arXiv subjects

Pier-Emmanuel Tremblay

Publications and source records attributed to Pier-Emmanuel Tremblay.

At least 55 records · Page 3Linked to original sources

arXiv2023

An HST COS ultra-violet spectroscopic survey of 311 DA white dwarfs.I. Fundamental parameters and comparative studies

White dwarf studies carry significant implications across multiple fields of astrophysics, including exoplanets, supernova explosions, and cosmological investigations. Thus, accurate determinations of their fundamental parameters (Teff and log g) are of utmost importance. While optical surveys have provided measurements for many white dwarfs, there is a lack of studies utilising ultraviolet (UV) data, particularly focusing on the warmer ones that predominantly emit in the UV range. Here, we present the medium-resolution far-UV spectroscopic survey of 311 DA white dwarfs obtained with Cosmic Origins Spectrograph (COS) onboard Hubble Space Telescope confirming 49 photometric Gaia candidates. We used 3D extinction maps, parallaxes, and hydrogen atmosphere models to fit the spectra of the stars that lie in the range 12 000 < Teff < 33 000 K, and 7 <= log g < 9.2. To assess the impact of input physics, we employed two mass-radius relations in the fitting and compared the results with previous studies. The comparisons suggest the COS Teff are systematically lower by 3 per cent on average than Balmer line fits while they differ by only 1.5 per cent from optical photometric studies. The mass distributions indicate that the COS masses are smaller by approximately 0.05 Msol and 0.02 Msol than Balmer lines and photometric masses, respectively. Performing several tests, we find that the discrepancies are either arising due to issues with the COS calibration, broadening theories for hydrogen lines, or interstellar reddening which needs further examination. Based on comparative analysis, we identify 30 binary candidates drawing attention for follow-up studies to confirm their nature.

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arXiv2023

A rotating white dwarf shows different compositions on its opposite faces

White dwarfs, the extremely dense remnants left behind by most stars after their death, are characterised by a mass comparable to that of the Sun compressed into the size of an Earth-like planet. In the resulting strong gravity, heavy elements sink toward the centre and the upper layer of the atmosphere contains only the lightest element present, usually hydrogen or helium. Several mechanisms compete with gravitational settling to change a white dwarf's surface composition as it cools, and the fraction of white dwarfs with helium atmospheres is known to increase by a factor ~2.5 below a temperature of about 30,000 K; therefore, some white dwarfs that appear to have hydrogen-dominated atmospheres above 30,000 K are bound to transition to be helium-dominated as they cool below it. Here we report observations of ZTF J203349.8+322901.1, a transitioning white dwarf with two faces: one side of its atmosphere is dominated by hydrogen and the other one by helium. This peculiar nature is likely caused by the presence of a small magnetic field, which creates an inhomogeneity in temperature, pressure or mixing strength over the surface. ZTF J203349.8+322901.1 might be the most extreme member of a class of magnetic, transitioning white dwarfs -- together with GD 323, a white dwarf that shows similar but much more subtle variations. This new class could help shed light on the physical mechanisms behind white dwarf spectral evolution.

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arXiv2023

An emerging and enigmatic spectral class of isolated DAe white dwarfs

Two recently discovered white dwarfs, WDJ041246.84$+$754942.26 and WDJ165335.21$-$100116.33, exhibit H$α$ and H$β$ Balmer line emission similar to stars in the emerging DAHe class, yet intriguingly have not been found to have detectable magnetic fields. These white dwarfs are assigned the spectral type DAe. We present detailed follow-up of the two known DAe stars using new time-domain spectroscopic observations and analysis of the latest photometric time-series data from TESS and ZTF. We measure the upper magnetic field strength limit of both stars as $B < 0.05$ MG. The DAe white dwarfs exhibit photometric and spectroscopic variability, where in the case of WDJ041246.84$+$754942.26 the strength of the H$α$ and H$β$ emission cores varies in anti-phase with its photometric variability over the spin period, which is the same phase relationship seen in DAHe stars. The DAe white dwarfs closely cluster in one region of the Gaia Hertzsprung-Russell diagram together with the DAHe stars. We discuss current theories on non-magnetic and magnetic mechanisms which could explain the characteristics observed in DAe white dwarfs, but additional data are required to unambiguously determine the origin of these stars.

