# Orbital decay in an accreting and eclipsing 13.7 minute orbital period
binary with a luminous donor

Kevin B. Burdge Department of Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Kareem El-
Badry Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  Harvard Society of Fellows, 78 Mount Auburn Street,
Cambridge, MA 02138  Max-Planck Institute for Astronomy, Königstuhl 17,
D-69117 Heidelberg, Germany  Saul Rappaport Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Kavli
Institute for Astrophysics and Space Research, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Tin Long Sunny Wong Department of
Physics, University of California, Santa Barbara, CA 93106, USA  Evan B. Bauer
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  Lars Bildsten Department of Physics, University of
California, Santa Barbara, CA 93106, USA  Kavli Institute for Theoretical
Physics, University of California, Santa Barbara, CA 93106, USA  Ilaria
Caiazzo Division of Physics, Mathematics and Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA  Deepto Chakrabarty Department of
Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute
of Technology, Cambridge, MA 02139, USA  Emma Chickles Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Kavli
Institute for Astrophysics and Space Research, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Matthew J. Graham Division of Physics,
Mathematics and Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  Erin Kara Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Kavli Institute for Astrophysics and
Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  S. R. Kulkarni Division of Physics, Mathematics and Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  Thomas R. Marsh Department
of Physics, University of Warwick, Coventry CV4 7AL, UK  Melania Nynka Kavli
Institute for Astrophysics and Space Research, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Thomas A. Prince Division of Physics,
Mathematics and Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  Robert A. Simcoe Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Kavli Institute for Astrophysics and
Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  Jan van Roestel Anton Pannekoek Institute for Astronomy, University of
Amsterdam, 1090 GE Amsterdam, NL  Division of Physics, Mathematics and
Astronomy, California Institute of Technology, Pasadena, CA 91125, USA  Zach
Vanderbosch Division of Physics, Mathematics and Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  Eric C. Bellm DIRAC
Institute, Department of Astronomy, University of Washington, 3910 15th Avenue
NE, Seattle, WA 98195, USA  Richard G. Dekany Caltech Optical Observatories,
California Institute of Technology, Pasadena, CA, USA  Andrew J. Drake
Division of Physics, Mathematics and Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA  George Helou IPAC, California Institute
of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA  Frank J.
Masci IPAC, California Institute of Technology, 1200 E. California Blvd,
Pasadena, CA 91125, USA  Jennifer Milburn  Caltech Optical Observatories,
California Institute of Technology, Pasadena, CA, USA  Reed Riddle Caltech
Optical Observatories, California Institute of Technology, Pasadena, CA, USA
Ben Rusholme IPAC, California Institute of Technology, 1200 E. California
Blvd, Pasadena, CA 91125, USA  Roger Smith Caltech Optical Observatories,
California Institute of Technology, Pasadena, CA, USA

###### Abstract

We report the discovery of ZTF J0127+5258, a compact mass-transferring binary
with an orbital period of 13.7 minutes. The system contains a white dwarf
accretor, which likely originated as a post-common envelope carbon-oxygen (CO)
white dwarf, and a warm donor ($T_{\rm eff,\,donor}=16,400\pm 1000\,\rm K$).
The donor probably formed during a common envelope phase between the CO white
dwarf and an evolving giant which left behind a helium star or helium white
dwarf in a close orbit with the CO white dwarf. We measure gravitational wave-
driven orbital inspiral with $\sim 35\sigma$ significance, which yields a
joint constraint on the component masses and mass transfer rate. While the
accretion disk in the system is dominated by ionized helium emission, the
donor exhibits a mixture of hydrogen and helium absorption lines. Phase-
resolved spectroscopy yields a donor radial-velocity semi-amplitude of $771\pm
27\,\rm km\,s^{-1}$, and high-speed photometry reveals that the system is
eclipsing. We detect a Chandra X-ray counterpart with $L_{X}\sim 3\times
10^{31}\,\rm erg\,s^{-1}$. Depending on the mass-transfer rate, the system
will likely evolve into either a stably mass-transferring helium CV, merge to
become an R Crb star, or explode as a Type Ia supernova in the next million
years. We predict that the Laser Space Interferometer Antenna (LISA) will
detect the source with a signal-to-noise ratio of $24\pm 6$ after 4 years of
observations. The system is the first _LISA_ -loud mass-transferring binary
with an intrinsically luminous donor, a class of sources that provide the
opportunity to leverage the synergy between optical and infrared time domain
surveys, X-ray facilities, and gravitational-wave observatories to probe
general relativity, accretion physics, and binary evolution.

Gravitational waves (687) — Semi-detached binary stars (1443)

††journal: ApJL††facilities: PO:1.2m (ZTF), Hale (CHIMERA), Keck:I (LRIS), CXO
(ACIS), Swift(XRT and UVOT)

##  1 Introduction

Our understanding of how the shortest orbital period binary systems in the
Galaxy form is undergoing a renaissance due to discoveries facilitated by
optical time domain surveys such as the Zwicky Transient Facility (ZTF) (Masci
et al., 2019; Graham et al., 2019; Dekany et al., 2020; Bellm et al., 2019).
We have known for decades that there exists a substantial population of
helium-rich cataclysmic variables (helium CVs, also known as AM CVns) with
orbital periods under the canonical period minimum of 80 minutes of hydrogen-
rich cataclysmic variables. About a hundred of these systems are currently
known (Ramsay et al., 2018), and they have a few defining characteristics:

  * •

Systems with orbital periods below about 30 minutes exhibit high-state
accretion disks (Duffy et al., 2021), which consistently maintain a high rate
of mass transfer, whereas systems with longer orbital periods (30-70 minutes)
have quiescent phases, and undergo irregular outbursts with a recurrence
timescale that lengthens at longer orbital periods (Levitan et al., 2015).
Until recently, these longer-period systems, which are most easily detected
via outbursts, have dominated the known population of ultracompact binaries.

  * •

The donors in these systems remain largely undetected, except when they
transit the accreting white dwarf (van Roestel et al., 2022). This is because
the donors are quite cool, likely because they have undergone adiabatic
expansion as a result of mass loss after reaching a minimum orbital period.
Thus, it is expected that most of the known population of helium CVs, which
host cold donors, are post-period-minimum objects.

