Preamble

Vol. 43, No. 2

June 2025

Progress in Astronomy Vol. 43, No. 2 June 2025 doi: 10.3969/j.issn.1000-8349.2025.02.05

Mass Measurements of Black Hole X-ray Transients

ZHOU Ao, WU Jianfeng
(College of Physical Science and Technology, Xiamen University, Xiamen 361005, China)

Abstract

To date, a total of 72 black hole X-ray transients and candidates have been detected, among which 19 have been dynamically confirmed. These transients are X-ray binary systems composed of stellar-mass black holes and low-mass companion stars. Measurements of the fundamental parameters of stellar-mass black holes help us better understand binary evolution and black hole formation, while also providing evidence for important questions in the field, such as whether a strict mass gap exists between stellar-mass black holes and neutron stars. This paper introduces the basic theory of black hole X-ray binaries, describes in detail the common methods and software used to measure stellar-mass black hole masses, analyzes possible sources of error in dynamical modeling, and finally summarizes the statistical properties of the existing sample of black hole X-ray transients, with an outlook on future research directions.

Keywords: X-rays; black hole binaries; transients; dynamical studies

1 Introduction

One of the most famous predictions of Einstein's general relativity (GR) is the existence of black holes (BH). In black holes, gravity completely dominates, creating a spacetime singularity within them [1]. The essential attribute of a black hole is its event horizon, an immaterial surface that confines the interior region that cannot communicate with external spacetime. The extremely strong gravitational field of black holes provides astronomers with a unique laboratory to observe and study high-energy physical phenomena in the universe, such as accretion, relativistic jets, and gamma-ray bursts. Consequently, black holes play important roles in many areas of astrophysical research, including massive star evolution, X-ray binaries, and active galactic nuclei.

Black holes are the simplest macroscopic objects in the universe. GR's no-hair theorem states that when an isolated black hole forms, the final object's properties are completely characterized by its mass, angular momentum, and charge [2, 3]. It is generally believed that charged black holes in realistic cosmic environments would be neutralized by their surroundings, so astrophysical black holes are defined solely by mass and angular momentum. Accurate measurement of these two parameters is fundamental for probing spacetime structure in strong gravitational fields. For black holes, mass is the basic parameter, and black hole identification is achieved by constraining their mass. Reliable black hole mass values are prerequisites for measuring black hole spin through X-ray continuum spectral fitting methods [4]. Confirmed black holes in the universe include: stellar-mass black holes (mass range 3 M⊙ ∼ 100 M⊙), supermassive black holes (mass range 10^6 M⊙ ∼ 10^10 M⊙), and intermediate-mass black holes (mass range 10^2 M⊙ ∼ 10^5 M⊙) that lie between the former two categories.

Dynamical modeling is one of the main pathways for black hole identification, with its basic principle being to constrain the mass of the central object through the observed Keplerian motion of stars or gas. Currently, all stellar-mass black holes confirmed in the Milky Way exist in binary systems. By observing the companion star in the binary, we can obtain the system's orbital solution, thereby precisely constraining the mass of the compact object. In 1939, Oppenheimer and Snyder [5] performed the first rigorous calculation of black hole formation. When a compact object's mass exceeds the Tolman-Oppenheimer-Volkoff limit, it can only collapse into a black hole. While the exact value of this limit remains an active research area, it is generally believed to be no more than 3 M⊙. Therefore, when a compact object's mass is reliably measured to be greater than 3 M⊙, it can be identified as a stellar-mass black hole. For supermassive black holes, mass lower limits are similarly determined by studying the orbits of stars around them, leading to eventual black hole confirmation. The radio source Sgr A* at the Galactic center was the first supermassive black hole confirmed through this method, with the latest mass measurement yielding M_BH = (4.297 ± 0.012) × 10^6 M⊙ [6]. For active galactic nuclei, reverberation mapping (RM) is the primary method for measuring supermassive black hole masses. Its basic principle involves measuring the velocity of broad-line region gas around the black hole and its distance from the black hole through spectroscopic and photometric observations, making it essentially a dynamical method [7, 8]. In recent years, reverberation mapping studies of dwarf galaxies have found evidence for intermediate-mass black holes at galactic centers, such as NGC 4395, whose central black hole has a mass on the order of 10^4 M⊙. In addition to dynamical methods, gravitational wave methods have detected nearly a hundred stellar-mass binary black hole merger events [9], and found evidence for intermediate-mass black holes formed through stellar-mass black hole mergers (such as in event GW190521, where the product of the binary black hole merger is a black hole with mass 142 M⊙ [10, 11]). Furthermore, gravitational lensing methods can be used to probe isolated black holes in the universe [12].

The physical processes around black holes of different masses show high similarity. Meanwhile, compared to supermassive black holes, stellar-mass black holes have much shorter variability timescales, allowing astronomers to conduct in-depth studies of their properties through a series of accretion mechanisms [13]. Stellar-mass black holes are important for many areas of astronomy. For example, they are one of the final products of massive star evolution, and the collapse of their progenitor stars enriches the universe with heavy elements [14]. Black hole mass can also reflect information about the progenitor star's mass, stellar wind loss, and other related properties. A rich sample of stellar-mass black holes can help us understand how massive stars end their lives in the universe and the special process of transforming into black holes. The mass distribution of stellar-mass black holes in galaxies has complex and close relationships with the number and evolution of massive stars, energy sources of supernova explosions, and the boundary between neutron stars and black holes. Moreover, complete binary orbital parameters are crucial for understanding dense binary evolution theory and comparing the merits of various models [15-17]. Known X-ray binary systems with three-dimensional space velocities can provide further constraints on black hole formation mechanisms, allowing us to explore the kicks experienced by black holes during formation and providing evidence for binary evolution and supernova models [18].

