The Gaia DR 2 catalogue of white dwarfs

The $Gaia$ Data Release 2 catalogue of white dwarfs and a comparison with SDSS

Abstract

We present a catalogue of white dwarf candidates selected from the second data release of Gaia (DR2). We used a sample of spectroscopically confirmed white dwarfs from the Sloan Digital Sky Survey (SDSS) to map the entire space spanned by these objects in the Gaia Hertzsprung-Russell diagram. We then defined a set of cuts in absolute magnitude, colour, and a number of Gaia quality flags to remove the majority of contaminating objects. Finally, we adopt a method analogous to the one presented in our earlier SDSS photometric catalogues to calculate a probability of being a white dwarf () for all Gaia sources which passed the initial selection. The final catalogue is composed of  stars with calculated  from which it is possible to select a sample of high-confidence white dwarf candidates in the magnitude range . We derive atmospheric parameters for most high-confidence degenerate stars using Gaia data and find them to be in good agreement with previously available photometric data sets. By comparing this catalogue with an independent sample of SDSS white dwarf candidates we estimate an upper limit in completeness of 81 per cent for white dwarfs with  mag and   K, at high Galactic latitudes (). However, the completeness drops at low Galactic latitudes, and the magnitude limit of the catalogue varies significantly across the sky as a function of Gaia’s scanning law. We also provide the list of objects within our sample with available SDSS spectroscopy. We use this spectroscopic sample to characterise the observed structure of the white dwarf distribution in the H-R diagram.

keywords:
white dwarfs - surveys - catalogues

1 Introduction

All stars with main sequence masses (Iben et al., 1997; Dobbie et al., 2006a) share the same common fate: they will one day evolve into white dwarfs, dense stellar embers destined to cool over billions of years (Fontaine et al., 2001; Althaus et al., 2010). This broad mass range includes over 90 per cent of all stars in the Galaxy. This makes white dwarfs significant contributors to the global stellar population and, thanks to their well defined cooling rates, accurate tracers of the formation and evolution of the Milky Way (e.g., Winget et al. 1987; Torres et al. 2005; Tremblay et al. 2014). The diagnostic potential of the Galactic white dwarf population can only be fully exploited once we have large, homogeneous, and well-defined samples of white dwarfs. Given the intrinsic low luminosities and relatively high proper motions of stellar remnants, these samples have been historically challenging to assemble.

The fundamental properties of white dwarfs (mass, cooling age, atmospheric and internal composition) can be determined from spectroscopic, photometric or asteroseismic analyses (Bergeron et al., 1992, 2001; Koester et al., 2009; Bergeron et al., 2011; Tremblay et al., 2013; Romero et al., 2017; Giammichele et al., 2018). These parameters are essential to constrain and calibrate stellar evolution theory. Important examples are the mass loss on the AGB (intimately linked to the initial-to-final mass relation, e.g., Weidemann 1977; Dobbie et al. 2006b; Williams et al. 2009; Kalirai et al. 2014; Romero et al. 2015; Cummings et al. 2016), internal rotation profiles and loss of angular momentum (Charpinet et al., 2009; Hermes et al., 2017), and fundamental nuclear reaction rates (Kunz et al., 2002). If the fundamental parameters of stellar remnants are accurately constrained for large and well understood samples, Galactic evolution can be derived from the space density (Holberg et al., 2002; Holberg et al., 2008; Giammichele et al., 2012; Sion et al., 2014; Hollands et al., 2018), kinematic properties (Wegg & Phinney, 2012), mass distribution (Bergeron et al., 1992; Liebert et al., 2005; Falcon et al., 2010; Tremblay et al., 2013; Tremblay et al., 2016), and age or luminosity distributions (Catalán et al., 2008; Giammichele et al., 2012; Tremblay et al., 2014; Rebassa-Mansergas et al., 2015; Kilic et al., 2017).

Large, well-defined samples are also the necessary starting point in searching for rare sub-types of white dwarfs like: magnetic white dwarfs (Gänsicke et al., 2002; Schmidt et al., 2003; Külebi et al., 2009; Kepler et al., 2013; Hollands et al., 2015), pulsating stars (Castanheira et al. 2004; Greiss et al. 2014; Gentile Fusillo et al. 2016), white dwarfs at the extremes of the mass distribution (Vennes & Kawka, 2008; Brown et al., 2010; Hermes et al., 2014), stellar remnants with unresolved low mass companions (Farihi et al., 2005; Girven et al., 2011; Steele et al., 2013), exotic atmospheric compositions (Schmidt et al., 1999; Dufour et al., 2010; Gänsicke et al., 2010; Kepler et al., 2016a), close double-degenerates (Marsh et al., 2004; Parsons et al., 2011), metal polluted white dwarfs (Sion et al., 1990; Zuckerman & Reid, 1998; Dufour et al., 2007b; Koester et al., 2014; Raddi et al., 2015) or degenerate stars with dusty or gaseous planetary debris discs (Gänsicke et al., 2006; Farihi et al., 2009; Debes et al., 2011; Wilson et al., 2014; Manser et al., 2016). Each one of these exotic sub-classes has extremely powerful applications in diverse areas of astronomy, from exo-planetary science to type Ia SN and cosmology.

Historic methods to identify white dwarfs include searches for UV-excess objects (e.g., the Palomar Green Survey; Green et al. 1986, and the Hamburg/ESO survey; Wisotzki et al. 1996, Hamburg Quasar ), which are restricted to the detection of blue and thus relatively hot and young white dwarfs, and the use of reduced proper motion as a proxy for their distance (e.g., Luyten 1979; Lépine & Shara 2005; Gentile Fusillo et al. 2015a), which allows the recovery of the faint end of the luminosity function. The vast majority of the spectroscopically confirmed white dwarfs known to date were discovered in the last 20 years thanks to large area spectroscopic surveys, most notably the Sloan Digital Sky Survey (SDSS; York et al. 2000, Eisenstein et al. 2006, Kleinman et al. 2013, Kepler et al. 2016b). While this sample has been of fundamental importance for many white dwarf population studies to date, it suffers from severe selection biases, is largely incomplete, and is dominated by relatively hot ( 10000 K) and young stars (cooling age 1 Gyr). Furthermore, the full extent of the selection effects in the SDSS white dwarfs (non-static observing strategy, colour bias, magnitude limits, etc) are very difficult to quantify (De Gennaro et al., 2008; Gentile Fusillo et al., 2015a; Tremblay et al., 2016), as the vast majority of these objects were serendipitous discoveries. Consequently, the southern hemisphere remains a largely unexplored territory with 15 per cent of the white dwarfs known prior to Gaia DR2 located below the celestial equator. While large catalogues of white dwarf candidates based on colours and reduced proper motion compiled by, e.g., Harris et al. 2003, Gentile Fusillo et al. 2015a, Munn et al. 2017, and Gentile Fusillo et al. 2017a circumvent many of the biases of the spectroscopic samples, they are still limited by the availability of deep multi-band photometry and accurate proper motions.

The European Space Agency (ESA) astrometric mission Gaia is the successor of the Hipparcos mission. Gaia determined positions, parallaxes, and proper motions for per cent of the stars in the Galaxy and it aims to be complete across the full sky down to Gaia magnitudes (Perryman et al., 2001; Gaia Collaboration et al., 2016). The final data release is expected to include between 250 000 and 500 000 white dwarfs and among those, 95 per cent will have a parallax precision better than 10 per cent (Torres et al., 2005; Carrasco et al., 2014).

