Black hole binaries and microquasars

Abstract

This is a general review on the observations and physics of black hole X-ray binaries and microquasars, with the emphasize on recent developments in the high energy regime. The focus is put on understanding the accretion flows and measuring the parameters of black holes in them. It includes mainly two parts: i) Brief review of several recent review article on this subject; ii) Further development on several topics, including black hole spin measurements, hot accretion flows, corona formation, state transitions and thermal stability of standard think disk. This is thus not a regular bottom-up approach, which I feel not necessary at this stage. Major effort is made in making and incorporating from many sources useful plots and illustrations, in order to make this article more comprehensible to non-expert readers. In the end I attempt to make a unification scheme on the accretion-outflow (wind/jet) connections of all types of accreting BHs of all accretion rates and all BH mass scales, and finally provide a brief outlook.

Graphical abstract

Keywords

accretion and accretion disks / neutron stars / black holes / jets and bursts / X-ray sources / -ray sources / X-ray binaries

1 Synopsis

I will start by defining what I mean by black hole binaries (BHBs) and microquasars in this article. I decide to restrict myself to only a subclass of BHBs, namely, BH X-ray binaries (BHXBs), since these are the only class of BHBs known observationally. I will then simply refer microquasars as BHXBs for reasons discussed in Section 2.

Since many excellent, comprehensive and quite up-to-date review articles on BHXBs are readily available in literature, I feel it is not necessary to write another bottom-up and comprehensive review article on the same subjects at this stage. I will thus take quite an unusual approach in this article. I will first give some concise guides on several representative review articles [1–4], with some necessary updates. I will then focus on the further developments on several topics I feel deserve more discussions, i.e., BH spin measurements (Section 4), hot accretion flows (Section 5), corona formation (Section 5), state transitions (Section 7) and thermal stability of SSD (Section 8). The emphasis is thus put on understanding the accretion flows and measuring the parameters of black holes in them. Some rather general issues on BH astrophysics, such as what astrophysical BHs are and how to identify them observationally, are not discussed here but can be found from my recent book chapter entitled “Astrophysical Black Holes in the Physical Universe” [5].

The usual practice of writing a review article is to end by listing some outstanding issues and major unsolved problems, and then to propose some possible approaches to them. I initially did not do this in the first draft. The history of astronomy tells us that major progress is almost always made by unexpected discoveries and research results; unpredictability is an essential nature of astronomy. This article is not intended to be read by funding agencies or proposal reviewers, so I thought I did not have to do it. In astronomy, knowing what has happened, but looking and doing it differently are far more important and effective than following other people’s advises. However, the editors of this book suggested me to write a brief “outlook” in the end. I thus did it nevertheless.

2 Acronyms and terminology

In Table 1, I list all acronyms used in this article; most of these are quite commonly used in this community.

A BH binary (BHB) is a gravitationally bound binary system in which one of the objects is a stellar mass BH with mass from several to tens of solar masses (M_{\odot }); the other object, i.e. its companion, can be either a normal star, a white dwarf, or a neutron star (NS). In case a binary system consists of two BHs, it is is referred to as a binary BH system, which is not covered in this article. When the companion in a BHB is a normal star, the gas from the star may be accreted to the BH and X-rays are produced, as a consequence of the heating by converting the gravitational potential energy into the kinetic energy of the gas, and a BH X-ray binary (BHXB) is referred to as such a binary system, as shown in Fig. 1. The possible existence of BHXBs was first suggested by Zel’dovich & Novikov [6, 7]. The first BHXB found is Cygnus X–1 [8], now a well-studied system among many others found subsequently in the Milky Way and nearby galaxies.

The terminology of microquasar has some twisting in it. Historically it was first referred to the BHXB 1E1740.7-2942 in the Galactic center region, because a double-sided jet was detected from it, mimicking some quasars with similar radio lobes, which have much larger scales [9]. Soon after, superluminal jets are observed from a BHXB GRS 1915+105 [10], which is also referred to as a microquasar. Nevertheless both BHXBs are quite unusual compared to many others, thus microquasars were considered quite unusual. However it became clear that microquasars may be quite common among BHXBs, since the discovery of a normal BHXB GRO J1655–40 [11], whose superluminal jets were observed [12] and showed some correlations with its X-ray emission [13]. Subsequently several more BHXBs have been observed with superluminal jets. At this stage microquasars began to be referred to as BHXBs with relativistic jets (with bulk motion of about or larger than 90% of the speed of light), to distinguish them from NS X-ray binaries (NSXBs) that only have mildly relativistic jets (with bulk motion of about or smaller than 50% of the speed of light) [14]. However the discovery of relativistic jets from a NSXB Circinus X–1 made the situation complicated: relativistic jets are no longer uniquely linked to BHs [15].

Now in retrospect, a microquasar can be literally and easily understood as the micro version of a quasar; however a quasar may or may not be observed with collimated jets. A quasar has been already understood as a special galaxy centered by an actively accreting supermassive BH with a mass from millions to billions of M_{\odot }, and thus its total light output is dominated by the BH’s accretion process, in a similar way as in BHXBs. The production or lack of relativistic jets may have similar or even the same underlying physical mechanisms in BHXBs and AGNs, though their surrounding environments may modify their observed morphologies [16]. It is therefore more natural to simply refer microquasars as BHXBs; in the rest of this article, microquasar and BHXB are used interchangeably.

