Electromagnetic Characterization of the LISA Verification Binary ZTF J0526+5934

We present an analysis of new and archival data to the 20.506 minute LISA verification binary J052610.42+593445.32 (J0526+5934). Our joint spectroscopic and photometric analysis finds that the binary contains an unseen M 1 = 0.89 ± 0.11 M ⊙ CO-core white dwarf primary with an M 2 = 0.38 ± 0.07 M ⊙ post-core-burning subdwarf, or low-mass white dwarf, companion. Given the short orbital period and relatively large total binary mass, we find that LISA will detect this binary with signal-to-noise ratio 44 after 4 yr of observations. J0526+5934 is expected to merge within 1.8 ± 0.3 Myr and likely result in a D6 scenario Type Ia supernova or form a He-rich star that will evolve into a massive single white dwarf.


INTRODUCTION
White dwarfs represent a relatively simple final evolutionary stage for most single-star stellar evolution.Interactions in a binary system complicate this evolution and can result in a wide range of astrophysically interesting systems.For binary evolution, the more massive star will evolve first, potentially leading to a phase common-envelope evolution as it evolves onto the asymptotic giant branch.This process strips the primary of its outer layers and leaves behind a CO-core white dwarf in a compact binary with orbital period ranging from hours to days.Depending on the mass ratio of the resulting compact binary, a second commonenvelope phase may occur as the companion fills its Roche lobe near the base of the red giant branch.This double-common-envelope evolutionary process results in a double-degenerate binary with orbital period ranging from less than an hour to a only a few hours (Li et al. 2019).Compact post-common-envelope binaries are excellent systems for studying binary evolution.Recent work by Scherbak & Fuller (2023) used compact eclipsing white dwarf binaries to place constraints on the common envelope ejection efficiency.
Compact binaries with periods less than about 6 h are considered to be merging binaries since the rate of their orbital angular momentum loss caused by gravitational wave emission is sufficient to result in a binary merger within a Hubble time.The merging binaries observed today therefore represent a population of progenitor binaries to merger products, such as AM CVn binaries (Kilic et al. 2016), He-rich stars (Zhang et al. 2014), massive single white dwarfs (Cheng et al. 2020;Kilic et al. 2023), and Type Ia supernovae (Woosley et al. 1986;Fink et al. 2007;Liu et al. 2018;Shen et al. 2018a).Characterization of merging white dwarf binaries provides constraints on the formation rates and potential formation channels of these merger products.Many compact white dwarf binaries have been discovered through targeted spectroscopic surveys, such as the ELM Survey (Brown et al. 2010(Brown et al. , 2022;;Kosakowski et al. 2023), and through large-scale systematic searches for photometric variability in time-domain surveys (Burdge et al. 2020;van Roestel et al. 2022;Ren et al. 2023).
White dwarf binaries are expected to be the dominant source of gravitational wave signal for the Laser Interferometer Space Antenna (LISA; Amaro-Seoane et al. 2017).The shortest period binaries, with P ≲ 1 h, emit gravitational waves at mHz frequencies that may be detected by LISA.LISA is expected to detect O(10 4 ) of these ultra-compact binaries, but only O(10 2 ) are expected to also be detectable through their electromag- netic radiation, allowing for a multi-messenger approach to studying binary evolution (Nelemans et al. 2001;Korol et al. 2017;Li et al. 2020;Amaro-Seoane et al. 2023).The strongest gravitational wave emitters will act as "verification binaries," which can be used to calibrate the LISA data set in the first few months of operation.So far, about 40 LISA detectable binaries have been characterized through their electromagnetic radiation (see Finch et al. 2023;Kupfer et al. 2023, and references therein).
Here we present an independent discovery and analysis of a new LISA verification binary with orbital period P = 20.506min, J052610.42+593445.32 (J0526+5934).J0526+5934 was originally reported as a candidate ultra-compact binary by Ren et al. (2023) based on periodic photometric variability seen in the Zwicky Transient Facility (ZTF; Bellm et al. 2019;Graham et al. 2019;Masci et al. 2019) data archive.The authors find that J0526+5934 will be detected by LISA with an expected signal-to-noise ratio S/N = 35.788after 4 years of observation.
Throughout this work, we adopt the convention that the unseen massive star, which evolved first, is the primary star, while the relatively low mass companion is the secondary, such that M 1 > M 2 .In Section 2 we describe our target selection criteria.In Sections 3 and 4 we provide the details of our spectroscopic and photometric analysis.In Sections 5 and 6 we discuss the expected rate of orbital decay of J0526+5934, prospects for LISA detection, and its potential merger outcomes.Finally, we summarize our results in Section 7.  (Lomb 1976;Scargle 1982;VanderPlas 2018).We searched for periodic signals with periods between P min = 3 min and P max = 684 min, split into 10-million evenly-spaced trial frequencies.To increase temporal sampling of the ZTF light curves with multiple measurements in different filters, we median-combined the light curves across each filter by artificially shifting the r-and i-band data such that their median magnitude values matched the median g-band magnitude.Our search made use of the Texas Tech University High Performance Computing Center to efficiently process each light curve.We manually inspected the output light curve images to identify objects with periodic photometric variability based on their peak power spectrum value with respect to the local noise level.
J0526+5934 (Gaia DR3 282679289838317184) was identified in our search as an ultra-compact binary, with dominant frequency f peak ≈ 140.445 cycles d −1 (P peak ≈ 10.253 min) and amplitude A ≈ 0.05 mag, suggesting ellipsoidal modulation at true orbital period P ≈ 20.506 min.We estimated the uncertainty in the orbital period through a bootstrapped analysis with 10,000 periodograms of the ZTF DR16 light curve data focused on the surrounding 40-seconds of the mostprobable period, split into 20-million frequency bins, and find P = 1230.37467± 0.00007 s. Figure 1  Our algorithm failing to recover other well-characterized binaries may be a consequence of our evenly-spaced frequency grid, which may be undersampled at higher frequencies and over-sampled at lower frequencies.However, some of these binaries, such as ZTF J1539+5027 (P = 6.9 min; Burdge et al. 2019a), are not included in the Gaia eDR3 white dwarf catalog and therefore are not identified in our search, but are otherwise easily recovered with our algorithm when targeted.
We downloaded the blue optical spectra and their associated calibration data from the Keck Observatory Archive and reduced the data using using standard iraf (Tody 1993) procedures including image correction, spectral extraction, dispersion correction using HgNeArCdZn arc-lamps, and wavelength calibration using BD28 • 4211 standard star observations taken with the same setup.
The optical spectrum of J0526+5934 is dominated by hydrogen absorption features and has relatively shallow He I absorption features at 4912 Å, 4471 Å, and 4026 Å, giving it the DAB classification.We see no evidence of the companion in the Keck spectroscopy.

