Mining for Candidates of Galactic Stellar-mass Black Hole Binaries with LAMOST
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Mining for Candidates of Galactic Stellar-mass Black Hole Binaries with LAMOST
Tuan Yi, Mouyuan Sun, and Wei-Min Gu Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, P. R. China (Received; Revised; Accepted)
ABSTRACTWe study the prospects of searching for black hole (BH) binary systems with a stellar-mass BH anda non-compact visible companion, by utilizing the spectroscopic data of
Large Sky Area Multi-ObjectFiber Spectroscopic Telescope (LAMOST). We simulate the Galactic BH binary population and deter-mine its optical visibility by considering the stellar synthetic population model and the distributionsof binary orbital parameters. By convolving the visibility of BH binaries with the LAMOST detectionsensitivity, we predict that (cid:38)
400 candidate BH binaries can be found by the low-resolution, non-time-domain survey, and ∼ ∼
47% have a mass function (the lower limit of the BH mass) largerthan 3 M (cid:12) . By complementing the LAMOST spectroscopic data with other photometric/spectroscopicsurveys or follow-up observations, these candidates could be confirmed. Therefore, by exploring theLAMOST data, we can enlarge the sample of dynamically confirmed BH binaries significantly, whichcan improve our understanding of the mass distribution of BHs and the stellar evolution model. Keywords: binaries: general — stars: black holes — stars: kinematics and dynamics — radial velocities INTRODUCTIONStellar-mass black holes (BHs) are the ultimate fate of massive stars at the end of their life. According to stellarevolution model, there are around 10 -10 stellar-mass BHs reside in the Galaxy (van den Heuvel 1992; Brown & Bethe1994; Timmes et al. 1996; Agol et al. 2002). A significant amount of these dark objects are thought to exist in binarysystems (i.e., a system with a BH and a non-compact companion star). So far, only about 60 of such systems werefound (Corral-Santana et al. 2016). Most of the candidates are interacting X-ray binaries (McClintock, & Remillard2006; Remillard & McClintock 2006), i.e., the BH can accrete gas from its closely orbiting donor companion.If the BH and its non-compact companion are detached (Karpov, & Lipunov 2001; Yungelson et al. 2006), namely,they are still gravitationally bounded but the companion is not filling the Roche-lobe, no mass-transfer-induced X-rayemissions will arise. Examples of discovered isolated and detached BH binaries are rare (Casares et al. 2014; Minniti etal. 2015; Giesers et al. 2018), since these systems are hard to detect through the X-ray window. Recently, the potentialof detecting BHs by the mission of Gaia was discussed intensively (Breivik et al. 2017; Mashian & Loeb 2017; Yalinewichet al. 2018; Yamaguchi et al. 2018).
Gaia provides a dedicated way for hunting BHs by directly resolving the orbitastrometric signature (Gaia Collaboration et al. 2016). Mashian & Loeb (2017) estimated that 2 × BHs could behunted by
Gaia , while Breivik et al. (2017) predicted a roughly consistent result of 3800-12000 BHs by synthetic stellarevolution simulation. Later, Yamaguchi et al. (2018) raised a key point that the effects of interstellar extinction oughtto be considered, which leads to a more moderate estimation of discovering 200-1000 BHs. Another feasible way toidentify short period candidates is looking for photometric data that present characteristics of microlensing and tidal
Corresponding author: Wei-Min [email protected] a r X i v : . [ a s t r o - ph . S R ] O c t Yi et al.
Table 1.
The distributions adopted in this paper.
Quantity Distribution Reference M ∗ ( M (cid:12) ) Initial Mass Function (IMF, subscript 0 means ‘initial’):Ψ M ( M ∗ ) ∝ (cid:40) ( M ∗ / . − . M (cid:12) (cid:54) M ∗ < . M (cid:12) ( M ∗ / . − . M (cid:12) (cid:54) M ∗ (cid:54) M (cid:12) Kroupa (2001) M BH ( M (cid:12) ) BH mass distributions: fiducial model(a) Ψ M BH ( M BH ) = (cid:110) A ( M BH ) n + (cid:2) B ( M BH ) − n + C ( M BH ) − n (cid:3) − (cid:111) /n ¨Ozel et al. (2010)where: n = − . A ( M BH ) = 4 . − . M BH + 0 . M , ¨Ozel et al. (2012) B ( M BH ) = 14 .
