分类： 天文学 >> 天文学 提交时间： 2023-02-19
Hailong Yuan
Zhenwei Li
Zhongrui Bai
Yiqiao Dong
Mengxin Wang
Sicheng Yu
Xuefei Chen
Yongheng Zhao
Yaoquan Chu
Haotong Zhang
摘要： The Extremely Low Mass White Dwarfs (ELM WDs) and pre-ELM WDs are helium core white dwarfs with mass $<\sim 0.3M_{\odot}$. Evolution simulations show that a lower mass limit for ELM WDs exists at $\approx0.14M_{\odot}$ and no one is proposed by observation to be less massive than that. Here we report the discovery of a binary system, LAMOST J224040.77-020732.8 (J2240 in short), which consists of a very low mass hot star and a compact companion. Multi-epoch spectroscopy shows an orbital period $P_{orb} =$0.219658$\pm0.000002$ days and a radial velocity semi-amplitude $K1=318.5\pm3.3km/s$, which gives the mass function of 0.74$M_{\odot}$, indicating the companion is a compact star. The F-type low resolution spectra illustrate no emission features, and the temperature ($\sim 7400K$) is consistent with that from Spectral Energy Distribution fitting and multi-color light curve solution. The optical light curves, in ZTF g, r and i bands and Catalina V band, show ellipsoidal variability with amplitudes $\sim30\%$, suggesting that the visible component is heavily tidally distorted. Combining the distance from Gaia survey, the ZTF light curves are modeled with Wilson-Devinney code and the result shows that the mass of the visible component is $M1=0.085^{+0.036}_{-0.024}M_{\odot}$, and the mass of the invisible component is $M2=0.98^{+0.16}_{-0.09}M_{\odot}$. The radius of the visible component is $R1=0.29^{+0.04}_{-0.03}R_{\odot}$. The inclination angle is approximately between 60$^{\circ}$ and 90$^{\circ}$. The observations indicate the system is most likely a pre-ELM WD + WD/NS binary, and the mass of pre-ELM is possibly lower than the $0.14M_{\odot}$ theoretical limit.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： The stability criteria of rapid mass transfer and common envelope evolution are fundamental in binary star evolution. They determine the mass, mass ratio and orbital distribution of many important systems, such as X-ray binaries, Type Ia supernovae and merging gravitational wave sources. We use our adiabatic mass-loss model to systematically survey the intermediate-mass stars' thresholds for dynamical-timescale mass transfer. The impact of metallicity on the stellar responses and critical mass ratios is explored. Both tables ($Z=0.001$) and fitting formula ($Z=0.001$ and $Z=0.02$) of critical mass ratios of intermediate-mass stars are provided. An application of our results to intermediate-mass X-ray binaries (IMXBs) is discussed. We find that the predicted upper limit to mass ratios, as a function of orbital period, is consistent with the observed IMXBs that undergo thermal or nuclear timescale mass transfer. According to the observed peak X-ray luminosity $L_\mathrm{X}$, we predict the range of $L_\mathrm{X}$ for IMXBs as a function of the donor mass and the mass transfer timescale.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： The evolution and stability of mass transfer of CO+He WD binaries are not well understood. Observationally they may emerge as AM CVn binaries and are important gravitational wave (GW) emitters. In this work, we have modeled the evolution of double WD binaries with accretor masses of $0.50 - 1.30\;M_{\odot}$ and donor masses of $0.17\; - 0.45\;M_{\odot}$ using the detailed stellar evolution code MESA. We find that the evolution of binaries with same donor masses but different accretor masses is very similar and binaries with same accretor masses but larger He donor masses have larger maximum mass transfer rates and smaller minimum orbital periods. We also demonstrate that the GW signal from AM CVn binaries can be detected by space-borne GW observatories, such as LISA, TianQin. And there is a linear relation between the donor mass and gravitational wave frequency during mass transfer phase. In our calculation, all binaries can have dynamically stable mass transfer, which is very different from previous studies. The threshold donor mass of Eddington-limited mass transfer for a given accretor WD mass is lower than previous studies. Assuming that a binary may enter common envelope if the mass transfer rate exceeds the maximum stable burning rate of He, we provide a new criterion for double WDs surviving mass transfer, which is below the threshold of Eddington-limit. Finally, we find that some systems with ONe WDs in our calculation may evolve into detached binaries consisting of neutron stars (NSs) and extremely low mass He WDs and further ultra-compact X-ray binaries.