arXiv subjects

Jiao Li

Publications and source records attributed to Jiao Li.

At least 55 records · Page 3Linked to original sources

arXiv2024

TYC 3340-2437-1: A Quadruple System with A Massive Star

Hierarchical massive quadruple systems are ideal laboratories for examining the theories of star formation, dynamical evolution, and stellar evolution. The successive mergers of hierarchical quadruple systems might explain the mass gap between neutron stars and black holes. Looking for light curves of O-type binaries identified by LAMOST, we find a (2+2) quadruple system: TYC 3340-2437-1, located in the stellar bow-shock nebula (SBN). It has a probability of over 99.99\% being a quadruple system derived from the surface density of the vicinity stars. Its inner orbital periods are 3.390602(89) days and 2.4378(16) days, respectively, and the total mass is about (11.47 + 5.79) + (5.2 + 2.02) = 24.48 $M_{\odot}$. The line-of-sight inclinations of the inner binaries, B$_1$ and B$_2$, are 55.94 and 78.2 degrees, respectively, indicating that they are not co-planar. Based on observations spanning 34 months and the significance of the astrometric excess noise ($D>2$) in Gaia DR3 data, we guess that its outer orbital period might be a few years. If it were true, the quadruple system might form through the disk fragmentation mechanism with outer eccentric greater than zero. This eccentricity could be the cause of both the arc-like feature of the SBN and the noncoplanarity of the inner orbit. The outer orbital period and outer eccentric could be determined with the release of future epoch astrometric data of Gaia.

astro-ph.SR↗

arXiv2023

The Common Envelope Evolution Outcome. II. Short Orbital Period Hot Subdwarf B Binaries Reveal a Clear Picture

The common envelope evolution (CEE) is vital in forming short orbital period compact binaries. It covers many objects, such as double compact merging binaries, type Ia supernovae progenitors, binary pulsars, and X-ray binaries. Knowledge about the common envelope (CE) eject efficiency still needs to be improved, though progress has been made recently. Short orbital period hot subdwarf B star plus white dwarf binaries are the most straightforward samples to constrain CEE physics. We apply the known orbital period-white dwarf relation to constrain the sdB progenitor of seven sdB+WD binaries with a known inclination angle. The average value of the CE efficiency parameter is 0.32, which is consistent with previous studies. However, the CE efficiency might not be a constant but is a function of the initial mass ratio based on well-constrained sdB progenitor mass and evolutionary stage. Our results can be used as physical inputs for binary population synthesis simulations on related objects. A similar method can also be applied to study other short orbital period WD binaries.

astro-ph.SR↗

arXiv2023

PSF-based Analysis for Detecting Unresolved Wide Binaries

Wide binaries play a crucial role in analyzing the birth environment of stars and the dynamical evolution of clusters. When wide binaries are located at greater distances, their companions may overlap in the observed images, becoming indistinguishable and resulting in unresolved wide binaries, which are difficult to detect using traditional methods. Utilizing deep learning, we present a method to identify unresolved wide binaries by analyzing the point-spread function (PSF) morphology of telescopes. Our trained model demonstrates exceptional performance in differentiating between single stars and unresolved binaries with separations ranging from 0.1 to 2 physical pixels, where the PSF FWHM is ~2 pixels, achieving an accuracy of 97.2% for simulated data from the Chinese Space Station Telescope. We subsequently tested our method on photometric data of NGC 6121 observed by the Hubble Space Telescope. The trained model attained an accuracy of 96.5% and identified 18 wide binary candidates with separations between 7 and 140 au. The majority of these wide binary candidates are situated outside the core radius of NGC 6121, suggesting that they are likely first-generation stars, which is in general agreement with the results of Monte Carlo simulations. Our PSF-based method shows great promise in detecting unresolved wide binaries and is well suited for observations from space-based telescopes with stable PSF. In the future, we aim to apply our PSF-based method to next-generation surveys such as the China Space Station Optical Survey, where a larger-field-of-view telescope will be capable of identifying a greater number of such wide binaries.

astro-ph.SR↗

arXiv2023

Chemical reaction mechanism of pre-curing process of two-component adhesive based on deformation behavior for automobile hood

