11institutetext: AEGORA Research Group - Joint Center for Ultraviolet
Astronomy, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040
Madrid, Spain 11email:  22institutetext: Departamento de Fisica de la Tierra y
Astrofisica, Fac. de CC. Matematicas, Plaza de Ciencias 3, 28040 Madrid, Spain
33institutetext: Departamento de Estadística e Investigación Operativa, Fac.
de CC. Matemáticas, Plaza de Ciencias 3, 28040 Madrid, Spain
33email<EMAIL_ADDRESS>††thanks:

# Uncovering a new group of T Tauri stars in the Taurus-Auriga molecular
complex from Gaia and GALEX data111Table 2 is only available in electronic
form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or
via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/

Ana Inés Gómez de Castro 112 2  Raúl de la Fuente Marcos 112 2  Ada Canet 112
2  Leire Beitia-Antero  1133 Javier Yáñez-Gestoso 11 3 3 Juan Carlos Vallejo
112 2

(Received ; accepted )

###### Abstract

Context. The determination of the complete census of young stars in any star
forming region is a challenge even for the nearest and best observed molecular
clouds such as Taurus-Auriga (TAMC). Deep surveys at infrared (IR) and X-ray
wavelengths and astrometric surveys using Gaia DR2 and DR3 have been carried
out to detect the sparse population and constrain the low mass end of the
initial mass function. These compilations have resulted in lists of more than
500 sources including reliable members of the association and candidates.The
astrometric information provided by the Gaia mission has proven fundamental to
evaluate these candidates.

Aims. In this work, we examine the list of 63 candidates to T Tauri star (TTS)
in the TAMC identified by their ultraviolet (UV) and infrared colours (IR)
measured from data obtained by the Galaxy Evolution Explorer all sky survey
(GALEX-AIS) and the Two Microns All Sky Survey (2MASS), respectively. These
sources have not been included in previous studies and the objective of this
work is twofold: evaluate whether they are pre-main sequence (PMS) stars and
evaluate the goodness of the UV-IR colour-colour diagram to detect PMS stars
in wide-fields.

Methods. The kinematic properties and the parallax of these sources have been
retrieved from the Gaia DR3 catalogue and used to evaluate their membership
probability. Several classification algorithms have been tested to search for
the kinematical groups but the final classification has been made with
$k$-means$++$ algorithms. Membership probability has been evaluated by
applying Logistic Regression. In addition, spectroscopic information available
in the archive of the Large Sky Area Multi Object Fiber Spectroscopic
Telescope (LAMOST) has been used to ascertain their PMS nature when available.

Results. About 20% of the candidates share the kinematics of the TAMC members.
Among them, HD 281691 is a G8-type field star located in front of the cloud
and HO Aur is likely a halo star given the very low metallicity provided by
Gaia. The rest are three known PMS stars (HD 30171, V600 Aur and
J04590305$+$3003004), two previously unknown accreting M-type stars
(J04510713$+$1708468 and J05240794$+$2542438) and, five additional sources,
which are very likely PMS stars. Most of these new sources are concentrated at
low galactic latitudes over the Auriga-Perseus region.

###### Key Words.:

stars: variables: T Tauri, Herbig Ae/Be, stars: low-mass, surveys, Galaxy:
stellar content

##  1 Introduction

The accurate determination of the low-mass end of the initial mass function
(IMF) is challenging but fundamental to understand the efficiency of star
formation and to constrain the significance of the low-mass leftovers in the
evolution of galaxies over cosmic scales. There have been systematic attempts
to make a complete accounting of this ”hidden” population in the solar
neighbourhood where the sensitivity for this investigation is optimal (see i.e
Luhman 2018 and references therein). The nearest stellar nurseries are located
within 200 pc, in a rim of molecular clouds that includes prominent star
forming regions such as Taurus-Auriga, Lupus, Chamaleon, Ophiuchus or Sco-Cen.
As a result, the Solar System is in a privileged location within the Galaxy to
investigate the formation of loose stellar associations and determine the low
mass end of their IMF.

Figure 1: Map of the dust distribution in the Taurus, Auriga, Perseus star
forming complexes obtained by the Far Infrared Surveyor on board the AKARI
satellite. The location of the main young associations is indicated as well as
that of T Tau and AB Aur, two prominent pre-main sequence stars in the region.
The yellow frame marks the area surveyed by GALEX and searched for young stars
by GdC2015.

The Taurus-Auriga molecular complex (TAMC) is the best studied region in the
area. The TAMC is part of the prominent ridge of molecular gas observed
towards the anticenter of the Galaxy; large-scale maps of the region date back
to the CO survey carried with the Columbia millimeter-wave telescope by
Ungerechts & Thaddeus (1987). The TAMC roughly extends over $20^{\rm o}\times
20^{\rm o}$ in the sky (see Fig. 1) and it is constituted by several filaments
and cores (Goldsmith et al., 2008; Garufi et al., 2021; Roccatagliata et al.,
2020; Long et al., 2019). There are more than 500 sources sparsely distributed
over the complex that could be pre-main sequence (PMS) stars (see Luhman 2022
for a recent accounting). The very young population (age $\leq 1$Myr) is well
identified by its significant near-infrared excess, its proximity to the
parent molecular gas and the presence of jets of molecular outflows; however,
the census of the most evolved sources, in particular, the so-called weak line
T Tauri stars (WTTSs), is still to be completed, especially at the very low
mass end towards the brown dwarfs frontier.

The release of the Gaia DR2 catalogue (Gaia Collaboration et al., 2018)
enabled for the first time an unbiased search for WTTSs in the region based on
the kinematical properties of the stars. The Gaia survey is magnitude-limited
to $G\simeq 21$ and thus, it is sensitive to the low mass stellar population
(late M types) in the TAMC which is located at $\sim 140$ pc. Galli et al.
(2019) generated a compilation of all the possible T Tauri stars (TTSs) in the
region including both, confirmed spectroscopic sources and possible
candidates, which were identified by the surveys carried out with the Spitzer
infrared telescope (Luhman et al., 2010), the Sloan Digital Sky Survey and the
2-Microns All Sky Survey (Luhman et al., 2017). Using a Hierarchical Mode
Association Clustering (HMAC) algorithm, they classified these sources into 21
distinct kinematical groups. A recent re-evaluation based on Gaia DR3 (Gaia
Collaboration et al., 2022) data has resulted in a grand total of 532 adopted
members distributed in 13 kinematical groups (Luhman, 2022). This kinematical
search does not provide a complete accounting of the young stars in the region
since young stars with peculiar proper motions such as stars ejected in close
encounters or stars in multiple systems, will be rejected as outliers by the
classification algorithms. Also, no information on their youth and
evolutionary state can be obtained in this manner. An additional problem is
the scarce and uncertain information provided by Gaia DR3 on the radial
velocity of these stars which makes difficult an accurate determination of the
three components of the LSR velocity $(U,V,W)$ for a large fraction of the
sources.