We therefore will focus on BHXBs and thus will not discuss binary systems producing gamma-rays and sometimes radio jets, which are most likely high-mass NSXBs and in which jets or pulsars’ winds interact with the wind of its high-mass companion to produce the observed gamma-rays [17–23]. Such systems show very different observational characteristics, e.g. the long (1667 days) super-orbital modulation with phase offset (about 280 days) between its X-ray and radio light curves found in LS I+{61}^{\circ }303 [24].

3 Review of reviews

Here I attempt to review the four recent review articles [1–4] I consider most useful to readers. Additional information and updates are provided when necessary. Some overlaps exists between these review articles, as expected and inevitable in bottom-up review articles. To avoid repetitions as much as possible in this article, I thus put different emphasizes on different articles, with of course my personal tastes and perceptions.

3.1 The most recent Science collection

The recent collection of perspectives and reviews in the Science magazine provides excellent introductions to and concise summaries of the current state of our understanding of BH physics and astrophysics [1, 25–27]. To the subjects of this article, the most relevant article in this collection is the one by Fender and Belloni entitled “Stellar-Mass Black Holes and Ultraluminous X-ray Sources”, which is focused on the observational characteristics of BHXBs [1]. In particular, the article provides an excellent description of the general picture of spectral evolutions of BHXBs with the Hardness-Intensity-Diagram (HID), which is found to be well correlated with the observed jets from BHXBs, as shown in Fig. 2. Actually this cycle is also well tracked by their flux variability, represented by the measured root-mean-squares (rms) above its average flux, as shown by the RMS-Intensity-Diagram (RID) [28] of the BHXB GX 339–4 in Fig. 3, which also shows additional horizontal tracks at intermediate intensities. Sometimes, a full HID cycle does not go into the soft state at all (Fig. 4), perhaps due to a failed outburst [29].

The basic scenario is as follows. During the initial stage of an X-ray outburst of a BHXB (A\toB), which is triggered by a sudden increase of accretion rate onto the BH, its spectrum is normally hard and steady jets are always observed. After reaching its peak luminosity, its spectrum begins to soften in a chaotic way and transient jets are normally observed (B\toC\toD). After this transition, the system calms down with a soft spectrum and no jets are present (D\toE). Finally the system returns to its quiescent state with a hard spectrum accompanied with the reappearance of jets (E\toF). Throughout this cycle, the presence of hot accretion disk winds appears to be anti-correlated with its spectral hardness and jet ejection. This empirical pattern appears to be quite universal for all BHXBs with very few exceptions, though the underlying physics is still not well understood yet. Nevertheless putting together the above scenario is a very significant progress in this field over the last ten years. Due to the conciseness of this article, some other important subjects on BHXBs are not fully discussed and many original references are also missing.

Other articles in this collection [25–27] are less relevant to the subjects of this article, but are still quite interesting to read, within the context of BH astrophysics and physics. The only other type of astrophysical BHs known to exist in the physical universe are supermassive BHs in the center of almost each galaxy. Volonteri [27] concisely summarized our current understanding on how they are formed and grow over the cosmic time; merging of two BHs is a key process here. Thorne [25] focused on what happens when BHs merge together to produce gravitational waves, which might be used as a new laboratory for studying gravitational physics and a new window for exploring the universe. Witten [26] then explained the quantum properties of BHs, in particular the basic ideas behind Hawking radiation, which might not be important at all for astrophysical BHs. Nevertheless the understanding gained through studying the quantum mechanics of BHs actually plays very important roles in developing other theories of physics, such as that in heavy ion collisions and high-temperature superconductors, as vividly described by Witten [26].

3.2 The most recent ARA&A article on BHXBs, with some updates

The most recent ARA&A article on BHXBs by Remillard and McClintock entitled “X-Ray Properties of Black-Hole Binaries” [2] (referred to as RM06 hereafter) provides the most complete, comprehensive and accurate review of BHXBs, which actually covers subjects much beyond just the X-ray properties of BHXBs. The Introduction of RM06 highlights the initial theoretical and observational studies of BHXBs, followed by brief comments on several main review articles on BHXBs preceding this one and the basic properties of BHs within the context of general relativity (GR). All main properties of BHXBs known at the time are summarized in the first table and figure there. In Table 2, I compile the most updated data on all BHXBs currently known, including three BHXBs outside the Galaxy and not in the Large Magellanic Cloud (LMC); the currently available spin measurements for these BHs are also included for completeness. In Table 2,

\begin{array}{}\text{(1)}& f(M)\equiv P_{\mathrm{orb}}{K}_{\mathrm{C}}^3/(2\pi G)=M_{\mathrm{BH}}{\sin }^{3}i/{(1+q)}^2\end{array}

where P_{\mathrm{orb}} is the orbital period, K_{\mathrm{C}} is the semi–amplitude of the velocity curve of the companion star, M_{\mathrm{BH}} is BH mass, i is the the orbital inclination angle, and q\equiv M_{\mathrm{C}}/M_{\mathrm{BH}} is the mass ratio. In Fig. 5, I show the updated graphical representation of most of the BHXBs listed in Table 2.