Shearing test is carried out on the joint which bonded under different pre curing processes with two component adhesives of acrylic and epoxy resin respectively. The pre curing strength is obtained, which used to analyze the relationship between the pre curing strength and time. The hoods with different pre curing strength are baking with high temperature. The deformation of different areas of the hood is measured with gauges, and the deformation characteristics of the hood after baking are acquired with comparative analysis. Combining the components of two component adhesives and DSC test, the pre curing mechanism of different adhesive systems are studied. Therefore, the key factors and regular pattern of the deformation for the hood are obtained. The results indicate that, finally deformation of the hood after high temperature baking varies with the pre curing time. The key of the pre curing time of acrylic adhesives lies in the chain initiation stage of the free radical polymerization reaction. Due to the influence of the chemical properties of methyl acrylate and its initiator, the pre curing reaction induced by free radical polymerization is very fast. The shear strength of this joint can reach to 3.67 MPa with a pre curing time of 1 h, which quickly achieving the pre cure strength required for deformation control. For epoxy adhesives, the rate determining step in pre curing process is the esterification reaction. Due to the influence of the structure of carboxylic acid, the esterification process is relatively long. Shear strength of this joint can only reach 0.87 MPa after pre curing for 4 h without external heating.

physics.chem-ph↗

arXiv2023

Survey on Controlable Image Synthesis with Deep Learning

Image synthesis has attracted emerging research interests in academic and industry communities. Deep learning technologies especially the generative models greatly inspired controllable image synthesis approaches and applications, which aim to generate particular visual contents with latent prompts. In order to further investigate low-level controllable image synthesis problem which is crucial for fine image rendering and editing tasks, we present a survey of some recent works on 3D controllable image synthesis using deep learning. We first introduce the datasets and evaluation indicators for 3D controllable image synthesis. Then, we review the state-of-the-art research for geometrically controllable image synthesis in two aspects: 1) Viewpoint/pose-controllable image synthesis; 2) Structure/shape-controllable image synthesis. Furthermore, the photometrically controllable image synthesis approaches are also reviewed for 3D re-lighting researches. While the emphasis is on 3D controllable image synthesis algorithms, the related applications, products and resources are also briefly summarized for practitioners.

cs.CV↗

arXiv2023

Graph based Label Enhancement for Multi-instance Multi-label learning

Multi-instance multi-label (MIML) learning is widely applicated in numerous domains, such as the image classification where one image contains multiple instances correlated with multiple logic labels simultaneously. The related labels in existing MIML are all assumed as logical labels with equal significance. However, in practical applications in MIML, significance of each label for multiple instances per bag (such as an image) is significant different. Ignoring labeling significance will greatly lose the semantic information of the object, so that MIML is not applicable in complex scenes with a poor learning performance. To this end, this paper proposed a novel MIML framework based on graph label enhancement, namely GLEMIML, to improve the classification performance of MIML by leveraging label significance. GLEMIML first recognizes the correlations among instances by establishing the graph and then migrates the implicit information mined from the feature space to the label space via nonlinear mapping, thus recovering the label significance. Finally, GLEMIML is trained on the enhanced data through matching and interaction mechanisms. GLEMIML (AvgRank: 1.44) can effectively improve the performance of MIML by mining the label distribution mechanism and show better results than the SOTA method (AvgRank: 2.92) on multiple benchmark datasets.

cs.CV↗

arXiv2023

Stellar Initial Mass Function Varies with Metallicities and Time

Most structural and evolutionary properties of galaxies strongly rely on the stellar initial mass function (IMF), namely the distribution of the stellar mass formed in each episode of star formation. As the IMF shapes the stellar population in all stellar systems, it turns out to become one of the most fundamental concepts of modern astronomy. Both constant and variable IMFs across different environments have been claimed despite a large number of theoretical and observational efforts. However, the measurement of the IMF in Galactic stellar populations has been limited by the relatively small number of photometrically observed stars, leading to high uncertainties. Here we report a star-counting result based on ~93,000 spectroscopically observed M-dwarf stars, an order of magnitude more than previous studies, in the 100--300 parsec (pc) Solar neighbourhood. We find unambiguous evidence of a variable IMF that depends on both metallicity and stellar age. Specifically, the stellar population formed at the early time contains fewer low-mass stars compared to the canonical IMF, independent of stellar metallicities. In present days, on the other hand, the proportion of low-mass stars increases with stellar metallicity. The variable abundance of low-mass stars in our Milky Way establishes a powerful benchmark for models of star formation and can heavily impact results in Galactic chemical enrichment modelling, mass estimation of galaxies, and planet formation efficiency.

