A Benchmark White Dwarf–Ultracool Dwarf Wide Field Binary

We present the discovery and multi-wavelength characterisation of VVV J1438-6158 AB, a new field wide-binary system consisting of a 4 . 6 + 5 . 5 − 2 . 4 Gyr and 𝑇 eff = 9500 ± 125 𝐾 DA white dwarf (WD) and a 𝑇 eff = 2400 ± 50 𝐾 M8 ultracool dwarf (UCD). The projected separation of the system is 𝑎 = 1236 . 73 au ( ∼ 13 . 8”), and although along the line-of-sight towards the Scorpius-Centaurus (Sco-Cen) stellar association, VVV J1438-6158 AB is likely to be a field star, from a kinematic 6D probabilistic analysis. We estimated the physical, and dynamical parameters of both components via interpolations with theoretical models and evolutionary tracks, which allowed us to retrieve a mass of 0 . 62 ± 0 . 18 𝑀 ⊙ for the WD, and a mass of 98 . 5 ± 6 . 2 𝑀 Jup ( ∼ 0 . 094 ± 0 . 006 𝑀 ⊙ ) for the UCD. The radii of the two components were also estimated at 0 . 01309 ± 0 . 0003 𝑅 ⊙ and 1 . 22 ± 0 . 05 𝑅 Jup , respectively. VVV J1438-6158 AB stands out as a benchmark system for comprehending the evolution of WDs and low-mass companions given its status as one of the most widely separated WD+UCD systems known to date, which likely indicates that both components may have evolved independently of each other, and also being characterised by a large mass-ratio ( 𝑞 = 0 . 15 ± 0 . 04), which likely indicates a formation pathway similar to that of stellar binary systems.


INTRODUCTION
Binary systems involving a white dwarf (WD) and a companion of extremely low-mass, such as an ultracool dwarf (UCD) or brown dwarf (BD), present a unique opportunity to investigate the evolution of low-mass stars and sub-stellar objects.The working definition and distinction between BDs ( eff ≤ 2200 ), and UCDs ( eff ≤ 2700 ) are based on their masses and temperatures.Both surpass the planetary threshold for deuterium burning in their cores (∼ 14 − 16  Jup ; Spiegel, Burrows, & Milsom 2011, but BDs fall short of initiating hydrogen nuclear fusion (∼ 80 − 85  Jup ; Kumar 1963;Burrows et al. 2001).In the upper span of their mass spectrum (⪆ 60  Jup ), their sizes are predominantly regulated by electrondegeneracy pressure, alike WDs, while at the lower limit of this range (∼ 14 − 20  Jup ), their sizes are mainly determined by Coulomb ★ E-mail: thiago.dossantos@yale.edupressure (Basri & Brown 2006).WDs, on the other hand, are remnants of stars with masses in the Main Sequence ranging from 0.8 to 7.5  ⊙ (Weidemann 1968;Saumon, Blouin, & Tremblay 2022), composed primarily of degenerate matter, and are supported against gravitational collapse by electron-degeneracy pressure.
Significant progress has been achieved in the investigation of cataclysmic variable stars (CVs) involving stellar remnants and substellar objects (e.g., Longstaff et al. 2019;Zorotovic & Schreiber 2022, and references therein), however, many of these systems are distinguished by their close proximity, often leading to common envelope interactions and profound influence on each other component's evolution.
VVV J1438-6158 AB was discovered from the investigation of low-mass stars and free-floating planets (FFPs) towards the young Lower Centaurus Crux stellar association (LCC; de Zeeuw et al. 1999) by Mejías et al. (2022).Differing from previously known WD+UCD/BD binaries (e.g., Farihi & Christopher 2004;Farihi, Becklin, & Zuckerman 2005;Schmidt et al. 2005;Steele et al. 2009;Casewell et al. 2012;Steele et al. 2013;Beuermann et al. 2013;Catalán 2013;Littlefair et al. 2014;Longstaff et al. 2019;Casewell et al. 2020), and also from the first BD discovered by the VVV survey not belonging to a binary system (VVV BD001; Beamín et al. 2013), in terms of orbital separation, VVV J1438-6158 AB resides at the opposite end of the spectrum, exhibiting a notably loose configuration wherein both components appear to have evolved independently from one another, i.e., without exerting any discernible influence on each other's evolutionary paths.Its significance relies upon the scarcity, albeit non-absence, of systems with well-established ages similar to it.A similar WD+UCD binary system, WD0837+185, was also discovered within the Praesepe cluster (Casewell et al. 2012), however, it differs from the system reported here with its much closer projected separation of ∼ 0.006 au and age of 625 Myr, which likely implies interaction, making unlikely the independent evolution of its components.As of the current writing, there are 8 known WD+UCD/BD binaries, spanning a range from relative proximity between the components ( < 100 au; Steele et al. 2013), or exhibiting an intermediate configuration (e.g., Becklin & Zuckerman 1988;Meisner et al. 2020;French et al. 2023), to loose separations ( > 1000 au; e.g., Rodriguez et al. 2011;Luhman, Burgasser, & Bochanski 2011;Day-Jones et al. 2011;Deacon et al. 2014).
In this paper, we present the characterisation of VVV J1438-6158 AB, a new WD+UCD binary system towards the young Scorpius-Centaurus (Sco-Cen) stellar association.In Section 2, we describe the observational data used in the analysis, focusing on the multicolour photometry obtained for both components: , , and  magnitudes from Gaia DR3 (Gaia Collaboration et al. 2021),  magnitudes from DECam Plane Survey (DECaPS; Schlafly et al. 2018), and   s magnitudes from the VISTA Variables in the Via Lactea survey (VVV; Minniti et al. 2010).In Section 3, we present the physical parameters of WD and its companion, including effective temperatures, masses, radii, surface gravities, and ages from interpolation with theoretical models.In Section 4, we briefly discuss the likelihood of membership to Sco-Cen by VVV J1438-6158 AB.A light curve analysis of the system in the VVV- s pass-band is discussed in Section 5, and in Section 6, we present our conclusions, summarising the key findings from our analysis.

