Characterisation of high velocity stars in the S-PLUS internal fourth data release

In general, the atypical high velocity of some stars in the Galaxy can only be explained by invoking acceleration mechanisms related to extreme astrophysical events in the Milky Way. Using astrometric data from Gaia and the photometric information in 12 filters of the S-PLUS, we performed a kinematic, dynamical, and chemical analysis of 64 stars with galactocentric velocities higher than 400 $\mathrm{km\,s}^{-1}$. All the stars are gravitationally bound to the Galaxy and exhibit halo kinematics. Some of the stars could be remnants of structures such as the Sequoia and the Gaia-Sausage/Enceladus. Supported by orbital and chemical analysis, we identified Gaia DR3 5401875170994688896 as a star likely to be originated at the centre of the Galaxy. Application of a machine learning technique to the S-PLUS photometric data allows us to obtain very good estimates of magnesium abundances for this sample of high velocity stars.


INTRODUCTION
After the second data release of the Gaia mission (Gaia DR2), Marchetti et al. (2019) reported stars with high velocities, including some with a high probability of escaping the Milky Way (MW).In Furthermore, the search for high-velocity stars extended to the subset of stars lacking complete phase-space data in Gaia's records.In this case, the astrometric information is complemented with radial velocities from spectroscopic surveys (Du et al. 2018a,b;Li et al. 2021;Quispe-Huaynasi et al. 2022;Li et al. 2023).The benefit of having stars in common between Gaia and spectroscopic surveys lies in the ability to employ chemical data to restrict their origin, using chemical tagging (Freeman & Bland-Hawthorn 2002).
As expected, interest in this population of stars has grown in recent years with the advent of Gaia.
But the interest is not new.For example, some of the earliest studies related to high-velocity stars include the work of Oort (1922Oort ( , 1924Oort ( , 1926Oort ( , 1930)).These stars were identified by their high radial velocity or high proper motion, and several mechanisms were proposed for their origin (Tutukov & Fedorova 2009).Among the main mechanisms, we may cite: the ejection of stars at high velocity due to the rupture of a binary system by tidal forces exerted by the black hole at the centre of the Galaxy (Hills mechanism, Hills 1988); Type II and Type Ia supernova explosion in binary systems (Blaauw 1961;Geier et al. 2015); and gravitational interaction in dense systems as young clusters (Poveda et al. 1967).Depending on whether the velocity of these stars surpasses or falls short of (Starkenburg et al. 2017), J-PLUS (Cenarro et al. 2019), or miniJPAS (Bonoli et al. 2021), among others, equipped with filters strategically positioned on distinctive absorption and emission lines, are being employed to acquire stellar parameters for a vast array of stars and to identify peculiar astronomical objects for subsequent spectroscopic investigations.
In this work, we use deep learning algorithms and exploit the photometric information in 12 filters from the S-PLUS (Southern Photometric Local Universe Survey, Mendes de Oliveira et al.

2019
)1 , to estimate effective temperature  eff , surface gravity log , and metallicity [Fe/H], for 64 high-velocity stars, as well as magnesium abundance [Mg/Fe], for the stars with [Fe/H] > -1, with the aim of constraining their possible origins.
The article is structured as follows: the data set considered for our analysis is described in Section 2. Section 3 describes the high-velocity stars selection process.The kinematic and dynamical analysis, based on the stars' orbital parameters, dynamical variables, and kinematic spaces, is described in Section 4. The determination of stellar parameters using deep learning algorithms is presented in Section 5. Finally, in Section 6, we present the conclusions.