astro-ph.GA↗

arXiv2022

LAMOST medium-resolution spectroscopic survey of binarity and exotic star (LAMOST-MRS-B): Observation strategy and target selection

LAMOST-MRS-B is one of the sub-surveys of LAMOST medium-resolution (R~7500) spectroscopic survey. It aims at studying the statistical properties (e.g., binary fraction, orbital period distribution, mass ratio distribution) of binary stars and exotic stars. We intend to observe about 30000 stars (10 mag <= G <= 14.5 mag) with at least 10 visits in five years. We first planned to observe 25 plates around the galactic plane in 2018. Then the plates were reduced to 12 in 2019 because of the limitation of observation. At the same time, two new plates located at the high galactic latitude were added to explore binary properties influenced by the different environments. In this survey project, we set the identified exotic and low-metallicity stars with the highest observation priorities. For the rest of the selected stars, we gave higher priority to the relatively brighter stars in order to obtain high-quality spectra as many as possible. Spectra of 49129 stars have been obtained in LAMOST-MRS-B field and released in DR8, of which 28828 and 3375 stars have been visited more than twice and ten times with SNR >= 10, respectively. Most of the sources are B-, A-, and F-type stars with 0.6 < [Fe/H] < 0.4 dex. We also obtain 347 identified variable and exotic stars and about 250 stars with [Fe/H] < 1 dex. We measure radial velocities (RVs) by using 892233 spectra of the stars. The uncertainties of RV achieve about 1 km/s and 10 km/s1 for 95% of late- and early-type stars, respectively. The datasets presented in this paper are available at http://www.doi.org/10.57760/sciencedb.j00113.00035.

astro-ph.SR↗

arXiv2022

The Eclipsing Binaries from the LAMOST Medium-resolution Survey.III. A High-precision Empirical Stellar Mass Library

High-precision stellar mass and radius measured directly from binaries can effectively calibrate the stellar models. However, such a database containing full spectral types and large range of metallicity is still not fully established. A continuous effort of data collecting and analysis are requested to complete the database. In this work, we provide a catalog containing 184 binaries with independent atmospheric parameters and accurate masses and radii as the benchmark of stellar mass and radius. The catalog contains 56 new detached binaries from LAMOST Medium-resolution spectroscopic (MRS) survey and 128 detached eclipsing binaries compiled from previous studies. We obtain the orbital solutions of the new detached binaries with uncertainties of masses and radii smaller than 5%. These new samples densify the distribution of metallicity of the high-precision stellar mass library and add 9 hot stars with Teff>8000 K. Comparisons show that these samples well agree with the PARSEC isochrones in Teff-logg-mass-radius-luminosity space. We compare mass and radius estimates from isochrone and SED fitting, respectively, with those from the binary orbital solution. We find that the precision of the stellar-model dependent mass estimates is >10% and the precision of the radius estimates based on atmospheric parameters is >15%. These give a general view of the uncertainty of the usual approaches to estimate stellar mass and radius.

astro-ph.SR↗

arXiv2022

A Roche Lobe-filling hot Subdwarf and White Dwarf Binary: possible detection of an ejected common envelope

Binaries consisting of a hot subdwarf star and an accreting white dwarf (WD) are sources of gravitational wave radiation at low frequencies and possible progenitors of type Ia supernovae if the WD mass is large enough. Here, we report the discovery of the third binary known of this kind: it consists of a hot subdwarf O (sdO) star and a WD with an orbital period of 3.495 hours and an orbital shrinkage of 0.1 s in 6 yr. The sdO star overfills its Roche lobe and likely transfers mass to the WD via an accretion disk. From spectroscopy, we obtain an effective temperature of $T_{\mathrm{eff}}=54\,240\pm1\,840$ K and a surface gravity of $\log{g}=4.841\pm0.108$ for the sdO star. From the light curve analysis, we obtain a sdO mass of $M_{\mathrm{sdO}}=0.55$ ${\mathrm{M_{\odot}}}$ and a mass ratio of $q=M_{\mathrm{WD}}/M_{\mathrm{sdO}}=0.738\pm0.001$. Also, we estimate that the disk has a radius of $\sim 0.41R_\odot$ and a thickness of $\sim 0.18R_\odot$. The origin of this binary is probably a common envelope ejection channel, where the progenitor of the sdO star is either an RGB star or, more likely, an early AGB star; the sdO star will subsequently evolve into a WD and merge with its WD companion, likely resulting in an R CrB star. The outstanding feature in the spectrum of this object is strong Ca H&K lines, which are blueshifted by $\sim$200 km/s and likely originate from the recently ejected common envelope, and we estimated that the remnant CE material in the binary system has a density $\sim 6\times 10^{-10} {\rm g/cm^3}$.