There are three main proposed progenitor channels for helium CVs: the
interaction of double white dwarfs (Marsh et al., 2004), the interaction of a
white dwarf accretor with a helium-burning star (Kupfer et al., 2020a), and
transitional cataclysmic variables, in which a white dwarf interacts with an
evolved main-sequence donor. The discovery of ZTF J1813+4251, a 51-minute
orbital period transitional CV with an F-type donor (Burdge et al., 2022a)
cemented the role of the transitional CV channel in producing at least some of
these systems. Several mass-transferring white dwarf plus helium star pairs
have been discovered in recent years (Burdge et al., 2020a; Kupfer et al.,
2020a, b, 2022), though it is unclear whether these systems will evolve into
the post-period-minimum helium CVs we see, or undergo a thermonuclear
supernova well before reaching such orbital periods. Finally, we know of many
detached short-orbital-period double-white-dwarf systems, the shortest of
which are the eclipsing binaries ZTF J1539+5027 (Burdge et al., 2019a) and ZTF
J2243+5242 (Burdge et al., 2020b) (with periods of $\sim 6.9$ and $\sim 8.8$
minutes, respectively), and at least two interacting white dwarf systems which
are undergoing direct impact accretion (Marsh et al., 2004), HM Cancri (Ramsay
et al., 2002) and V407 Vul (Marsh & Steeghs, 2002) (with orbital periods of
$\sim 5.4$ and $\sim 9.5$ minutes, respectively). HM Cancri hosts a carbon-
oxygen core (CO) white dwarf accretor, with a likely helium core white dwarf
donor, and V407 Vul likely consists of the same configuration, though it is
difficult to study the characteristics of these systems because HM Cancri’s
flux is dominated by the accretor (with some hint of reprocessed radiation
from the donor), while V407 Vul’s optical flux output is dwarfed by a
spatially coincident G type main sequence star (it is unclear whether it is
associated with the ultracompact binary). Both systems exhibit a negative
period derivative, indicating that they are likely pre-period minimum;
however, recent work has suggested that HM Cancri is exhibiting a slowing in
its orbital decay rate, and may reach a period minimum in the near future
(Strohmayer, 2021; Munday et al., 2023). There is ongoing debate whether these
examples of double white dwarfs indeed contribute to the population of post-
period minimum stably mass transferring helium CVs which have been observed
over the last few decades, as Shen (2015) suggest that even systems with low
mass ratios in which mass-transfer would normally be stable are driven towards
merger due to nova shells expelled by the accretor interacting with the donor.
Additionally, Brown et al. (2016) found that observationally, the space
density of close CO+He core WD binaries exceeds the population of stably mass-
transferring helium CVs by a factor of 40, suggesting that the vast majority
of double white dwarfs merge, or that we are significantly underestimating the
space density of helium CVs, most of which have been selected on the basis of
outbursting behavior.

Here, we report the discovery and characterization of ZTF J0127+5258, an
accreting binary system with an orbital period of $13.7$ minutes. Remarkably,
the system hosts a luminous donor which is clearly visible in the optical,
contributing comparable luminosity to the high state accretion disk
surrounding its white dwarf companion. The donor exhibits primarily helium
lines, but still manifests hydrogen absorption as well, an atmosphere very
similar to the pre-mass transfer 20-minute orbital period ellipsoidal variable
PTF J0533+0209 (Burdge et al., 2019b). We detect the presence of an accretion
disk, which manifests itself both in the form of prominent He II emission in
the optical spectrum, as well as in a broad triangular-shaped eclipse of the
warm donor star. Using archival photometric observations, we detect that the
system is undergoing rapid orbital decay, suggesting that it has not yet
reached a period minimum. Because of the elevated temperature of its donor,
the system likely did not originate from the continuously mass-transferring
transitional CV channel described in Burdge et al. (2022a), but instead
probably represents the final inspiral phase of either the double white dwarf
channel, in which an ELM white dwarf recently overflowed its Roche lobe, or
the helium star channel, though we disfavor this scenario due to the presence
of hydrogen at the surface of the donor. In this letter, we discuss the
observational properties of this system and its implications for compact
binary evolution.

##  2 Observations

###  2.1 Discovery in ZTF data

We discovered ZTF J0127+5258 using archival photometry from the Zwicky
Transient Facility, as part of an ongoing campaign to identify short orbital
period binaries (Burdge et al., 2020a). The object was discovered in a search
for periodic photometric variations in 20 million objects selected as being
underluminous relative to the main sequence of stars in _Gaia_ DR2 data (Gaia
Collaboration et al., 2018); this selection and the period search are
described in further detail in Burdge et al. (2022b, 2020a). We note that ZTF
J0127+5258 falls in a chip gap in ZTF’s primary field grid, and thus only
contains photometry from the secondary field grid. This grid has much poorer
sampling than the primary field grid, with ZTF J0127+5258 averaging less than
90 epochs per year in the ZTF g and r filters combined, demonstrating the
viability of discovering ultracomapct binaries at even relatively sparse
cadences.

###  2.2 High-speed photometry

After discovering the source in ZTF data, we observed ZTF J0127+5258 using the
Caltech HIgh-speed Multi-color camERA (CHIMERA) (Harding et al., 2016) on the
200-inch Hale telescope at Palomar observatory. We obtained these observations
on four separate occasions, in the $g^{\prime}$ filter on the blue channel of
the instrument, and the $r^{\prime}$ and $i^{\prime}$ filters on the red
channel of the instrument. These observations took place on July 23, 2020,
November 14, 2020, Oct 10, 2021, and Oct 31, 2021. The absolute timestamps of
exposures were time-tagged to better than millisecond accuracy using a GPS
receiver.

###  2.3 Spectroscopic observations

On October 15, 2020, we obtained phase-resolved spectroscopic observations of
ZTF J0127+5258 using the Low-Resolution Imaging Spectrometer (LRIS) (Oke et
al., 1995) on the 10-m W. M. Keck I Telescope on Mauna Kea. We adopted a
2-minute exposure time and obtained a total of 39 exposures over approximately
96 minutes, covering 7 orbital cycles of the system. We binned these exposures
$4\times 4$ in the blue channel and $2\times 2$ in the red channel to minimize
readout time and readout noise and used the 600/4000 grism as the dispersive
element on the blue channel, and the 400/8500 grating as the dispersive
element for the red channel.

##  3 Analysis and interpretation

###  3.1 Lightcurve

Figure 1 shows the CHIMERA $g^{\prime}$ light curve of ZTF J0127+5258 and a
representative model light curve. We used the LCURVE modeling code
(Copperwheat et al., 2010) because in modeling the lightcurve, we discovered
that we could only account for the complex structure by including an accretion
disk and a bright impact spot of the accretion stream onto the outer edge of
the disk.