24 exp( − . M BH ), C ( M BH ) = 3 .
322 exp( − . M BH )BH mass distributions: alternative model(b) Ψ M BH ( M BH ) ∝ exp( − kM BH ) Fryer & Kalogera (2001) a ( R (cid:12) ) Logarithmically-flat distributed separation: Abt (1983)Ψ A ( a ) ∝ /a , 3 R (cid:12) (cid:54) a (cid:54) R (cid:12) i Randomly-distributed orbital orientation: − Ψ I ( i ) = Constant , (cid:54) i (cid:54) π/ n ( r, θ, φ ) n ( r, θ, φ ) = n [exp( − r sin φ + R Sun . − r cos φ + Z Sun . ) +0 .
04 exp( − r sin φ + R Sun . ) − r cos φ + Z Sun . ] Mashian & Loeb (2017)where: n ≈ − , R Sun = 8 kpc, Z Sun = 25 pc distortion effects (Wyrzykowski et al. 2016; Masuda & Hotokezaka 2018). From the spectroscopic perspective, thesystem of a BH plus a non-compact companion is a single-lined spectroscopic binary (SB1). The stellar spectra provideinformation of stellar parameters and radial velocities ( V R ) for the visible companion. The mass function equation,derived from the radial velocity curve of the bright star, puts a robust lower limit to the mass of the unseen star (e.g.Casares & Jonker 2014). If the orbital inclination is obtained through fitting the photometric light curves by syntheticmodels (e.g. Orosz, & Hauschildt 2000; Beer, & Podsiadlowski 2002), and the binary mass ratio constrained through,for instance, resolving the rotational broadening of the photospheric lines from the visible companion (e.g. Marsh etal. 1994), the mass of the unseen object can be well determined.The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST, Su & Cui 2004; Cui et al. 2012) is aunique astronomical instrument that has 4000 optical fibers assembled in the focal plane, allowing the spectrographsto take thousands of spectra simultaneously. The LAMOST Experiment for Galactic Understanding and Exploration(LEGUE; Deng et al. 2012) survey is mobilized with highly automated pipelines for sky subtraction (Bai et al. 2017b),cosmic ray removal (Bai et al. 2017a), data reduction and calibration (Song et al. 2012; Luo et al. 2014), and stellarparameters determination (Wu et al. 2011, 2014; Xiang et al. 2015). Since its first data release in 2013 (Luo etal. 2015), the LAMOST spectroscopic survey has provided more than 10 million stellar spectra to date, making itthe largest stellar spectral database ever. The ongoing LAMOST medium-resolution survey has released data ofaround 3,124,000 stellar spectra with S/N R >
10 at its first and second observing campaign. The medium-resolutionspectrographs have spectral resolution of R ∼ http://dr7.lamost.org/ ining black holes with LAMOST
31 km s − for S/N R ∼
20 (Liu et al. 2019). In the future, the ongoing LAMOST medium-resolution time-domaincampaign will provide high-quality multi-epoch spectroscopic observations, i.e., each target will have more than 60exposures. Thus the LAMOST spectroscopic database is an invaluable resource for mining BH binaries through thespectroscopic point of view, i.e., with radial velocity measurements that provide dynamical constraints for binarysystems.In this paper, we focus on the prospects of mining candidate BH binaries with LAMOST. This manuscript isorganized as follows. In Section 2, we present our model. In Section 3, we show our results. Summary and discussionsare made in Section 4. THEORETICAL MODELIn order to estimate the number of detectable BH candidates with LAMOST, we must solve the following threequestions:(a)
Population : how many BH binary systems with stellar companions reside in our Galaxy?(b)
Observability : how many of these BH binaries are within the detection limits of LAMOST?(c)
Identifiability : how many of these BH binaries have a signature of being identified as candidates?The first question concerns the population and the evolution paths of the BH binaries. We start with some definitionsthat describe the orbital parameters of a binary system: M min =0 . M (cid:12) : minimum mass of a hydrogen burning star. M max =100 M (cid:12) : maximal mass of a star that remains stable under the balance of stellar gravity and its radiationpressure. M ( M (cid:12) ): mass of the primary, the most massive component in the initially formed binary system. M ( M (cid:12) ): mass of the companion, the least massive component in the initially formed binary system ( M min (cid:54) M (cid:54) M (cid:54) M max are satisfied) . M BH ( M (cid:12) ): mass of the BH as the remnant of a progenitor that ends its life. q ≡ M / M BH : mass ratio of BH binary system. a ( R (cid:12) ): orbital semi-major axis (binary separation). P orb (days): orbital period of the binary. i : orbital inclination angle. 2.1. Population and Evolution Paths
