Preamble

Vol. 66 No. 4

July 2025

Acta Astronomica Sinica

doi: 10.15940/j.cnki.0001-5245.2025.04.005

An Overview of the Insight-HXMT Scientific Satellite Measuring Fundamental Parameters of Black Holes*

WANG Wei† and ZHU Hai-fan

(Department of Astronomy, School of Physics and Technology, Wuhan University, Wuhan 430072)

Abstract

This paper systematically reviews the primary methods for measuring the fundamental parameters of black holes—mass and spin—and briefly introduces techniques for determining the distances to black hole binary systems. Through case studies, it highlights the importance of precise distance measurements in refining black hole fundamental parameters. We explore three common approaches to black hole mass measurement from the perspectives of dynamical measurements, spectral fitting, and quasi-periodic oscillations (QPOs), while discussing their practical applications and inherent limitations based on observational results. In the aspect of spin measurement, the focus is on the research achievements of the Insight-HXMT satellite. The paper provides an in-depth analysis of two methods: thermal continuum fitting and reflection component fitting, emphasizing the crucial role of the Insight-HXMT satellite in spin research. It meticulously dissects the theoretical foundations and model assumptions of these two methods and presents their measurement processes through examples. Furthermore, we showcase the role and advantages of the Insight-HXMT satellite in spin measurement using these methods.

Keywords: black hole physics, X-rays: binaries, accretion disk, radiation mechanism

1 Introduction

Black holes serve as unique laboratories for physics and astronomy, offering us opportunities to glimpse the mysteries of the universe and opening doors to observe a series of fundamental astrophysical phenomena. These include critically important and complex processes such as accretion, relativistic jets, and gamma-ray burst generation, all occurring in the vicinity of black holes and representing key focal points in astrophysical research. From binary systems to ultra-luminous X-ray sources, galaxies, and quasars across various scales, the study of black holes constitutes a vital component. Stellar-mass black holes are considered the most ideal research targets because they provide unique and relatively reliable samples that facilitate in-depth exploration and understanding of these astrophysical phenomena. In astrophysics, black holes are characterized by only two parameters: mass and spin. This makes a deep understanding of black hole mass and spin crucial for deciphering spacetime structure in strong gravitational fields. These parameters not only determine the properties of spacetime around black holes but also play key roles in how various gravitational phenomena manifest. Therefore, precise knowledge of these two critical parameters enables us to more comprehensively understand black hole behavior in the universe and significantly enhances our understanding of stellar evolution.

Accurate black hole mass measurements are essential for validating massive protostars, Type Ibc supernova explosions, and binary evolution models. Among the notable issues is the mass gap problem: a significant deficit has been found between the maximum neutron star mass and the lower end of the black hole mass distribution. However, the existence of this mass gap remains controversial. Kreidberg et al. argue that an important systematic error source was neglected in the analysis, and the mass gap is likely caused by underestimation of inclination angles. By correcting the black hole mass in GRO J0422+32 based on inclination angle estimation biases, they found that this black hole might fall within the mass gap. However, they also note that if this source is excluded from the analysis, previous conclusions still hold. On the other hand, Belczynski et al. suggest the mass gap might be real and could reveal new theories about supernova explosion models. Kochanek proposed an alternative explanation based on the fact that red supergiants in the 16–25 M⊙ range do not serve as progenitors of Type IIp supernovae: due to their weak hydrogen envelopes, these massive stars eject their outer layers, leaving behind black holes with masses equivalent to the star's helium core (5–8 M⊙). This could explain both the absence of supernova progenitors in the 16–25 M⊙ range and the existence of the mass gap. Precise black hole mass measurements are key to resolving the mass gap problem.

Spin is also a crucial parameter for tracking black hole formation and evolution. The stellar-mass black holes discovered today in Galactic X-ray binary systems formed through the core collapse of massive stars. In this context, black hole spin information reflects the physical processes occurring during stellar core collapse. Black hole evolution, particularly its spin, begins with gravitational collapse of the star, imparting non-zero spin to the black hole. The initial spin state depends on the angular momentum of the progenitor star and the magnetohydrodynamics of rotating stellar core collapse. Black hole spin can also change through mergers. Additionally, spin can grow by accreting nearby matter (such as plasma and gas). Gammie et al., based on relativistic collapse calculations, obtained black hole spin ranges of approximately 0.7–0.9, while relativistic magnetohydrodynamic models describing accretion onto rotating black holes show that accretion does not necessarily lead to extreme spins, and even thin-disk accretion may fail to produce spins approaching unity. In contrast, all current spin measurement methods (except those involving gravitational waves, which represent possible future techniques) utilize accretion disks to infer black hole spin. The two most commonly used methods include reflection component fitting and continuum fitting. Both methods assume a geometrically thin disk with high radiative efficiency, and the fitted radiation essentially terminates within the inner stable circular orbit. Yan et al. compiled current black hole spin measurement results and found a clear bimodal distribution (peaks at spin values of 0.17 and 0.8). They suggest that low-spin black holes (a < 0.3) resemble neutron star low-mass X-ray binaries, where compact objects increase their spin through low-level accretion, while high-spin black holes (a > 0.5) have experienced a brief supercritical accretion state that rapidly increased their spin. These two scenarios ultimately produce the observed bimodal distribution. Precise and more extensive spin measurements will enable us to better understand black hole formation and evolution processes.