DATA SOURCES
The selection and kinematic characterisation of the sample of high-velocity stars have been performed using only the information provided by the Gaia astrometric mission (Gaia Collaboration et al. 2016).The sample has been selected from the sample of stars with full phase-space information in the third release of data (Gaia DR3, Gaia Collaboration et al. 2022).
Effective temperature, surface gravity and metallicity for the high velocity stars were estimated using deep learning algorithms trained with sources in common between the S-PLUS and the LAMOST (Large Sky Area Multi-Object Fiber Spectroscopic Telescope, Cui et al. 2012), and also with sources in common between the S-PLUS and the APOGEE (Apache Point Observatory Galactic Evolution Experiment Data Release, Abdurro'uf et al. 2022) 2 .Specifically, to train and test the machine learning algorithms, we use data from the APOGEE DR17, and the Stellar Parameter Catalog of A, F, G, and K Stars, determined from low resolution spectra ( = 1 800), provided in LAMOST DR8.S-PLUS is a photometric survey that collects data using the 12 optical filters listed in Table 1.
Central wavelengths are located in important regions of emission and absorption lines, that provide valuable information about galaxies and stellar populations.The survey is being conducted from the Cerro Tololo Inter-American Observatory, in Chile, using a 0.8 m robotic telescope (T80-South) equipped with a camera (T80Cam) of 9.2k × 9.2k pixels, that provides a 2 deg 2 field of view (FoV), with a pixel scale of 0.55 arsec pix −1 .Observational strategy, image reduction, and main scientific goals are presented in Mendes de Oliveira et al. (2019).The S-PLUS iDR4 (internal Data Release 4), used in this study, includes 1 629 fields, covering about 3 000 square degrees of the southern sky, reduced and calibrated in all the survey bands.
Along the text, we identify the stars using their Gaia ID rather than their S-PLUS ID, because this facilitates the cross-match with other surveys.

