arXiv subjects

P. -E. Tremblay

Publications and source records attributed to P. -E. Tremblay.

At least 37 records · Page 2Linked to original sources

arXiv2021

A catalogue of white dwarfs in Gaia EDR3

We present a catalogue of white dwarf candidates selected from Gaia early data release three (EDR3). We applied several selection criteria in absolute magnitude, colour, and Gaia quality flags to remove objects with unreliable measurements while preserving most stars compatible with the white dwarf locus in the Hertzsprung-Russell diagram. We then used a sample of over 30 000 spectroscopically confirmed white dwarfs and contaminants from the Sloan Digital Sky Survey (SDSS) to map the distribution of these objects in the Gaia absolute magnitude-colour space. Finally, we adopt the same method presented in our previous Gaia DR2 work to calculate a probability of being a white dwarf (Pwd) for $\simeq$1.3 million sources which passed our quality selection. The Pwd values can be used to select a sample of $\simeq$359 000 high-confidence white dwarf candidates in the magnitude range 8< G <21. We calculated stellar parameters (effective temperature, surface gravity, and mass) for all these stars by fitting Gaia astrometry and photometry with synthetic models. We estimate an upper limit of 93 per cent for the overall completeness of our catalogue for white dwarfs with G $\leq$20 mag and effective temperature (Teff)>7000K, at high Galactic latitudes (|b|>20°). Alongside the main catalogue we include a reduced-proper-motion extension containing $\simeq$10 200 white dwarf candidates with unreliable parallax measurements which could, however be identified on the basis of their proper motion. We also performed a cross-match of our catalogues with SDSS DR16 spectroscopy and provide spectral classification based on visual inspection for all resulting matches.

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arXiv2021

Constraining the solar neighbourhood age-metallicity relation from white dwarf-main sequence binaries

The age-metallicity relation is a fundamental tool for constraining the chemical evolution of the Galactic disc. In this work we analyse the observational properties of this relation using binary stars that have not interacted consisting of a white dwarf - from which we can derive the total age of the system - and a main sequence star - from which we can derive the metallicity as traced by the [Fe/H] abundances. Our sample consists of 46 widely separated, but unresolved spectroscopic binaries identified within the Sloan Digital Sky Survey, and 189 white dwarf plus main sequence common proper motion pairs identified within the second data release of Gaia. This is currently the largest white dwarf sample for which the metallicity of their progenitors have been determined. We find a flat age-metallicity relation displaying a scatter of [Fe/H] abundances of approximately 0.5 dex around the solar metallicity at all ages. This independently confirms the lack of correlation between age and metallicity in the solar neighbourhood that is found in previous studies focused on analysing single main sequence stars and open clusters.

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arXiv2020

A white dwarf bound to the transiting planetary system WASP-98

WASP-98 is a planetary system containing a hot Jupiter transiting a late-G dwarf. A fainter star 12 arcsec distant has previously been identified as a white dwarf, with a distance and proper motion consistent with a physical association with the planetary system. We present spectroscopy of the white dwarf, with the aim of determining its mass, radius and temperature and hence the age of the system. However, the spectra show the featureless continuum and lack of spectral lines characteristic of the DC class of white dwarfs. We therefore fitted theoretical white dwarf spectra to the ugriz apparent magnitudes and Gaia DR2 parallax of this object in order to determine its physical properties and the age of the system. We find that the system is old, with a lower limit of 3.6 Gyr, but theoretical uncertainties preclude a precise determination of its age. Its kinematics are consistent with membership of the thick disc, but do not allow us to rule out the thin-disc alternative. The old age and low metallicity of the system suggest it is subject to an age-metallicity relation, but analysis of the most metal-rich and metal-poor transiting planetary systems yields only insubstantial evidence of this. We conclude that the study of bound white dwarfs can yield independent ages to planetary systems, but such analysis may be better-suited to DA and DB rather than DC white dwarfs.