The Insight-HXMT satellite was successfully launched on June 15, 2017, from the Jiuquan Satellite Launch Center in northwestern China, becoming China's first X-ray astronomy satellite. Its Chinese name honors Academician He Zehui, while its English name is Insight Hard X-ray Modulation Telescope (Insight-HXMT). It operates in a low Earth orbit at an altitude of 550 km with an inclination of 43°. The primary scientific objectives of the Insight-HXMT satellite include: searching for new transients in the Galactic plane and monitoring known variable sources; observing X-ray binaries to study motion and radiation mechanisms in strong gravitational or magnetic fields; and monitoring and studying gamma-ray bursts and electromagnetic counterparts of gravitational wave events. To meet scientific observation requirements, the satellite features three attitude control modes. The first is all-sky survey mode: the sunshade is perpendicular to the solar direction, and the satellite rotates slowly around the solar direction to keep Earth outside the telescope's field of view. In this mode, the entire sky can be covered within half a year. The second is pointed observation mode: the satellite operates in three-axis stabilization mode, with the telescope's optical axis pointing at the observation target for a sustained period. The third is small-area scanning mode: the telescope's optical axis changes slowly according to a planned trajectory to cover specific sky regions. The Insight-HXMT satellite carries three telescopes: the Low Energy Telescope with a total detection area of 384 cm², observing in the 1–12 keV band with a time resolution of 1 ms; the Medium Energy Telescope with a total detection area of 952 cm², observing in the 5–30 keV band with a time resolution of 255 μs; and the High Energy Telescope with a total detection area of 5100 cm², observing in the 20–350 keV band with a time resolution of 4 μs.

The main advantages of Insight-HXMT observations include broad energy coverage, large effective area in the high-energy X-ray band, high time resolution, minimal detector dead time, and absence of photon pile-up effects for bright sources. Since its stable operation in orbit, the satellite has achieved fruitful results in multiple fields, particularly making significant progress in the study of X-ray binaries, such as constructing and improving the outburst evolution picture of black hole binary systems, studying quasi-periodic oscillations (QPOs) and timing properties during black hole outbursts, and investigating fundamental black hole properties and accretion disk corona structures. This paper primarily introduces measurement methods for black hole fundamental parameters—mass and spin—combined with research results from the Insight-HXMT satellite. Black hole mass measurement mainly employs three methods: dynamical methods, spectral fitting, and quasi-periodic oscillations, while spin measurement primarily uses thermal continuum fitting and reflection component fitting.

2 Distance Measurement

Distance determination is crucial for measuring black hole fundamental parameters, such as mass and luminosity calculations, as well as spin measurement through the thermal continuum method. Therefore, before discussing black hole mass and spin measurements, we briefly overview distance measurement methods and introduce distance measurement results for several sources mentioned later. The distances to black hole X-ray binary systems are typically estimated by studying the optical/infrared spectra of the companion star. Additionally, jet proper motion can provide upper limits on distance, while lower limits can be estimated through interstellar extinction or HI absorption lines. X-ray dust scattering halos have also been used to constrain distances to some black hole X-ray binary systems. However, these methods all depend on models and involve certain assumptions.

The only model-independent distance determination method is high-precision trigonometric parallax measurement. However, since the typical distances to these systems are several kiloparsecs, high-precision observations can only be achieved through Very Long Baseline Interferometry (VLBI) or instruments like the Gaia satellite. Gaia's high-precision measurements of black hole X-ray binary systems in the Galactic plane may be limited by high extinction and low optical brightness outside outburst periods. Furthermore, Gaia's parallax measurements suffer from a global zero-point offset whose magnitude remains controversial. Therefore, targeted VLBI astrometric observations of black hole X-ray binary systems during outburst are still essential. Among currently observed black hole X-ray binary system candidates, only systems such as V404 Cyg, Cygnus X-1, Cygnus X-3, and GRS 1915+105 have relatively accurate parallax measurements.

Atri et al. used the Very Long Baseline Array (VLBA) and the European VLBI Network (EVN) to precisely measure the parallax of the black hole X-ray binary system MAXI J1820+070, providing a model-independent distance estimate. The measured parallax was 0.348 ± 0.033 mas, corresponding to a distance of 2.96 ± 0.33 kpc. This distance indicates that the source reached 15% ± 3% of the Eddington luminosity at its outburst peak. Additionally, using this distance to revise previous estimates of jet inclination, jet velocity, and black hole mass yielded results of 63° ± 3°, (0.89 ± 0.09)c, and (9.2 ± 1.3) M⊙, respectively. Chauhan et al. used the Australian Square Kilometre Array Pathfinder (ASKAP) and MeerKAT to observe neutral hydrogen absorption spectra of the black hole X-ray binary MAXI J1348–630, obtaining results indicating a most probable distance of 2.2^{+0.5}_{-0.6} kpc for MAXI J1348–630, with a strong upper limit of 5.3 ± 0.1 kpc. Further distance estimates show that MAXI J1348–630 reached 17% ± 10% of the Eddington luminosity at its outburst peak, and the luminosity during the soft-to-hard state transition was 2.5% ± 1.5% of the Eddington luminosity.