To estimate the BH binary population, we develop a similar methodology to Mashian & Loeb (2017). It is widelybelieved that more than ∼
50% of stars reside in binary systems (Duchˆene, & Kraus 2013; Yuan et al. 2015). Formassive early-type stars, the binary fraction even reaches ∼
70% in the open clusters (Sana et al. 2012). BHs withstellar companions can be formed by different stellar evolution channels (Belczynski et al. 2002; Heger et al. 2003).Typically, a massive progenitor with mass (cid:38) M (cid:12) at its final evolution stage will explode as a supernova (SN) andgive birth to a remnant BH by fallback accretion onto the collapsing core. SN could lead to the destruction of thebinary by the natal kick that unbinds the remnant compact object and its companion (Fryer et al. 2012). Accordingto a recent study (Kochanek et al. 2019) on surviving star plus remnant binaries in a sample of 49 SN remnants, (cid:46) ∼
5% of the star plus remnant binaries eventually survive the SN. More massive progenitor with mass (cid:38) M (cid:12) maydirectly collapse into a BH without experiencing an SN explosion. If we consider a Kroupa (2001) initial mass function(IMF, Table 1, first row), the fraction of binaries that end up as a BH and a visible stellar companion is: f BHB = 0 . × ( f s (cid:90) M (cid:12) M (cid:12) Ψ M ( M ∗ ) dM ∗ + (cid:90) M (cid:12) M (cid:12) Ψ M ( M ∗ ) dM ∗ ) , (1)where the factor 0.5 is assumed as the fraction of the binaries in the Galaxy, namely, there are 50 binaries and 50single stars in 150 stars. f s is the fraction of binaries that survived the SN explosion, and Ψ M ( M ∗ ) is the IMF with M ∗ denoting the stellar mass. We assume f s = 0 .
05. The integration yields f BHB ≈ . Yi et al.
In order to detect the BH binaries via LAMOST, the non-compact companion should be still shinning. FollowingMashian & Loeb (2017), we calculate the fraction of stars that are still shinning as follows, f shinning = 1 − ρ ∗ ( z ( t LB = t age ( M ∗ ))) ρ ∗ (0) , (2)where M ∗ , ρ ∗ ( z ), t LB and t age ( M ∗ ) are the stellar mass, the comoving mass density , the look-back time , and the ageof the star, respectively. Following Mashian & Loeb (2017), we also integrate the star formation history (Madau, &Dickinson 2014; Madau, & Fragos 2017) of the Universe to obtain ρ ∗ ( z ); the age of the star is estimated by using theanalytical formula (Equation (4)) in Hurley et al. (2000). Hence the stellar mass distribution (SMF) Ψ M ( M ∗ ) of thevisible companion at present ( z =0) is: Ψ M ( M ∗ ) ∝ f shinning ( M ∗ ) Ψ M ( M ∗ ) . (3)Shown in the left panel of Figure 1 are the IMF (dashed line) and the SMF (solid line).The observability is also affected by the spatial distribution of BH binaries. For simplicity, we assume that thespatial distribution of the binaries traces the distribution of the stars in the Milky Way, which can be modeled asa double exponential thick and thin disk model (Gilmore, & Reid 1983; Juri´c et al. 2008). The adopted normalizednumber density profile is shown in Table 1 (last row; Mashian & Loeb 2017).It should be noted that during the evolution, binary interactions (e.g. Hurley et al. 2002) can change the componentmasses and the orbital separation. Processes such as mass loss by the stellar wind, mass transfer, mass accretion,and common envelope (CE) evolution (e.g. Paczynski 1976; Iben, & Livio 1993; Ivanova et al. 2013) are not includedin the current work for simplicity. Binaries that have evolved through the CE phase may have an orbital period(separation) orders of magnitude smaller than the initial period (separation), since most of the angular momentum istaken away by the envelope (Paczynski 1976). We take a detour by adopting a logarithmically flat distribution for theorbital separation (Table 1, row 3; Abt 1983). The distribution can approximate the real cases, as it is a statisticallysummarized result from observations. 2.2. Mass Distributions of BHs