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Mass transfer stability is an essential issue in binary evolution. Ge et al. studied critical mass ratios for dynamically stable mass transfer by establishing adiabatic mass loss model and found that the donor stars on the giant branches tend to be more stable than that based on the composite polytropic stellar model. We would investigate the influence of mass transfer stability on the formation and properties of DWD populations. We performed a series of binary population synthesis, where the critical mass ratios from adiabatic mass loss model (Ge's model) and that from the composite polytropic model are adopted, respectively. For Ge's model, most of the DWDs are produced from the stable non-conservative Roche lobe overflow plus common envelope (CE) ejection channel (RL+CE channel) regardless of the CE ejection efficiency $\alpha_{CE}$. While the results of the polytropic model strongly depend on the adopted value of $\alpha_{ CE}$. We find DWDs produced from the RL+CE channel have comparable WD masses and the mass ratio distribution peaks at around 1. Based on the magnitude-limited sample of DWDs, the space densities for the detectable DWDs and those with extremely low-mass WD (ELM WD) companions in Ge's model is $1347$ and $473 kpc^{-3}$, respectively, close to observations. While the polytropic model overpredicts space density of DWDs by a factor of about $2-3$. We also find that the results of DWD merger rate distribution in Ge's model reproduce the observations better than that of the polytropic model, and the merger rate of DWDs with ELM WD companions in the Galaxy is about $1.8\times 10^{-3} yr^{-1}$ in Ge's model, which is comparable to the observation estimation. We confirm that the mass transfer stability plays important roles in the formation and properties of DWD populations, and then in the progenitors of SNe Ia and detectable GW sources.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
Hongwei Ge
Christopher A Tout
Xuefei Chen
Matthias U Kruckow
Hailiang Chen
Dengkai Jiang
Zhenwei Li
Zhengwei Liu
Zhanwen Han
摘要： Common envelope evolution (CEE) physics plays a fundamental role in the formation of binary systems, such as mergering stellar gravitational wave sources, pulsar binaries and type Ia supernovae. A precisely constrained CEE has become more important in the age of large surveys and gravitational wave detectors. We use an adiabatic mass loss model to explore how the total energy of the donor changes as a function of the remnant mass. This provides a more self-consistent way to calculate the binding energy of the donor. For comparison, we also calculate the binding energy through integrating the total energy from the core to the surface. The outcome of CEE is constrained by total energy conservation at the point at which both component's radii shrink back within their Roche lobes. We apply our results to 142 hot subdwarf binaries. For shorter orbital period sdBs, the binding energy is highly consistent. For longer orbital period sdBs in our samples, the binding energy can differ by up to a factor of 2. The CE efficiency parameter $\beta_\mathrm{CE}$ becomes smaller than $\alpha_\mathrm{CE}$ for the final orbital period $\log_{10} P_{\mathrm{orb}}/\mathrm{d} > -0.5$. We also find the mass ratios $\log_{10} q$ and CE efficiency parameters $\log_{10} \alpha_{\mathrm{CE}}$ and $\log_{10} \beta_{\mathrm{CE}}$ linearly correlate in sdBs, similarly to De Marco et al. (2010) for post-AGB binaries.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
Hailong Yuan
Zhenwei Li
Zhongrui Bai
Yiqiao Dong
Mengxin Wang
Sicheng Yu
Xuefei Chen
Yongheng Zhao
Yaoquan Chu
Haotong Zhang
