arXiv subjects

Jingkun Zhao

Publications and source records attributed to Jingkun Zhao.

At least 37 records · Page 2Linked to original sources

arXiv2025

Mapping the Milky Way with Gaia Bp/Rp spectra I: Systematic flux corrections and atmospheric parameters for 68 million stars

Gaia Bp/Rp spectra for over two hundred million stars have great potential for mapping metallicity across the Milky Way. We aim to construct an alternative catalog of atmospheric parameters from Gaia Bp/Rp spectra by fitting them with synthetic spectra based on model atmospheres, and provide corrections to the Bp/Rp fluxes according to stellar colors, magnitudes, and extinction. We use GaiaXPy to obtain calibrated spectra and apply FERRE to match the corrected Bp/Rp spectra with models and infer atmospheric parameters. We train a neural network using stars in APOGEE to predict flux corrections as a function of wavelength for each target. Based on the comparison with APOGEE parameters, we conclude that our estimated parameters have systematic errors and uncertainties in $T_{\mathrm{eff}}$, $\log g$, and [M/H] about $-38 \pm 167$ K, $0.05 \pm 0.40$ dex, and $-0.12 \pm 0.19$ dex, respectively, for stars in the range $4000 \le T_{\mathrm{eff}} \le 7000$ K. The corrected Bp/Rp spectra show better agreement with both models and Hubble Space Telescope CALSPEC data. Our correction increases the precision of the relative spectrophotometry of the Bp/Rp data from $3.2\% - 3.7\%$ to $1.2\% - 2.4\%$. Finally, we have built a catalog of atmospheric parameters for stars within $4000 \le T_{\mathrm{eff}} \le 7000$ K, comprising $68,394,431$ sources, along with a subset of $124,188$ stars with $\mathrm{[M/H]} \le -2.5$. Our results confirm that the Gaia Bp/Rp flux calibrated spectra show systematic patterns as a function of wavelength that are tightly related to colors, magnitudes, and extinction. Our optimization algorithm can give us accurate atmospheric parameters of stars with a clear and direct link to models of stellar atmospheres, and can be used to efficiently search for extremely metal-poor stars.

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arXiv2025

The Stellar Abundances and Galactic Evolution Survey (SAGES). II. Machine Learning-Based Stellar parameters for 21 million stars from the First Data Release

Stellar parameters for large samples of stars play a crucial role in constraining the nature of stars and stellar populations in the Galaxy. An increasing number of medium-band photometric surveys are presently used in estimating stellar parameters. In this study, we present a machine-learning approach to derive estimates of stellar parameters, including [Fe/H], logg, and Teff, based on a combination of medium-band and broad-band photometric observations. Our analysis employs data primarily sourced from the SAGE Survey , which aims to observe much of the Northern Hemisphere. We combine the $uv$-band data from SAGES DR1 with photometric and astrometric data from Gaia EDR3, and apply the random forest method to estimate stellar parameters for approximately 21 million stars. We are able to obtain precisions of 0.09 dex for [Fe/H], 0.12 dex for logg, and 70 K for Teff. Furthermore, by incorporating 2MASS and WISE infrared photometric and GALEX ultraviolet data, we are able to achieve even higher precision estimates for over 2.2 million stars. These results are applicable to both giant and dwarf stars. Building upon this mapping, we construct a foundational dataset for research on metal-poor stars, the structure of the Milky Way, and beyond. With the forthcoming release of additional bands from SAGE Survey such DDO51 and H-alpha, this versatile machine learning approach is poised to play an important role in upcoming surveys featuring expanded filter sets

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arXiv2025

Detection of Ubiquitous Circumbinary Matter in Hot Subdwarfs Formed from Common-Envelope Ejections

The formation of compact binary systems is largely driven by their evolution through a common envelope (CE) phase, crucial for understanding phenomena such as type Ia supernovae and black hole mergers. Despite their importance, direct observational evidence for CE material has been elusive due to the transient nature of these envelopes. Numerical simulations suggest that some envelope material may persist post-ejection. In this study, we investigate circumstellar material (CSM) surrounding hot subdwarf (sdB) stars, focusing on material ejected during the CE phase of binary evolution. We analyze Ca II K absorption lines in 727 sdB candidates from the LAMOST-LRS survey, selecting 145 stars with strong absorption features, indicating the presence of CSM. We compare the velocities of the Ca II K lines with the systemic velocities of sdB binaries, confirming that the material originates from ejected common-envelope material. The results show that the CSM persists long after the CE event, suggesting the formation of a stable, long-lived circumstellar environment around sdB stars. This study enhances our understanding of the role of CSM in post-CE evolution and provides new insights into the physical processes shaping the evolution of sdB binaries.

