSS footprint and have $ugriz$-colours,
although none of them were found in the search for AM CVn systems using SDSS-
colours by Carter et al. (2013a) and Carter et al. (2014). ZTFJ1637+49 was
observed during the eclipse which mostly affected the $r$-band measurement and
caused the system to be rejected as a candidate. ZTFJ0003+14 was in outburst
when SDSS observed it, and the colours fall outside of the selection criteria
(we used the Pan-STARRS $g$ and $r$ measurements in Figure 11). The primary
SDSS measurement of ZTFJ0407−00 also falls outside of the selection. However,
it has been observed 14 times by SDSS, and only 2 out of these 14 do not pass
the selection criteria. ZTFJ0220+21 and ZTFJ2252−05 both pass the selection
criteria, but no spectra were obtained; the spectroscopic completeness of
Carter et al. (2013a) is 70%. While the fact that none of these systems were
found by Carter et al. (2013a) initially suggested that they underestimated
their recovery efficiency, but there is a plausible explanation for why all of
theses systems were not discovered. We conclude that the space density
estimate by Carter et al. (2013a) is not significantly affected by any
systematic bias or uncertainty.

###  7.2 Outbursts and accretion state

The overall trend for AM CVn systems is that the average accretion rate
decreases with the orbital period. At orbital periods of 22 minutes and
longer, the disk is generally in a low state with outbursts semi-regularly.
Levitan et al. (2015) showed that there is an exponential correlation
($\tau\propto P_{\mathrm{orb}}^{7.35}$) between the orbital period and
outburst frequency, with a super-outburst frequency of one year at an orbital
period of 34 minutes. The behaviour of the systems we found show behaviour
consistent with this correlation, with a few detected outbursts for the two
short-period systems over the last decade, and no detected outbursts for the
longer period systems.

Because these systems are eclipsing we can determine the relative luminosity
of the disk and because the inclinations are similar also directly compare the
spectra. Lightcurve modelling shows that the relative contribution by the disk
is $\approx 10$–$30\%$ for the short period systems, just $\approx 5\%$ for
the 50 minute systems, and seems to be non-existent for the longest period
system (see Table 4). A useful aspect of our sample is that there are two pair
of systems with almost identical periods. A direct comparison between the
systems in each pair shows interesting differences. As can be seen in both the
lightcurves and the spectra (Figure 10), the contribution by the disk varies
significantly between the members of each pair. This is possibly due to the
recent outburst, which would empty the disk and heat the white dwarf, but
could also be due to the more compact nature of the donor. Continued
monitoring of the systems using high-speed photometry and high SNR lightcurves
to more precisely measure the mass and radius of the donor (Green et al.,
2018a) will enable us to resolve these issues.

For ZTFJ0407−00, we observe one outburst in the ZTF data, which showed
multiple short ’echo’ outbursts days to weeks after the main superoutburst,
also seen by Green et al. (2020), Duffy et al. (2021), and Pichardo Marcano et
al. (2021). Well sampled lightcurves of the systems also show a brighter
baseline level and up to 20 echo-outbursts after a superoutburst. In the case
of ZTFJ0407−00, we also observe a change in the eclipse depth; the eclipse is
total in the ZTF data (see Figure 1) but only 60% deep in the Chimera data.
These three observations suggest a change in the state of the system, possibly
an increased temperature of the white dwarf, a brighter disk, or both.

Because ZTFJ0407−00 is eclipsing, we can potentially measure the changes in
the systems more directly using the eclipses. By obtaining high-speed
photometry before, during, and after a superoutburst, we can measure changes
both in the white dwarf temperature as well as the size and brightness of the
disk (e.g. Copperwheat et al., 2011).

Figure 12: The mass and radius of donor stars in AM CVn systems compared to
model predictions, adapted from Green et al. (2018b). Stars indicate
measurements of eclipsing systems, the 5 new systems from this paper, YZ LMi
from Copperwheat et al. (2011), and Gaia14aae from Green et al. (2018a). Solid
lines show the white dwarf channel models (Deloye et al. 2007 in grey, Wong et
al. in prep in black), dashed lines show the He-star channel models (Yungelson
2008 in grey, Wong et al. in prep. in black), and dotted lines show the
evolved CV channel models (Goliasch & Nelson, 2015). The white dwarf and
helium star tracks include multiple levels of entropy, with the lower entropy
models at the bottom. The lowest line shows a zero-entropy donor. Models by
Wong et al. diverge; donors that are allowed to cool shrink at long orbital
periods, while donors that are not allowed to cool and lose entropy as they
lose mass stay large.