We do not present parameter estimates for the component radii and masses based
on the example lightcurve model, because including an accretion disk and
bright spot introduces a significant number of free parameters, many of which
are degenerate with physical parameters of interest. The model illustrated in
Figure 1 is based on a plausible set of parameters, with the accreting white
dwarf radius fixed to that expected for a $\sim 0.75\rm M_{\odot}$ WD, a mass
ratio of $\sim 0.2$, a donor which is Roche-lobe filling, and an accretion
disk with an inner edge at the radius of the accreting WD, and outer edge at
the radius of a hot spot, fixed to approximately 70 percent the Roche lobe
radius of the accretor based on the maximum radius of a stable accretion disk
Paczyński (1971). In the future, a higher signal-to-noise lightcurve obtained
with an instrument such as HiPERCAM may allow for a clear delineation of the
ingress/egress structure in the eclipse arising from the two bodies transiting
each other from structure that originates from the accretion disk and hot spot
eclipse; if the substructure of the eclipse can be characterized, then this
system could be robustly constrained on the basis of Kepler’s laws and Roche
geometry alone, like the case of ZTF J1813+4251 (Burdge et al., 2022a).

Figure 1: A model fit of the binned CHIMERA $g^{\prime}$ lightcurve of ZTF
J0127+5258. We used the LCURVE modeling code to construct a physical model for
the system. In order to account for the behavior in the lightcurve,
particularly the geometry of the eclipses, as well as the increased flux at
the quadrature phase at $0.75$, we had to include an accretion disk with a
bright spot, indicating that the disk and bright spot contribute an
appreciable fraction of the optical flux.

###  3.2 Orbital inspiral

After constructing the lightcurve model illustrated in Figure 1, we fit this
model to archival PTF (Law et al., 2009) and ATLAS (Tonry et al., 2018; Heinze
et al., 2018) lightcurves, as well as our four CHIMERA epochs, in order to
investigate whether any measurable orbital evolution was occurring in the
system–see Burdge et al. (2019a) for further details on this procedure. As
illustrated by the polynomial fit to the data in Figure 2, we clearly detect
orbital decay is occurring, at a rate of $-6.53\pm 0.19\times 10^{-12}\,\rm
s\,s^{-1}$. Because ZTF J1027+5258 is mass transferring, translating this
period decay into a direct constraint on the component masses is nontrivial;
we discuss this further in Section 3.7. Nevertheless, because of the orbital
decay and warm donor, we can deduce that the system is almost certainly pre-
period minimum, and in the final phase of its inspiral.

Figure 2: The measured evolution of eclipse times in ZTF J0127+5258. The red
fit of a quadratic clearly indicates that the system is undergoing orbital
decay, as the coefficient of the quadratic term of the polynomial is clearly
negative. A precise measurement of the orbital evolution was made possible by
the eclipsing nature of the source (allowing for precise timing), and the long
baseline resulting from archival PTF and ATLAS observations.

###  3.3 Spectroscopy

In order to measure the radial velocity semi-amplitude of the donor, we fit
Voigt profiles to the Balmer series of hydrogen absorption lines, which appear
to originate primarily from the donor’s atmosphere. Figure 3 illustrates the
radial velocity measurements resulting from this analysis, which are
consistent with a semi-amplitude of $771\pm 27\,\rm km\,s^{-1}$. We also tried
to fit the numerous helium I absorption lines in the spectrum, and found a
smaller semi-amplitude, with a phase strongly inconsistent with the mid-
eclipse time (whereas the radial velocities from the Balmer lines yielded a
phase in good agreement with the eclipse time in the lightcurve). We attribute
the strange behavior of the helium I lines to contamination and dilution by
emission originating from the accretion disk/bright spot. This accretion-
associated emission manifests itself even more strongly in the helium II
lines, as indicated in Figure 4.

Figure 3: The measured radial velocities of the hydrogen absorption lines in
the spectrum of ZTF J0127+5258. The orbital phases are referenced to those
determined from the mid-eclipse time in the lightcurve model. The poor fit
near inferior conjunction (phase $0.5$) may originate from a
Rossiter–McLaughlin effect as the accretion disk and accreting white dwarf
transit the donor.  Figure 4: The extinction-corrected optical spectrum of ZTF
J0127+5258 obtained with LRIS on the Keck I telescope. Much like the spectrum
of PTF J0533+0209 (Burdge et al., 2019b), the atmosphere of the donor exhibits
a mixture of hydrogen and helium absorption lines, indicating that it is
helium-rich but still contains traces of hydrogen. Because of the contribution
of the accretion disk to both the continuum and the dilution of the absorption
lines by emission, we do not fit a model atmosphere to the object. Instead, we
use the spectral energy distribution to estimate the temperature of the
object.

###  3.4 SWIFT Observation

We observed ZTF J0127+5258 using the XRT and UVOT instruments aboard the Swift
observatory with a 3 kilosecond exposure time (observation ID: 14103001). We
do not detect the source in the XRT observations, however, we see an
ultraviolet counterpart in the UVOT image (taken in the UVW2 filter). We
obtain a source brightness of $21.74\pm 0.15\,\mathrm{(stat)}\,\pm
0.03\,\mathrm{(sys)}\,m_{AB}$ in this filter, which we estimate corresponds to
a de-reddened apparent magnitude of approximately $19.37\,\rm m_{AB}$.
However, the uncertainties are large in the ultraviolet luminosity of the
source, due to the large degree of sensitivity of ultraviolet flux to
extinction corrections.

###  3.5 SED analysis and distance/temperature estimate

Because of its faint apparent magnitude, ZTF J0217+5258 lacks a well-
constrained parallax in _Gaia_ DR3, with a reported value of $\pi=0.44\pm
0.47$. If we treat the Gaia parallax, $\pi$, as having a Gaussian probability
distribution with a most probable value of 0.44 mas and an rms scatter of 0.47
mas, at least for $\pi>0$, then the corresponding distance distribution has a
maximum at 1100 pc, and $10\%$ limits of 600 pc on the close side and 4200 pc
on the distant side. In addition to these basic constraints we can infer from
the parallax, we also explore alternative approaches to estimating the
distance to the source. Our first approach is based on Figure 3 of Ramsay et
al. (2018), which illustrates the absolute magnitude of various accreting
ultracompact binaries as a function of the orbital period. The dynamic range
of luminosity in this figure is large, especially at the short orbital periods
relevant to ZTF J0127+5258, but we estimate that the absolute magnitude of the
source should fall in the range of 5-8 in _Gaia_ G given its orbital period.
We compute the posterior distribution of our distance estimates by assuming a
uniform prior in the absolute magnitude of the source in the range of 5-8,
applying an extinction correction of $-0.75$ magnitudes to the _Gaia_ G filter
based on the observed reddening of $E(g-r)=0.29$ at distances of $>3\rm\,kpc$
Green et al. (2019). Based on its apparent magnitude and this reddening and
the typical absolute magnitude of interacting ultracompact binaries in this
period range, we estimate that the source is approximately
$3.5^{+1.7}_{-1.5}\,\rm kpc$ from the Solar System.

