# The Gaia DR2 halo white dwarf population: the luminosity function, mass
distribution and its star formation history

Santiago Torres1,2, Alberto Rebassa-Mansergas1,2, María E. Camisassa1,3,4 and
Roberto Raddi1
1 Departament de Física, Universitat Politècnica de Catalunya, c/Esteve
Terrades 5, 08860 Castelldefels, Spain
2 Institute for Space Studies of Catalonia, c/Gran Capita 2–4, Edif. Nexus
104, 08034 Barcelona, Spain
3Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La
Plata, Paseo del Bosque s/n, 1900 La Plata, Argentina
4 Instituto de Astrofísica de La Plata, UNLP-CONICET, Paseo del Bosque s/n,
1900 La Plata, Argentina E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We analyze the volume-limited nearly complete 100 pc sample of 95 halo white
dwarf candidates identified by the second data release of Gaia. Based on a
detailed population synthesis model, we apply a method that relies on Gaia
astrometry and photometry to accurately derive the individual white dwarf
parameters (mass, radius, effective temperature, bolometric luminosity and
age). This method is tested with 25 white dwarfs of our sample for which we
took optical spectra and performed spectroscopic analysis. We build and
analyse the halo white dwarf luminosity function, for which we find for the
first time possible evidences of the cut-off at its faintest end, leading to
an age estimate of $\simeq 12\pm 0.5\,$Gyr. The mass distribution of the
sample peaks at $0.589\,M_{\odot}$, with $71\%$ of the white dwarf masses
below $0.6\,M_{\odot}$ and just two massive white dwarfs of more than
$0.8\,M_{\odot}$. From the age distribution we find three white dwarfs with
total ages above 12 Gyr, of which J1312-4728 is the oldest white dwarf known
with an age of $12.41\pm 0.22\,$Gyr. We prove that the star formation history
is mainly characterised by a burst of star formation that occurred from 10 to
12 Gyr in the past, but extended up to 8 Gyr. We also find that the peak of
the star formation history is centered at around 11 Gyr, which is compatible
with the current age of the Gaia-Enceladus encounter. Finally, $13\%$ of our
halo sample is contaminated by high-speed young objects (total age<7 Gyr). The
origin of these white dwarfs is unclear but their age distribution may be
compatible with the encounter with the Sagittarius galaxy.

###### keywords:

stars: white dwarfs – Galaxy: stellar content – stars: luminosity function,
mass function – Galaxy: halo

††pubyear: 2015††pagerange: The Gaia DR2 halo white dwarf population: the
luminosity function, mass distribution and its star formation history–The Gaia
DR2 halo white dwarf population: the luminosity function, mass distribution
and its star formation history

White dwarfs are long-lived objects whose evolutionary characteristics are
reasonably well understood (e.g. Althaus et al., 2010, and references
therein). They represent the vast majority of low- and intermediate-mass stars
remnants. Thus, their ensemble properties carry valuable information about the
past history and evolution of the different components of the Galaxy. In
particular, white dwarfs are reliable cosmochronometers and, consequently,
they have been used for studying several Galactic populations. As illustrative
examples of this, we can mention the analysis of the Galactic thin and thick
disk (e.g. García-Berro et al., 1999; Torres et al., 2002; Rowell & Hambly,
2011; Kilic et al., 2017), the halo (e.g. Mochkovitch et al., 1990; Isern et
al., 1998; García-Berro et al., 2004; Cojocaru et al., 2015) the bulge (e.g
Calamida et al., 2014; Torres et al., 2018) or precise studies of Galactic
open and globular clusters (e.g. García-Berro et al., 2010; Jeffery et al.,
2011; Hansen et al., 2013; Torres et al., 2015).

Regarding the population of white dwarfs in the Galactic stellar halo, it has
been the focus of attention in this field since the first observational and
theoretical studies (Mochkovitch et al., 1990; Liebert et al., 1989; Isern et
al., 1998). Shortly after, and since the MACHO collaboration experiment for
the microlensing detection (e.g. Alcock et al., 2000), halo white dwarfs have
been suggested as natural candidates to contribute to the dark matter content
of the Galaxy (e.g. Oppenheimer et al., 2001). An intense debate arose on this
issue, although later studies demonstrated that the white dwarf contribution
to dark matter was rather limited (e.g. Torres et al., 2002; Flynn et al.,
2003; García-Berro et al., 2004; Kilic et al., 2004; Bergeron et al., 2005).
In any case, the search for white dwarfs in the Galactic halo has proven to be
a difficult endeavour. In this sense, the intrinsic faintness and the low
space density of halo white dwarfs, along with their high surface gravity,
which erases any trace on metal content from their atmospheres and disables
accurate radial velocity measurements due to the broadening of the Balmer
lines, are some of the main factors that have hindered the discovery of
suitable candidates. Consequently, during decades, the identification of halo
members has relied on the search of cool objects in high proper motion surveys
(e.g. Monet et al., 2000; Nelson et al., 2002; Hall et al., 2008; Catalán et
al., 2012; Munn et al., 2016). It was not until the ESO SNIa Progenitor surveY
(SPY) project (see Napiwotzki et al., 2020, and references therein) that
radial velocities were measured for the first time with enough precision to
recover the 3D kinematics of white dwarfs. That permitted to identify high
eccentric retrograde orbits as strongly indicators of their belonging to the
halo population (Pauli et al., 2006). Although the number of halo members was
sparse, an estimate of the age of the inner halo has been possible ($\sim
11.4\,$Gyr; Kalirai, 2012).

A key ingredient of these studies is the white dwarf luminosity function.
Defined as the number of white dwarfs per bolometric magnitude unit and cubic
parsec, the white dwarf luminosity function is a valuable tool to derive the
age, history and evolution of the components of our Galaxy. Moreover, the
luminosity function holds all the information about the cooling process of the
white dwarfs, being then an excellent tool for testing the physics of
evolutionary models (see García-Berro & Oswalt 2016, for a comprehensive
review about these issues). However, the effectiveness of the white dwarf
luminosity function requires the existence of a complete, well-defined volume
sample. In this sense, the lack of accurate distances previous to the Gaia
era, joint to the selection biases inherited from magnitude-limited samples,
have hampered the achievement of more conclusive results in such an elusive
sample as the halo white dwarf population.

Figure 1: Gaia HR-diagram for our halo white dwarf sample within 100 pc.
Marked as red circles are those white dwarfs for which we obtained optical
spectra. In the left panel we plot the theoretical cooling tracks for DA
(solid lines) and DB (dashed lines) white dwarfs for different masses ranging
from 0.5 to $1.1\,M_{\odot}$ in steps of $0.2\,M_{\odot}$. In the right panel
a typical simulation (gray dots) taking into account photometric and
astrometric errors is shown. For illustrative purposes and for the sake of
clarity we only plot some $1\sigma$ ellipses. See text for details.

The first observational halo white dwarf luminosity function (Liebert et al.,
1989) contained only five objects, and no more than 20 were proposed as halo
candidates through a neural network artificial intelligent classification
algorithm (Torres et al., 1998). Based on the SuperCOSMOS Sky Survey
observations towards the southern Galactic Cap, Oppenheimer et al. (2001)
identified 38 high-tangential velocity white dwarfs, from which they deduced
that white dwarfs may represent 2% of the local dark matter density. An
intense debate began in this regard, mainly based on the difficulty of
unequivocally identifying halo members. Currently, several halo white dwarf
luminosity functions have been proposed (e.g. Rowell & Hambly, 2011; Munn et
al., 2016; Lam et al., 2018). These samples, based on reduced-proper motion
surveys, are magnitude limited. Thus, different statistical approaches,
starting from the well-known maximum volume estimator method, have been
applied in order to correct the observational biases of the intrinsic
incompleteness of magnitude limited samples. Parallel to the search for
statistical significant halo white dwarf samples, a major effort has been
devoted to identify individual halo white dwarf members. The list includes
from hot and warm objects up to cool and ultracool white dwarfs (e.g. Hambly
et al., 1997; Ibata et al., 2000; Pauli et al., 2006; Kalirai, 2012; Kawka &
Vennes, 2012; Gianninas et al., 2015; Si et al., 2017).

However, as stated before, achieving a statistical complete sample of halo
white dwarfs has become an arduous task, at least during the pre-Gaia era. A
first estimate of the number of halo white dwarfs showed that up to magnitude
$G<20$, around $85$ objects within 100 pc and $\sim 500$ objects within 400 pc
are expected to be accessible by Gaia (Torres et al., 2005). Since Gaia Data
Release 2 was published only a few studies of the Galactic halo white dwarf
population have been published. In Kilic et al. (2018), a sample of 142
objects inconsistent with disk kinematics has been analyzed. In particular, an
age estimate has been provided for many of the members of the sample,
obtaining, from the coolest white dwarfs, an age estimate of the inner halo of
10.9 Gyr. However, the sample, which contains objects as far as 540 pc, is far
from being complete. On the other hand, Torres et al. (2019a) focused on the
search of halo white dwarfs in the near-complete sample within 100 pc from the
Sun. With the help of artificial intelligence algorithms, Torres et al.
(2019a) categorized the membership of white dwarfs in the different Galactic
components, identifying the largest volume-complete halo white dwarf sample,
consisting in 95 stars.

A renewed interest on the halo population has appeared since the astrometric
Gaia mission has provided accurate parallax and proper motion measurements for
approximately 1.4 billion stars of our Galaxy (Gaia Collaboration et al.,
2018). In the new Gaia-Enceladus paradigm, a major impact event took place in
our Galaxy around 10.5-11.5 Gyr in the past, being at the origin of the
formation of the thick disk and inner stellar halo (Helmi et al., 2018;
Gallart et al., 2019). The precise age, intensity and effects of this past
event on the kinematics and properties of the stars in the solar neighborhood
are not yet well understood.

In this paper, we analyze the halo white dwarf sample identified in Torres et
al. (2019a). In particular, we obtain the halo white dwarf luminosity
function, the mass distribution of the sample and an estimate of its star
formation history. To achieve this goal, we complement our analysis with
spectroscopic observation of 27 stars of the sample. The stellar parameters of
each white dwarf (i.e. luminosity, age and mass) are obtained from a robust
method which makes use of Gaia photometry and astrometry together with a
detailed population synthesis code, based on Monte Carlo techniques and which
incorporates the most up-to-date evolutionary sequences of white dwarfs.
Finally, it is important to emphasise that we use the term halo to describe
that sample whose characteristics are different from the average thin or thick
disk white dwarf sample. If these objects really belong to an ancient halo,
inner spheroid or are the remnants of a major merger event, it is something
that will be analyzed along the paper.

The paper is organized as follows. In the first section we describe our halo
white dwarf sample. That includes how we identified the halo candidates, the
fundamental parameters that were derived from those spectroscopically observed
and the main characteristics of our population synthesis modeling. In the
second section we present our strategy and testing for deriving the
fundamental parameters extended to the whole halo white dwarf sample. Then,
the halo white dwarf luminosity function, mass distribution and star formation
history are presented. Finally, in Section 3 we analyze the results achieved
and summarize them in the concluding remarks.