Black holes detected through gravitational waves follow different mass distributions from those confirmed dynamically, suggesting they may represent different black hole populations. To date, stellar-mass black holes confirmed dynamically are mainly located in X-ray binary systems [19]. Most black hole X-ray binaries were first discovered through X-ray outbursts, and then their compact objects were identified as stellar-mass black holes through dynamical modeling. After such outbursts, these binary systems typically return to a weak X-ray radiation state on timescales of months to years, i.e., the "quiescent state." Therefore, these X-ray binaries are called black hole X-ray transients (X-ray transient, XRT). Black hole X-ray transients constitute the main body of stellar-mass black holes confirmed dynamically to date. This paper will mainly discuss the dynamical modeling and mass measurement of black hole X-ray transients.

2.1 History of X-ray Binary Research

Isolated black holes are difficult to produce observable radiation, making binary systems composed of black holes and ordinary stars an effective way to find and observe black holes. When a black hole is in a close binary system and the process of accreting matter from the companion star is relatively active, bright X-ray radiation is produced, making it easy to detect in the X-ray band. Therefore, the first strong evidence for a stellar-mass black hole came from observations of the X-ray binary Cygnus X-1 [20, 21]. X-rays are high-energy radiation released when companion star material is heated to extremely high temperatures in the accretion disk around the black hole, providing extremely valuable information for deeply exploring the properties of black holes themselves and the accretion behavior of surrounding matter. Through detailed analysis of these X-ray signals, we can constrain multiple parameters of black hole binary systems based on their light curves and energy spectra. X-rays also help reveal the structure, temperature distribution, thickness, and mass transfer processes of accretion disks, while observed rapid X-ray variability and burst activity can help scientists understand many aspects such as instabilities and turbulence processes inside accretion disks [19]. The optical radiation of Cygnus X-1 mainly comes from its companion star. Through observations of optical band spectra, we can measure the companion star's radial velocity and its physical parameters (such as mass), thereby estimating the black hole's mass through dynamical modeling.

X-ray binaries are binary systems with a neutron star or black hole as the primary star and an ordinary star as the companion star. Based on the companion star's mass, they are divided into high-mass X-ray binaries (HMXB) and low-mass X-ray binaries (LMXB). The companion stars of HMXBs are typically massive O/B-type stars with mass M > 10 M⊙. Strong stellar winds from the companion star directly blow onto the compact object, which accretes matter through wind accretion, producing X-ray radiation as persistent bright X-ray sources. Cygnus X-1 and the second black hole X-ray binary discovered later—LMC X-3 in the Large Magellanic Cloud [22]—both belong to this category. However, currently only Cygnus X-1 has been confirmed as a black hole HMXB system in the Milky Way, with the vast majority of known HMXBs being neutron star binaries. The companion stars of LMXBs are mostly G/K/M-type stars (mass less than about 1 M⊙). After the companion star fills its Roche lobe, material is transported to the accretion disk through the Lagrangian point. The third black hole binary system discovered, A 0620-00, is a representative source of black hole LMXBs. Unlike Cygnus X-1, it is not a persistent X-ray source. A 0620-00 was first detected through an X-ray outburst event, with peak brightness on the order of 10^-9 W·m^-2 [23]. After the outburst, over a period of about 1 year, A 0620-00 gradually returned to quiescent X-ray levels, eventually stabilizing at 10^-17 W·m^-2, becoming a relatively faint X-ray source [24]. For the X-ray outburst events lasting several months produced by black hole LMXBs, the driving force is widely believed to be some physical instability mechanism within the accretion disk [25, 26]. In this process, when the accretion material flow provided by the companion star to the black hole is insufficient, material in the outer regions of the accretion disk begins to accumulate, reaching a critical point where the disk's equilibrium is disrupted, triggering a large-scale release of matter and energy—this is the X-ray outburst event we observe [27]. This outburst mechanism helps us understand many astrophysical processes such as accretion disk dynamics, matter transport, and interactions between compact objects and their companion stars. Therefore, black hole LMXBs exist in the form of X-ray transients [19]. Such systems are in X-ray quiescent states for most of the time (with typical X-ray luminosities below 10^25 J·s^-1).

The spatial distribution differences of these two types of X-ray binaries in the Milky Way reflect their associations with different stellar populations. HMXBs tend to be associated with young stars and are commonly found in the galactic disk regions, which often have abundant gas and dust and active star formation. LMXBs are more concentrated in the central bulge region of the Milky Way and within globular clusters, which contain large populations of old stars that more easily form LMXB systems over long evolutionary timescales.

2.2 X-ray Spectral Properties of Black Hole Binaries

The X-ray spectra of black hole binaries can typically be decomposed into thermal and non-thermal components. The thermal component mainly comes from blackbody radiation of the accretion disk with characteristic temperatures near 1 keV. The non-thermal component mainly comes from inverse Compton scattering, appearing as a power-law spectrum. X-ray photons with power-law spectra irradiating the accretion disk will be reflected by it. Although this accretion disk reflection component is ubiquitous, it is more easily observed in black hole binaries with smaller accretion disk inclination angles (i.e., the angle between the accretion disk normal and line of sight) [28]. In addition to the above continuum spectral features, black hole X-ray spectra also contain some line features, such as the relativistically broadened Fe Kα emission line. Current research is exploring the feasibility of using Fe Kα fluorescence lines to measure black hole mass [29]. In black hole binary systems, the companion star may produce narrow Fe Kα emission lines when irradiated by X-rays. If the narrow line component can be separated from the observed Kα emission line, the companion star's radial velocity curve can be generated. The feasibility of this method depends on other system parameters such as mass ratio q and orbital inclination i. Additionally, there are practical limitations such as shielding by complex stellar wind configurations. Nevertheless, in the era of high-resolution X-ray astronomy represented by microcalorimeters, if the feasibility of such methods can be observationally confirmed, it will significantly improve the precision of stellar-mass black hole mass measurements.