As for the mass distribution of BHs, we adopt the distribution of ¨Ozel et al. (2010, 2012) as a fiducial one (hereafterreferred to
Fiducial model). ¨Ozel et al. (2010) derived a BH mass distribution from dynamical mass measurementsof 16 BHs in transient low-mass X-ray binaries. The distribution peaks at around 7 M (cid:12) , and presents a mass gap(2-5 M (cid:12) ; see also Bailyn et al. 1998; Farr et al. 2011; Belczynski et al. 2012) between the population of BH andthe most massive neutron star. We also propose an alternative distribution for the purpose of comparison. Ourdistribution is motivated by Fryer & Kalogera (2001), who studied the compact remnants masses by analyzing thebalance between the stellar bounding energy and supernova explosion energy. Fryer & Kalogera (2001) found that theBH mass distribution falls off exponentially. Thus, we assume the BH mass distribution takes the form:Ψ M BH ( M BH ) ∝ exp( − kM BH ) (4)where the exponential factor k is taken to be 0.1, 0.2, 0.3 in the calculation. Both distributions are shown in Table 1(second row) and plotted in the right panel of Figure 1.2.3. Observational Cuts
The low-resolution spectrographs of LAMOST are capable of observing stars with an average limiting magnitude m LV ∼
18 mag in V -band and a precision V preR =5 km s − for radial velocity ( V R ) measurements (Deng et al. 2012).The detection limits for medium-resolution spectrographs are: m LV ∼
15 mag and V preR =1 km s − (Liu et al. 2019).The sky footprint of LAMOST is − π/ < δ < π/ δ : the declination) in the equatorial coordinate (Zhao et al. 2012).To estimate the apparent brightness of binaries, we assume the mass-luminosity relation L = M . for zero-agemain sequence (ZAMS) star. Therefore, for a fixed distance d (in units of pc), there is a lower limit of the mass of thestars that are bright enough to be captured by LAMOST, i.e., this lower mass limit is: M d min = (cid:18) ( d
10 pc ) M SunV − m LV+ A V( d )2 . (cid:19) . , (5) comoving mass density: the stellar mass density at redshift z. look-back time: the cosmological time at redshift z. ining black holes with LAMOST Kroupa IMFSMF at z = - - - - - - M * ( M ⊙ ) d N / d l og M * FiducialExp01Exp02Exp03 M BH ( M ⊙ ) F r ac ti on Figure 1.
Left panel: the IMF (dashed line) and the SMF (solid line). Right panel: The fiducial BH mass distribution adoptedfrom ¨Ozel et al. (2010, black), and the alternative exponential distributions motivated by Fryer & Kalogera (2001):
Exp01 (green),
Exp02 (blue), and
Exp03 (red) correspond to the exponential factor k =0.1, 0.2, and 0.3, respectively. All distributionsare normalized to make sure the integration over the mass range equals unity. where M SunV = 4 .
83 is the absolute magnitude of the Sun, and A V ( d ) is the interstellar extinction. Following Yamaguchiet al. (2018), we adopt the average extinction of the Galactic disk: A V ( d ) = d/ (1 kpc) (Spitzer 1978).There is also an accessible range of the radial velocity curve semi-amplitude K , given the precision of V R measure-ments. From Kepler’s third law, the semi-amplitude K is: K = (cid:18) GM BH a (1 + q ) (cid:19) sin i . (6)For the orbital inclination i , we assume that the binary orbits are randomly distributed (Table 1, row 4). The circularorbit assumption (zero eccentricity) is also assumed.Let us define Ψ K ( k ) as the probability distribution function (PDF) and Φ K ( k ) as the cumulative distributionfunction (CDF) of K . By definition,Φ K ( k ) = Prob ( K (cid:54) k ) = (cid:90) K (cid:54) k Ψ A ( a )Ψ M BH ( m )Ψ Q ( q )Ψ I ( i )d a d m d q d i Ψ K ( k ) = dd k Φ K ( k ) , (7)where Prob ( K (cid:54) k ) is the probability of a specific binary system whose amplitude K is less than or equal to ameasured value k .We assure the reliability of K measurements by putting a lower cut K min2 of 10 times V preR . In other words, thereliable measured K should be >
50 km s − for low-resolution spectrograph, and >
10 km s − for medium-resolutionspectrograph. So to acquire a proper error estimation of the amplitude K , i.e., for a radial velocity curve with semi-amplitude K > K min2 = 50 km s − taken by the low-resolution spectrograph, the uncertainty of K is < − provided that the average V R uncertainty = 5 km s − .Table 2 summarizes the observational cuts for the LAMOST surveys. Yi et al.