The X-ray properties of BHXBs are characterized by their X-ray light curves, timing and spectra. Essentially all known BHXBs were discovered initially as bright X-ray sources, and the majority of them were detected as transient X-ray sources with X-ray all-sky monitors. The transient properties of some of BHXBs can be interpreted by the disk instability model (DIM) (see Ref. [49] and references therein), which assumes a constant mass transfer rate from the mass donor to the accretion disk. However the accretion rate from the disk to the compact object, i.e., a white dwarf (WD), a neutron star (NS), or a BH, is normally lower than the mass supply rate in the disk, so mass is accumulated in the disk. When the accumulated mass exceeds a certain critical value, a sudden increase of accretion rate results in a nova-like outburst. DIM is most successful in explaining an outburst with fast rise and exponential decay (FRED). However as shown in RM06, many observed X-ray light curves of BHXBs are far more complicated than just FRED and their recurrent time scales are also not compatible with DIM. Perhaps disk truncation (a subject to be discussed extensively more later) and mass transfer instability are additional ingredients [50].

X-ray emission from a BHXB is variable at all time scales, from the free-fall or Keplerian orbital time scale of milliseconds near the BH, to the various oscillations (some are even related to GR) in the disk with time scales from milliseconds to minutes, to the viscous time scales of minutes, and to the various instabilities of different time scales. Therefore timing studies of BHXBs can probe the geometry and dynamics in BHXBs. However, lack of coherent signals, such as that observed from pulsars, makes it difficult to unambiguously identify the underlying mechanisms from the detected X-ray variabilities. Nevertheless power density spectra (PDS) and rms still allow us to make progress in understanding the general characteristics of a BHXB, in particular when combined with its spectral behaviors, as already discussed briefly above.

The most successful understanding of BHXBs so far is the description of their thermal X-ray spectral component by the classical Shakura–Sunyaev Disk (SSD) model [51]. After applying GR to SSD, the temperature distribution in the disk can be obtained [52]. Applying this model to BHXBs, one can even measure the spin of their BHs [47], by assuming that the inner accretion disk boundary is the inner-most stable circular orbit (ISCO) of the BH (a subject to be discussed more in Section 4). However a power-law (PL) component is almost ubiquitous in the spectra of BHXBs. The interplay between these two components results in various spectral states, which are also found to be well correlated with their timing properties.

Historically these spectral states have been named in many different ways, reflecting mostly how they were identified with the observations available at the times. In RM06, three states are defined, which show distinctively different spectral and timing behaviors, as shown in Fig. 6 based on the RXTE data on the BHXB GRO J1655–40. The thermal state has its X-ray spectrum dominated by the thermal disk component and very little variability. The hard state has its X-ray spectrum dominated by the a PL component and strong variability. The steep power-law (SPL) state is almost a combination of the above two states, but the PL is steeper. I will keep using these definitions throughout this article for consistency. Please refer to Table 2 in RM06 for quantitative descriptions of these three states. When a BHXB is in quiescence, i.e., not in an outburst, its spectral shape is similar to the hard state spectrum.

Referring back to Fig. 2, RM06 found that the HID can be reasonably well understood in terms of transitions between the above three states; I summarize this in Table 3. After examining the extensive RXTE data on spectral evolutions of six BHXBs, the above conclusion seems to be valid generally. Thus a coherent picture seems to emerge: jets are produced only when the PL component in the spectrum is strong and winds are present only when the PL component disappears; jets and winds appear to be mutually exclusive. This suggests that transient winds should appear in the SPL state, anti-phased with transient jets, if winds quench jets as suggested recently [53, 54].

The robust link between the PL component and jets provides an important clue to the jet production mechanism. The PL component is believed to be produced in an optically thin and geometrically thick corona, in which hot electrons up-scatter the thermal photons from the disk. As discussed in RM06, the inner region of the disk appears to be truncated quite far away from the BH in the hard state; this should be the primary reason for the low luminosity of this state, as evidence for the existence of BHs in BHXBs. However it is still not understood how the corona is formed; I will discuss this later in Section 6.

In the final part, RM06 discussed the exciting possibilities of using BHXBs as probes of strong gravity, including confirming these BHXBs contain true BHs, measuring the spins of BHs, relating BH spin to the Penrose process and other phenomena, and finally carrying out tests of the Kerr Metric. They also discussed in length how to measure the spins of BHs and commented the various methods of doing so. In Section 7 I will re-visit the BH spin measurements in details.

Lacking of a solid surface, a BHXB is not expected to produce and actually never observed to have coherent pulses. However quasi-periodic oscillations (QPOs) are frequently detected in their X-ray light curves; a QPO is defined as a “bump” feature in the PDS, if Q=\nu /\mathrm{\Delta }\nu >2, where ν and \mathrm{\Delta }\nu are the peak frequency and FWHM of the bump, respectively. The observed QPOs are further divided into low frequency (LF) (0.1–30 Hz) and high frequency (HF) (>30 Hz) QPOs. Typical LFQPOs can be found as the peak around several Hz in the top and bottom PDS of Fig. 6. In literature, LFQPOs are further divided into several subclasses and their underlying mechanisms are far from clear at this stage, though many models have been proposed and briefly discussed in RM06. HFQPOs have been detected from several sources at 40–450 Hz. Of particular interests are their stable nature when detected, which may be linked to either the mass or spin or both of the BH in a BHXB.