3 Mass Measurement of Black Hole X-ray Transients

Dynamical modeling of black hole binaries is mainly accomplished through optical spectroscopic and photometric observations of the companion star. Unlike conventional observations, dynamical observations are unique in their focus on time-domain analysis, requiring multiple measurements over long periods to ensure coverage of most orbital phases, thereby enabling accurate modeling of periodic radial velocity curves and light curves. Photometric studies measure periodic variations in the system's optical brightness, while spectroscopic studies mainly measure the line center shifts of absorption lines from the companion star's atmosphere (corresponding to radial velocity) and their broadening (corresponding to the companion star's rotational velocity). The companion stars of black hole HMXBs are O/B-type stars that dominate the system's optical radiation. For black hole LMXBs, during outbursts, the optical radiation from the low-mass companion star is completely overwhelmed by the bright accretion disk. Only during X-ray quiescent states can the companion star's optical radiation be detected, enabling dynamical modeling through optical spectroscopic and photometric observations of the companion star to measure the black hole's mass. Therefore, optical observations for dynamical studies of black hole X-ray transients need to be conducted during their X-ray quiescent states.

3.1 Method Introduction

Dynamical studies of binary systems rely on Kepler's third law, expressed in the form of a mass function defined by equation (1):

$$f(M) = \frac{PK_2^3}{2\pi G} = \frac{M\sin^3 i}{(1+q)^2}$$

This equation naturally applies to compact object binary systems, where M represents the mass of the compact object. The mass function f(M) defined by this equation represents the lower limit of the compact object's mass. If this lower limit exceeds 3 M⊙, the compact object is identified as a black hole.

Note that equation (1) implicitly assumes a circular orbit, which is a reasonable assumption because the orbital circularization timescale of X-ray binaries is much shorter than their expected lifetime [30]. Starting from the definition of the mass function, the process for measuring black hole mass in X-ray transients follows these steps: (1) Identify features in the companion star's optical spectrum, determine its spectral type, and select the appropriate stellar spectral template; (2) Fit the companion star's radial velocity measured from optical spectra covering multiple orbital phases to obtain its semi-amplitude K2 and orbital period P, then calculate the compact object's mass function f(M); (3) Use optical spectra to measure the companion star's projected rotational velocity v sin i, then calculate the mass ratio q = M_c/M; (4) Obtain the optical light curve through photometry, model the companion star's ellipsoidal modulation to derive the binary orbital plane inclination i, and finally calculate the black hole mass M.

The period P can also be obtained from the optical light curve. However, in some special cases (see Section 3.2.2), the photometric period from the light curve does not reflect the true orbital period, requiring precise constraints from fitting the companion star's radial velocity curve. We detail these steps below.

3.1.1 Measuring P and K2 via Companion Radial Velocity Curve

We first need to select an appropriate template spectrum with absorption line features almost identical to those of the companion star for measuring radial and rotational velocities. The most effective technique is the cross-correlation method in spectroscopy. Specifically, we can use telescopes to obtain a series of stellar spectra with known spectral types, or directly select spectra from the PHOENIX stellar spectral template library published by Husser et al. in 2013 [32]. This work must first perform cross-correlation analysis between the average spectrum of the black hole binary system and numerous template spectra to select the spectrum with the highest correlation as the template for subsequent processing. Then, each spectrum of the black hole binary is cross-correlated with the template spectrum to measure the radial velocity corresponding to different spectra. Many software packages can be used for cross-correlation analysis between spectra, such as IRAF/fxcor [33], Python/laspec [34, 35], etc., all based on the method proposed by Tonry and Davis [36]. The specific process is: first set a series of velocity values as the velocity difference between the template spectrum and the actual spectrum, then correct the wavelength according to the Doppler effect, and finally use the cross-correlation function to calculate the correlation between the two spectra. The velocity value with the best correlation is the relative radial velocity between the two spectra. For black hole binaries, the radial velocity obtained directly from cross-correlation between the observed spectrum and the template spectrum is the companion star's velocity relative to the ground-based telescope, which must be corrected to the heliocentric reference frame. We typically use the pyasl.helcorr module in the PyAstronomy library [37] to implement this correction, which inputs telescope coordinates and observation times to correct for Earth's motion. Finally, the thejoker package [38] in Python can be used to fit the companion star's radial velocity curve. This is a Monte Carlo sampler specifically designed for simulating radial velocity curves of binary systems, selecting an optimal set of parameters (i.e., model curves) through χ^2 comparisons between different model curves and actual data points. The fitted parameters include the orbital period P and companion star radial velocity semi-amplitude K2 needed for calculating the mass function. Figure 1 [FIGURE:1] shows an example of radial velocity fitting, with the horizontal axis representing phase distribution after period folding and the vertical axis representing radial velocity data and residuals relative to the model curve. The black line is the optimal radial velocity model curve obtained from fitting. This example shows the radial velocity curve of the black hole X-ray transient Nova Muscae 1991, consisting of radial velocity values measured from 72 spectra obtained over two observing nights, covering complete orbital phases [31]. The mass function f(M) is the basis for identifying stellar-mass black holes and can typically be measured with a precision of a few percent. This requires the spectrograph's resolving power to be higher than λ/Δλ ≈ 1500 [13], where λ represents wavelength and Δλ is the minimum resolvable wavelength interval of the spectrograph. Meanwhile, to more accurately fit the radial velocity curve, the orbital phases corresponding to spectroscopic observation times should uniformly cover the complete orbital period.

