11institutetext: Departament de Física, Universitat Politècnica de Catalunya,
c/Esteve Terrades 5, 08860 Castelldefels, Spain 22institutetext: Institute for
Space Studies of Catalonia, c/Gran Capità 2–4, Edif. Nexus 104, 08034
Barcelona, Spain 33institutetext: Centro de Astrobiología (CAB), CSIC-INTA,
Camino Bajo del Castillo s/n, campus ESAC, 28692, Villanueva de la Cañada,
Madrid, Spain 44institutetext: École d’ingénieurs (EPF), 55 Av. du Président
Wilson, 94230 Cachan, France

# White dwarf spectral type–temperature distribution from Gaia-DR3 and the
Virtual Observatory

S. Torres Email<EMAIL_ADDRESS>P. Cruz 33 R. Murillo-Ojeda 33 F.
M. Jiménez-Esteban 33 A. Rebassa–Mansergas 1122 E. Solano 33 M. E. Camisassa
11 R. Raddi 11 J. Doliguez Le Lourec 44

###### Abstract

Context. The characterization of white dwarf atmospheres is crucial for
accurately deriving stellar parameters such as effective temperature, mass,
and age. However, the inclusion of physical processes like convective mixing
and convective dilution in current white dwarf atmospheric models predicts a
spectral evolution of these objects. To constrain these models, accurate
observational data and analysis are necessary.

Aims. To classify the population of white dwarfs up to 500 pc into hydrogen-
rich or hydrogen-deficient atmospheres based on Gaia spectra and to derive an
accurate spectral type-temperature distribution, i.e., the ratio between the
number of non-DAs to the total number of white dwarfs as a function of the
effective temperature for the largest observed unbiased sample of these
objects.

Methods. We took advantage of the recent Gaia low-resolution spectra available
for 76 657 white dwarfs up to 500 pc. We calculated synthetic J-PAS narrow-
band photometry and fitted the spectral energy distribution of each object
with up-to-date models for hydrogen-rich and helium-rich white dwarf
atmospheres. We estimated the probability for a white dwarf to have a
hydrogen-rich atmosphere and validated the results using the Montreal White
Dwarf Database. Finally, precise effective temperature values were derived for
each object using La Plata evolutionary models.

Results. We have successfully classified a total of 65 310 white dwarfs (57
155 newly classified objects) into DAs and non-DAs with an accuracy of 94%. An
unbiased subsample of nearly 34 000 objects was built, from which we computed
a precise spectral distribution spanning an effective temperature range from 5
500 to 40 000 K, while accounting for potential selection effects.

Conclusions. Some characteristic features of the spectral evolution, such as
the deficit of helium-rich stars at $T_{\rm eff}\approx 35\,000-40\,000\,$K
and in the range $22\,000\lesssim T_{\rm eff}\lesssim 25\,000\,$K, as well as
a gradual increase from $18\,000\,$K to $T_{\rm eff}\approx 7\,000\,$K, where
the non-DA stars percentage reaches its maximum of 41%, followed by a decrease
for cooler temperatures, are statistically significant. These findings will
provide precise constraints for the proposed models of spectral evolution.

###### Key Words.:

stars: white dwarfs — stars: atmospheres – Virtual observatory tools –
Astronomical databases: catalogs