##  1 The halo white dwarf sample

###  1.1 Identification of halo white dwarfs

Gaia Data Release 2 has provided a wealth of unprecedented information
concerning the Galactic white dwarf population (Jiménez-Esteban et al., 2018;
Gentile Fusillo et al., 2019). The high astrometric accuracy and the
photometry provided by Gaia has allowed us to build a clean color-magnitude
diagram to select a large sample of white dwarf candidates. With the aid of a
detailed population synthesis simulator, we analysed the main properties of
the white dwarf population available from Gaia (Jiménez-Esteban et al., 2018).
In this study, we showed that the largest and most complete sample of white
dwarfs available by Gaia is up to 100 pc from the Sun. For larger distances,
the completeness of the sample decays dramatically and biases and magnitude-
selection effects begin to grow in importance.

With the aid of advanced intelligent algorithms, we studied the main
properties of the Gaia 100 pc white sample (Torres et al., 2019a). In
particular, we have been able to disentangle the white dwarf populations from
the different components of the Galaxy, i.e., thin and thick disk, and halo.
The artificial intelligent method used consisted in a supervised method based
on Random Forest techniques. This Random Forest algorithm was applied to an
8-dimensional space formed by equatorial coordinates, parallax, proper motion
components and photometric magnitudes. This 8-dimensional space permits the
algorithm to maximize the information in order to classify its different
components. Our results showed that the algorithm presents an accuracy of
85.3%. In particular, our analysis indicates that $80\%$ of possible halo
white dwarfs in the 100 pc sample are expected to be correctly identify, and
only a low $5\%$ contamination is expected. These scores are higher than any
other obtained by tangential velocity cuts, Toomre diagram criteria or reduced
proper motion criteria usually applied for selecting the different kinematic
populations. Moreover, practically all white dwarfs within 100 pc that were
previously identified in the literature as halo members are also included in
our sample. Thus, the sample found represents the most complete and largest
volume-limited sample of the halo white dwarf population to date. However,
these facts do not exclude the need to perform a detailed analysis of the
completeness of the sample, in particular, for the faint region of the sample.
This is done in Section 2.2.

The sample consists of 95 halo white dwarfs candidates representing an old and
high velocity population. Its space density is $(4.8\pm 0.4)\times
10^{-5}\,{\rm pc^{-3}}$, accounting for $1\%$ of the whole white dwarf
population within 100 pc. A complete list of the 95 halo members with their
Gaia source ID, coordinates and main parameters is shown in Table 3 of Torres
et al. (2019a). In Figure 1 we plot our 95 halo candidates in the Gaia
Hertzsprung-Russell (HR) diagram. For illustrative purposes we also plot the
cooling tracks belonging to hydrogen-rich and pure helium atmosphere models
for different masses. As shown in Fig. 1, many objects lie near the blue-hook
of the hydrogen-rich atmosphere tracks, thus representing cold and old
objects. At first sight, however, it seems to reveal that many of these
objects have low masses,
$\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$}0.5\,M_{\odot}$,
even when compared with He-pure atmosphere tracks. This effect was analyzed by
Bergeron et al. (2019) claiming that probably most of the possible non-DA
objects in this region are DA or a large fraction of unresolved double
degenerates populates this region. However, we shall see in Section 1.3 that
this apparent shift towards lower masses is nicely resolved when taking into
account photometric and astrometric errors in the observed data.

###  1.2 Spectroscopic observation of halo white dwarfs

Figure 2: Spectral model fit to the DA white dwarf J0148–1712. Top left panel:
best-fit (black lines) to the normalised Balmer line profiles sampled by the
FORS2 spectrum (gray lines). Top right panel: 3, 5, and 10$\sigma$ $\chi^{2}$
contour plots in the T${}_{\mathrm{eff}}-\log g$ plane. The black contours
refer to the best line profile fit, the red contours to the fit of the whole
spectrum. The dashed line indicates the occurrence of maximum H$\beta$
equivalent width. The best “hot” (not visible in the figure) and “cold” line
profile solutions are indicated by black dots, the best fit to the whole
spectrum is indicated by a red dot. Bottom panels: the white dwarf spectrum
(gray line) along with the best-fit white dwarf model (black line) (top) and
the residuals of the fit (gray line, bottom). Figure 3: Spectral model fit to
the DC white dwarf J2117–4156. Top panel: the white dwarf spectrum (gray line)
along with the best-fit white dwarf model (black line). Bottom panel: the
residuals of the fit (gray line).

We obtained low-resolution spectra for 27 of our 95 halo white dwarf
candidates with the Very Large Telescope UT1 equipped with the FOcal
Reducer/low-dispersion Spectrograph (FORS2) (Appenzeller et al., 1998). We
used the GRIS_1200+97 grism and the 1" slit width, which resulted in spectra
covering the $\simeq$3800-5200 Å wavelength range at a resolving power of
$R\simeq$1400\. We used the Pamela software (Marsh, 1989) to subtract the sky
contribution and to extract the one dimensional spectra. The data were then
wavelength and flux calibrated using arc lamps and flux standard stars taken
at the same time of the observations within the molly package111Tom Marsh’s
molly package is available at
http://deneb.astro.warwick.ac.uk/phsaap/software. Visual inspection of the
flux-calibrated spectra revealed that 24 targets were DC white dwarfs, a
result which is not surprising given that these objects are expected to be
very old and cold. The remaining three white dwarfs were classified as two DQs
(J0248-3001 and J1159-4629) and one DA (J0148-1712). The Gaia source ID, short
name, Gaia absolute magnitude $M_{\rm G}$, colour $G_{\rm{\tiny
BP}}\\!-\\!G_{\rm{\tiny RP}}$ and spectral type of the 24 DC+1 DA observed
white dwarfs are, respectively, provided in the first five columns of Table 1.

In order to derive the stellar parameters of the white dwarfs with available
spectra we used the fitting routine of Rebassa-Mansergas et al. (2007).
Briefly, this procedure uses a grid of model spectra (Koester, 2010)222The
grid includes 1260 spectra of effective temperatures between 3000 and 40,000 K
in steps of 250 K up to 20,000 K, steps of 1000 K up to 30,000 K and steps of
2000 K up to 40,000 K; and surface gravities between 6 and 9.5 dex in steps of
0.25 dex for each effective temperature. Prior to the fit, the model spectra
were folded at the resolving power of the observed spectra. to fit both the
continuum and the normalised Balmer lines of the spectra to derive the
effective temperatures and surface gravities. By adopting then a white dwarf
mass-radius relation (e.g. Camisassa et al., 2016), the mass and radius (hence
bolometric luminosity since the temperature is known) are also obtained. This
routine is only valid if the white dwarfs are hydrogen-rich DAs. In this case,
the fit to the continuum is only used to differentiate between the so-called
hot and cold solutions obtained from the Balmer line fitting. The fit to our
only DA halo white dwarf candidate with available FORS2 spectrum is shown in
Figure2. However, for our DC white dwarfs, which are the vast majority of the
spectroscopically observed sample, we could only rely on the fits to the
continuum since these objects are absent of any lines. Thus, only the
effective temperatures were obtained in these cases. A fit to one of our DC
white dwarf halo candidates can be seen in Figure3.

We note that another possibility exists for deriving the white dwarf radii of
our DC white dwarfs by using the flux scaling factors between the observed
spectra and the best fit models, since the distances are known from the Gaia
parallaxes. Hence, a spectroscopic bolometric luminosity could be obtained
from the effective temperatures and radii via the Stefan-Boltzmann equation.
We performed a first attempt to follow this procedure by first re-scaling the
observed spectra to the Gaia fluxes (and PanStarrs fluxes when available too)
of our stars.

The radii, effective temperatures and bolometric luminosities thus derived for
our DA and the rest of DC halo white dwarf candidates are shown, respectively,
in the last three columns of Table 1. In Section 2.1 we discuss the validity
of the spectroscopic parameter values thus obtained.

Table 1: Spectroscopic determinations of the parameters of the halo white dwarf candidates for which the spectrum is available. The stellar parameters derived, i.e., radius, effective temperature and bolometric luminosity, are shown, respectively, in the last three columns. Gaia | Short | $M_{\rm G}$ | $G_{\rm{\tiny BP}}\\!-\\!G_{\rm{\tiny RP}}$ | Spectral | Radius | $T_{\rm eff}$ | Luminosity
---|---|---|---|---|---|---|---
source ID | name | (mag) | (mag) | type | $(R_{\odot}/10^{3})$ | $(K)$ | $(L_{\odot}/10^{5})$
5042228731477861888 | J0129-2257 | 15.55 | 1.41 | DC | $11.76\pm 0.97$ | $4094\pm 100$ | $3.49\pm 0.83$
5142197118950177280 | J0148-1712 | 13.12 | 0.47 | DA | $15.90\pm 1.09$ | $7144\pm 50$ | $59.15\pm 10.56$
2490975272405858048 | J0205-0517 | 15.59 | 1.53 | DC | $10.64\pm 0.69$ | $4094\pm 100$ | $2.86\pm 0.58$
4616895783694397184 | J0237-8445 | 15.77 | 1.57 | DC | $13.25\pm 2.31$ | $3955\pm 100$ | $3.85\pm 1.64$
5188044687948351872 | J0301-0044 | 15.30 | 1.55 | DC | $10.89\pm 1.05$ | $4287\pm 100$ | $3.60\pm 0.95$
4862884499360563968 | J0340-3301 | 15.79 | 1.32 | DC | $10.58\pm 0.89$ | $4238\pm 100$ | $3.24\pm 0.77$
3249657094642979840 | J0342-0344 | 15.50 | 1.11 | DC | $10.90\pm 0.91$ | $4541\pm 100$ | $4.54\pm 1.05$
2989049057626796416 | J0518-1155 | 15.70 | 1.74 | DC | $9.45\pm 1.02$ | $4001\pm 100$ | $2.05\pm 0.60$
5228861484450843648 | J1049-7400 | 16.09 | 1.55 | DC | $8.22\pm 0.87$ | $4047\pm 100$ | $1.63\pm 0.47$
3801499128765222400 | J1053-0307 | 15.39 | 1.44 | DC | $11.19\pm 0.98$ | $4141\pm 100$ | $3.31\pm 0.82$
5348874243767794304 | J1123-5150 | 15.61 | 1.34 | DC | $11.89\pm 1.19$ | $4865\pm 100$ | $2.83\pm 1.87$
5377861317357370240 | J1159-4630 | 15.19 | 0.42 | DC | $8.32\pm 0.74$ | $4489\pm 100$ | $2.52\pm 0.62$
6085402414245451520 | J1312-4728 | 15.58 | 1.54 | DC | $10.58\pm 0.79$ | $4094\pm 100$ | $2.83\pm 0.63$
6165095738576250624 | J1342-3415 | 14.66 | 1.05 | DC | $10.88\pm 0.85$ | $5095\pm 100$ | $7.16\pm 1.54$
5824436284328653312 | J1517-6645 | 16.01 | 1.31 | DC | $9.71\pm 1.33$ | $3865\pm 100$ | $1.89\pm 0.67$
6007140379167609984 | J1518-3803 | 15.68 | 1.56 | DC | $10.51\pm 1.11$ | $4189\pm 100$ | $3.05\pm 0.87$
5827557213731539328 | J1539-6124 | 15.23 | 1.33 | DC | $11.62\pm 0.89$ | $4336\pm 100$ | $4.29\pm 0.96$
5817295536128445568 | J1707-6319 | 15.12 | 1.41 | DC | $11.25\pm 0.94$ | $4437\pm 100$ | $4.40\pm 1.04$
6647162730439433984 | J1936-4913 | 15.60 | 1.51 | DC | $10.77\pm 0.99$ | $4189\pm 100$ | $3.21\pm 0.82$
6471523921227261056 | J2042-5218 | 14.70 | 0.94 | DC | $11.65\pm 0.86$ | $4755\pm 100$ | $6.23\pm 1.31$
6580458035746362496 | J2117-4156 | 15.16 | 1.34 | DC | $11.80\pm 0.84$ | $4437\pm 100$ | $4.85\pm 1.02$
6580551872194787968 | J2129-0034 | 15.39 | 1.58 | DC | $12.60\pm 1.53$ | $4287\pm 100$ | $4.81\pm 1.51$
2687584757658775424 | J2230-7515 | 15.58 | 1.55 | DC | $10.62\pm 0.82$ | $4141\pm 100$ | $2.98\pm 0.67$
6357629089412187648 | J2319-0613 | 15.07 | 1.38 | DC | $7.05\pm 0.74$ | $4979\pm 100$ | $2.75\pm 0.74$
2631967439437024384 | J2349-0124 | 15.01 | 1.23 | DC | $13.41\pm 1.08$ | $4287\pm 100$ | $5.45\pm 1.26$