OBSERVATIONS OF VVV J1438-6158 AB
VVV J1438-6158 AB was identified by El-Badry, Rix, & Heintz (2021) as a co-moving pair with a small chance alignment probability R = 3.015×10 −5 , who classified it as a co-moving pair composed of a WD and a main-sequence (MS) star.Independently, we classified the system as a field binary consisting of a WD and a UCD towards the Sco-Cen stellar association from data of the VVV survey and Gaia DR3.In the absence of radial velocity measurements for both components of VVV J1438-6158 AB, we cannot estimate the 3D space velocity, nor the difference, for the system.Nevertheless, the high-bound probability assumption for the system relies on R ≪ 0.1, which corresponds approximately to ≥90% bound probability (e.g., El-Badry, Rix, & Heintz 2021).Table 1 presents the Gaia DR3 measurements for both components, encompassing locations, parallaxes, proper motions in equatorial components, the renormalised unit weight errors (RUWEs), and distances derived from the parallax measurements.
We adopt a distance of 89.6 ± 0.5 pc for this system based on the more accurately measured parallax value of the WD component, which has an uncertainty of 0.07 mas.Its Galactic space veloci-ties components are  = −0.508  −1 ,  = 4.392   −1 and  = −1.569  −1 from Torres et al. (2019), which suggests that the system belongs to the thin disc.The tangential velocity of both components,  t = 4.74 •  total • Δ, was estimated at  t,WD = 14.62 ± 0.02   −1 and  t,UCD = 17.87±0.15  −1 .The projected separation between the WD and UCD is 13.8", which corresponds to a physical separation of ∼ 1240 au at that distance, and note that the RUWE values for both components are relatively low, which indicates reliable astrometric estimations from Gaia (Lindegren et al. 2021), and that both components do not likely possess close stellar companions (Fitton, Tofflemire, & Kraus 2022).
Although the WD companion was too faint to be detected during observations of the 2MASS survey (Skrutskie et al. 2006), the UCD was detected (source ID 2MASS J14383082-6159022).For both components, however, near-infrared (near-IR)   s photometry was obtained by the VVV survey (Minniti et al. 2010).They are situated within the overlapping region of the Galactic disc VVV tiles d014 and d015, and despite the crowded nature of the field, the optical and near-IR images suggest that they remain relatively uncontaminated (see Figure 1).
Figure 2 depicts the optical and near-IR colour-magnitude (CMD) and colour-colour diagrams (CCD) of the surrounding field (5 arcmin of the central location of the WD) containing the targets.In both diagrams, the two components are separated.Solar metallicity isochrones 1 (PARSEC v1.2S and COLIBRI S 35 tracks; Chen et al. 2014;Pastorelli et al. 2019) for ages of 1 − 10 Gyr, extinctions   ranging from 0 − 8 mag, shifted to the distance of the Galactic centre, ≈ 8.7 kpc (Vanhollebeke, Groenewegen, & Girardi 2009), are overlaid to the CMDs and CCD.VVV J1438-6158 AB is located in a region of relatively high reddening2 , with a measured colour excess E(B−V) SFD = 1.75 ± 0.06 mag, and visual extinction  V,SFD = 5.42 ± 0.19 mag (Schlegel, Finkbeiner, & Davis 1998;Schlafly & Finkbeiner 2011), while the colour excess at the distance of the system (E(B−V) Δ ) is 0.012 ± 0.017 mag and the visual extinction also at the distance of the system ( V,Δ ) is 0.037 ± 0.052 mag (Capitanio et al. 2017) 3 .
Table 2 presents the photometric data for the WD and UCD obtained from various optical and near-IR catalogues, including the Gaia DR3, DECaPS, VVV, and 2MASS (Skrutskie et al. 2006).The absolute magnitudes in Gaia−, and VVVX−/ s bands, assuming a distance of Δ = 89.6 pc for this system, are provided in the last rows.Conservative errors were considered, primarily influenced by the measured parallax difference between both components.It is worth noting the absence of near-IR excess in the WD's colours, which suggests the absence of an additional close companion, or a circumstellar disc.Moreover, a thorough examination of the near-IR CCD diagram presented in Steele et al. (2011) (refer to their Figure 1, which also relies on the findings of Wellhouse et al. 2005), confirms that the WD occupies a distinct position within the solitary WD region.