Besides the usual disk emission line, we note that a small but significant He-
II–4686Å central spike is detected. A high temperature is required to ionise
helium, which suggests that there is a hot, but small boundary layer in all
five systems. This features is also seen in other long period systems (e.g.
Kupfer et al., 2015) and speculate that this feature is present to some extent
in all AM CVn systems, but they can only be detected with high SNR spectra.

###  7.3 The properties of the donor stars

####  7.3.1 Chemical composition

The spectra show many different spectral lines, see Figure 10 and Table 5.
Modelling the spectra to determine elemental abundances is challenging and
beyond the scope of this paper. We limit this discussion to qualitative
estimates only.

Nelemans et al. (2010) showed that abundances can be used to determine the
evolutionary channel the donor formed through. First, the presence of any
hydrogen unambiguously points to an evolved main-sequence donor (e.g. Breedt
et al., 2014; Green et al., 2020). We do not detect any hydrogen in the
spectra of any of the systems. Since even a trace amount of hydrogen should be
visible in the spectra, we conclude that none of the systems are formed
through the ’evolved’ CV channel.

When the donors have experienced significant helium burning, as is the case
for helium star donors, the N abundance decreases to almost undetectable
levels. Detailed helium star models for the donor stars in AM CVn systems
predict typical surface abundance ratios N/C=0.001–1 while for white-dwarf
donor stars N/C$\approx 0.1$ if no CNO burning took place, or N/C$>>1$ if CNO
burning did take place.

We have detected N in both ZTFJ1637+49 and ZTFJ0220+21 and do not detect any
C. This is typical for many AM CVn systems and suggests that these systems are
formed through the He-WD model.

In ZTFJ0220+21 we also detect oxygen, which suggests a O/N$\approx 1$. The O/N
ratio in the WD model is set by the progenitor mass. Models by Nelemans et al.
(2010) predict a ratio of O/N of close to 1 for a 1 $\mathrm{M_{\sun}}$
progenitor mass, and a O/N ratio of 0.1 for a 2 $\mathrm{M_{\sun}}$ initial
mass. This suggests that ZTFJ1637+49 evolved from WD donor with a higher
initial mass, while ZTFJ0220+21 evolved from a lower initial mass WD donor.

For systems ZTFJ0003+14, ZTFJ2252−05, and ZTFJ0407−00, there is no clear
detection of N lines, which makes any inference of the formation channel
impossible.

Besides the CNO lines, we also observe a variety of metal absorption lines.
The line strengths depend on the element in question and a combination of
sedimentation timescales Koester (2009), temperature, surface gravity,
accretion rate, and abundances of the accreted material.

The expectation is that the abundances of metals are primordial. However,
sedimentation and mixing during the donor star lifetime affect the current
abundance of metals. We have measured the temperature and surface gravity of
the white dwarf, and by modelling the spectra using atmosphere models, we can
in principle constrain the abundance of the metals but this is beyond the
scope of this paper. This can shed light on both the primordial abundance in
these systems as well as constrain the amount of sedimentation and/or mixing
that took place during the formation of these systems.

(a) The absolute $g$-band magnitude versus the $g$-$r$ colour of the white
dwarf in eclipsing AM CVn systems. The measurements have been de-reddened
based on the $g-r$ reddening (see Table 1), indicated with lines with the same
colour as the markers. Model temperature are indicated below the model in kK.
(b) The temperature and mass of the white dwarf in eclipsing AM CVn systems
versus the orbital period. The horizontal lines in the bottom panels show the
average mass and the one standard deviation range.

Figure 13: Properties of the accreting white dwarfs in eclipsing AM CVn
systems. Models for a 0.65 $\mathrm{M_{\sun}}$ (dashed line) and 0.85
$\mathrm{M_{\sun}}$ (solid line) white dwarf and are from Wong et al. in prep.
Values for YZ LMi and Gaia14aae have been taken from Copperwheat et al. (2011)
and Green et al. (2018a) and Green et al. (2019b).