In some cases a 3:2 frequency ratio is found. Although the 3:2 HFQPO pairs could be interpreted in some epicyclic resonance models [55, 56], there remain serious uncertainties as to whether epicyclic resonance could overcome the severe damping forces and emit X-rays with sufficient amplitude and coherence to produce the HFQPOs. A revised model applies epicyclic resonances to the magnetic coupling (MC) of a BH’s accretion disk to interpret the HFQPOs [57]. This model naturally explains the association of the 3:2 HFQPO pairs with the steep power-law states and finds that the severe damping can be overcome by transferring energy and angular momentum from a spinning BH to the inner disk in the MC process.

3.3 Two comprehensive and long articles on modeling accretion flows in BHXBs

Page limitations to the above review articles did not allow in depth discussions on the detailed processes and models of accretion flows in BHXBs, which are responsible for the above described energy spectra, PDS, state transitions, jets, and wind. Here I introduce two comprehensive and long articles just on this, by Done [3] and Done, Gierliński & Kubota [4].

3.3.1 A beginner’s guide

The first is intended to readers who just start research in this field [3]. It started with the basic tools in plotting spectra and variability, and then described the basic ideas and methods used to infer the inner accretion disk radius when a BHXB is in a thermal state, in order to measure the BH spin [47]. A useful introduction is made on how to make various corrections to account for various effects, including color correction, special and general relativistic effects, starting from the original work [47]. Several commonly use fitting models, i.e., DISKBB, BHSPEC and KERRBB, in the XSPEC package are also briefly introduced. An example is given to demonstrate that the expected relation L_{\mathrm{disk}}\propto T_{\mathrm{in}}^4 [47] agrees with the data from the BHXB GX 339–4 [58], where L_{\mathrm{disk}} and T_{\mathrm{in}} are the disk’s total luminosity and temperature at the inner disk boundary, respectively.

The hard state is then briefly touched upon, using the Advection Dominated Accretion Flow (ADAF) model [59]. ADAF naturally explains the hot corona required by the observed PL component, especially at very low accretion rate. Evaporation at low accretion rate in the inner disk region has been proposed as the mechanism producing a geometry with a radially truncated disk and a hot inner flow; the latter might be the ADAF [60]. This also means that when a significant PL component is present in the spectrum, BH spin measurement cannot be done with the inferred inner disk boundary (see, however, counter evidence discussed in Sections 4 and 6). This geometry is considered a paradigm that can account for many of the observed diverse phenomena from spectral evolution to timing properties.

A brief, yet interesting discussion is given on scaling up the above models to Active Galactic Nuclei (AGNs), which host actively accreting supermassive BHs in the centers of galaxies. Applying the insights learnt from BHXBs on their spectral evolution, changing disk-temperature with accretion rate and BH mass, and disk-jet connections, one might be able to understand many phenomena beyond the simple AGN unification scheme, in which the different observational appearance is all attributed to an viewing angle difference.

The continuum emissions of both the disk thermal and PL components are modified by absorptions and added by additional spectral features along the line of sight (LOS). Absorptions in neutral media produce various photoelectric absorption edges, but in ionized media result in both absorption edges and lines. In addition to recovering the original X-ray emission of a BHXB, modeling the absorption features is important in learning the physical properties of the absorption media, such as column density, ionization and velocity along LOS. Winds from BHXBs discussed above are always detected this way. Several XSPEC fitting models for various kinds of absorption edges and lines are also introduced here, i.e. TBABS, ZTHABS, TBVARABS, ZVPHABS, TBNEW, ABSORI and WARMABS. Figure 7 shows the calculated absorption structures with ABSORI and WARMABS; note that ABSORI does not include absorption lines, but WARMABS does.

X-rays interacting with the surrounding medium can ionize it and heat the free electrons up by Compton scattering; hot electrons can also loose energy by interacting with lower energy photons. A Compton temperature of the plasma is reached when the above two processes reaches an equilibrium. The temperature is determined by only the spectral shape of the continuum, so the heated plasma can escape as winds from the disk if the velocity of the ions exceed that of the escape velocity at that radius, which is usually very far away from the inner disk boundary. As the continuum luminosity approaches the Eddington limit, the radiation pressure reduces the escape velocity substantially in the inner disk region such that continuum driven winds can be launched almost everywhere in the disk of a BHXB, forming a radiation driven wind, which is also called thermal wind; alternatively winds may also be driven magnetically in the inner disk region, but this is much less understood yet. In contrast, in an AGN the peak continuum emission is in the UV band, which has a much larger opacity than Compton scattering in neutral or weakly ionized medium during both photoelectric and line absorptions. This means the effective Eddington luminosity is reduced by large factors and UV line driven wind is easily produced at high velocity. This explains the relatively lower velocity (hundreds km/s) and highly ionized winds from BHXBs, but much higher velocity (thousands km/s to 0.2c) and weakly ionized winds from AGNs.