The most common and successful ways to search for WTTSs in molecular clouds
are based on their strong magnetic activity which results in an enhancement of
the atmospheric flux: coronal emission at X-ray, the radiation from the
transition region at ultraviolet (UV) wavelengths and the chromospheric
radiation measurable at UV but also at optical wavelengths. In fact, the first
catalogues of TTSs were built from stars displaying enhanced chromospheric
emission at the H$\alpha$ line (see e.g. Herbig & Kameswara Rao 1972). Many
WTTSs candidates were identified in the TAMC from the ROSAT all-sky X-ray
survey (Neuhaeuser et al., 1995) and subsequent observations (especially of
the Li I absorption) proved that many were indeed, WTTSs. The release of the
”all sky UV survey” carried out by the Galaxy Evolution Explorer (GALEX)
enabled to search for TTSs by their UV excess; a list of 63 new TTSs
candidates was elaborated by comparing the UV color and the IR color excesses
in the TAMC (Gómez de Castro et al. 2015, hereafter GdC2015). Gaia was
launched in December 2013 and its subsequent data releases came much later
than the publication of this list thus, neither parallax nor kinematical
information was available at the time to narrow down this large list of
candidates. It is the purpose of this research to carry out such analysis, to
identify the bona fide TTSs in the sample. For this purpose, we use the Gaia
DR3 parallax and kinematical information as well as spectra from the
spectroscopic survey carried out by the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope (LAMOST) when available, to determine the spectral
type and the Li I equivalent width.

This article is structured as follows. In section 2, the characteristics of
the GdC2015 sources are described. Sources with Gaia DR3 parallaxes
incompatible with TAMC membership are disregarded reducing the working sample
of viable candidates to 13 sources. In section 3, the membership probability
of these candidates is evaluated using Gaia DR3 data. In the process, a list
of trusted TTSs in the TAMC for kinematical studies is produced; this
compilation is significantly shorter than that produced by Luhman (2022). In
section 4, LAMOST spectra are analyzed; two new accreting late M-type stars
are identified in the GdC2015 sample. Finally, a short discussion (Sect. 5) on
the radial velocity distribution of the TTSs and the coupling with the
molecular gas is included. A short summary of the work is at the end, in
Section 6. Technicalities concerning the error analysis of the samples are
compiled in the Appendix to facilitate a comprehensive reading of the work.

##  2 UV candidates to T Tauri stars in the Taurus-Auriga region

Figure 2: Candidates to TTSs in the TAMC from GdC2015. Top panel, the location
of the TTSs and candidates on the sky is overlaid on the density of GALEX NUV
sources, in galactic coordinates; stellar densities are color coded. The
density of molecular gas is outlined from the 2MASS extinction map by Lombardi
et al. 2010). Bottom panel, color-color diagram used for the selection of the
TTSs candidates in GdC2015. Candidates are marked with black crosses, and
known CTTSs and WTTSs from the qualification sample are represented by blue
squares and red circles, respectively. The regression line marking the
location of the WTTSs in the diagram is plotted (solid green line) as well as
the uncertainty band from the fit (dashed green lines); most of the TTSs
candidates to be analyzed in this work are within this strip.

The GALEX All Sky Survey (AIS) (Martin et al., 2005) provided for the first
time an unbiased view of the TAMC at UV wavelengths. Though the area was not
mapped completely due to GALEX sensitivity constraints, 380 square degrees in
the sky (equivalent to 197 GALEX fields) were imaged and as many as
$\sim$163,000 UV sources were detected in the near ultraviolet (NUV) band (see
GdC2015). Through a comparison of UV and IR colours, 63 new candidates to TTSs
were identified; their location is shown in the top panel of Fig. 2 and their
UV-IR colours in the bottom panel.

These 63 sources were selected using a qualification sample of TTSs that was
generated from two sets of sources: [1] the 31 known TTSs detected in the
GALEX AIS survey of the TAMC and [2] the 21 TTSs observed with the
International Ultraviolet Explorer (IUE) in the low dispersion mode with high
signal-to-noise ratio spectra (see Table 1 in GdC2015); synthetic GALEX
photometry was calculated from the IUE spectra for these sources using
Morrisey et al. (2007) conversion. 2MASS photometry was available for all the
sources in the qualification sample. All these stars are marked with blue
squares (if CTTSs) and red circles (if WTTSs) in the bottom panel of Fig. 2.
The 63 candidates to be evaluated in this work appear as crosses in the plot.

Note that the WTTSs define a clear regression line with the hardest UV colours
being observed in the sources displaying the smallest J-K color. CTTSs
however, are scattered in the diagram. Most of the candidates studied in this
work were identified in the WTTSs strip (see Fig. 2, bottom panel) close to
the location of reddened massive cool stars and many are located close to
Galactic plane (see Fig. 2, top panel). Thus, further observables such as
distance or kinematics are needed to confirm the PMS nature of these sources.

Gaia DR3 counterparts have been searched and found for all candidates using a
2 arcsecond search radius employing the cross match services provided by the
Centre de Données Stellaires (CDS).

Figure 3: Density of Gaia sources in the TAMC. The densities are color coded
in stars per 6 arcmin2 (see lateral bar). The shadow produced by the TAMC
filaments over the stellar background is readily identified. The two over
dense regions correspond to NGC 1747 (an open cluster at 550 pc) and NGC 1746,
an asterism or apparent concentration of stars (Cantat-Gaudin & Anders, 2020),
are easily identifiable. The location of the GdC2015 candidates with Gaia
counterparts within 1200 pc is marked by circles and asterisks, color coded by
distance. Asterisks represent sources with high quality Gaia measurements
(RUWE $\leq 1.4$) and circles sources with RUWE larger than this threshold.

In Fig. 3, we display the projected location on the plane of the sky of all
sources within 1,200 pc. Only 13 among them have parallaxes compatible with
TAMC membership; i.e. 5 mas $-$ 9 mas, according to Roccatagliata et al.
(2020). Seven among them are concentrated in the north of the region, close to
the Galactic plane, but are separated in two distance groups at $\sim 110$ pc
and $\sim 160$ pc.