摘要： The Extremely Low Mass White Dwarfs (ELM WDs) and pre-ELM WDs are helium core white dwarfs with mass $<\sim 0.3M_{\odot}$. Evolution simulations show that a lower mass limit for ELM WDs exists at $\approx0.14M_{\odot}$ and no one is proposed by observation to be less massive than that. Here we report the discovery of a binary system, LAMOST J224040.77-020732.8 (J2240 in short), which consists of a very low mass hot star and a compact companion. Multi-epoch spectroscopy shows an orbital period $P_{orb} =$0.219658$\pm0.000002$ days and a radial velocity semi-amplitude $K1=318.5\pm3.3km/s$, which gives the mass function of 0.74$M_{\odot}$, indicating the companion is a compact star. The F-type low resolution spectra illustrate no emission features, and the temperature ($\sim 7400K$) is consistent with that from Spectral Energy Distribution fitting and multi-color light curve solution. The optical light curves, in ZTF g, r and i bands and Catalina V band, show ellipsoidal variability with amplitudes $\sim30\%$, suggesting that the visible component is heavily tidally distorted. Combining the distance from Gaia survey, the ZTF light curves are modeled with Wilson-Devinney code and the result shows that the mass of the visible component is $M1=0.085^{+0.036}_{-0.024}M_{\odot}$, and the mass of the invisible component is $M2=0.98^{+0.16}_{-0.09}M_{\odot}$. The radius of the visible component is $R1=0.29^{+0.04}_{-0.03}R_{\odot}$. The inclination angle is approximately between 60$^{\circ}$ and 90$^{\circ}$. The observations indicate the system is most likely a pre-ELM WD + WD/NS binary, and the mass of pre-ELM is possibly lower than the $0.14M_{\odot}$ theoretical limit.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： The stability criteria of rapid mass transfer and common envelope evolution are fundamental in binary star evolution. They determine the mass, mass ratio and orbital distribution of many important systems, such as X-ray binaries, Type Ia supernovae and merging gravitational wave sources. We use our adiabatic mass-loss model to systematically survey the intermediate-mass stars' thresholds for dynamical-timescale mass transfer. The impact of metallicity on the stellar responses and critical mass ratios is explored. Both tables ($Z=0.001$) and fitting formula ($Z=0.001$ and $Z=0.02$) of critical mass ratios of intermediate-mass stars are provided. An application of our results to intermediate-mass X-ray binaries (IMXBs) is discussed. We find that the predicted upper limit to mass ratios, as a function of orbital period, is consistent with the observed IMXBs that undergo thermal or nuclear timescale mass transfer. According to the observed peak X-ray luminosity $L_\mathrm{X}$, we predict the range of $L_\mathrm{X}$ for IMXBs as a function of the donor mass and the mass transfer timescale.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Neutron star-helium white dwarf (NS+He WD) binaries are important evolutionary products of close-orbit binary star systems. They are often observed as millisecond pulsars and may continue evolving into ultra-compact X-ray binaries (UCXBs) and continuous gravitational wave (GW) sources that will be detected by space-borne GW observatories, such as LISA, TianQin and Taiji. Nevertheless, the stability of NS+He WD binaries undergoing mass transfer is not well studied and still under debate. In this paper, we model the evolution of NS+He WD binaries with WD masses ranging from 0.17-0.45 $M_{\odot}$, applying the detailed stellar evolution code mesa. Contrary to previous studies based on hydrodynamics, we find that apparently all NS+He WD binaries undergo stable mass transfer. We find for such UCXBs that the larger the WD mass, the larger the maximum mass-transfer rate and the smaller the minimum orbital period during their evolution. Finally, we demonstrate numerically and analytically that there is a tight correlation between WD mass and GW frequency for UCXBs, independent of NS mass.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
Luqian Wang
Jiao Li
You Wu
Douglas R. Gies
Jin Zhong Liu
Chao Liu
Yanjun Guo
Xuefei Chen
Zhanwen Han
摘要： 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.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： We present a study of eclipsing binary J064726.39+223431.6 using spectra from the LAMOST-MRS and TESS photometry. We use full-spectrum fitting to derive radial velocities and spectral parameters: ${T_{\rm eff}}_{A,B}=6177,\,5820$ K, $v \sin{i}_{A,B}=59,\,50~\kms$ and ${\rm [Fe/H]}_{A,B}=-0.19$ dex. The orbital solution and light curve analysis suggest that it is a close pair of fast rotating stars on circular orbit. We measure their masses to be $M_{A,B}=1.307\pm0.007,\, 1.129\pm0.005\,M_\odot$ and their radii to be $R_{A,B}=1.405\pm0.052,\, 1.219\pm0.060\,R_\odot$ resulting in surface gravities of $\log{g}_{A,B}=4.259\pm0.033,\,4.319\pm0.042$ (cgs). Theoretical models cannot match all of these properties, predicting significantly higher ${T_{\rm eff}}$ for a given metallicity. Derived age of the system 1.56 Gyr indicates that both components are younger than Sun, which contradicts to much longer orbit's circularisation timescale.