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arXiv2020

Doppler beaming factors for white dwarfs, main sequence stars, and giant stars Limb-darkening coefficients for 3D (DA and DB) white dwarf models

We computed Doppler beaming factors for DA, DB, and DBA white dwarf models, as well as for main sequence and giant stars covering the transmission curves of the Sloan, UBVRI, HiPERCAM, Kepler, TESS, and Gaia photometric systems. The calculations of the limb-darkening coefficients for 3D models were carried out using the least-squares method for these photometric systems. The beaming factor calculations, which use realistic models of stellar atmospheres, show that the black body approximation is not accurate, particularly for the filters $u$, $u'$, $U$, $g$, $g'$, and $B$. The black body approach is only valid for high effective temperatures and/or long effective wavelengths. Therefore, for more accurate analyses of light curves, we recommend the use of the beaming factors presented in this paper. Concerning limb-darkening, the distribution of specific intensities for 3D models indicates that, in general, these models are less bright toward the limb than their 1D counterparts, which implies steeper profiles. To describe these intensities better, we recommend the use of the four-term law (also for 1D models) given the level of precision that is being achieved with Earth-based instruments and space missions such as Kepler and TESS (and PLATO in the future).

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arXiv2020

$\textit{Gaia}$ white dwarfs within 40 pc I: spectroscopic observations of new candidates

We present a spectroscopic survey of 230 white dwarf candidates within 40 pc of the Sun from the William Herschel Telescope and Gran Telescopio Canarias. All candidates were selected from $\textit{Gaia}$ Data Release 2 (DR2) and in almost all cases had no prior spectroscopic classifications. We find a total of 191 confirmed white dwarfs and 39 main-sequence star contaminants. The majority of stellar remnants in the sample are relatively cool ($\langle T_{\rm eff} \rangle$ = 6200 K), showing either hydrogen Balmer lines or a featureless spectrum, corresponding to 89 DA and 76 DC white dwarfs, respectively. We also recover two DBA white dwarfs and 9--10 magnetic remnants. We find two carbon-bearing DQ stars and 14 new metal-rich white dwarfs. This includes the possible detection of the first ultra-cool white dwarf with metal lines. We describe three DZ stars for which we find at least four different metal species, including one which is strongly Fe- and Ni-rich, indicative of the accretion of a planetesimal with core-Earth composition. We find one extremely massive (1.31 $\pm$ 0.01 M$_{\odot}$) DA white dwarf showing weak Balmer lines, possibly indicating stellar magnetism. Another white dwarf shows strong Balmer line emission but no infrared excess, suggesting a low-mass sub-stellar companion. High spectroscopic completeness ($>$99%) has now been reached for $\textit{Gaia}$ DR2 sources within 40 pc sample, in the northern hemisphere ($δ>$ 0 deg) and located on the white dwarf cooling track in the Hertzsprung-Russell diagram. A statistical study of the full northern sample is presented in a companion paper.

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arXiv2020

TESS first look at evolved compact pulsators: Known ZZ Ceti stars of the southern ecliptic hemisphere as seen by TESS

Context. We present our findings on 18 formerly known ZZ Ceti stars observed by the TESS space telescope in 120s cadence mode during the survey observation of the southern ecliptic hemisphere. Aims. We focus on the frequency analysis of the space-based observations, comparing the results with the findings of the previous ground-based measurements. The frequencies detected by the TESS observations can serve as inputs for future asteroseismic analyses. Methods. We performed standard pre-whitening of the data sets to derive the possible pulsation frequencies of the different targets. In some cases, we fitted Lorentzians to the frequency groups that emerged as the results of short-term amplitude/phase variations that occurred during the TESS observations. Results. We detected more than 40 pulsation frequencies in seven ZZ Ceti stars observed in the 120s cadence by TESS, with better than 0.1 microHz precision. We found that HE 0532-5605 may be a new outbursting ZZ Ceti. Ten targets do not show any significant pulsation frequencies in their Fourier transforms, due to a combination of their intrinsic faintness and/or crowding on the large TESS pixels. We also detected possible amplitude/phase variations during the TESS observations in some cases. Such behaviour in these targets was not previously identified from ground-based observations.