####  7.3.2 Mass and radius

Figure 12 shows the mass and radius of the donors in eclipsing AM CVn systems
and four AM CVn systems for which the mass-ratio has been measured using
spectroscopy (Green et al., 2018a). Because the donor is filling its Roche-
lobe, the density of the donor depends only on the orbital period, indicated
by diagonal lines in the figure (Faulkner et al., 1972; Green et al., 2020).
Donors lose mass as the orbital period of AM CVn systems increase, and evolve
from right to left in this diagram. The WD and He-star donors are degenerate,
and they increase in size as they lose mass.

The donor-models for each formation channel predict a different entropy, and
therefore a different mass and radius for a given period. In the figure, we
show tracks for evolved CVs (Goliasch & Nelson, 2015), the white dwarf track
Deloye et al. (2007), and tracks for the He-star channel from (Yungelson,
2008). We also show WD-tracks and He-star tracks by Wong et al. in prep. The
tracks by Wong et al. diverge; the models that curve downwards include
cooling, the tracks that continue on to the top-left do not include cooling.

First of all, the measurements indicate that masses are 1.5–2.5 times more
massive than a zero-entropy donor for the same orbital period. This is similar
to what has been measured for YZ LMi and Gaia14aae.

The short-period systems seem to be consistent with either white dwarf models
or He-star models. However, current WD and He-star models predict that the
donors cool and shrink at orbital periods of 30-40 minutes, which is not
consistent with the observations.

The long period systems seem to be very large, similar to Gaia14aae. As noted
by Green et al. (2018a), the measurements are consistent with CV-channel
donors, but this channel is excluded by the lack of hydrogen in the spectra
(Nelemans et al., 2010). In addition, this channel is also predicted to be the
least common, although more recent work by Liu et al. (2021) suggest this
might not be the case.

The alternative solution is that AM CVn systems do evolve through the WD
and/or He-star channel but for some reason do not cool and shrink at long
orbital periods (indicated by the models by Wong et al.). To answer this
question, the donor properties of the new ZTF systems need to be measured with
similar accuracy as for Gaia14aae. In addition, precise characterization of
donors of to-be discovered short period eclipsing systems ($\lesssim 25$ min)
are needed to constrain the models before adiabatic cooling is predicted to
become important. In addition, models need to be updated to make sure any
source of potential heating of the donor is taken into account.

###  7.4 Properties of the white dwarf

Figure 13 shows the properties of the white dwarf in eclipsing AM CVn systems.
We first compare the absolute magnitude versus the colour of the white dwarf
with model values. We calculate these values by using the Pan-STARRS
measurement, Gaia eDR3 distances, the estimated white dwarf contribution from
table 4, and the reddening towards the system (see table 1). A general
prediction is that the effective temperature of white dwarfs in AM CVn systems
are determined by the average accretion rate and therefore correlated with the
orbital period (Bildsten et al., 2006). Our measurements generally agree with
the model values calculated by Wong et al. (in prep.). Although the errorbars
are too large to estimate the mass, the general trend is that the long period
systems are massive $\mathord{\sim}0.85$ $\mathrm{M_{\sun}}$, while the
shorter period systems seem more consistent with lower masses
$\mathord{\sim}0.65$ $\mathrm{M_{\sun}}$.

We have measured the temperature from the SEDs and the mass using the
eclipses. As we have already seen from the colour of the white dwarf, long
period systems are colder while short period systems contain a hotter white
dwarf However, the temperature for the longer period systems seem
systematically larger than models predict at those particular periods.

This could be due to a larger than expected donor entropy which results in a
higher mass-accretion rate and therefore higher temperature. However, it could
also be due to an additional source of energy in the white dwarf that causes
the white dwarf to remain hot for a longer time (e.g. Bauer et al., 2020). It
is possible that the temperature estimates of the short period systems are
systematically underestimated, possibly by imperfect corrections to the
overall luminosity. Alternatively, metals in the white dwarfs atmosphere can
significantly suppress the UV-luminosity, which results in an underestimate of
the temperature. Since the SED of hot white dwarfs peaks in the UV, short-
period systems would be more affected.