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arXiv2023

The ubiquity of carbon dredge-up in hydrogen-deficient white dwarfs as revealed by GALEX

The convective dredge-up of carbon from the interiors of hydrogen-deficient white dwarfs has long been invoked to explain the presence of carbon absorption features in the spectra of cool DQ stars ($T_{\rm eff} < 10{,}000\,{\rm K}$). It has been hypothesized that this transport process is not limited to DQ white dwarfs and also operates, albeit less efficiently, in non-DQ hydrogen-deficient white dwarfs within the same temperature range. This non-DQ population is predominantly composed of DC white dwarfs, which exhibit featureless optical spectra. However, no direct observational evidence of ubiquitous carbon pollution in DC stars has thus far been uncovered. In this Letter, we analyze data from the Galaxy Evolution Explorer (GALEX) to reveal the photometric signature of ultraviolet carbon lines in most DC white dwarfs in the $8500\,{\rm K} \leq T_{\rm eff} \leq 10{,}500\,{\rm K}$ temperature range. Our results show that the vast majority of hydrogen-deficient white dwarfs experience carbon dredge-up at some point in their evolution.

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arXiv2023

Carbon dredge-up required to explain the Gaia white dwarf colour-magnitude bifurcation

The Gaia colour--magnitude diagram reveals a striking separation between hydrogen-atmosphere white dwarfs and their helium-atmosphere counterparts throughout a significant portion of the white dwarf cooling track. However, pure-helium atmospheres have Gaia magnitudes that are too close to the pure-hydrogen case to explain this bifurcation. To reproduce the observed split in the cooling sequence, it has been shown that trace amounts of hydrogen and/or metals must be present in the helium-dominated atmospheres of hydrogen-deficient white dwarfs. Yet, a complete explanation of the Gaia bifurcation that takes into account known constraints on the spectral evolution of white dwarfs has thus far not been proposed. In this work, we attempt to provide such a holistic explanation by performing population synthesis simulations coupled with state-of-the-art model atmospheres and evolutionary calculations that account for element transport in the envelopes of white dwarfs. By relying on empirically grounded assumptions, these simulations successfully reproduce the bifurcation. We show that the convective dredge-up of optically undetectable traces of carbon from the deep interior is crucial to account for the observations. Neither the convective dilution/mixing of residual hydrogen nor the accretion of hydrogen or metals can be the dominant drivers of the bifurcation. Finally, we emphasize the importance of improving theoretical models for the average ionization level of carbon in warm dense helium, which governs the shape of the diffusive tail of carbon and in turn the predicted amount of dredged-up carbon.

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arXiv2023

Systematic uncertainties in the characterisation of helium-dominated metal-polluted white dwarf atmospheres

White dwarf photospheric parameters are usually obtained by means of spectroscopic or photometric analysis. These results are not always consistent with each other, with the published values often including just the statistical uncertainties. The differences are more dramatic for white dwarfs with helium-dominated photospheres, so to obtain realistic uncertainties we have analysed a sample of 13 of these white dwarfs, applying both techniques to up to three different spectroscopic and photometric data sets for each star. We found mean standard deviations of < $σT_{\mathrm{eff}}$ > = 524 K, < $σ\log g$ > = 0.27 dex and < $σ\log(\mathrm{H/He})$ > = 0.31 dex for the effective temperature, surface gravity and relative hydrogen abundance, respectively, when modelling diverse spectroscopic data. The photometric fits provided mean standard deviations up to < $σT_{\mathrm{eff}}$ > = 1210 K and < $σ\log g$ > = 0.13 dex. We suggest these values to be adopted as realistic lower limits to the published uncertainties in parameters derived from spectroscopic and photometric fits for white dwarfs with similar characteristics. In addition, we investigate the effect of fitting the observational data adopting three different photospheric chemical compositions. In general, pure helium model spectra result in larger $T_{\mathrm{eff}}$ compared to those derived from models with traces of hydrogen. The $\log g$ shows opposite trends: smaller spectroscopic values and larger photometric ones when compared to models with hydrogen. The addition of metals to the models also affects the derived atmospheric parameters, but a clear trend is not found.