astro-ph.SR↗

arXiv2022

Mass-Ratio Distribution of Binaries From the LAMOST-MRS Survey

Binary evolution leads to the formation of important objects crucial to the development of astrophysics, but the statistical properties of binary populations are still poorly understood. The LAMOST-MRS has provided a large sample of stars to study the properties of binary populations, especially for the mass ratio distributions and the binary fractions. We have devised a Peak Amplitude Ratio (PAR) approach to derive the mass ratio of a binary system based on results obtained from its spectrum. By computing a cross-correlation function (CCF), we established a relationship between the derived mass ratio and the PARs of the binary systems. By utilizing spectral observations obtained from LAMSOT DR6 & DR7, we applied the PAR approach to form distributions of the derived mass ratio of the binary systems to the spectral types. We selected the mass ratio within the range of $0.6-1.0$ for investigating the mass-ratio distribution. Through a power-law fitting, we obtained the power index $γ$ values of $-0.42\pm0.27$, $0.03\pm0.12$, and $2.12\pm0.19$ for A-, F-, and G-type stars identified in the sample, respectively. The derived $γ$-values display an increasing trend toward lower primary star masses, and G-type binaries tend to be more in twins. The close binary fractions (for $P\lesssim 150\,{\rm d}$ and $q\gtrsim 0.6$) in our sample for A, F and G binaries are $7.6\pm 0.5 \%$, $4.9\pm 0.2 \%$ and $3.7 \pm 0.1 \%$, respectively. Note that the PAR approach can be applied to large spectroscopic surveys of stars.

astro-ph.SR↗

arXiv2022

Identification of new classical Be stars from the LAMOST MRS survey

Be stars are B-type main-sequence stars that display broad Balmer emission lines in their spectra. Identification of Be population is essential to further examine the formation and evolutionary models. We report the detection of classical Be (CBe) stars from observations with the Large sky Area Multi-Object fiber Spectroscopic Telescope Medium Resolution Survey of Date Release 7 (LAMOST MRS DR7). We used a deep convolutional neural network, the ResNet, with an 18-layer module to examine the morphology of the H alpha profile. We identified 1,162 candidate Be stars from the collection of 2,260,387 spectra for 789,918 stars in the database. The ResNet network achieves a Be star classification accuracy of 99.5%. Among the detections, 151 of these are prior known Be stars cross-matched from the literature. By applying a three-step test, we identified 183 new CBe stars. We find that 41 CBe stars are members of known open clusters. Based upon an investigation of the kinematics of the identified CBe stars from the Gaia EDR3 astrometric solutions, we identified 16 new runaways. These new identifications will provide a reference for future follow-ups to further investigate their physical properties.

astro-ph.SR↗

arXiv2022

Sentence Alignment with Parallel Documents Facilitates Biomedical Machine Translation

Objective: Today's neural machine translation (NMT) can achieve near human-level translation quality and greatly facilitates international communications, but the lack of parallel corpora poses a key problem to the development of translation systems for highly specialized domains, such as biomedicine. This work presents an unsupervised algorithm for deriving parallel corpora from document-level translations by using sentence alignment and explores how training materials affect the performance of biomedical NMT systems. Materials and Methods: Document-level translations are mixed to train bilingual word embeddings (BWEs) for the evaluation of cross-lingual word similarity, and sentence distance is defined by combining semantic and positional similarities of the sentences. The alignment of sentences is formulated as an extended earth mover's distance problem. A Chinese-English biomedical parallel corpus is derived with the proposed algorithm using bilingual articles from UpToDate and translations of PubMed abstracts, which is then used for the training and evaluation of NMT. Results: On two manually aligned translation datasets, the proposed algorithm achieved accurate sentence alignment in the 1-to-1 cases and outperformed competing algorithms in the many-to-many cases. The NMT model fine-tuned on biomedical data significantly improved the in-domain translation quality (zh-en: +17.72 BLEU; en-zh: +17.02 BLEU). Both the size of the training data and the combination of different corpora can significantly affect the model's performance. Conclusion: The proposed algorithm relaxes the assumption for sentence alignment and effectively generates accurate translation pairs that facilitate training high quality biomedical NMT models.