###  1.3 The synthetic halo white dwarf population

We complement our analysis of the 95 halo white dwarfs candidates with the aid
of a detailed population synthesis code. Our code, based on Monte Carlo
techniques, has been widely used in the study of the white dwarf population of
the different Galactic components, i.e., disk, halo and bulge, as well as in
globular and open clusters (e.g. García-Berro et al., 1999, 2004, 2010; Torres
et al., 2001, 2002, 2015; Torres & García-Berro, 2016; Torres et al., 2018).
Here we will only mention the main ingredients of our halo simulation, while a
thorough description of the code, as well as its Gaia performances, can be
found in Torres et al. (2005); Jiménez-Esteban et al. (2018); Torres et al.
(2019a).

The main objective of our halo white dwarf simulation is to over-populate the
entire white dwarf region of the HR-diagram space. Synthetic white dwarfs will
be used then to extract the physical parameters at the HR-diagram loci where
observed white dwarfs have been found. For this reason, it is not necessary to
take into account a meaningful star formation history, nor an exact age of the
halo population. Consequently, we just adopt a constant star formation rate
and we generate stars with an upper-limiting age fixed at the age of the
Universe of 13.7 Gyr (Planck Collaboration et al., 2016). Stars generated are
drawn form a Salpeter initial mass function with a standard slope value of
$\alpha=-2.35$. We considered that white dwarfs are formed only through single
evolution. Additionally, as done in the halo white dwarf analysis by Kilic et
al. (2018), a constant metallicity value of $[{\rm Fe/H}]=-1.5$ is adopted.
Main-sequence lifetimes are drawn from BaSTI models (Hidalgo et al., 2018) and
the semi-empirical initial-to-final mass relationship of Catalán et al. (2008)
is applied. White dwarf cooling times are derived from a complete set of
cooling sequences, which encompass the full range of masses (Althaus et al.,
2015; Camisassa et al., 2017, 2019). It is worth noting that these cooling
sequences are specifically calculated for the adopted metallicity and are the
result of the full previous progenitor evolution, starting at the zero-age
main sequence, all the way through central hydrogen and helium burning,
thermally pulsing asymptotic giant branch (AGB) and post-AGB phases. The
lifetime of the white dwarf progenitors along with the initial-to-final mass
relationship obtained this way are in completely agreement with those input
models previously stated. This fact guarantees us the use of a coherent set of
evolutionary sequences. Hydrogen-rich and hydrogen-deficient atmosphere models
are generated according to the canonical distribution of $80\%$ and $20\%$,
respectively. For the spatial distribution an isothermal model is adopted,
which is practically equivalent to an isotropic distribution for the local 100
pc neighborhood. Magnitudes are interpolated in the corresponding cooling
sequences and calculated in the Gaia filters (R. Rohrmann’s private
communication) using the appropriate atmosphere models. Finally, photometric
and astrometric errors are added following Gaia’s
performance333http://www.cosmos.esa.int/web/gaia/science-performance.

A representative HR-diagram of our simulated halo white dwarf sample is
presented in the right panel of Fig. 1. The number of objects generated is
large enough to ensure statistical significance in all the regions around the
observed objects. As it can be seen in Fig. 1, the synthetic sample populates
all the regions where our 95 halo white dwarf candidates are located. Only one
object (J0055+3847) is outside of the simulated space, and this is probably
because it is either a He-core white dwarf or a double degenerate system. It
is also worth noting that those objects previously discussed in Section 1.1 as
cold and abnormally shifted in the HR-diagram towards very low masses are now
naturally recovered in the simulated space, once photometric and astrometric
errors are taken into account.

Figure 4: Spectroscopic effective temperatures (top left panel), radii (top
right panel) and luminosities (bottom left panel) for the 24 DC (black dots)
and one DA (red dot) observed objects (see Section 1.2) compared to the
respective photometric parameters derived as the average value within
$1\sigma$ ellipses. Also shown (bottom right panel) is the photometric
luminosity obtained in this work with other estimations also photometrically
obtained. The equality line is plotted in all panels as a dashed line. The
photometric effective temperatures agree with those spectroscopically derived.
That is not the case of the radii, where intrinsic errors when applying flux
calibration induce a systematic error in the radius calculation in the
spectroscopic method, hence in the bolometric luminosities also. On the
contrary, the luminosities derived by our photometric procedure are in perfect
agreement with those published in the literature (see text for details).

##  2 Results

###  2.1 Methodology and testing

The population synthesis sample described in the previous section allows us to
derive the representative white dwarf parameters at each locus of the HR-
diagram by just taking into account the Gaia photometric and astrometric
errors. For each of the 95 objects of our halo sample we first derive the
theoretical errors in $G$ magnitude and $G_{\rm BP}-G_{\rm RP}$ colour,
$\sigma_{G}$ and $\sigma_{G_{\rm BP}-G_{\rm RP}}$, respectively. We build then
a region centered in the location of each object in the HR-diagram that
contains all the synthetic stars within $1\sigma$. The ellipses thus formed
are shown in the right panel of Fig. 1. For the sake of clarity we only plot
some of the most representatives. The $G$-magnitude error is mainly due to the
photometric error in the $G$ filter, given that the astrometric error in
parallax is practically negligible. On the contrary, the error in $G_{\rm
BP}-G_{\rm RP}$ colour is the addition of the photometric errors from $G_{\rm
BP}$ and $G_{\rm RP}$ filters. Thus, in general terms, the size of the
$1\sigma$-ellipses increases for dimmer objects, being the error in the
absolute $G$ magnitude substantially smaller than the error in colour.

For each of the $1\sigma$ ellipses we obtain the average luminosity, mass,
age, effective temperature and radius of all synthetic white dwarfs within it.
We can now compare the photometric parameters thus obtained with the
spectroscopic ones derived for the 25 observed objects presented in Section
1.2. The results are plotted in Figure 4. On the top panels we show the
comparison between the photometric versus the spectroscopic effective
temperature (top left panel) and white dwarf radius (top right panel). The
agreement in effective temperatures is good, since most of the objects are
within the $1\sigma$ errors. However, that is not the case for the white dwarf
radii. Those obtained from the spectroscopic analysis are systematically lower
than those obtained photometrically. This effect is most likely due to
intrinsic errors when applying the flux calibration to our FORS2 spectra,
which directly affect the radii measurements through the flux scaling factors.

Consequently, the luminosity thus obtained from the spectroscopic method
suffers from this imprecision. In the bottom left panel of Fig. 4 we compare
the photometric versus the spectroscopic luminosity for the objects in which
the spectrum is available. Photometric luminosities are shifted towards larger
(brighter) luminosities than those derived spectroscopically. That is a
consequence of systematically lower spectroscopic radii, as previously stated.
For this reason, we do not consider the spectroscopic radii (hence bolometric
luminosities) to be as reliable as the photometric ones and hence decided not
to make use of these values.

Finally, we carry out an additional verification comparing the photometric
luminosity of our work with other values published in the literature and also
derived photometrically (Kilic et al., 2018; Tremblay et al., 2020). The
results are shown in the bottom right panel of Fig. 4. As it can be seen, the
agreement is excellent. Thus, our photometric procedure provides effective
temperatures which are in agreement with those derived spectroscopically and
yields luminosities which are in perfect agreement with those also derived
photometrically in the literature. Consequently, we can conclude that the rest
of parameters are robustly obtained and derived in a consistent way by our
photometric method.

Once we have demonstrated that our strategy is feasible to compute the white
dwarf luminosities we extend it to the whole halo white dwarf sample.
Analogously, we also derive the rest of parameters: mass, age, effective
temperature and radius. The only exception is for object J0055+3847, which is
located, as previously stated, outside the region of the HR-diagram covered by
the single white dwarf population (see Fig. 1). In Table 2 we present the
parameters thus derived from our photometric method. Along with the Gaia
source ID (first column) and the short name (second column), we show the
absolute magnitude and colour and their respective errors used in building our
ellipsoids (columns third to sixth) and the mass, effective temperature,
bolometric luminosity and age, with their respective errors (columns seventh
to tenth). It is worth recalling that our photometric method assumes a mixed
population of $80\%\,$DA and $20\%\,$DB white dwarfs, and then just counts the
physical parameters not caring what the atmospheric composition is.