SPECTRAL ENERGY DISTRIBUTION (SED) ANALYSIS
We integrated the observed fluxes from the optical ( min = 4673.24Å) to IR-region ( max = 21376 Å) using the Virtual Ob-   servatory Spectral Energy Distribution (SED) Analyser 4 web tool (VOSA; Bayo et al. 2008) to estimate the physical and atmospheric parameters of both components of VVV J1438-6158 AB system.Figure 3 depicts their multi-frequency SED collected from the Gaia, DECaPS, and VVV surveys, and Table 3 summarises the main properties of both components of VVV J1438-6158 AB binary system.

White Dwarf's characterisation
In the Gaia EDR3 catalogue, VVV J1438-6158 A is classified as a WD with 99.47% probability (Gentile Fusillo et al. 2021), and have 79.3% probability of being a DA-type WD (with a predominantly hydrogen-rich atmosphere; Jiménez-Esteban et al. 2023).Pure-hydrogen and pure helium atmospheric models from The best fit to the available photometric measurements of VVV J1438-6158 A (Table 2) was obtained with the WD spectra and atmosphere model library presented in Koester (2010), yielding an effective temperature of  eff = 9500 ± 125  and surface gravity log() = 8.00 ± 0.12, with  2  = 1.38 (11 free parameters).The bolometric luminosity for the adopted distance is  bol = (1.257±0.005)×10−3  ⊙ , and applying the Stefan-Boltzmann law led to an estimation of the object's radius as  = 0.01309 ± 0.0003  ⊙ .From the best-fit surface gravity and estimated radius, the WD's mass was determined as  = 0.63 ± 0.18  ⊙ .Within their uncertainties, these WD parameters appear to be similar to what has been derived for similar objects (e.g., Chandrasekhar 1935;Schwarzschild 1958; Landau  & Lifshitz 1969).Alternatively, a direct black body fit resulted in  eff,BB = 9900 ± 25 , and  bol,BB = 1.257 × 10 −3 ± 1.6 × 10 −6  ⊙ .

WD's Age and Progenitor mass
To estimate the age of VVV J1438-6158 A, we used the wdwarfdate software (Kiman et al. 2022), which adopts a Bayesian approach for thick hydrogen (DA) and non-DA layer, taking into account the remnant's effective temperature and surface gravity.This approach allowed us to estimate the final mass of the WD and the initial mass of its progenitor.For a DA WD, the main-sequence age and mass were derived as Γ DA = 3.88 +5.57−2.51 Gyr and  init,DA = 1.37 +0.63 −0.32  ⊙ , respectively, from an initial-final mass relation model of Cummings et al. (2018), a cooling model from Bédard et al. (2020), and evolutionary tracks from Choi et al. (2016) and Dotter (2016).
Figure 4 depicts the optical  vs.  BP −  RP CMD for nearby objects presented in the Gaia catalogue, and the positioning of VVV J1438-6158 A in such a diagram.Overlaid to the CMD are indicated evolutionary cooling tracks5 for pure-hydrogen DA-and pure-helium DB-type WDs with log() = 7.5 − 9.0 from Holberg & Bergeron (2006) and Bédard et al. (2020).
Using Kepler's third law, by considering the masses of the WD and UCD along with their projected separation and face-on orientation, we estimated a minimum orbital period of the VVV J1438-6158 AB system as 141 ± 4 Kyr.Despite the lengthy orbital period, the system is expected to remain gravitationally bound, with a comparison with Figure 7 of Mužić et al. (2012) -stating empirical limits for the stability of a sample of binary systems -, which in turn is based on Burgasser, Sheppard, & Luhman (2013), supporting such a conclusion.