By modelling the eclipse lightcurves and assuming a white dwarf M-R relation
we have also determined the mass of the white dwarfs. These are similar to the
white dwarf masses in the other eclipsing AM CVn systems (Gaia14aae and YZ
Lmi). The unweighted average and variance of the mass is $0.81\pm 0.05$
$\mathrm{M_{\sun}}$. This is more massive than the typical isolated white
dwarfs (Kepler et al., 2007), and similar to white dwarfs in hydrogen-rich CVs
which have an average mass of $\approx 0.83$ $\mathrm{M_{\sun}}$ (Littlefair
et al., 2008; Savoury et al., 2011; Pala & Gänsicke, 2017).

However, the explanation for this is different. For hydrogen CVs, angular
momentum loss causes systems with low-mass white dwarfs to merge during the
early stages of the CV phase (Schreiber et al., 2016; Nelemans et al., 2016;
Zorotovic & Schreiber, 2020). In the case of AM CVn systems, accretion in
systems with low-mass white dwarf is unstable and causes a merger during the
common-envelope (e.g. Shen, 2015).

Kilic et al. (2016) show that the number of known white dwarfs with $M\gtrsim
0.8$ $\mathrm{M_{\sun}}$ and low mass white dwarf companions (WD-channel) is
sufficient to explain the AM CVn birth rate. However, the low-mass white
dwarfs have a low entropy which would result in cool and small donors (Deloye
et al., 2007), inconsistent with our results (Figure 12).

##  8 Conclusion and summary

We searched the ZTF lightcurves for deep eclipsing white dwarfs and identified
five new eclipsing AM CVn systems with periods ranging from 35 to 62 minutes.
Using this sample, we estimated the local space density which is consistent
with previous observational estimates. This is again two order of magnitude
less than model predictions and excludes the possibility that this is due to a
hidden population of faint, long period AM CVn systems.

We obtained phase-resolved spectra and high cadence lightcurves to
characterize the systems. The high SNR averaged spectra of the longer period
systems revealed many broad metal absorption lines, including potassium and
zinc which have not previously been detected in any other AM CVn systems.
Doppler maps show the presence of a ’central spike’ for the longest period
system.

We modelled the high-candence lightcurves and the spectral energy
distributions and measured the binary parameters (masses, radii, inclination,
and white dwarf temperature) for all five systems. First, we showed that the
effective accreting white dwarf temperatures of long-period systems are higher
than models predict. This suggest a delay in cooling of the white dwarf.
Second, the average accreting white dwarf mass is $\approx 0.8$
$\mathrm{M_{\sun}}$, more massive than typical single white dwarfs. This is
suggests that AM CVn systems can only be formed with massive accreting white
dwarfs. Third, the donor stars have a high entropy, and are a factor of
1.5-2.5 times more massive than a zero-temperature donor at the same orbital
period. The high observed entropy (radius) is consistent with white dwarf or
He-star models for the two short period systems. The long period systems also
have a large entropy (radius), while both WD and He models predict that AM CVn
donor should decrease in entropy (radius) as they evolve to periods longer
than $\approx 40$ min. More accurate donor mass-radius measurements and more
complete models are needed to resolve this inconsistency.

##  9 Future work

We will continue to obtain data on the systems presented in this paper to
fully utilize their potential. We aim to obtain high SNR, high-cadence
lightcurves in order to measure the donor mass and radius to a precision of
$\mathord{\sim}$1% (similar to that of Gaia14aae). Continued monitoring of
these systems with high-speed cameras will also allow us to measure changes in
the eclipse arrival time.

As estimated in this paper, we expect there to be another hand-full of
eclipsing AM CVn systems brighter than $\approx\\!$20 mag. We will continue to
search for new ZTF data for more eclipsing systems. Other survey telescope
lightcurves (ATLAS, Gaia, BlackGEM) can also be used to find eclipsing AM CVn
systems. This does require the ability to reliably detect deep eclipses,
either by implementing image differencing or by using forced photometry.

The Vera C. Rubin observatory (Ivezić et al., 2019) is starting observations
in the near future. With a single epoch limiting magnitude of $r\approx 24$,
it will observe thousands of AM CVn systems. However, spectroscopic follow-up
of this faint population is expensive, and identifying AM CVn systems by
measuring the period eclipses can be done with just Vera C. Rubin observatory
data. This will limit the identification to eclipsing AM CVn systems only, but
there should be tens to a hundred of these in the Vera C. Rubin observatory
footprint. Such a large sample of eclipsing AM CVn systems will provide a much
clearer picture of the donor properties and with that the formation channels
of AM CVn systems. We have shown that at least 5 in-eclipse measurements are
needed to identify a period, which implies that a few hundred epochs will be
needed. Vera C. Rubin observatory cadence will be sparse compared to ZTF and
the hundreds of epochs required might be spread over $>5$ years which does
make it more challenging to identify the period using eclipses.