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arXiv2024

Moving Groups in the Solar Neighborhood with Gaia, APOGEE, GALAH, and LAMOST: Dynamical Effects Gather Gas and the Ensuing Star Formation Plays an Important Role in Shaping the Stellar Velocity Distributions

With Gaia, APOGEE, GALAH, and LAMOST data, we investigate the positional, kinematic, chemical, and age properties of nine moving groups in the solar neighborhood. We find that each moving group has a distinct distribution in the velocity space in terms of its metallicity, $α$ abundance, and age. Comparison of the moving groups with their underlying background stars suggests that they have experienced the enhanced, prolonged star formation. We infer that any dynamical effects that gathered stars as a moving group in the velocity space also worked for gas. We propose for the first time that the ensuing newborn stars from such gas inherited the kinematic feature from the gas, shaping the current stellar velocity distributions of the groups. Our findings improve the understanding of the origins and evolutionary histories of moving groups in the solar neighborhood.

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arXiv2024

Chemical abundances of 20 barium stars from the OHP spectra

Based on the high resolution and high signal-to-noise spectra, we derived the chemical abundances of 20 elements for 20 barium (Ba-) stars. For the first time, the detailed abundances of four sample stars, namely HD 92482, HD 150430, HD 151101 and HD 177304 have been analyzed. Additionally, Ba element abundance has been measured using high resolution spectra for the first time in six of the other 16 sample stars. Based on the [s/Fe] ratios, the Ba-unknown star HD 115927 can be classified as a strong Ba-star, while the Ba-likely star HD 160538 can be categorized into a mild Ba-star. Consequently, our sample comprises three strong and 17 mild Ba-stars. The light odd-Z metal elements and Fe-peak elements exhibit near-solar abundances. The [α/Fe] ratios demonstrate decreasing trends with increasing metallicity. Moreover, the abundances of n-capture elements show significant enhancements in different degrees. Using a threshold of the signed distances to the solar r-process abundance pattern ds = 0.6, we find that all of our sample stars are normal Ba-stars, indicating that the enhancements of s-process elements should be attributed to material transfer from their companions. We compare the observed n-capture patterns of sample stars with the FRUITY models, and estimate the mass of the Thermally-Pulsing Asymptotic Giant Branch stars that previously contaminated the Ba-stars. The models with low masses can successfully explain the observations. From a kinematic point of view, we note that most of our sample stars are linked with the thin disk, while HD 130255 may be associated with the thick disk.

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arXiv2024

The Stellar Abundances and Galactic Evolution Survey (SAGES) III -- The g/r/i-band Data Release

The Stellar Abundances and Galactic Evolution Survey (SAGES) is a multi-band survey that covers the northern sky area of ~12000 deg2. Nanshan One-meter Wide-field Telescope (NOWT) of Xinjiang Astronomical Observatory (XAO) carried out observations on g/r/i bands. We present here the survey strategy, data processing, catalog construction, and database schema. The observations of NOWT started in 2016 August and was completed in 2018 January, total 17827 frames were obtained and ~4600 deg2 sky areas were covered. In this paper, we released the catalog of the data in the g/r/i bands observed with NOWT. In total, there are 109,197,578 items of the source records. The catalog is the supplement for the SDSS for the bright end, and the combination of our catalog and these catalogs could be helpful for source selections for other surveys and the Milky Way sciences, e.g., white dwarf candidates and stellar flares.