##  1 Introduction

The Gaia mission has revealed important features of the white dwarf population
with unprecedented precision. That is the case, for instance, of the existence
of two well-defined branches in the color-magnitude diagram (roughly between
$0.0\lesssim G_{\rm BP}-G_{\rm RP}\lesssim 0.5$) referred to as the A and B
branches, which correspond to the majority of white dwarfs with hydrogen-rich
and helium-rich atmospheres (see Gaia Collaboration et al., 2018). The most
plausible explanation to reproduce the B branch invokes the presence of small
amount of hydrogen or carbon into helium-dominated atmospheres (Bergeron et
al., 2019; Camisassa et al., 2023; Blouin et al., 2023). These models are
based on well-studied physical processes that can alter the composition of the
outer layers, such as convective mixing and convective dilution, among others
(e.g. Rolland et al., 2018, and references therein). The specific
characteristics of each model, such as the hydrogen content, the depth of the
convective zone as a function of the effective temperature, or even the
possibility of accreting material from surrounding asteroids, give rise to
different channels of formation and evolution of white dwarf spectral
types111White dwarfs are classified as DAs or non-DAs based on the presence or
absence of hydrogen lines in their spectra, respectively. This last group is
formed by those who exhibits helium lines (DB), metal lines (DZ), carbon lines
(DQ), or no lines at all (DC), among others (Sion et al., 1983). (e.g. Rolland
et al., 2018; Ourique et al., 2018; Cunningham et al., 2019, 2020; Bédard et
al., 2020). Therefore, accurate observational data is required to constrain
these models.

Figure 1: Distance distribution (left panel) and apparent $G$ magnitude
cumulative distribution (right panel) for the entire sample of white dwarfs
that fulfil our selection criteria (red histogram) and for the subsamble of
objects that have Gaia spectra (blue histogram). A constant cumulative slope
is shown (black line) as indicative of the completeness of the sample.

The proper explanation for the formation of the Gaia A and B branches extends
beyond the effective temperature range of these branches, encompassing a
broader issue. A crucial observational factor in analyzing the spectral
evolution of white dwarfs is the ratio of non-DA to DA stars as a function of
effective temperature222Throughout the paper, we use the term ’spectral type-
temperature distribution’ or simply ’spectral distribution’ to refer to the
ratio of non-DA to the total number of objects as a function of the effective
temperature. We prefer it to the term ’spectral evolution’ as we consider this
case specifically refers to the physical processes that result in a change in
spectral type.. Extensive efforts have been made since the pioneering work of
Sion (1984) to obtain statistically significant spectral distributions (e.g.
Tremblay & Bergeron, 2008). However, the advent of large spectroscopic and
photometric surveys such as the Sloan Digital Sky Survey (SDSS; York et al.
2000), Galaxy Evolution Explorer (GALEX; Morrissey et al. 2007), and Gaia
(Gaia Collaboration et al., 2016) has significantly increased both the
quantity and quality of available data (e.g. Ourique et al., 2018; Genest-
Beaulieu & Bergeron, 2019; Blouin et al., 2019; Cunningham et al., 2020;
McCleery et al., 2020; López-Sanjuan et al., 2022). Even though, complete
spectroscopic samples have been limited to a distance of up to 40 pc, or in
other cases, magnitude-selection effects introduce significant biases in the
final distribution.

Nevertheless, we can leverage the exceptional quality of astrometric and
photometric data provided by the Gaia mission. The third data release (DR3) of
Gaia includes low-resolution spectra for nearly 100 000 white dwarfs (Gaia
Collaboration et al., 2022), making it the largest sample of white dwarfs
available for analysis. In our recent study (Jiménez-Esteban et al., 2023), we
classified 8 150 white dwarfs within a nearly volume-complete 100 pc sample
into DA or non-DA categories. The achieved accuracy of 90% was remarkable and
allowed us to derive a detailed spectral distribution within the range of
temperatures from $5\,500\,$K up to $23\,000\,$K.

In this paper, we extend our previous analysis to a distance of 500 pc,
significantly increasing the expected number of white dwarfs, particularly for
hotter effective temperatures. Our goal is to derive for the first time the
spectral distribution in the entire range of temperatures between $5\,500\,$K
up to $40\,000\,$K, where spectral evolution is significant.

The paper is structured as follows: in Section 2, we describe the selection
procedure used to obtain our white dwarf sample from Gaia data. We provide a
summary of the main steps of our classification methodology in Section 3. Once
our sample is classified and validated, Section 4 focuses on analyzing
selection effects and addressing the completeness correction of the sample. In
Section 5, we present our spectral distribution, discuss our findings, and
compare them to previous works. We summarize our key results and draw our main
conclusions in Section 6.