HIGH VELOCITY STARS SELECTION
Most of the stars with high velocity in the past were not identified by total space velocity, but by their high radial velocity or high proper motions.This was mainly due to the lack of information on stellar distances.However, with the Gaia data releases providing parallaxes and proper motions for a large number of stars, the search for stars with high velocity using total velocity became possible.In this work, we select stars with galactocentric velocity  GC greater than 400 km s −1 .We take this limit to avoid the high velocity tail of the velocity distribution of disc stars.The distances used to compute the  GC are the photo-geometric distances estimated by Bailer-Jones et al. ( 2021), using a Bayesian approach.Our choice of this photo-geometric distances instead of the geometric distances (also estimated in Bailer-Jones et al. 2021), is because they are more precise, especially for distant sources.But even if we consider the  GC calculated using the geometric distances, it would not change the conclusions of this work.On the other hand, because  GC is very sensitive to stellar distances, we discuss in the Appendix A the effect of the photo-astrometric distances estimated by Anders et al. (2022) on the values of  GC for our sample.
In order to select sources with a reliable astrometric and photometric parameters in Gaia, we only consider stars with positive parallax  > 0, and stars flagged with the following labels in Gaia DR3: ruwe < 1.4 and −3 < astrometric_gof_al < 3, to ensure good astrometric solutions and to avoid binary systems, respectively.We also consider sources with fidelity_v2 > 0.5, classified as good astrometric solution according to Rybizki et al. (2022).Additionally, using norm_dg < −3, defined in Rybizki et al. (2022), we remove stars with potential colour contamination from nearby sources3 .
The transformation from the International Celestial Reference System (ICRS) coordinates to the Galactocentric reference frame (GRF) was performed using the Pyia package (Price-Whelan 2018).The parameters of the transformation are the ones set by default in version 4.0 of the Astropy Galactocentric frame: Sun's distance to the GC 8.122 kpc (Gravity Collaboration et al. 2018), Sun's height over the Galactic mid-plane 20.8 pc (Bennett & Bovy 2019), and Sun's Cartesian velocity around the GC (12.9, 245.6, 7.78) km s −1 (Drimmel & Poggio 2018).The uncertainty propagation of the transformation is performed using 1 000 Monte Carlo (MC) realisations, selected from multivariate normal distributions, N (, Σ), where  = (, , ,   ,   ,  rad ) are the observables and Σ is the covariance matrix provided by Gaia4 .Finally, we consider the median and the 16th and 84th percentiles over the distributions to get the final positions and velocities in the GRF.
After this process, we cross-match that sample with the S-PLUS iDR4 catalogue, resulting in a set of 64 stars.From this sample, all the stars have fidelity_v2 ∼ 1, therefore with a good astrometric solution.On the other hand, as warned by Babusiaux et al. (2022), sources with grvs_mag − phot_g_mean_mag < −3 may present problems in radial velocity measurements due to nearby bright sources.We verify that all stars in our sample have these values > −3.
Figure 1 shows the Galactocentric velocity as a function of the Galactocentric radial distance  GC , in spherical coordinates.The solid lines in blue, red, and green in the figure represent the escape velocity curves calculated using the model I potential described in Irrgang et al. (2013), the potential of McMillan ( 2017), and the MWPotential2014 potential of Bovy (2015), respectively.
The green dashed line is the escape velocity curve assuming a higher mass of ∼ 1.2 × 10 12  ⊙ for the halo in the MWPotential2014 potential (MWPotential2014_heavy), since its default value is underestimated compared to more recent estimates given in the literature (Bland-Hawthorn & Gerhard 2016;Wang et al. 2020).The yellow dots are stars with /  ≥ 10 (sources with good quality parallax measurements), the rest of the dots are stars with /  < 10 (sources with poor parallax).Except for the star Gaia DR3 2690227738799604224, with /  ∼ 6, all other stars are bound to the Galaxy according to the potentials considered (taking into account that the halo mass for the MWPotential2014 is underestimated).Looking at the astrometric parameters of the unbound candidate, we verify that the contribution to its high Galactocentric velocity comes from a radial velocity of 492 km s −1 .As high radial velocities in Gaia DR3 can represent spurious measurements due to very low signal to noise ratio (rv_expected_sig_to_noise, SNR), as explained in Katz et al. (2023), we verify that this star has rv_expected_sig_to_noise = 3.2 < 5, and therefore there is a high chance of its radial velocity being spurious.It is worth mentioning that this is the most metal-poor star in the sample, thus the star with the weakest spectral lines.
To check if the stars in the sample have radial velocity information in other catalogues, we performed a cross-match with spectroscopic catalogues.Unfortunately, the unbound candidate star  has no spectrum available in archives of observed data, and there is currently no way to obtain a better estimate of its radial velocity.But we found radial velocity information for 7 stars in GALAH DR3, and for 4 stars in the LAMOST DR8 LR catalogue.Astrometric parameters from Gaia and radial velocities from spectroscopic surveys are reported in Table 2. Comparing the radial velocities from spectroscopic surveys with the ones provided by Gaia DR3, we find that the mean difference is ∼ −1 km s −1 , with a dispersion of 7 km s −1 , showing a good agreement between the different catalogues.
Table 2 shows that the spectra of some targets have very low SNR, and therefore no reliable radial velocity in Gaia.In this case, the typical approach involves a cutoff in the SNR, but the threshold in SNR is not well defined.For instance, if we set a SNR > 5, following Marchetti et al.
(2022b), we would exclude various stars, including Gaia DR3 3753526229559762176 and Gaia DR3 3696393857329932672, whose radial velocities in Gaia are consistent with radial velocities from spectroscopic surveys.Therefore, we will not adopt any cut in SNR, but conclusions about these stars will be discussed in the light of their stellar parameters in Section 5.5.After these considerations, our final sample consists of 64 stars, and hereafter we will refer to it as the HiVel sample.
Orbital parameters, kinematic variables and dynamical variables are fundamental to understand the kinematic nature of stars.This section describes how the orbital parameters are calculated and then presents three of the most important kinematic spaces for the HiVel sample.

Orbit integration
With position and velocity information, calculated from the astrometric parameters given by Gaia, and considering a gravitational potential for the Galaxy, we calculate the orbital parameters applying backwards orbital integration over 10 Gyr in the past, using the Galpy package (Bovy 2015) 5 .The orbit integration is performed for each of the MC realisations of each star (described in the previous section), and the final values of the parameters are given by the median, the 16th and 84th percentiles.We adopted the gravitational potential of McMillan (2017).This model is composed of a bulge, a dark matter halo, a thin disc, a thick disc, and an HI and molecular gas disc.
All the components are axisymmetric and are implemented in Galpy.
The left panel of Fig. 2  We also investigate the possible influence of the Galactic bar in the orbital properties derived for our sample, which might be specially important for orbits crossing the disc close to the centre.We integrate orbits for the same stars, but in the barred potential of Sormani et al. (2022, also Hunter et al. in prep.).This potential is a good fit to both the inner and outer parts of the MW potential and it is available in the AGAMA (Action-based Galaxy Modelling Architecture, Vasiliev 2019) repository6 .
We assume a pattern speed Ω  = 35.5 km s −1 kpc −1 , a bar inclination angle of 30 • , and again we integrate for −10 Gyr.From the analysis of the orbital trajectories in this gravitational potential, we see that the distance from the GC to the point of the last disc crossing, for the two stars that might be originated in the GC, is ∼ 1 kpc, as in the previous simulations without a bar.In Section 5.5, we will re-discuss the possible origin of these two stars taking into account their position in the [Mg/Fe] vs. [Fe/H] diagram.