3.1.2 Measuring Mass Ratio q via v sin i

The mass function f(M) only provides a lower limit to the compact object's mass. The accurate value of the compact object's mass also requires measurements of the mass ratio q and binary orbital plane inclination i. The main method for determining the mass ratio is measuring the rotational broadening (v sin i) of absorption lines from the companion star's photosphere. This method exploits the fact that for short-period binary systems undergoing mass transfer, the companion star often fills its Roche lobe and is tidally locked, with its rotation period equal to its orbital period, making its absorption lines significantly broader than those of slowly rotating single stars. Under spherical approximation, the relationship between rotational broadening and binary mass ratio can be quantified by [39]:

$$v\sin i \approx 0.462K_2q^{1/3}(1+q)^{2/3}$$

Therefore, by measuring v sin i and combining it with K2 obtained from the previous step, we can constrain the range of q.

The optimal subtraction method implemented by the molly software can be used to measure v sin i [40]. The basic idea is to process the template spectrum (such as broadening absorption lines) to simulate the real companion star spectrum. Comparing the optical spectrum of the black hole binary with the corresponding template (i.e., a field star spectrum of the same spectral type) reveals that absorption lines in the black hole binary spectrum are broader and shallower. The broadening of absorption lines is caused by the faster rotation of the tidally locked companion star as described above. In black hole binary systems, the continuous spectrum in the optical band often contains an additional component from the accretion disk, which dilutes the companion star's absorption line features, making them appear shallower in the spectrum. The optimal subtraction method can simultaneously measure the companion star's rotational velocity v sin i and the companion star's fractional contribution fs (0 < fs < 1) to the spectrum, with the accretion disk's contribution being fd = 1 - fs. The specific approach is: set a series of rotational velocity v sin i values, broaden the template spectrum's absorption lines according to the v sin i values, multiply by the fs factor, and subtract this from the black hole binary's spectrum. Ideally, if the set v sin i and fs values match the actual parameters of the black hole binary system, the resulting residual spectrum will no longer contain absorption lines from the companion star spectrum, only noise. We perform a χ^2 test on the residual spectrum, and through χ^2 minimization, we can find the optimal broadening velocity v sin i that matches the two spectra while also obtaining the companion star's fractional contribution fs. Based on the measured v sin i value, the mass ratio q can be calculated through equation (2). Figure 2 [FIGURE:2] shows a schematic diagram of measuring v sin i for the black hole X-ray transient Nova Muscae 1991 using the optimal subtraction method. The middle spectrum is the template spectrum. After processing this template through the steps described above and subtracting it from the upper black hole binary spectrum, the residual spectrum shown at the bottom contains only noise (plus non-companion-star spectral features such as accretion disk emission lines and interstellar medium absorption) at the optimal v sin i and fs values.

Note: The three spectra from top to bottom are the target source Nova Muscae 1991's average normalized spectrum, the template spectrum with the same spectral type as the companion star, and the residual obtained by subtracting the processed template spectrum from Nova Muscae 1991's spectrum.

Several technical details need attention when measuring v sin i and fs using the optimal subtraction method. Black hole binary spectra often also contain interstellar medium absorption features and accretion disk spectral features (such as Balmer emission lines), requiring these wavelength ranges to be masked. Additionally, when placing the template spectrum and target average spectrum in the same rest frame for rotational broadening measurement, we need to apply limb darkening correction to the template spectrum and, to simulate the blurring effect caused by absorption line center movement during the exposure, also blur the template spectrum according to the exposure time and instantaneous radial velocity. In black hole X-ray transients, typical companion star rotational broadening ranges from 30 ∼ 150 km·s^-1, requiring high spectral resolution to obtain accurate v sin i values; otherwise, measurement errors will be large, affecting the accurate calculation of the mass ratio.

3.1.3 Constraining Inclination i via Ellipsoidal Modulation of Light Curves

Binary system orbital inclinations are typically determined by fitting ellipsoidal modulation models to optical or near-infrared band light curves. First, to obtain the target source's magnitude on photometric images, aperture photometry can be performed using the IRAF/DAOPHOT tool. Generally, 1.5 times the point source image's full width at half maximum (FWHM) can be chosen as the photometric aperture size. In some special cases, such as when nearby sources contaminate photometry, image subtraction can more accurately measure variability. This is a widely used data processing method in galaxy research that allows us to focus on variable components by subtracting constant parts from images. Common software includes SFFT (saccadic fast Fourier transform) [41], HOTPANTS, and ZOGY. These methods match point spread functions across images taken at different times and subtract pixels between images based on the matched PSF characteristics. SFFT performs these operations in Fourier space, making processing faster. After obtaining the light curve, periodicity analysis of photometric results is needed. We recommend using the Lomb-Scargle method [42] to find the time frequency with the strongest periodic signal. This method, based on Fourier transform algorithms, first calculates power at different temporal frequencies, then folds the data according to the obtained period to nicely display the periodic light curve.