Table 2: Stellar parameters derived by our photometric method for the whole halo white dwarf sample within 100 pc. Along with the Gaia Source ID and short name (first and second columns, respectively), the location within the HR-diagram (third and fourth columns) and the size of the adopted ellipses (fifth and six columns) are shown. Radius, effective temperature, bolometric luminosity and total age are shown, respectively, in the last four columns. Gaia | Short | $M_{\rm G}$ | $G_{\rm{\tiny BP}}\\!-\\!G_{\rm{\tiny RP}}$ | $\sigma_{M_{\rm G}}$ | $\sigma_{G_{\rm{\tiny BP}}\\!-\\!G_{\rm{\tiny RP}}}$ | Mass | $T_{\rm eff}$ | Luminosity | Age
---|---|---|---|---|---|---|---|---|---
Source ID | name | (mag) | (mag) | (mag) | (mag) | $M_{\odot}$ | (K) | $\log(L/L_{\odot})$ | (Gyr)
420531621029108608 | J0013+5438 | 15.47 | 1.47 | 0.007 | 0.041 | $0.551\pm 0.027$ | $4306\pm 73$ | $-4.273\pm 0.007$ | $11.27\pm 1.83$
5006232026455470848 | J0045-3329 | 16.05 | 1.51 | 0.011 | 0.072 | $0.613\pm 0.051$ | $3829\pm 128$ | $-4.544\pm 0.019$ | $12.22\pm 1.48$
367799116372410752 | J0055+3847 | 13.51 | 0.89 | 0.015 | 0.037 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
2583365245917474816 | J0106+1141 | 15.76 | 1.43 | 0.039 | 0.169 | $0.599\pm 0.065$ | $4112\pm 163$ | $-4.404\pm 0.013$ | $11.43\pm 1.39$
5042228731477861888 | J0129-2257 | 15.54 | 1.41 | 0.086 | 0.282 | $0.589\pm 0.067$ | $4333\pm 170$ | $-4.302\pm 0.022$ | $10.56\pm 1.56$
96095735719745280 | J0132+1941 | 14.85 | 1.13 | 0.049 | 0.145 | $0.556\pm 0.049$ | $4941\pm 130$ | $-4.033\pm 0.015$ | $8.36\pm 2.41$
5142197118950177280 | J0148-1712 | 13.11 | 0.46 | 0.011 | 0.025 | $0.529\pm 0.023$ | $7286\pm 124$ | $-3.311\pm 0.010$ | $6.52\pm 2.66$
343356212681211392 | J0157+3932 | 15.64 | 1.44 | 0.046 | 0.187 | $0.585\pm 0.054$ | $4205\pm 135$ | $-4.350\pm 0.013$ | $11.03\pm 1.44$
521409549427243904 | J0158+6904 | 15.47 | 1.14 | 0.103 | 0.298 | $0.630\pm 0.108$ | $4516\pm 279$ | $-4.272\pm 0.024$ | $9.80\pm 1.60$
2490975272405858048 | J0205-0517 | 15.58 | 1.52 | 0.008 | 0.046 | $0.544\pm 0.019$ | $4165\pm 57$ | $-4.325\pm 0.009$ | $11.81\pm 1.42$
4616895783694397184 | J0237-8445 | 15.74 | 1.55 | 0.039 | 0.176 | $0.579\pm 0.047$ | $4086\pm 117$ | $-4.394\pm 0.012$ | $11.62\pm 1.38$
5065611697758431360 | J0248-3001 | 13.91 | 0.71 | 0.019 | 0.050 | $0.573\pm 0.040$ | $6143\pm 162$ | $-3.669\pm 0.011$ | $4.85\pm 2.02$
5188044687948351872 | J0301-0044 | 15.24 | 1.51 | 0.041 | 0.157 | $0.543\pm 0.033$ | $4516\pm 86$ | $-4.181\pm 0.012$ | $10.60\pm 1.79$
4862884499360563968 | J0340-3301 | 15.79 | 1.31 | 0.032 | 0.143 | $0.636\pm 0.087$ | $4177\pm 220$ | $-4.414\pm 0.012$ | $11.17\pm 1.23$
3249657094642979840 | J0342-0344 | 15.46 | 1.09 | 0.057 | 0.194 | $0.680\pm 0.112$ | $4646\pm 285$ | $-4.272\pm 0.016$ | $9.21\pm 1.28$
4857106909354185344 | J0345-3611 | 15.37 | 1.47 | 0.061 | 0.212 | $0.559\pm 0.043$ | $4422\pm 110$ | $-4.235\pm 0.015$ | $10.54\pm 1.74$
66837563803594880 | J0346+2455 | 15.58 | 1.50 | 0.013 | 0.069 | $0.553\pm 0.031$ | $4197\pm 83$ | $-4.322\pm 0.009$ | $11.49\pm 1.51$
4864861112027378944 | J0431-3816 | 15.75 | 1.77 | 0.133 | 0.456 | $0.584\pm 0.053$ | $4089\pm 143$ | $-4.399\pm 0.029$ | $11.54\pm 1.37$
4864752883148064512 | J0432-3902 | 14.86 | 1.25 | 0.005 | 0.025 | $0.527\pm 0.021$ | $4846\pm 57$ | $-4.037\pm 0.013$ | $9.86\pm 2.08$
2989049057626796416 | J0518-1155 | 15.70 | 1.74 | 0.079 | 0.306 | $0.568\pm 0.040$ | $4102\pm 104$ | $-4.377\pm 0.017$ | $11.69\pm 1.32$
192454873200555392 | J0559+4248 | 13.42 | 0.52 | 0.022 | 0.049 | $0.578\pm 0.035$ | $6992\pm 172$ | $-3.445\pm 0.010$ | $3.71\pm 1.60$
977441274176008192 | J0711+4607 | 14.76 | 1.16 | 0.033 | 0.105 | $0.536\pm 0.034$ | $4983\pm 93$ | $-3.996\pm 0.014$ | $9.03\pm 2.42$
5613373001468333696 | J0732-2558 | 15.11 | 1.27 | 0.097 | 0.265 | $0.568\pm 0.059$ | $4714\pm 155$ | $-4.130\pm 0.023$ | $9.20\pm 2.14$
874900643675606912 | J0745+2626 | 15.76 | 1.60 | 0.014 | 0.081 | $0.572\pm 0.046$ | $4046\pm 112$ | $-4.406\pm 0.012$ | $11.80\pm 1.40$
1110759459929880704 | J0748+7141 | 15.74 | 1.62 | 0.062 | 0.251 | $0.579\pm 0.048$ | $4078\pm 124$ | $-4.398\pm 0.016$ | $11.67\pm 1.37$
3144837318276010624 | J0750+0711a | 15.07 | 1.25 | 0.002 | 0.013 | $0.530\pm 0.023$ | $4664\pm 56$ | $-4.108\pm 0.006$ | $9.54\pm 2.07$
3144837112117580800 | J0750+0711b | 15.31 | 1.41 | 0.002 | 0.010 | $0.565\pm 0.025$ | $4512\pm 59$ | $-4.202\pm 0.006$ | $10.79\pm 1.19$
5726927573083821440 | J0822-1249 | 15.54 | 1.30 | 0.037 | 0.148 | $0.597\pm 0.069$ | $4346\pm 170$ | $-4.305\pm 0.012$ | $10.40\pm 1.48$
5742629217603133056 | J0912-0953 | 15.27 | 1.15 | 0.094 | 0.265 | $0.606\pm 0.090$ | $4656\pm 230$ | $-4.192\pm 0.023$ | $9.17\pm 1.86$
5215833263797633664 | J0913-7553 | 15.37 | 1.37 | 0.038 | 0.147 | $0.562\pm 0.046$ | $4439\pm 116$ | $-4.231\pm 0.013$ | $10.39\pm 1.80$
3840846114438361984 | J0925+0018 | 15.12 | 1.15 | 0.032 | 0.115 | $0.578\pm 0.061$ | $4729\pm 155$ | $-4.134\pm 0.013$ | $8.83\pm 2.02$
1064978578888570496 | J0941+6511 | 15.14 | 1.34 | 0.013 | 0.059 | $0.539\pm 0.032$ | $4610\pm 83$ | $-4.139\pm 0.011$ | $10.49\pm 2.04$
3836593100382315904 | J1005+0254 | 15.26 | 1.29 | 0.065 | 0.207 | $0.574\pm 0.058$ | $4580\pm 148$ | $-4.187\pm 0.018$ | $9.62\pm 1.91$
746045096445123968 | J1012+3233 | 15.70 | 1.39 | 0.131 | 0.399 | $0.615\pm 0.091$ | $4224\pm 240$ | $-4.374\pm 0.034$ | $11.02\pm 1.45$
5192296911732427904 | J1036-8225 | 15.67 | 1.54 | 0.040 | 0.176 | $0.573\pm 0.044$ | $4150\pm 109$ | $-4.361\pm 0.011$ | $11.37\pm 1.37$
3862858165427681536 | J1036+0732 | 15.22 | 1.49 | 0.023 | 0.098 | $0.540\pm 0.027$ | $4530\pm 75$ | $-4.171\pm 0.010$ | $10.63\pm 1.69$
1076941716370493696 | J1036+7110 | 15.39 | 1.24 | 0.002 | 0.012 | $0.582\pm 0.071$ | $4461\pm 165$ | $-4.241\pm 0.009$ | $10.52\pm 1.65$
855361055035055104 | J1045+5904 | 13.89 | 0.23 | 0.009 | 0.029 | $1.074\pm 0.033$ | $9211\pm 481$ | $-3.527\pm 0.024$ | $2.69\pm 0.25$
5228861484450843648 | J1049-7400 | 16.07 | 1.54 | 0.021 | 0.122 | $0.613\pm 0.054$ | $3813\pm 134$ | $-4.551\pm 0.020$ | $12.23\pm 1.54$
3801499128765222400 | J1053-0307 | 15.37 | 1.42 | 0.041 | 0.156 | $0.556\pm 0.042$ | $4422\pm 107$ | $-4.232\pm 0.013$ | $10.62\pm 1.77$
3865951435233552896 | J1055+0816 | 15.46 | 1.67 | 0.043 | 0.181 | $0.556\pm 0.042$ | $4324\pm 108$ | $-4.271\pm 0.014$ | $11.14\pm 1.62$
1055533400343235456 | J1101+6333 | 15.33 | 1.44 | 0.068 | 0.226 | $0.560\pm 0.045$ | $4464\pm 115$ | $-4.219\pm 0.017$ | $10.31\pm 1.81$
831946229073235200 | J1107+4855 | 15.19 | 1.27 | 0.015 | 0.066 | $0.552\pm 0.041$ | $4588\pm 104$ | $-4.162\pm 0.011$ | $9.97\pm 1.94$
5348874243767794304 | J1123-5150 | 15.58 | 1.32 | 0.069 | 0.242 | $0.604\pm 0.080$ | $4325\pm 204$ | $-4.321\pm 0.018$ | $10.57\pm 1.47$
856513235846126720 | J1123+5742 | 14.74 | 0.65 | 0.061 | 0.159 | $0.893\pm 0.100$ | $6163\pm 418$ | $-3.997\pm 0.017$ | $5.15\pm 0.24$
5224999346778496128 | J1147-7457 | 15.66 | 1.58 | 0.003 | 0.021 | $0.575\pm 0.037$ | $4159\pm 95$ | $-4.357\pm 0.005$ | $11.58\pm 1.20$
