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A&A
Volume 693, January 2025
Article Number L11
Number of page(s) 5
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202453100
Published online 10 January 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1. Introduction

Double white dwarf (WD) binaries that evolve through periods of mass transfer and end up with small separations potentially form the dominant population of supernovae Type Ia progenitor systems (Webbink 1984; Fink et al. 2007; Ferrand et al. 2022) and will dominate the detectable gravitational wave signal at low frequencies (Nelemans et al. 2001a; Korol et al. 2020). Despite their huge importance for modern astrophysics, the formation of close double WD binaries is not yet fully understood.

Nelemans et al. (2000) showed that two common envelope phases described by simple energy conservation relations are unable to reproduce the observed population of double helium-core WDs, and this problem remains largely unsolved. Possible solutions are alternative prescriptions of common envelope evolution based on angular momentum conservation (Nelemans et al. 2001b) or evolutionary scenarios that include stable mass transfer (Webbink 2008; Woods et al. 2012). Recent results on the progenitors of double WDs seem to support the latter (Lagos et al. 2022; Li et al. 2023), but conclusive evidence for one or the other scenario is currently not available. A particular system whose existence highlights the failure of our current concepts of double WD binary formation is SDSS J1257+5428.

Badenes et al. (2009) speculated that SDSS J1257+5428 consists of a WD with a neutron star or a black hole companion, but this guess was based on overestimating the mass of the WD that dominates the optical spectrum. Kulkarni & van Kerkwijk (2010) showed that instead the system is composed of two WDs (see also Marsh et al. 2011). Later, using observations from the Hubble Space Telescope (HST), Bours et al. (2015) refined the WD parameters and found that the low-mass WD (∼0.1 − 0.24 M⊙) is significantly older (1.6 − 5 Gyr) than its close and massive WD companion (1.06 ± 0.05 M⊙), which cooled for only ∼1 Gyr. In addition, the broad lines of the massive WD indicate that it is either magnetic or rapidly rotating.

The age difference between the ∼1 M⊙ and very low-mass WD contradicts a formation scenario through two common envelope phases as in that case the more massive star in the initial binary (which should produce the more massive WD) should be the first to evolve off the main sequence. In addition, given that the low-mass WD formed first, the initial binary separation must have been relatively short (≲80 days) as otherwise the helium core of the progenitor would have grown to higher masses before the first phase of mass transfer (Rappaport et al. 1995). This short initial orbital period excludes common envelope evolution during the first mass transfer phase as the binary would have merged or, at the very least, ended up with a period too short for the binary to survive a second common envelope. Observed samples of post common envelope binaries consisting of WDs with M dwarf companions lack WDs with masses below 0.3 M⊙, which indicates that systems with periods short enough to produce such a low-mass WD do not survive common envelope evolution (Rebassa-Mansergas et al. 2011).

However, assuming stable mass transfer for the first phase of mass transfer does not offer an obvious explanation either. While the binary could have survived stable mass transfer, it is impossible to keep the WD mass resulting from the first mass transfer as low as observed (0.1 − 0.24 M⊙) and at the same time widen the orbit enough to allow the core of the second star to grow to the high mass of the carbon–oxygen (C/O) WD (1.06 ± 0.05 M⊙) that we observe today.

This failure of understanding the formation of SDSS J1257+5428 indicates that either our cooling models for WDs or our prescriptions for mass transfer need to be revised, that perhaps one of the observationally determined parameters is incorrect, or that we are missing an essential ingredient of the initial configuration. Further constraining the formation of SDSS J1257+5428 therefore holds the potential to progress with our general understanding of close WD binary formation and evolution. Here we present the identification of a distant tertiary WD to SDSS J1257+5428. The discovery of this tertiary makes the system the third ever detected triple WD, and also provides strong constraints on the total age of the system, an important parameter for understanding the past of SDSS J1257+5428.

2. The Gaia detection of the tertiary

Recent studies of EL CVn binaries (Lagos et al. 2020) and of binaries containing a WD and a main sequence (AFGK) star companion (Parsons et al. 2023; Lagos-Vilches et al. 2024) show that close binaries containing extremely low-mass WDs (< 0.3 M⊙) with nondegenerate companions are frequently the inner binaries of hierarchical triple stars. As the extremely low-mass WD in SDSS J1257+5428 formed first, the system should have appeared as an extremely low-mass WD plus a nondegenerate companion for some time. This motivated us to search for a tertiary companion.