In black hole X-ray transients, the companion star fills a teardrop-shaped Roche lobe. At different orbital phases, the facing surface area toward the observer varies, along with non-uniform surface brightness distribution, resulting in a characteristic double-peaked modulation phenomenon in the light curve (called "ellipsoidal modulation"). Figure 3 [FIGURE:3] shows the light curve of the black hole X-ray transient GRO J1655-40, with the horizontal axis representing orbital phase and the vertical axis representing V-band magnitude. The solid line is the best-fitting light curve model, and black dots represent data with error bars. This is a very typical example of ellipsoidal modulation. Within one orbital period, two equal brightness peaks represent the phases when the companion star's facing surface area is maximum (i.e., when the companion star's radial velocity absolute value is maximum), while two unequal brightness troughs represent phases when the companion star is in front of or behind the black hole from the observer's perspective. The amplitude of this modulation has a close functional relationship with the binary system's orbital inclination i: the larger the inclination i, the closer the binary orbital plane's normal is to perpendicular to our line of sight, meaning the orbital plane faces the observer "edge-on," resulting in larger ellipsoidal modulation amplitude. GRO J1655-40 is a black hole X-ray transient with an F6 IV-type intermediate-mass companion star, whose system orbital inclination has been precisely measured [43]. However, the more common situation is black hole X-ray transients with fainter K-M type companion stars, where light curves may be severely contaminated by other non-stellar light sources (see Section 3.2).

To obtain accurate light curve models, researchers need to consider the local intensity at each point on the companion star's surface, affected by factors such as limb darkening and gravity darkening. Limb darkening refers to the phenomenon where regions near the edge of a star's surface appear darker than the central region. Gravity darkening refers to the phenomenon where a star's equatorial region expands due to centrifugal force, decreasing density and causing reduced surface temperature and brightness. By comprehensively considering the distribution of local intensities over the Roche lobe geometry and correcting for effects from limb darkening and gravity darkening, we can obtain light curve models. Kurucz and NEXTGEN atmospheric models [44] are often used for best results. There are many methods to measure binary system orbital inclinations through ellipsoidal modulation, using software packages such as ELC (eclipsing light curve) [44], W-D (Wilson-Devinney) [45], or PHOEBE [46].

3.1.4 Other Feasible Methods

In some systems with highly active accretion disks, black hole X-ray binaries may exhibit "superhump" modulation in optical light curves, where the period from the light curve slightly deviates from the orbital period (see Section 3.2.2 for explanation). For such systems, if the mass ratio q is in the range 0.04 ∼ 0.30, there exists a relationship between period and q:

$$\frac{\Delta P}{P} = \frac{P_{sh} - P_{orb}}{P_{orb}} \simeq (0.216 \pm 0.018)q$$

[47, 48]. This empirical formula was obtained through fitting strongly correlated data and is physically reasonable because when mass ratio q = 0, orbital precession-induced superhump modulation should not occur. This method can both verify the reasonableness of q values and directly estimate their values.

In addition to fitting ellipsoidal modulation of light curves, binary orbital inclination i can also be constrained through several other methods. For black hole binary systems where radio jets can be detected, the angle between the jet and observer's line of sight is used to replace the orbital inclination, based on the assumption that the jet is perpendicular to the binary orbital plane. However, evidence suggests that jets are not necessarily perpendicular to the accretion disk direction. Poutanen et al. [49] demonstrated, using the black hole binary MAXI J1820+070 as an example, that the line of sight, jet, and orbital plane normal point in different directions.

Additionally, Hα emission line features from the accretion disk around the black hole can be used to measure mass ratio q and orbital inclination i. Based on existing results from 11 black hole X-ray transients, Casares [50] performed a linear fit using the least squares method in 2016, deriving the following relationship:

$$\lg q = -(6.88 \pm 0.52) - (23.2 \pm 2.0)\lg W_{FWHM}$$

where DP and W_{FWHM} represent the double-peak separation and full width at half maximum of the Hα emission line profile, respectively. There is also a correlation between Hα double-peak valley depth T and orbital inclination i [51]:

$$i = (93.5 \pm 6.5)T + (23.7 \pm 2.5); \quad T = 1 - 2^{1-(DP/W)}$$

where W represents the FWHM value of a single Gaussian curve when fitting the Hα emission line with a double-Gaussian symmetric model. However, parameters in these methods are fitted from limited data points and may require more data to verify their accuracy.

3.2.1 Accretion Disk Optical Radiation Contamination

As seen in equation (1), black hole mass calculation depends on sin^3 i, so uncertainties in black hole mass measurements mainly come from errors in measuring the system's orbital plane inclination. The primary source of error for orbital inclination i is contamination from the accretion disk's optical radiation in the system's light curve, causing rapid aperiodic variations. High time-resolution optical monitoring of black hole X-ray transients shows that quiescent black hole X-ray transients can exhibit significant optical variability on timescales as short as minutes [52, 53]. This variability shows no obvious regularity and interferes with light curve modeling (both periodic analysis and ellipsoidal modulation analysis). This irregular variability is more prominent in black hole binary systems with cooler companion stars, and its characteristic timescale increases with longer orbital periods. These characteristics suggest this variability may originate from the accretion disk. Lower companion star temperature means its thermal radiation is weaker than the accretion disk's radiation, while in systems with hotter companion stars, the companion's radiation increases the system's total luminosity, making accretion disk radiation variations relatively less significant. Although there is no definitive evidence identifying the physical mechanism of such rapid variations, evidence shows that the amplitude of irregular variability is positively correlated with the proportion of accretion disk radiation measured spectroscopically: the higher the proportion of optical radiation contributed by the accretion disk in black hole X-ray transient systems, the larger the amplitude of irregular variability [54], proving that this irregular variability mainly comes from the accretion disk.