Gaia Data Release 1 only included six directly detected degenerate stars (Tremblay et al., 2017). By contrast, Gaia Data Release 2 (DR2; Gaia Collaboration et al. 2018a) is more complete by orders of magnitude, and it provides precise astrometry (Lindegren et al., 2018) as well as (330–680 nm), (640–1000 nm), and (330–1000 nm) passband photometry (Evans et al., 2018). We note that Gaia low-resolution spectrophotometry is not yet available in DR2. Furthermore, Radial Velocity Spectrometer (RVS) measurements in the region of the Ca triplet around 860 nm (Katz et al., 2018) are of little relevance for white dwarfs as most of them are featureless in this region or too faint.

Gaia DR2 allows for the first time the identification of field white dwarfs in an absolute magnitude versus colour (Hertzsprung-Russell, H-R) diagram, a method that has successfully been employed in the past 20 years to identify white dwarfs in clusters (see, e.g., Renzini et al., 1996; Richer et al., 1997). This represents the greatest opportunity to identify a large catalogue of white dwarfs over the entire sky independently of colours or proper motions.

Following on from our overview of Gaia stellar remnants in the local 20 pc sample (Hollands et al., 2018), we present a catalogue of high-confidence white dwarf candidates selected from Gaia DR2 based on their Gaia parallax and photometry. This catalogue is meant to include all single and double Gaia white dwarfs that have a colour and a reliable parallax in DR2. Our catalogue includes a number of unresolved white dwarf plus main-sequence binaries as well as extremely-low-mass (ELM) white dwarfs but it is not in any way complete for these stellar types. We compare this Gaia catalogue with a new, carefully constructed sample of SDSS white dwarf candidates and assess the robustness of our Gaia selection in terms of the sky completeness of the resulting magnitude-limited sample.

2 White dwarf selection

In order to assess the total parameter space spanned by stellar remnants in the Gaia H-R diagram, we began by retrieving all available Gaia data for the spectroscopically confirmed SDSS white dwarfs and contaminant objects contained in the catalogue of Gentile Fusillo et al. (2015a). We used this sample to define a set of broad cuts in H-R space which exclude the vast majority of the main sequence, subdwarfs and quasars while preserving the confirmed white dwarfs (Fig. 1).

 parallax_over_error >1 (1)
 and Gabs >5 (2) and Gabs >5.93+5.047×(GBP−GRP) (3) and Gabs >6×(GBP−GRP)3 −21.77×(GBP−GRP)2+ +27.91×(GBP−GRP)+0.897 (4) \textscand (GBP−GRP) <1.7 , (5)

where  is defined as:
phot_g_mean_mag. For objects with low-accuracy parallax measurements, absolute magnitude values calculated in this way can be significantly different from the true absolute magnitudes. However, for the purpose of our selection, we are only interested in the overall distribution of different objects in the H-R diagram. We do not use the values to infer the properties of any single object. The initial cuts still include 8 144 735 Gaia sources and serve primarily to limit the space within which to carry out further selections (Table 1). The resulting Gaia DR2 sample contains large numbers of objects with bad astrometry and/or photometry, and unreliable detections and hence an additional refined selection within this sample is necessary to establish a clean catalogue of white dwarf candidates. We cross-matched the positions of all objects within our initial cuts with all 4 851 200 spectra currently available within SDSS DR14. We found a total of objects with spectra and proceeded to separate white dwarfs and contaminants by visual inspection. The contaminant objects are dominated by main sequence stars and subdwarfs, but a small number of quasars also clear our initial selection. This SDSS spectroscopic sample is described in more details in Section 5. These spectroscopically confirmed white dwarfs and contaminants can be used to test the effect that any additional quality filtering will have on the completeness of our white dwarf selection. Unless otherwise stated, we use for our filtering the measurements and flags provided in DR2.

The flags with the largest impact on our selection are phot_bp_rp_excess_factor, astrometric_sigma5d_max and astrometric_excess_noise. The value of phot_bp_rp_excess_factor , where is the flux) indicates whether the three Gaia photometric bands are consistent with the assumption of an isolated source and can be used to identify objects with unreliable colours or a bright sky background (see Section 8 of Evans et al., 2018). astrometric_excess_noise is a measure of the residuals in the astrometric solution for the source, and can therefore be used to identify objects with unreliable parallax measurements (Lindegren et al., 2018). astrometric_sigma5d_max is a five-dimensional equivalent to the semi-major axis of the Gaia position error ellipse and is useful for filtering out cases where one of the five parameters, or some linear combination of several parameters, is particularly bad (Lindegren et al., 2018). The quality cuts in astrometric_excess_noise and phot_bp_rp_excess_factor proposed in Gaia Collaboration et al. (2018b) do indeed provide a very clean sample of objects, but they also exclude over 15 per cent of the known SDSS white dwarfs brighter than . Striving to construct a sample as complete as possible we defined the following set of quality cuts based on Gaia flags and measurements, which exclude non-white dwarf contaminants and objects with poor measurements, while preserving most of the degenerate stars.

 \textscsqrt(\textscpmra2+\textscpmdec2) >2 (6) \textscand \textscphot_bp_rp_excess_factor <(1.7+0.06 ×\textscbp_rp2) (7) \textscand (\textscastrometric_sigma5d_max <1.5 \textscor (8) (\textscastrometric_excess_noise <1 \textscand \textscparallax_over_error >4 \textscand \textscsqrt(\textscpm_ra2+\textscpm_dec2) >10))

These quality cuts exclude only per cent of the known SDSS white dwarfs with , while removing 25 per cent of the similarly bright SDSS contaminants. Using this selection also eliminates the vast majority of Gaia objects with poor measurements, bringing the sample size to 774 090 (Fig. 2). Beyond , however, the quality of Gaia photometric and astrometric data quickly deteriorates and only 56 per cent of the known SDSS white dwarfs with are retrieved by our quality cuts. Gaia measurements get significantly worse in areas of the sky with high stellar density and main-sequence stars located in the Galactic plane have such large scatter in the H-R diagram that they contaminate most of the white dwarf locus. In order to filter these sources we calculated a density parameter for all 774 090 objects included by Eqs. 1-8. This was done by dividing up the sky into approximately 10’10’ tiles, with sides defined by lines of constant RA and Dec. The number of Gaia DR2 targets in each tile was calculated and converted to a density of targets per square degree. Then each catalogue object was assigned the density corresponding to the tile that it fell into. We then proceeded to apply stricter quality filters on objects in the plane as

 |b| <25∘ (9) and density >100000 (10) and phot_bp_rp_excess_factor >(1.0+0.015 (11) ×\textscbp_rp2) and phot_bp_rp_excess_factor <(1.3+0.06 (12) ×\textscbp_rp2)

This additional filtering removes objects from the sample. Visual inspection of the distribution in H-R space of our sample of SDSS spectroscopic objects and of the remaining 644 790 Gaia objects (Table 1), reveals an over-density of Gaia sources in areas scarcely populated by SDSS objects. These over-abundant Gaia objects are almost exclusively located in the Magellanic clouds. This over-density is therefore spurious and we need to remove extra-galactic sources from our sample as efficiently as possible while attempting to preserve foreground stars. We select two broad areas which encompass the Magellanic clouds defined as two rectangles, the first one centred on extends in right ascension and in declination; the second one centred on extends in right ascension and in declination. Within this space we adopt the following further filtering on objects in crowded areas:

 density >11000 (13) and parallax_over_error >10 (14)

This final filtering brings the size of our Gaia sample to the final value of   (Fig. 2, Table 1).