##  3 Kinematical properties of the candidates

In addition to distance, stellar kinematics needs to be analyzed. This has
been done in two steps. Firstly, the kinematics of known and trusted TTSs in
the TAMC with high quality Gaia measurements is
studied.222https://gea.esac.esa.int/archive/documentation/GDR2/Gaia_archive/chap_datamodel/sec_dm_main_tables/ssec_dm_ruwe.html
Kinematical groups are identified within this sample using the $k$-means ++
algorithm and then, the probability of the 13 candidates being kinematical
members is evaluated. Only trusted PMS sources with high quality astrometric
data will be considered for this qualification sample. The Gaia Data
Processing and Analysis Consortium states that a value of the re-normalised
unit weight error (RUWE) $\leq$1.4 is indicative of a good astrometric
solution.333https://gea.>el/sec_dm_main_tables/ssec_dm_ruwe.html[gea.esac.esa.int]
thus, only Gaia observations with RUWE $\leq$1.4 have been considered for the
qualification of the kinematics of the TTSs in the TAMC. As an aside, note
that according to Stassun & Torres (2021) RUWE values even slightly above 1.0
may be signaling unresolved binaries in Gaia data.

###  3.1 Kinematics of known TTSs in the TAMC

The initial sample of trustable TTSs in the TAMC has been taken from Joncour
et al. (2017); more recent catalogues are available, e.g. Galli et al. (2019)
and Luhman (2022), but they include together with bona fide TTSs, candidates
to TTSs from various surveys. Joncour et al. (2017) compilation includes 338
sources and it is mainly based in the mid-infrared survey of the TAMC carried
out with the Spitzer telescope to perform a census of the disc population in
the region that was completed with H$\alpha$ data for the few TTSs that were
not detected (Luhman et al., 2010); only 274 sources of the catalogue have
GAIA DR3 counterpart and only 170 have measurements with RUWE $\leq 1.4$.

Thus, the initial qualification sample consisted of these 170 TTS with high
quality astrometric measurements; however, after cross-checking Gaia
parallaxes three sources were found no to belong to the TAMC and have not been
considered (see Table 1). The final qualification sample consists of 92
classical TTSs (CTTSs) and 75 weak-line TTSs (WTTSs) (see Table 2).

Table 1: Sources misclassified as TTSs in the TAMC Name | RUWE | Parallax(mas) | Distance
---|---|---|---
2MASS J04375670$+$2546229 | 1.056 | $0.9734\pm 0.0877$ |  $\sim 1030$ pc
2MASS J04163048$+$3037053 | 1.003 | $3.0552\pm 0.094$ |  $317-338$ pc
2MASS J04080782$+$2807280 | 1.201 | $4.3569\pm 0.0543$ |  $227-232$ pc
Table 2: Qualification Sample for the Kinematics of the TTSs in the TAMC∗

Name |  lgal |  bgal | Parallax | Parallax Err. | PM (RA) | err. PM(RA) | PM (DEC) | err. PM(DEC) | RUWE
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | mas | mas |  mas yr-1 |  mas yr-1 |  mas yr-1 |  mas yr-1 |
$2MASSJ04161210+2756385$ | 168.728127 | -16.2045517 | 7.426 | 0.0989 | 8.735 | 0.139 | -24.99 | 0.089 | 0.88
$2MASSJ04202555+2700355$ | 170.105104 | -16.1639581 | 6.0362 | 0.1102 | 11.175 | 0.132 | -17.696 | 0.09 | 0.921

∗ | Full table is available on-line
---|---

Gaia proper motions in the ICRS system have been converted into the Galactic
system using the procedure described in the Gaia EDR3
documentation.444https://gea.esac.esa.int/archive/documentation/GEDR3/Data_processing/chap_cu3ast/sec_cu3ast_intro/ssec_cu3ast_intro_tansforms.html
In Fig 4, these proper motions are overlaid on the map obtained by the Plank
mission of the area where the distribution of the warm dust and the
orientation of the magnetic field are shown (Planck Collaboration et al.,
2016). The TTSs motion is very coherent as expected in a young association
that still keeps memory of the motion of the parent molecular cloud; the young
gravitationally bound open cluster contains hundreds of members with similar
ages and composition; however, the kinematical memory of this common origin
will be lost over time due to the interaction with the large scale
gravitational field of the Galaxy. As pointed out back in the early 90’s
(Gomez de Castro & Pudritz, 1992) the TTSs star move roughly perpendicularly
to the direction of the magnetic field traced by the polarization of the dust.

There are mild differences between WTTSs and CTTSs. This is best shown by the
$\mu_{b}/\mu_{l}$ ratio (see Figs. 4, 5); while CTTSs have a single peak
distribution with a smooth tail to small ratios, the WTTSs display a double
peaked distribution. As CTTSs are younger stars, it is expected that they are
better coupled to the gas and keep a common kinematics. This effect is also
observed in the spatial distribution of the accreting TTSs and the WTTSs;
CTTSs are located close to the molecular cloud while WTTSs are more sparsely
distributed (Luhman et al., 2010). Two outliers are detected: the CTTSs,
LkH$\alpha$ 332 ($2MASSJ04420777+2523118$) and the WTTSs
$2MASSJ04555288+3006523$; LkH$\alpha$ 332 is at 11.3 arcsec of V1000 Tau,
another TTSs.

Figure 4: Proper motions (velocity projected in the plane of the sky) of the
known TTSs in the TAMC with high quality astrometric measurements
($RUWE<1.4$). CTTSs and WTTSs are represented with blue and red arrows,
respectively. Note that the motion of the stars is not parallel to the
filaments and it is approximately perpendicular to the direction of the
magnetic field.  Figure 5:  Histogram of the distribution of the ratio
$\mu_{b}/\mu_{l}\cos(b)$ obtained for the known CTTSs (blue) and WTTSs (red)
in the TMC.

####  3.1.1 k-cluster analysis

---
Figure 6:  Location of the two groups identified by the decision tree
algorithm are represented. Top: sources location, middle: proper motions of
the sources in Galactic coordinates and bottom: inclination of the proper
motion with respect to the Galactic plane versus distance. The two groups are
identified by orange squares (group 0) and violet circles (group 1).