To obtain a more precise distance estimate (and also a temperature estimate),
we fit the spectral energy distribution (SED) of the source. Although the
source is reddened, the gradient of the reddening is small at distances
$>1.5\,\rm kpc$, and thus we fix the reddening to $E(g-r)=0.29$. We use the
Kernel density estimator Ultranest (Buchner, 2014, 2017, 2021) to fit the SED
photometry of the source, including the Swift UVOT UVW2 measurement. In
modeling the SED, we sample over the masses of the accretor, which accounts
for luminosity contributions from the white dwarf and the disk, and donor
(giving both a uniform prior of $0-1.4\,M_{\odot}$), fix the radius of the
donor to its Roche lobe, and allow the effective radius of the accretor to
vary between 0 up to $95$ percent of its Roche lobe (a limit set by the
instability criterion for accretion disks). Using our CHIMERA lightcurve, we
also determine that on average, the donor contributes 50 percent of the flux
in g-band, and thus require that the ratio of flux contributed by the two
components in this filter be confined within the range of $1\pm 0.1$. We
sample over effective temperatures for both the donor and accretor. We find an
effective donor temperature of $16400^{+1000}_{-900}\,\rm K$, and a distance
of $3.1^{+0.5}_{-0.6}\,\rm kpc$, consistent with the distance predicted based
on the luminosity of other accreting binaries at similar orbital periods.

###  3.6 Chandra Observation

We observed ZTF J0127+5258 using the ACIS-I instrument aboard the Chandra
X-ray observatory (Proposal ID: 23300558) for $16362\,\rm s$. In this
exposure, we recorded 26 photons within $2\arcsec$, one of which we omitted as
likely a background photon due to its high energy of $\sim 13\,\rm keV$. We
binned the photons into five spectral bins, and used the XSPEC tool to fit for
the flux and photon index of a power law model of the spectrum. We fixed the
column density to $n_{H}=1.6\times 10^{21}$ based on the estimated reddening
of $\mathrm{(}E(g-r))=0.29$. We obtain an unabsorbed flux value of
$3.0^{+0.8}_{-0.6}\times 10^{-14}\,\rm erg\,cm^{-2}\,s^{-1}$ in the
$0.2-8.0\,\rm keV$ bandpass, and a power law index of $\gamma=1.33\pm 0.44$.
Given the source distance and the column density, we estimate that the X-ray
luminosity of the source in the $0.2-8.0\,\rm keV$ band is $3.5\pm 1.3\times
10^{31}\,\rm erg\,s^{-1}$. This luminosity is comparable to the $10.4\rm\,min$
accreting AM CVn binary ES Ceti, which exhibits an X-ray luminosity of
approximately $8.3\times 10^{30}\,\rm erg\,s^{-1}$ in the $0.2-5.0\,\rm keV$
band (Strohmayer, 2004). We note that because of the edge-on inclination of
the system, this X-ray flux represents only a fraction of the bolometric X-ray
luminosity of the system, and likely arises from the boundary layer of the
accretion disk, which is at least partially obscured at near edge-on
inclinations.

Table 1: Measured parameters

Quantity: | Measured value
---|---
$\rm RA$ | 01:27:47.62 (h:m:s)
$\rm Dec$ | 52:58:13.03 (d:m:s)
$\pi$ | $0.44\pm 0.47\,\rm mas\,yr^{-1}$
$\mu_{\rm RA}$ | $-1.14\pm 0.41\,\rm mas\,yr^{-1}$
$\mu_{\rm Dec}$ | $-2.57\pm 0.42\,\rm mas\,yr^{-1}$
$\rm G_{mag}$ | $19.81\pm 0.01\,\rm(Vega\,\,magnitude)$
$K_{\mathrm{Donor}}$ | $771\pm 27\,\rm km\,s^{-1}$
$T_{0}$ | $59518.2991305\pm 0.0000081\,\rm MBJD_{TDB}$
$P_{b}$ | $822.68031456\pm 0.000043\,\rm s$
$\dot{P}_{b}$ | $-6.53\pm 0.19\times 10^{-12}\,\rm s\,s^{-1}$
$T_{\mathrm{Donor}}$ | $16400^{+1000}_{-900}\,\rm K$

Note. — Measured parameters for ZTF J0127+5258. Coordinates are given at epoch
J2000.0, using J2000 equinox.

Table 2: Inferred component parameters

Quantity: | $\log(\dot{M}/(M_{\odot}\,\rm yr^{-1}))\sim\mathcal{N}(-8.5,0.5)$ | $\log(\dot{M}/(M_{\odot}\,\rm yr^{-1}))\sim\mathcal{N}(-7.3,0.5)$
---|---|---
$M_{\mathrm{accretor}}$ | $0.75\pm 0.06\,M_{\odot}$ | $0.87\pm 0.11\,M_{\odot}$
$M_{\mathrm{donor}}$ | $0.19\pm 0.03\,M_{\odot}$ | $0.31\pm 0.11\,M_{\odot}$
$R_{\mathrm{donor}}$ | $0.051\pm 0.003\,R_{\odot}$ | $0.059\pm 0.008\,R_{\odot}$
$L_{\mathrm{donor}}$ | $0.17\pm 0.05\,L_{\odot}$ | $0.23\pm 0.08\,L_{\odot}$

Note. — Inferred component parameters for ZTF J0127+5258 based on the measured
parameters. Because of the uncertainty in the mass transfer rate and its
implications for the parameter estimates, we report two columns illustrating
how the inferred parameters change assuming the high vs low mass transfer rate
cases.