Assuming the accretion disk's optical radiation follows a negative power-law spectrum [55], previous work chose to observe and analyze in the near-infrared band, ignoring accretion disk radiation in this band to facilitate precise extraction of binary orbital parameters through ellipsoidal fitting. However, observations of some black hole X-ray transients in the near-infrared band show that the proportion contributed by the accretion disk in near-infrared spectra remains high, and its contamination of light curves cannot be ignored [56, 57], challenging the validity and reliability of using only near-infrared band light curves for pure ellipsoidal fitting [31].

Research proposing solutions to the accretion disk contamination problem originated from over a decade of optical variability monitoring of the quiescent black hole X-ray transient A 0620-00. Cantrell et al. [58] identified three main states in A 0620-00's optical light curve: "active state," "loop state," and "passive state," as shown in Figure 4 [FIGURE:4]. These three states can be distinguished based on average magnitude, color, and variability amplitude at different times, and also apply to other black hole binary systems. In the passive state, the binary system's radiation flux is lowest and shows minimal irregular variability, with the light curve most closely approximating ideal ellipsoidal modulation. As the system evolves toward loop and active states, it becomes brighter but irregular variability increases significantly, making ellipsoidal modulation signals less obvious. Therefore, light curves in the passive state are most suitable for ellipsoidal modulation fitting to determine binary system orbital inclinations. Cantrell et al. [59] reanalyzed A 0620-00's V, I, and H-band light curves in the passive state, finding after ellipsoidal fitting that the system's orbital inclination was about 10° higher than previous results. Earlier studies directly performed ellipsoidal fitting on near-infrared light curves without considering contamination from light sources other than the companion star (such as the accretion disk). Therefore, neglecting accretion disk contamination effects could lead to overestimation of black hole mass in A 0620-00 by about a factor of 2. This study emphasizes that to accurately measure binary system orbital inclinations, it is crucial to use light curves with minimal flickering activity during passive states. Another study of black hole X-ray transients also shows that quiescent black hole binary systems have a long-term gradual brightening trend in the optical band [60], indicating gradual accumulation of accretion disk density and increasing radiation [26, 27]. The period immediately after an X-ray outburst when the system has just entered quiescence has the dimmest optical radiation and lowest accretion disk contamination, making it the optimal time for dynamical studies of black hole X-ray transients through optical spectroscopy and photometry [60].

Note: The left panel shows the passive state, the middle panel shows the loop state (which can be considered a transitional state), and the right panel shows the active state. It is easy to see that the intensity of aperiodic variations gradually increases from passive to active states.

3.2.2 Superhump Modulation Phenomenon

Superhump modulation is another effect that can introduce systematic errors in binary orbital inclination. This is a period offset caused by precession of the accretion disk. The disk's precession period is much longer than the binary orbital period. The mixing of these two periodic signals causes us to detect a signal in the light curve with a period slightly longer than the orbital period, which we call the superhump period (P_sh). The relationship between these three periods can be expressed as: P_pr = (P_sh^-1 - P_orb^-1)^-1 [61-63], where P_pr represents the disk precession period, P_orb and P_sh are the orbital and superhump periods, respectively. Superhump modulation causes long-term variations in the shape and amplitude of ellipsoidal light curves, potentially creating offsets between the orbital inclination constrained from such ellipsoidal modulation curves and the true inclination value. The period measured in this case is inaccurate and should not be used to constrain inclination. The period from radial velocity curves is not affected by superhump modulation and represents the true orbital period.

3.2.3 Systematic Errors in Measuring v sin i

When modeling data to fit orbital parameters, models often cannot perfectly reproduce binary motion. Therefore, researchers usually make reasonable approximations that have minimal impact on results in areas beyond observational and theoretical reach. When a star fills its Roche lobe, it is very close to the compact object and no longer maintains a perfect spherical shape, with the side near the compact object being stretched. However, when measuring rotational broadening, a model spectrum with fixed velocity broadening is typically used without considering the star's actual shape variation with phase, resulting in v sin i values that often underestimate the mass ratio q [40]. Meanwhile, commonly used limb darkening and gravity darkening laws [64, 65] may lead to underestimation of true rotational broadening, resulting in smaller calculated mass ratios [66]. Despite these many possible sources of systematic error, statistical errors in measuring v sin i are usually larger than systematic errors [13]. Furthermore, considering that companion stars in black hole X-ray transients are much less massive than black holes (mass ratio q << 1), systematic errors in q have relatively small impact on the final black hole mass estimate.

4 Statistical Properties of Black Hole X-ray Transients

The statistical properties of black hole X-ray transients can provide important clues about black hole formation mechanisms. Over the past decades, using dynamical studies of these systems, researchers have constructed models for the mass and spatial distributions of stellar-mass black holes [67, 68], which can be compared and validated against supernova theory predictions and gravitational wave source observations [15, 69].