The TAMC occupies a large area in the sky and several kinematic groups have
been identified within the association (see e.g. Galli et al. 2019; Luhman
2023). In this work, we have used the $k$-means++ clustering algorithm to
determine which groups of stars in our qualification sample could be
kinematically related. Given their young age, kinematically coherent sources
may have been born out of the same gas cloudlet within the TAMC. This
algorithm is an unsupervised machine-learning technique that performs a
centroid-based analysis using iterative refinement (Lloyd, 2006; Arthur &
Vassilvitskii, 2007). In our case, we carried out a multiparametric analysis
that included both positional information (parallax) and kinematic information
(the projections of the proper motion along the Galactic coordinates), i.e.
our data set has three parameters. The $k$-means++ algorithm is not density-
based like DBSCAN or OPTICS, but partitional so every object in the sample is
assigned to one and only one cluster. This algorithm implicitly assumes that
clusters are convex and isotropic and it performs best when applied to a
mixture of Gaussian distributions with the same variances but perhaps
different means. This mathematical precondition matches well the physical
context of the TAMC sample when parallaxes and proper motions are considered.
As our analysis is not including the coordinates of the sources but parallaxes
and proper motions, the actual distributions of the data are rather isotropic
and normally distributed.

For each one of the 167 TTSs (92 CTTSs and 75 WTTSs), a three-dimensional
vector was generated. An initial set of $k$ points (centroids) was defined in
this three-dimensional space and $k$ clusters were created by associating
every vector with the nearest centroid. In a second step, a new centroid was
computed for each cluster by calculating the mean among the vectors. Later,
the association of each vector to a given cluster was re-evaluated for the new
centroid. The process was repeated until convergence was reached.

Here, we used the k-means++ algorithm as implemented by the Python library
Scikit-learn (Pedregosa et al., 2011). Before applying the $k$-means++
algorithm, we scaled the data set using $Z$-score normalization: found the
mean and standard deviation for each set of variables used in the analysis,
subtracted the relevant mean from each value, and then divided by its
corresponding standard deviation. Although the variables may originally have
different variances, $Z$-score normalization standardizes variances and avoids
placing more weight on variables with smaller variance when applying
$k$-means++. Distance assignment between the data points in our sample assumed
a Euclidean metric (other metrics gave consistent results) and we used the
elbow method to determine the optimal value of clusters, $k$, that minimized
the sum of the distances of all data points to their respective cluster
centres. Test analyses using OPTICS on the same data set led to similar
results.

This procedure was applied to the list of bona-fide TTSs (Table 2), but in
order to take into account the associated uncertainties and their
correlations, we generated 104 instances of the sample as explained in
Appendix A and for each synthetic instance, we determined the clusters through
the $k$-means++ algorithm. In almost all the cases, our clustering analysis
produced two statistically significant clusters which are represented in Fig.
6. Sources in the first group are located at an average distance of 160 pc
while those in the second group are found at 130 pc. This clean classification
in two groups differs from the results of the previous works by Galli et al.
(2019) and Luhman (2023). Galli et al. (2019) uses Hierarchical Association
Clustering Mode (HACM) which is a non-parametric statistical approach used for
clustering analysis. HACM finds the modes of a kernel-based estimate of the
density of points in the working space and groups the data points associated
with the same modes into one cluster with arbitrary shape. Clustering by mode
identification requires only the space and the bandwidth of the kernel to be
defined. In particular, Galli et al. (2019) make use of a 5-dimensional space
consisting in the equatorial coordinates of the sources, their proper motions
in equatorial coordinates and the parallax
$(\alpha,\delta,\mu_{\delta},\mu_{\alpha}\cos\delta,\pi)$. The algorithm
identifies 21 clusters hierarchically linked in the dendrogram only 4 of them
having more than 10 sources. This result is natural since the coordinates are
taken into account and the stellar population is sparsely distributed in the
TAMC thus, stars with similar proper motions and parallaxes located in
different areas are identified as independent groups. Luhman (2023) follows a
different approach. Gaia astrometry is analyzed in terms of proper motions
offsets defined as the differences between the observed proper motion of the
star and the the motion expected at the celestial coordinates and parallactic
distance of the star for a specified space velocity. This procedure is set to
minimize the projection effects given the large extent of the TAMC and the
offsets are calculated with respect to the expected LSR velocity given in term
of the velocity vector (U,V,W) = (-16, -12, -9) km s-1 , which according to
Luhman (2018) approximates the median velocity of Taurus members. 9 groups are
identified in this manner (see Fig. 1 and Table 6 in Luhman (2023) ). These
groups do not coincide with those in Galli et al. (2019) but some of the
levels in the hierarchy defined by Galli et al. (2019) contain some of these
groups. In our work, the classification is made with a $k$-means++ clustering
algorithm which does not provide a hierarchical classification, neither a
dendrogram suitable for comparison with Galli et al. (2019). Moreover, the
analysis in Sect. 3.1, as well as previous works (Gomez de Castro & Pudritz,
1992), show that galactic coordinates are better suited for the kinematical
analysis of the region (see Fig. 4). Proper motions are converted from
equatorial to galactic coordinates and the cluster analysis is carried in a
3-dimensional space including these proper motions and the parallax
$(\mu_{b},\mu_{l}cosb,\pi)$. The number of groups identified by the algorithm
is reduced to 2 (groups 0 and 1) with a clear difference between them in the
proper motion space and distance. Groups L1544 and L1517 in Luhman (2023)
belong to group 0 and 1 in this work, respectively. However, the many groups
identified by Luhman (2023) in the center of the TAMC belong to one of the two
underlying groups identified in this work. Our method does not assume a common
LSR velocity for the full TAMC, as in Luhman (2023) and does nor require to
know the radial velocity of the stars to evaluate the differences in the
(U,V,W) because there very few sources with accurate determinations of the
radial velocity (see, Galli et al. (2019) ). We just use high quality data
from Gaia DR3 and the analysis of the properties of the TAMC to use optimal
dynamical tracers for the classification. This strategy shows to be very
efficient in the description of the region.

###  3.2 Kinematics of the TTS candidates and membership probability

So far, we have discussed the kinematics in the TAMC based on the sample of
167 ’bona fide’ TTS. In this section, we use the kinematic classification
performed in Sec. 3.1 in order to assign a membership probability to each
group of each of the 13 TTS candidates and to that purpose, we build a
logistic regression model using the Scikit-learn package (Pedregosa et al.,
2011). A detailed explanation of this method with an application to
astrophysics is provided in Beitia-Antero et al. (2018); in this work, we only
provide a simple explanation with the basics of the methodology as a guide to
the reader for our particular case.