We inspected the Gaia DR3 archive and discovered that Gaia DR3 1570271757456852096 is indeed a common proper motion companion with an angular separation of 67.27902 ± 5.3 × 10−4 arcsec, which translates to a projected separation of 8086 ± 49 au. Table 1 lists the proper motion and parallaxes as well as the SDSS and Gaia magnitudes of both the inner binary and the tertiary object. Even at these large projected separations, the agreement between the parallax and proper motion measurements is convincing and chance alignment is still one to two orders of magnitude less likely than a triple star configuration (see El-Badry et al. 2021a, their Fig. 3). We therefore conclude that the double WD binary star SDSS J1257+5428 is very likely the inner binary of a hierarchical triple system.

Table 1.

Positions, proper motions, and separation measured by Gaia, as well as SDSS and Gaia photometry of the inner binary and the tertiary.

The existence of a tertiary object offers the potential to progress with our understanding of the formation of the inner binary SDSS J1257+5428 as it might allow us to constrain otherwise unknown system parameters. In this context, the nature of the tertiary is of fundamental importance. Remarkably, the tertiary is listed in Gentile Fusillo et al. (2021) as a WD candidate that is likely both massive (1.2 ± 0.2 M⊙) and cool (6500 ± 1500 K). If confirmed, this tertiary WD to the double WD binary SDSS J1257+5428 would convert the system to only the third known triple WD.

The other two known triple WD systems are WD 1704+481 and SDSS J195336.00−101931.6 (hereafter SDSS J1953−1019). WD 1704+481 represents a hierarchical triple system with the inner binary consisting of a helium (He) core WD with a C/O WD companion. As the more massive C/O WD is older than the lower mass He-core component, this system is consistent with having formed through two common envelope phases (Maxted et al. 2000). SDSS J1953−1019 is a resolved triple systems in which all three components evolved unaffected by their companions and with all three stellar components being C/O WDs with masses between 0.6 and 0.63 M⊙ (Perpinyà-Vallès et al. 2019).

In contrast to these other triple WDs, the origin of SDSS J1257+5428, the inner binary of our triple WD system, is difficult to understand. In addition to the intriguing discovery of a rare triple WD system, the tertiary to SDSS J1257+5428 might therefore offer the potential to constrain formation theories for the inner binary. By determining the cooling age of the tertiary WD and the nuclear timescale of its progenitor using initial-to-final mass relations for WDs, it is possible to determine the age of the tertiary. As all three stars were born at the same time, this age of the tertiary would also correspond to the total age of the mysterious inner binary system, thereby providing a key parameter for understanding its evolutionary history.

3. Observations

On 25 March 2023 we observed the tertiary with the Low Resolution Imaging Spectrometer (LRIS) at the W. M. Keck Observatory using the grating 400/8500 and 400/3400 respectively for the red and blue part of the spectrum. We took two exposures in the blue and red arm of 960 and 1200 seconds, respectively. The data were reduced using LPipe (Perley 2019) and the spectra were coadded.

Flux calibration of the data obtained with Keck/LRIS can be uncertain by a factor of a few, due to variations in airmass or seeing between observations of the source and the standard star. This clearly was the case for the obtained spectrum of the tertiary. We therefore forced the spectrum to roughly match the photometry obtained by Gaia and SDSS by adding a constant flux. The resulting spectrum is shown in Fig. 1 together with two model spectra and the photometric fluxes. The spectrum does not show clear absorption lines, but some broad features of uncertain origin at ∼4200 and ∼4800 Å (highlighted in the inset of Fig. 1) and some missing flux in the wavelength range between 4000 and 6000 Å. These characteristics lead to two possible interpretations.

thumbnail Fig. 1.

Spectrum of the tertiary taken with (LRIS) at the Keck observatory. The flux calibration of the data was affected by variations in airmass or seeing between observations of the source and standard star. We therefore forced the spectrum to fit the photometry obtained by Gaia and SDSS. Balmer lines are absent in the spectrum, which might indicate either that the WD atmosphere is helium dominated or that the WD is magnetic. The broad features at ∼4200 and 4700 Å (highlighted in the inset), as well as the missing flux in the range between 4000 and 6000 Å, might be consistent with the latter scenario. Fitting the photometry assuming a hydrogen-rich atmosphere, we obtain an effective temperature of 6401 ± 13 K and log(g) = 8.967 ± 0.002, while for a helium-dominated atmosphere we obtained Teff = 6211 ± 14 K and log(g) = 8.883 ± 0.001 (errors are purely statistical here; see text for a rough estimate of the systematics). The model spectra calculated with these parameters are shown as dashed lines. Taking into account systematic errors, the corresponding cooling ages are ≳4.0 Gyr, which represents a lower limit on the total age of the triple WD.