ON THE MEMBERSHIP LIKELIHOOD OF VVV J1438-6158 AB TO THE SCO-CEN ASSOCIATION
To assess the likelihood of VVV J1438-6158 AB pair being a member of the Sco-Cen stellar association, we used the BANYAN Σ algorithm based on a 6D XYZUVW kinematic analysis (Gagné et al. 2018), applying both to the LCC stellar association, the Upper Centaurus Lupus (UCL), and Upper Scorpius (US); branches within the Sco-Cen group.This analysis indicates a null probability for the system to belong to the LCC group or any sub-structures within it.This conclusion is also supported by the fact that the inferred ages of the WD component (as shown in Table 3) do not align with the ages typically associated with the LCC stellar association; previous studies, for instance, by Goldman et al. (2018) and Pecaut & Mamajek (2016) suggest ages of 10 Myr and 17 Myr, respectively.Unlike massive stars that evolve into WDs within a few million years, the formation of a WD from relatively low-mass star progenitors generally requires a significant amount of time for a star to exhaust its nuclear fuel, shed its outer layers, and gradually cool down (Kippenhahn, Weigert, & Weiss 2013).This process typically takes billions of years, making it improbable for a WD like VVV J1438-6158 A to have an age as young as 17 Myr.This inconsistency between the estimated ages and the established age range adds to evidence that contradicts VVV J1438-6158 AB's association with the LCC, supporting that the system is likely to be a field object.

NEAR-IR VVV LIGHT CURVES
Both targets are situated in the overlapping areas of two VVV tiles (d014, centred at  = 314.40• ,  = −1.64974• , and d015, centred at  = 315.83• ,  = −1.64972• ), yielding almost twice as many observations concerning single-tile objects.However, in some of these images, the targets are too close to the edge, rendering impossible photometric measurements.Consequently, the final  s band light curves consist of 115 photometric data points for the WD (46 in d014 and 69 in d015), and 285 for the UCD companion (104 in d014 and 181 in d015), respectively.In addition, there are 7 measurements in the − and  − bands for both components, plus 11 in the −band and 6/5 measurements in the −band for both the WD/BD (see Table 2).The VVV  s −filter light curves for the UCD and WD are presented in Figure 6.
We examined periodic patterns in the VVV  s −band light curves of both the companions of the VVV J1438-6158 AB pair within a range of 0.   Minimisation (PDM; Stellingwerf 1978), and Multi-Harmonic Analysis of Variance (MHAoV; Schwarzenberg-Czerny 1996) algorithms through the astrobase software (Bhatti et al. 2021).VVV J1438-6158 A displays variability with a median absolute variation of 0.08 mag and root mean square of 0.13 mag at a mean magnitude of 16.88 mag.The period search analysis reveals a prominent modulation at 0.112 days (∼161.2minutes) with the GLS method, which could be associated with periodic patterns caused by surface magnetic variations on the WD that become visible as it rotates, e.g., Kilic et al. (2015Kilic et al. ( , 2021)); Williams, Hermes, & Vanderbosch (2022).The other period-search algorithms yielded similar periods of 0.126 and 0.109 days for the PDM and BLS al- gorithms, while for the MHAoV a prominent 2.076-day period was found.Given the relatively noisy nature of the resulting periodograms (see Appendix A), further investigation is needed to unveil the nature of these modulations.For VVV J1438-6158 B, slight variability was noticed in the  s −band, suggesting two potential modulation periods: 1.288 days with the LS method and 1.729 days with the PDM method.None of these appears significant given the noisy nature of the resulting periodogram (see Appendix A) and no clear sine-like variations at these periods.The UCD phase diagram recovered with MHAoV at a period of 7.73 days does not exhibit any discernible temporal variation, while applying the BLS algorithm, a modulation period of  BLS,UCD = 0.111 days was recovered, albeit also with low-significance due to inherent noise in the periodogram.These observed modulations might be influenced by various factors, such as errors in measurements or issues with the measuring instruments, resulting in stochastic fluctuations in the data.We also note the possibility of false periods in the periodogram, potentially arising from systematic effects introduced during the data windowing process, which could involve hidden periods or non-stationary behaviour.Further investigation is also needed to unveil the nature of these modulations.