We work with a qualification sample of $N=167$ stars (the ’bona fide’ TTS)
that are classified into two groups based on their values of
$(\mu_{l^{*}},\mu_{b},\pi)$ as explained in the previous section, so the
classification depends on three features and is binary (groups $k=\\{1,0\\}$).
In this case, the logistic regression model gives the probability that a given
star $x_{i}$ belongs to Group 1, since the probability of belonging to Group 0
is the complementary:

$p(x_{i}\in
1)=\frac{1}{1+\exp{-(\beta_{0}+\mu_{l^{*}}\beta_{1}+\mu_{b}\beta_{2}+\pi\beta_{3})}}$
(1)

where the $\beta_{m}$ are the logistic regression parameters fitted using the
qualification sample, that have a median value of $\beta_{0}=17.052$,
$\beta_{1}=-0.085$, $\beta_{2}=0.616$, $\beta_{3}=-1.575$ (see Appendix B for
more details on how these parameters were computed taking into account the
associated errors on proper motions and parallax). We analyzed the confusion
matrix and only one star is misclassified, so this choice of parameters
(proper motions in galactic coordinates and parallax) gives a good separation
of the groups.

Once the logistic regression model is trained, we apply it to the sample of
TTS candidates, and following the methodology explained in Appendix B, we have
taken into account the errors in the measurements and computed a median
probability of belonging to any of the two groups; these probabilities are
shown in Table 3. The location of the candidates in the proper motions-
distance diagram is shown in Fig. 7. Only sources No. 3 (J04423721$+$3401492)
and No. 12 (V600 Aur) display properties very different from those of the
qualification sample. We want to highlight that membership probability has
been computed even for those sources with astrometric measurements of moderate
quality: J04290082$+$3152597 (RUWE: 1.413), J05000310$+$3001074 (RUWE: 1.615)
and J05005485$+$3229168 (RUWE: 1.773) and even bad quality J04455129$+$1555496
(RUWE: 4.6815).

Table 3: Classification of the 13 TTs candidates after applying the Logistic Regression model with variables ( $\pi$ , $\mu_{l_{gal}}\cos{b_{gal}}$, $\mu_{l_{gal}}$). No. | Name | RA | DEc | P(Group 0) | P(Group 1) | RUWE
---|---|---|---|---|---|---
| 2MASS |  |  |  |  |
1 | J04090973+2901306 | 62.29073244819 | 29.0249285705 | 0.9974 | 0.0026 | 1.011
2 | J04290082+3152597 | 67.25351322368 | 31.88312957994 | 0.9919 | 0.0081 | 1.413
3 | J04423721+3401492 | 70.65496319697 | 34.03025801994 | 0.1347 | 0.8653 | 1.482
4 | J04455129+1555496 | 71.46378376348 | 15.93038180969 | 0.0026 | 0.9974 | 4.685
5 | J04510713+1708468 | 72.77977787635 | 17.14620505952 | 0.6846 | 0.3154 | 1.122
6 | J04590305+3003004 | 74.76271085992 | 30.04998394972 | 0.8744 | 0.1256 | 1.242
7 | J04595003+3049082 | 74.958677841 | 30.81864271002 | 1.0 | 0.0 | 23.043
8 | J05000310+3001074 | 75.01300900515 | 30.01899690108 | 0.9288 | 0.0712 | 1.615
9 | J05005485+3229168 | 75.22852314723 | 32.48790302855 | 0.7567 | 0.2433 | 1.773
10 | J05050770+2923477 | 76.28212032867 | 29.39648322479 | 0.9993 | 0.0007 | 1.04
11 | J05074697+3120186 | 76.94572807385 | 31.33837744853 | 0.8431 | 0.1569 | 1.211
12 | J05102234+3126402 | 77.59293165054 | 31.44435544281 | 1.0 | 0.0 | 3.529
13 | J05240794+2542438 | 81.03309628183 | 25.71205212925 | 0.8749 | 0.1251 | 1.368

Figure 7:  As in the bottom panel of Fig. 6. The location of the 13 candidates
to TTSs examined in this work is indicated and labeled according to their
entry in Table 3. The candidates are marked according to their RUWE value:
green diamonds (RUWE $\leq$ 1.4) and black dartboards (RUWE $>1.4$).

##  4 Spectral information on the candidates

In addition to the kinematical information, Gaia DR3 provides data on the
effective temperature, surface gravity and metallicity for seven of the
thirteen candidate sources (see Table 4). These data are also indicated in the
table and compatible with the candidates being late type stars as expected for
the TAMC stellar population.

A peculiar source is HO Aur. The metallicity derived by Gaia for this source
is very low (-3.9707) and given the high quality of the measurements (RUWE $<$
1.4), we feel inclined to rely on the data provided by the mission and
tentatively identify HO Aur as a Population II field star.

LAMOST spectra are available for few sources and are compatible with the
effective temperatures derived by Gaia. In particular, the spectra of
$J04510713+1708468$ and $J05240794+2542438$ are compatible with late M
spectral type and display strong TiO bands. We have downloaded the spectra
from the Vizier server and measured the TiO-5 index (Cruz & Reid, 2002) for
these sources obtaining values of 1.28 and 0.85, respectively which confirms
their M-type classification. The spectra also display a strong Na I absorption
feature at 8183/8199 Å (see Fig. 8). which is gravity sensitive (Martín et
al., 2010) unfortunately, the spectral resolution is too low to cross-check
the effective gravity determined by Gaia for these sources. The spectra also
show Li I absorption at 6708 Å but again, equivalent widths are very uncertain
and thus, it is difficult to provide an age estimate of the sources.

The most noticeable characteristics of the spectra of these stars are the
prominent emission lines, in particular the Balmer series and the Ca II
doublet (see Fig. 8). This indicates that the two M dwarfs are young and
possibly accreting (see Basri 2000 and references therein). This is consistent
with the method used by GdC2015 to search for TTSs candidates that was based
on selecting sources that was based of the UV excess of the sources; 2MASS
$J04510713+1708468$ has $FUV-NUV=1$ and $J-K=0.9$ and 2MASS
$J05240794+2542438$ has $FUV-NUV=1.4$ and $J-K=0.97$.