cs.CL↗

arXiv2021

The Spectroscopic Binaries from LAMOST Medium-Resolution Survey (MRS). I. Searching for Double-lined Spectroscopic Binaries (SB2s) with Convolutional Neural Network

We developed a convolutional neural network (CNN) model to distinguish the double-lined spectroscopic binaries (SB2s) from others based on single exposure medium-resolution spectra ($R\sim 7,500$). The training set consists of a large set of mock spectra of single stars and binaries synthesized based on the MIST stellar evolutionary model and ATLAS9 atmospheric model. Our model reaches a novel theoretic false positive rate by adding a proper penalty on the negative sample (e.g., 0.12\% and 0.16\% for the blue/red arm when the penalty parameter $Λ=16$). Tests show that the performance is as expected and favors FGK-type Main-sequence binaries with high mass ratio ($q \geq 0.7$) and large radial velocity separation ($Δv \geq 50\,\mathrm{km\,s^{-1}}$). Although the real false positive rate can not be estimated reliably, validating on eclipsing binaries identified from Kepler light curves indicates that our model predicts low binary probabilities at eclipsing phases (0, 0.5, and 1.0) as expected. The color-magnitude diagram also helps illustrate its feasibility and capability of identifying FGK MS binaries from spectra. We conclude that this model is reasonably reliable and can provide an automatic approach to identify SB2s with period $\lesssim 10$ days. This work yields a catalog of binary probabilities for over 5 million spectra of 1 million sources from the LAMOST medium-resolution survey (MRS), and a catalog of 2198 SB2 candidates whose physical properties will be analyzed in our following-up paper. Data products are made publicly available at the journal as well as our Github website.

astro-ph.SR↗

arXiv2021

The Binarity of Early-type Stars from LAMOST Medium-resolution Spectroscopic Survey

Massive binaries play significant roles in many fields. Identification of massive stars, particularly massive binaries, is of great importance. In this paper, by adopting the technique of measuring the equivalent widths of several spectral lines, we identified 9,382 early-type stars from LAMOST medium-resolution survey and divided the sample into four groups, T1 ($\sim$O-B4), T2 ($\sim$B5), T3 ($\sim$B7), and T4 ($\sim$B8-A). The relative radial velocities $RV_{\rm rel}$ were calculated using the Maximum Likelihood Estimation. The stars with significant changes of $RV_{\rm rel}$ and at least larger than 15.57km s$^{-1}$ were identified as spectroscopic binaries. We found that the observed spectroscopic binary fractions for the four groups are $24.6\%\pm0.5\%$, $20.8\%\pm0.6\%$, $13.7\%\pm0.3\%$, and $7.4\%\pm0.3\%$, respectively. Assuming that orbital period ($P$) and mass ratio ($q$) have intrinsic distributions as $f(P) \propto P^π$ (1\textless$P$\textless1000 days) and $f(q) \propto q^κ$ (0.1\textless$q$\textless1), respectively, we conducted a series of Monte-Carlo simulations to correct observational biases for estimating the intrinsic multiplicity properties. The results show that the intrinsic binary fractions for the four groups are 68$\%\pm8\%$, 52$\%\pm3\%$, 44$\%\pm6\%$, and 44$\%\pm6\%$, respectively. The best estimated values for $π$ are -1$\pm0.1$, -1.1$\pm0.05$, -1.1$\pm0.1$, and -0.6$\pm0.05$, respectively. The $κ$ cannot be constrained for groups T1 and T2 and is -2.4$\pm0.3$ for group T3 and -1.6$\pm0.3$ for group T4. We confirmed the relationship of a decreasing trend in binary fractions towards late-type stars. No correlation between the spectral type and the orbital period distribution has been found yet, possibly due to the limitation of observational cadence.