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arXiv2022

Spectral analysis of ultra-cool white dwarfs polluted by planetary debris

We identify two ultra-cool ($T_\mathrm{eff} < 4000$ K) metal-polluted (DZ) white dwarfs WDJ2147$-$4035 and WDJ1922$+$0233 as the coolest and second coolest DZ stars known to date with $T_\mathrm{eff} \approx 3050$ K and $T_\mathrm{eff} \approx 3340$ K, respectively. Strong atmospheric collision-induced absorption (CIA) causes the suppression of red optical and infra-red flux in WDJ1922$+$0233, resulting in an unusually blue colour given its low temperature. WDJ2147$-$4035 has moderate infra-red CIA yet has the reddest optical colours known for a DZ white dwarf. Microphysics improvements to the non-ideal effects and CIA opacities in our model atmosphere code yields reasonable solutions to observations of these ultra-cool stars. WDJ2147$-$4035 has a cooling age of over 10 Gyr which is the largest known for a DZ white dwarf, whereas WDJ1922$+$0233 is slightly younger with a cooling age of 9 Gyr. Galactic kinematics calculations from precise Gaia EDR3 astrometry reveal these ultra-cool DZ stars as likely members of the Galactic disc thus they could be pivotal objects in future studies constraining an upper age limit for the disc of the Milky Way. We present intermediate-resolution spectroscopy for both objects, which provides the first spectroscopic observations of WDJ2147$-$4035. Detections of sodium and potassium are made in both white dwarfs, in addition to calcium in WDJ1922$+$0233 and lithium in WDJ2147$-$4035. We identify the magnetic nature of WDJ2147$-$4035 from Zeeman splitting in the lithium line and also make a tentative detection of carbon, so we classify this star as DZQH. WDJ1922$+$0233 likely accreted planetary crust debris, while the debris composition that polluted WDJ2147$-$4035 remains unconstrained.

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arXiv2022

When Do Stars Go BOOM?

The maximum mass of a star that can produce a white dwarf (WD) is an important astrophysical quantity. One of the best approaches to establishing this limit is to search for WDs in young star clusters in which only massive stars have had time to evolve and where the mass of the progenitor can be established from the cooling time of the WD together with the age of the cluster. Searches in young Milky Way clusters have not thus far yielded WD members more massive than about 1.1$~M_{\odot}$, well below the Chandrasekhar mass of $1.38~M_{\odot}$, nor progenitors with masses in excess of about $6~M_{\odot}$. However, the hunt for potentially massive WDs that escaped their cluster environs is yielding interesting candidates. To expand the cluster sample further, we used HST to survey four young and massive star clusters in the Magellanic Clouds for bright WDs that could have evolved from stars as massive as 10$~M_{\odot}$. We located five potential WD candidates in the oldest of the four clusters examined, the first extragalactic single WDs thus far discovered. As these hot WDs are very faint at optical wavelengths, final confirmation will likely have to await spectroscopy with 30-metre class telescopes.

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arXiv2022

A white dwarf accreting planetary material determined from X-ray observations

The atmospheres of a large proportion of white dwarf stars are polluted by heavy elements that are expected to sink out of visible layers on short timescales. This has been interpreted as a signature of ongoing accretion of debris from asteroids, comets, and giant planets. This scenario is supported by the detection of debris discs and transits of planetary fragments around some white dwarfs. However, photospheric metals are only indirect evidence for ongoing accretion, and the inferred accretion rates and parent body compositions heavily depend on models of diffusion and mixing processes within the white dwarf atmosphere. Here we report a 4.4$σ$ detection of X-rays from a polluted white dwarf, G29$-$38, using a 106 ks exposure with the Chandra X-ray Observatory, demonstrating directly that the star is currently accreting. From the measured X-ray luminosity, we find an instantaneous accretion rate of $\dot{M_{\rm X}}=1.63^{+1.29}_{-0.40}\times 10^{9}\mathrm{\,g\,s^{-1}}$. This is the first direct measurement of the accretion rate onto the white dwarf, which is independent of stellar atmosphere models. This rate exceeds estimates based on past studies of the photospheric abundances by more than a factor two, and implies that convective overshoot has to be accounted for in modelling the spectra of debris-accreting white dwarfs. We measure a low plasma temperature of $kT=0.5\pm0.2\,\mathrm{keV}$, corroborating the predicted bombardment solution for white dwarfs accreting at low accretion rates. Offering a new method for studying evolved planetary systems, these observations provide the opportunity to independently measure the instantaneous accretion rate of planetary material, and therefore investigate the timescale of accretion onto white dwarfs, and the evolution and replenishment of debris disks.