HD 281691 is a G8 star located in front of the TAMC, at a distance of 108.7
pc, according to the Gaia parallax and its proper motions are significantly
different from those of the TTSs in the system. HD 281691 has a nearby
companion located at projected distance of 6.78 arcsec (738 au) (Janson et
al., 2013) and it is often included in surveys for debris disk (Meshkat et
al., 2017) . The infrared excess from the disk and the UV emission from the
active star resulted in its detection by the GdC2015 survey.

HD 30171 (J04455129$+$1555496) was detected as a TTS candidate in the GdC2015
survey and it is known to be a WTTS since the early searches for Li I
absorption of the X-ray sources detected by the ROSAT satellite in TAMC
(Wichmann et al., 2000). As such, it is included in the catalogue used for
qualification of the TTSs sample (Joncour et al., 2017) and it is a clear
member of Group 2.

V600 Aur is an active X-ray source which was included in the sample studied by
Xing (2010) for the determination of the Li I equivalent width (see Table 1 in
the article, [LH98] 173). Its rapid rotation period (2.201 days) and high Li I
abundance support its pre-main sequence nature and TAMC membership. Note that
though its kinematics deviates apparently from that of the TAMC members, this
may be caused by the low quality of the Gaia measurements (RUWE$=$3.529).

J04590305$+$3003004 and J05000310$+$3001074 are included in the Catalogue of
Stars in the Northern Milky Way Having H$\alpha$ in Emission (Kohoutek &
Wehmeyer, 1999) and display infrared excess emission (Luhman, 2023). Recently,
J04590305$+$3003004 has been reported as a 8.8 Myr old dipper star, a subgroup
among the young stellar objects that exhibit dimming variability in their
light curves (drops in brightness by 10%$-$50%), attributed to the occultation
of the star by the circumstellar disk material (Capistrant et al., 2022).

Little or no information is available about J04290082$+$3152597,
J04423721$+$3401492, J04595003$+$3049082, J05005485$+$3229168 and
J05050770$+$2923477\. Two of them, which according to our work are very good
candidates to being members of the sparse young stellar population of the TAMC
however, J04423721$+$3401492 kinematics differs significantly from that of the
TAMC members and the quality of the Gaia measurements is quite good
(RUWE$=$1.482).

In summary, 3 out the 13 candidates identified in the GdC2015 survey are known
pre main sequence stars namely, HD 30171, V600 Aur and J04590305$+$3003004\.
Another two are not TTSs associated to the TAMC; HD 281691 is a field G8-type
star located in front of the TAMC with a nearby companion at a projected
distance of 738 AU and HO Aur has a too low metallicity to belong to the TAMC.
The rest (excluding J04423721$+$3401492) are very reliable candidates and two
of them, in particular, have been confirmed as new late type TTSs from their
LAMOST spectrum: $J04510713+1708468$ and $J05240794+2542438$.

Table 4: Stellar properties Name | Teff | log(g) | [Fe]/[H] ratio | LAMOST spectrum | Comments
---|---|---|---|---|---
2MASS | (K) |  [cm2/s] |  |  |
J04090973+2901306 | 5035 | 4.3171 | -0.5184 | Y | HD 281691, Debris Disk
J04290082+3152597 | 4735.8 | 4.5392 | -0.1402 | N |
J04423721+3401492 |  |  |  | N |
J04455129+1555496 |  |  |  | N | HD 30171
J04510713+1708468 | 3385.4 | 4.2684 | 0.1421 | Y | Late type dwarf with Emission Lines
J04590305+3003004 |  |  |  | N |
J04595003+3049082 |  |  |  | N |
J05000310+3001074 |  |  |  | N | GZ Aur
J05005485+3229168 |  |  |  | N |
J05050770+2923477 | 4133.7 | 4.4843 | 0.2099 | N |
J05074697+3120186 | 5718.9 | 4.1804 | -3.9707 | N | HO Aur, Population II
J05102234+3126402 | 4899.5 | 4.3814 | -0.5804 | Y | V600 Aur, LiI (Xing 2010)
J05240794+2542438 | 3529.2 | 4.044 | 0.1507 | Y | ate type dwarf with Emission Lines

Figure 8:  LAMOST spectrum of the two newly identified accreting brown dwarfs
in the TAMC. Top: LAMOST spectrum; the Balmer series and the TiO bands are
readily identified. Bottom: Zoom around the Li I feature.

##  5 Discussion: dynamical coupling between the TAMC and the TTSs in the
region

The radial velocity of the TTSs also provides fundamental information to
determine the coupling between the molecular gas in the TAMC and the young
stellar population (see e.g. Gomez de Castro & Pudritz 1992). This
determination is relevant for two purposes. Firstly, it indicates the degree
of dynamical relaxation of the young stellar association. Also, if the
coupling is strong, this information can be used to estimate the motion of the
molecular gas in the plane of the sky which otherwise, is unfeasible; if the
dynamical coupling between the gas and the stars is strong (if the radial
velocities are similar) then, the motion of the clouds in the plane of the sky
can be inferred from the stellar motion and the 3D velocity vector of the
molecular gas can be restored.

There have been several attempts to examine this coupling, in particular that
by Galli et al. (2019) based on Gaia DR2 data. In their approach, they
selected several TTSs (28) and compared their radial velocity with that of the
molecular gas in the nearest area within the complex (see Fig. 15 in Galli et
al. 2019). There is a well known velocity gradient in the TAMC which is
observed in the 12CO and 13CO maps (Goldsmith et al., 2008; Narayanan et al.,
2008); the radial (VLSR) velocity decreases from 8.5 to 5.5 km s-1 roughly as
Galactic longitude increases (see Fig. 7 to 16 in Narayanan et al. 2008).
Galli et al. (2019) attempted to search traces of this gradient in the stellar
radial velocity with uncertain results. In their work, they selected the TTSs
with good radial velocity measurements from Gaia DR2 and extracted from the
13CO data cubes the velocity of the nearby (in projection) molecular gas (the
beam size of the radio maps is 45” and the Nyquist-samplex pixel $\sim$20”).
They found that stars near L1495, B213 and B216 (which are at
$l_{gal}\leavevmode\nobreak\ \sim 170^{o}$) have larger $V_{LSR}$ than stars
at L1536 ($l_{gal}\leavevmode\nobreak\ \sim 175^{o}$) but the correlation is
not good further than these two extreme groups. Moreover, the uncertainties of
the 13CO $V_{LSR}$ measurements are high given the large broadening of the
13CO profiles and the extent of the area within the beam. In summary, though
the attempt is to be commended the results are non-conclusive.