##  2 The Gaia-DR3 white dwarf sample

We selected our objects from Gaia-DR3
catalogue333http://gea.esac.esa.int/archive/ following the criteria used in
Jiménez-Esteban et al. (2023) but extended up to 500 pc:

  * •

$\omega-3\sigma_{\omega}\geq 2$ mas and $\omega/\sigma_{\omega}\geq 10$

  * •

$F_{\rm BP}/\sigma_{F_{\rm BP}}\geq 10$ and $F_{\rm RP}/\sigma_{F_{\rm
RP}}\geq 10$

  * •

RUWE¡1.4; where RUWE stands for Renormalised Unit Weight Error preventing
against poor astrometric solutions (Lindegren et al., 2021).

  * •

$|C^{*}|<3\sigma_{C^{*}}$; where $|C^{*}|$ is an estimate of the BP and RP
flux excess factor and $\sigma_{C^{*}}$ its scatter following the prescription
by Riello et al. (2021).

Selected objects were corrected from extinction following the 3D interstellar
Galactic extinction maps from Lallement et al.
(2022)444https://stilism.obspm.fr/. In principle, we selected only those
objects falling below the 0.45 ${\rm M}_{\sun}$ cooling track on the
Hertzsprung-Russell (HR) diagram. Additionally, as the atmospheric models we
used (see Section 3) provide a reliable estimate of the effective temperature
in the range from 5 500 K up to 40 000 K, we chose those objects with
unreddened color between $-0.5<{\rm BP-RP}<0.86$. A total number of 100 173
objects were selected, from which 76 657 have Gaia low-resolution spectra
available.

Figure 2: Top panel: probability distribution of being DA for the white dwarf
labelled as DA or non-DA (blue and red histograms, respectively) in the MWDD.
Bottom panel: confusion matrix of our estimator of being DA. Displayed values
represent the total number of objects, while in brackets the percentages with
respect to the total population.

Figure 1 displays the distance distribution (left panel) and cumulative
distribution of apparent magnitude $G$ (right panel) for our entire sample
(red histogram) and white dwarfs with Gaia spectra (blue histogram). The use
of inverse parallax as a distance estimator, combined with a parallax error
threshold of less than 10%, introduces negligible discrepancies (less than
$\approx 4\%$) compared to other distance estimators (Bailer-Jones et al.,
2021). Most of our selected white dwarfs (73%) are within 250 pc, with a long
tail extending up to 500 pc. The cumulative $G$ magnitude distribution reveals
a deficit of objects starting at $G\sim 20$ mag for the entire sample and
around $G\sim 19.5$ mag for white dwarfs with Gaia spectra. Although the
nominal limiting Gaia magnitude is $\sim 21.0$, we adopted a conservative
value of $G_{\rm lim}=19.5$ mag for our completeness analysis (see Section 4).

##  3 White dwarf spectral classification

For those sources of our selected sample with available Gaia spectra, we
followed the same procedure as described in Jiménez-Esteban et al. (2023). A
brief description of the methodology for classifying white dwarfs into DA and
non-DA types used in that work is provided as follows.

First, for each white dwarf of our sample with available Gaia spectrum we
determined, by means of the Python package
GaiaXPy555https://www.cosmos.esa.int/web/gaia/gaiaxpy and taking into account
all the coefficients of the Gaia spectrum, the synthetic Javalambre-Physics of
the Accelerating Universe Astrophysical Survey (J-PAS; Benitez et al. 2014)
filter system (Marín-Franch et al., 2012) photometry. We focused on those
filters covering the range from 4000 to 9590.54 Å, and discarding those
filters with effective wavelength shorter than 4000 Å. Second, for each object
we built a spectral energy distribution (SED) using the derived J-PAS
photometry. Although most of the SEDs have 56 photometric points, in some
noisy spectra the number of points is lower, due to our threshold in the
photometric error of 10% to each individual photometric measurement obtained
with GaiaXPy.