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arXiv2020

Gravity and limb-darkening coefficients for compact stars: DA, DB, and DBA eclipsing white dwarfs

The main objective of the present work is to extend these investigations by computing the gravity and limb-darkening coefficients for white dwarf atmosphere models with hydrogen, helium, or mixed compositions (types DA, DB, and DBA). We computed gravity and limb-darkening coefficients for DA, DB, and DBA white dwarfs atmosphere models, covering the transmission curves of the Sloan, UBVRI, Kepler, TESS, and Gaia photometric systems. Specific calculations for the HiPERCAM instrument were also carried out. For all calculations of the limb-darkening coefficients we used the least-squares method. Concerning the effects of tidal and rotational distortions, we also computed for the first time the gravity-darkening coefficients $y(λ)$ for white dwarfs using the same models of stellar atmospheres as in the case of limb-darkening. A more general differential equation was introduced to derive these quantities, including the partial derivative $\left(\partial{\ln I_o(λ)}/{\partial{\ln g}}\right)_{T_{\rm eff}}$. Six laws were adopted to describe the specific intensity distribution: linear, quadratic, square root, logarithmic, power-2, and a more general one with four coefficients. The computations are presented for the chemical compositions log[H/He] = $-$10.0 (DB), $-$2.0 (DBA) and He/H = 0 (DA), with log g varying between 5.0 and 9.5 and effective temperatures between 3750 K-100\,000 K. For effective temperatures higher than 40\,000 K, the models were also computed adopting nonlocal thermal equilibirum (DA). The adopted mixing-length parameters are ML2/$α= 0.8$ (DA case) and 1.25 (DB and DBA). The results are presented in the form of 112 tables. Additional calculations, such as for other photometric systems and/or different values of log[H/He], $\log g,$ and T$_{\rm eff}$ can be performed upon request.

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arXiv2019

Calibration of the mixing length theory for structures of helium-dominated atmosphere white dwarfs

We perform a calibration of the mixing length parameter at the bottom boundary of the convection zone for helium-dominated atmospheres of white dwarfs. This calibration is based on a grid of 3D DB (pure-helium) and DBA (helium-dominated with traces of hydrogen) model atmospheres computed with the CO5BOLD code, and a grid of 1D DB and DBA envelope structures. The 3D models span a parameter space of hydrogen-to-helium abundances between -10.0 and -2.0, surface gravities between 7.5 and 9.0 and effective temperatures between 12000 K and 34000 K. The 1D envelopes cover a similar atmospheric parameter range, but are also calculated with different values of the mixing length parameter, namely ML2/alpha between 0.4 and 1.4. The calibration is performed based on two definitions of the bottom boundary of the convection zone, the Schwarzschild and the zero convective flux boundaries. Thus, our calibration is relevant for applications involving the bulk properties of the convection zone including its total mass, which excludes the spectroscopic technique. Overall, the calibrated ML2/alpha is smaller than what is commonly used in evolutionary models and theoretical determinations of the blue edge of the instability strip for pulsating DB and DBA stars. With calibrated ML2/alpha we are able to deduce more accurate convection zone sizes needed for studies of planetary debris mixing and dredge-up of carbon from the core. We highlight this by calculating examples of metal-rich 3D DBAZ models and finding their convection zone masses. Mixing length calibration represents the first step of in-depth investigations of convective overshoot in white dwarfs with helium-dominated atmospheres.

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arXiv2019

J-PLUS: photometric calibration of large area multi-filter surveys with stellar and white dwarf loci

We present the photometric calibration of the twelve optical passbands observed by the Javalambre Photometric Local Universe Survey (J-PLUS). The proposed calibration method has four steps: (i) definition of a high-quality set of calibration stars using Gaia information and available 3D dust maps; (ii) anchoring of the J-PLUS gri passbands to the Pan-STARRS photometric solution, accounting for the variation of the calibration with the position of the sources on the CCD; (iii) homogenization of the photometry in the other nine J-PLUS filters using the dust de-reddened instrumental stellar locus in (X - r) versus (g - i) colours, where X is the filter to calibrate. The zero point variation along the CCD in these filters was estimated with the distance to the stellar locus. Finally, (iv) the absolute colour calibration was obtained with the white dwarf locus. We performed a joint Bayesian modelling of eleven J-PLUS colour-colour diagrams using the theoretical white dwarf locus as reference. This provides the needed offsets to transform instrumental magnitudes to calibrated magnitudes outside the atmosphere. The uncertainty of the J-PLUS photometric calibration, estimated from duplicated objects observed in adjacent pointings and accounting for the absolute colour and flux calibration errors, are ~19 mmag in u, J0378 and J0395, ~11 mmag in J0410 and J0430, and ~8 mmag in g, J0515, r, J0660, i, J0861, and z. We present an optimized calibration method for the large area multi-filter J-PLUS project, reaching 1-2% accuracy within an area of 1 022 square degrees without the need for long observing calibration campaigns or constant atmospheric monitoring. The proposed method will be adapted for the photometric calibration of J-PAS, that will observe several thousand square degrees with 56 narrow optical filters.