3 Black Hole Mass Measurement

3.1 Dynamical Methods

Black hole mass measurement is a complex yet critical problem in astronomy. For stellar-mass black holes, the most easily observable systems are those forming binaries with ordinary stars. The most common method for measuring their mass utilizes the motion characteristics of the visible star to obtain the mass of the primary (black hole), generally requiring optical band observations to determine the visible star's motion. In such systems, the visible star's radial velocity exhibits periodic variations as it orbits the black hole. According to Kepler's laws, we can derive the mass function:

$$f(M) = \frac{(M_1 \sin i)^3}{(M_1 + M_2)^2} = \frac{P_{\rm orb} K_2^3}{2\pi G}$$

where $M_1$ and $M_2$ are the masses of the primary and visible star, respectively, $i$ is the binary system inclination, $P_{\rm orb}$ is the orbital period, $K_2$ is the velocity semi-amplitude, and $G$ is the gravitational constant. For visible stars with multi-band observational data, we can use spectral energy distribution fitting to obtain the visible star's parameters and consequently derive more accurate black hole mass estimates. However, in most cases, sufficient observational data are unavailable, requiring many assumptions for estimation. Assuming accretion occurs through Roche lobe overflow, we can calculate the visible star's mass and radius.

Taking MAXI J1820+070 as an example, we briefly introduce the estimation process. [FIGURE:1] shows the observed periodic radial velocity curve of the visible star. Fitting these data yields information such as a period of $0.68549 \pm 0.00001$ d and a velocity semi-amplitude of $417.7 \pm 3.9$ km s⁻¹, giving a mass function of $f(M) = \frac{P_{\rm orb} K_2^3}{2\pi G} = 5.18 \pm 0.15$ M⊙. Spectral analysis of the observations reveals the visible star's spectral type as K3–K5. According to stellar accretion theory, when the visible star fills its Roche lobe, the orbital period and stellar mean density relate as:

$$\bar{\rho} = \frac{3M}{4\pi R^3} = \frac{35}{8G} \frac{P_{\rm orb}^2}{110\pi}$$

where $R$ is the stellar radius. K3-type dwarfs typically have a mean density of $2.7$ g cm⁻³, yielding a period of about 6.4 h. Given MAXI J1820+070's orbital period of $0.68549 \pm 0.00001$ d, its companion must clearly be an evolved star to fill its Roche lobe. Evolved companions are not uncommon in accreting binaries. Patterson et al.'s photometric results show a period of $0.703 \pm 0.003$ d; the difference between photometric and orbital periods arises from a long hump period caused by precession of the binary's accretion disk. Using the relationship between these two periods yields a mass ratio of 0.12. While Torres et al. discuss many scenarios in their estimation, we provide a simpler calculation here. According to Frank et al., the relationship between visible star mass and period gives a visible star mass of approximately $1.72$ M⊙. Assuming an inclination of 77°, substituting into the mass function yields a compact object mass of about $8.27$ M⊙, consistent with Torres et al.'s value of 7–8 M⊙.

[FIGURE:1] illustrates optical observations of MAXI J1820+070's companion star using the 10.4 m Gran Telescopio Canarias (GTC) and 4.2 m William Herschel Telescope (WHT). The radial velocity (Vel) shows clear periodic variations. The figure is from Torres et al.

The above example using MAXI J1820+070 briefly introduces one method for estimating black hole X-ray binary masses, which primarily relies on optical observations of the visible star to obtain required parameters. In another scenario, if periodic attenuation is observed in X-ray light curves, this period can be considered the orbital period, and similar methods can be applied for estimation, though more assumptions are required and results have greater uncertainty. For example, when estimating MAXI J1803–298's mass using this method, the X-ray light curve shows periodic variations with a 7 h period, with velocities derived from optical observations. Using this information yields a mass range of 3–10 M⊙; specific details can be found in Sánchez et al.

Dynamical methods can yield relatively precise black hole mass measurements because they directly utilize the motion characteristics of visible stars in binary systems. By measuring periodic variations in radial velocity, we can obtain the primary star's (black hole) mass. This process is based on Kepler's laws using parameters like the mass function for estimation. However, despite theoretical validity, practical observations present difficulties that limit its application. A major limitation is that many X-ray binary systems lack high-quality optical data for their counterparts. In many cases, strong accretion disk radiation or obscuration effects make accurate detection of the visible star's optical characteristics difficult. This prevents obtaining sufficient information through traditional optical observations, affecting accurate measurement of the visible star's motion. Although dynamical methods are theoretically robust, practical applications must overcome observational challenges. To compensate for these limitations, researchers seek alternative methods to estimate black hole mass, such as using periodic features in X-ray light curves. However, these methods' applicability and accuracy depend on various factors, including accretion disk properties and system geometry, resulting in greater mass estimation uncertainty and lower reliability. In summary, while dynamical methods provide an effective approach for black hole mass measurement, optical counterpart observations of many X-ray binary systems are limited, making this method's application significantly constrained. This motivates researchers to continuously explore other observational techniques and methods.

3.2 Spectral Fitting

In addition to radial velocity measurements, X-ray telescopes alone can provide approximate black hole mass ranges. In X-ray astronomy, the main information we can obtain includes X-ray variability and spectra. Fitting spectra to derive various parameters is a common and widely used method, particularly for black hole binaries lacking dynamical studies. Using simple models can yield black hole mass, but results often have large errors. For example, using a combination of nthcomp and kerrbb models to study MAXI J1803–298 gives a mass range of 7.9–9.9 M⊙, which is significantly larger than the 8.5–16 M⊙ value obtained by Sánchez et al., mainly due to uncertainties in distance determination and different assumed inclination ranges used in model fitting.