We analyzed $67\,340$ new white dwarfs spectra, not previously studied in our
100 pc sample (Jiménez-Esteban et al., 2023). For those objects with more than
4 photometric points ($57\,155$), their SEDs were fitted using either pure
hydrogen white dwarf atmospheric models (DA) or models with helium and a small
trace of hydrogen (non-DA, log N(H)/N(He) = -6). Both DA and non-DA models
covered the temperature range of interest for this study ($5\,500\,$ to
$40\,000\,$K) and surface gravities from 7 to 9 dex (see Section 3.1 in
Jiménez-Esteban et al. 2023 for detailed model information). The fitting
process was performed using the Virtual Observatory Spectral energy
distribution Analyzer666http://svo2.cab.inta-csic.es/theory/vosa (VOSA; Bayo
et al. 2008), a powerful tool developed by the Spanish Virtual Observatory.
Among the new $57\,155$ analyzed objects, only 3 had Vgfb777Vgfb: Modified
reduced $\chi^{2}$ calculated by forcing $\sigma(F_{\rm obs})$ to be larger
than $0.1\times F_{\rm obs}$, where $\sigma(F_{\rm obs})$ is the error in the
observed flux ($F_{\rm obs}$). Vgfb smaller than 10–15 is often perceived as a
good fit. greater than 15 and were disregarded. For each object, two reduced
chi-squared values ($\chi^{2}_{\rm DA}$ and $\chi^{2}_{\rm non-DA}$) were
obtained. Finally, the estimator of the probability of being a DA white dwarf
($P_{\rm DA}$) was defined as

$P_{\rm DA}=\frac{1}{2}\left(\frac{\chi^{2}_{\rm non-DA}-\chi^{2}_{\rm
DA}}{\chi^{2}_{\rm non-DA}+\chi^{2}_{\rm DA}}+1\right),$ (1)

where we classified an object as DA if $P_{\rm DA}\geq 0.5$, otherwise as a
non-DA.

We validated our classification procedure by means of the spectroscopically
labelled white dwarf sample of the Montreal White Dwarf
Database888https://www.montrealwhitedwarfdatabase.org/ (MWDD; Dufour et al.
2017). A total of 8 482 objects from the MWDD were used in our validating test
(including those at distances closer than 100 pc). We adopted as DA class all
MWDD objects whose primary spectral type is DA regardless of secondary types.
The rest of the objects were considered as non-DA. In the left panel of figure
2 we show the distribution of the probability of being a DA, $P_{\rm DA}$, for
white dwarfs labelled as DA (blue histogram) or non-DA (red histogram) in the
MWDD. The distribution confirms that the adopted threshold at $P_{\rm DA}=0.5$
effectively separates both populations and that the percentage of
missclassified object is reasonable small, $\lesssim 7\%$. In the right panel
of Fig. 2 we show the confusion matrix, where rows represent the number of
already labelled spectral objects, while columns are the prediction of our
classification (in parentheses the percentages with respect to the total
population). The results of the confusion matrix reveal that the performance
of our spectral type estimator is excellent, as it is corroborated by the
derived metrics999For a definition of these parameters, please refer for
instance to Appendix A from Echeverry et al. (2022): accuracy of 0.94,
F1-score of 0.96, recall of 0.94 and precision of 0.98.

Figure 3: Distribution of $G$ apparent magnitude for the entire sample of
white dwarfs with Gaia spectra (blue histogram), those with spectral
classification in MWDD (gray histogram) and those of the previous sample
missclassfied by our method (red histogram).

A last check was performed before applying our classification method to the
observed Gaia white dwarf sample. In Figure 3 we depicted the distribution of
the $G$ apparent magnitude for the entire sample of white dwarfs with Gaia
spectra (blue histogram) and those with spectral classification in MWDD (gray
histogram). We verified that the MWDD sample covers the full range of
magnitudes and closely resemble the observed Gaia sample distribution. These
facts guarantee the proper use of the MWDD sample for testing our
classification method. Moreover, in Fig. 3, we present the magnitude
distribution of the white dwarfs misclassified by our method (red histogram).
As expected, the fainter the magnitude the larger the fraction of
missclassified objects. Considering the percentage of error as a function of
magnitude and extrapolating it to the Gaia sample, we estimated that the error
in the final classification should not exceed 10%.