3892524535332945280 | J1151+0159 | 15.13 | 1.24 | 0.061 | 0.188 | $0.564\pm 0.054$ | $4683\pm 139$ | $-4.138\pm 0.017$ | $9.35\pm 2.11$
5377861317357370240 | J1159-4630 | 15.17 | 1.24 | 0.029 | 0.109 | $0.561\pm 0.046$ | $4630\pm 118$ | $-4.154\pm 0.013$ | $9.54\pm 2.00$
5377861592235273856 | J1159-4629 | 13.44 | 0.41 | 0.009 | 0.023 | $0.705\pm 0.051$ | $7589\pm 241$ | $-3.437\pm 0.016$ | $2.32\pm 0.37$
1573358945589364608 | J1205+5502 | 15.84 | 1.42 | 0.106 | 0.360 | $0.623\pm 0.089$ | $4091\pm 233$ | $-4.439\pm 0.030$ | $11.50\pm 1.39$
3905186270720273152 | J1217+0830 | 15.72 | 1.31 | 0.075 | 0.264 | $0.619\pm 0.089$ | $4219\pm 227$ | $-4.380\pm 0.019$ | $10.99\pm 1.43$
1533950318546008448 | J1235+4109 | 15.47 | 1.59 | 0.041 | 0.193 | $0.563\pm 0.053$ | $4333\pm 136$ | $-4.272\pm 0.008$ | $10.66\pm 1.21$
1570514066627694336 | J1250+5446 | 18.01 | 1.75 | 0.004 | 0.024 | $0.563\pm 0.038$ | $4033\pm 97$ | $-4.402\pm 0.014$ | $11.93\pm 1.25$
1531097433767946240 | J1255+4655 | 15.57 | 1.17 | 0.011 | 0.055 | $0.728\pm 0.112$ | $4641\pm 286$ | $-4.322\pm 0.010$ | $9.52\pm 0.99$
1459546263999675264 | J1303+2603 | 15.51 | 1.45 | 0.011 | 0.058 | $0.562\pm 0.041$ | $4293\pm 106$ | $-4.290\pm 0.009$ | $11.06\pm 1.56$
6085402414245451520 | J1312-4728 | 15.58 | 1.53 | 0.002 | 0.014 | $0.542\pm 0.002$ | $4126\pm 19$ | $-4.337\pm 0.006$ | $12.41\pm 0.22$
3607725941130742528 | J1316-1536 | 11.09 | -0.20 | 0.002 | 0.004 | $0.551\pm 0.015$ | $15778\pm 71$ | $-1.991\pm 0.015$ | $2.37\pm 0.60$
6188655210447329792 | J1338-2747 | 15.68 | 1.50 | 0.085 | 0.302 | $0.594\pm 0.064$ | $4190\pm 162$ | $-4.367\pm 0.021$ | $11.09\pm 1.46$
6165095738576250624 | J1342-3415 | 14.66 | 1.05 | 0.002 | 0.009 | $0.521\pm 0.013$ | $5055\pm 39$ | $-3.956\pm 0.010$ | $9.45\pm 1.86$
3714266139665215488 | J1348+0527 | 15.03 | 1.20 | 0.086 | 0.233 | $0.566\pm 0.058$ | $4790\pm 153$ | $-4.099\pm 0.022$ | $8.89\pm 2.25$
Table 3: continued Gaia | Short | $M_{\rm G}$ | $G_{\rm{\tiny BP}}\\!-\\!G_{\rm{\tiny RP}}$ | $\sigma_{M_{\rm G}}$ | $\sigma_{G_{\rm{\tiny BP}}\\!-\\!G_{\rm{\tiny RP}}}$ | Mass | $T_{\rm eff}$ | Luminosity | Age
---|---|---|---|---|---|---|---|---|---
Source ID | name | (mag) | (mag) | (mag) | (mag) | $M_{\odot}$ | (K) | $\log(L/L_{\odot})$ | (Gyr)
1174809276422844160 | J1442+1003 | 15.51 | 1.40 | 0.105 | 0.324 | $0.593\pm 0.071$ | $4366\pm 183$ | $-4.292\pm 0.026$ | $10.43\pm 1.58$
1161215296909017728 | J1450+0733 | 11.14 | -0.17 | 0.004 | 0.006 | $0.537\pm 0.015$ | $14638\pm 180$ | $-2.069\pm 0.015$ | $3.46\pm 1.36$
1294793345366747776 | J1500+3600 | 14.65 | 1.17 | 0.034 | 0.102 | $0.530\pm 0.031$ | $5077\pm 89$ | $-3.956\pm 0.013$ | $9.19\pm 2.50$
1600259390916467072 | J1502+5409 | 15.33 | 1.42 | 0.060 | 0.204 | $0.559\pm 0.044$ | $4466\pm 113$ | $-4.217\pm 0.016$ | $10.33\pm 1.82$
1612339420228653440 | J1503+5509 | 15.76 | 1.36 | 0.073 | 0.265 | $0.615\pm 0.083$ | $4163\pm 210$ | $-4.399\pm 0.020$ | $11.23\pm 1.38$
5824436284328653312 | J1517-6645 | 15.99 | 1.30 | 0.102 | 0.359 | $0.661\pm 0.114$ | $4029\pm 307$ | $-4.506\pm 0.029$ | $11.70\pm 1.56$
6007140379167609984 | J1518-3803 | 15.66 | 1.55 | 0.040 | 0.173 | $0.571\pm 0.043$ | $4151\pm 108$ | $-4.359\pm 0.011$ | $11.39\pm 1.36$
1277219369981634432 | J1522+3146 | 15.70 | 1.54 | 0.103 | 0.351 | $0.596\pm 0.066$ | $4171\pm 170$ | $-4.376\pm 0.025$ | $11.18\pm 1.45$
1277232907719022464 | J1523+3152 | 15.59 | 1.26 | 0.021 | 0.098 | $0.626\pm 0.079$ | $4364\pm 197$ | $-4.326\pm 0.010$ | $10.30\pm 1.35$
5827557213731539328 | J1539-6124 | 15.18 | 1.31 | 0.026 | 0.104 | $0.550\pm 0.039$ | $4593\pm 100$ | $-4.157\pm 0.012$ | $10.05\pm 1.97$
5817295536128445568 | J1707-6319 | 15.10 | 1.40 | 0.019 | 0.079 | $0.533\pm 0.029$ | $4637\pm 74$ | $-4.123\pm 0.012$ | $10.68\pm 2.00$
5802598780807649920 | J1715-7323 | 15.43 | 1.09 | 0.111 | 0.310 | $0.640\pm 0.116$ | $4586\pm 304$ | $-4.255\pm 0.025$ | $9.50\pm 1.63$
1711005951573009792 | J1749+8247 | 13.15 | 0.47 | 0.001 | 0.003 | $0.547\pm 0.000$ | $7257\pm 0$ | $-3.331\pm 0.000$ | $4.33\pm 0.00$
6363668569344689408 | J1812-8028 | 15.55 | 1.04 | 0.143 | 0.390 | $0.659\pm 0.131$ | $4520\pm 353$ | $-4.302\pm 0.029$ | $9.85\pm 1.54$
6653858618815379328 | J1814-5305 | 15.68 | 1.08 | 0.130 | 0.376 | $0.669\pm 0.132$ | $4396\pm 356$ | $-4.361\pm 0.027$ | $10.41\pm 1.47$
4484289866726156160 | J1824+1213 | 15.57 | 1.63 | 0.012 | 0.069 | $0.548\pm 0.025$ | $4195\pm 74$ | $-4.316\pm 0.011$ | $11.72\pm 1.25$
2146619161278293248 | J1852+5333 | 15.72 | 1.39 | 0.134 | 0.410 | $0.617\pm 0.093$ | $4216\pm 245$ | $-4.379\pm 0.035$ | $11.06\pm 1.44$
6663268308043562112 | J1926-4627 | 15.02 | 1.46 | 0.060 | 0.190 | $0.533\pm 0.029$ | $4711\pm 79$ | $-4.095\pm 0.016$ | $10.25\pm 2.06$
6647162730439433984 | J1936-4913 | 15.58 | 1.50 | 0.057 | 0.219 | $0.574\pm 0.047$ | $4251\pm 121$ | $-4.320\pm 0.015$ | $10.94\pm 1.47$
2301882675705225472 | J1940+8348 | 15.51 | 1.23 | 0.010 | 0.049 | $0.608\pm 0.065$ | $4413\pm 155$ | $-4.290\pm 0.012$ | $9.78\pm 1.39$
2082254987541672960 | J2006+4544 | 15.21 | 1.22 | 0.037 | 0.136 | $0.574\pm 0.057$ | $4629\pm 143$ | $-4.169\pm 0.013$ | $9.32\pm 1.93$
6471523921227261056 | J2042-5218 | 14.68 | 0.94 | 0.043 | 0.121 | $0.601\pm 0.070$ | $5249\pm 199$ | $-3.973\pm 0.015$ | $6.22\pm 1.96$
1737588947276271744 | J2052+0709 | 15.89 | 1.56 | 0.059 | 0.250 | $0.605\pm 0.061$ | $3980\pm 158$ | $-4.468\pm 0.020$ | $11.78\pm 1.38$
6580458035746362496 | J2117-4156a | 15.15 | 1.33 | 0.024 | 0.095 | $0.542\pm 0.033$ | $4608\pm 84$ | $-4.143\pm 0.012$ | $10.28\pm 1.96$
6580551872194787968 | J2117-4156b | 15.34 | 1.30 | 0.028 | 0.115 | $0.572\pm 0.052$ | $4493\pm 130$ | $-4.220\pm 0.013$ | $9.94\pm 1.78$
1783614400935169408 | J2127+1545 | 14.30 | 0.82 | 0.023 | 0.066 | $0.599\pm 0.058$ | $5700\pm 191$ | $-3.827\pm 0.012$ | $5.06\pm 1.86$
2687584757658775424 | J2129-0034 | 15.37 | 1.57 | 0.112 | 0.341 | $0.562\pm 0.045$ | $4427\pm 120$ | $-4.236\pm 0.022$ | $10.46\pm 1.76$
6465689878168451328 | J2139-5058 | 15.85 | 1.55 | 0.084 | 0.320 | $0.607\pm 0.067$ | $4025\pm 175$ | $-4.450\pm 0.025$ | $11.68\pm 1.33$
2205493129867600256 | J2225+6357 | 14.53 | 1.05 | 0.006 | 0.024 | $0.525\pm 0.020$ | $5190\pm 63$ | $-3.912\pm 0.012$ | $8.98\pm 2.20$
6357629089412187648 | J2230-7515 | 15.57 | 1.55 | 0.002 | 0.012 | $0.575\pm 0.059$ | $4275\pm 157$ | $-4.313\pm 0.011$ | $10.61\pm 0.99$
2709539840202060800 | J2237+0636 | 15.85 | 1.54 | 0.110 | 0.396 | $0.617\pm 0.081$ | $4058\pm 211$ | $-4.446\pm 0.031$ | $11.60\pm 1.37$
1941133391670459648 | J2314+4545 | 15.96 | 1.41 | 0.099 | 0.358 | $0.640\pm 0.095$ | $4008\pm 248$ | $-4.492\pm 0.029$ | $11.74\pm 1.44$
2631967439437024384 | J2319-0613 | 15.06 | 1.37 | 0.007 | 0.035 | $0.530\pm 0.023$ | $4661\pm 52$ | $-4.112\pm 0.013$ | $10.40\pm 1.68$
2641576685735609472 | J2349-0124 | 14.99 | 1.22 | 0.028 | 0.099 | $0.544\pm 0.038$ | $4771\pm 97$ | $-4.083\pm 0.013$ | $9.43\pm 2.17$
2310942857676734848 | J2354-3634 | 11.29 | -0.16 | 0.003 | 0.005 | $0.574\pm 0.018$ | $14636\pm 121$ | $-2.134\pm 0.005$ | $1.68\pm 0.49$