## Acknowledgements

JvR is partially supported by NASA-LISA grant 80NSSC19K0325. MJG was supported
by the European Research Council (ERC) under the European Union’s FP7
Programme, Grant No. 833031 (PI: Dan Maoz). V.S.D and HiPERCAM are supported
by STFC.

This research benefited from interactions at the ZTF Theory Network Meeting
that were funded by the Gordon and Betty Moore Foundation through Grant
GBMF5076 and support from the National Science Foundation through PHY-1748958.

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

Some of the data presented herein were obtained at the W.M. Keck Observatory,
which is operated as a scientific partnership among the California Institute
of Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W.M. Keck Foundation. The authors wish to recognize
and acknowledge the very significant cultural role and reverence that the
summit of Mauna Kea has always had within the indigenous Hawaiian community.
We are most fortunate to have the opportunity to conduct observations from
this mountain.

The research leading to these results has received funding from the European
Research Council under the European Union’s Horizon 2020 research and
innovation programme numbers 677706 (WD3D) and 340040 (HiPERCAM).

Based on observations made with the Gran Telescopio Canarias (GTC) installed
in the Spanish Observatorio del Roque de los Muchachos of the Instituto de
Astrofísica de Canarias, in the island of La Palma.

Based on observations obtained at the international Gemini Observatory, a
program of NSF’s NOIRLab, which is managed by the Association of Universities
for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation on behalf of the Gemini Observatory partnership:
the National Science Foundation (United States), National Research Council
(Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio
de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência,
Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space
Science Institute (Republic of Korea).

This research made use of matplotlib, a Python library for publication quality
graphics (Hunter, 2007); NumPy (Harris et al., 2020); Astroquery (Ginsburg et
al., 2019); Astropy, a community-developed core Python package for Astronomy
(Astropy Collaboration et al., 2018, 2013); and the VizieR catalogue access
tool, CDS, Strasbourg, France.

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. Funding for the Sloan
Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the
U.S. Department of Energy Office of Science, and the Participating
Institutions.

SDSS-IV acknowledges support and resources from the Center for High-
Performance Computing at the University of Utah. The SDSS web site is
www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for
the Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science, Carnegie
Mellon University, the Chilean Participation Group, the French Participation
Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica
de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley
National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-
Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für
Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik
(MPE), National Astronomical Observatories of China, New Mexico State
University, New York University, University of Notre Dame, Observatário
Nacional / MCTI, The Ohio State University, Pennsylvania State University,
Shanghai Astronomical Observatory, United Kingdom Participation Group,
Universidad Nacional Autónoma de México, University of Arizona, University of
Colorado Boulder, University of Oxford, University of Portsmouth, University
of Utah, University of Virginia, University of Washington, University of
Wisconsin, Vanderbilt University, and Yale University. This publication makes
use of data products from the Wide-field Infrared Survey Explorer (Wright et
al., 2010), which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration. The Pan-STARRS1
Surveys (PS1) have been made possible through contributions of the Institute
for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the
Max-Planck Society and its participating institutes, the Max Planck Institute
for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial
Physics, Garching, The Johns Hopkins University, Durham University, the
University of Edinburgh, Queen’s University Belfast, the Harvard-Smithsonian
Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation under Grant No.
AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE).

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##  Appendix A ZTF detection efficiency

We determined the detection efficiency for ZTFJ1637+49 as a function of the
number of epochs, see Figure 14. We did this by randomly removing a set
fraction of points from the lightcurve, and tested if the BLS algorithm
recovered the period. We model the curve with the function:

$P(k,p)=1-\sum_{0}^{k}\binom{N}{k}p^{k}(1-p)^{N-k}$ (1)

with $k$ the number of in-eclipse points and $p$ the effective eclipse duty-
cycle. For ZTF data of ZTFJ1637+49, the values of the best fit are $k=5$ and
$p=0.0127$. This shows that the simple approximation is a good description of
the actual recovery efficiency, and 5 in-eclipse points are enough to recover
the orbital period.