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arXiv2022

The Atmospheric Response to High Nonthermal Electron Beam Fluxes in Solar Flares. II. Hydrogen Broadening Predictions for Solar Flare Observations with the Daniel K. Inouye Solar Telescope

Red-shifted components of chromospheric emission lines in the hard X-ray impulsive phase of solar flares have recently been studied through their 30 s evolution with the high resolution of IRIS. Radiative-hydrodynamic flare models show that these redshifts are generally reproduced by electron-beam generated chromospheric condensations. The models produce large ambient electron densities, and the pressure broadening of hydrogen Balmer series should be readily detected in observations. To accurately interpret upcoming spectral data of flares with the DKIST, we incorporate non-ideal, non-adiabatic line broadening profiles of hydrogen into the RADYN code. These improvements allow time-dependent predictions for the extreme Balmer line wing enhancements in solar flares. We study two chromospheric condensation models, which cover a range of electron beam fluxes ($1-5 \times 10^{11}$ erg s$^{-1}$ cm$^{-2}$) and ambient electron densities ($1 - 60 \times 10^{13}$ cm$^{-3}$) in the flare chromosphere. Both models produce broadening and redshift variations within 10 s of the onset of beam heating. In the chromospheric condensations, there is enhanced spectral broadening due to large optical depths at H$α$, H$β$, and H$γ$, while the much lower optical depth of the Balmer series H12$-$H16 provides a translucent window into the smaller electron densities in the beam-heated layers below the condensation. The wavelength ranges of typical DKIST/ViSP spectra of solar flares will be sufficient to test the predictions of extreme hydrogen wing broadening and accurately constrain large densities in chromospheric condensations.

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arXiv2021

Spectral analysis of cool white dwarfs accreting from planetary systems: from the UV to the optical

The accretion of planetary debris into the atmospheres of white dwarfs leads to the presence of metal lines in their spectra. Cool metal-rich white dwarfs, which left the main-sequence many Gyr ago, allow the study of the remnants of the oldest planetary systems. Despite their low effective temperatures ($T_\mathrm{eff}$), a non-neglible amount of their flux is emitted in the near ultraviolet (NUV), where many overlapping metal lines can potentially be detected. We have observed three metal-rich cool white dwarfs with the Space Telescope Imaging Spectrograph (STIS) onboard the Hubble Space Telescope (HST), and compare the results determined from the NUV data with those previously derived from the analysis of optical spectroscopy. For two of the white dwarfs, SDSSJ1038-0036 and SDSSJ1535+1247, we find reasonable agreement with our previous analysis and the new combined fit of optical and NUV data. For the third object, SDSSJ0956+5912, including the STIS data leads to a ten percent lower $T_\mathrm{eff}$, though we do not identify a convincing explanation for this discrepancy. The unusual abundances found for SDSSJ0956+5912 suggest that the accreted parent-body was composed largely of water ice and magnesium silicates, and with a mass of up to $\simeq 2\times 10^{25}$g. Furthermore SDSSJ0956+5912 shows likely traces of atomic carbon in the NUV. While molecular carbon is not observed in the optical, we demonstrate that the large quantity of metals accreted by SDSSJ0956+5912 can suppress the C$_2$ molecular bands, indicating that planetary accretion can convert DQ stars into DZs (and not DQZs/DZQs).

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arXiv2021

A young spectroscopic binary in a quintuple system part of the Local Association

Double-lined spectroscopic binaries (SB2) allow us to determine a lower limit of the masses of their components directly to test stellar models. In this work, our aim is to derive the orbital and physical parameters of GJ1284, a young SB2. We also revise the membership of this system and its two wide co-moving companions, GJ898 and GJ897AB, to a young moving group to assess, along with other youth indicators, their age. Afterwards, we compare the results from these analyses and the photometry of these systems with several pre-main-sequence evolutionary models. We determine the orbit of the GJ1284 system alongside its systemic velocity from high resolution spectra. Additionally, we use TESS photometry to derive the rotational period of the GJ1284 and its two wide companions. GJ1284 is a binary system located at approximately 16 pc with an eccentric orbit ($ e = 0.505 $) of 11.83 d period made up of an M2-M2.5 + M3-M3.5. The revised systemic velocity of $ γ= 0.84 \pm 0.14\,\mathrm{km\,s}^{-1} $ suggests that it is a member of the Local Association. The kinematics together with other activity and youth indicators imply an age of 110-800 Myr for this system and its two companions. The isochronal ages derived from the comparison of the photometry with several evolutionary models are younger than the age estimated from the activity indicators for the three co-moving systems. The masses for the components of GJ1284, derived from their luminosity and age using the different models, are not consistent with the masses derived from the photometry, except for the PARSEC models, but are compatible with dynamical masses of double-lined eclipsing binaries with similar ages and spectral types. The effect of magnetic activity in the form of spots can reconcile to some extent the photometric and dynamical masses, but is not considered in most of the evolutionary models.