Figure 3 shows that, even after applying all the filtering on Gaia parameters described above, there remains an overlap between white dwarfs and other stars. Consequently selecting white dwarfs with any cut in H-R space alone would result in an incomplete, inhomogeneous, and contaminated sample. To overcome this problem, we developed a selection analogous to the one presented in Gentile Fusillo et al. (2015a), i.e. we use the sample of spectroscopically confirmed SDSS white dwarfs and contaminants we developed, to calculate probabilities of being a white dwarf () for all objects in our Gaia sample. We used a total of 20 759 spectroscopically confirmed white dwarfs and 2449 contaminants (1481 stars, 965 QSOs and 3 Galaxies) still included in the Gaia sample after all the quality filtering described above. We used these objects to map the distribution of white dwarfs and contaminants in H-R (, ) space (Fig. 3). In order to create a continuous map, every object was treated as a 2D Gaussian, the width of which reflects the uncertainty in vs.  space of the object. These Gaussians were normalised so that their volume equals unity and therefore the sum of the integrals of all the Gaussians in the map is equal to the number of objects in the training sample. This results in two continuous smeared-out density maps for white dwarfs and for contaminants. We then defined a probability map as the ratio of the white dwarf density map to the sum of both density maps and then use this map to calculate the  of any given object by integrating the product of its Gaussian distribution in H-R space with the underlying probability map, giving a direct indication of how likely it is for the source to be a white dwarf. Regions outside our H-R cuts (Eqs. 1-4) are considered to have zero probability of being a white dwarf. Our SDSS training sample contained only some objects with very blue ( ) colours or large absolute magnitudes () resulting in a patchy probability map with large areas with no information.  values calculated for objects in these regions are not reliable. These regions cover areas of the H-R diagram that we assume should be populated only by white dwarfs. Therefore we defined two polynomial lines (Eqs. 15-16) in H-R space,

 Gabs >GBP−GRP×68.42+59.50 (15) Gabs >GBP−GRP)5×0.25 −(GBP−GRP)4×1.3 +(GBP−GRP)3×2.14 −(GBP−GRP)2×0.98 +(GBP−GRP)×1.37+13.98 (16)

below which any object has as default. In the final catalogue we include a flag to indicate which objects have been assigned a  in this way (Fig. 4). Selecting sub-samples of white dwarf candidates based on  allows for a flexible compromise between completeness and level of potential contamination. As a generic guideline selecting objects with recovers 95 per cent of the spectroscopically confirmed SDSS white dwarfs in the catalogue and only 4 per cent of the contaminant objects. Cleaner, but significantly less complete, white dwarf subsets can be obtained by combining our  values with additional cuts in Gaia quality parameters (e.g., astrometric_excess_noise) stricter than those already adopted in our selection. In total we estimate the final catalogue to contain genuine white dwarfs, nearly an eight-fold increase in sample size compared to the number of white dwarfs known before the release of Gaia DR2 (Fig. 5).

3 The catalogue of white dwarfs

The selection described in Section 2 results in a sample of   sources for which we calculated  values based on the distribution of spectroscopically confirmed SDSS white dwarfs and contaminants in colour and  magnitude. We define this sample as our final catalogue of Gaia DR2 white dwarf candidates. With this catalogue we aim to be as complete as possible in recovering single white dwarfs and double degenerate binaries with reliable Gaia data. However, as illustrated in Fig. 6 a number of subdwarfs, white dwarf plus main sequence binaries and cataclysmic variables (CVs) could also be included in our selection. We are not able to completely exclude these objects from our catalogue nor do we aim to be fully inclusive of them. While the vast majority of subdwarfs included in the final selection will have relatively low  values, some white dwarf main sequence binaries, CVs and extremely low mass white dwarfs (ELM) will have  values comparable to those of typical single white dwarfs.

The format of the catalogue is described in detail in Table 2. Column of the catalogue contains the WD J name we assigned to the objects following the new convention described in Section 3.1. Columns 3-28 are directly acquired from the DR2 gaia_source table (Gaia Collaboration et al., 2018a). We also include the SDSS name of all white dwarfs with a reliable SDSS match. We find that the SDSSDR9_best_neighbour table provided in Gaia DR2 does not include 2375 SDSS objects which have reliable matches in Gaia. We were not able to determine any specific selection effect which caused these objects to be excluded. Therefore we performed our own cross match between Gaia and SDSS using a matching radius of two arcseconds and accounting for the difference in epoch of observation and the proper motion of each object. Some SDSS photometric objects which appear in SDSS DR7 were not included again in subsequent data releases (Gentile Fusillo et al., 2015a), so our cross-match with SDSS was carried out both on DR7 and DR14.

The entries in the remaining columns (43-56) result from our model fit to the data for a subset of the sample and are fully described in Section 4.

The Gaia catalogue does not include the error on magnitude but this can be easily calculated from the relative error on the electron flux per second as

 σ(GS)=2.5ln(10)σ(fS)fS (17)

where refers to any Gaia passband, is the magnitude, and is the flux.

3.1 A new white dwarf naming convention

As in many areas of astronomy, individual white dwarfs are often known by more than one name, e.g. vMa2 = Wolf 28 = EGGR 5 = WD 0046+051 or GD362 = G 204–14 = NLTT 44986 = EGGR 545 = WD 1729+371, which is often source of confusion. McCook & Sion (1987) introduced the “WD number” as a unifying identifier, which is composed of the first four digits of right ascension (hours and minutes), the sign of the declination, the first two digits of the declination (degrees) and a third digit which expresses the minutes of the declination as a truncated fraction of a degree, where the coordinates are in the 1950 equinox. Over the last two decades, the number of known white dwarfs has rapidly grown from just over 2 000 (McCook & Sion, 1999) to well over 30 000 (Eisenstein et al., 2006; Kleinman et al., 2013; Gentile Fusillo et al., 2015a; Kepler et al., 2015, 2016c; Gentile Fusillo et al., 2017b; Raddi et al., 2017), and it is clear that with the next order-of-magnitude increase in the sample size presented here, the historically used naming convention will no longer be suitable. We therefore propose to adopt a new naming convention, which shall account for proper motions. WD JHHMMSS.SSDDMMSS.S will be defined as the white dwarf coordinates in IRCS, at equinox 2000 and epoch 2000. This definition should be sufficient to avoid duplicate names even in the era of LSST (estimated to identify over 13 million white dwarfs; LSST Science Collaboration et al. 2009), except in the densest environments, such as globular clusters. We have computed the white dwarf names in Table 2 following this convention, using the Gaia IRCS coordinates and proper motions, which are provided at equinox and epoch 2015.5.