When studying MAXI J1535–571, Shang et al. adopted the Two-Component Advective Flow (TCAF) model to fit the spectrum. The TCAF model originates from solutions of radiation hydrodynamics equations, where the "hot" Compton cloud in previous models is replaced by a region formed by the accumulation of low-viscosity (subcritical), low angular momentum, and optically thin material behind a centrifugal barrier. This region is called CENBOL (CENtrifugal pressure-supported BOundary Layer), and this material is referred to as the sub-Keplerian or halo accretion component. The other component of the accretion flow is high-viscosity, high angular momentum, geometrically thin, and optically thick Keplerian disk material immersed within the halo component. According to this model, the Keplerian disk is naturally truncated at the shock location, which is also the outer boundary of CENBOL. Low-energy (soft) thermal photons from the Keplerian disk interact with CENBOL (composed of hot electrons), emitting high-energy (hard) photons through repeated inverse Compton scattering, thereby cooling CENBOL. Some emitted hard photons are reflected by the Keplerian disk, and this iterative process self-consistently produces a so-called reflection component. This model can provide relatively accurate accretion parameters in many black hole X-ray binary sources and has thus been used to study MAXI J1535–571's properties.

For a black hole binary system, if sufficient dynamical studies have determined the mass, this parameter serves as an input in the model. However, MAXI J1535–571 lacks relevant dynamical studies, making mass an unknown quantity treated as a free parameter in model fitting. [FIGURE:2] shows the evolution of relevant parameters throughout the outburst, where black hole mass is a free parameter but the fitting results are roughly 8–11 M⊙, with an average mass of 9.1^{+1.6}_{-1.2} M⊙ throughout the outburst.

[FIGURE:2] presents the spectral fitting results for the outburst of MAXI J1535–571. Panel (a) displays the fitting outcomes with black hole mass as a free parameter. Panel (b) illustrates the ratio of mass accretion rates (ARR = \dot{m}_h/\dot{m}_d), where \dot{m}_d represents the mass accretion rate of the Keplerian disk and \dot{m}_h corresponds to the mass accretion rate of the sub-Keplerian or halo component. Panel (c) shows the frequency variation of QPOs during the outburst, while panel (d) depicts the power-law spectral index. The vertical dashed lines represent transition points between different states, marking the transitions from the hard state (HS) to the hard intermediate state (HIMS) and further to the soft intermediate state (SIMS). These results are sourced from Shang et al.

Another well-studied source is MAXI J1348–630. Jana et al. also used the TCAF model to fit X-ray observational data of this source throughout its outburst, treating mass as a free parameter to obtain the best-fit value, with results shown in [FIGURE:3]. The mass fitting results show little variation throughout the outburst, remaining between 8.44 and 9.72 M⊙. Averaging these fitting results yields a black hole mass of (8.9 ± 1.0) M⊙. Tominaga et al. used the simpler kerrbb model to fit this source. Since many parameters in this model are coupled, the mass cannot be accurately determined. With distance fixed at 4 kpc, the masses obtained for several given spin and inclination sets were 7 M⊙ (a=0; i=0°), 14 M⊙ (a=0; i=60°), 18 M⊙ (a=0.998; i=0°), and 76 M⊙ (a=0.998; i=60°). Soria et al. used optical and X-ray observational data of MAXI J0637–430 to well constrain black hole parameters, employing radiation characteristics of inner and outer disks combined with optical He II λ4686 features to constrain the black hole mass range to 4–7 M⊙.

[FIGURE:3] illustrates the spectral fitting results for the outburst of MAXI J1348–630, where (a) presents the fitting outcomes with black hole mass as a free parameter, (b) indicates the shock position (X_s in units of the Schwarzschild radius R_s), and (c) displays the shock compression ratio (R = ρ_+/ρ_-, the ratio of post-shock to pre-shock density) during the outburst. The vertical dashed lines represent temporal transition points between different states, delineating the complete evolutionary sequence from HS → HIMS → SIMS → Soft State (SS) → SIMS → HIMS → HS. These results are sourced from Jana et al.

3.3 Quasi-Periodic Oscillations

Quasi-periodic oscillations are commonly observed in black hole X-ray binaries. They appear as narrow peaks in Fourier power density spectra calculated from rapidly varying light curves, indicating quasi-periodic variations in the light curve. Studying this phenomenon helps us understand black hole accretion processes, construct physical pictures of the geometry around black holes, and measure fundamental black hole parameters. Many methods have been developed for QPO research. Chen et al. developed a wavelet analysis method for QPO analysis based on Insight-HXMT observations of MAXI J1535–571, finding a positive correlation between QPO signal frequency and photon count rate in energy bands above 10 keV, implying that QPO signals are related to coronal activity. Jin et al. used Insight-HXMT data to conduct detailed studies of QPO phenomena during outbursts of GX 339–4. Zhu et al. performed timing analysis of MAXI J1803–298 based on Insight-HXMT observations, studying the relationship between QPO phenomena and energy. Numerous additional studies have used QPOs to investigate black hole surrounding geometry and disk-corona evolution. This paper focuses on detailing the basic process of using QPOs to measure black hole mass.