Table 2.
Observational cutssurvey
R m
LV ( ∗ ) K min2 δ (mag) (km s − ) (radians)(1) (2) (3) (4) (5)low-resolution ∼ π ∼ π medium-resolution ∼ π ∼ π Note —Column (1): the LAMOST surveys. Column (2): the resolution of the spectrographs. Column (3): the detection limit.Column (4): the detection limit of the V R semi-amplitude. Column (5): the equatorial latitude. (*): The LAMOST lowresolution survey can capture spectra for stars brighter than r (cid:46)
19 ( r -band) during dark/grey time, and r (cid:46)
17 or J (cid:46)
16 onnights that are moonlit or have low transparency (Deng et al. 2012).
The Total Number of Candidates
Based on the knowledge discussed above, the number of detectable BH binary candidates can be estimated by amulti-dimensional integral: N = f BHB × (cid:90) M (cid:12) Min[100 ,M d min ] f shinning ( M ∗ ) Ψ M ( M ∗ ) dM ∗ × (cid:90) (cid:90) − π/ <δ<π/ sin φdφdθ (cid:90) d r drn ( r, θ, φ ) × (cid:90) ∞ V preR Ψ K ( k )d k × f cad , (8)where the coefficient f BHB is introduced in Equation (1). The first integral constrains the fraction of visible starsthat do not exceed LAMOST’s detection threshold, the integrand is introduced at Equation (3), which calculates thefraction of stars that are still shining today by evolving the stellar population. The lower limit of the first integral iscalculated by using Equation (5). The second, third, and fourth integrals sum up the stellar number density in theGalactic coordinate system, with the constraint of the equatorial declination: − π/ < δ < π/
3. The transformationof this LAMOST visible sky region from equatorial to Galactic coordinate system is calculated by using Equations(1)-(4) in Poleski (2013). The fifth integral is the fraction of the accessible range of K . The last term f cad in theequation is defined as the fraction of sources with no less than three observations (spectra). In our opinion, threeobservations is a necessary condition to set constraints on K .2.5. Discovering and Confirming a BH
So far we have covered the first (population) and the second (observability) motivating questions given at thebeginning of Section 2. Now, discovering a potential candidate BH is one thing, confirming (identifying) is another.Strategies are required to search for candidate BHs from large spectroscopic surveys. Notably, Gu et al. (2019) adoptedthe relation of the stellar radius and the Roche-lobe radius to constrain the binary separations, and used a few ( (cid:62) V R measurements to constrain the lower limit of V R excursions. Their method was applied to search for BH candidateswith LAMOST DR6. Thompson et al. (2018) developed a strategy based on the maximum acceleration of the system,to select potential candidates in SDSS APOGEE survey which has in average 2-4 measurements per system. If onehas sufficient V R measurements, the radial velocity curve and hence the mass function can be obtained. However, onlya fraction of sources have intense observations in the LAMOST low-resolution survey. For example, ∼
6% of targetshave three or more low-resolution visits in the LAMOST
Kepler field (spanning five years from 2012 to 2017; Zong etal. 2018, Table1). In this case, complementary follow-up spectroscopic and photometric observations are required tomeasure the orbital periods. For instance, Zheng et al. (2019) searched for BH candidates with orbital periods revealedby the ASAS-SN photometry. RESULTS AND ANALYSESFigure 2 shows the distribution (PDF) and cumulative distribution (CDF) of visible companion’s mass and massfunction for the detectable BH binaries by LAMOST. The left panel is the PDF and CDF of the mass of detectable ining black holes with LAMOST Low - ResolutionMed - Resolution C D F M ( M ⊙ ) P D F M K G F A
FiducialExp01Exp02Exp03 C D F f ( M BH ) ( M ⊙ ) P D F Figure 2.