astro-ph.SR↗

arXiv2021

The Early-type Stars from LAMOST survey: Atmospheric parameters

Massive stars play key roles in many astrophysical processes. Deriving atmospheric parameters of massive stars is important to understand their physical properties and thus are key inputs to trace their evolution. Here we report our work on adopting the data-driven technique Stellar LAbel Machine ({\tt SLAM}) with the non-LTE TLUSTY synthetic spectra as the training dataset to estimate the stellar parameters of LAMOST optical spectra for early-type stars. We apply two consistency tests to verify this machine learning method and compare stellar labels given by {\tt SLAM} with that in literature for several objects having high-resolution spectra. We provide the stellar labels of effective temperature ($T_\mathrm{eff}$), surface gravity ($\log{g}$), metallicity ([M/H]), and projected rotational velocity ($v\sin{i}$) for 3,931 and 578 early-type stars from LAMOST Low-Resolution Survey (LAMOST-LRS) and Medium-Resolution Survey (LAMOST-MRS), respectively. To estimate the average statistical uncertainties of our results, we calculated the standard deviation between the predicted stellar label and the pre-labeled published values from the high-resolution spectra. The uncertainties of the four parameters are $σ(T_\mathrm{eff}) = 2,185 $K, $σ(\log{g}) = 0.29$ dex, and $σ(v\sin{i}) = 11\, \rm km\,s^{-1}$ for MRS, and $σ(T_\mathrm{eff}) = 1,642 $K, $σ(\log{g}) = 0.25$ dex, and $σ(v\sin{i}) = 42\, \rm km\,s^{-1}$ for LRS spectra, respectively. We notice that parameters of $T_\mathrm{eff}$, $\log{g}$ and [M/H] can be better constrained using LRS spectra rather than using MRS spectra, most likely due to their broad wavelength coverage, while $v\sin{i}$ is constrained better by MRS spectra than by LRS spectra, probably due to the relatively accurate line profiles of MRS spectra.

astro-ph.SR↗

arXiv2021

LAMOST Time-Domain Survey: First Results of four $K$2 plates

From Oct. 2019 to Apr. 2020, LAMOST performs a time-domain spectroscopic survey of four $K$2 plates with both low- and med-resolution observations. The low-resolution spectroscopic survey gains 282 exposures ($\approx$46.6 hours) over 25 nights, yielding a total of about 767,000 spectra, and the med-resolution survey takes 177 exposures ($\approx$49.1 hours) over 27 nights, collecting about 478,000 spectra. More than 70%/50% of low-resolution/med-resolution spectra have signal-to-noise ratio higher than 10. We determine stellar parameters (e.g., $T_{\rm eff}$, log$g$, [Fe/H]) and radial velocity (RV) with different methods, including LASP, DD-Payne, and SLAM. In general, these parameter estimations from different methods show good agreement, and the stellar parameter values are consistent with those of APOGEE. We use the $Gaia$ DR2 RV data to calculate a median RV zero point (RVZP) for each spectrograph exposure by exposure, and the RVZP-corrected RVs agree well with the APOGEE data. The stellar evolutionary and spectroscopic masses are estimated based on the stellar parameters, multi-band magnitudes, distances and extinction values. Finally, we construct a binary catalog including about 2700 candidates by analyzing their light curves, fitting the RV data, calculating the binarity parameters from med-resolution spectra, and cross-matching the spatially resolved binary catalog from $Gaia$ EDR3. The LAMOST TD survey is expected to get breakthrough in various scientific topics, such as binary system, stellar activity, and stellar pulsation, etc.

astro-ph.SR↗

arXiv2021

Binary fraction of O and B-type stars from LAMOST data

Binary stars plays important role in the evolution of stellar populations . The intrinsic binary fraction ($f_{bin}$) of O and B-type (OB) stars in LAMOST DR5 was investigated in this work. We employed a cross-correlation approach to estimate relative radial velocities for each of the stellar spectra. The algorithm described by \cite{2013A&A...550A.107S} was implemented and several simulations were made to assess the performance of the approach. Binary fraction of the OB stars are estimated through comparing the uni-distribution between observations and simulations with the Kolmogorov-Smirnov tests. Simulations show that it is reliable for stars most of whom have $6,7$ and $8$ repeated observations. The uncertainty of orbital parameters of binarity become larger when observational frequencies decrease. By adopting the fixed power exponents of $π=-0.45$ and $κ=-1$ for period and mass ratio distributions, respectively, we obtain that $f_{bin}=0.4_{-0.06}^{+0.05}$ for the samples with more than 3 observations. When we consider the full samples with at least 2 observations, the binary fraction turns out to be $0.37_{-0.03}^{+0.03}$. These two results are consistent with each other in $1σ$.

astro-ph.SR↗