Radial Velocity and Kinematics
We estimated the radial velocity for each of the ten blue optical spectra against a zero-velocity low-mass DA white dwarf template spectrum using the crosscorrelation package rvsao.xcsao(Kurtz & Mink 1998) within iraf.We then shifted each of the ten component spectra of J0526+5934 to zero-velocity and coadded them into a single high-quality, zero-velocity spectrum, which we later use to estimate atmospheric parameters.Finally, we obtained precise radial velocity estimates for each component spectrum by using the co-added spectrum as a template for another round of cross-correlation.Our individual radial velocity measurements are presented in Table 1.
We fit a circular orbit to the radial velocity measurements to estimate the orbital period (P ), velocity semiamplitude (K), and systemic velocity (γ) of the binary.We find best-fitting parameters P RV = 20.54 ± 0.12 min, K = 549.7 ± 4.7 km s −1 , and γ = −40.7 ± 4.1 km s −1 , roughly twice the orbital period identified through our Lomb-Scargle analysis of the ZTF light curve.However, because the exposure time used for each spectrum covers a significant fraction of the orbital period (9.8%), we corrected the orbital solution by fitting an average integrated sine curve to the observed data, taking into account the exposure time at each observed orbital phase.We find smearing-corrected velocity semiamplitude K = 558.3± 4.8 km s −1 , corresponding to mass function 0.255 ± 0.007 M ⊙ .Our best-fitting average integrated sine curve orbital solution is presented in Figure 2.
We estimated Galactic space velocities for J0526+5934 by using our best-fitting systemic velocity and the Gaia DR3 astrometry measurements.We find U = 47.6 ± 1.9 km s −1 (U positive toward the Galactic center), V = −7.3± 1.7 km s −1 , and W = 3.8 ± 1.1 km s −1 , corrected for the motion of the local standard of rest (Schönrich et al. 2010), consistent with a Galactic disk population based on the average velocity and dispersion distributions for the Galactic disk and halo from Chiba & Beers (2000).

Atmospheric Parameters
We estimated the atmospheric parameters of J0526+5934 by fitting a grid of hot subdwarf model atmospheres (Saffer et al. 1994) to the co-added blue optical spectrum and find best-fitting parameters T eff = 27300 ± 260 K, log g = 6.37 ± 0.03, and log N (He)  N (H) = −2.45± 0.06, which suggest that J0526+5934 is a post-core-burning subdwarf, or an inflated He-core low-mass white dwarf.We summarize our best-fitting parameters in Table 2. Our best-fitting model is over-plotted onto the Keck blue optical spectrum in Figure 3.