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arXiv2024

Candidate Members of the VMP/EMP Disk System of the Galaxy from the SkyMapper and SAGES Surveys

Photometric stellar surveys now cover a large fraction of the sky, probe to fainter magnitudes than large-scale spectroscopic surveys, and are relatively free from the target-selection biases often associated with such studies. Photometric-metallicity estimates that include narrow/medium-band filters can achieve comparable accuracy and precision to existing low-resolution spectroscopic surveys such as SDSS/SEGUE and LAMOST. Here we report on an effort to identify likely members of the Galactic disk system among the very metal-poor (VMP; [Fe/H] $\leq$ --2) and extremely metal-poor (EMP; [Fe/H] $\leq$ --3) stars. Our analysis is based on an initial sample of $\sim11.5$ million stars with full space motions selected from the SkyMapper Southern Survey (SMSS) and Stellar Abundance and Galactic Evolution Survey (SAGES). After applying a number of quality cuts to obtain the best available metallicity and dynamical estimates, we analyze a total of $\sim$5.86 million stars in the combined SMSS/SAGES sample. We employ two techniques that, depending on the method, identify between 876 and 1,476 VMP stars (6.9%-11.7% of all VMP stars) and between 40 and 59 EMP stars (12.4%-18.3% of all EMP stars) that appear to be members of the Galactic disk system on highly prograde orbits (v$_ϕ > 150$ km/s). The total number of candidate VMP/EMP disk-like stars is 1,496, the majority of which have low orbital eccentricities, ecc $\le 0.4$; many have ecc $\le 0.2$. The large fractions of VMP/EMP stars associated with the Milky Way disk system strongly suggest the presence of an early forming ``primordial" disk.

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arXiv2024

Four-hundred Very Metal-poor Stars Studied with LAMOST and Subaru. III. Dynamically Tagged Groups and Chemodynamical Properties

Very metal-poor (VMP) stars record the signatures of early accreted galaxies, making them essential tools for unraveling the early stages of Galaxy formation. Understanding the origin of VMP stars requires comprehensive studies of their chemical compositions and kinematics, which are currently lacking. Hence, we conduct a chemodynamical analysis of 352 VMP stars selected from one of the largest uniform high-resolution VMP star samples, jointly obtained from LAMOST and Subaru. We apply a friends-of-friends clustering algorithm to the master catalog of this high-resolution sample, which consists of 5778 VMP stars. It results in 131 dynamically tagged groups with 89 associated with known substructures in the Milky Way, including Gaia-Sausage-Enceladus (GSE), Thamnos, Helmi streams, Sequoia, Wukong, Pontus, and the very metal-poor disk (VMPD). Our findings are: (i) the VMPD shows lower Zn abundances than the rest, which indicates that it could be a relic of small stellar systems; (ii) Sequoia shows moderately high r-process abundances; (iii) Helmi streams show deficiencies in carbon and light neutron-capture elements; (iv) the fraction of carbon-enhanced metal-poor stars with no enhancement in heavy elements (CEMP-no stars) seems low in the VMPD and the Helmi streams; and (v) a subgroup in GSE exhibits a very high fraction of r-process enhanced stars, with four out of five showing [Eu/Fe]> +1.0. The abundance patterns of other elements in VMP substructures largely match the whole VMP sample. We also study large-scale correlations between abundance ratios and kinematics without classifying stars into substructures, but it does not yield significant correlations once the overall chemical evolution is considered for most elements.

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arXiv2024

Variable white dwarfs in TMTS: Asteroseismological analysis of a ZZ Ceti star, TMTS J17184064+2524314

The Tsinghua University-Ma Huateng Telescope for Survey (TMTS) has been constantly monitoring the northern sky since 2020 in search of rapidly variable stars. To find variable white dwarfs (WDs), the TMTS catalog is cross-matched with the WD catalog of Gaia EDR3, resulting in over 3000 light curves of WD candidates. The WD TMTS J17184064+2524314 (hereafter J1718) is the second ZZ~Ceti star discovered among these common sources. Based on the light curves from TMTS, follow-up photometric observations, and TESS, 10 periods and 3 combination periods are detected. A rotation period of $25.12\pm0.18$ hr is derived, according to the identified rotational splitting. Our spectroscopic observation indicates that this WD belongs to DA type with $T_{\rm eff}=11,670\pm604$ K, log $g=8.16\pm0.36$, $M = 0.70\pm0.23$ M$_{\odot}$, and age=$0.51\pm0.34$ Gyr. Based on core-parameterized asteroseismological model grids ($\geqslant$ 14 million), we derive a best-fit solution of $T_{\rm eff}=11,640\pm20$ K, log $g=8.267\pm0.008$, and $M = 0.750\pm0.005$ M$_{\odot}$ for J1718, consistent with the spectral fitting results. For this WD, the corresponding carbon and oxygen abundances in the core are 0.43 and 0.57, respectively. The distance derived from the intrinsic luminosity given by asteroseismology is $64\pm15$ pc, in accord with the distance of $70.1\pm0.2$ pc from Gaia DR3 within the uncertainties.