First, according to the hydrogen-rich atmosphere models by Koester (2010), Balmer lines should be detectable at ∼6500 K. However, the absence of Balmer absorption lines and the simultaneous presence of broad absorption features (see inset of Fig. 1) with an unclear origin might indicate that the tertiary WD is strongly magnetic. A strong magnetic field might split the energy levels through the Zeeman effect, which could cause broad absorption features instead of well-defined Balmer absorption lines. This scenario roughly fits with the high mass and cool temperature of the WD as Bagnulo & Landstreet (2021) showed that strongly magnetic WDs tend to be more massive and old. The missing flux and the generally rather unusual spectrum might also indicate the presence of a strong magnetic field (compare with Gentile Fusillo et al. 2019, their Fig. 13, especially WD J075227.93+195314.41, a known magnetic WD). Assuming that the tertiary is indeed magnetic, the origin of the magnetic field could be the recently proposed crystallization-driven dynamo (e.g., Isern et al. 2017; Schreiber et al. 2021, 2022); a late-appearing field generated in the convective core of the main sequence progenitor (Camisassa et al. 2024); or, especially because of the very massive WD, the result of a merger (García-Berro et al. 2012).

The alternative to the magnetic hydrogen atmosphere WD is that the tertiary is a WD with a helium-dominated atmosphere. Helium atmosphere WDs are not expected to show absorption lines at low temperatures, which would fit the absence of clear absorption lines in the spectrum, but does not offer an explanation for the broad features that seem to be real. Given the uncertainty in the atmospheric composition of the tertiary WD, we determined the cooling age using both hydrogen and helium atmosphere models.

4. Determining the system’s age

We retrieved theoretical cooling tracks for cool massive WDs from Bédard et al. (2020) and Althaus et al. (2023), which we interpolated linearly. Using the parallax of the inner binary and the synthetic Gaia and SDSS magnitudes provided for both cooling models, we applied the Levenberg–Marquardt algorithm to fit the observed data. Assuming a hydrogen-rich atmosphere we derived Teff = 6401 ± 13 K, log(g) = 8.967 ± 0.002 (Bédard et al. 2020) and Teff = 6425 ± 15 K, log(g) = 8.883 ± 0.002 (Althaus et al. 2023). Assuming a helium-dominated atmosphere, only the models of Bédard et al. (2020) cover a sufficiently large parameter range and we obtained Teff = 6211 ± 14 K, log(g) = 8.939 ± 0.003. The hydrogen-rich models suggest ages of 5.13 ± 0.01 Gyr and 4.322 ± 0.008 Gyr, respectively. For a helium-rich atmosphere we estimate an age of 4.60 ± 0.01 Gyr.

The uncertainties given above are purely statistical. Based on the two independent measurements for hydrogen atmosphere models, we estimate a systematic uncertainty of ∼0.57 Gyr for the cooling age. For typically quoted systematic uncertainties on Teff and log(g) of five and one percent, respectively (Izquierdo et al. 2023), we obtain a similar systematic uncertainty on the WD age (∼0.45 Gyr). We thus conclude that, independent of the cooling model and its atmospheric composition, the WD is older than ∼4 Gyr. The possibility that the tertiary might be a massive magnetic WD that formed through a merger event implies that the tertiary, and therefore the entire system, might in fact be much older than 4 Gyr. In Fig. 1 we show the SDSS and Gaia photometry, the spectrum taken with Keck/LRIS, and the synthetic spectra calculated with the WD atmosphere code from Koester (2010) using the parameters obtained from interpolating the tables of Bédard et al. (2020).

5. Binary evolution scenario for SDSS J1257+5428 and remaining problems

The tertiary WD companion to SDSS J1257+5428 allowed us to derive a lower limit on the age of this triple system, which provides additional constraints on the evolutionary history of the inner binary. The age limit of ∼4 Gyr clearly implies that the progenitor of the more massive WD in the inner binary or the WD itself must have gained significant amounts of mass as the total formation of a ∼1 Gyr old ∼1 M⊙ WD takes just 1.1 Gyr, which is much less time than the age of the system. A plausible scenario is therefore that the first mass transfer phase was conservative stable mass transfer. The initial binary must therefore have had an orbital period of just a few days to allow stable mass transfer to end before the core of the primary grew in mass beyond the low value we observe today.