###  2.2 Completeness analysis of the halo white dwarf sample

Figure 5: Top panel: Halo white dwarf luminosity functions built with the
classical estimator $\phi=N/V$ (red line and dots), with the
$1/\mathcal{V}_{\rm max}$ method, $\phi_{\rm max}$ (blue line and dots), and
with the modified volume, $\phi_{\rm mod}$ (black line and dots). Middle
panel: average $\langle V/V_{\rm max}\rangle$ as a function of the bolometric
magnitude for different magnitude cuts, $G_{c}\in[20.25,21.0]\,$mag. Also
shown as a dashed line the value representing an uniform complete sample,
$\langle V/V_{\rm max}\rangle=0.5$. Bottom panel: completeness of the whole
Gaia 100 pc sample of white dwarfs (red line, $1\sigma$ pink area) and that of
the halo subsample (black line, $1\sigma$ gray area).

As a preliminary step for building the halo white dwarf luminosity function,
we analyze in detail the completeness of our selected sample. First, we need
to take into account that our ‘halo sample’ is not a ‘direct’ observed sample.
That is, we first obtain a sample of white dwarfs within 100 pc, and from that
we apply a classifying method based on a Random Forest algorithm in order to
obtain the ‘halo sample’. Then, the incompleteness raises from two sources:
from the selection process of the observed 100 pc sample and from the
classification algorithm.

With respect to the first item, the Gaia mission is expected to be (understood
as an end-of-mission goal) complete up to $G=21\,$mag. Recent estimates based
on the EDR3-Gaia catalogue of nearby stars provide a completeness of 97%, 95%
and 91% for $G$ magnitudes 19.9, 20.2 and 20.5, respectively (Gaia
Collaboration et al., 2020). Taking this into account, we extend our
completeness analysis of the 100 pc sample (see Figure 5 from Jiménez-Esteban
et al. (2018)) up to $G_{BP}-G_{RP}=2.0$. The resulting completeness as a
function of the bolometric magnitude for the whole Gaia 100 pc sample is shown
in the bottom panel (red line) of Figure 5. A completeness above 90% is
achieved for most of the sample up to $M_{\rm bol}\sim 14.0$. At this
magnitude, we note a decreasing trend reaching a 50% completeness at magnitude
$M_{\rm bol}\sim 16.6\,$mag.

Secondly, we analyze the incompleteness introduced by our Random Forest
classification algorithm. We recall that in our classification process we used
an 8-dimensional space (equatorial coordinates, parallax, proper motion
components and photometric magnitudes), where the algorithm estimates the
entropy function, evaluated on each splitting branch of the different
decisions trees of the Random Forest algorithm (see Torres et al. 2019a). For
each individual object the minimum entropy found, $S_{i}$, is used to classify
the object in a certain Galactic component. Consequently, this entropy value
$S_{i}$ can be logically understood as an inverse of the probability that an
object belongs to a certain group. Using our Monte Carlo simulator we generate
several samples of $\sim 95$ halo white dwarfs and calculate, through our
Random Forest algorithm, the individual probability of being classified as
such. We compute, then, the average value as a function of the bolometric
magnitude taking into account the completeness of the whole 100 pc sample. The
resulting distribution is shown in the bottom panel of Figure 5 as a black
line (gray area corresponding to $1\sigma$ dispersion). We observe an
irregular pattern for the bright region ($M_{\rm
bol}\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}14\,$mag), while for fainter
magnitudes the halo white dwarf completeness distribution resembles that of
the whole sample. According to our analysis, the noisy appearance of the
bright region is due to two factors: mainly, the low number of objects
expected in these bins, which are dominated by Poisson counting error
statistics, and secondly, and to a minor extent, to the intrinsic difficulty
to classify a bright object as belonging to the halo population. High-speed
and cool objects are closer to what we expect to be a halo white dwarf than
hot and high-speed objects. In the first case, the automatic classification
algorithm seems to relax the kinematic condition to accept moderately fast
objects if they are cool enough, while the reverse condition happens for hot
halo candidates. Besides, more hot high-speed disk objects may contaminate the
hot region and the difficulty to disentangle halo objects increases in that
region. On the other hand, for fainter magnitudes the halo sample completeness
resembles that of the whole 100 pc sample, indicating that the probability to
correctly classifying a halo object in this region is very high. It should be
noted that the completeness of the halo sample for magnitudes around $M_{\rm
bol}=16.0$ is $\sim 60\%$, which can be considered as an acceptable value,
while for the rest of the sample is around $\sim 50\%$.

We are in position now to estimate the luminosity function. Among the
different estimators proposed (see, for instance, Geijo et al. 2006 and
reference therein) the $1/\mathcal{V}_{\rm max}$ method (Schmidt, 1968;
Felten, 1976) is the most commonly used to estimate the white dwarf luminosity
function. However, this method, based on the assumption of a complete
magnitude-limited sample, is not applicable in our case, since our
classification method introduces an incompleteness factor which is not only
magnitude dependent. Consequently, we will use the $1/\mathcal{V}_{\rm max}$
method for comparative purposes just as a first guess in our analysis. In the
middle panel of Figure 5 we show the average $\langle V/V_{\rm max}\rangle$ as
function of $M_{\rm bol}$ adopting different $G$ magnitude cuts, $G_{c}$, of
the whole 100 pc sample in the range $G_{c}\in[20.25,21.0]\,$mags. We also
recall that the $1/\mathcal{V}_{\rm max}$ provides an estimation of the
completeness of the sample, once assumed a complete uniformly distributed
sample, resulting in a average value of $\langle V/V_{\rm max}\rangle=0.5$. In
our case, adopting a conservative value of $G_{c}=20.5\,$mags, the $\langle
V/V_{\rm max}\rangle$ distribution is closer to the value 0.5 (dashed line)
for magnitudes $M_{\rm bol}\sim 15.7\,$mags. Specifically, a value of $\langle
V/V_{\rm max}\rangle=0.476\pm 0.049$ (adopting for $N$ objects a deviation of
$1/\sqrt{12N}$, Rowell & Hambly 2011) thus indicating an acceptable degree of
completeness for that particular magnitude bin. However, as we mentioned
earlier, brighter magnitudes appear to be far from being complete as shown by
the low $\langle V/V_{\rm max}\rangle$ value.

In the top panel of Figure 5, we compare the different halo white dwarfs
luminosity functions built from our previous completeness analysis. First, we
use the classical estimator of the luminosity function for volume-limited
samples, $\phi=N/V$. The resulting luminosity function is shown in red.
Second, the $1/\mathcal{V}_{\rm max}$ provides an estimation of the luminosity
function, $\phi_{\rm max}$, binning the sample in $i\in(1,N)$ magnitude bins
and weighting the contribution of each object as inversely proportional to its
maximum volume available within the selection cuts: $\phi_{\rm
max}=\sum_{j=1}^{N_{i}}1/\mathcal{V}^{j}_{\rm max}$, where the deviation is
calculated as the sum in quadrature of the individual errors. Adopting a
magnitude cut of $G_{c}=20.5\,$mag, the luminosity function thus obtained is
shown in blue in the top panel of Fig. 5. However, as previously stated, both,
classical and $1/\mathcal{V}_{\rm max}$ method, can be only understood as a
first guess for building a proper luminosity function.

In order to properly take into account all possible sources of incompleteness
in our sample, as previously discussed, we use a generalization of the
$1/\mathcal{V}_{\rm max}$ method (Lam et al. 2015 and references therein). A
modified volume is defined as

$\mathcal{V}^{i}_{{\rm mod}}=\Omega_{i}\int^{r_{\rm max}}_{r_{\rm
min}}\chi\frac{\rho(r)}{\rho_{\sun}}r^{2}dr$ (1)

where $\Omega_{i}$ is the solid angle covered by object $i$, $r_{\rm max}$ and
$r_{\rm min}$ are, respectively the maximum and minimum distances and
$\rho(r)/\rho_{\sun}$ the density ratio along the line of side. This volume
integral is modified by the function $\chi$, traditionally named as the
discovery fraction, and typically used for taking into account the number of
objects that pass a certain tangential velocity threshold. In our case, we
generalize the $\chi$ function to incorporate our 8-dimensional space of
equatorial coordinates, proper motion components, parallax and photometric
magnitudes,
$\chi=\chi(\alpha,\delta,\mu^{*}_{\alpha},\mu_{\delta},\varpi,G,G_{RP},G_{BP})$.
Thus, the $\chi$ function becomes equivalent to the probability derived from
our Random Forest algorithm of being correctly classified as an halo member.
Adopting a constant density profile, which is a natural assumption for a halo
distribution in a small volume, we can compute the luminosity function as a
generalization of the $1/\mathcal{V}_{\rm max}$ method by using the previously
modified volume: $\phi_{\rm mod}=\sum_{j=1}^{N_{i}}1/\mathcal{V}^{j}_{\rm
mod}$. The corresponding luminosity function, $\phi_{\rm mod}$ is shown in
black in the top panel of Fig. 5.

The analysis of the three estimators studied here reveals a similar trend of
the corresponding luminosity functions for the faint region, i.e. $M_{\rm
bol}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$}15\,$mag,
being this fact a consequence of a reasonable completeness ($\sim 60\%$) for
this low-luminosity region. On the contrary, the brighter region suffers from
a larger degree of incompleteness, mainly due to a low number statistic. In
this situation, the $1/\mathcal{V}_{\rm max}$ is markedly inefficient at
recovering the original distribution, greatly underestimating the brightest
bins. Given that our modified estimator is the only one that properly takes
into account the different sources of incompleteness, in what follows we will
use the luminosity function derived from it as representative of the halo
white dwarf population.