LMC origin
Since the ejection mechanisms mentioned before can also operate in the Large Magellanic Cloud (LMC), stars with high velocities may have formed in this system (Przybilla et al. 2008;Lennon et al. 2017;Erkal et al. 2019).To determine if any of the stars in the HiVel sample might have come from the LMC, we calculate the orbit of the LMC as a test particle under dynamical friction.To compute the effect of dynamical friction, we consider the mass of 1.5 × 10 11  ⊙ for the LMC, as adopted in Erkal et al. (2019), together with the McMillan17 potential, and the half-mass radius (rhm) of 6 kpc in the dynamical friction formula implemented in Galpy (ChandrasekharDynamicalFrictionForce).
The rhm value was selected in order to match the orbit of the LMC to the orbit given by Vasiliev (2023).However, as detailed in that paper, the trajectory of the LMC is very sensitive to the mass of the MW, the mass of the LMC, the shape of the halo, and uncertainties in the present-day position and velocity measurements.In this work, the phase-space information for the LMC was calculated using astrometric data from Gaia Collaboration et al. (2021) and the distance given by Pietrzyński et al. (2019).To determine the probability of the origin of a star in the LMC, we calculate the mutual distance  star−LMC between the orbit of the star and the orbit of the LMC at each integration step, and check if they have a close encounter,  star−LMC < 5 kpc, in the past.This is done for the 1 000 MC realisations of each star.We find that  star−LMC > 5 kpc in all cases, and conclude that all the stars have a very low probability of having been ejected from the LMC.

Kinematical spaces
Three of the most commonly used kinematic spaces in the literature to understand the kinematic nature of stars are described below for the HiVel sample.Integrals of motion used to construct these kinematic planes were calculated with Galpy, using the McMillan's gravitational potential.
The first kinematic space is the Toomre diagram (upper panel of Fig. 3).Considering that the F. Quispe-Huaynasi et al.
integrals of motion calculated by assuming a gravitational potential for the MW, the Toomre space is built purely from velocity information, considering the galactocentric velocities in cylindrical coordinates.The azimuthal galactocentric velocity (  ) is plotted on the x-axis and the quantity √︃  2  +  2  is plotted on the y-axis.Because the stars in the disc have nearly circular orbits and low velocity dispersion, they lie in the shaded region defined for stars with | GC −  LSR | < 210 km s −1 , with  LSR = 232 km s −1 being the velocity of the Local Standard of Rest (LSR), adopted from Koppelman et al. (2018).On the other hand, with a larger contribution of the vertical and radial velocities, the halo stars will be outside this region, | GC −  LSR | > 210 km s −1 .It is worth noting, however, that the boundary between these two populations is not well defined.With the exception of a few stars, which lie close to the separation boundary between these two populations, the majority of our HiVel stars clearly exhibit the kinematic behaviour of the halo stars.This is further supported by the fact that the distribution of   has approximately equal numbers of prograde and retrograde stars.
The second kinematical space (left bottom panel of Fig. 3) is the energy vs. angular momentum plane, also known as Lindblad diagram.The two quantities ,   used in this diagram, are constants in axisymmetric potentials.This space concentrates disc and halo stars into specific regions.Unlike halo stars, which are mainly located in regions of low angular momentum and high orbital energy, disc stars will occupy regions of positive angular momentum (  > 0) and moderate orbital energy (Lane et al. 2022).However, as in Toomre's diagram, there is a region of overlap between these two populations.With complete phase space data from a substantial sample of stars, this space was also used to identify stellar structures in the Galactic halo (Helmi 2020).However, the regions occupied by these structures in the Lindblad diagram are not well defined, and some of them may overlap (see Naidu et al. 2020;Bonaca et al. 2021).We can see that some HiVel stars are in regions expected for structures such as Sequoia (magenta shaded area), that has stars with retrograde motion, and the Gaia-Enceladus/Sausage (GES) structure (green shaded region), which is composed of stars with prograde and retrograde motion.The borders of the shaded regions were extracted from Koppelman et al. (2019).As expected from the previous section, star Gaia DR3 2690227738799604224 has positive orbital energy ( > 0), indicating an unbound candidate.
The third kinematical space is the so-called action diamond (right-bottom panel of Fig. 3).The actions   ,   ,   are integrals of motion of the system.In axisymmetric potentials, in particular,   =   , but   and   need to be estimated numerically.The actions for our HiVel stars have been calculated using the Stäckel approximation (Binney 2012).This diagram allows to segregate stellar populations (Lane et al. 2022), and to identify substructures in the Galactic halo (Myeong et al. 2019;Naidu et al. 2020).As shown in Fig. 3, stars with eccentric orbits tend to group in the lower corner of the diamond, while stars with retrograde and prograde motions concentrate in the left and right corners, respectively.The region in magenta in this diagram corresponds to the location of the remnants of the Sequoia, with highly retrograde stars, and the region in green corresponds to the location of the remnants of the GES merger, with stars in highly eccentric orbits (Myeong et al. 2019).The limits of these regions are not well defined and we adopt the ones given in Monty et al. (2020).We can see that some stars of the HiVel sample lie within or very close to these two regions.In principle, a detailed analysis of the chemical abundances may help to confirm whether these stars belong to these populations, although this is still an open problem (Koppelman et al. 2019;Aguado et al. 2021;Horta et al. 2023)