A simpler, statistical approach may help to examine the stars-gas coupling.
There are about 6.2 million stars with galactic coordinates
($l_{gal}\in[164^{o},185^{o}],b_{gal}\in[-24^{o},-4^{o}]$) in the Gaia DR3
survey. This number reduces to 697,990, if only sources with accurate radial
velocity (relative error $<10$%) are included. Among them, only 20,806 are
located at a distance compatible with TAMC membership. We have evaluated the
radial velocity distribution ($V_{LSR}$) for these field stars and compared it
with that obtained for the TTSs with high quality Gaia DR3 radial velocity
measurements (66 TTSs of those 40 are WTTSs and 26 are CTTSs). The velocity
distribution of the field stars in the area is very broad with a slight
asymmetry towards bluewards shifted velocities however, the velocity
distribution of the TTSs (both WTTSs and CTTSs) peaks at redwardshifted
velocities of $\sim$ 2.5 km s-1 (see Fig. 9). The Kolmogorov-Smirnov test
rejects the null hypothesis (both distributions are the same); the $p$-value
is 1.32$\times 10^{-12}$. Thus, as otherwise expected, the TTSs differ from
field stars also in radial velocity. The relevant issue for the sake of this
discussion is that the peak frequency of the TTSs velocity is smaller than the
VLSR of the TAMC thus, the TTSs as a whole seem to be already dynamically
decoupled from the cloud.

Unfortunately, the statistic is too poor to seek for differences between CTTSs
and WTTSs. In spite of the apparent slight bias of the CTTSs towards a higher
$V_{LSR}$ hinted at in Fig. 10, the Kolmogorov-Smirnov test clearly indicates
that both CTTSs and WTTS distributions are independent samples of the same
underlying distribution with $p$-value 0.998.

Figure 9:  Histogram of the frequencies of the distribution of stars in the
TAMC field in terms of VLSR. The distribution of field stars is compared with
that of the bona fide TTSs.
Figure 10:  Histogram of the frequencies of the distribution of TTSs stars in
terms of VLSR. The distribution of the CTTSs is compared with that of the
WTTSs.

##  6 In summary

In this work, Gaia DR3 astrometric data have been used to review the list of
candidates to TTSs selected by GdC2015 on the basis of their UV and IR
colours. From the initial compilation of 63 sources, only 10 (16%) have been
found to be reliable members (shared kinematics and location with well known
members of the association). Most of these sources are located at low galactic
latitudes, in the Auriga-Perseus area. Gaia spectroscopic information has
shown that one of candidates pinpointed in the GdC2015 list, HO Aur, is a very
low metallicity, population II star which cannot be member of this young
association. All the rest of the sources are very likely TTSs; in particular,
two of them have been identified as accreting late-type dwarfs from their
LAMOST spectra: 2MASS J04510713+1708468 and 2MASS J05240794+2542438.

This work has also produced some additional results. Firstly, a clean,
reliable list of TTSs for astrometric studies has been produced. In addition,
the analysis of the radial velocity ($V_{LSR}$) of the TTSs in comparison to
those of the field stars and the molecular cloud has shown that the
association has clearly distinct kinematics and that the TTSs are no longer
coupled to the molecular gas in the TAMC. As the sample is dominated by WTTSs,
this suggests that the decoupling is already significant $10-100$ Myr after
star formation begun.

###### Acknowledgements.

This work has been partially financed by the Ministry of Science and
Innovation through grants: ESP2017-87813-R and PID2020-116726RB-I00. This
research has made use of the SIMBAD database, operated at CDS, Strasbourg,
France and the computational facilities of the Joint Center for Ultraviolet
Astronomy in the campus of the Universidad Complutense de Madrid. L.B.-A.
acknowledges the receipt of a Margarita Salas postdoctoral fellowship from
Universidad Complutense de Madrid (CT31/21), funded by the ’Ministerio de
Universidades’ with Next Generation EU funds. 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.

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##  Appendix A Impact of uncertainties on our clusterings analysis

In order to account for the data uncertainties, we used a Monte Carlo sampling
technique to generate 104 synthetic datasets. For each one, we applied the
k-means++ algorithm, keeping track of the number of groups generated and the
positions of the centroids (the nearest neighbour was used as proxy to obtain
physical coordinates). Therefore, for each source in our list of bona-fide
TTSs, we generated 104 realizations of its astrometric parameters. Gaia DR3
provides the correlations between each pair of astrometric parameters. Given
two parameters, $x$ and $y$, with standard deviations $\sigma_{x}$ and
$\sigma_{y}$, their respective covariance and correlation coefficients,
$\sigma_{xy}$ and $\rho_{xy}$, can be written as
$\sigma_{xy}=\rho_{xy}\,\sigma_{x}\,\sigma_{y}$. If C is the covariance matrix
at a given epoch associated with the astrometric solution that is symmetric
and positive-semidefinite, then C = A A${}^{\textbf{{T}}}$ where A is a lower
triangular matrix with real and positive diagonal elements,
A${}^{\textbf{{T}}}$ is the transpose of A. In the particular case studied
here, these matrices are $5\times 5$. If the elements of C are written as
$c_{\rm ij}=\rho_{\rm ij}\,\sigma_{\rm i}\,\sigma_{\rm j}$ and those of A as
$a_{\rm ij}$, where those are the entries in the $i$-th row and $j$-th column,
and if $\@vec{r}$ is a vector made of univariate Gaussian random numbers
(components $r_{i}$ with $i=1,5$), the required multivariate Gaussian random
samples are given by the expressions:

$\displaystyle\alpha_{\rm c}$ $\displaystyle=\alpha+a_{11}\,r_{1}$ (2)
$\displaystyle\delta_{\rm c}$
$\displaystyle=\delta+a_{22}\,r_{2}+a_{21}\,r_{1}$ $\displaystyle\pi_{\rm c}$
$\displaystyle=\pi+a_{33}\,r_{3}+a_{32}\,r_{2}+a_{31}\,r_{1}$
$\displaystyle\mu_{\alpha\ {\rm c}}$
$\displaystyle=\mu_{\alpha}+a_{44}\,r_{4}+a_{43}\,r_{3}+a_{42}\,r_{2}+a_{41}\,r_{1}$
$\displaystyle\mu_{\delta\ {\rm c}}$
$\displaystyle=\mu_{\delta}+a_{55}\,r_{5}+a_{54}\,r_{4}+a_{53}\,r_{3}+a_{52}\,r_{2}+a_{51}\,r_{1}\,,$

where $\alpha$, $\delta$, $\pi$, $\mu_{\alpha}$, and $\mu_{\delta}$ are the
values of right ascension, declination, absolute stellar parallax and proper
motions in right ascension and declination directions provided by Gaia DR3 and
the $a_{ij}$ coefficients are given by:

$\displaystyle a_{11}$ $\displaystyle=\sigma_{\alpha}$ (3) $\displaystyle
a_{21}$ $\displaystyle=\rho_{\alpha\ \delta}\,\sigma_{\delta}$ $\displaystyle
a_{31}$ $\displaystyle=\rho_{\alpha\ \pi}\,\sigma_{\pi}$ $\displaystyle
a_{41}$ $\displaystyle=\rho_{\alpha\ \mu_{\alpha}}\,\sigma_{\mu_{\alpha}}$
$\displaystyle a_{51}$ $\displaystyle=\rho_{\alpha\
\mu_{\delta}}\,\sigma_{\mu_{\delta}}$ $\displaystyle a_{22}$
$\displaystyle=\sqrt{\sigma_{\delta}^{2}-a_{21}^{2}}$ $\displaystyle a_{32}$
$\displaystyle=(\rho_{\delta\
\pi}\,\sigma_{\delta}\,\sigma_{\pi}-a_{21}\,a_{31})/a_{22}$ $\displaystyle
a_{42}$ $\displaystyle=(\rho_{\delta\
\mu_{\alpha}}\,\sigma_{\delta}\,\sigma_{\mu_{\alpha}}-a_{21}\,a_{41})/a_{22}$
$\displaystyle a_{52}$ $\displaystyle=(\rho_{\delta\
\mu_{\delta}}\,\sigma_{\delta}\,\sigma_{\mu_{\delta}}-a_{21}\,a_{51})/a_{22}$
$\displaystyle a_{33}$
$\displaystyle=\sqrt{\sigma_{\pi}^{2}-a_{31}^{2}-a_{32}^{2}}$ $\displaystyle
a_{43}$ $\displaystyle=(\rho_{\pi\
\mu_{\alpha}}\,\sigma_{\pi}\,\sigma_{\mu_{\alpha}}-a_{31}\,a_{41}-a_{32}\,a_{42})/a_{33}$
$\displaystyle a_{53}$ $\displaystyle=(\rho_{\pi\
\mu_{\delta}}\,\sigma_{\pi}\,\sigma_{\mu_{\delta}}-a_{31}\,a_{51}-a_{32}\,a_{52})/a_{33}$
$\displaystyle a_{44}$
$\displaystyle=\sqrt{\sigma_{\mu_{\alpha}}^{2}-a_{41}^{2}-a_{42}^{2}-a_{43}^{2}}$
$\displaystyle a_{54}$ $\displaystyle=(\rho_{\mu_{\alpha}\
\mu_{\delta}}\,\sigma_{\mu_{\alpha}}\,\sigma_{\mu_{\delta}}-a_{41}\,a_{51}-a_{42}\,a_{52}-a_{43}\,a_{53})/a_{44}$
$\displaystyle a_{55}$
$\displaystyle=\sqrt{\sigma_{\mu_{\delta}}^{2}-a_{51}^{2}-a_{52}^{2}-a_{53}^{2}-a_{54}^{2}}\,,$

where $\sigma_{\alpha}$, $\sigma_{\delta}$, $\sigma_{\pi}$,
$\sigma_{\mu_{\alpha}}$, and $\sigma_{\mu_{\delta}}$ are the standard errors
in right ascension, declination, parallax and proper motions from Gaia DR3,
and $\rho_{\alpha\ \delta}$, $\rho_{\alpha\ \pi}$, $\rho_{\alpha\
\mu_{\alpha}}$, $\rho_{\alpha\ \mu_{\delta}}$, $\rho_{\delta\ \pi}$,
$\rho_{\delta\ \mu_{\alpha}}$, $\rho_{\delta\ \mu_{\delta}}$, $\rho_{\pi\
\mu_{\alpha}}$, $\rho_{\pi\ \mu_{\delta}}$, and $\rho_{\mu_{\alpha}\
\mu_{\delta}}$ their respective correlation coefficients, also from Gaia DR3.
For each $\pi_{\rm c}$, we computed the value of the distance by applying the
usual relationship, $d_{\rm c}=1/\pi_{\rm c}$.

For each set of synthetic sources statistically compatible with the
qualification sample, we compute the proper motions along the Galactic
coordinates as pointed out above and use their values together with the
parallaxes to apply the $k$-means++ algorithm. The resulting centroids are
shown in Fig. 11 in red.

Figure 11: Effect of the uncertainties on the clustering analysis. Results in
physical parameters. In black we have the sources and in red the 104
centroids.

##  Appendix B Fitting the logistic regression model with errors

From the mathematical point of view, the logistic regression model is trained
on a set of qualification sources with well-defined groups, where all the
values are considered to be free of errors. However, the proper motions and
parallax of our training sample of bona fide TTS have inherent errors and
depend one to each other – the parameters are correlated. In order to take
this constraint into account, we produce a set of $10^{4}$ mock realizations
of the sample using the covariance matrix for generating random samples of
($\mu_{\alpha^{*}},\mu_{\delta},\pi$) that we then convert into proper motions
in Galactic coordinates as explained in the Gaia
documentation555https://gea.esac.esa.int/archive/documentation/GEDR3/Data_processing/chap_cu3ast/sec_cu3ast_intro/ssec_cu3ast_intro_tansforms.html.
As a result, we obtain a distribution for each value of the logistic
regression model ($\beta_{m}$, $m=0,1,2,3$) as shown in Fig. 12. In these
plots we have included the median value for each coefficient in black, which
is taken as the best-fit value, as well as the results if we fit the original
sample without taking the errors into account.

Figure 12: Final distributions for the logistic regression coefficients after
$10^{4}$ Monte Carlo realizations. The solid black line corresponds to the
median value adopted as the best-fit, while the red dashed line corresponds to
the value obtained from the fit of the original variables without taking
errors into account.

For the classification of the TTS candidates, we perform an analogous Monte
Carlo analysis and generate again 104 samples taking into account the errors,
and for each realization we assign a membership probability. The values given
in the main text are the median values of the probabilities of the $10^{4}$
random samples.