Black hole mass estimates obtained through spectral fitting are largely correlated with the accuracy of other parameters. Titarchuk et al. developed a scaling technique for determining black hole mass based on Shaposhnikov et al. GRO J1655–40 is a Galactic black hole binary source whose mass, distance, and orbital inclination have been very precisely determined through dynamical methods. Fitting the spectrum of this source's outburst using the BMC (bulk-motion Comptonization) model to describe the Compton component, this model describes the Compton scattering of soft photons by relativistically moving matter. Model parameters include the characteristic blackbody temperature of the soft photon source, spectral (energy) index, and irradiation parameter characterizing the illumination of the moving matter flow by the thermal photon source. It must be emphasized that this model is not a simple additive combination of power spectra and thermal sources but represents a self-consistent convolution. Bulk-motion upscattering and Compton recoil combine to produce a hard spectral tail, which combined with the thermal source produces the classic high/soft state spectrum of black hole accretion. The scaling method primarily requires parameter information including the spectral index and model normalization. Many QPO phenomena were discovered during this source's outburst, and fitting QPO frequencies revealed strong correlations with spectral index and normalization parameters. Shaposhnikov et al. found similar correlations in black hole binary sources including Cygnus X-1, GX 339–4, 4U 1543–47, XTE J1550–564, XTE J1650–500, H 1743–322, and XTE J1859–226, with results shown in [FIGURE:4].

[FIGURE:4] shows the correlation and fitting results between spectral index/model normalization parameter and QPO frequency for several sources. The figure with frequency on the horizontal axis presents data for mass determination using the scaling method, while the figure with BMC normalization on the horizontal axis presents data for distance determination using the scaling method.

The core idea of the scaling method is based on using a source with determined mass and clear evolutionary process as a reference to compare with new data and obtain the mass of new sources. As mentioned earlier, GRO J1655–40's parameters have been precisely determined through dynamical methods, making it the reference source, with other sources as targets. The specific operation method is briefly introduced as follows. First, define a scaling factor:

$$s = \frac{\nu_r}{\nu_t} = \frac{M_t}{M_r}$$

where $\nu$ represents QPO frequency and $M$ represents mass, with subscripts r and t denoting reference and target sources, respectively. This definition is based on the inverse relationship between low-frequency QPO frequency and central black hole mass. The results presented in [FIGURE:4] all show similar functional relationships: monotonic increase before reaching a specific frequency, the cutoff frequency $\nu_{\rm tr}$; and smoothly transitioning to a horizontal line near the cutoff frequency. Based on this characteristic, select the following function for fitting:

$$f(\nu) = A - D \ln[\exp((\nu_{\rm tr} - \nu)/B) + 1]$$

where $A$, $B$, and $D$ are parameters. On the other hand, for $\nu \ll \nu_{\rm tr}$, the slope equals $D/B$, from which we can understand the meaning of parameter $B$ as the slope. For $\nu \gg \nu_{\rm tr}$, $f(\nu) = A$, so parameter $A$ is the value where the spectral index remains constant. Parameter $B$ is introduced to control the speed of functional form transition. Considering the defined scaling factor, the transformation yields:

$$M_t = M_r \frac{\nu_{\rm tr;r}}{\nu_{\rm tr;t}}$$

where $M_t$ and $M_r$ are the masses of target and reference sources, respectively, and $\nu_{\rm tr;t}$ and $\nu_{\rm tr;r}$ are the cutoff frequencies of target and reference sources, respectively. This method can determine target source mass based on the observed relationship between QPO frequency and spectral index, given a reference source. As mentioned earlier, there are relationships not only between spectral index and frequency but also between model normalization parameter and frequency; the latter can be used to estimate distance, with the same conceptual approach, which will not be repeated here. [TABLE:1] compares some results with dynamical results.

[TABLE:1] Black hole masses and distances derived with scaling technique

Source Dynamical mass/ M⊙ Inclination/ ° Distance/ kpc Scaled mass/ M⊙ Scaled distance/kpc References GRO J1655–40 6.3 ± 0.3 70 ± 2 3.2 ± 0.2 >67.5 ± 1.6 12.3 ± 1.4 Greene et al. GX 339–4 5.75 ± 0.8 >50 2.6 ± 0.7 9.4 ± 1.0 9.4 ± 1.4 Muñoz-Darias et al. 4U 1543–47 9.4 ± 1.0 20.7 ± 1.5 7.5 ± 1.0 9.4 ± 1.4 9.4 ± 1.8 Park et al. XTE J1550–564 9.5 ± 1.1 72 ± 5 2.5^{+1.6}_{-0.9} 10.7 ± 1.5 3.3 ± 0.5 Orosz et al. XTE J1650–500 2.7^{+0.7}_{-0.3} >50 2.6 ± 0.7 9.7 ± 1.6 3.3 ± 0.7 Orosz et al. H 1743–322 3.3 ± 0.7 >50 8.5^{+2.0}_{-1.8} 14.8 ± 1.0 35 ± 5 McClintock et al. XTE J1859+226 2.5^{+0.6}_{-0.4} >50 7.9 ± 1.0 2.2 ± 0.3 14.8 ± 0.9 Yanes-Rizo et al. Cyg X1 14.8 ± 1.0 27.1 ± 0.5 2.5 ± 0.1 3.4 ± 0.4 2.5 ± 0.3 Orosz et al. MAXI J1348–630 9.1 ± 1.5 50 ± 2 2.96 ± 0.33 9.5 ± 1.1 2.5 ± 0.3 Titarchuk et al.

¹ Parameter values derived from dynamical methods, serving as reference data.

² The values are given assuming the dynamical measurements of XTE J1550–564 serve as the reference source.

4 Black Hole Spin Measurement

Spin is a fundamental physical quantity of black holes that affects spacetime around them. Spin magnitude influences the innermost stable orbit around the black hole, thereby affecting the accretion process. Therefore, spin is also an important physical quantity to consider when studying black hole accretion disk properties and evolution. Currently, two methods are widely used to measure black hole spin: thermal continuum fitting and X-ray reflection feature fitting. Additionally, there is a method using QPOs to measure spin, but this method has high requirements for observed QPOs, limited applicability, and has not been widely used. Therefore, the following sections mainly introduce the two mainstream methods.