The distribution (PDF) and cumulative distribution (CDF) of the visible companion’s mass and mass functionfor the detectable BH binaries by LAMOST. Left panel: The PDF and CDF of the visible companion’s mass, for low- andmedium-resolution spectrographs, respectively. Most of the expected candidates have a visible companion of M-, K-, G-, orF-type star. Right panel: The PDF and CDF of the BH binary mass function. Around 47% of the BH binaries have a massfunction larger than 3 M (cid:12) (gray shallow line) for Fiducial model. visible companions. The distributions are calculated with equally spaced sampling points in the linear space of M ,with a bin size of 0.05 M (cid:12) . By using Equation (8), we first calculate the number of BH binaries and the fraction of theobservable companions per mass bin at different distances, ranging from 20 pc to 5 kpc, then we sum up the number ofeach mass bin at all distances to find the fractions. The red and blue lines represent the low- and medium-resolutionspectrographs, respectively. As suggested in the CDF, most of the detectable candidates are low-mass BH binarieswith M-, K-, G- or F-type stars (MK system; Morgan, & Keenan 1973), i.e., for low-resolution spectrographs, thefractions for M-, K-, G- and F-type stars are ∼ ∼ ∼ ∼ ∼ f BHB is evaluated to be ∼ is useful to constrain a lower mass limit of the unseen objectin the SB1 system (Casares & Jonker 2014). The binary mass function for a BH system is given by: f ( M BH ) = M BH sin i (1 + q ) = K P orb πG . (9)The right-hand side of the equation hints that f ( M BH ) can be calculated by K and P orb , which are measurablequantities from the radial velocity curve. Note the difference between the initial mass function (IMF) and the binary mass function: the IMF describes the mass distribution ofthe Galactic stellar population, while the binary mass function is a measurable quantity for a specific SB1 system, the lower mass limit ofthe unseen companion.
Yi et al. F r ac ti on — Fiducial — Exp01 — Exp02 — Exp03 — BlackCAT ●● ●●●● ▲▲ ▲▲▲▲ ⊕⊕⊕⊕ ⊕⊕△△ △△ △△○○ ○○○○ ★★ ★★★★ P orb ( day ) K ( k m s - ) ● Swift J1357.2â ● XTE J1650â ● XTE J1118 + ▲ XTE J1859 + ▲ SAXJ1819.3 - ▲ XTE J1550â ⊕ GRO J1655â ⊕ GRS 1009 - ⊕ GRS 1915 + △ GRO J0422 + △ GRS 1124 - △ GS 2023 + ○ GS 2000 + ○ GS 1354 - ○ H 1705 - ★ ★
1H J1659 - ★
4U 1543â f ( M B H ) = . . Fraction
Figure 3.
The distributions of orbital period P orb and radial velocity amplitude K for visible companion. Gray histogramsdenote the distributions of 18 dynamically confirmed BHs. Data of K and P orb are adapted from BlackCAT (Corral-Santanaet al. 2016, Table A.4., columns 3 and 4). The contours corresponding to f ( M BH ) = 0.03, 0.3, 3, 30, and 300 are also draw inthe figure. The right panel of Figure 2 shows the PDF and CDF of the mass function. The distributions are calculated withequally spaced sampling points in the logarithmic space of f ( M BH ), with a step size of 0.1 M (cid:12) . Comparing to theresults derived from Exp model, the one from
Fiducial model rises more rapidly when passing 3 M (cid:12) , peaks at around5 M (cid:12) , and falls off more rapidly with increasing mass. The vertical shallow gray line shows the BH binary system with f ( M BH ) = 3 M (cid:12) . Note that BH binaries may have a f ( M BH ) < M (cid:12) . For instance, the mass function f ( M BH ) ofSAXJ1819.3-2525 is 2 . ± . M (cid:12) (V4641 Sgr, Orosz et al. 2001; MacDonald et al. 2014). In a practical perspective,if a SB1 system has mass function significantly larger than the mass of the visible star, the unseen star must be acompact object. In the calculation, we find that ∼ ∼ ∼ ∼
32% of the BH binaries have massfunctions larger than 3 M (cid:12) , for the Fiducial , Exp01 , Exp02 , and