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arXiv2021

Forever young white dwarfs: when stellar ageing stops

White dwarf stars are the most common end point of stellar evolution. Of special interest are the ultramassive white dwarfs, as they are related to type Ia Supernovae explosions, merger events, and Fast Radio Bursts. Ultramassive white dwarfs are expected to harbour oxygen-neon (ONe) cores as a result of single standard stellar evolution. However, a fraction of them could have carbon-oxygen (CO) cores. Recent studies, based on the new observations provided by the {\it Gaia} space mission, indicate that a small fraction of the ultramassive white dwarfs experience a strong delay in their cooling, which cannot be attributed only to the occurrence of crystallization, thus requiring an unknown energy source able to prolong their life for long periods of time. In this study we find that the energy released by $^{22}$Ne sedimentation in the deep interior of ultramassive white dwarfs with CO cores and high $^{22}$Ne content is consistent with the long cooling delay of these stellar remnants. On the basis of a synthesis study of the white dwarf population, based on Monte Carlo techniques, we find that the observations revealed by {\it Gaia} can be explained by the existence of these prolonged youth ultramassive white dwarfs. Although such a high $^{22}$Ne abundance is not consistent with the standard evolutionary channels, our results provide sustain to the existence of CO-core ultramassive white dwarfs and to the occurrence of $^{22}$Ne sedimentation.

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arXiv2021

Weighing stars from birth to death: mass determination methods across the HRD

The mass of a star is the most fundamental parameter for its structure, evolution, and final fate. It is particularly important for any kind of stellar archaeology and characterization of exoplanets. There exists a variety of methods in astronomy to estimate or determine it. In this review we present a significant number of such methods, beginning with the most direct and model-independent approach using detached eclipsing binaries. We then move to more indirect and model-dependent methods, such as the quite commonly used isochrone or stellar track fitting. The arrival of quantitative asteroseismology has opened a completely new approach to determine stellar masses and to complement and improve the accuracy of other methods. We include methods for different evolutionary stages, from the pre-main sequence to evolved (super)giants and final remnants. For all methods uncertainties and restrictions will be discussed. We provide lists of altogether more than 200 benchmark stars with relative mass accuracies between $[0.3,2]\%$ for the covered mass range of $M\in [0.1,16]\,\msun$, $75\%$ of which are stars burning hydrogen in their core and the other $25\%$ covering all other evolved stages. We close with a recommendation how to combine various methods to arrive at a "mass-ladder" for stars.

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arXiv2021

Horizontal spreading of planetary debris accreted by white dwarfs

White dwarfs with metal-polluted atmospheres have been studied widely in the context of the accretion of rocky debris from evolved planetary systems. One open question is the geometry of accretion and how material arrives and mixes in the white dwarf surface layers. Using the 3D radiation-hydrodynamics code CO$^5$BOLD, we present the first transport coefficients in degenerate star atmospheres which describe the advection-diffusion of a passive scalar across the surface-plane. We couple newly derived horizontal diffusion coefficients with previously published vertical diffusion coefficients to provide theoretical constraints on surface spreading of metals in white dwarfs. Our grid of 3D simulations probes the vast majority of the parameter space of convective white dwarfs, with pure-hydrogen atmospheres in the effective temperature range 6000-18000 K and pure-helium atmospheres in the range 12000-34000 K. Our results suggest that warm hydrogen-rich atmospheres (DA; $\gtrsim$13000 K) and helium-rich atmospheres (DB, DBA; $\gtrsim$30000 K) are unable to efficiently spread the accreted metals across their surface, regardless of the time dependence of accretion. This result may be at odds with the current non-detection of surface abundance variations at white dwarfs with debris discs. For cooler hydrogen- and helium-rich atmospheres, we predict a largely homogeneous distribution of metals across the surface within a vertical diffusion timescale. This is typically less than 0.1 per cent of disc lifetime estimates, a quantity which is revisited in this paper using the overshoot results. These results have relevance for studies of the bulk composition of evolved planetary systems and models of accretion disc physics.