4.1 Thermal Continuum Fitting

The thermal continuum fitting method is based on the fact that black hole spin affects the inner disk temperature, which is highest during black hole accretion. Since spin influences the position of the innermost stable circular orbit, it affects the location of the accretion disk's inner edge. Under the context of specific accretion disk models, this influence can be quantified, enabling spin determination. This method is particularly suitable for systems with geometrically thin, optically thick, steady accretion disks and weak X-ray coronae. The structure of such accretion disks was calculated by Novikov & Thorne, with main assumptions including steady state, constant inward mass flow (i.e., negligible mass loss due to disk winds), angular momentum loss through internal disk stresses (i.e., negligible external torque due to large-scale magnetic fields), and local radiation of energy released in the flow. Finally, an inner boundary condition must be specified for the disk model, typically assuming internal stress vanishes at the innermost stable circular orbit (the so-called zero-torque boundary condition).

Applying the thermal continuum fitting method to measure spin requires several elements. First, the system description must conform well to the disk model proposed by Novikov & Thorne, meaning no (or weak) disk winds and spectra dominated by thermal radiation from the optically thick accretion disk. Then, models with a color correction factor $f_{\rm col}$ are used for fitting, a function derived from detailed radiative transfer calculations. Black hole X-ray binaries cycle through different states during outburst, one of which is the so-called thermal dominant state, very close to the aforementioned disk model. Therefore, for stellar-mass black holes, we can monitor and wait for the system to enter the thermal dominant state before using the obtained spectrum to determine spin.

The Insight-HXMT satellite's exceptional detection capabilities across a broad energy range provide significant advantages for spectral research, particularly in studying black hole X-ray binary spins through spectroscopy. MAXI J1820+070 is a Galactic black hole X-ray binary that outburst on May 11, 2018, exceeding 4 Crab units in brightness and becoming one of the brightest X-ray transients. Guan et al. analyzed the spectrum of MAXI J1820+070 during its soft state (from MJD 58310 to MJD 58380), utilizing 49 observations from the Insight-HXMT satellite for detailed spectral analysis. [TABLE:2] shows the models and energy bands used in the study. Continuum fitting used both low-energy and medium-energy bands, demonstrating the advantage of Insight-HXMT's broad energy range in spectral research. The main parameters to be specified in the model include black hole mass $M = 8.48$ M⊙, inclination $i = 63°$, and distance $D = 2.96$ kpc.

[TABLE:2] The models and energy bands utilized by Guan et al.

Model Energy bands M1: constant × TBabs × (diskbb+powerlaw) 2–5 keV (LE), 10–20 keV (ME) M2: constant × TBabs × (simpl × kerrbb2) 2–5 keV (LE), 10–20 keV (ME) M3¹: constant × TBabs × (simpl × kerrbb) 2–5 keV (LE), 10–20 keV (ME) M4: constant × TBabs × (simplr × kerrbb2 + kerrconv × (ireflect × simplc)) 2–10 keV (LE), 10–35 keV (ME)

¹ M3 is employed solely for comparison against M2.

We present the fitting results in [FIGURE:5]. The results show a clear evolutionary trend in spin during the selected time period. However, since the physical spin evolution timescale for stellar-mass black holes due to accretion is about $3 \times 10^5$ yr, we infer that MAXI J1820+070's spin evolution is clearly non-physical and must be related to problems in estimating the innermost stable circular orbit radius, i.e., related to $R_{\rm in}$ in M1. [FIGURE:5] indeed shows $R_{\rm in}$ decreasing around MJD 58330, then remaining stable in epoch 3. Therefore, during epoch 3 the disk extends to the inner stable circular orbit radius, and the spin values obtained during this period ($a_* = 0.2^{+0.2}_{-0.3}$) are reliable.

Zhao et al. also measured MAXI J1820+070's spin using continuum fitting, studying spectra in intermediate and soft states with both non-relativistic models (constant × TBabs × simpl × diskbb) and relativistic models (constant × TBabs × simpl × kerrbb2). Here, TBabs describes interstellar medium absorption of X-rays, simpl is an empirical model describing Compton scattering, and kerrbb2 is an improved multi-blackbody disk model that more accurately simulates radiation from rotating black hole accretion disks through a hardening factor. The final spin parameter range gave a spin value of $a_* = 0.14 \pm 0.09$, which is basically consistent with Guan et al.'s results.

In addition to MAXI J1820+070, spin measurements have been performed for multiple sources using Insight-HXMT spectral observations, such as GRS 1915+105, MAXI J1348–630, and Cygnus X-1. We list only the results for these sources below.