Exp03 models, respectively.The distributions of the spectroscopic observables P orb and K are predicted and compared to 18 dynamicallyconfirmed BHs (Corral-Santana et al. 2016). The distributions of the K values are calculated by Equation (7). Byimplementing Kepler’s third law, the distribution of P orb values, for the fiducial and exponential M BH models, isderived from the distribution of orbital separations a (Table 1, row 3). Figure 3 shows the smoothed PDFs of P orb and K . Gray histogram denotes the corresponding distribution (fractional counts in the logarithmic space with auniform step size = 0.1) of the 18 dynamically confirmed BHs. Data of K and P orb are adapted from BlackCAT(Corral-Santana et al. 2016, Table A.4., columns 3 and 4). The contours corresponding to f ( M BH ) = 0.03, 0.3, 3, 30,and 300 are also draw in the figure.The orbital period distributions peak at around 0.23 ( Fiducial )-0.3 (
Exp01 ) days, with ∼
76% (
Exp01 )-78% (
Fidu-cial ) binaries in the range 0.2-2 days, indicating that short period BH candidates are quite common in the observational ining black holes with LAMOST Low - resolution f cad = f cad = f cad = N u m b e r Total ~ ~ ~ - resolution f cad = f cad = d ( kpc ) N u m b e r Total ~ ~ Figure 4.
The number of detectable candidates as a function of distance. The blue, red and black lines represent the expectedresults by LAMOST low-resolution, non-time-domain survey, with f cad = 0 .
1, 0 .
06, and 0 .
01, respectively. The orange andgreen lines represent the expected results by medium-resolution, time-domain survey, with f cad = 0 .
5, and 0 .
1, respectively. Thetotal predicted numbers of each case are rounded to the nearest 50s. space of parameters. This implies that a large fraction will be quiescent BH transients, i.e., interacting binaries withaccretion disks. Quiescent BH transients can be easily identified from the presence of broad emission lines, chiefly H α . In some cases, such as Swift J1357-0922 (Mata S´anchez et al. 2015), the companion star is overwhelmed bythe accretion disk light and, consequently, no radial velocity information can be obtained. In these cases, surveysexploiting the detection and properties of the H α line (Casares 2015, 2018) can be very useful.0 Yi et al.
Table 3.
The distributions of K in linear space.Model Fractions of K at different range (km s − ) <
200 200-400 400-600 600-800 800-1000 > Fiducial
Exp01
Exp02
Exp03
Regarding the radial velocity semi-amplitude K distribution, the results show a peak at around 500 - 600 km s − .We also present the calculated fractions in linear space, as presented in Table 3. It shows that the fraction of K values under 400 km s − is larger than the fraction above, indicating that there is a good chance of detecting BHbinaries with K <
400 km s − (even for K <
200 km s − , the odds are still good). Note that extremely large K ∼ − might be possible. These are close pairs that contain a low-mass dwarf (M dwarf) star and a BH with M BH (cid:38) M (cid:12) . The BHs could form through the direct collapse of massive O stars or Wolf-Rayet Stars, withoutexperiencing a supernova explosion and thus barely losing any mass.We calculate Equation (8) by setting f cad =0 .
06 (see Section 2) for the low-resolution survey; f cad =0 . f cad =0 .
01, toincluding less optimistic estimation. As for the medium-resolution time-domain survey, sufficient observational epochsare guaranteed ( ∼
60 visits for every single target), leading to a tentative value of f cad =1. However, the value of f cad is unknown, so we take f cad = 0.5 and 0.1 for illustrative purposes. Shown in the Figure 4 is the expected numberof detectable candidates as a function of distance. The calculation step size is 0.05 in logarithmic space of distance d . We present only the results derived from Fiducial distribution as
Exp one gives same results. The blue, red, andblack lines represent the results by LAMOST low-resolution, non-time-domain survey, for f cad = 0 .
1, 0 .
06, and 0 . ∼ ∼ f cad . The orange and green linesrepresent the expected results by the medium-resolution, time-domain survey, for f cad = 0 .
5, and 0 .