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arXiv2021

Alkali metals in white dwarf atmospheres as tracers of ancient planetary crusts

White dwarfs that accrete the debris of tidally disrupted asteroids provide the opportunity to measure the bulk composition of the building blocks, or fragments, of exoplanets. This technique has established a diversity in compositions comparable to what is observed in the solar system, suggesting that the formation of rocky planets is a generic process. Whereas the relative abundances of lithophile and siderophile elements within the planetary debris can be used to investigate whether exoplanets undergo differentiation, the composition studies carried out so far lack unambiguous tracers of planetary crusts. Here we report the detection of lithium in the atmospheres of four cool (<5,000 K) and old (cooling ages 5-10 Gyr) metal-polluted white dwarfs, where one also displays photospheric potassium. The relative abundances of these two elements with respect to sodium and calcium strongly suggest that all four white dwarfs have accreted fragments of planetary crusts. We detect an infrared excess in one of the systems, indicating that accretion from a circumstellar debris disk is on-going. The main-sequence progenitor mass of this star was $4.8\pm0.2 M_\odot$, demonstrating that rocky, differentiated planets may form around short-lived B-type stars.

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arXiv2020

3D spectroscopic analysis of helium-line white dwarfs

In this paper, we present corrections to the spectroscopic parameters of DB and DBA white dwarfs with -10.0 < log(H/He) < -2.0, 7.5 < log(g) < 9.0 and 12000 K < T_eff < 34000 K, based on 282 3D atmospheric models calculated with the CO5BOLD radiation-hydrodynamics code. These corrections arise due to a better physical treatment of convective energy transport in 3D models when compared to the previously available 1D model atmospheres. By applying the corrections to an existing SDSS sample of DB and DBA white dwarfs, we find significant corrections both for the effective temperature and surface gravity. The 3D log(g) corrections are most significant for T_eff < 18000 K, reaching up to -0.20 dex at log(g) = 8.0. However, in this low effective temperature range, the surface gravity determined from the spectroscopic technique can also be significantly affected by the treatment of the neutral van der Waals line broadening of helium and by non-ideal effects due to the perturbation of helium by neutral atoms. Thus, by removing uncertainties due to 1D convection, our work showcases the need for improved description of microphysics for DB and DBA model atmospheres. Overall, we find that our 3D spectroscopic parameters for the SDSS sample are generally in agreement with Gaia DR2 absolute fluxes within 1-3σ for individual white dwarfs. By comparing our results to DA white dwarfs, we have determined that the precision and accuracy of DB/DBA atmospheric models are similar. For ease of user application of the correction functions, we provide an example Python code.

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arXiv2020

Single magnetic white dwarfs with Balmer emission lines: A small class with consistent physical characteristics as possible signposts for close-in planetary companions

We report the identification of SDSS J121929.45+471522.8 as the third apparently isolated magnetic (B~18.5+/-1.0,MG) white dwarf exhibiting Zeeman-split Balmer emission lines. The star shows coherent variability at optical wavelengths with an amplitude of ~0.03mag and a period of 15.26h, which we interpret as the spin period of the white dwarf. Modelling the spectral energy distribution and Gaia parallax, we derive a white dwarf temperature of 7500+/-148K, a mass of 0.649+/-0.022Msun, and a cooling age of 1.5+/-0.1Gyr, as well as an upper limit on the temperature of a sub-stellar or giant planet companion of ~250K. The physical properties of this white dwarf match very closely those of the other two magnetic white dwarfs showing Balmer emission lines: GD356 and SDSS J125230.93$-$023417.7. We argue that, considering the growing evidence for planets and planetesimals on close orbits around white dwarfs, the unipolar inductor model provides a plausible scenario to explain the characteristics of this small class of stars. The tight clustering of the three stars in cooling age suggests a common mechanism switching the unipolar inductor on and off. Whereas Lorentz drift naturally limits the lifetime of the inductor phase, the relatively late onset of the line emission along the white dwarf cooling sequence remains unexplained.

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