Figure 6: Left panel: Gaia halo white dwarf luminosity function within 100 pc
obtained in this work (black dots and lines) compared to the Rowell & Hambly
(2011) spheroid white dwarf luminosity function (blue squares and lines), and
the luminosity function for high speed white dwarfs obtained by Munn et al.
(2016) (green squares and lines) and Lam et al. (2018) (magenta squares and
lines). Right panel: Gaia halo white dwarf luminosity function obtained in
this work (black dots and lines), compared with the synthetic populations
simulated in this work that best fit the observed features. This is, synthetic
populations considering a star formation burst from 11 to 12 Gyrs ago, and a
standard proportion of DA to DB white dwarfs of 80-20% (red squares and lines)
and one in which all synthetic white dwarfs are DA (blue squares and lines).

###  2.3 The halo white dwarf luminosity function

In Figure 6 (left panel) we plot the luminosity function derived in this work
(black) compared to some of the most recent white dwarf luminosity functions
obtained for a high speed or equivalently halo population. In particular, we
show the spheroid white dwarf luminosity function obtained from the
SuperCOSMOS Sky Survey via the effective volume technique from Rowell & Hambly
(2011) –blue squares–; the luminosity function for high speed white dwarfs,
$200<v_{\rm tan}<500\,$${\rm\,km\,s^{-1}}$, derived from SDSS deep proper
motion survey by Munn et al. (2016) – green squares –; and the most recent
high speed white dwarf luminosity function derived by Lam et al. (2018) from
the Pan–STARRS 1 3$\pi$ Steradian Survey – magenta squares.

First of all, the space density previously derived in Torres et al. (2019a)
for our halo candidate sample, $(4.8\pm 0.4)\times 10^{-5}\,{\rm pc^{-3}}$, is
in agreement with the value reported by Lam et al. (2018), $(5.291\pm
0.2)\times 10^{-5}\,{\rm pc^{-3}}$. Moreover, our space density estimate is
slightly larger than that of Munn et al. (2016), $(3.5\pm 0.7)\times
10^{-5}\,{\rm pc^{-3}}$, but it is well below the upper limit of $(1.9\times
10^{-4}\,{\rm pc^{-3}}$ presented by Rowell & Hambly (2011).

Second, our white dwarf luminosity function is defined in fewer magnitude bins
than the rest of samples. That paucity in the number of objects is a
consequence of the relative small size of our sample, which is limited within
100 pc. However, we need to recall that our sample has been extracted from a
nearly volume limited – and thus practically complete – sample, while the rest
of samples are magnitude-limited (see Section 2.2 for a detailed analysis of
the completeness of our sample). Consequently some biases are expected, in
particular for the dimmer intervals of the luminosity function. In this sense,
our luminosity function shows a clear peak at $M_{\rm bol}\sim 15.5$ mag and a
marked drop-off for the fainter bins. The location of the peak agrees with the
one presented by the Lam et al. (2018)’s luminosity function. At the same
time, the slight depression shown at $M_{\rm bol}\sim 14.5$ mags by our
luminosity function is also present in that of Rowell & Hambly (2011) but
seems not to appear in any of the other two distributions. However, the drop-
off beyond 15.5 mags is the most relevant feature of our luminosity function,
which is not present in any other of the distributions. The only white dwarf
luminosity function which extends beyond the cut-off is that of Rowell &
Hambly (2011). Unfortunately, their faintest bins are poorly constrained and
no cut-off is observed. Based on the completeness analysis of our sample (see
Section 2.2), we can claim that this drop-off is real and thus we are
observing for the first time the cut-off of the halo white dwarf luminosity
function.

Fitting the cut-off of the white dwarf luminosity function has been
extensively used as a consistent technique for estimating ages (e.g García-
Berro & Oswalt, 2016). For this purpose we build a set of synthetic white
dwarf luminosity functions derived from a 1 Gyr burst of star formation
applied at different ages. Our best fit, that is, the one that best reproduces
the cut-off of our luminosity function, corresponds to a burst that happened
between 11.0-12.0 Gyr in the past. In the right panel of Fig. 6 we show our
results for the best fit model when a population of 80:20, DA to non-DA ratio,
is considered (red) and when only DAs are taken into account (blue). Both
models are able to correctly fit the peak and the cut-off bins. However, the
depression previously commented at $M_{\rm bol}\sim 14.5$ mags seems not to be
reproduced by a burst model. The paucity of objects in that bin prevents us
from drawing further conclusion. Finally, the possible effects of non-DA stars
(in particular those with He-rich atmospheres) are only apparent for the
faintest bins of the luminosity function, but these are beyond our observed
sample.

###  2.4 The mass and age distributions and the star formation history

The photometric method outlined in Section 2.1 has also allowed us to derive
some other important stellar parameters such as the mass and the age of the
white dwarfs. In Figure 7 we show the mass distribution (left panel) for our
halo white dwarf sample. The mean mass $\langle M_{\rm
WD}\rangle=0.589\,M_{\odot}$ is smaller than the mean value of $\sim 0.65\,$
generally reported for the single white dwarf disk population (e.g. Tremblay
et al., 2016; Bergeron et al., 2019; McCleery et al., 2020). The majority of
white dwarfs ($71\%$) present a mass smaller than $0.6\,M_{\odot}$, which is
expected for an old population, as more low-mass progenitors have had enough
time to evolve and become white dwarfs of slightly lower masses than the
canonical value. On the other hand, we have found just two massive white
dwarfs (J1045+5904 and J1123+5742) with masses $1.074\pm 0.033\,M_{\odot}$ and
$0.893\pm 0.100\,M_{\odot}$, respectively.

In the right panel of Fig. 7 we show the age distribution for our sample of
white dwarfs. We recall that the age represented corresponds to the total age
of the white dwarfs, that is, the cooling time plus the progenitor lifetime. A
first glance to the age distribution reveals that the vast majority of stars
($87\%$) have total ages between 8 to 12 Gyr. In particular, most of them
between 10-12 Gyr, being compatible with the 11-12 Gyr burst we applied for
fitting the cut-off of the luminosity function (See section 2.3). Three
objects found in our sample (J0045-3329, J1049-7559 and J1312-4728) are older
than 12 Gyr, being the last of them the oldest with an age of $12.41\pm
0.22\,$Gyr. These stars appear older than some of the previously published
oldest white dwarfs in the Solar neighborhood. For instance, the objects SDSS
J1102+4113 and WD 0346+246 have, respectively, age estimates of 11 and 11.5
Gyr (Kilic et al., 2012). It is worth mentioning that this last object, WD
0346+246 – which was firstly discover by Hambly et al. (1997) and extensively
studied by e.g. Oppenheimer et al. (2001); Bergeron (2001); Kilic et al.
(2012) – is also present in our sample with an estimate total age of $11.49\pm
1.51\,$Gyr that is in perfect agreement with these previous works.

On the other hand, 12 white dwarfs have age estimates younger than 7 Gyr. In
principle, they may be ruled out as genuine halo members but, at the same
time, their high speed kinematics are indicative that their origin is
different from the bulk of the disk population. They represent $13\,$% of the
sample, a value that can be considered as a general estimate of the
contamination of any possible halo white dwarf sample. Moreover, these objects
seemed to be clustered at around 2.5 and 5 Gyr, respectively. With the aid of
our Monte Carlo simulator, we analyze the statistical significance of this
apparent clustering. We uniformly distribute 12 objects in the range 0-8 Gyr
in intervals of 0.5 Gyr. The probability of obtaining a peak (of three or more
stars) is rather high, 0.42. However, the probability of recovering two of
these peaks is considerably low, 0.05. In other words, the observed
distribution has a statistical significance of $2\sigma$ of rejecting that
they come from a uniform distribution. In the forthcoming analysis some
hypothesis about their origin are presented.

Figure 7: Left panel: Gaia halo white dwarf mass function within 100 pc
obtained in this work. Right panel: total age (cooling age plus progenitor
lifetime) distribution for our halo white dwarf sample. Figure 8: Star
formation rate (gray histogram) obtained in this work for the Gaia halo white
dwarfs within 100 pc. For comparative purposes we also plot the star formation
history for the thin disk (magenta line) and thick disk (blue line) population
of the Milky Way determined by Ruiz-Lara et al. (2020). These last two
distributions are arbitrarily normalized.

Figure 9: Action space corresponding to the integral of motion $V_{\Delta
E}\equiv(U^{2}+2V^{2})^{1/2}$ as a function of $V$ (left panel) and the Toomre
diagram (right panel) for the white dwarfs of our halo candidate sample
(coloured scaled as a function of the age). Also shown, for illustrative
purposes, are the thin (light red dots) and thick (light blue dots) velocities
for the 100 pc white dwarf sample from Torres et al. (2019a). Dashed lines
represent curves for constant $U^{2}+V^{2}+W^{2}$ at values 50, 100, 150 and
200${\rm\,km\,s^{-1}}$.

A subproduct of our photometric procedure is the mass of the white dwarf
progenitor. Given that we also know the age distribution, we can easily
estimate the star formation rate, $\Psi$, as the quantity of stellar mass
produced per unit time and unit volume. In Figure 8 we show as a gray
histogram our SFR thus computed in units of $M_{\odot}$/Gyr/pc3. The SFR
resembles, as logically expected, the age distribution previously shown in
Fig. 7, although the peaks, in particular for the younger bins, are more
pronounced. We observe that the peak of the star formation history is centered
at around 11 Gyr, which is compatible with the current age of the Gaia-
Enceladus encounter. It is worth noting here that our estimate of the halo SFR
is restricted to the mass range of those progenitors that, according to our
population synthesis model, are able to form a white dwarf, i.e., masses in
the range $\sim 1-5.5\,M_{\odot}$.

For a comparative purpose we also show in Fig. 8 the recent star formation
history from Ruiz-Lara et al. (2020) corresponding to the thin disk (magenta
line) and thick disk (blue line) of our Galaxy. In their work, Ruiz-Lara et
al. (2020) thoroughly analyzed the HR-diagram derived from the precise
astrometric and photometric data provided by Gaia DR2. They found clear
evidences that the close encounter of the Milky Way with Saggitarius dwarf
galaxy has enhanced the star formation rate at epochs 5.7, 1.9 and 1 Gyr in
the past. The location of the two older peaks seems to resemble the location
of the peaks for the younger white dwarfs found in our sample at 5 and 2.5
Gyr. It can be expected that the close encounter with the Saggitarius dwarf
galaxy may have an effect not only in the enhancement of star formation but
also in heating the kinematics of the affected stars. Consequently, it is
reasonable to link the origin of these high-speed and young white dwarfs to
the gravitational effects induced by the pass of the Saggitarius galaxy.