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arXiv2023

On the Migration Origin of the Hercules Moving Group with GAIA, LAMOST, APOGEE, and GALAH Surveys

Using Gaia DR3 data and the wavelet transformation technique, we study the substructures of the Hercules moving group (HMG): Hercules 1 (H1) and Hercules 2 (H2). Spectroscopic survey data from LAMOST, APOGEE, and GALAH are used to obtain metallicities and ages of stars belonging to the HMG. Our analysis leads to several key findings as follows: ($a$) the HMG is on average richer in metallicity than the Galactic disk, with H2 being metal richer than H1; ($b$) the HMG likely has a radial metallicity gradient distinct from that of the disk; ($c$) the HMG is on average older than the disk, with H2 being older than H1; ($d$) the HMG likely has a radial age gradient distinct from that of the disk; and ($e$) the metallicity and age distributions of the HMG depend mainly on the Galactic radius but show no dependence on the azimuthal velocity. Taken all together, we conclude that the HMG is composed primarily of stars undergoing radial migration. We suggest that the HMG is associated with a higher-order dynamical resonance of the bar of the Galaxy.

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arXiv2023

High-precision chemical abundances of Galactic building blocks. II. Revisiting the chemical distinctness of the Helmi streams

Context: The Helmi streams are a kinematic substructure whose progenitor is likely a dwarf galaxy. Although 20 years have passed since their discovery, it is still unclear whether their members are chemically distinguishable from other halo stars in the Milky Way. Aim: We aim to precisely characterize the chemical properties of the Helmi streams. Methods: We analyzed high-resolution, high signal-to-noise ratio spectra for 11 Helmi stream stars through a line-by-line abundance analysis. We compared the derived abundances to homogenized literature abundances of the other halo stars, including those belonging to other kinematic substructures, such as Gaia-Enceladus and Sequoia. Results: Compared to typical halo stars, the Helmi stream members clearly show low values of [X/Fe] in elements produced by massive stars, such as Na and $α$-elements. This tendency is seen down to metallicities of at least [Fe/H]$\sim -2.2$, suggesting type~Ia supernovae already started to contribute to the chemical evolution at this metallicity. We find that the [$α$/Fe] ratio does not evolve significantly with metallicity, making the Helmi stream stars less distinguishable from Gaia-Enceladus stars at [Fe/H]$\gtrsim -1.5$. The almost constant but low value of [$α$/Fe] might be indicative of quiescent star formation with low efficiency at the beginning and bursty star formation at later times. We also find extremely low values of [Y/Fe] at low metallicity, providing further support for the claim that light neutron-capture elements are deficient in Helmi streams. While Zn is deficient at low metallicity, it shows a large spread at high metallicity. The origin of the extremely low Y abundances and Zn variations remains unclear. Conclusion: The Helmi stream stars are distinguishable from the majority of the halo stars if homogeneously derived abundances are compared.

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arXiv2023

The Stellar Abundances and Galactic Evolution Survey (SAGES) -- -- I. General Description and the First Data Release (DR1)

The Stellar Abundances and Galactic Evolution Survey (SAGES) of the northern sky is a specifically-designed multi-band photometric survey aiming to provide reliable stellar parameters with accuracy comparable to those from low-resolution optical spectra. It was carried out with the 2.3-m Bok telescope of Steward Observatory and three other telescopes. The observations in the $u_s$ and $v_s$ passband produced over 36,092 frames of images in total, covering a sky area of $\sim9960$ degree$^2$. The median survey completeness of all observing fields for the two bands are of $u_{\rm s}=20.4$ mag and $v_s=20.3$ mag, respectively, while the limiting magnitudes with signal-to-noise ratio (S/N) of 100 are $u_s\sim17$ mag and $v_s\sim18$ mag, correspondingly. We combined our catalog with the data release 1 (DR1) of the first of Panoramic Survey Telescope And Rapid Response System (Pan-STARRS1, PS1) catalog, and obtained a total of 48,553,987 sources which have at least one photometric measurement in each of the SAGES $u_s$ and $v_s$ and PS1 $grizy$ passbands, which is the DR1 of SAGES and it will be released in our paper. We compare our $gri$ point-source photometry with those of PS1 and found an RMS scatter of $\sim2$% in difference of PS1 and SAGES for the same band. We estimated an internal photometric precision of SAGES to be on the order of $\sim1$%. Astrometric precision is better than $0^{\prime\prime}.2$ based on comparison with the DR1 of Gaia mission. In this paper, we also describe the final end-user database, and provide some science applications.