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arXiv2019

4MOST Consortium Survey 3: Milky Way Disc and Bulge Low-Resolution Survey (4MIDABLE-LR)

The mechanisms of the formation and evolution of the Milky Way are encoded in the orbits, chemistry and ages of its stars. With the 4MOST MIlky way Disk And BuLgE Low-Resolution Survey (4MIDABLE-LR) we aim to study kinematic and chemical substructures in the Milky Way disc and bulge region with samples of unprecedented size out to larger distances and greater precision than conceivable with Gaia alone or any other ongoing or planned survey. Gaia gives us the unique opportunity for target selection based almost entirely on parallax and magnitude range, hence increasing the efficiency in sampling larger Milky Way volumes with well-defined and effective selection functions. Our main goal is to provide a detailed chrono-chemo-kinematical extended map of our Galaxy and the largest Gaia follow-up down to $G = 19$ magnitudes (Vega). The complex nature of the disc components (for example, large target densities and highly structured extinction distribution in the Milky Way bulge and disc area), prompted us to develop a survey strategy with five main sub-surveys that are tailored to answer the still open questions about the assembly and evolution of our Galaxy, while taking full advantage of the Gaia data.

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arXiv2018

Discovery of the first resolved triple white dwarf

We report the discovery of J1953-1019, the first resolved triple white dwarf system. The triplet consists of an inner white dwarf binary and a wider companion. Using Gaia DR2 photometry and astrometry combined with our follow-up spectroscopy, we derive effective temperatures, surface gravities, masses and cooling ages of the three components. All three white dwarfs have pure-hydrogen (DA) atmospheres, masses of 0.60-0.63 Msun and cooling ages of 40-290 Myr. We adopt eight initial-to-final mass relations to estimate the main sequence progenitor masses (which we find to be similar for the three components, 1.6-2.6 Msun) and lifetimes. The differences between the derived cooling times and main sequence lifetimes agree for most of the adopted initial-to-final mass relations, hence the three white dwarfs in J1953-1019 are consistent with coeval evolution. Furthermore, we calculate the projected orbital separations of the inner white dwarf binary (303.25 +- 0.01 au) and of the centre of mass of the inner binary and the outer companion (6398.97 +- 0.09 au). From these values, and taking into account a wide range of possible configurations for the triplet to be currently dynamically stable, we analyse the future evolution of the system. We find that a collision between the two inner white dwarfs due to Lidov-Kozai oscillations is unlikely, though if it occurs it could result in a sub-Chandrasekhar Type Ia supernova explosion.

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arXiv2018

Fundamental parameter accuracy of DA and DB white dwarfs in Gaia Data Release 2

We report on a comparison of spectroscopic analyses for hydrogen (DA) and helium atmosphere (DB) white dwarfs with Gaia Data Release 2 (DR2) parallaxes and photometry. We assume a reddening law and a mass-radius relation to connect the effective temperatures (Teff) and surface gravities (log g) to masses and radii. This allows the comparison of two largely independent sets of fundamental parameters for 7039 DA and 521 DB stars with high-quality observations. This subset of the Gaia white dwarf sample is large enough to detect systematic trends in the derived parameters. We find that spectroscopic and photometric parameters generally agree within uncertainties when the expectation of a single star is verified. Gaia allows the identification of a small systematic offset in the temperature scale between the two techniques, as well as confirming a small residual high-mass bump in the DA mass distribution around 11,000-13,000 K. This assessment of the accuracy of white dwarf fundamental parameters derived from Gaia is a first step in understanding systematic effects in related astrophysical applications such as the derivation of the local stellar formation history, initial-to-final mass relation, and statistics of evolved planetary systems.