DISCUSSION AND CONCLUSIONS
We present the characterisation of VVV J1438-6158 AB, a wide binary system consisting of a WD and a UCD first discovered with VVV/VVVX data analysis during searches for companions to UCDs, BDs and FFPs towards the Sco-Cen/LCC stellar association El-Badry, Rix, & Heintz (2021) and independently by Mejías et al. (2022).The SED analysis of this system has been conducted from data of the Gaia DR3, VVV and DECaPS survey within VOSA, to derive the physical, atmospheric, and dynamical parameters of both components.Space velocities demonstrate that the system belongs to the thin disc and is gravitationally bounded from the considerably small chance alignment probability.The DA-type WD has an effective temperature of 9500 ± 125 , a surface gravity (cgs) of 8.00 ± 0.12, a mass of 0.62 ± 0.18  ⊙ , and a radius of 0.01309 ± 0.0003  ⊙ .The M8-type UCD, on the other hand, have an effective temperature of 2400 ± 50 , surface gravity (cgs) of 5.20 ± 0.04, mass of 98.5 ± 6.2  Jup , and radius of 1.22 ± 0.05  Jup .These parameters are consistent with the expected characteristics of objects of their respective kinds.Spectroscopic confirmation for both members of the VVV J1438-6158 AB pair is encouraged and necessary, especially to accurately determine the spectral type of VVV J1438-6158 B beyond its photometric classification, and to assess its precise age.VVV J1438-6158 AB stands out as a benchmark system for comprehending the evolution of WDs and their low-mass companion objects given several notable features, along with the scarcity of such reported binary types.Considering the WD's nominal mass to present 0.62  ⊙ , the mass ratio ( ≡  UCD / WD ) of VVV J1438-6158 AB can be estimated as 0.15 ± 0.04, which suggests a binary-like formation scenario for this system, i.e., the two objects form independently, and their masses might not be closely related (Chabrier et al. 2014;Bowler, Blunt, & Nielsen 2020).The co-natal status of the pair, as well as for other WD+UCD/MS and low-mass double binaries -both wide and close -contingent upon their gravitational binding and co-mobility, remains a subject of ongoing debate and will be investigated in a future study.Nelson et al. (2021), for instance, demonstrated that ∼ 73% of the far (2 × 10 5 − 10 7 au; 31 pairs) co-moving MS+MS binaries are co-natal, which is in-line with theoretical predictions from Kamdar et al. (2019).The total age of VVV J1438-6158 A was estimated at 4.6 +5.5  −2.4 Gyr (cooling age of 0.73 +0.09 0.06 Gyr), and given the long orbital period of the system (141.8± 4.5 Kyr) plus loose separation (∼ 1236.73 au), it indicates a detached coexistence, with minimal perturbations or interchanges, inviting further scrutiny into the intricate interplay of forces that have governs the evolution of WD+UCD/BD/MS' evolution.

Figure 1 .
Figure 1.Near-IR finding charts from the VVV DR5 of the WD (component A; blue circles), and its UCD companion (component B; red circles) in    s pass-bands in 2010.Despite the crowded nature of the field, the binary components are well-separated (13.8"), and visibly, the UCD is brighter than the WD at longer near-IR wavelengths.All images have the same orientation as indicated in the first panel.

Figure 2 .
Figure 2. Optical Gaia  vs.  −  (first panel), near-IR VVV  s vs.  −  s CMD (second panel), and  −  vs.  −  s CCD (third panel) diagrams for a 5 arcmin field centred on the WD+UCD binary pair.The WD is shown with a blue circle and the UCD with a red circle.Evolutionary tracks for distinct   in each panel are from PARSEC v1.2S + COLIBRI S 35 .

Figure 3 .
Figure3.SED fitting of VVV J1438-6158 AB pair.The circles correspond to the observed flux for both components (red for the UCD and blue for the WD), while the dashed lines plus crosses correspond to the respective best-fit models.

Figure 6 .
Figure 6.Observed  s −band near-IR light curves for the WD (top panel) and UCD companions (bottom panel) spanning 10 years.The red dots depict the moving averages for 500 days of steps.

Table 1 .
Gaia DR3 astrometric measurements and derived parameters for VVV J1438-6158 AB.Note: Component A (first row) corresponds to the WD, while component B (second row) corresponds to the UCD.

Table 3 .
Fundamental properties of VVV J1438-6158 AB binary retrieved with SED analysis.

Table 4 .
VVV near-IR photometry of VVV J1438-6158 AB pair.The full version of this table is available in the supplementary material, where VVV ID 11846281028924 stands for the WD companion, and VVV ID 11846281000738 stands for the UCD companion.