Once we have validated the reliability of our classification method, we
applied it to the sample of white dwarfs with Gaia spectra within 500 pc. A
total of 65 310 white dwarfs (including those previously classified in
Jiménez-Esteban et al. 2023) have been classified into the spectral types DA
(50 189; 77%) and non-DA (15 121; 23%) with an accuracy of 0.94. The catalogue
with the spectral classification is available online as supplementary material
hosted by the journal and at _The SVO archive of Gaia white dwarfs_
101010http://svocats.cab.inta-csic.es/wdw/index.php at the Spanish Virtual
Observatory portal111111https://svo.cab.inta-
csic.es/docs/index.php?pagename=Archives.

Figure 4: Gaia HR diagram for the population of white dwarfs classified by our
probability estimator as DA (left panel) and non-DA (right panel). As a visual
reference, we plotted the cooling sequence of a 0.58 ${\rm M}_{\sun}$ DA white
dwarf according to La Plata models.

In Figure 4, we showed the HR diagram of the corresponding DA and non-DA
populations. For visual reference we also showed the cooling sequence of a
0.58 ${\rm M}_{\sun}$ DA white dwarf (Camisassa et al., 2016). The
distribution of DA and non-DA white dwarfs clearly follows different tracks,
with the latter group being, on average, less luminous for a certain color (in
particular for ${\rm BP-RP}>0$) than the first one. It is worth mentioning
that our classification into DA and non-DA groups is less model dependent than
H- versus He-rich classification. However, such a classification is still
needed for a correct astrophysical interpretation. Thus, higher resolution
spectroscopy is required to flag misclassified objects, identify He-rich DAs,
magnetic DAH with distorted Balmer jumps, as well as DZ and DQ with unusual
colors, among other cases.

##  4 Completeness correction

Figure 5: Left panel: Gaia color-magnitude diagram of our population of DA and
non-DA white dwarfs (yellow dots). Highlighted in blue are those objects at
distances $d>250\,$pc. Assuming a limiting magnitude of $G_{\rm lim}=19.5$,
only those brighter than $M_{G}<12.5$ (horizontal red line) are observable.
The cooling track for a 1.05 ${\rm M}_{\sun}$white dwarf (black line) is
adopted as our lower selection function limit. For a given distance, only
objects (marked in dark blue) at the left of the corresponding color value
(vertical red line) contribute to the final spectral distribution. Right
panel: highlighted in red are the white dwarfs selected for building our
spectral distribution. Objects above or below the cooling track for a 0.51
${\rm M}_{\sun}$ helium-rich and a 1.05 ${\rm M}_{\sun}$ hydrogen-rich white
dwarf, respectively, were discarded.

We analyzed the different selection effects and how they can be corrected or
at least mitigated. First of all, as our sample is initially built as a
magnitude-selected sample, objects fainter than a certain magnitude limit
would be absent from our sample. A standard $1/\mathcal{V}_{\rm max}$ method
(Schmidt, 1968) will provide an unbiased estimate of the space density.
However, it requires extra conditions of completeness and homogeneity to be
fulfilled (see Geijo et al., 2006, and references therein). These conditions
are not guaranteed in our sample, as the requirement to have a Gaia spectrum
adds a new selection effect and, mainly because DA and non-DA populations
have, as previously stated, a different distribution in the color-magnitude
diagram (see Fig.4).