Parameters determined by Convolutional Neural Network
With the development and implementation of machine learning algorithms, and access to large amounts of data from space telescopes and ground-based telescopes in astronomy, the application in the different areas of astronomy was expected (see Djorgovski et al. 2022;Smith & Geach 2023, and references therein).Here we describe the process to obtain stellar parameters for the HiVel sample using a Convolutional Neural Network (CNN) architecture, the photometric information in 12 filters from S-PLUS, and the stellar parameters from LAMOST.CNN is a deep learning algorithm frequently used in computer vision.However, CNNs can also be used for one-dimensional sequential data (Kiranyaz et al. 2021).As an application in astronomy, Gebran et al. (2022, and references therein) shows how CNN architectures can be used for the determination of stellar parameters through stellar spectra.
Since the order of the features is important in the CNN architecture for pattern recognition, we order the magnitudes according to increasing wavelength.The labels of the data (output parameters) are the values of  eff , [Fe/H], and log .

Data training and testing
The data set used to train and test the CNN comes from the data in common between S-PLUS iDR4 and LAMOST DR8 (∼ 60 000 stars).Before this process we perform the following cuts: (i) For the S-PLUS data, we consider stars with apparent magnitudes mag_auto < 20 and associated uncertainties mag_err_auto < 0.15, in all the photometric bands.This is a compromise between the quality and the quantity of the data for training and testing the CNN.In particular, it guarantees enough training/testing data in the [Fe/H] < −1 region (∼ 3 500 stars).Imposing a more strict condition, like mag_err_auto < 0.05, reduces the data sample in the [Fe/H] < −1 region by about 20%.
(ii) For the LAMOST data, we select stars with temperatures in the range 3 500 ≤  eff ≤ 8 000 K, with uncertainties of < 150 K.In addition, we consider sources with uncertainties < 0.2 dex both in log  and [Fe/H].
By applying these quality cuts, we obtain a sample consisting of about 55 000 stars, of which 80% is used to train and 20% to test the CNN.
The initial CNN architecture for training the algorithm is described in Quispe-Huaynasi et al. (2023).We use three one-dimensional convolutional layers to build the features map from the input data.A one-dimensional maxpooling layer is associated to each of the convolutional layers, in order to reduce the number of inputs that will feed the neural network.The network itself is composed of an input layer, three hidden layers, and an output layer.The Relu activation function is used in both the convolutional layers and the network layers.We select the Adam optimisation algorithm (Kingma & Ba 2014), and the mean absolute error (mae) as loss function.
The final hyperparameters of the CNN, learning rate, number of kernels, kernel size, and number of neurons in each hidden layer are obtained via the keras tuner (O'Malley et al. 2019), using the Bayesian search optimiser.