Spectral Energy Distribution
A spectral energy distribution (SED) fit was performed to measure the radius and mass of J0526+5934.The angular diameter of the star is measured and, in combination with the Gaia DR3 parallax, we derive the radius of the visible component in J0526+5934.The luminosity and mass are calculated using the atmospheric parameters measured from spectroscopy.This method is described in detail by Heber et al. (2018).Because J0526+5934 is missing archival GALEX UV and SDSS u-band photometry, we fixed the effective temperature and surface gravity to our spectroscopic values.
Using the functions of Fitzpatrick et al. ( 2019), we account for interstellar reddening.The color excess E (44−55) is treated as a free parameter and the the extinction parameter R(55) was fixed to the standard value of 3.02.To estimate the radius we apply R = Θ/2ϖ, where Θ is the angular diameter derived from the SED fit and ϖ is the parallax extracted from Gaia DR3.The mass follows from the M = gR 2 /G, where g is the surface gravity and G is the gravitational constant.Our fit to the available SED, including Gaia G, G BP , and G RP , PanSTARRS grizy (Chambers et al. 2016), and (un)WISE W 1 (Schlafly et al. 2019), finds R 2 = 0.061 +0.006 −0.005 R ⊙ , corresponding to mass M 2 = 0.32 +0.06 −0.05 M ⊙ .

Light Curve Modeling
We obtained high-speed g ′ , r ′ , and i ′ -band follow-up light curves of J0526+5934 using the McDonald 2.1meter telescope on 2022 September 30, 2022 October 01, and 2022 October 02, respectively.
We used lcurve (Copperwheat et al. 2010) to perform simultaneous g ′ -, r ′ -, and i ′ -band modeling to our McDonald light curves.We fit for the mass ratio (q = M 2 /M 1 < 1.0), orbital inclination (i), scaled companion radius (r 2 = R 2 /a), time of primary conjunction (t 0 ), and the filter-dependent gravity-darkening and quadratic limb-darkening coefficients.We included Gaussian priors on the surface gravity, effective temperature, velocity semi-amplitude, and radius of the lowmass companion based on the values obtained from our fits to the optical spectroscopy and available SED.We used gravity and quadratic limb-darkening coefficients from Claret et al. (2020) for DA white dwarfs with atmospheric parameters T eff,2 = 27, 500 K, log g 2 = 6.37 and T eff,1 = 10, 000 K, log g 1 = 8.00.We marginalized over the limb and gravity-darkening coefficients by assigning Gaussian priors based on the 2σ uncertainties of our spectroscopic atmospheric parameters.Mass Function (M⊙) 0.255 ± 0.007 We find most-probable model parameters q = 0.426 +0.052 −0.051 , i = 57.1 +4.3 −4.1 • , and R 2,vol = 0.070 ± 0.005 R ⊙ , where R vol is the volumetric radius.These parameters correspond to stellar masses M 2 = 0.378 +0.066 −0.060 M ⊙ and M 1 = 0.887 +0.110 −0.098 M ⊙ , in agreement to within 1σ of the mass and radius estimates from our SED fitting.We adopt the light curve modeling solution as the true mass and radius and summarize these parameters in Table 2. Figure 4 presents a corner-plot of our parameter distributions with the the most-probable model over-plotted onto our McDonald light curves.

ORBITAL DECAY
The orbit of compact binaries decays due to the loss of orbital angular momentum through the emission of gravitational waves.We estimated the magnitude of this effect for J0526+5934 using Equation 1 (Landau & Lifshitz 1975;Piro 2019 where a is the binary separation.We find that the expected rate of orbital decay in J0526+5934 due to the emission of gravitational waves is ṖGW = −(8.25 ± 2.88) × 10 −12 s s −1 , where the large uncertainties are dominated by our uncertainty in the component masses.Tidal interactions contribute to the total orbital decay in ultra-compact binaries as orbital energy is used to spin-up the stars in the binary.We ignore the effects of tidal heating and assumed that the stars are tidally locked and estimated the contribution from tidal interactions to the orbital decay of J0526+5934 using Equation 2 (see Equation 6 in Piro 2019) where i is the moment of inertia of each star.Burdge et al. (2019b) finds k 2 = 0.066 and k 1 = 0.14 based on white dwarf models for less massive stellar components in an ultra-compact binary, while Marsh et al. (2004) finds that k ≈ 0.2 is an appropriate estimate for white dwarfs based on the Eggleton zero-temperature mass-radius relation.We used k 1 = k 2 = 0.15 ± 0.05 and find total orbital decay ṖGW + Ṗtides = −(8.66± 3.03) × 10 −12 s s −1 , corresponding to tidal contribution Ṗtides Ṗtotal = 4.8 ± 1.9%.The orbital decay of compact binaries can be directly measured as an observable through timing offsets in periodic photometric variability, such as with precise eclipse timing measurements over multi-year baselines (see Hermes et al. 2012;Burdge et al. 2019aBurdge et al. ,b, 2020Burdge et al. , 2023)).However, while we find a baseline of ≈ 2500 d in archival data from ZTF and the Asteroid Terrestrialimpact Last Alert System (ATLAS; Tonry et al. 2018;Heinze et al. 2018), the orbital period precision in our McDonald data is insufficient to measure the effects of orbital decay in J0526+5934.Future long-term monitoring of J0526+5934 will provide precise orbital period and ephemeris timings which will be used to directly measure the orbital decay and place observational constraints on the chirp mass of J0526+5934.