4.1 Statistics of Confirmed Black Holes

Table 1 [TABLE:1] summarizes the basic parameters of the 19 currently known dynamically confirmed black hole X-ray transients (sorted by discovery time). Typically, uncertainties listed in the table are 1σ standard deviations, though for orbital inclination errors, 90% or 95% confidence level data are sometimes provided. Since some sources have been detected multiple times over the past decades, multiple measurement results often exist for the same parameter, with varying credibility due to observational limitations. Parameter values in Table 1 favor results with cleaner, more reliable data—for example, higher spectral resolution and signal-to-noise ratio for measuring radial velocity and companion star rotational broadening, and lower irregular variability amplitude in light curves used to constrain system orbital inclination. Mass functions are obtained from companion star radial velocity curves during X-ray quiescent states. However, for the GX 339-4 system, although the companion star had not been detected in quiescence in the earliest studies, researchers derived a lower limit for the mass function using emission lines excited from the irradiated hemisphere of the companion star during outburst [70]. Later, Heida et al. [71] in 2017 used the companion star's spectrum to provide more complete parameter constraints for this system. Binary system orbital inclinations mainly come from ellipsoidal light curve model fitting results during quiescent states, but for GRS 1915+105, its inclination was inferred from radio jet direction [72]. For a few sources without obvious X-ray eclipses in their light curves but with measured mass ratios, upper limits on orbital inclination can be given. Figure 5 [FIGURE:5] shows the historical changes in the number of discovered and confirmed black hole X-ray transients, also indicating the operational periods of major X-ray satellites. In the past decade, more than 10 black hole X-ray transient candidates have been added, yet only two have been dynamically confirmed: MAXI J1305-704 and MAXI J1820+070. MAXI J1820+070 exhibits relatively special properties, still producing smaller outbursts three to four years after its X-ray outburst, indicating its accretion disk may remain in a relatively active state.

From Table 1, we can see that none of the confirmed black hole X-ray transients have particularly high orbital inclinations. In fact, other black hole candidates show the same characteristic. This may implicitly reflect observational selection effects or methodological biases. Some studies suggest that due to outward-curving, highly obscuring accretion disks around black holes, X-ray sources in high-inclination binary systems are effectively blocked, making them difficult to observe directly [74]. This warped disk phenomenon may be related to the accretion process and surrounding environment, such as heat distribution within the accretion disk and magnetic field structure, which collectively cause the warped morphology of accretion disks and affect observations of such high-inclination binary systems. The importance of discovering high-inclination black hole X-ray transients lies in the fact that for these systems, the uncertainty in black hole mass caused by inclination errors is small, promising relatively precise black hole mass measurements. Therefore, they will play a key role in constructing mass distributions of compact objects. Swift J1357.2-0933, with its very broad Hα emission line profile, extremely low peak X-ray brightness, and brightness trough signals in its optical light curve, suggests it may be a high orbital inclination black hole X-ray transient [75]. However, it shows no orbital modulation in the X-ray band (neither eclipses nor X-ray dimming features), and no strong emission or absorption lines were found in X-ray spectra from XMM-Newton satellite, casting doubt on its high-inclination hypothesis [76].

4.2 Spatial Distribution in the Milky Way

Figure 6 [FIGURE:6] shows the spatial distribution of 35 black hole X-ray transients and candidates with distance estimates in a Galactic polar view [73]. Dynamically confirmed black holes are marked with solid orange circles, while yellow star symbols represent massive black hole candidates not yet dynamically confirmed. Notably, about half of the dynamically confirmed black holes are distributed within a small region about 4.5 kpc from the Sun, indicating that interstellar extinction severely affects dynamical confirmation of more distant black holes. Interstellar extinction refers to intensity attenuation and color changes of starlight when passing through Milky Way dust clouds, posing severe challenges to methods relying on optical spectroscopy and photometry to determine black hole dynamical masses. Furthermore, an obvious regular distribution pattern is observed (see Figure 6): confirmed black holes (orange symbols) are not uniformly distributed throughout the Milky Way but concentrated near specific spiral arm structures. This discovery provides new clues for further exploring the distribution patterns of black holes in Milky Way evolution and their relationship with star formation environments. Based on this pattern, for black hole binary systems with large distance uncertainties, we can set credible distance intervals according to the approximate distance ranges of their located spiral arms, effectively improving the accuracy and reliability of distance parameter determination for such black hole systems. Black hole X-ray transient candidates (yellow symbols) are mainly distributed in the bulge, possibly related to higher stellar densities in this region.

Note: Dynamically confirmed black holes are marked with orange circles, while black hole candidates are marked with yellow stars. Error bars on each symbol indicate distance ranges.

Figure 7 [FIGURE:7] shows the distribution of black hole X-ray transients perpendicular to the Galactic plane, where the horizontal axis represents the distance from the transient source to the Galactic plane, and the vertical axis represents its space-time density. The vertical distribution of black hole binary systems should theoretically follow a similar exponential law as ordinary stars, i.e., spatial density decreases exponentially with increasing height above the Galactic plane [97]. As shown in Figure 7, the black solid line is the fitting result of exponential decay law. Furthermore, modern X-ray astronomy detection techniques show no obvious bias or selection effect in the vertical Galactic plane direction when searching for and identifying black hole binary systems [67]. That is, as long as certain X-ray radiation intensity conditions are met, the probability of detection is relatively high regardless of which vertical layer of the Milky Way the black hole binary system is located in. Therefore, the current sample can reflect the distribution characteristics of black hole X-ray transients perpendicular to the Galactic plane in the Milky Way. Gandhi et al. [98] explored the relationship between this spatial distribution characteristic and the orbital period of black hole X-ray transients, finding an anti-correlation: the farther from the Galactic plane, the shorter the system's orbital period. They proposed two possible physical mechanisms to explain this phenomenon: (1) Black hole X-ray binary systems originate in the Galactic disk, and their spatial scattering is caused by birth kicks, with only compact binary systems surviving strong kicks; (2) Binaries originate in the Galactic halo, with interactions in globular clusters shortening system orbital periods. We expect more correlations between spatial distribution and binary parameters to be discovered in the future, providing important observational constraints on binary system formation and evolution.