Figure 14: The period recovery efficiency as function of number of epochs. We
used the ZTF lightcurve of ZTFJ1637+49. We randomly removed a fixed fraction
of epochs and tested if our search algorithm recovered the known period. We
fitted the curve with a simple probabilistic model (equation 1), and is a good
approximation of the data. We use the model to rapidly evaluate in what
fraction of the lightcurve an eclipse would be recovered.

##  Appendix B Follow-up observations

Table 6 shows an overview of all spectroscopic and photometric followup
observations.

Table 6: Summary of the followup observations Object | Date | UT | Tele./Inst. |  Nexp | Exp. time (s) | Wavelength
---|---|---|---|---|---|---
Photometry |
ZTFJ1637+49 | 2019-05-29 | 06:37 - 08:09 | P200/CHIMERA | 1000 |  03.0 | $g$
ZTFJ1637+49 | 2019-05-29 | 06:37 - 08:09 | P200/CHIMERA | 1200 |  03.0 | $r$
ZTFJ1637+49 | 2019-05-30 | 04:03 - 05:52 | P200/CHIMERA | 1300 |  03.0 | $g$
ZTFJ1637+49 | 2019-05-30 | 04:03 - 05:52 | P200/CHIMERA | 1300 |  05.0 | $r$
ZTFJ1637+49 | 2019-06-29 | 04:27 - 11:30 | P200/CHIMERA | 4300 |  03.0 | $g$
ZTFJ1637+49 | 2019-06-29 | 04:27 - 11:30 | P200/CHIMERA | 4300 |  03.0 | $r$
ZTFJ1637+49 | 2019-09-06 | 20:42 - 21:54 | GTC/HiPERCAM | 335 | 12.9 | $u$
ZTFJ1637+49 | 2019-09-06 | 20:42 - 21:54 | GTC/HiPERCAM | 1340 |  03.2 | $g$
ZTFJ1637+49 | 2019-09-06 | 20:42 - 21:54 | GTC/HiPERCAM | 1340 |  03.2 | $r$
ZTFJ1637+49 | 2019-09-06 | 20:42 - 21:54 | GTC/HiPERCAM | 670 |  06.4 | $i$
ZTFJ1637+49 | 2019-09-06 | 20:42 - 21:54 | GTC/HiPERCAM | 335 | 12.9 | $z$
ZTFJ1637+49 | 2020-01-24 | 11:17 - 11:35 | P200/CHIMERA | 350 |  03.0 | $g$
ZTFJ1637+49 | 2020-01-24 | 11:17 - 11:35 | P200/CHIMERA | 175 |  06.0 | $r$
ZTFJ0220+21 | 2020-01-23 | 02:15 - 03:41 | P200/CHIMERA | 1700 |  03.0 | $g$
ZTFJ0220+21 | 2020-01-23 | 02:15 - 03:41 | P200/CHIMERA | 1600 |  03.0 | $r$
ZTFJ0220+21 | 2020-02-16 | 06:17 - 06:33 | Gemini/Alopeke | 320 |  03.0 | $g$
ZTFJ0220+21 | 2020-02-16 | 06:17 - 06:33 | Gemini/Alopeke | 320 |  03.0 | $i$
ZTFJ0220+21 | 2020-02-17 | 05:25 - 05:41 | Gemini/Alopeke | 320 |  03.0 | $g$
ZTFJ0220+21 | 2020-02-17 | 05:25 - 05:41 | Gemini/Alopeke | 320 |  03.0 | $i$
ZTFJ0220+21 | 2020-02-17 | 06:21 - 06:37 | Gemini/Alopeke | 320 |  03.0 | $g$
ZTFJ0220+21 | 2020-02-17 | 06:21 - 06:37 | Gemini/Alopeke | 320 |  03.0 | $i$
ZTFJ0220+21 | 2020-02-18 | 06:21 - 06:37 | Gemini/Alopeke | 320 |  03.0 | $g$
ZTFJ0220+21 | 2020-02-18 | 06:21 - 06:37 | Gemini/Alopeke | 320 |  03.0 | $i$
ZTFJ0220+21 | 2020-02-18 | 05:29 - 05:37 | Gemini/Alopeke | 320 |  03.0 | $g$
ZTFJ0220+21 | 2020-02-18 | 05:29 - 05:37 | Gemini/Alopeke | 320 |  03.0 | $i$
ZTFJ2252−05 | 2020-10-17 | 02:12 - 06:58 | P200/CHIMERA | 3400 |  05.0 | $g$
ZTFJ2252−05 | 2020-10-17 | 02:12 - 06:58 | P200/CHIMERA | 3200 |  05.0 | $i$
ZTFJ2252−05 | 2020-10-21 | 02:52 - 08:00 | P200/CHIMERA | 2300 |  05.0 | $g$