LISA Detection
We used the parallel tempered Markov Chain Monte Carlo algorithm gbmcmc within ldasoft (Littenberg et al. 2020) to simulate the expected gravitational wave signal of J0526+5924 from LISA.We fixed the sky position and orbital period, placed a Gaussian prior on the distance based on the Gaia DR3 parallax, and placed a uniform prior on the orbital inclination based on the 2σ uncertainties from our light curve modeling.Our simulations find that LISA will recover the orbital inclination and chirp mass with similar or better precision than our electromagnetic analysis after a 2-year mission with inclination i = 56.3+3.7 −5.2 • , chirp mass M = 0.49±0.03M ⊙ , and gravitational wave amplitude A = (2.9 +0.3 −0.2 )×10 −22 , corresponding to signal to noise ratio S/N ≈ 27 after a 2-year mission, and S/N ≈ 44 after a 4-year mission, using the Galactic foreground noise model of Cornish & Robson (2017).

Merger Outcome
We find that J0526+5934 will merge within τ GW = 1.8 ± 0.3 Myr due to loss of orbital angular momentum from a combination of gravitational wave emission and tidal interaction.However, given our large mass uncertainties, the merger outcome of J0526+5934 is uncertain.
On the median and upper-end of our mass estimates, our data suggests that the most likely merger out-come of J0526+5934 is a "dynamically driven doubledegenerate double-detonation" (D 6 ) scenario in which unstable mass transfer ignites a Helium detonation near the surface of the accretor, which triggers a CO-core detonation and results in a sub-Chandrasekhar Type Ia supernova explosion of the accretor (Dan et al. 2012(Dan et al. , 2015;;Shen et al. 2018a;Wong & Bildsten 2023).In this double-detonation scenario, the low-mass donor may survive its companion's explosion as a hyper-velocity star, retaining its orbital speed from before the explosion (see Shen et al. 2018b;Bauer et al. 2021;El-Badry et al. 2023).
On the lower-end of our mass estimates, our data suggests that the merger of J0526+5934 is likely to result in a stable He-rich star (Zhang et al. 2014), such as an R Coronae Borealis type star (Webbink 1984).This would naturally evolve into a massive CO white dwarf over time, contributing to the large fraction of merger products in the population of massive single white dwarfs (see Cheng et al. 2020;Kilic et al. 2023).

SUMMARY & CONCLUSIONS
In this work, we have presented our spectroscopic and photometric analysis of a new P = 20.506min ultracompact LISA verification binary, independently discovered in the ZTF data archive and first reported in Ren et al. (2023).
We used archival Keck LRIS spectroscopy to estimate the atmospheric parameters of the visible component and find that, with log g 2 = 6.37 ± 0.03, the low-mass visible star is a post-core-burning hot subdwarf or an inflated low-mass He-core white dwarf.We performed light curve modeling to new multi-band high-speed photometry from the McDonald Observatory and find mass ratio q = 0.426 +0.052 −0.050 , mass M 2 = 0.378 +0.066 −0.060 M ⊙ , and volumetric radius R 2,vol = 0.070 ± 0.005 R ⊙ , consistent with the estimates from our best-fitting SED model, M 2,SED = 0.32 +0.06 −0.05 M ⊙ and R 2,SED = 0.061 +0.006 −0.005 R ⊙ .We estimated the rate of orbital decay based on our most-probable system parameters and find that J0526+5934 will merge within 1.8 ± 0.3 Myr and most likely result in a D 6 scenario supernova explosion or form a He-rich star that eventually evolves into a massive single white dwarf.While our mass estimates are uncertain, which results in large uncertainties in potential merger outcome, future timing measurements will provide a precise estimate to the chirp mass of J0526+5934 and help characterize the expected LISA gravitational wave signal, providing a clear solution to the eventual fate of J0526+5934.

Figure 1 .
Figure 1.Left: ZTF DR16 light curve of J0526+5934 (top), its Lomb-Scargle power spectrum (middle), and phase-folded ZTF DR16 light curve (bottom).Data points are colored based on the filter used.Green data points represent ZTF g, red data points represent ZTF r.Right: Gaia DR3 color-magnitude diagram.The location of J0526+5934 is marked with a red symbol.

Figure 3 .
Figure 3. Best-fitting model atmosphere to the co-added optical spectrum of ZTF J0526+5934.

Table 1 .
Radial velocity measurements for J0526+5934 based on our cross-correlation fit.