Zhao et al. used Insight-HXMT observations of Cygnus X-1 from August 24, 2017, to August 5, 2018, fitting spectra with both non-relativistic (constant × TBabs × simpl × diskbb) and relativistic (constant × TBabs × simpl × kerrbb2) models. They reported a spin of $a_ > 0.967$ (under different mass-distance parameters), confirming this as an extreme spin black hole. Wang et al. used Insight-HXMT observations to study GRS 1915+105's spectral characteristics throughout the entire observation period. The results show that spectra can be well fitted with a combined model containing disk components (diskbb/kerrbb) and power-law components (simpl). The lower limit on black hole spin given by the kerrbb model is 0.9990, close to the maximum value of 1, thus verifying that GRS 1915+105 is an extreme spin black hole. Wu et al. used Insight-HXMT observations in the 2–20 keV range from MJD 58588 to MJD 58596 to measure the spin of the stellar-mass black hole in MAXI J1348–630 through continuum fitting, confirming this source's spin value as $a_ = 0.42^{+0.13}{-0.50}$. Yorgancioglu et al. used Insight-HXMT and Neutron star Interior Composition Explorer (NICER) spectral data to measure the spin of 4U 1543–47 through thermal continuum fitting, obtaining a spin result of $a = 0.65^{+0.14}{-0.24}$. Subsequently, Chen et al. fitted Insight-HXMT spectra of this source in the soft state, selecting 11 observations and considering uncertainties in black hole mass, distance, and orbital inclination on spectral fitting. They performed fits using 50 mass values in the 8–11 M⊙ range, 50 distance values in the 6–9 kpc range, and 100 uniformly distributed inclination values in the 20°–42° range. [FIGURE:6] shows the final spin distribution obtained from fitting 11 observed spectra, yielding a spin result of $a = 0.456 \pm 0.126$ (68% confidence level).

[FIGURE:6] Summed histogram of $a_*$ for 11 spectra. The figure is from Chen et al.

4.2 Reflection Component Fitting

The X-ray reflection model applies to optically thick, geometrically thin accretion disk systems, measuring spin through gravitational redshift of spectral lines near the innermost stable orbit. Therefore, it is suitable for systems with accretion rates of 0.01–0.3 times the Eddington rate, though this limit can extend to near the Eddington limit when accretion energy is partially transferred to the corona. Most Seyfert galaxies, moderate-luminosity quasars, and black hole X-ray binaries in high-luminosity hard states are believed to be in this accretion rate range.

In active galactic nuclei and some black hole X-ray binaries, their X-ray power-law components can extend to 100 keV and constitute a large fraction of the total luminosity. For active galactic nuclei, optically thick accretion disks can produce quasi-blackbody spectra; for stellar-mass black holes, quasi-blackbody spectrum temperatures are generally 0.01–0.1 keV. Early studies of black holes recognized that a high-temperature, high-energy corona must exist above the accretion disk, producing hard X-rays through inverse Compton scattering of disk thermal radiation. X-rays generated in the corona subsequently irradiate the accretion disk, causing photoionization of some material in the optically thick accretion disk. Therefore, the disk's surface layer radiates an energy spectrum rich in X-ray fluorescence and recombination emission lines, commonly called the X-ray reflection spectrum, featuring many soft X-ray emission lines, iron emission line features, and a hump-like feature formed by iron photoionization and Compton scattering.

The observed X-ray reflection spectrum is distorted by Doppler effects from orbital motion of matter in the accretion disk and gravitational redshift from the black hole. These effects are stronger closer to the black hole. On the side of the accretion disk moving toward the observer, emission lines form a sharp blue-shifted peak, while matter near the black hole forms an extended red-shifted wing. Assuming a black hole spin of 0.9, the results are shown in [FIGURE:7]. The X-ray reflection spectrum is truncated at the innermost stable orbit; inside this orbit, the density of accreting matter drops sharply due to radial acceleration, leading to complete photoionization of electrons in the plasma. Therefore, the innermost stable orbit and black hole spin manifest through the strength of Doppler and gravitational broadening effects in the X-ray reflection spectrum.

By convolving the X-ray reflection spectrum with relativistic (Doppler or gravitational) broadening and redshift effects, precise models of accretion disk reflection spectra have been constructed. Overall, disk reflection models mainly include the following parameters: accretion disk inclination to the observer, inner disk ionization parameter, accretion disk metallic element abundance (primarily iron abundance), spectral index of X-ray continuum irradiating the accretion disk, and black hole spin. Black hole mass is unrelated to the disk reflection spectral model because the accretion disk's velocity field and gravitational potential are dimensionless when expressed in gravitational radii.

Liu et al. studied Insight-HXMT observations of the X-ray binaries MAXI J1535–571 and 4U 1630–472 during their 2017 and 2020 outbursts. They first fitted spectral data using constant × tbabs × nthcomp, finding clear structures at 5–7 keV. As shown in [FIGURE:8], obvious iron emission line features exist between 5–8 keV, confirming the presence of reflection components in these two sources. They then used the relxillCp model for final fitting. relxillCp is a model widely used to describe reflection spectra, giving fitting results at 90% confidence level as follows: MAXI J1535–571 spin is $a_ = 0.9916 \pm 0.0012$, and 4U 1630–472 spin is $a_ = 0.817 \pm 0.014$. Song et al. used reflection models to fit Insight-HXMT observations of MAXI J1348–630, with best-fit results showing this source's spin as $a_* = 0.82^{+0.04}_{-0.03}$.

Finally, we present in [TABLE:3] the black hole spin parameter measurement results obtained using both continuum fitting and reflection component fitting methods based on Insight-HXMT observations, along with information on energy bands used.

[FIGURE:7] Illustrative profiles of a single emission line from the surface of a geometrically thin accretion disk around a Kerr black hole. The line is broadened and skewed through the combined action of relativistic Doppler shifts, relativistic beaming, and gravitational redshifts. We assume a rest-frame line energy of 6.4 keV, a black hole spin parameter of 0.9, a viewing inclination for the disk of $i = 30°$ from face-on, and a line emissivity as a function of radius corresponding to excitation by a point source located on the black hole spin axis at a distance of 5 $r_g$ (a lamppost geometry). The figure is from Reynolds et al.