Note: The solid line represents the fitting result using the Levenberg-Marquardt least squares method.

4.3 Black Hole Mass Distribution

Based on data in Table 1, the black hole mass distribution in X-ray transients is obtained (as shown in Figure 8 [FIGURE:8]), with errors for most black hole masses within 2 M⊙. For black holes in Table 1 that only provide a mass range (often with large error bars), we use the midpoint of that range as the black hole mass value. The neutron star mass distribution [17] is also marked in Figure 8 for comparison. It can be seen that there is an obvious gap in the (2 ∼ 5) M⊙ interval between the mass distributions of neutron stars and black holes, which is the "mass gap" problem [67, 99]. Whether this mass gap truly exists has been widely studied. This issue is important not only because it is a blank region but also because some theoretical predictions of supernova explosions generating stellar-mass black holes suggest more black holes should exist in this interval [15]. Özel et al. [67] proved that selection effects from detection sensitivity to X-ray outbursts are insufficient to fully explain the existence of the "mass gap" when considering black hole mass measurement errors. This provides constraints for supernova explosion models (e.g., references [16, 17, 100, 101]). From an observational perspective, Kreidberg et al. [102] pointed out that accretion disk contamination of black hole X-ray transients in optical/infrared bands (see Section 3.2) can cause underestimation of binary system orbital inclinations, leading to overestimation of black hole masses. After considering this issue, black hole masses in some black hole X-ray transients, such as GRO J0422+32, may fall within the mass gap region. Precise mass measurements of more stellar-mass black holes can improve statistics of black hole mass distribution, provide more samples for mass gap research, and further constrain supernova explosion mechanisms and binary evolution.

5 Summary and Outlook

Black hole X-ray transients are X-ray binary systems composed of stellar-mass black holes and low-mass stars. This paper introduces the background of such systems and their dynamical research methods. We mainly search for low-mass black hole X-ray binaries through detected X-ray outbursts. Generally, binary systems return to X-ray quiescent states within months after outbursts, allowing us to study the entire binary system's orbital parameters through companion star spectroscopy and photometry. Combining Kepler's third law and stellar atmosphere models, we derive the mass range of the central compact object. If the measured mass exceeds 3 M⊙, we can identify the compact object as a black hole. For different systems, the measurement process inevitably has more or less error, often determined by the nature of observational data itself. We must carefully identify and select the most reliable data, analyzing reasonable error ranges to ensure result reliability. Additionally, we conducted statistical analysis of the black hole X-ray transient sample, studying patterns in their spatial distribution within the Milky Way and black hole mass distribution. Stellar-mass black holes and their candidates tend to be distributed near spiral arms and the bulge region of the Milky Way, with most systems concentrated near the disk plane. For the existing sample's mass distribution, the mass gap between neutron stars and black holes remains significant. However, whether this mass gap is caused by sample incompleteness and selection effects remains to be determined.

For the existing sample, many black hole or candidate masses in X-ray transient systems have not been sufficiently constrained, requiring further photometric monitoring to select phases with less accretion disk contamination for precise dynamical modeling. Regarding X-ray outburst frequency, we find that the vast majority of black hole transients have only been observed to have one outburst event, with repeated outbursts being rare. Studies have also investigated the relationship between X-ray outburst periods and black hole fundamental parameters. Lin et al. [103] found that a 12 h orbital period is a turning point for outburst frequency: transient systems with periods below this show only one outburst in X-ray observational history, while sources with multiple outbursts generally have orbital periods longer than 12 h. This phenomenon may be related to different mass transfer rates from companion stars at different orbital periods. Therefore, future continuous X-ray monitoring is of great importance for studying the evolution of black hole X-ray transients.

Stellar evolution theory predicts that the number of stellar-mass black holes that may exist in the Milky Way is greater than 10^8 [104]. In comparison, the proportion of discovered stellar-mass black holes and their candidates in the Milky Way is extremely small. Traditional methods for detecting stellar-mass black holes rely on X-ray observations. X-ray satellites with higher detection sensitivity and wider field coverage, such as China's Einstein Probe satellite, are expected to discover more black hole X-ray binaries. Meanwhile, given that most black hole binaries are in X-ray quiescent states (or even lack mass transfer), directly searching for and confirming black holes from a dynamical perspective, independent of X-ray observations, is an effective means to substantially expand the stellar-mass black hole sample. More and more telescopes are now in operation, such as China's Large Area Multi-Object Spectroscopic Telescope (LAMOST), China Space Station Telescope (CSST), Multiplexed Survey Telescope (MUST), and Europe's Gaia satellite, which can provide large-scale stellar spectroscopic, photometric, and astrometric databases through survey modes, thereby discovering more black holes and candidates and potentially producing representative samples of stellar-mass black holes to provide important evidence for black hole formation and evolution studies. Algorithms and software for processing large-scale survey data are also developing vigorously. Several black hole binaries or candidates have already been discovered based on these survey databases, such as LB-1 [105, 106], 2MASS J05215658+4359220 [107], V723 Mon [108], Gaia BH1 [109], BH2 [110], and BH3 [111]. These systems typically have long orbital periods (hundreds of days to decades) and high orbital eccentricities (up to 0.5). Comprehensive analysis of stellar-mass black holes discovered through optical surveys, combined with samples obtained through gravitational wave detection (nearly 100 cases), gravitational lensing, X-ray transients, and other methods, can greatly expand the stellar-mass black hole sample and obtain more accurate stellar-mass black hole mass distributions, providing better constraints on black hole formation mechanisms.

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