4 Atmospheric Parameters

Only a small fraction of the white dwarfs in our catalogue have available spectroscopy and we must therefore use another technique to characterise the atmospheric parameters ( and ) of the sample as a whole. Photometric surveys, such as Pan-STARRS, 2MASS, SDSS, and GALEX, only offer a partial coverage, either in magnitude range or sky area. The advantage of adding near-ultraviolet, near-infrared, or narrow-band photometry to the Gaia data set is also limited by the fact that we do not know the atmospheric compositions. Therefore, fitting additional data sets with either pure-H or pure-He model atmospheres may not necessarily improve the accuracy of the atmospheric parameters. As a consequence, we use Gaia DR2 data only to determine the atmospheric parameters, and we compare our results to Pan-STARRS and SDSS photometry for bright DA white dwarfs to understand possible systematic effects in Section 4.1.

There is a degeneracy between the atmospheric parameters and reddening when using Gaia data alone. The first step of our photometric analysis is therefore to derive an estimate for the amount of reddening. We have queried the Schlegel et al. (1998) reddening maps and incorporated the correction proposed by Schlafly & Finkbeiner (2011). The resulting extinction coefficients in the Gaia passband are given in column 31 of our catalogue (Table 2). The reddening as a function of distance is parametrised assuming that the absorbing material along the line of sight is concentrated along the plane of the Galactic disc with a scale height of 200 pc. Under this assumption the dereddened magnitudes are given by

 (18)
 GBP,star=GBP,obs−1.364×AG(1−exp(−sin(b)200ϖ)) (19)
 GRP,star=GRP,obs−0.778×AG(1−exp(−sin(b)200ϖ)) (20)

where is the vertical Galactic coordinate and the parallax is in arcsec. This parameterisation is slightly different to the one used in Harris et al. (2006), Tremblay et al. (2011), and Genest-Beaulieu & Bergeron (2014), where interstellar absorption was assumed to be negligible within 100 pc and to vary linearly from zero to a maximum value for between 100 and 250 pc. We have verified that our new parameterisation provides a slightly better empirical agreement between the observed hot white dwarf cooling sequences () for distances in the ranges 75 and 75–250 pc, respectively. In this colour range the Gaia sample is expected to be fairly complete up to 250 pc and the properties of the dereddened white dwarfs are not expected to depend on the distance. From this experiment we could clearly rule out a gas scale height that is either two times larger or two times smaller than 200 pc. Improved Gaia DR2 reddening maps in three dimensions will eventually supersede our simple parameterisation but it is currently outside the scope of this work.

We have employed the Gaia DR2 revised quantum efficiency for the , and passbands (Evans et al., 2018) to calculate synthetic absolute magnitudes using the relation

 M=−2.5log(∫S(λ)F(λ)λdλ∫S(λ)λdλ1(10pc)2)+CS , (21)

where 10 pc is expressed in cm (1 pc = cm), is the zero point, and is the integrated stellar flux in erg s Å relating to the emergent monochromatic Eddington flux as

 F(λ)=4πR2Hλ(Teff,logg) , (22)

where is the white dwarf radius. We employ standard H-atmosphere spectral models (Tremblay et al., 2011) including the L red wing absorption of Kowalski & Saumon (2006) and covering the range (K) and . For , we use the evolutionary sequences with thick hydrogen layers ( of Fontaine et al. (2001,  K, C/O-core 50/50 by mass fraction mixed uniformly) and Wood (1995, 30 000 K, pure C-core). For lower masses, we use the He-core cooling sequences of Serenelli et al. (2001). We have also computed synthetic magnitudes for He-atmosphere models (Bergeron et al., 2011) using a mass-radius relation for thin hydrogen layers (Fontaine et al., 2001, ). For the Gaia passbands Vega has a magnitude of +0.03 (Jordi et al., 2010), and the zero points defined with this reference are given in Table LABEL:zeropoints along with nominal wavelengths. The values for the pre-launch nominal Gaia filters are also given (Jordi et al., 2010; Carrasco et al., 2014; Tremblay et al., 2017).

The dereddened observed Gaia flux in the passband in units of erg cm s can be computed from the dereddened apparent Gaia magnitude in the same passband as

 GS=−2.5log(fS)+CS , (23)

which is related to the passband and stellar disc integrated flux in erg s as

 fS=ϖ2 FS . (24)

Our fitting technique relies on the non-linear least-squares method of Levenberg-Marquardt (Press et al., 1992). The value of is taken as the sum over all passbands of the difference between both sides of Eq. (24), weighted by the corresponding Gaia flux and parallax uncertainties. Only and are free parameters as the stellar radius in Eq. (22) is fixed by our adopted theoretical mass-radius relation. The uncertainties on both parameters are obtained directly from the covariance matrix of the fit.

4.1 The precision of Gaia atmospheric parameters

For the 20 pc sample of white dwarfs, Gaia DR2 photometry and astrometry have been used to derive effective temperatures and surface gravities (Hollands et al., 2018). One advantage of this volume-complete sample is that reddening is negligible. Despite the wide range of spectral types for the local sample, with 39 per cent or more of the remnants having a magnetic field, carbon, or metals, Hollands et al. (2018) found the Gaia photometric parameters to agree with previous photometric and spectroscopic analyses with no significant systematic offset. They found standard deviations of 3.1 per cent in  and 0.10 dex in with respect to individual published parameters. The precision of Gaia atmospheric parameters is likely better than these values owing to the inhomogeneity and lower precision of previously available ground-based observations. It was shown that as a result of the broad Gaia passbands, using either the pure-H or pure-He atmosphere approximation does not result in a significant offset for the parameters of yet unclassified DZ or DQ white dwarfs.

The degenerate stars in the local volume-complete sample have an average temperature of 8000 K, significantly cooler than in our Gaia magnitude-limited catalogue. As a consequence, it is also important to verify the precision of Gaia atmospheric parameters for white dwarfs with  K, especially since the Gaia colours become increasingly less sensitive to temperature as the latter increases (Carrasco et al., 2014). We have therefore employed as a reference our Gaia-SDSS spectroscopic sample (Section 5) of DA white dwarfs, restricting the comparison to single, non-magnetic stars with a spectroscopic signal-to-noise ratio larger than 20, the latter to ensure that we do not have undetected subtypes (e.g., magnetic, binaries). In addition to the available SDSS photometry, we have also cross-matched this subsample with the Pan-STARRS catalogue (Chambers et al., 2016). We additionally use the bright, single and non-magnetic DA stars from Gianninas et al. (2011) that we have cross-matched with both our Gaia DR2 catalogue and Pan-STARRS.

For the comparison of these photometric data sets we have applied additional Gaia quality cuts (astrometric_excess_noise and phot_bp_rp_excess_factor 1.3 + ), similar to those employed in Gaia Collaboration et al. (2018b). The Pan-STARRS and SDSS photometry, both in the AB magnitude system, was fitted along with Gaia parallaxes using the photometric method described in Section 4. The models were integrated over the Pan-STARRS (Tonry et al., 2012) and SDSS passbands (Fukugita et al., 1996). We have applied the same reddening law as a function of wavelength and distance to all photometric data sets. At first order, reddening effects and model atmosphere systematics should not be a concern as we perform a differential comparison of multiple data sets for the same objects using the same models and reddening law. However, the Gaia passbands are considerably broader than those of SDSS and Pan-STARRS, and therefore we can not rule out these residual model effects.