###  3.7 Effect of mass transfer on period decay

In detached binaries, a measurement of the period derivative yields a direct
constraint on the chirp mass of the system. Mass transfer complicates this
constraint since the immediate effect of transferring mass from the lower-mass
component to the higher-mass component is widening of the orbit. This can be
seen by considering the orbital angular momentum of a binary with a circular
orbit:

$J_{{\rm orb}}=\frac{M_{{\rm wd}}M_{{\rm donor}}}{\left(M_{{\rm wd}}+M_{{\rm
donor}}\right)^{1/3}}P_{{\rm
orb}}^{1/3}\left(\frac{G^{2}}{2\pi}\right)^{1/3}.$ (1)

Solving for $P_{\rm orb}$ and taking a logarithmic derivative yields:

$\frac{\dot{P}_{{\rm orb}}}{P_{{\rm orb}}}=3\frac{\dot{J}_{{\rm orb}}}{J_{{\rm
orb}}}+\frac{\dot{M}_{{\rm tot}}}{M_{{\rm tot}}}-3\frac{\dot{M}_{{\rm
wd}}}{M_{{\rm wd}}}-3\frac{\dot{M}_{{\rm donor}}}{M_{{\rm donor}}},$ (2)

where $M_{\rm tot}=M_{\rm wd}+M_{\rm donor}$. For conservative mass transfer,
we expect $\dot{M}_{{\rm tot}}=0$ and $\dot{M}_{\rm wd}=-\dot{M}_{\rm donor}$.
If we then define $\dot{M}\equiv\dot{M}_{\rm wd}$ (a positive quantity), we
obtain

$\frac{\dot{P}_{{\rm orb}}}{P_{{\rm orb}}}=3\frac{\dot{J}_{\rm orb}}{J_{\rm
orb}}+3\frac{\dot{M}}{M_{\rm donor}}\left(1-q\right),$ (3)

where $q=M_{\rm donor}/M_{\rm wd}$. Assuming the only significant angular
momentum loss comes from gravitational waves, we have

$\frac{\dot{J}_{{\rm orb}}}{J_{{\rm
orb}}}=-\frac{32G^{5/3}}{5c^{5}}\frac{M_{{\rm wd}}M_{{\rm
donor}}}{\left(M_{{\rm wd}}+M_{{\rm
donor}}\right)^{1/3}}\left(\frac{2\pi}{P_{{\rm orb}}}\right)^{8/3}.$ (4)

To constrain the masses of both components in the presence of mass transfer,
we fit the observed period derivative and donor RV semi-amplitude, while
leaving the component masses, inclination, and mass transfer rate as free
parameters. The likelihood function compares the observed and predicted semi-
amplitudes and period derivatives:

$\begin{split}\ln L=-\frac{1}{2}\frac{\left(K_{{\rm donor,\,pred}}-K_{{\rm
donor,\,obs}}\right)^{2}}{\sigma_{K_{{\rm donor}}}^{2}}\\\
-\frac{1}{2}\frac{\left(\dot{P}_{{\rm orb,\,pred}}-\dot{P}_{{\rm
orb,\,obs}}\right)^{2}}{\sigma_{\dot{P}_{{\rm orb}}}^{2}},\end{split}$ (5)

where $\dot{P}_{\rm orb,\,pred}$ is the period derivative predicted by
Equation 3 (assuming mass transfer is conservative), and $K_{{\rm
donor,\,pred}}=\left[\frac{2\pi G}{\left(M_{{\rm donor}}+M_{{\rm
wd}}\right)^{2}P_{{\rm orb}}}\right]^{1/3}M_{{\rm wd}}\sin i$, with $i$ the
inclination.

Motivated by the models of Wong & Bildsten (2021), we adopt two different
priors on the mass transfer rate. The first is a relatively low value,
$\log(\dot{M}/(M_{\odot}\,\rm yr^{-1}))\sim\mathcal{N}(-8.5,0.5)$ (this
notation indicates a normal distribution centered at $-8.5$ with standard
deviation $0.5$, which matches their models for a low-mass white dwarf donor.
However, these models only undergo Roche-lobe overflow at shorter periods and
are already evolving toward longer periods by the time they reach 13.7
minutes. Their models in which the donor is a helium star begin mass transfer
at longer periods ($\sim 20$ minutes) and thus do pass through 13.7 minutes
while the period is decreasing (see also Brooks et al., 2015). These models
have significantly higher mass transfer rates, and for these, we adopt
$\log(\dot{M}/(M_{\odot}\,\rm yr^{-1}))\sim\mathcal{N}(-7.3,0.5)$.

Since the system is eclipsing, we use a prior on the inclination,
$i\sim\mathcal{U}(75,90)$. We use flat priors on $M_{\rm wd}$ and $M_{\rm
donor}$. The results of this fitting are shown in Figure 5.

Figure 5: Constraints from joint fitting of the donor’s RV semi-amplitude and
the observed period derivative. Black and red contours adopt priors of
$\dot{M}\sim\mathcal{N}(-8.5,0.5)$ and $\dot{M}\sim\mathcal{N}(-7.3,0.5)$,
respectively. The upper right inset shows how, for two different combinations
of $M_{\rm WD}$ and $M_{\rm donor}$ that match the observed RV amplitude, the
predicted period derivative depends on the mass transfer rate. Mass transfer
widens the orbit, so a larger $\dot{M}$ requires higher masses to match the
observed $\dot{P}_{\rm orb}$.

In general, we infer higher masses for both components with the high-$\dot{M}$
prior. Since mass transfer widens the orbit and counteracts the GR-driven
orbital inspiral, a larger value of $\dot{M}$ requires higher component masses
to produce the same net period derivative.

For the low-$\dot{M}$ prior, the orbit-widening effects of mass transfer are
relatively modest for all values of $\dot{M}$ allowed by the prior, and as a
result, the inferred component masses depend only weakly on $\dot{M}$.
However, for the high-$\dot{M}$ prior, large enough values of $\dot{M}$ are
supported that they can significantly change the expected $\dot{P}$, and so
the inferred component masses have larger uncertainties. For values of
$\dot{M}$ above $10^{-7}\,M_{\odot}\,\rm yr^{-1}$, the total mass required to
reproduce the observed $\dot{P}$ and $K_{\rm donor}$ even exceeds the
Chandrasekhar mass.