1, respectively.Compared to the low-resolution survey that can go deeper, the medium-resolution survey can only catch stars thatare brighter than 15 mag in the V-band, hence most of the candidates are expected to be located within 2 kpc. Weexpect ∼ CONCLUSIONS AND DISCUSSIONIn this paper, we study the prospects of searching for binary systems with a stellar-mass BH and a non-compactvisible companion, by utilizing the spectroscopic data of LAMOST. Our results can be summarized as follows. • Most of the expected candidates have a visible companion of M-, K-, G-, or F-type star. • About 47 % of these BH binaries have a mass function larger than 3 M (cid:12) . • A majority of candidates have an orbital period in between 0.2-2 days, suggesting that the discovery of short periodBH binaries is favored. • Most of the detectable BH candidates are located within 2 kpc of the solar neighborhood. • We predict that (cid:38)
400 candidate BH binaries can be found by the low-resolution, non-time-domain survey, and ∼ ining black holes with LAMOST Gaia satellite. The
Gaia astrometric measurementsresolve the orbital motion of the visible star, such robust observations allow one to solve the mass of unseen objectsimply by Kepler’s third law. Since
Gaia has a unique strategy of sky-scanning that is designed for optimizing theastrometric accuracy, it is better at detecting long period binaries ( (cid:38)
30 days) over short period ones. At this point,LAMOST has the advantage of observing binary systems with a wider range of orbital periods. For each target,LAMOST takes three (or more) exposures in a single observation, and each exposure takes 10-30 minutes. Each targetwill be covered ∼
60 times during the course of the entire time-domain survey. Hence LAMOST is capable of trackingshort period binaries down to a few hours, or hunting long period binaries up to years.As mentioned in Section 2, the potential of mining BH binaries from spectroscopic observations can be reinforcedwith follow-up spectroscopic and photometric observations. In fact, dynamically confirmed BHs are studied throughboth their spectroscopy and photometry (e.g. Orosz, & Bailyn 1997; McClintock et al. 2001; Wu et al. 2016). On onehand, spectra provide information for the visible star’s radial velocities, spectral types, and possible signature froman accretion disk (broad H α emission line); on the other hand, light curves from photometric measurements providevaluable information for the orbital period, orbital inclination, and the mass ratio of the binary (in the case of extrememass ratio, q can be best obtained from resolving the rotational broadening ( V sin i ) of the donor star (e.g. Marshet al. 1994)). It implies a novel approach for hunting a BH via the optical point of view: (a) Start from picking outSB1 sources with large radial velocity excursions, inspect whether broad emission lines are present. (b) Cross-matchsuspected sources with other spectroscopic and photometric surveys, set constraints upon the orbital period or thebinary separation. (c) Calculate the mass function of the unseen companion and select candidates that have a massfunction f ( M BH ) > M (cid:12) . (d) Collect data from follow-up observations and measure the full set of orbital parametersthat shall confirm candidates and put final constraints on the BH masses.We thank Hao-Tong Zhang, Zhong-Rui Bai, Wei-Kai Zong, Xuefei Chen, and Hailiang Chen for beneficial discussion,and thank the anonymous referee for giving numerous detailed, helpful suggestions that improved the manuscript.This work was supported by the National Natural Science Foundation of China under grants 11573023, 11603022 and11973002, and the Fundamental Research Funds for Xiamen University under grants 20720190122 and 20720190115.Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a NationalMajor Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by theNational Development and Reform Commission. LAMOST is operated and managed by the National AstronomicalObservatories, Chinese Academy of Sciences. REFERENCES Abt, H. A. 1983, ARA&A, 21, 343Agol, E., Kamionkowski, M., Koopmans, L. V. E., &Blandford, R. D. 2002, ApJL, 576, L131Bai, Z., Zhang, H., Yuan, H., et al. 2017, PASP, 129, 024004Bai, Z.-R., Zhang, H.-T., Yuan, H.-L., et al. 2017, Researchin Astronomy and Astrophysics, 17, 091 Bailyn, C. D., Jain, R. K., Coppi, P., et al. 1998, ApJ, 499,367.Beer, M. E., & Podsiadlowski, P. 2002, MNRAS, 331, 351.Belczynski, K., Kalogera, V., & Bulik, T. 2002, ApJ, 572,407.Belczynski, K., Kalogera, V., Rasio, F. A., et al. 2008,ApJS, 174, 223. Yi et al.
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