In order to avoid this bias that would distort the spectral distribution, we
developed a strategy in which we consider objects contributing to the spectral
distribution only if they are brighter than a certain magnitude and hotter
than a certain temperature (color). We adopted the cooling sequence for a
$1.05\,$${\rm M}_{\sun}$ white dwarf as our faint limiting region. For a given
distance of the white dwarf, the Gaia limiting magnitude we adopted $G_{\rm
lim}=19.5$ (see Section 2) will fix the absolute observable magnitude limit.
The corresponding color at this magnitude for the 1.05 ${\rm M}_{\sun}$ white
dwarf cooling sequence delimits the possible contribution to the spectral
distribution. Only objects with a bluer color than this value will contribute,
while redder objects will be disregarded. In the left panel of Figure 5, we
present an example of this strategy. White dwarfs beyond a distance of 250 pc
are highlighted in the Gaia HR diagram. Only those objects (marked in blue)
above the horizontal line are observable, and those to the left of the
vertical line and above the 1.05 ${\rm M}_{\sun}$ track are included in the
final sample to construct our spectral distribution. Thus, we prevent non-DA
objects (which, on average, are fainter than DAs for a given color; see Fig.
4) from being underestimated in the spectral distribution. It should be noted
that this procedure automatically eliminates any massive white dwarf from our
sample. However, their contribution is estimated to be less than $\approx 3$%
of the entire population (e.g. Kilic et al., 2020; Jiménez-Esteban et al.,
2023).

Figure 6: Spectral distribution presented in this work (black lines and solid
circles) and when no selection criteria and correction function is applied at
all (blue open triangles). Also shown the one obtained for the 100 pc sample
(Jiménez-Esteban et al., 2023, red circles).

A second important source of incompleteness comes from the fact that not all
white dwarf sources have an available Gaia spectrum. Moreover, even those
sources that have it may not have a good VOSA determination of the probability
of being DA or non-DA. It is expected that the number of sources without a
determination of this probability increases for fainter and distant objects.
We take into account this fact by introducing a weight function that depends
on the distance, $d$, of the object and its specific location within the HR
diagram, $w(G_{\rm BP}-G_{\rm RP},\,M_{G},\,d)$. For a given source with
parameters $(G_{\rm BP}-G_{\rm RP},\,M_{G},\,d)_{0}$ we computed the number of
sources, $n_{\rm sources}$, inside a volume $\mathcal{V}=\Delta(G_{\rm
BP}-G_{\rm RP})\times\Delta M_{G}\times\Delta d$ centered at the previous
value and with $\Delta(G_{\rm BP}-G_{\rm RP})=0.1$, $\Delta M_{G}=0.1$ and
$\Delta d=50\,$pc. Besides the number of sources inside $\mathcal{V}$, we also
computed the number of objects with available Gaia spectra, $n_{Gaia-sp}$, and
the number of those who have a VOSA estimation of the probability of being DA,
$n_{VOSA-PDA}$. Assuming that the completeness weight function should be
inversely proportional to the probability of an object of belonging to the
final sample, a straightforward application of the Bayes’ theorem for
conditional probability leads to:

$w(G_{\rm BP}-G_{\rm RP},\,M_{G},\,d)=\left(\frac{n_{Gaia-
sp}}{n_{sources}}\times\frac{n_{VOSA-PDA}}{n_{Gaia-
sp}}\right)^{-1}=\frac{n_{sources}}{n_{VOSA-PDA}}.$ (2)

Finally, in addition to the selection function previously described (that is,
we selected objects hotter than a given color determined by the $1.05\,$${\rm
M}_{\sun}$ evolutionary track for a given distance), we considered objects
located in the HR diagram below the cooling track for a $0.51\,$${\rm
M}_{\sun}$ helium-rich white dwarf. This way, we avoid unresolved binary white
dwarf systems and the contribution of white dwarfs evolved from binary
evolution (Jiménez-Esteban et al., 2023). Furthermore, the existence of low-
mass white dwarfs with helium-rich atmospheres has not been proven (see, for
instance, Genest-Beaulieu & Bergeron, 2019; Battich et al., 2020). Thus, we
adopted the $0.51\,$${\rm M}_{\sun}$low-mass limit as a conservative
criterion. For each object of our final sample we determined the effective
temperature by interpolating the Gaia photometry in the La Plata models. Those
objects classified as DA were interpolated in the models of Camisassa et al.
(2016), while for those labelled as non-DA, we used the hydrogen-deficient
cooling models of Camisassa et al. (2017), in both cases for carbon-oxygen-
core white dwarfs (see Jiménez-Esteban et al., 2023, for details). Atmospheric
models where those used in the SED analysis (see Section 3.1 from Jiménez-
Esteban et al., 2023), i.e., Koester’s models with pure hydrogen composition
for DAs and helium with a small trace of hydrogen (log N(H)/N(He) = -6) for
non-DAs (Koester, 2010). It is worth noting here that recent studies have
emphasized the importance of considering the carbon content in non-DA
atmospheres (Camisassa et al., 2023; Blouin et al., 2023). Assuming the
maximum non-observable carbon enrichment prescription from Camisassa et al.
(2023), i.e. carbon sequence -1 dex, the difference with respect to a pure
helium model is $\sim 10$% at 12 000 K, and approximately 5% at $6\,000\,$K.
The differences are not larger than $1\,500\,$K, which corresponds to the bin
width of our spectral distribution. Thus, no major effects are expected in
this regard.