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Based on the two-minutes TESS data, we analyzed intrinsic oscillations of the primary component and identified seven confident independent $\delta$ Scuti frequencies ($f_1$, $f_2$, $f_3$, $f_4$, $f_7$, $f_{11}$, and $f_{12}$). Both of single-star evolutionary models and mass-accreting models are computed to reproduce the $\delta$ Scuti freqiencies. Fitting results of them match well with each other. The stellar parameters of the primary star yielded by asteroseismology are $M$ = $1.92^{+0.10}_{-0.02}$ $M_{\odot}$, $Z$ = 0.011$^{+0.006}_{-0.001}$, $R$ = $2.068^{+0.050}_{-0.007}$ $R_{\odot}$, $\log g$ = $4.090^{+0.010}_{-0.002}$, $T_{\rm eff}$ = $8346^{+244}_{-320}$ K, $L$ = $18.65^{+3.31}_{-2.82}$ $L_{\odot}$, which match well with the dynamic ones by the binary model. Furthermore, our asteroseismic results show that OO Dra is another Algol system that has just undergone the rapid mass-transfer stage. Fitting results of single-star evlutionary models indicate that the pulsator is helium-poor star with an age of 8.22$^{+0.12}_{-1.33}$ Myr, and the further mass-accreting models show that the primary star looks like an almost unevolved star formed by an extremely helium-poor mass accretion in Case A evolutionary scenario.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Double white dwarf (DWD) binaries are important for studies of common-envelope (CE) evolution, Type Ia supernova progenitors and Galactic sources of low-frequency gravitational waves (GWs). PTF J0533+0209 is a DWD system with a short orbital period of Porb ~ 20 min and thus a so-called LISA verification source. The formation of this system and other DWDs is still under debate. In this paper, we discuss the possible formation scenarios of this binary and argue that it is not likely to have formed through CE evolution. Applying a new magnetic braking prescription, we use the MESA code to model the formation of this system through stable mass transfer. We find a model which can well reproduce the observed WD masses and orbital period but not the effective temperature and hydrogen abundance of the low-mass He WD component. We discuss the possibility of using H flashes to mitigate this discrepancy. Finally, we discuss the future evolution of this system into a AM CVn binary such as those that will be detected by space-borne GW observatories like LISA, TianQin and Taiji.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： The maximum mass of neutron stars (NSs) is of great importance for constraining equations of state of NSs and understanding the mass gap between NSs and stellar-mass black holes. NSs in X-ray binaries would increase in mass by accreting material from their companions (known as recycling process), and the uncertainties in the accretion process give challenge to study the NS mass at birth. {In this work, we investigate the NS accreted mass with considering the effect of NS spin evolution and give the maximum accreted mass for NSs in the recycling process. By exploring a series of binary evolution calculations, we obtain the final NS mass and the maximum accreted mass for a given birth mass of NS and a mass transfer efficiency. Our results show that the NSs can accrete relatively more material for binary systems with the donor masses in the range of $1.8\sim 2.4M_\odot$, the NSs accrete relatively more mass when the remnant WD mass is in the range of $\sim 0.25-0.30M_\odot$, and the maximum accreted mass is positively correlated with the initial NS mass. For a $1.4M_\odot$ NS at birth with a moderate mass transfer efficiency of 0.3, the maximum accreted mass could be $0.27M_\odot$. The results can be used to estimate the minimum birth mass for systems with massive NSs in observations.