An accurate measurement of the system luminosity can constrain $\dot{M}$. For
a high state system like J0127, the expected accretion luminosity is $L_{{\rm
acc}}\sim G\dot{M}M_{{\rm wd}}/\left(2R_{{\rm wd}}\right)\sim
11\,L_{\odot}\left(\frac{\dot{M}}{10^{-8}\,M_{\odot}\,{\rm
yr}^{-1}}\right)\left(\frac{M_{{\rm
wd}}}{0.7\,M_{\odot}}\right)\left(\frac{R_{{\rm
wd}}}{0.01\,R_{\odot}}\right)$. The high-$\dot{M}$ prior thus implies an
accretion luminosity of $L_{\rm acc}\sim 50\,L_{\odot}$. The low-$\dot{M}$
prior implies $L_{\rm acc}\lesssim 5\,L_{\odot}$. The presence of hydrogen in
the donor spectrum suggests a recent onset of mass transfer, in better
agreement with the evolutionary scenario leading to the low-$\dot{M}$ prior.
Additionally, the low bolometric luminosity of the donor (see Table 2) also
favors the low mass transfer rate scenario, since the eclipse depth indicates
roughly equal luminosity of the donor and the accretor in the optical–however,
because most of the accretion luminosity is likely emitted in the ultraviolet,
this observation cannot definitively rule out the high mass transfer rate
case.

###  3.8 He WD and He Star models

We compare ZTF J0127+5258 to Modules for Experiments in Stellar Astrophysics
(${\tt MESA}$; Paxton et al., 2011, 2013, 2015, 2018, 2019; Jermyn et al.,
2022) models. The He WD models are made in a similar fashion to Wong &
Bildsten (2021), assuming a non-rotating donor with orbital angular momentum
loss due to gravitational wave emission. The initial models are made by
stripping the envelope of an evolved star once a He core is formed, down to a
hydrogen mass fraction of $X=0.1$, and allowing it to cool to the desired
central specific entropy, $s$, with element diffusion enabled. For comparison,
we also include a MESA model with a $0.37\,\mathrm{M_{\odot}}$ He star donor
based on Bauer & Kupfer (2021), which can evolve through a similar orbital and
$T_{\mathrm{eff}}$ configuration as the highest entropy He WD donor models,
but cannot satisfy the constraint of having some residual hydrogen at its
surface by the time $P_{\mathrm{orb}}=13.7\,\mathrm{min}$. The He star model
represents roughly a maximum radius/entropy configuration relative to a He WD
model because the He star comes into contact while core He burning is ongoing,
then evolves adiabatically once it donates enough mass to quench core He
burning ($M\lesssim 0.3\,\mathrm{M_{\odot}}$).

Higher-entropy (less degenerate) He WDs have larger radii and therefore
initiate mass transfer at longer periods. As their non-degenerate outer layers
become lost, their radii and thus the orbit shrink in response, and $\dot{M}$
reaches a peak at period minimum (Kaplan et al., 2012). Since
$\dot{M}\propto\dot{J}_{\mathrm{gr}}$ and longer periods have smaller
$|\dot{J}_{\mathrm{gr}}|$, higher-entropy He WDs have smaller peak $\dot{M}$
(Deloye et al., 2007; Wong & Bildsten, 2021). This is shown by the top panel
of Figure 6 for several models with an initially $0.23\,\mathrm{M_{\odot}}$
donor and $0.75\,\mathrm{M_{\odot}}$ accretor (chosen to fit the donor orbital
velocity), and different initial donor entropies
($s/(N_{\mathrm{A}}k_{\mathrm{B}})=3.6-4.0$, where $N_{\mathrm{A}}$ is
Avogadro’s number and $k_{\mathrm{B}}$ is Boltzmann constant). Except for the
highest entropy model, we show only the evolution prior to period minimum. We
also omit a portion of the evolution not relevant to this paper where model
convergence is difficult due to an equation of state blend between Skye
(Jermyn et al., 2021) and CMS (Chabrier et al., 2019).

Based on the requirement that the donor has to begin Roche-filling at
$P_{\mathrm{orb}}>13.7\,\mathrm{min}$, a low-entropy He WD can be ruled out. A
He WD with $s/(N_{\mathrm{A}}k_{\mathrm{B}})\approx 4.0$ and an initial mass
of $0.23\,\mathrm{M_{\odot}}$ can furthermore yield good agreement with the
measured $\dot{P}$ at $13.7\,\mathrm{min}$, though other combinations of donor
mass, accretor mass, and donor entropy may also provide a good fit.

In addition, the high measured effective temperature of ZTF J0127+5258 favors
a high-entropy He WD donor. As shown in the middle panel of Figure 6, a
higher-entropy donor has higher $T_{\mathrm{eff}}$ at contact. Upon mass
transfer, $T_{\mathrm{eff}}$ decreases, and the
$s/(N_{\mathrm{A}}k_{\mathrm{B}})=4.0$ model appears slightly too cool for ZTF
J0127+5258 at $13.7\,\mathrm{min}$, while the
$s/(N_{\mathrm{A}}k_{\mathrm{B}})=3.8$ model shows a good fit. A donor
initially with mass $\gtrsim 0.2\,\mathrm{M_{\odot}}$ is also favored, since
lower mass donors never reach an effective temperature of 16400 K. However,
irradiation by the accretor and tidal heating, which are ignored in our
modeling, may be at work.

Following Burdge et al. (2019b), we estimate the effects of tidal heating on
the donor surface temperature. Assuming that the donor is spinning
synchronously with the orbit, and is rigidly rotating, the energy flux in the
donor due to tides is $\dot{E}_{\mathrm{tide}}\approx 4\pi^{2}I\dot{P}/P^{3}$,
where $I$ is the moment of inertia of the donor. Fuller & Lai (2013) found
that below a critical period $P_{\mathrm{c}}\approx 1\,\mathrm{hr}$, a
detached He WD becomes nearly but not fully synchronized. The tidal energy
flux contributes mostly to spinning up the He WD, and the rest to heat
dissipation. The tidal heating rate is reduced to
$\dot{E}_{\mathrm{heat}}\approx\dot{E}_{\mathrm{tide}}(P/P_{\mathrm{c}})^{19/18}$.
We assume that tidal heat is dissipated near the surface of the He WD, such
that its surface luminosity, $L_{\mathrm{s}}$, is obtained by adding
$\dot{E}_{\mathrm{heat}}$ to the intrinsic luminosity of the model. The
estimated surface temperature is then given by
$T_{\mathrm{eff}}=\left[L_{\mathrm{s}}/(4\pi
R^{2}\sigma_{\mathrm{sb}})\right]^{1/4}$. This is shown as a dot-dashed line
in Figure 6, for the $s/(N_{\mathrm{A}}k_{\mathrm{B}})=4.0$ model. If we
choose to use $\dot{E}_{\mathrm{tide}}$ instead, as an upper limit to the
effects of tidal heating, the surface temperature is increased to the dashed
line. While tidal heating may raise the temperature of the donor to that
observed, an intrinsically hot donor is likely required.