Our final sample to estimate the spectral fraction, consisting of 33 997 white
dwarfs, is shown in Figure 5, with 25 984 (76.4%) classified as DAs and 8 013
(23.6%) as non-DAs.

##  5 The spectral type-temperature distribution

The spectral type-temperature distribution, $f$, is defined as the ratio of
weighted non-DA white dwarfs, to the total number of weighted objects,
$N_{w}$, per effective temperature interval. We adopted that the contribution
of each object to its temperature bin depends on its weight function, $w$, and
the probability of classification, $P_{i}$ (that is $P_{\rm DA,i}$ for DAs,
$1-P_{\rm DA,i}$ for non-DAs). Hence, the weighted number of objects is
defined as:

$N_{w}=\sum_{i}^{N}w(G_{\rm BP}-G_{\rm RP},\,M_{G},\,d)_{i}\times P_{i},$ (3)

where $N$ is the number of objects in that interval, $w$ the weighted function
and $P_{i}$ the probability aforementioned. Error bars were estimated taking
into account Poissonian error and the corresponding object weight, that is
$\sigma_{f}=\sqrt{f\times(1-f)/N_{W}}$.

In Figure 6 we displayed our final spectral distribution (black solid circles
and black line). To evaluate the extent of the completeness correction
introduced in our sample, we included the ratio distribution of the entire
classified white dwarf population (raw sample consisting of $65\,310$ objects)
when no selection function is applied at all (blue open triangles). For
comparative purposes, we also plotted the spectral distribution for the sample
of 100 pc (red circles; Jiménez-Esteban et al., 2023). The analysis of the
different spectral distributions revealed that the effects due to completeness
correction are of minor order. The general trend is practically coincident,
and only small discrepancies appear for the coolest bin, where error bars
underestimate the fact that the classification is less certain. In any case,
we can conclude that our weighted spectral distribution provides a robust
estimate of the ratio of DA versus non-DA white dwarfs.

Figure 7: Spectral distribution for our final white dwarf sample (black points
and lines). For comparative purposes we also show the spectral distributions
from other works found in literature.

In Figure 7, we show a comparison of the spectral distribution found in this
work (black solid circles) with some of the most recent ratio distributions
found in the literature. The first remarkable characteristic of the spectral
distribution found in this work is the wider range of effective temperatures
covered by it, thus constituting a solid estimate of the spectral evolution of
white dwarfs. While a detailed analysis of the implications for spectral
evolution is outside the scope of this work, in what follows, we make a brief
analysis of the most relevant points found:

  * -

At the hottest end, i.e., $T_{\rm eff}\approx 35\,000-40\,000\,$K, the lowest
ratio of non-DAs was found. As previously reported, there is not a complete
absence of non-DAs in this region, but an average $\sim 5\%$ is indicative of
the presence of the so called DB-gap (e.g. Bergeron et al., 2011; Koester &
Kepler, 2015, and references therein).

  * -

A statistical significant deficit of non-DAs was also found for effective
temperatures in the range
$22\,000\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}T_{\rm
eff}\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}25\,000\,$K. It is also in
perfect agreement with the ratios found by Ourique et al. (2018) and López-
Sanjuan et al. (2022).