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Most subdwarf B (sdB) + Helium white dwarf (He WD) binaries are believed to be formed from a particular channel. In this channel, the He WDs are produced first from red giants (RGs) with degenerate cores via stable mass transfer and sdB stars are produced from RGs with degenerate cores via common envelope (CE) ejection. They are important for the studies of CE evolution, binary evolution, and binary population synthesis. However, the relation between WD mass and orbital period of sdB + He WD binaries has not been specifically studied. In this paper, we first use a semi-analytic method to follow their formation and find a WD mass and orbital period relation. Then we use a detailed stellar evolution code to model their formation from main-sequence binaries. We find a similar relation between the WD mass and orbital period, which is in broad agreement with observations. For most sdB + He WD systems, if the WD mass (orbital period) can be determined, the orbital period (WD mass) can be inferred with this relation and then the inclination angle can be constrained with the binary mass function. In addition, we can also use this relation to constrain the CE ejection efficiency and find that a relative large CE ejection efficiency is favoured. If both the WD and sdB star masses can be determined, the critical mass ratios of dynamically unstable mass transfer for RG binaries can also be constrained.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Mass transfer stability is an essential issue in binary evolution. Ge et al. studied critical mass ratios for dynamically stable mass transfer by establishing adiabatic mass loss model and found that the donor stars on the giant branches tend to be more stable than that based on the composite polytropic stellar model. We would investigate the influence of mass transfer stability on the formation and properties of DWD populations. We performed a series of binary population synthesis, where the critical mass ratios from adiabatic mass loss model (Ge's model) and that from the composite polytropic model are adopted, respectively. For Ge's model, most of the DWDs are produced from the stable non-conservative Roche lobe overflow plus common envelope (CE) ejection channel (RL+CE channel) regardless of the CE ejection efficiency $\alpha_{CE}$. While the results of the polytropic model strongly depend on the adopted value of $\alpha_{ CE}$. We find DWDs produced from the RL+CE channel have comparable WD masses and the mass ratio distribution peaks at around 1. Based on the magnitude-limited sample of DWDs, the space densities for the detectable DWDs and those with extremely low-mass WD (ELM WD) companions in Ge's model is $1347$ and $473 kpc^{-3}$, respectively, close to observations. While the polytropic model overpredicts space density of DWDs by a factor of about $2-3$. We also find that the results of DWD merger rate distribution in Ge's model reproduce the observations better than that of the polytropic model, and the merger rate of DWDs with ELM WD companions in the Galaxy is about $1.8\times 10^{-3} yr^{-1}$ in Ge's model, which is comparable to the observation estimation. We confirm that the mass transfer stability plays important roles in the formation and properties of DWD populations, and then in the progenitors of SNe Ia and detectable GW sources.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
Jiangdan Li
Christopher A. Onken
Christian Wolf
Péter Németh
Mike Bessell
Zhenwei Li
Xiaobin Zhang
Jiao Li
Luqian Wang
Lifang Li
Yangping Luo
Hailiang Chen
Kaifan Ji
Xuefei Chen
Zhanwen Han
摘要： 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 $\sim 6\times 10^{-10} {\rm g/cm^3}$.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
Jiangdan Li
Jiao Li
Chao Liu
Chunqian Li
Yanjun Guo
Luqian Wang
Xuefei Chen
Lifeng Xing
Yonghui Hou
Zhanwen Han
摘要： 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 $\gamma$ 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 $\gamma$-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.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
Yanjun Guo
Jiao Li
Jianping Xiong
Jiangdan Li
Luqian Wang
Heran Xiong
Feng Luo
Yonghui Hou
Chao Liu
Zhanwen Han
Xuefei Chen
摘要： 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 $f(P) \propto P^\pi$ (1\textless$P$\textless1000 days) and $f(q) \propto q^\kappa$ (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 $\pi$ are -1$\pm0.1$, -1.1$\pm0.05$, -1.1$\pm0.1$, and -0.6$\pm0.05$, respectively. The $\kappa$ 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.
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