ZTFJ2252−05 | 2020-10-21 | 02:49 - 08:00 | P200/CHIMERA | 2350 |  05.0 | $r$
ZTFJ0407−00 | 2020-01-23 | 03:49 - 06:07 | P200/CHIMERA | 2726 |  03.0 | $g$
ZTFJ0407−00 | 2020-01-23 | 03:49 - 06:07 | P200/CHIMERA | 2720 |  03.0 | $r$
ZTFJ0407−00 | 2020-01-24 | 01:59 - 03:14 | P200/CHIMERA | 1500 |  03.0 | $g$
ZTFJ0407−00 | 2020-01-24 | 01:59 - 03:14 | P200/CHIMERA | 1500 |  03.0 | $r$
ZTFJ0407−00 | 2020-08-20 | 10:48 - 12:26 | P200/CHIMERA | 1900 |  03.0 | $g$
ZTFJ0407−00 | 2020-08-20 | 10:49 - 12:29 | P200/CHIMERA | 1950 |  03.0 | $r$
ZTFJ0407−00 | 2020-08-27 | 11:29 - 11:44 | P200/CHIMERA | 150 |  06.0 | $g$
ZTFJ0407−00 | 2020-08-27 | 11:29 - 11:44 | P200/CHIMERA | 150 |  06.0 | $r$
ZTFJ0407−00 | 2020-10-17 | 02:12 - 06:58 | P200/CHIMERA | 1500 |  05.0 | $g$
ZTFJ0407−00 | 2020-10-17 | 02:12 - 06:58 | P200/CHIMERA | 1500 |  05.0 | $i$
ZTFJ0407−00 | 2020-10-21 | 02:52 - 08:00 | P200/CHIMERA | 1000 |  05.0 | $g$
ZTFJ0407−00 | 2020-10-21 | 02:49 - 08:00 | P200/CHIMERA | 1000 |  05.0 | $r$
ZTFJ0407−00 | 2020-11-17 | 05:50 - 06:32 | P200/CHIMERA | 500 |  05.0 | $g$
ZTFJ0407−00 | 2020-11-17 | 05:50 - 06:36 | P200/CHIMERA | 550 |  05.0 | $r$
ZTFJ0003+14 | 2020-12-15 | 02:01 - 02:18 | P200/CHIMERA | 100 | 10.0 | $g$
ZTFJ0003+14 | 2020-12-15 | 02:01 - 02:18 | P200/CHIMERA | 100 | 10.0 | $i$
ZTFJ0003+14 | 2020-12-16 | 03:50 - 06:13 | P200/CHIMERA | 1650 |  05.0 | $g$
ZTFJ0003+14 | 2020-12-16 | 05:51 - 06:13 | P200/CHIMERA | 800 | 10.0 | $r$
Spectroscopy |
ZTFJ1637+49 | 2019-05-07 | 11:00 - 12:45 | Keck/LRIS/GR600 | 8 | 600 | 3200 - 5600Å
ZTFJ1637+49 | 2019-05-07 | 11:00 - 12:45 | Keck/LRIS/R600 | 8 | 600 | 5600 - 8700Å
ZTFJ1637+49 | 2019-07-05 | 06:09 - 10:22 | Keck/LRIS/GR600 | 47 | 300 | 3200 - 5600Å
ZTFJ1637+49 | 2019-07-05 | 06:09 - 10:22 | Keck/LRIS/R600 | 41 | 300 | 5600 - 8700Å
ZTFJ0220+21 | 2019-12-03 | 10:33 - 10:43 | Keck/LRIS/GR600 | 1 | 600 | 3200 - 5600Å
ZTFJ0220+21 | 2019-12-03 | 10:33 - 10:43 | Keck/LRIS/R600 | 1 | 600 | 5600 - 8700Å
ZTFJ0220+21 | 2020-01-25 | 05:22 - 07:30 | Keck/LRIS/GR600 | 20 | 300 | 3200 - 5600Å
ZTFJ0220+21 | 2020-01-25 | 05:22 - 07:30 | Keck/LRIS/R600 | 23 | 300 | 5600 - 8700Å
ZTFJ0407−00 | 2020-01-25 | 07:47 - 09:17 | Keck/LRIS/GR600 | 20 | 200 | 3200 - 5600Å
ZTFJ0407−00 | 2020-01-25 | 07:47 - 09:17 | Keck/LRIS/R600 | 23 | 200 | 5600 - 8700Å
ZTFJ2252−05 | 2020-09-16 | 12:24 - 13:10 | Keck/LRIS/GR600 | 11 | 220 | 3200 - 5600Å
ZTFJ2252−05 | 2020-09-16 | 12:25 - 13:10 | Keck/LRIS/R600 | 8 | 220 | 5600 - 8700Å
ZTFJ0003+14 | 2020-12-10 | 08:19 - 08:24 | Keck/LRIS/GR600 | 1 | 300 | 3200 - 5600Å
ZTFJ0003+14 | 2020-12-10 | 08:19 - 08:24 | Keck/LRIS/R300 | 1 | 300 | 5600 - 8700Å
ZTFJ0003+14 | 2020-12-17 | 07:29 - 08:36 | Keck/LRIS/GR600 | 12 | 300 | 3200 - 5600Å
ZTFJ0003+14 | 2020-12-17 | 07:29 - 08:36 | Keck/LRIS/R600 | 8 | 420 | 5600 - 8700Å