Following the analysis by Ruiz-Lara et al. (2020) we also found an enhancement
in the star formation of the thick disk population centered at $\sim
10.5\,$Gyr. This enhancement is not associated to the encounters with the
Saggitarius dwarf galaxy, but rather to the formation of the thick disk
itself. However, this issue is under an intense debate, since it has been also
found that in the current scenario of the Milky Way, the formation of the
inner stellar halo seems to be triggered by the major merger collision with
the Gaia-Enceladus Galaxy (Helmi et al., 2018). In this sense, it has been
hypothesized that part of the halo may be associated to the Gaia-Enceladus
encounter, and part to an in situ halo (e.g. Gallart et al., 2019). It is
beyond of the scope of the present work to unravel the origin of the local
halo white dwarf population. Be that as it may, the formation history found
for the oldest white dwarfs of our sample is in agreement with this enhacement
of the star formation occurred $\simeq 11-12\,$Gyr in the past.

Nevertheless, the hypothesis above mentioned of the origin of these young halo
white dwarfs does not exclude other possible scenarios. For instance, that is
the case of the well known LP 93-21 white dwarf, which is also present in our
sample as J1045+4509. Firstly discovered in the high-proper motion Luyten
Palomar survey (Luyten, 1968), this white dwarf has been found to have a hot
DQ atmosphere, a mass in the range $1.029-1.10\,M_{\odot}$ and cooling age
estimate of $2.28-2.81\,$Gyr (Kawka et al., 2020, and references therein). It
is worth noting that these values are in perfect in agreement with our
estimates of $1.074\pm 0.033\,M_{\odot}$ for the mass and $2.69\pm 0.25$ for
the total age. In a recent paper, Kawka et al. (2020) thoroughly analyze the
kinematics of LP 93-21 reaching the conclusion that this DQ white dwarf is the
likely merger product of two other white dwarfs, which rejuvenated leading to
a shorter cooling age estimate. Furthermore, the kinematics of LP 93-21 is
compatible with a dwarf galaxy merger event, suggesting that this white dwarf
was either captured by the Milky Way or its orbit was affected by the galaxy
merger, hence claiming a total age of $\gtrsim 10$ Gyr.

On the other hand, we also note that a small fraction of halo white dwarfs
could even form as walk-away or runaway secondaries, which are ejected
following the disruption of binary systems that underwent either core-collapse
or thermonuclear supernova explosions. As an example, Renzo et al. (2019)
reported that simulations of systems with primary stars more massive than
$7.5\,M_{\odot}$, produce ejected main-sequence stars that will eventually
become white dwarfs with velocities in the range of 10–90
${\rm\,km\,s^{-1}}$(8 percent of simulated binaries). These ejected main-
sequence stars would also be rejuvenated due to mass-transfer occurring before
the core-collapse supernova of the companion, hence making the future white
dwarf appears younger. Similar or even faster ejection velocities are achieved
through the thermonuclear supernova channel (Shen et al., 2018, and refereces
therein), where the ejected companions could achieve ejection velocities up to
500 ${\rm\,km\,s^{-1}}$(main-sequence and helium-core donors) or more than 1
000 ${\rm\,km\,s^{-1}}$(white dwarf donors).

It is beyond the scope of the present work to ascertain the origin of those
young halo white dwarf candidates found here. However, along with the
possibility that the age distribution of these objects is compatible with the
Sagittarius galaxy encounters, an individual analysis is required to fully
unravel their origin.

###  2.5 Kinematics of the halo white dwarf sample

The stellar parameter analysis done in the previous sections can be
complemented with a kinematic study of the halo white dwarf sample. However,
we should be cautious given that only proper motions are provided for the
majority of objects of our sample. Consequently, we adopt the standard
assumption of null radial velocity in deriving the Galactic components of the
velocity $U$, $V$ and $W$. Despite the biases that this assumption may induce,
(e.g. Pauli et al., 2006; Torres et al., 2019a) some of the clustering
properties of the sample can remain unaltered (Fuchs & Dettbarn, 2011; Torres
et al., 2019b). In this sense, we use the integral of motion space as an
appropriate tool for analyzing our sample (e.g Helmi et al., 2006). Since the
volume of our sample is relatively small in terms of the size of our Galaxy,
we can safely approximate the component of angular momentum perpendicular to
the Galactic plane, $L_{z}$, by the $V$-velocity component, as well as the
radial and azimuthal Galactic components, $V_{R}$ and $V_{\phi}$, by the
components $U$ and $V$, respectively.

In Figure 9 we show the action space corresponding to the integral of motion
$V_{\Delta E}\equiv(U^{2}+2V^{2})^{1/2}$ as a function of $V$ (left panel) and
the Toomre diagram (right panel) for the white dwarfs of our halo candidate
sample. For helping the analysis we divided the sample in three groups
attending to the total age: young, $T_{\rm age}<8\,$Gyr (yellow dots), middle
$8<T_{\rm age}<10\,$Gyr (red dots) and old objects $T_{\rm age}>10\,$Gyr
(black dots). Also shown, for illustrative purposes, are the thin (light red
dots) and thick disc (light blue dots) velocities for the 100 pc white dwarf
sample from Torres et al. (2019a). The integral of motion $V_{\Delta E}$ can
be understood as a measure of the eccentricity, $e$, of the orbit (see Fuchs &
Dettbarn, 2011, and references therein). Thus, the left panel shown in Fig. 9
is equivalent to a $(L_{z},e)$ diagram. In this sense, we observe that the
bulk of old objects of our sample appear with moderately eccentric and
retrograde orbits in the range $\approx-100$ to $-200$ ${\rm\,km\,s^{-1}}$. A
value slightly above that of $\approx-220\,$${\rm\,km\,s^{-1}}$generally
adopted for the local standard of rest with respect of the center of the
Galaxy, but in agreement for what is considered a classical halo population.
On the contrary, those with the highest eccentric and highly retrograde orbits
are some of the youngest objects of our sample. These facts suggest to discard
these objects as belonging to a typical thick disk population, hence to be
thick disk contaminats. Besides, some prograde and relatively middle age
objects appear in our sample. However, we should be cautious about this fact,
given that one of the effects of the null radial velocity assumption is the
misleading prograde orbits, specially for high speed objects, as pointed out
in Pauli et al. (2006). Regarding the Toomre diagram (right panel of Fig. 9),
a certain clustering seems to be found for the oldest objects of our sample
while a greater dispersion is found for the middle and younger ones. In
particular, the youngest objects exhibit, on average, the largest speeds.

##  3 Conclusions

The Gaia space mission on its DR2 has provided an unprecedented wealth of
photometric and astrometric data. In particular, the white dwarf population
has found to be nearly complete up to 100 pc around the Sun. In this sample,
95 white dwarfs have been identified as belonging to a halo population. In the
present work, this subsample has been analyzed and the stellar parameters of
the objects has been derived.

Observations made with the Very Large Telescope UT1 equipped with the FORS2
spectrograph allowed us to obtain low-resolution spectra for 27 of our 95 halo
white dwarf candidates. By applying a fitting routine to the observed spectra
we derive effective temperatures for 24 DCs and 1 DA of the observed sample.
We also derived the surface gravity of the DA white dwarf via fitting its
Balmer lines, which results also in obtaining its mass, radius and luminosity
via interpolating the observed effective temperature and surface gravity in
the appropriate cooling sequences. On the other hand, we apply a procedure
based on Gaia astro-photometry and on a detailed population synthesis code,
that permitted to derive the stellar parameters of the white dwarfs. The
parameters thus derived were tested with those available from spectroscopy.
The agreement of both methods, in particular in the effective temperature
parameter, guarantees us the reliability of our photometric routine to derive
the stellar parameters, i.e., mass, radius, bolometric luminosity and age, for
the whole sample of 95 halo white dwarf candidates.

The major results found for the different parameter distributions are
summarized as follows:

  * •

A generalized version of the modified volume technique, which allow us to take
into account all kind of incompleteness sources, has been used to build the
100 pc halo white dwarf luminosity function. The halo luminosity function
reasonably presents the first ever detected evidence of a clear cut-off at
faint bolometric magnitudes. Although the number of objects of our sample is
relatively small compared to other published halo white dwarf samples, the
completeness analysis reveals $\sim 60\%$ completeness for the faint region
thus guarantying an acceptable statistical significance of the cut-off found
in our sample. The corresponding fitting of the cut-off leads to an age of
$\approx 12\pm 0.5\,$Gyr.

  * •

The halo white dwarf mass distribution peaks at $0.589\,M_{\odot}$, with most
of the stars ($71\%$) having masses below $0.6\,M_{\odot}$. Only two objects
have been found to be more massive than $0.8\,M_{\odot}$.

  * •

The majority of white dwarfs ($60\%)$ have total ages (cooling time plus
progenitor lifetime) older than $10\,$Gyr. In particular, we found 3 objects
with total ages above 12 Gyr, being the object J1312-4728 the oldest white
dwarf found so far with an age of $12.41\pm 0.22\,$Gyr .

  * •

The star formation history is basically reproduced by a burst of star
formation occurring from 10 to 12 Gyr and extended up to 8 Gyr. The peak of
the star formation history is centered at around 11 Gyr, which is compatible
with the current age of the Gaia-Enceladus encounter.

  * •

$13\%$ of our halo sample is contaminated by high-speed young objects (total
age<7 Gyr). The origin of these white dwarfs is unclear but their age
distribution may be compatible with the pass of the Sagittarius galaxy. An
individual analysis is required to unravel the origin of each object.

  * •

Finally, the kinematics analysis of the halo white dwarf sample reveals that
there is some clustering of the oldest ($T_{\rm age}>10\,Gyr$) objects of the
sample suggesting a common origin, whereas on the contrary, youngest objects
exhibit larger eccentric and retrograde orbits. Some prograde orbits have also
been found, however, the lack of radial velocity observations prevents us to
obtain definitive conclusions.

In this work we have demonstrated the utility of white dwarfs to address
important open questions in astronomy such as the age of the Galactic halo and
its star formation history. These objects can also potentially help to reveal
the past history and evolution of our Galaxy, an issue that we will analyse in
the future with the forthcoming data releases of Gaia.

## Acknowledgements

We wish to thank the suggestions and comments of our anonymous referee that
strongly improved the original version of this paper. This work was partially
supported by the MINECO grant AYA2017-86274-P and the Ramón y Cajal programme
RYC-2016-20254 and by the AGAUR (SGR-661/2017). ST wants to thank Marco Lam
for kindly providing the set of luminosity functions used in this work. ST
also wants to thank Carme Gallart for useful discussions on the star formation
history of the Galaxy. 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.

Based on observations collected at the European Southern Observatory under ESO
programme(s) 60.A-9203(E) and 0103.D-0271(A,B,C,D,E).

RR has received funding from the postdoctoral fellowship programme Beatriu de
Pinós, funded by the Secretary of Universities and Research (Government of
Catalonia) and by the Horizon 2020 programme of research and innovation of the
European Union under the Maria Skłodowska-Curie grant agreement No 801370.

## Data Availability Statement

The data underlying this article are available in the article. Supplementary
material will be shared on reasonable request to the corresponding author.

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