Material illuminated by X-rays produce both fluorescence lines and reflection features, which depends on both the continuum spectral shape and ionization state of the material, as shown in Fig. 8 calculated with the XSPEC ATABLE model REFLIONX.MOD, which includes the self-consistent line and recombination continuum emission. Note that for highly ionized reflection, the Compton heated upper layers of the disk broadens the spectral features; these effects are not included in the simpler XSPEC model PEXRIV. Replacing the stationary slab by a disk around a BH, both the special and general relativistic effects will further smear (broaden) the spectral features, as shown in Fig. 9 for the iron line region with different values of ξ, inner disk radius r_{\mathrm{in}}, and viewing angle i; these are calculated with the REFLIONX.MOD and then convolved with KDBLUR in the XSPEC package. Other similar XSPEC fitting models are DISKLINE, LAOR, and KY. The inner disk radius can in principle be determined by modeling the observed broad iron line features, that in turn can be used to measure BH spins, as will be discussed briefly in Section 4. Figure 10 shows all components of a broad band spectrum of a BHXBs, including interstellar absorption.

We thus have two ways discussed so far to determine r_{\mathrm{in}} of BHXBs by either fitting its thermal continuum or the fluorescent iron line feature. However results are not always consistent, obtained with different methods, even using the same data by different authors. The author believes that r_{\mathrm{in}} increases at low accrete rate, to make room for the corona occupying this space. Nevertheless a fairly good summary on the current conflicts and confrontations is made on this issue, which I will discuss further in Section 4.

3.3.2 An expert’s handbook

The second one can be considered as a handbook on accretion flows on BHXBs, and really lives up to its subtitle “Everything you always wanted to know about accretion but were afraid to ask” [4] (DGK07 hereafter). Besides its much longer length of 66 pages, the main difference from the above review articles is that it is focused on confrontations between theories and observations and intends to depict a coherence picture of the accretion physics in BHXBs. In the following I will summarize briefly the main points and conclusions reached in DGK07; those I have reviewed above and will discuss more later will be skipped for brevity.

The underlying physics of DIM for triggering the outbursts in BHXBs discussed above is the hydrogen-ionization instability, which produces the so-called “S”’-curve, as shown in Fig. 11; irradiation by the inner hot disk to keep the outer disk hot is required to produce the slow flux decays, e.g. the exponential decays, frequently observed in them. The outer disk radius is obviously another key parameter, which is determined by the tidal instability in a binary system. This mechanism can explain why BHXBs with high mass companions are all persistently bright, since their outer disks are always in the upper branch due to the combination of their higher average mass transfer rate and inner disk irradiation. Similarly it also explains the differences and similarities between the light curve properties of neutron star X-ray binaries (NSXBs) and BHXBs, since an NS has a lower mass. It should be noted that the additional surface emission from the NS may also help to maintain the outer disk hot and stay in the upper stable branch [62].

The SSD prescription assumes that the stress is proportional to pressure. Because the gas pressure is P_{\mathrm{gas}}\propto T but radiation pressure is P_{\mathrm{rad}}\propto T^4, so a small temperature increase causes a large pressure increase when P_{\mathrm{rad}}⩾P_{\mathrm{gas}}, and thus large stress increase, which in turn heats the disk even more. The opacity cannot decrease effectively to cool it down, so the disk becomes unstable when L⩾0.06L_{\mathrm{Edd}}, where L_{\mathrm{Edd}} is the Eddington luminosity of a BHXB. However in most BHXBs their disk emissions appear to be stable up to around 0.5L_{\mathrm{Edd}}. One way out is to assume that the stress is proportional to \sqrt{P_{\mathrm{rad}}{P}_{\mathrm{gas}}}\propto T^{5/2}, so the stress increases slower in the radiation pressure dominated regime. Beyond this the disk should become unstable, as evidenced by the sometimes “heart-beat” bursts of the super-Eddington BHXB GRS 1915+105; other BHXBs with L_{\max }>L_{\mathrm{Edd}} (e.g. V404 Cyg and V4641) were not observed with such instabilities, perhaps due to the combination of insufficient observational coverage and sensitivity. However similar “heart-beat” bursts were also observed recently from IGR J17091–3624, which is likely substantially sub-Eddington unless it is located much beyond 20 kpc and/or its BH mass is quite small [63].

Super-Eddington accretion flow (SEAF) can become stable again, if the radiation instability is overcome by an optically thick ADAF, i.e., the slim disk model, in which the trapped photons in the flow is advected inwards, thus balancing the heating generated by viscosity. Strong radiation driven winds can be easily produced; this can happen in BHXBs and NSXBs (e.g. Z-sources). Evidence exists that truncated inner disk (TID) is quite common in SEAF and outflow even dominates over inflow in SEAF [64].

The observed PL component in BHXBs cannot be explained by the SSD-like models and thus requires a hot accretion flow (HAF). At low accretion luminosity (e.g. the quiescent state), the HAF may be the advection dominated accretion flow (ADAF), the convection dominated accretion flow (CDAF), or the advection dominated accretion inflow/outflow solution (ADIOS). At higher luminosity (e.g. the hard state), the original hot and optically thin disk solution (i.e., the SLE solution) is unstable, because the electron heating efficiency by the Coulomb coupling between protons and electrons is too low. The luminous HAF (LHAF), however, has the advection as a heating source to electrons, so the heating efficiency increases and thus electrons can cool the flow more effectively. Outflows can be produced in ADAF/ADIOS; collimated jets can also be produced if magnetic fields are involved, so an accretion flow may even be jet dominated (i.e., JDAF).