The comparison of Gaia and Pan-STARRS temperatures is presented in Fig. 7. A large fraction of the brightest known DA stars are recovered in both surveys, resulting in 1128 objects from Gianninas et al. (2011) compared in the top panel of Fig. 7. We also show the comparison for 4878 objects which are among the brightest DA white dwarfs in the SDSS. Since we use Gaia parallaxes in all photometric fits, the shifts in surface gravities correlate with those in and do not bring additional information. Our results suggest that the data sets are in good agreement across the full range in with no clear systematic trends. Possible small differences visible in Fig. 7 could be caused by residual effects from model atmospheres or reddening given the different passbands and therefore our results suggest that the Gaia photometric calibration is accurate within the combined Pan-STARRS and Gaia uncertainties. The comparison of Gaia and SDSS photometric temperatures in Fig. 8 confirms this behaviour. The overlap of the Gianninas et al. (2011) sample with the SDSS catalogue is very small and therefore the comparison is omitted. Overall there is no evidence of any systematic offset in the Gaia photometric calibration over the 14 19 magnitude range covered by our comparison samples. Our results also suggest that the SDSS and Pan-STARRS photometric data sets agree well on average with the predictions of their respective nominal passbands without the need of any empirical correction.

4.2 Limitations

The best fit and values, corresponding uncertainties and values, and implied masses from our adopted theoretical mass-radius relations, are given in columns 43-56 of our catalogue (Table 2) for both pure-H and pure-He atmospheres. The parameters are only given for a subset of the full catalogue where all of the following conditions apply

 PWD>0.75 (25) σTeff/Teff<0.75 (26) σlogg<2.0 (27) σMWD/M⊙<1.0 (28) 0.1

with the following additional restriction for pure-He atmospheres

 3000

The restriction to high probability white dwarf candidates () is the most significant. The additional cuts remove a further 13.1 per cent of the high probability white dwarfs (see Table 1) with unreliable atmospheric parameters, resulting in a final subset of 226 734 degenerate candidates with at least one set of atmospheric parameters. The first category of cuts (Eqs. 26-30) reflects the limitation in the precision of Gaia at the fainter magnitudes and larger distances, which results in a bias predominantly against the hotter white dwarfs in the sample, but can also remove some of the most peculiar stars, such as ultra-cool white dwarfs. The second category of cuts (Eqs. 31-33) removes strong outliers in Gaia quality flags for which the atmospheric parameters are clearly offset from the remaining objects in the sample, and therefore we have no reason to believe they are reliable. We note that it may be justified or even necessary to use further cuts on Gaia quality flags when employing the atmospheric parameters for subsamples of our catalogue.

We do not yet have spectral types for the vast majority of the Gaia DR2 white dwarfs. For the overall sample, our atmospheric parameters derived from pure-H and pure-He atmospheres agree 95.6 percent of the time within 1, which suggests that the pure-H approximation may be sufficient for many applications. The values given in the table should be used with caution and we have no evidence that they can help to discriminate between spectral types. The distribution of values is fairly smooth and the tail containing large values could include both peculiar white dwarfs and objects with underestimated Gaia uncertainties. Considering the extremely small error bars of some Gaia measurements, a large value does not mean there is an obvious discrepancy with the input model atmospheres.

We find an average surface gravity of = 8.00 assuming pure-H atmospheres. We remind the reader that while most objects with atmospheric parameters are in agreement with single star evolution, the catalogue contains a large number of sources with inferred masses below , which would imply a main-sequence lifetime larger than the Hubble time. For the vast majority of them, the mass error is too large to rule out single star evolution. Nevertheless, this issue creates a noticeable artefact at 0.46  in our mass distribution, where we switch from C/O-core cooling tracks to He-core white dwarfs at lower masses.

5 The Gaia-SDSS spectroscopy sample

Although limited in sample size compared to the full Gaia catalogue, the 21 325 white dwarfs with SDSS spectroscopy currently represent the largest sample of spectroscopically confirmed Gaia white dwarfs, and, combined with the Gaia data, allow us to explore their global properties as well as to further characterise our catalogue. In our classification of SDSS spectra mentioned in Section 2 we adopted 17 spectral types for single white dwarfs (DA, DB, DBA, DAO, DC, DAZ, DBZ, DZ, DQ, hotDQ, DQpec, DAH, DBH, DZH, MWD, WD and WDpec, Fig. 9) and six additional classes for white dwarfs in binaries, and contaminants (DB+MS, DA+MS, DC+MS, STAR, QSO, Unreliable). Objects classified as “MWD” are magnetic white dwarfs where line splitting is so large that we were unable to identify the atmospheric composition. While spectra marked as “Unreliable” have a signal-to-noise ratio too low to attempt any visual classification, objects simply classified as “WD” have spectra too poor for detailed classification, but still recognizable as those of degenerate stars. We also include a new classification for some peculiar white dwarfs as “WDpec” (see later in this section).

In Fig. 10, we display the locus of the individual white dwarf sub-classes in the Gaia H-R diagram separately, along with the general distribution of 16 723 white dwarf candidates within 100 pc selected from our catalogue, adopting (see also Section 6.3 for a discussion of the local sample). Several noticeable structures are present in the white dwarf cooling sequence, some of which have been discussed already in Gaia Collaboration et al. (2018b). The dominant feature is a bifurcation into two sequences, which are easily distinguishable at 0.0 0.5. As illustrated in the top-left and middle-left panels, the upper one of these two tracks is easily explained as the cooling sequence of the most common DA white dwarfs (, see Section 4 for details on the adopted evolutionary models1). In contrast, the middle-right panel in Fig. 10 shows that the majority of He-atmosphere white dwarfs (DB, DC2, DZ) are located on the second narrow track just below the DA cooling sequence. This lower branch of the bifurcation has been interpreted by Kilic et al. (2018) as the signature of a sub-population of high-mass white dwarfs. Even though a relatively small number of DAs also occupy this space (see Section 5.1), this second track is most likely explained as the cooling sequence of canonical mass He-atmosphere white dwarfs and not as a second higher-mass sequence of H-atmosphere white dwarfs. We note that separating the two families of objects in a clean way for the overall 100 pc sample is practically impossible without spectroscopic IDs. More critically, the theoretical cooling sequences for pure-He atmospheres displayed in the top-right diverge from the Gaia observations of He-atmosphere white dwarfs for 7000 K  K (0.0 0.5). This discrepancy therefore does not impact the DB white dwarfs which largely fall onto the 0.6 pure-He cooling sequence, but only the cooler DC, DQ and DZ stars. El-Badry et al. (2018) have speculated that uncertainties from additional sources of opacity in cool white dwarfs may be the cause of the diverging observed He cooling sequence, which we also conclude is the most likely explanation. However, it does not seem to have an obvious link with the presence of metals according to Fig. 10. An additional note concerns an apparent dearth of DA white dwarfs around (Fig. 10, middle-left panel). Matching this colour range with the SDSS photometry (Table 2) shows that this under-density corresponds to objects with , which is a region in colour space in which the spectroscopic completeness of SDSS is significantly reduced compared to the rest of the colour space occupied by  K white dwarfs (see Fig. 11 of Gentile Fusillo et al. 2015a). We hence conclude that this particular structure in the distribution of SDSS DA white dwarfs is an artefact of the SDSS spectroscopic target selection strategy.