The He star model starts with $T_{\mathrm{eff}}>20{,}000\,\rm K$ before it
begins donating He-dominated material. After donating enough mass to fall
below the threshold for core He burning, it evolves adiabatically and its
$T_{\mathrm{eff}}$ decreases into the range observed for the donor in ZTF
J0127+5258. However, a He star donor can only donate enough mass to reach this
lower temperature after completely exhausting its residual H envelope, which
always resides at mass coordinates outside $m>0.3\,\mathrm{M_{\odot}}$.
Therefore, a model with an He star donor cannot be made consistent with the
observation of residual H in the donor atmosphere at the observed
$T_{\mathrm{eff}}$ and $P_{\mathrm{orb}}$ of ZTF J0127+5258.

The period derivative is shown in the bottom panel of Figure 6. As discussed
in Section 3.7, this is influenced by $\dot{M}$. The models that are explored
show $\log(\dot{M}/(\mathrm{M_{\odot}}\,{\rm yr}^{-1}))$ between $-8.0$ and
$-8.5$ when they shrink to $P_{\mathrm{orb}}=13.7\,\mathrm{min}$ from a wider
orbit, in agreement with the low-$\dot{M}$ prior. However, we caution that
these models give a wide range of predictions for $\dot{M}$ and $\dot{P}$. It
is possible that ZTF J0127+5258 is only coming into contact at
$13.7\,\mathrm{min}$. Near the onset of mass transfer, $\dot{M}$ can vary by
several orders of magnitude. For $\dot{M}\lesssim
10^{-9.5}\,\mathrm{M_{\odot}}\,{\rm yr}^{-1}$, the effects of mass transfer
can be negligible to $\dot{P}$. In this case, the measured donor orbital
velocity and $\dot{P}$ constrain the donor mass to be $\approx
0.16-0.18\,\mathrm{M_{\odot}}$. A high-entropy $0.18\,\mathrm{M_{\odot}}$ He
WD model in ${\tt MESA}$ can reach an effective temperature of $\approx
13000\,\mathrm{K}$, so this scenario can match observations only if the donor
surface is significantly heated by tidal dissipation or irradiation. We
further note that a H-rich envelope can modify $\dot{P}$ and the mass transfer
history (Kaplan et al., 2012).

These high-entropy He WD donors have been predicted by population synthesis
(see Figure 1 of Deloye et al., 2007, using the results of Nelemans et al.
2001)111For comparison, our $0.23\mathrm{M_{\odot}}$ He WDs with
$s/(N_{\mathrm{A}}k_{\mathrm{B}})=3.8-4.0$ have
$\log_{10}(T/\mathrm{K})\approx 7.8$ and
$\log_{10}(\rho/\mathrm{g}\,\mathrm{cm}^{-3})\approx 4.7-4.9$ at contact., and
require a common envelope origin. Little time ($\approx 10^{7}\,\mathrm{yr}$)
is allowed for the donor to cool to lower entropies before coming into contact
again, so the post-common envelope period has to be short (30-50 min).
Interestingly, extremely low-mass (ELM) WD binaries have been suggested to
form at short periods (lognormal distribution with mean of 30 min, albeit with
high uncertainty; Brown et al., 2016).

Figure 6:  Mass transfer rate (top), donor effective temperature (middle), and
period derivative (bottom) as a function of the orbital period. Colored lines
show He WD models labeled by the initial central specific entropy of the donor
in units of $N_{\mathrm{A}}k_{\mathrm{B}}$, whereas the grey line shows a He
star model similar to the $0.37\,\mathrm{M_{\odot}}$ model in Bauer & Kupfer
(2021). The observed properties of ZTF J0127+5258 are shown in blue lines with
uncertainties bracketed by blue regions. For the He WD $4.0$ model, we show
the full evolution past period minimum, with arrows in the top panel
indicating the flow of time. For this model, we also show the estimated
effects of tidal heating, by adding $\dot{E}_{\mathrm{tide}}$ (dashed line) or
$\dot{E}_{\mathrm{heat}}$ (dot-dashed line) to the intrinsic luminosity of the
model and re-calculating the effective temperature via the Stefan-Boltzmann
law.

###  3.9 Fate of system

ZTF J0127+5258 is currently undergoing rapid orbital decay and will either
merge or reach a period minimum and begin evolving to longer orbital periods
in the next one to two million years.

Whether it merges depends in part on whether direct impact accretion can be
avoided, as it increases angular momentum loss from the orbit to the accretor
and tends to destabilize the system. Direct impact accretion likely occurs if
the minimum distance that the accretion stream passes from the accretor,
$R_{\mathrm{min}}$, is smaller than the accretor radius, $R_{\mathrm{WD}}$.
For ${\tt MESA}$ He WD models (made with no H but same donor entropy as those
presented in Sec 3.8) with an initially $0.75\,\mathrm{M_{\odot}}$ accretor,
we compare $R_{\mathrm{min}}$ (Nelemans et al., 2001, fit to Lubow & Shu 1975)
to $R_{\mathrm{WD}}$ assuming a cold WD mass-radius relation for the accretor
(Verbunt & Rappaport, 1988). A binary with an initially
$0.23\,\mathrm{M_{\odot}}$ He WD donor can avoid direct impact accretion if
the initial donor entropy $s/(N_{\mathrm{A}}k_{\mathrm{B}})\gtrsim 3.6$, since
higher-entropy donors allow for wider binary separations; with an initially
$0.20\,\mathrm{M_{\odot}}$ He WD donor, the parameter space widens to
$s/(N_{\mathrm{A}}k_{\mathrm{B}})\gtrsim 3.2$. The He star donor models also
generically avoid direct impact accretion due to their larger radii and wider
orbital configurations at contact (Bauer & Kupfer, 2021). Furthermore, we
model the accretor in ${\tt MESA}$ for the $0.23\,\mathrm{M_{\odot}}$,
$s/(N_{\mathrm{A}}k_{\mathrm{B}})=3.9\ \&\ 4.0$ models. The accretor radius is
slightly inflated compared to that of a cold WD, but direct impact accretion
is still avoided (though $R_{\mathrm{min}}$ is only $\approx 10\%$ larger than
$R_{\mathrm{WD}}$ at period minimum for the
$s/(N_{\mathrm{A}}k_{\mathrm{B}})=3.9$ model). Thus, for He WD models that
provide decent matches to observations of ZTF J0127+5258, the system may
narrowly avoid direct impact accretion in the future.