The interplays between the SSD and HAF may be responsible for the observed different states in BHXBs discussed above. If the PL component is produced by Compton up-scattering, then the combination of the optical depth τ and {\mathcal{L}}_{\mathrm{h}}/{\mathcal{L}}_{\mathrm{s}}, where {\mathcal{L}}_{\mathrm{h}} is the heating power in electrons and {\mathcal{L}}_{\mathrm{s}} is the cooling power in seed photons, can describe the observed variety of spectra. For example, the hard state has {\mathcal{L}}_{\mathrm{h}}/{\mathcal{L}}_{\mathrm{s}}\gg 1, but the thermal and SPL state have {\mathcal{L}}_{\mathrm{h}}/{\mathcal{L}}_{\mathrm{s}}⩽1. The location of r_{\mathrm{in}} is proposed to be closely related to {\mathcal{L}}_{\mathrm{h}}/{\mathcal{L}}_{\mathrm{s}}, as illustrated in Fig. 12. As discussed above, the steeper PL in the thermal and SPL states is mostly non-thermal in nature, so non-thermal Comptonization is required. Detailed comparisons with data suggest thermal Comptonization also cannot be ignored even in the thermal and SPL states.

In Fig. 12, the hot inner flow (HIF) and patchy corona are responsible for thermal and non-thermal Comptonization, respectively. When a source transits from hard to SPL state, r_{\mathrm{in}} decreases, HIF is reduced but the patchy corona becomes dominant. The transition from SPL to soft state is then marked by the disappearance of HIF and significant reduction of the patchy corona. The above scenario obviously depends on two fundamental assumptions: i) slab-HIF produces the non-thermal PL component; ii) slab-HIF is mostly located between the TID and the BH. Both assumptions are examined exhaustively based on the existing observations and their spectral modeling. Slab-HIF is found to be consistent with essentially all data. An alternative to slab-HIF is that the PL is produced from the jet base and beamed away from the disk; however observations suggest the PL emission is quite isotropic, thus in conflict with this. TID is found also consistent with data when the PL component becomes important, since the observed L_{\mathrm{disk}}\sim T_{\mathrm{in}}^4 deviates significantly from a linear relation, suggesting r_{\mathrm{in}} is larger when the PL component becomes important. However some studies showed that r_{\mathrm{in}} is unchanged in the initial hard state, if the Compton up-scattering process is treated properly in Monte-Carlo simulations to recover the lost disk photons [65]; this issue will be further discussed in Section 4.

The above developed HIF/TID model based on the spectral evolutions of BHXBs can also be applied to explain the majority of time variability PDS of BHXBs consistently. The TID acts as a low pass filter as it cannot response effectively to variations in the HIF, so r_{\mathrm{in}} controls the low frequency break of the PDS: {\nu }_{\mathrm{LFB}}\sim 0.2{(r/6)}^{-3/2}{(m/10)}^{-1} Hz, where r=r_{\mathrm{in}}/r_{\mathrm{g}} and m=M_{\mathrm{BH}}/M_{\odot }; this predicts that {\nu }_{\mathrm{LFB}} changes from 0.03 to 0.2 Hz as observed during transitions from the hard to SPL and soft states if r decreases from 20 to 6. The HIF/TID model can also explain the observed LFQPO variations with luminosity, if the LFQPOs are some kinds of characteristic frequencies related to r_{\mathrm{in}}. To some extend, the shape of the PDS can also be explained by this model. However the observed variability rms \sim flux linear correlation and log-normal distribution of fluctuations may need additional ingredient, such as the proposed propagation fluctuation model. However I point out that such correlations and distributions are also observed in the light curves of gamma-ray bursts and solar flares [66, 67], which can be produced by the generic self-organized criticality mechanism [66]. Similar rms \sim flux correlation has also been found for blazars, in agreement with the minijets-in-a-jet statistical model [68].

The spectral and timing properties of weakly magnetized NSXBs are known to have many similarities and differences from BHXBs; this is particularly true for the atoll sources that have similar ranges of L/L_{\mathrm{Edd}} to BHXBs. The essential distinction is that an NS has a solid surface, but a BH does not. Observationally the continuum spectra of atoll sources can be modeled as composed of SSD, PL emission, and blackbody emission from the NS surface (or the boundary layer between the TID and NS surface); the former two components are quite similar to BHXBs. The spectral evolutions of these NSXBs are thus driven similarly by the combinations of τ and {\mathcal{L}}_{\mathrm{h}}/{\mathcal{L}}_{\mathrm{s}}. However the blackbody emission is an additional source of {\mathcal{L}}_{\mathrm{s}}, so the PL is not as hard as that in BHXBs and r_{\mathrm{in}} variations are less effective in changing {\mathcal{L}}_{\mathrm{h}}/{\mathcal{L}}_{\mathrm{s}}; the latter means it is more difficult to find evidence of TID from modeling only the spectral evolutions in NSXBs. On the other hand, the TID/HIF model in NSXBs can produce timing behaviors in the same way as in BHXBs discussed above, consistent with observations; additional timing behaviors, such as coherent X-ray pulsations and kilo-Hz QPOs observed in these NSXBs, are caused by the rapid spins of the hard surfaces of the NSs.