Comparing the cooling sequences of DA white dwarfs (middle-left panel) and those with He-dominated atmospheres (middle-right panel), it is apparent that the DA white dwarfs have a larger spread in absolute magnitudes at any given colour. The very tight sequence of the DB and DZ stars suggests that the scatter seen in the DA sequence is not a result of the larger sample size of the DA white dwarfs. On the one hand, the low-mass tail is likely to be linked to binary evolution preferentially forming DA stars (Gianninas et al., 2014; Parsons et al., 2017). The confirmed double degenerates from Breedt et al. (2017) and Rebassa-Mansergas et al. (2017) are located above the DA cooling sequence, which is expected as many double-degenerates contain at least one low-mass He-core white dwarf (Marsh et al., 1995; Rebassa-Mansergas et al., 2011), and as the luminosity of these unresolved binaries adds up. On the other hand, the mass-dependence of the mechanisms that determine the total amount of hydrogen in the envelope of white dwarfs, and how the hydrogen convectively mixes with the underlying helium layer, could explain the high-mass DA tail and the lack of massive degenerate stars with He-atmospheres (Kalirai et al., 2005). The initial-to-final-mass relation can also be invoked to describe the shape of that high-mass tail (Tremblay et al., 2016; El-Badry et al., 2018). Following upon the investigation of Kalirai et al. (2005), we note that 64 out of the 65 white dwarfs that are confirmed young open cluster members (cluster age 700 Myr) in Cummings et al. (2016) are DA stars (the one DBA star is the Hyades member WD 0437+138). These objects cover the range and , with 69 per cent of the sample below the so-called DB gap or deficiency ( K; Bergeron et al., 2011; Koester & Kepler, 2015) for field white dwarfs. This provides strong evidence that single star evolution can explain the lack of massive DB stars.

The DA sample also appears to have an over-density of under-luminous stars below the cooling track forming a separate third sequence (distinguishable at 0.0 1.0 in the middle left panel of Fig. 10), located below the cooling track of He-rich white dwarfs discussed above. This “transversal” sequence, also seen in the overall 100 pc sample (top panels), does not run parallel to the DA cooling sequences and is therefore not a constant mass track, ruling out a straightforward astrophysical explanation such as binary evolution or effects from the initial-to-final mass relation. Explaining the origin of this feature is beyond the scope of this paper but we speculate that it could be the result of a mass-dependent cooling effect (Tremblay et al. 2018, in prep.).

In the bottom left panel of Fig. 10 we show the location of a representative number of magnetic white dwarfs. It appears that these objects span a relatively large range of absolute magnitudes for a given colour, but on average they are under-luminous compared to typical DA white dwarfs. This finding seems to corroborate the long standing theory that these white dwarfs are more massive and so smaller than their non-magnetic counterparts (Liebert, 1988; Ferrario et al., 2015). In our visual inspection of SDSS spectra we also identified four objects that despite having parallax and colours consistent with those of white dwarfs, have a unique spectral appearance among the 21 325 white dwarfs spectroscopically confirmed by SDSS (Fig. 11). These peculiar white dwarfs (classified as WDpec in our catalogue) all exhibit one broad absorption feature and a number of smaller “satellite” absorption lines. The main broad absorption features appear to be shifted by hundreds of Å from star to star. In the H-R diagram these four stars line up below the DA cooling sequence much like most of the magnetic white dwarfs. Two of these stars (WD J033320.57+000720.65 and WD J075227.93+195314.41) are already known as magnetic degenerates with unidentifiable features (Reimers et al., 1998; Kepler et al., 2015), so we speculate that these peculiar objects may all be members of the same family of magnetic white dwarfs. We are however unable to venture any hypothesis on their atmospheric composition.

Fig. 10 (bottom-left panel) also illustrates the location of spectroscopically confirmed ultracool white dwarfs (  K) (Harris et al., 2001; Gates et al., 2004; Harris et al., 2008; Hollands et al., 2017). These objects, still rare even in the very large Gaia sample, migrate to bluer colours (and so a distinct location compared to hotter white dwarfs) as a result of collision-induced absorption (Hansen, 1998). Many of these white dwarfs occupy areas of the H-R diagram in which we apply the _correction (see Sect.2, Table 2), so particular care should be taken when attempting to select these objects from the catalogue. Finally the bottom right panel shows a distinction in the distribution of DQ (Dufour et al., 2005) and hot DQ white dwarfs (Dufour et al., 2007a). Cooler DQ stars roughly line up with the cooling sequence of other He-atmosphere white dwarfs, while hot DQs appear distinctly under-luminous and occupy the same locus as many magnetic white dwarfs. This is not surprising, as a large fraction, if not all, hot DQs are thought to harbour magnetic fields (Dufour et al., 2008; Lawrie et al., 2013; Williams et al., 2013; Williams et al., 2016). We note that hot DQ stars, and in general the entire lower branch of the magnetic white dwarf cooling track, overlap with the “transversal” sequence observed for DA white dwarfs.

5.1 He-atmosphere DA white dwarfs

A small number of stars with the spectroscopic appearance of DAZ white dwarfs are known to actually have He-dominated atmospheres with unusually large H components and metal pollution. In Fig. 12 we show the position of the currently known five members of this family: GD 16, GD 17, GD 362, PG 1225079, and SDSS J124231.07+522626.6 (Koester et al., 2005; Gentile Fusillo et al., 2017a; Gianninas et al., 2004; Kawka & Vennes, 2005; Kilkenny, 1986; Raddi et al., 2015) on the observed Gaia H-R diagram. All five objects broadly lie on the observed Gaia cooling sequence of He atmosphere white dwarfs. This could indicate that these peculiar objects evolve in a similar way as average mass He-atmosphere white dwarfs. Alternatively, these objects could behave as thin-H layer DA white dwarfs without suffering from the bifurcation problem of He-atmospheres, and therefore have masses slightly higher than the canonical . If the location on the He-atmosphere sequence were to be confirmed for other He-rich DA white dwarfs, this property could be exploited to identify more of these objects, and help to unravel the question of the origin of the H in their atmosphere. Indeed as shown in Fig. 10 a number of spectroscopically confirmed SDSS DA white dwarfs occupy the same region on the H-R diagram and analogously to these five metal polluted stars, some may actually have He dominated atmospheres, especially if their spectroscopic masses assuming pure-H atmospheres are unusually large (Tremblay et al., 2010; Rolland et al., 2018).

6 Discussion

6.1 Sky density and limiting magnitude

Using a   reference sample we can attempt to estimate the overall sky density of white dwarfs in Gaia DR2. Contrary to what is expected from simple Galactic structure, the sky density of white dwarfs in our catalogue does not significantly increase at lower Galactic latitudes (Fig. 5). This is a consequence of the stricter selection we apply to the areas with high stellar densities at lower Galactic latitudes (Section 2). Additionally, virtually no white dwarfs are found in the most central regions of the plane where crowding is highest (, ). We find significant structure in the density of white dwarfs across the entire sky, as a result of the non-uniform limiting magnitude of Gaia observations. As shown in Fig. 13 the Gaia limiting magnitude can vary by more than 1 mag across the sky in a pattern that closely follows that of Gaia scanning law. It would therefore not be meaningful to estimate the sky average density of all white dwarf candidates in our catalogue. In DR2, a limiting magnitude of least 20 is reached for 75 per cent of the sky, and we estimate the sky density of white dwarfs in these regions with to be . We can assume that with future Gaia data releases the effective limiting magnitude will become more uniform across the sky, and in subsequent versions of our catalogue we will be able to identify more faint white dwarfs.