During this first and stable mass transfer phase the orbital period increased. Therefore, and because the mass ratio of the resulting post mass transfer binary must have been huge, the second mass transfer phase must have been common envelope evolution. Based on the hypothesis of stable mass transfer followed by common envelope evolution, our aim is to develop an evolutionary scenario. Directly derived from observations are the following parameters: the cooling age and the mass of the low-mass WD are 1.6 − 5 Gyr and 0.1 − 0.24 M⊙ (Bours et al. 2015); the cooling age and mass of the younger WD are ∼1 Gyr and 1.06 ± 0.05 M⊙ (Bours et al. 2015). This implies that the difference in cooling age between the two WDs of the inner binary is ≳0.6 Gyr. The cooling age (≳4 Gyr) of the massive tertiary WD (∼1.2 M⊙) provides a lower limit to the total age of the triple system. The orbital period of the inner close double WD is 4.6 h.

These parameters, directly derived from measurements, have the following implications for the formation of the inner double WD. Using the initial-to-final mass ratio incorporated in the SSE code (Hurley et al. 2000), the mass of the progenitor star of the massive WD in the inner binary must have been ∼5 M⊙. This implies that the evolutionary timescale of the progenitor of the massive WD in the inner binary was ∼0.1 Gyr, in clear contradiction to the minimum difference in the cooling ages of both WDs of 0.6 Gyr.

For the core of the ∼5 M⊙ star to grow to 1.06 ± 0.05 M⊙, the star must have evolved toward the end of the AGB without filling its Roche lobe. This implies an orbital period during the intermediate stage of ∼200 days. In contrast, the extremely low mass of the older WD in the inner binary implies an orbital period of ≲80 days according to the WD mass final orbital period relation for stable mass transfer (Joss et al. 1987; Rappaport et al. 1995; Tauris & Savonije 1999). We note that this relation is nearly independent of the mass transfer efficiency, the mass of the progenitor of the helium core WD and its companion, as well as the angular momentum loss mechanisms (e.g., De Vito & Benvenuto 2010; Chen & Liu 2013; Jia & Li 2014).

Given the minimum age of the older WD (1.6 Gyr) and the minimum total age of the system (4 Gyr), the main sequence lifetime of the initially more massive star of the inner binary must have been at least 2.4 Gyr. This implies that each of the main sequence stars forming the initial inner binary must have had masses ≲1.8 M⊙. This upper limit on the initial masses makes it impossible to form the mass of the progenitor of the massive WD in the inner binary (∼5 M⊙) through conservative stable mass transfer.

The observational constraints and their implications for the system are illustrated in Fig. 2. The parameters derived directly from observations are highlighted in blue, while the inferred parameters are in black. The values in bold contradict each other. It is very clear that either the evolutionary scenario is missing one or more fundamental ingredients or that one of the parameters derived from the observations is incorrect.

thumbnail Fig. 2.

Illustration of the problems generated by assuming stable mass transfer followed by common envelope evolution. The parameters shown in blue correspond to the values determined directly from observation, while those in black represent the values inferred from stellar and binary star evolution. Highlighted in bold are the measured and inferred parameters that contradict each other. The existence of the inner binary continues to contradict standard white dwarf binary formation models.

6. Concluding discussion

The inconsistencies described in the previous section are mostly related to the massive WD (∼1 M⊙) in the inner binary that must have had a progenitor star of ∼5 M⊙. If this WD were less massive (e.g., 0.7 M⊙), the evolutionary timescale of its progenitor would fit the difference between the cooling ages of the WDs, and the total evolutionary timescale of the binary could easily exceed 4 Gyr, resolving two fundamental problems of the suggested formation scenario. However, the HST spectra presented by Bours et al. (2015) clearly exclude a WD mass much lower than 1.0 M⊙: fits performed assuming different fixed values for the low-mass WD consistently provide ∼1.0 M⊙ for the C/O WD (see their Table 1).

As an alternative solution, one might be inclined to challenge the estimated range of the cooling age of the low-mass WD as cooling ages of low-mass WDs are notoriously uncertain. However, Bours et al. (2015) took into account all the available cooling models, and even with the most conservative models the low-mass WD is still at least 0.6 Gyr older than the massive WD, which is in disagreement with the evolutionary timescale of the progenitor of the massive WD.

Concerning the disagreement between the period predicted by the final period-to-WD mass relation, one might consider the extremely enhanced mass loss recently proposed by Gao & Li (2023), which indeed could produce a longer period. However, this strongly enhanced mass loss would make the stable mass transfer highly nonconservative, which would require very massive initial stellar masses with very short nuclear evolution timescales in disagreement with the derived total age.