4 Further developments on BH spin measurements

A BH predicted in GR can only possess three parameters, namely, mass, spin and electric charge, known as the so-called BH no hair theorem. Even if a BH was born with net electric charge, its electric charge can be rapidly neutralized by attracting the opposite charge around it in any astrophysical setting, because the strength of electromagnetic interaction is many orders of magnitude stronger than that of gravitational interaction. Therefore an astrophysical BH may only have two measurable properties, namely, mass and spin, making BHs the simplest macroscopic objects in the universe. Practically, only Newtonian gravity is needed in measuring the BH mass in a binary system. However, GR is needed in measuring the BH spin.

The mass and spin of a BH has different astrophysical meanings. Its mass can be used to address the question of “How much matter (and energy) has plunged into the BH?”. However its spin can be used to address the question of “How did the matter (and energy) plunge into the BH?”. This is because matter and energy plunged into a BH can carry angular momentum, which is a vector with respect to the spin axis of the BH. In order to increase the total gravitating mass-energy from M_{\mathrm{i}} with zero spin to M_{\mathrm{f}}, the added rest-mass must be [69, 70]

\begin{array}{}\text{(2)}& \mathrm{\Delta }M=3M_{\mathrm{i}}\left(\arcsin {\displaystyle \frac{M_{\mathrm{f}}}{3M_{\mathrm{i}}}}-\arcsin {\displaystyle \frac{1}{3}}\right)\end{array}

and its final spin becomes

\begin{array}{}\text{(3)}& a_{\ast }\equiv {\displaystyle \frac{cJ}{G{M}_{\mathrm{f}}^2}}={\left({\displaystyle \frac{2}{3}}\right)}^{1/2}{\displaystyle \frac{M_{\mathrm{i}}}{M_{\mathrm{f}}}}\left[4-{\left({\displaystyle \frac{18M_{\mathrm{i}}^2}{M_{\mathrm{f}}^2}}-2\right)}^{1/2}\right]\end{array}

where J is the BH’s angular momentum. Clearly we have a_{\ast }=1 when M_{\mathrm{f}}/M_{\mathrm{i}}=6^{1/2}; further accretion simply maintains this state [70]. Therefore the required additional rest-mass to spin a BH from zero to maximum spin is \mathrm{\Delta }M\simeq 1.85M_{\mathrm{i}}=0.75M_{\mathrm{f}}; this is a lower limit to the accreted mass [71]. Figure 13 shows a_{\ast } as a function of \mathrm{\Delta }M. Ignoring Hawking radiation of a macroscopic BH, the only way to extract the energy of a BH and thus reducing its gravitating mass is by extracting its spin energy. Recently, evidence of BH spin energy extraction to power relativistic jets has been found, from the observed correlation between the maximum radio luminosity and its BH spin of a microquasar [72, 73]; however the average jet power is not correlated with BH spin [74]. This indicates that jets may be produced by both Blandford–Payne (BP) [75] and Blandford–Znajek (BZ) [76] mechanisms; but the BZ mechanism is more powerful and responsible for producing the peak radio luminosity. This provides another possible way to estimate a BH’s spin [73], similar to a recent proposal of using the peak luminosity of the disk emission to estimate a BH’s spin [77]. However there is so far no independent demonstration of validness of either of the above two new methods, which are thus not discussed further in this section.

Figure 14 shows the mass reduction, \mathrm{\Delta }m=(M_{\max }-M_{\mathrm{f}})/M_{\max } (M_{\max } and M_{\mathrm{f}} are the BH’s maximum mass and final mass, respectively), as a function of extracting efficiency ϵ of an extreme Kerr BH. For a 10M_{\odot } BH with ϵ=1 (\mathrm{\Delta }m\sim 0.3), the total extracted energy is \mathrm{\Delta }m{M}_{\mathrm{f}}{c}^2\simeq {10}^{54} erg and its total gravitating mass is reduced to M_{\max }/\sqrt{2} [71]; this energy could be sufficient to power gamma-ray bursts (GRBs). It is thus plausible that supercritical accretion onto a newly born BH may spin it up and extract its spin energy to power ultra-relativistic jets; multiple spin-up and spin-down cycles may also happen during one GRB, if the collapsing material is clumpy.

As shown in Fig. 15, the radius of the ISCO of the BH, R_{\mathrm{ISCO}}, is a monotonic function of the BH spin [78], beyond which radius a test particle will plunge into the BH under any perturbation; however, in Newtonian gravity a stable circular orbit can be found at any radius. It is thus reasonable to assume that the accretion disk around a BH terminates at this radius, i.e., r_{\mathrm{in}}=R_{\mathrm{ISCO}}. Therefore, a_{\ast } can be inferred if one can measure R_{\mathrm{ISCO}} in units of its gravitational radius r_{\mathrm{g}}=GM/c^2. Currently three methods have been proposed to measure the BH spin in BHXBs, and all these methods rely essentially on measuring R_{\mathrm{ISCO}}. In case the radiative efficiency (\eta \equiv L/\dot{M}{c}^2) can be measured, a_{\ast } can also be determined this way, as shown in Fig. 16. Actually η is a very simple function of R_{\mathrm{ISCO}}, i.e, \eta \sim 1/R_{\mathrm{ISCO}}, as shown in Fig. 17.