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arXiv2023

Mass distribution for single-lined hot subdwarf stars in LAMOST

Masses for 664 single-lined hot subdwarf stars identified in LAMOST were calculated by comparing synthetic fluxes from spectral energy distribution (SED) with observed fluxes from virtual observatory service. Three groups of hot subdwarf stars were selected from the whole sample according to their parallax precision to study the mass distributions. We found, that He-poor sdB/sdOB stars present a wide mass distribution from 0.1 to 1.0 $\mathrm{M}_{\odot}$ with a sharp mass peak around at 0.46 $\rm{M}_{\odot}$, which is consistent with canonical binary model prediction. He-rich sdB/sdOB/sdO stars present a much flatter mass distribution than He-poor sdB/sdOB stars and with a mass peak around 0.42 $\mathrm{M}_{\odot}$. By comparing the observed mass distributions to the predictions of different formation scenarios, we concluded that the binary merger channel, including two helium white dwarfs (He-WDs) and He-WD + main sequence (MS) merger, cannot be the only main formation channel for He-rich hot subdwarfs, and other formation channels such as the surviving companions from type Ia supernovae (SNe Ia) could also make impacts on producing this special population, especially for He-rich hot subdwarfs with masses less than 0.44 $\mathrm{M}_{\odot}$. He-poor sdO stars also present a flatter mass distribution with an inconspicuous peak mass at 0.18 $\mathrm{M}_{\odot}$. The similar mass - $ΔRV_\mathrm{max}$ distribution between He-poor sdB/sdOB and sdO stars supports the scenario that He-poor sdO stars could be the subsequent evolution stage of He-poor sdB/sdOB stars.

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arXiv2023

Properties and Asteroseismological analysis of a new ZZ ceti discovered by TMTS

Tsinghua university-Ma Huateng Telescope for Survey (TMTS) aims to discover rapidly evolving transients by monitoring the northern sky. The TMTS catalog is cross-matched with the white dwarf (WD) catalog of Gaia EDR3, and light curves of more than a thousand WD candidates are obtained so far. Among them, the WD TMTS J23450729+5813146 (hereafter J2345) is one interesting common source. Based on the light curves from the TMTS and follow-up photometric observations, periods of 967.113 s, 973.734 s, 881.525 s, 843.458 s, 806.916 s and 678.273 s are identified. In addition, the TESS observations suggest a 3.39 h period but this can be attributed to the rotation of a comoving M dwarf located within 3". The spectroscopic observation indicates that this WD is DA type with Teff = 11778+/-617K,log g = 8.38+/-0.31,mass=0.84+/-0.20Msun and age=0.704+/-0.377 Gyrs. Asteroseismological analysis reveals a global best-fit solution of Teff =12110+/-10K and mass=0.760+/-0.005Msun,consistent with the spectral fitting results, and Oxygen and Carbon abundances in the core center are 0.73 and 0.27, respectively. The distance derived from the intrinsic luminosity given by asteroseismology is 93 parsec, which is in agreement with the distance of 98 parsec from Gaia DR3. Additionally, kinematic study shows that this WD is likely a thick disk star. The mass of its zero-age main-sequence mass is estimated to be 3.08 Msun and has a main-sequence plus cooling age of roughly 900 Myrs.