Model testing
The horizontal axes of the three panels in Fig. 4 are the values of  eff , [Fe/H], and log  provided by LAMOST for the test sample, while the vertical axes show the corresponding values predicted by the CNN model.Overall, the stellar parameters show a good agreement.The mean values of the offsets are close to zero in all the three cases, and the -scatter is 115 K, 0.2 dex and 0.2 dex, in the CNN−LAMOST direction, respectively.
Additionally, in order to test the CNN with an external catalogue, we apply the model to ∼ 8 000 stars that are found in common between the S-PLUS iDR4 and the APOGEE DR17 catalogues.
The comparison of the parameters given by the CNN and those given by APOGEE is illustrated in Fig. 5.In general, there is a good agreement between the parameters.However, for  eff ≲ 4 500 K and  eff ≳ 5 500 K, there is a systematic offset.For [Fe/H] the agreement is good, but with higher dispersion for [Fe/H] ≲ −1.For log , the agreement is good over the whole range, although  there is a slight systematic offset.Probably, an important contribution to the observed systematic offsets is related to the systematic effects between APOGEE and LAMOST (a discussion on these discrepancies found in previous data releases is given in Anguiano et al. 2018).

Comparison with SPHINX
Here we compare the results of the CNN to the results obtained with the SPHINX artificial neural network (Whitten et al. 2019).The current model of SPHINX (Whitten et al. 2021) has been trained using colours and magnitudes from the 12 filter photometry of S-PLUS.In particular, the model applied here is based on the S-PLUS iDR4.The training labels were taken from a combination of LAMOST DR7 and the SEGUE DR9 (Sloan Extension for Galactic Understanding  The application of the current SPHINX model to our HiVel sample produces the results reported in Table 3.The comparison with the CNN-estimated values is shown in Fig. 6, for sources with metallicity uncertainties < 0.3 dex and with uncertainty information in SPHINX.In the case of the effective temperature (left panel), we see that there is a slight positive offset of 10 K, with a dispersion of 100 K.In the case of the [Fe/H] (right panel), we clearly see a systematic offset of 0.1 dex, and a dispersion of 0.2 dex in the CNN−SPHINX direction.

The Hertzsprung-Russell diagram
In Fig. 7, we plot the Kiel diagram (left panel) using the CNN-estimated stellar parameters reported in Table 3, and the CMD diagram (right panel) using Gaia photometry.These diagrams aim to identify the evolutionary stage of the HiVel stars.The absolute magnitude in the CMD is calculated using the G-band apparent magnitudes, corrected by extinction using the SFD2D dust map (Schlafly & Finkbeiner 2011), through the dustmaps package (Green 2018)  The fact that some stars are inconsistent with the corresponding isochrones shown in Fig. 7 may be related to the large dispersion in the inferred log  values.On the other hand, according to their position in the [Mg/Fe] vs. [Fe/H] diagram, four stars are located between the thin disc and the halo.Two stars are in the highest density region, with a chance of being thin disc stars.This, in addition to the kinematic analysis, would imply in low mass disc stars with halo kinematics, which is something unexpected.Analysing the stellar properties, we see that in the H-R diagram these stars are located in the low mass main-sequence, with a G apparent magnitude around 15 and low signal to noise (SNR < 5).Therefore, it is possible that these stars have a spurious radial velocity, and consequently a fake halo star kinematic behaviour.However, a spectroscopic study would be required to confirm the radial velocities given by Gaia.
The remaining stars in our sample, with [Fe/H] < −1, can be classified as metal-poor stars, which is a characteristic expected of halo stars.Knowledge of -element abundances is essential to determine whether halo stars in the −1.8 < [Fe/H] < −0.8 dex metallicity range have an extra-galactic origin due to the merger of extra-galactic stellar systems with the MW (so called accreted stars, characterised by a low -elements abundance), or if they formed locally within the MW (so called in situ stars, characterised by a high -elements abundance).The limit between accreted and in situ stars is approximately given by [/Fe] = −0.15[Fe/H] + 0.07 (Nissen & Schuster 2010).Using this limit together with the -element values reported in Table 5, most of the HiVel stars fall in the region of accreted stars.Note, however, that this limit may be slightly off due to systematic differences between the spectra used by Nissen & Schuster (2010) and those from other spectroscopic surveys.
It is important to note that the analysis presented in this section is based on results obtained using machine learning algorithms and medium/low resolution spectroscopic surveys.Therefore, to confirm the results, it is mandatory to perform a chemical abundance analysis using high resolution spectra.