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arXiv2018

The White Dwarf Initial-Final Mass Relation for Progenitor Stars From 0.85 to 7.5 M$_\odot$

We present the initial-final mass relation (IFMR) based on the self-consistent analysis of Sirius B and 79 white dwarfs from 13 star clusters. We have also acquired additional signal on eight white dwarfs previously analyzed in the NGC 2099 cluster field, four of which are consistent with membership. These reobserved white dwarfs have masses ranging from 0.72 to 0.97 M$_\odot$, with initial masses from 3.0 to 3.65 M$_\odot$, where the IFMR has an important change in slope that these new data help to observationally confirm. In total, this directly measured IFMR has small scatter ($σ$ = 0.06 M$_\odot$) and spans from progenitors of 0.85 to 7.5 M$_\odot$. Applying two different stellar evolutionary models to infer two different sets of white dwarf progenitor masses shows that when the same model is also used to derive the cluster ages, the resulting IFMR has weak sensitivity to the adopted model at all but the highest initial masses ($>$5.5 M$_\odot$). The non-linearity of the IFMR is also clearly observed with moderate slopes at lower masses (0.08 M$_{\rm final}$/M$_{\rm initial}$) and higher masses (0.11 M$_{\rm final}$/M$_{\rm initial}$) that are broken up by a steep slope (0.19 M$_{\rm final}$/M$_{\rm initial}$) between progenitors from 2.85 to 3.6 M$_\odot$. This IFMR shows total stellar mass loss ranges from 33\% of M$_{\rm initial}$ at 0.83 M$_\odot$ to 83\% of M$_{\rm initial}$ at 7.5 M$_\odot$. Testing this total mass loss for dependence on progenitor metallicity, however, finds no detectable sensitivity across the moderate range of -0.15 $<$ [Fe/H] $<$ +0.15.

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arXiv2018

Pure-helium 3D model atmospheres of white dwarfs

We present the first grid of 3D simulations for the pure-helium atmospheres of DB white dwarfs. The simulations were computed with the CO5BOLD radiation-hydrodynamics code and cover effective temperatures and surface gravities between 12000 K < Teff < 34000 K and 7.5 < log(g) (cgs units) < 9.0, respectively. In this introductory work, synthetic spectra calculated from the 3D simulations are compared to appropriate 1D model spectra under a differential approach. This results in the derivation of 3D corrections for the spectroscopically derived atmospheric parameters of DB stars with respect to the 1D ML2/alpha = 1.25 mixing-length parameterisation. No significant Teff corrections are found for the V777 Her instability strip region, and therefore no 3D revision is expected for the empirical blue and red edges of the strip. However, large log(g) corrections are found in the range 12000 K < Teff < 23000 K for all log(g) values covered by the 3D grid. These corrections indicate that 1D model atmospheres overpredict log(g), reminiscent of the results found from 3D simulations of pure-hydrogen white dwarfs. The next step will be to compute 3D simulations with mixed helium and hydrogen atmospheres to comprehend the full implications for the stellar parameters of DB and DBA white dwarfs.

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arXiv2017

Destroying Aliases from the Ground and Space: Super-Nyquist ZZ Cetis in K2 Long Cadence Data

With typical periods of order 10 minutes, the pulsation signatures of ZZ Ceti variables (pulsating hydrogen-atmosphere white dwarf stars) are severely undersampled by long-cadence (29.42 minutes per exposure) K2 observations. Nyquist aliasing renders the intrinsic frequencies ambiguous, stifling precision asteroseismology. We report the discovery of two new ZZ Cetis in long-cadence K2 data: EPIC 210377280 and EPIC 220274129. Guided by 3-4 nights of follow-up, high-speed (<=30 s) photometry from McDonald Observatory, we recover accurate pulsation frequencies for K2 signals that reflected 4-5 times off the Nyquist with the full precision of over 70 days of monitoring (~0.01 muHz). In turn, the K2 observations enable us to select the correct peaks from the alias structure of the ground-based signals caused by gaps in the observations. We identify at least seven independent pulsation modes in the light curves of each of these stars. For EPIC 220274129, we detect three complete sets of rotationally split ell=1 (dipole mode) triplets, which we use to asteroseismically infer the stellar rotation period of 12.7+/-1.3 hr. We also detect two sub-Nyquist K2 signals that are likely combination (difference) frequencies. We attribute our inability to match some of the K2 signals to the ground-based data to changes in pulsation amplitudes between epochs of observation. Model fits to SOAR spectroscopy place both EPIC 210377280 and EPIC 220274129 near the middle of the ZZ Ceti instability strip, with Teff = 11590+/-200 K and 11810+/-210 K, and masses 0.57+/-0.03 Msun and 0.62+/-0.03 Msun, respectively.