##  Appendix C Lightcurve and binary parameters

Tables 7–11 show the lightcurve model parameters and the derived binary
parameter for each object and each band.

| $u$ | $g$ | $r$ | $i$ | $z$
---|---|---|---|---|---
$q$ | $0.026^{+0.010}_{-0.009}$ | $0.025^{+0.008}_{-0.006}$ | $0.024^{+0.012}_{-0.009}$ | $0.025^{+0.012}_{-0.010}$ | $0.026^{+0.013}_{-0.011}$
$i$ (∘) | $82.7^{+0.9}_{-0.8}$ | $82.8^{+0.6}_{-0.6}$ | $82.9^{+1.0}_{-1.0}$ | $82.8^{+1.1}_{-1.0}$ | $82.7^{+1.1}_{-1.0}$
$r_{1}$ | $0.020^{+0.003}_{-0.002}$ | $0.019^{+0.001}_{-0.002}$ | $0.020^{+0.003}_{-0.002}$ | $0.019^{+0.003}_{-0.002}$ | $0.021^{+0.004}_{-0.003}$
velocity scale ($\mathrm{kms^{-1}}$) | $583^{+16}_{-20}$ | $593^{+13}_{-13}$ | $585^{+18}_{-20}$ | $588^{+18}_{-24}$ | $575^{+23}_{-27}$
$t_{0}$ (s) | $-1.4^{+0.6}_{-0.7}$ | $-0.5^{+0.2}_{-0.2}$ | $-0.7^{+0.2}_{-0.2}$ | $-1.2^{+0.3}_{-0.4}$ | $-0.8^{+0.8}_{-0.8}$
$a$ ($\mathrm{R_{\sun}}$) | $0.492^{0.014}_{-0.017}$ | $0.501^{0.010}_{-0.011}$ | $0.494^{0.015}_{-0.017}$ | $0.496^{0.015}_{-0.019}$ | $0.485^{0.020}_{-0.022}$
$M_{1}$ ($\mathrm{M_{\sun}}$) | $0.85^{+0.07}_{-0.08}$ | $0.90^{+0.05}_{-0.05}$ | $0.87^{+0.07}_{-0.08}$ | $0.88^{+0.07}_{-0.09}$ | $0.82^{+0.09}_{-0.10}$
$M_{2}$ ($\mathrm{M_{\sun}}$) | $0.022^{+0.011}_{-0.009}$ | $0.023^{+0.009}_{-0.006}$ | $0.021^{+0.013}_{-0.009}$ | $0.022^{+0.013}_{-0.010}$ | $0.021^{+0.014}_{-0.011}$