6.2 Comparison with an SDSS sample of white dwarf candidates

With high-confidence candidates, our catalogue of white dwarfs is certainly the largest ever published, but in order to explore the full diagnostic potential of this vast sample, we need to evaluate the completeness of our selection. A number of factors within Gaia DR2 and/or in our selection method may cause some genuine white dwarfs to be excluded from this catalogue. In order to assess this issue it is necessary to compare the Gaia catalogue of white dwarfs with a sufficiently large and well characterised sample of stellar remnants. The spectroscopic samples of white dwarfs currently available (e.g. SDSS in Section 5) are ill-fitted for this task as they are severely incomplete and biased by the specific observing strategy adopted. Therefore we decided to rely on a sample of SDSS white dwarf candidates selected on the basis of their colour and reduced proper motion as described in Gentile Fusillo et al. (2015a). However, the original catalogue of Gentile Fusillo et al. (2015a) only included objects brighter than  mag as fainter sources did not have reliable proper motions in SDSS. In order to create a sample which better matches the magnitude limit of our Gaia catalogue we extended the Gentile Fusillo et al. (2015a) catalogue to  mag by making use of the more accurate proper motions from the Gaia-PS1-SDSS (GPS1) catalogue (Tian et al., 2017). Full details about the development and characterisation of this deep SDSS comparison sample are available in Appendix A.

From this deep photometric SDSS catalogue we select a sample of 60 739 white dwarf candidates, which only has seven per cent contamination while still including 97 per cent of all the white dwarfs in the full sample. However, it is important to notice that, because of the colour restrictions used, this sample only contains white dwarfs with   K, and an additional stellar remnants in the SDSS footprint are potentially missing because they have no proper motion measurement. For completeness, we note that the footprint of the SDSS photometry mostly covered high Galactic latitudes with . In conclusion, we estimate the deep SDSS sample to contain per cent of all the white dwarfs observed by SDSS, brighter than  mag and with   K.

We cross matched our Gaia catalogue of white dwarf candidates with the deep SDSS comparison sample and retrieved 47 503 of the SDSS white dwarf candidates. Accounting for the expected level of contamination of the deep SDSS sample (7 per cent), we can use the percentage of objects missing in the Gaia white dwarf candidate sample to estimate an upper limit in completeness of the Gaia catalogue of 81 per cent for white dwarfs with  mag and   K, at high Galactic latitudes (). Similarly, we can use the estimated completeness of the deep SDSS sample and the number of objects we retrieved in the cross-match with the Gaia catalogue to calculate a lower limit in completeness of 56 per cent. Additionally, we can use this comparison as a diagnostic of potential biases in our Gaia selection. As illustrated in Fig. 14 the completeness of our Gaia catalogue drops close to the Galactic plane and in these areas the upper limit on the overall completeness can be as low as 50 per cent. This effect is a direct consequence of the stricter quality selection we impose on crowded areas at low Galactic latitudes (see Section 2). Even in the era of Gaia the Galactic plane represents a challenging environment to be surveyed accurately, nonetheless the catalogue presented here still includes the largest sample of Galactic plane white dwarf candidates available to date. A potentially more complete selection of white dwarfs in the Galactic plane could be achieved combining Gaia observations with dedicated photometric surveys (e.g., IPHAS (Drew et al., 2005) or VPHAS+ (Drew et al., 2014)).

We also tested the relative completeness of our Gaia white dwarf selection as a function of magnitude and colour (Fig. 15). Since the level of contamination of our deep SDSS comparison sample selection is itself colour dependent, for this test we use a sample of high signal-to-noise ratio spectroscopically confirmed SDSS degenerate stars. This comparison does not reflect the absolute completeness of our Gaia catalogue and should only be used to explore any potential correlation with magnitude and/or colour. Fig. 15 (top panel) shows no obvious correlation with magnitude. The apparent drop in completeness at is most likely due to small number statistics as a consequence of SDSS reaching saturation.

The bottom panel in Fig. 15 clearly illustrates that there is no marked colour trend in the completeness of our catalogue with respect to the SDSS spectroscopic sample. However, our spectroscopic comparison sample only includes white dwarfs with   K and the completeness of our Gaia selection may vary for cooler (and redder) objects.

6.3 Volume completeness

Hollands et al. (2018) carefully determined the selection function and completeness of the Gaia DR2 sample of 139 white dwarfs within 20 pc and found the space density to be  pc. To recover these numbers with our catalogue, we must apply and owing to the relatively large number of main-sequence stars scattered in the local sample owing to erroneous parallaxes (Hollands et al., 2018).

This can be compared with the number of degenerate stars that we find at larger distances in our catalogue, and here we take a particular interest in the 100 pc sample. First, we must consider the quality cuts to apply. There are 16 723 white dwarf candidates in our catalogue with mas and , though a considerable number of those have poor quality flags. By applying the cut as in Hollands et al. (2018), we would remove 12.6 per cent of the 100 pc white dwarf candidates, many of them likely genuine, faint and cool stellar remnants. We employ the sample of white dwarfs with reliable atmospheric parameters (Section 4.2) as a compromise, resulting in 15 041 high-confidence members of the 100 pc sample, and a less drastic cut of 10.1 per cent compared to the initial sample. The inferred space density within 100 pc is 86.6 per cent of that found for the 20 pc sample, though at such large distances the approximation of constant space density is unlikely to hold because of the finite scale height of the Galactic disc. Furthermore, the white dwarf luminosity function is known to peak at before dropping off at fainter magnitudes owing to the finite age of the disc (Winget et al., 1987). Given the sky average limiting Gaia magnitude of 20 for our white dwarf catalogue, it is unlikely to be complete for distances larger than about 70 pc (see also Carrasco et al., 2014).

Fig. 16 presents the normalised Gaia white dwarf luminosity function for different distances. The similarity of the luminosity functions from 30 to 70 pc confirms that our catalogue is essentially a volume-limited sample up to that distance. We remind the reader that these local samples are not complete because they lack a largely volume-independent fraction of white dwarfs owing to the incompleteness of Gaia observations. At larger distances, the number of cool and/or massive degenerate stars with is clearly decreasing as a result of the Gaia limiting magnitude. Nevertheless, the drop in the luminosity function is still fairly small and it is not expected to impact the total number of white dwarfs within 100 pc by more than a few per cent.