[FIGURE:8] The results of fitting using the constant × tbabs × nthcomp model. The left panel represents the fitting results for MAXI J1535–571, and the right panel represents the fitting results for 4U 1630–472. The images are from Liu et al.

[TABLE:3] Black hole spin measurement results using Insight-HXMT

Source name Energy range/keV Method Result Reference MAXI J1820+070 2–10, 10–35 continuum-fitting method $a_* = 0.2^{+0.2}_{-0.3}$ Guan et al. MAXI J1820+070 2–10, 10–25 continuum-fitting method $a_* = 0.14 \pm 0.09$ Zhao et al. Cygnus X-1 2–7, 10–25 continuum-fitting method $a_* > 0.967$ Zhao et al. GRS 1915+105 2–10, 10–25 continuum-fitting method $a_* > 0.9990$ Wang et al. MAXI J1348–630 2–10, 10–25 continuum-fitting method $a_* = 0.42^{+0.13}_{-0.50}$ Wu et al. 4U 1543–47 2–8, 10–30 continuum-fitting method $a_* = 0.65^{+0.14}_{-0.24}$ Yorgancioglu et al. 4U 1543–47 2–10, 9–29, 27–100 continuum-fitting method $a_* = 0.456 \pm 0.126$ Chen et al. MAXI J1535–571 2–8, 10–20, 30–80 reflection method $a_* = 0.9916 \pm 0.0012$ Liu et al. 4U 1630–472 2–8, 10–20, 30–80 reflection method $a_* = 0.817 \pm 0.014$ Liu et al. MAXI J1348–630 2–8, 10–20, 30–80 reflection method $a_* = 0.82^{+0.04}_{-0.03}$ Song et al.

5 Summary and Outlook

We briefly introduced relevant research methods and progress in measuring the mass and spin of stellar-mass black holes. For black hole mass measurement, we described three methods. Currently, results obtained through dynamical methods are considered the most accurate. We also demonstrated the main steps and results using MAXI J1820+070 as an example. If sufficient optical observations can provide parameters for the visible star and orbital inclination, black hole mass can be well determined. In contrast, results obtained through spectral fitting and quasi-periodic oscillation methods have relatively large uncertainties, making precise measurements difficult. Particularly for the source MAXI J1348–630 we listed, spectral fitting gives a mass of about 9.1 M⊙, while the QPO-based method gives a mass of 14.8 M⊙. At present, our understanding of QPO phenomena remains incomplete, with many unresolved questions regarding the relationship between QPO origins and the black hole surrounding environment and its physical properties. First, the origin mechanism of QPOs is still unknown. This phenomenon may involve complex dynamical processes of matter in black hole accretion disks, observational effects, and other aspects. Second, QPO characteristics such as frequency and amplitude may be closely related to properties of the black hole environment. Understanding matter distribution near black holes, geometric structure, and the geometry of accretion disks and coronae is crucial for interpreting QPO phenomena. Meanwhile, QPO frequencies may have some relationship with black hole spin and mass. Through more accurate black hole mass measurements and large QPO observational datasets, we hope to establish more precise connections between them. In summary, comprehensive understanding of QPOs has potential importance for black hole mass measurement and other aspects. The Insight-HXMT satellite's time resolution is very suitable for studying QPO phenomena and conducting long-term monitoring, and it will play a significant role in this area in the future.

Black hole spin research has always been a very popular topic. We introduced the two mainstream methods for measuring black hole spin—thermal continuum fitting and reflection component fitting—and briefly described important measurement results obtained using these methods with the Insight-HXMT satellite. Insight-HXMT's broad energy band characteristics enable better constraints on many model fitting parameters, yielding more credible results. In the thermal continuum fitting method, we briefly introduced two results from MAXI J1820+070 studies, which obtained basically consistent results. Additionally, many works have used Insight-HXMT spectra that meet model assumptions to measure spins of other sources, such as GRS 1915+105, MAXI J1348–630, and 4U 1543–47, with basically similar steps. Reflection component fitting for spin measurement is also a field where Insight-HXMT data are widely applied. We mainly introduced the theoretical basis and briefly described fitting results for MAXI J1535–571 and 4U 1630–472.

Since Insight-HXMT officially began scientific observations on December 1, 2017, it has produced massive observational data, conducting long-term monitoring of multiple transient sources. Further in-depth mining of these data can help us better understand black hole physical properties. The future Einstein Probe satellite will be able to systematically discover and explore high-energy transient sources, even searching for X-ray signals from gravitational wave sources, greatly enhancing our understanding of extreme compact objects and their merger processes. Additionally, polarization observations of black hole transients can reveal properties of accretion disks and coronal structures around black holes. Combined with Insight-HXMT's broad energy band, this can provide a relatively complete picture of the geometry around black holes. Future astronomy is gradually entering the multi-messenger era, meaning astronomical research will no longer rely solely on single-type observational data but will require comprehensive multi-band observational information. In the past, X-ray observations have been one of the main means for studying transient sources. The development of new-generation astronomical observation equipment provides us with more observational means, including radio, optical, infrared, and gravitational waves. By integrating observational data from different bands, we can more comprehensively understand black hole properties and physical processes. Through this multi-band comprehensive observational approach, we hope to solve many outstanding questions in black hole research and gain deeper understanding of their formation, evolution, and interaction with surrounding environments. This will advance our scientific understanding of one of the most mysterious and fascinating objects in the universe to new levels.