We can also exclude that the tertiary had an impact on the evolution of the inner binary. We calculated the timescale of the von Zeipel-Lidov-Kozai (ZLK) oscillations during the main sequence using Eq. (41) from Antognini (2015) estimating the initial outer semimajor axis with the adiabatic mass loss approximation and assuming an outer eccentricity of 0.7. We found that the ZLK timescale is on the order of hundreds of Gyr, which is far longer than the system’s age (and the age of the Universe). Therefore, ZLK oscillations generated by the tertiray did not affect the evolution of the inner binary.

The lower limit of the age of the tertiary that we estimated here also excludes an idea outlined by Bours et al. (2015). These authors also found that the massive WD and its young age are the main problem for any formation scenario, and thus suggested that the low-mass WD is orbiting an unseen and old neutron star and that the massive WD detected with HST is an unresolved tertiary. According to this scenario, together with the discovery of the distant WD tertiary presented here, the system would consist of four stars. However, this scenario appears impossible: if the massive and young WD formed from the evolution of a single star, its total age would correspond to ∼1.1 Gyr, in strong disagreement with the age of the distant tertiary.

Given all the above, we see only one alternative binary star formation scenario. Belloni & Schreiber (2023) recently showed that cataclysmic variables with evolved donor stars might detach when the mass fraction of the convective envelope of the stripped donor star becomes negligible (see also El-Badry et al. 2021b). This leads to the formation of detached binaries consisting of a massive WD with a low-mass helium core WD companion, which could in principle resemble the inner binary of SDSS J1257+5428. According to this scenario, the first phase of mass transfer was common envelope evolution (during which the massive WD formed) followed by stable mass transfer from an evolved donor. The massive WD would have accreted at a high rate and therefore currently appear much younger than it actually is, and the low-mass WD would be the helium core of the evolved donor that lost its entire envelope. Whether the detailed parameters of SDSS J1257+5428 can be reproduced assuming this channel probably depends on the assumed strength of the magnetic braking. This is because the period at the onset of the detached phase depends on how evolved the donor star is at the beginning of stable mass transfer. How evolved a donor can be at this moment (onset of mass transfer) without diverging towards longer periods depends in turn on the strength of magnetic braking (see Belloni & Schreiber 2023, for more details). This channel also fits with the nature (either rapidly rotating or magnetic) of the massive WD in the inner binary. This idea should therefore be explored by performing dedicated binary star evolution simulations.

Data availability

The reduced Keck spectrum is available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/693/L11

Acknowledgments

CAB, MRS, and DB are supported by ANID (grant number 21241605) and FONDECYT (grant numbers 1221059 and 3220167). MRS also acknowledges support through eRO-STEP (SA 2131/15-2 project number 414059771). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement numbers 101020057 and 101078773). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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All Tables

Table 1.

Positions, proper motions, and separation measured by Gaia, as well as SDSS and Gaia photometry of the inner binary and the tertiary.

In the text

All Figures

thumbnail Fig. 1.

Spectrum of the tertiary taken with (LRIS) at the Keck observatory. The flux calibration of the data was affected by variations in airmass or seeing between observations of the source and standard star. We therefore forced the spectrum to fit the photometry obtained by Gaia and SDSS. Balmer lines are absent in the spectrum, which might indicate either that the WD atmosphere is helium dominated or that the WD is magnetic. The broad features at ∼4200 and 4700 Å (highlighted in the inset), as well as the missing flux in the range between 4000 and 6000 Å, might be consistent with the latter scenario. Fitting the photometry assuming a hydrogen-rich atmosphere, we obtain an effective temperature of 6401 ± 13 K and log(g) = 8.967 ± 0.002, while for a helium-dominated atmosphere we obtained Teff = 6211 ± 14 K and log(g) = 8.883 ± 0.001 (errors are purely statistical here; see text for a rough estimate of the systematics). The model spectra calculated with these parameters are shown as dashed lines. Taking into account systematic errors, the corresponding cooling ages are ≳4.0 Gyr, which represents a lower limit on the total age of the triple WD.
In the text
thumbnail Fig. 2.

Illustration of the problems generated by assuming stable mass transfer followed by common envelope evolution. The parameters shown in blue correspond to the values determined directly from observation, while those in black represent the values inferred from stellar and binary star evolution. Highlighted in bold are the measured and inferred parameters that contradict each other. The existence of the inner binary continues to contradict standard white dwarf binary formation models.
In the text

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