To understand further the properties of the 100 pc sample, we employ the white dwarf population simulation drawn from Tremblay et al. (2016). In brief, this simulation uses constant stellar formation history over the past 10 Gyr, the Salpeter initial mass function, main-sequence lifetime for solar metallicity from Hurley et al. (2000), the initial-to-final mass relation of Cummings et al. (2016), a uniform distribution in Galactic coordinates U and V (corresponding to the plane of the disc), and Galactic disc heating in the vertical coordinate W (Seabroke & Gilmore, 2007) starting with an initial scale height of 75 pc for a total age of 1 Gyr or less, resulting in an age-average scale height of 230 pc for the local sample. The simulation also assumes a limiting Gaia magnitude of . We have repeated the simulation by artificially multiplying and dividing the disc scale height by a factor of two, respectively. The results are presented in Fig. 17, where the observed Gaia space density agrees remarkably well with our standard model and an age average disc scale height of 230 pc. The simulation naturally explains the smaller space density within 100 pc compared to 20 pc without the need to invoke the completeness of Gaia, for which the selection function is not expected to change remarkably between these distances.

7 Conclusion

We retrieved the available Gaia DR2 data for spectroscopically confirmed white dwarfs from SDSS, and analysed the properties and distribution of these objects in the Hertzsprung-Russell diagram to define a reliable method to select high-confidence white dwarf candidates from Gaia DR2. After defining several quality cuts to remove objects with poor Gaia measurements, we find that no simple selection relying solely on Gaia colour and absolute magnitude can separate white dwarfs from contaminant objects without excluding a significant number of known white dwarfs. We therefore make use of the distribution in colour and  of a sample of spectroscopically confirmed white dwarfs and contaminants from SDSS to calculate probabilities of being a white dwarf () for all Gaia objects in our sample. This results in a total of  objects with calculated  from which it is possible to select a sample of high-confidence white dwarf candidates. The  values, coupled with Gaia quality flags, can be used to flexibly select samples of white dwarfs with varying degree of completeness and contamination according to one’s specific goals. For general purpose we recommend a cut at , which we estimate includes 95 per cent of all the white dwarfs in the total sample, with minimal level of contamination ( per cent). We also provide stellar parameters (, and mass) for a subsample of 226 734 candidates that have Gaia parallax and photometric measurements precise enough to achieve a reliable fit to our adopted models. We find the atmospheric parameters obtained fitting only Gaia observations to be in good agreement with those obtained using SDSS and Pan-STARRS photometry.

We further characterised the Gaia sample of white dwarfs by visually inspecting the observed cooling sequence in the H-R diagram of representative samples of spectroscopically confirmed stellar remnants from the SDSS. We identify a number of sub-structures in the white dwarf cooling tracks, some of which are the result of different spectral types and others that remain unexplained.

We have used a newly constructed sample of SDSS white dwarf candidates selected on the basis of their colours and proper motions to estimate the overall completeness of our Gaia catalogue of white dwarf candidates. We found the catalogue to be between 56 and 81 per cent complete for white dwarfs with  mag and   K, at high Galactic latitudes ().

The presented Gaia catalogue represents the first step towards a homogeneous all-sky census of all white dwarfs, and to fully explore the rich scientific potential of this sample, spectroscopic follow-up will ultimately be needed to study these objects in detail. The values that we derived allow to tailor future spectroscopic campaigns prioritising efficiency for single target observations or completeness in large scale surveys. With large multi-fibre spectroscopic facilities approaching first light in both hemispheres (e.g., WEAVE, 4MOST, DESI, SDSS-V, Dalton et al. 2014; de Jong et al. 2014; DESI Collaboration et al. 2016; Kollmeier et al. 2017), our catalogue represents a key resource for future white dwarf studies.

Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme n. 677706 (WD3D) and the European Union’s Seventh Framework Programme (FP/2007- 2013) / ERC Grant Agreements n. 320964 (WDTracer). Additional funding was provided by STFC via grant ST/P000495/1.

R. R. acknowledge funding by the German Science foundation (DFG) through grants HE1356/71-1 and IR190/1-1.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

The work presented in this article made large use of TOPCAT and STILTS Table/VOTable Processing Software (Taylor, 2005).

This work has made use of observations from the SDSS-III, funding for which has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.

This work was supported in part by Sonderforschungsbereich SFB 881 ”The Milky Way System” of the German Research Foundation (DFG).

Appendix A The deep SDSS-GPS1 comparison sample

Our original catalogue of SDSS white dwarf candidates published in Gentile Fusillo et al. (2015a) relied on SDSS DR10 proper motions and was limited to objects brighter than  mag. In order to create a comparison sample matching the depth of our Gaia catalogue we needed to extend the sample of SDSS white dwarfs candidates to fainter magnitudes. As SDSS proper motions quickly become unreliable past  mag we adopted proper motions from the Gaia-PS1-SDSS (GPS1) Catalog (Tian et al., 2017). Following the same colour selection as described in Gentile Fusillo et al. (2015a) and Gentile Fusillo et al. (2017a) we selected 263 944 blue SDSS point sources with  mag. This colour cut limits the sample to only white dwarfs with   K. Large areas of the sky at RA are entirely missing in the GPS1 catalogues and no proper motions could be retrieved for objects at these location. In order to circumvent the effects of these gaps in GPS1 we further limit our comparison sample to SDSS sources with RA before carrying out the cross match with GPS1. This brings the number of objects in the sample to 253 640. We cross matched the positions of these objects with GPS1 to retrieve their proper motions. Coordinates in GPS1 are provided in epoch J2010 while SDSS observations were collected between 2000 and 2008. Since high proper motions objects like white dwarfs can move significantly over these time scales, we carried out our cross-match accounting for this epoch difference following the method described in Gentile Fusillo et al. (2017a) and finally retrieved proper motions for 211 988 of the 253 640 SDSS objects. Using SDSS colours and GPS1 proper motions we then calculated a reduced-proper-motion based probability of being a white dwarf for all our objects using the method described in Gentile Fusillo et al. (2015a). This values are different and unrelated to the Gaia-based values presented in section 2. Using our training set of spectroscopically confirmed SDSS white dwarfs and contaminants, we can calculate completeness (ratio of the number of selected white dwarfs to the total number of white dwarfs) and an efficiency (ratio of the number of selected white dwarfs to the total number of objects selected) for different threshold values of . For example, selecting all objects with results in a sample 97 per cent complete with an efficiency of 93 per cent. This also allows us to estimate that the entire sample of objects for which we calculated contains white dwarfs. However, we could retrieve GPS1 proper motions (and so calculate values) only for  per cent of the SDSS objects within our initial colour and RA cut. Additionally, there appears to be a colour () dependence in the number of objects for which no proper motion was found (Fig. 18). When combining this effect with the distribution of white dwarfs in and the efficiency of our cut in different bins of colour-space, we find that on average up to 25 per cent of white dwarfs may not have been included in the sample as a result of not having a proper motion in the GPS1 catalogue (Fig. 18). We therefore conclude that our deep SDSS comparison sample of objects with calculated only includes 75 per cent of all the white dwarfs in the SDSS footprint with RA 12, and  K. Nonetheless, we can estimate that an additional white dwarfs are among the objects initially included in our SDSS colour-cut, but which have no proper motion in GPS1.

Footnotes

1. As discussed in Gaia Collaboration et al. (2018b) and Hollands et al. (2018), the observed cooling track diverges from the evolution models towards low masses for  K.
2. Strictly speaking, a DC classification only implies a featureless spectrum. In most cases, this is consistent with a cool He-dominated atmosphere, however, a small number of the objects classified as DC white dwarfs could have strongly magnetic H-atmospheres, wiping out the Balmer lines.

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