Moreover, if a He nova occurs, the accretor may undergo a double-detonation as
a thermonuclear supernova (e.g., Shen & Moore, 2014), or the system may merge
(Shen, 2015). In the latter case, due to the compact nature of the binary,
nova ejecta can easily engulf the binary; a merger may occur since dynamical
friction with the nova ejecta leads to additional orbital angular momentum
loss. For the $0.23\,\mathrm{M_{\odot}}$,
$s/(N_{\mathrm{A}}k_{\mathrm{B}})=3.9$ model, we find that the initially
$0.75\,\mathrm{M_{\odot}}$ accretor will undergo a He flash after accumulating
$\approx 0.06\,\mathrm{M_{\odot}}$ of He. However, whether a He detonation
will occur is not immediately clear. For the
$s/(N_{\mathrm{A}}k_{\mathrm{B}})=4.0$ model, a He flash is instead avoided
since the $\dot{M}$ is so low that accretion is unable to heat the He shell to
ignition. This model will evolve to become an AM CVn binary.

Figure 7: A diagram of the possible outcomes of double white dwarf mergers
presented in Shen (2015), with the dashed blue lines indicating the mass
constraints on ZTF J0127+5258 in the low mass transfer rate case, and the
dashed green lines indicating the constraints in the high mass transfer rate
case. If ZTF J0127+5258 fails to undergo stable mass transfer and evolve into
a helium CV, this diagram indicates that it will likely form an R Coronae
Borealis star which eventually cools into a white dwarf if its current mass
transfer rate is on the lower end of our estimates, or will explode as a Type
Ia supernova if its mass transfer rate is on the higher end of our estimates.
Note that parenthetical entries in the merger outcomes may not apply to all
objects in that mass range.

##  4 Conclusions

The discovery of ZTF J0127+5258 demonstrates the existence of an important new
class of sources: pre-period minimum accreting binary systems with luminous
donors that have orbital periods so short that they fall near the peak
sensitivity of _LISA_. The existence of this object suggests that at least
some post-period minimum He CVs originate from the double-white-dwarf or He
star channel, unless the mechanism proposed in Shen (2015) prevents the system
from undergoing stable mass transfer and is driven to merger instead.

As seen in Figure 7, if the system merges, it will likely form an R Crb Star
(Iben & Tutukov, 1984; Webbink, 1984; Shen, 2015), which will eventually cool
into a carbon-oxygen white dwarf star if its current mass transfer rate is on
the lower end of our estimates. However, if the system has a high mass
transfer rate, the masses we infer suggest that the system will explode in a
Type Ia supernova (Iben & Tutukov, 1984; Webbink, 1984; Shen, 2015), which
would make it the first gravitational wave-detectable Type Ia progenitor
known. This work demonstrates the value of constraining the influence of mass
transfer in the orbital evolution of these systems, as this can have major
implications for the inferred past evolutionary history and predicted merger
outcome. When such a source is detected by _LISA_ , we will still have
uncertainty in the contribution of mass transfer to the orbital evolution,
unless we are able to independently constrain it; however, by combining our
distance estimate with the measured gravitational wave strain amplitude, we
will obtain an independent constraint on the chirp mass and can infer the
degree to which mass-transfer is influencing the orbital evolution.

As demonstrated in this work, these systems can exhibit rich structure in
their optical lightcurves and are also sources of X-ray emission. Unlike their
post-period minimum counterparts which host cold donors, binaries like ZTF
J0127+5258 have donors luminous enough to directly detect in optical spectra
despite the presence of a high-state accretion disk, allowing us to
characterize their atmospheric composition, and measure radial velocity semi-
amplitudes using phase-resolved spectroscopy. This constraint, when combined
with an inclination constraint and inferred chirp mass, allows us to uniquely
determine the two component masses in the system. In the _LISA_ era, we expect
to discover many more such sources using gravitational radiation, and by then
localizing them to an X-ray and/or optical/IR counterpart, we will be able to
probe accretion physics in an unprecedented manner, at a large range of
inclinations (as _LISA_ will detect both edge-on and face-on systems, and will
give inclination constraints independent of the electromagnetic observations).
Proposed facilities such as the _Advanced X-ray Imaging Satellite_ (Mushotzky
et al., 2019), with its large effective area and small PSF over a moderately
large FOV, will be particularly powerful tools for identifying the
electromagnetic counterparts of this class of _LISA_ sources. This is because
many of these mass-transferring sources modulate their X-flux on the orbital
frequency, a frequency that will be known as a result of the _LISA_ detection,
enabling rapid and robust association of an X-ray source within the
gravitational wave error circle if there are enough X-ray photons from the
source to detect this modulation.

## acknowledgments

Kevin Burdge is a Pappalardo Postdoctoral Fellow in Physics at MIT and thanks
the Pappalardo fellowship program for supporting his research.

This work was supported in part by Chandra grant GO2-23014A.

This work was supported, in part, by the National Science Foundation through
grant PHY-1748958. T.L.S.W was supported by the Gordon and Betty Moore
Foundation through grant GBMF5076. T.L.S.W acknowledges use of computational
facilities at UC Santa Barbara funded by NSF grant CNS 1725797, and thanks the
Center for Scientific Computing for supporting this resource.

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

##  Appendix A _LISA_ signal-to-noise ratio

To estimate the _LISA_ SNR of the source, we follow the same procedure as
presented in Burdge et al. (2020a). In addition to this approach, we compared
our SNR estimates to those estimated by the LEGWORK code (Wagg et al., 2022b,
a). We found discrepancies between the two results, and corresponded with the
authors of LEGWORK, finding several errors in both the code used to estimate
SNRs in Burdge et al. (2020a), as well as in LEGWORK, and have now converged
on consistent SNR estimates after conducting this exercise. The result of
these corrections reduced the estimated signal-to-noise ratios by about a
factor of two in both codes (in the case of LEGWORK, the corrections applied
only to the position/polarization/inclination specific sources, not the sky-
averaged SNRs). The updated version of LEGWORK (0.4.0 and after) has resolved
these issues.

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