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arXiv2017

White Dwarf Rotation as a Function of Mass and a Dichotomy of Mode Linewidths: Kepler Observations of 27 Pulsating DA White Dwarfs Through K2 Campaign 8

We present photometry and spectroscopy for 27 pulsating hydrogen-atmosphere white dwarfs (DAVs, a.k.a. ZZ Ceti stars) observed by the Kepler space telescope up to K2 Campaign 8, an extensive compilation of observations with unprecedented duration (>75 days) and duty cycle (>90%). The space-based photometry reveals pulsation properties previously inaccessible to ground-based observations. We observe a sharp dichotomy in oscillation mode linewidths at roughly 800 s, such that white dwarf pulsations with periods exceeding 800 s have substantially broader mode linewidths, more reminiscent of a damped harmonic oscillator than a heat-driven pulsator. Extended Kepler coverage also permits extensive mode identification: We identify the spherical degree of 61 out of 154 unique radial orders, providing direct constraints of the rotation period for 20 of these 27 DAVs, more than doubling the number of white dwarfs with rotation periods determined via asteroseismology. We also obtain spectroscopy from 4m-class telescopes for all DAVs with Kepler photometry. Using these homogeneously analyzed spectra we estimate the overall mass of all 27 DAVs, which allows us to measure white dwarf rotation as a function of mass, constraining the endpoints of angular momentum in low- and intermediate-mass stars. We find that 0.51-to-0.73-solar-mass white dwarfs, which evolved from 1.7-to-3.0-solar-mass ZAMS progenitors, have a mean rotation period of 35 hr with a standard deviation of 28 hr, with notable exceptions for higher-mass white dwarfs. Finally, we announce an online repository for our Kepler data and follow-up spectroscopy, which we collect at http://www.k2wd.org.

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arXiv2017

Multi-band photometry and spectroscopy of an all-sky sample of bright white dwarfs

The upcoming NASA Transiting Exoplanet Survey Satellite (TESS) will obtain space-based uninterrupted light curves for a large sample of bright white dwarfs distributed across the entire sky, providing a very rich resource for asteroseismological studies and the search for transits from planetary debris. We have compiled an all-sky catalogue of ultraviolet, optical, and infrared photometry as well as proper motions, which we propose as an essential tool for the preliminary identification and characterisation of potential targets. We present data for 1864 known white dwarfs and 305 high-probability white dwarf candidates brighter than 17 mag. We describe the spectroscopic follow-up of 135 stars, of which 82 are white dwarfs and 25 are hot subdwarfs. The new confirmed stars include six pulsating white dwarf candidates (ZZ Cetis), and nine white dwarf binaries with a cool main-sequence companion. We identify one star with a spectroscopic distance of only 25 pc from the Sun. Around the time TESS is launched, we foresee that all white dwarfs in this sample will have trigonometric parallaxes measured by the ESA Gaia mission next year.

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arXiv2017

Two white dwarfs in ultrashort binaries with detached, eclipsing, likely substellar companions detected by K2

Using data from the extended Kepler mission in K2 Campaign 10 we identify two eclipsing binaries containing white dwarfs with cool companions that have extremely short orbital periods of only 71.2 min (SDSS J1205-0242, a.k.a. EPIC 201283111) and 72.5 min (SDSS J1231+0041, a.k.a. EPIC 248368963). Despite their short periods, both systems are detached with small, low-mass companions, in one case a brown dwarf, and the other case either a brown dwarf or a low-mass star. We present follow-up photometry and spectroscopy of both binaries, as well as phase-resolved spectroscopy of the brighter system, and use these data to place preliminary estimates on the physical and binary parameters. SDSS J1205-0242 is composed of a $0.39\pm0.02$M$_\odot$ helium-core white dwarf which is totally eclipsed by a $0.049\pm0.006$M$_\odot$ ($51\pm6$M$_J$) brown dwarf companion, while SDSS J1231+0041 is composed of a $0.56\pm0.07$M$_\odot$ white dwarf which is partially eclipsed by a companion of mass $\lesssim 0.095$M$_\odot$. In the case of SDSS J1205-0242 we look at the combined constraints from common-envelope evolution and brown dwarf models; the system is compatible with similar constraints from other post common-envelope binaries given the current parameter uncertainties, but has potential for future refinement.

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