Figure 1 .
Figure 1.Galactocentric velocity as a function of Galactocentric distance, coded according to /  .
shows the maximum height relative to the galactic plane  max , as a function of eccentricity .It is clear from the figure that the orbits are eccentric ( > 0.5) and that most stars have  max > 5 kpc.To evaluate a possible origin at the centre of the Galaxy, the right panel of the Fig.2plots the relation between the orbital energy  and the distance from the GC to the point of last disc crossing  dc (i.e. the galactocentric distance when  = 0 for the last time).We see that the stars Gaia DR3 3619582352969797888, with  dc < 1 kpc, and Gaia DR3 5401875170994688896, with  dc = 1.4 kpc, have the highest probability among the sample of having originated in the GC.The vertical dashed line represents the boundary between the bound ( < 0) and unbound ( > 0) stars.

Figure 2 .Figure 3 .
Figure 2. Left: Maximum height above the Galactic plane as a function of eccentricity.Right: Galactocentric distance to the point of the last crossing through the disc (R dc ,  = 0) as a function of orbital energy.The vertical line shows the limit between bound and unbound stars, and the horizontal line corresponds to 1 kpc.The colour code is the same as in Fig. 1.

Figure 4 .
Figure 4. Comparison between the stellar parameters predicted by the CNN and the parameters given by the LAMOST catalogue.The colour scale represents the density of points.

Figure 5 .
Figure 5.Comparison of the stellar parameters predicted by the CNN ans the parameters given by the APOGEE catalogue.The colour scale represents the density of points.

Figure 6 .
Figure 6.Comparison between the effective temperature (left panel) and the metallicity (right panel) using the CNN and SPHINX.

Figure 7 .
Figure 7. Location of the high velocity stars in the Kiel diagram (left), using the CNN values, and in the colour-magnitude diagram (right), using the Gaia magnitudes.The colour lines are isochrones computed for different metallicities (see text for details).The only unbound candidate star (Gaia DR3 2690227738799604224) is shown in orange.The two stars with a probable origin in the GC, identified through kinematic analysis, are denoted by a black dot (Gaia DR3 5401875170994688896) and a red dot (Gaia DR3 3619582352969797888).The colour code is the same as in Fig. 1.

aFigure 8 .
Figure 8.The left panel shows the comparison between the [Mg/Fe] ratio predicted by the ANN and the ratio given by APOGEE, for the test sample.The middle panel shows the density plot of the [Mg/Fe] vs. [Fe/H] distribution of the test sample, predicted by ANN.The right panel shows the [Mg/Fe] vs. [Fe/H] distribution, predicted by the ANN, for the APOGEE sample (density plot) and for the HiVel sample (dots).Stars with probable origin in the GC are denoted by a black dot (Gaia DR3 5401875170994688896) and a red dot (Gaia DR3 3619582352969797888).The dashed lines indicate the approximate boundary between the thin disc (lower region) and the thick disc (upper region).

Table 1 .
S-PLUS photometric bands

Table 2 .
Astrometric parameters from the Gaia DR3.The last column reports the radial velocity from other spectroscopic surveys.
a GALAH DR3 catalogue b LAMOST DR8 LR catalogue

Table 3 .
Stellar parameters derived with our CNN and with SPHINX

Table 4 .
Stellar radii of the high velocity stars.