  * -

For temperatures cooler than $\sim 18\,000$ K, coinciding with the onset of
convection mixing in DAs (e.g. Cunningham et al., 2020), a marked increase in
the ratio of non-DAs was found, leading from $\sim 10\%$ at $18\,000\,$K up to
$\sim 40\%$ at $8\,000\,$K and in agreement with most of the spectral
distributions found in the literature.

  * -

Our spectral distribution presented a peak around $T_{\rm eff}\approx
7\,000\,$K with a ratio of non-DA objects of $f\approx 0.41$. This maximum is
found at lower temperatures than that of our previous work (Jiménez-Esteban et
al., 2023) and others retrieved in literature such as Ourique et al. (2020),
probably as a consequence of an unweighted distribution in these cases.
However, the weighted distribution presented here is in agreement with the 40
pc spectroscopic complete sample analyzed in McCleery et al. (2020).

  * -

A statistically significant decrease in the ratio of non-DAs is found at the
coolest bin, $T_{\rm eff}\approx 6\,000\,$K, dropping to $f\approx 0.35$. This
behaviour was also reported in our previous work (Jiménez-Esteban et al.,
2023) and is also in agreement with Blouin et al. (2019) and McCleery et al.
(2020), although no known physical mechanism can be associated to it (Blouin
et al., 2019).

##  6 Conclusions

Following the methodology presented in Jiménez-Esteban et al. (2023), we have
expanded our white dwarf study sample up to 500 pc. A total of $65\,310$ white
dwarfs have been classified as DAs and non-DAs based on their Gaia spectra,
with an accuracy of 94%. This has allowed us to construct a statistically
significant and precise distribution of the DA versus non-DA ratio as a
function of effective temperature. Nearly $34\,000$ white dwarfs have
contributed to the final selected sample, making it the largest sample to date
in terms of the number of objects and the range of effective temperatures
analyzed, from $5\,500\,$K to $40\,000\,$K.

The comparative analysis of our distribution with others found in the
literature reveals statistically significant features such as: the deficit of
DBs within the effective temperature range of approximately
$35\,000-40\,000\,$K and between $22\,000--25\,000\,$K, along with a gradual
rise starting from $18\,000\,$K up to around $7\,000\,$K, where the proportion
of non-DA white dwarfs peaks at 41%, followed by a decline for lower
temperatures.

Finally, we can state that selection effects have been taken into account in
the construction of the final sample. This fact, along with the high number of
objects per interval in the sample, ensures that our spectral distribution can
be considered a robust and precise element in the analysis of the spectral
evolution of white dwarfs.

###### Acknowledgements.

We acknowledge support from MINECO under the PID2020-117252GB-I00 grant and by
the AGAUR/Generalitat de Catalunya grant SGR-386/2021. P.C. acknowledges
financial support from the Government of Comunidad Autónoma de Madrid (Spain)
via postdoctoral grant ‘Atracción de Talento Investigador’ 2019-T2/TIC-14760.
R.M.O. is funded by INTA through grant PRE-OBSERVATORIO. MC acknowledges grant
RYC2021-032721-I, funded by MCIN/AEI/10.13039/501100011033 and by the European
Union NextGenerationEU/PRTR. RR acknowledges support from Grant
RYC2021-030837-I funded by MCIN/AEI/ 10.13039/501100011033 and by “European
Union NextGenerationEU/PRTR”. F.J.E. acknowledges support from ESA through the
Faculty of the European Space Astronomy Centre (ESAC) - Funding reference
4000139151/22/ES/CM. 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. This work has made use of
the Python package GaiaXPy, developed and maintained by members of the Gaia
Data Processing and Analysis Consortium (DPAC) and in particular, Coordination
Unit 5 (CU5), and the Data Processing Centre located at the Institute of
Astronomy, Cambridge, UK (DPCI). This publication makes use of VOSA, developed
under the Spanish Virtual Observatory (https://svo.cab.inta-csic.es) project
funded by MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00.
We extensively made used of Topcat (Taylor, 2005). This research has made use
of the VizieR catalogue access tool, CDS, Strasbourg, France. We acknowledge
use of the ADS bibliographic services.

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