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arXiv2023

Detection of rubidium and samarium in the atmosphere of the ultra-hot Jupiter MASCARA-4b

Ultra-hot Jupiters (UHJs) possess the most extreme environments among various types of exoplanets, making them ideal laboratories to study the chemical composition and kinetics properties of exoplanet atmosphere with high-resolution spectroscopy (HRS). It has the advantage of resolving the tiny Doppler shift and weak signal from exoplanet atmosphere and has helped to detect dozens of heavy elements in UHJs including KELT-9b, WASP-76b, WASP-121b. MASCARA-4b is a 2.8-day UHJ with an equilibrium temperature of $\sim2250$ K, which is expected to contain heavy elements detectable with VLT. In this letter, we present a survey of atoms/ions in the atmosphere of the MASCARA-4b, using the two VLT/ESPRESSO transits data. Cross-correlation analyses are performed on the obtained transmission spectra at each exposure with the template spectra generated by petitRADTRANS for atoms/ions from element Li to U. We confirm the previous detection of Mg, Ca, Cr and Fe and report the detection of Rb, Sm, Ti+ and Ba+ with peak signal-to-noise ratios (SNRs) $>$ 5. We report a tentative detection of Sc+, with peak SNRs $\sim$6 but deviating from the estimated position. The most interesting discovery is the first-time detection of elements Rb and Sm in an exoplanet. Rb is an alkaline element like Na and K, while Sm is the first lanthanide series element and is by far the heaviest one detected in exoplanets. Detailed modeling and acquiring more data are required to yield abundance ratios of the heavy elements and to understand better the common presence of them in UHJ's atmospheres.

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arXiv2023

Thermal Emission from the hot Jupiter WASP-103b in J and Ks Bands

Hot Jupiters, particularly those with temperature higher than 2000 K are the best sample of planets that allow in-depth characterization of their atmospheres. We present here a thermal emission study of the ultra hot Jupiter WASP-103 b observed in two secondary eclipses with CFHT/WIRCam in J and Ks bands. By means of high precision differential photometry, we determine eclipse depths in J and Ks to an accuracy of 220 and 270 ppm, which are combined with the published HST/WFC3 and Spitzer data to retrieve a joint constraints on the properties of WASP-103 b dayside atmosphere. We find that the atmosphere is best fit with a thermal inversion layer included. The equilibrium chemistry retrieval indicates an enhanced C/O (1.35$^{+0.14}_{-0.17}$) and a super metallicity with [Fe/H]$=2.19^{+0.51}_{-0.63}$ composition. Given the near-solar metallicity of WASP-103 of [Fe/H]=0.06, this planet seems to be $\sim$100 more abundant than its host star. The free chemistry retrieval analysis yields a large abundance of FeH, H$^{-}$, CO$_2$ and CH$_4$. Additional data of better accuracy from future observations of JWST should provide better constraint of the atmospheric properties of WASP-103b.

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arXiv2022

Hot subdwarf stars identified in LAMOST DR8 with single-lined and composite spectra

222 hot subdwarf stars were identified with LAMOST DR8 spectra, among which 131 stars show composite spectra and have been decomposed, while 91 stars present single-lined spectra. Atmospheric parameters of all sample stars were obtained by fitting Hydrogen (H) and Helium (He) line profiles with synthetic spectra. Two long-period composite sdB binaries were newly discovered by combining our sample with the non-single star data from Gaia DR3. One of the new systems presents the highest eccentricity (i.e., 0.5 +/- 0.09) among known wide sdB binaries, which is beyond model predictions. 15 composite sdB stars fall in the high probability binary region of RUWE-AEN plane, and deserve priority follow-up observations to further study their binary nature. A distinct gap is clearly presented among temperatures of cool companions for our composite-spectra sample. But we could not come to a conclusion whether this feature is connected to the formation history of hot subdwarf stars before their binary natures are confirmed.

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arXiv2022

Revisit of open clusters UPK 39, UPK 41 and PHOC 39 : a new binary open cluster found

We investigate the three open clusters near Aquila Rift cloud, named as UPK 39 (\texttt{c1} hereafter), UPK 41 (\texttt{c2} hereafter) in Sim et al. (2019) and PHOC 39 (\texttt{c3} hereafter) in Hunt \& Reffert (2021), respectively. Using photometric passpands, reddening, and extinction from Gaia DR3, we construct the color-absolute-magnitude diagram (CAMD). Using isochrone fits their ages are estimated as $6.3\pm0.9$, $8.1\pm1.4$ and $21.8\pm2.2$ Myr, respectively. Their proper motions and radial velocities, estimated using data from Gaia and LAMOST are very similar. From their orbits, relative distances among them at different times, kinematics, ages, and metallicities, we conclude that \texttt{c1} and \texttt{c2} are primordial binary open cluster, which are likely to have been formed at the same time, and \texttt{c3} may capture \texttt{c1}, \texttt{c2} in the future.

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