4.1 The first black hole spin measurement with X-ray spectral continuum fitting

In 1997, I and my colleagues proposed a method of measuring the BH spin in BHXBs. It started when we tried to measure the mass of the BH in GRO J1655–40 [11] by measuring r_{\mathrm{in}} from its X-ray continuum spectral fitting and assuming the BH is not spinning; we found the mass of the BH is around 4M_{\odot } [79]. Coincidentally just when this paper was going to press, Orosz and Bailyn [80] announced their accurate measurement of the BH mass of GRO J1655–40, which is significantly larger than what we found. A co-author of our paper, Rashid Sunyaev urged me to resolve this apparent discrepancy. I immediately realized that a spinning BH of 7M_{\odot } in GRO J1655–40 would be consistent with the inferred with X-ray data, and added a note in proof in the paper suggesting this possibility [79]. This turns out to be the first BH spin measurement on record. I then invited two of my close friends and collaborators, Wei Cui and Wan Chen, to join me to apply this method systematically to other BHXBs; a new main conclusion in this work was that the first microquasar in the Milky Way, GRS 1915+105, also contains a spinning BH [47].

At the time DISKBB was the only available fitting model in the XSPEC package for determining r_{\mathrm{in}} from an observed X-ray continuum spectra, if the disk continuum is described by the SSD model: L_{\mathrm{disk}}=4\pi \sigma r_{\mathrm{in}}^2{T}_{\mathrm{in}}^2, where T_{\mathrm{in}} in DISKBB is the disk temperature at r_{\mathrm{in}} (i.e., the peak disk temperature in SSD) and L_{\mathrm{disk}} can be calculated from the disk flux (after the correction to absorption), the distance to the source and disk inclination (an issue to be discussed later). To determine the physical inner disk radius from the DISKBB parameter r_{\mathrm{in}}, several effects must be considered: i) electron scattering in the disk modifies the observed X-ray spectrum; ii) the temperature distribution in the disk is not accurately described by the Newtonian gravity as assumed in SSD; iii) the observed temperature distribution is different from the locally emitted one; iv) the observed flux is different from the locally emitted one. The latter three are all due to GR effects [52]. For each of the above effects, we introduced a correction factor, using the best available knowledge at the time. Since then, several improvements have been made to correct for these effects and this continuum fitting (CF) method is now quite mature in making accurate BH spin measurements, given sufficiently high quality X-ray continuum spectral measurements and accurate system parameters of the observed BHXB.

4.2 Further developments and applications of the continuum fitting method

This CF method of measuring BH spin has since been applied widely to essentially every BHXB with a well measured X-ray continuum spectrum showing a prominent thermal accretion disk component. In particular, this method has been improved and incorporated into the widely used X-ray spectral fitting package XSPEC, e.g., KERRBB [81], BHSPEC [82], and KERRBB2 [35, 37]. Both the KERRBB and BHSPEC are relativistic models, but they have their own drawbacks and advantages (See Ref. [35] for a detailed comparison). KERRBB includes all the relativistic effects, but it requires to fix the spectral hardening factor. In contrast, BHSPEC could calculate the spectral hardening factor on its own; however, it does not include the returning radiation effect, which turns out to be an important factor in BH spin determination in BHXBs. KERRBB2 combines both models by generating the spectral hardening factor table from BHSPEC and using the table as the input for KERRBB. The research group led by Ramesh Narayan of Harvard University, Jeffrey McClintock at Smithsonian Astronomical Observatory (SAO), and Ronald Remillard of Massachusetts Institute of Technology (MIT) [83] has since applied this method and contributed to most of the BH spin measurements available in the community, as shown in Table 2.

The CF method relies on two fundamental assumptions: i) The measured r_{\mathrm{in}} is uniquely related to R_{\mathrm{ISCO}} of the BH. ii) There is no or negligible X-ray radiation from the plunging matter onto the BH beyond R_{\mathrm{ISCO}}. The latter has been studied with numerical simulations that include the full physics of the magnetized flow, which predict that a small fraction of the disk’s total luminosity emanates from the plunging region [84]. However, in the context of BH spin estimation, it has been found that the neglected inner light in the CF method only has a modest effect, i.e., this bias is less than typical observational systematic errors [85, 86].

The first assumption above requires that the measured r_{\mathrm{in}} remains stable as a BHXB changes its spectral state and luminosity. However it was noticed that r_{\mathrm{in}} measured is usually much smaller, sometimes even smaller than R_{\mathrm{ISCO}} of a extreme Kerr BH in a prograde orbit, when the X-ray spectrum contains a significant hard PL component, which is believed to be produced by inverse Compton scattering of the thermal disk photons in a hot corona. We realized that the inferred smaller r_{\mathrm{in}} could be due to the lost thermal disk photons in the scattering process. We then investigated this problem and confirmed that the inferred r_{\mathrm{in}} can be made consistent with r_{\mathrm{in}} inferred from the thermal disk component dominated spectrum, if the scattered photons are recovered properly by doing detailed radiative transfer in the corona [65]; the same conclusion was also reached by the Harvard/SAO/MIT group independently without knowing our much earlier results [87]. Therefore the method of BH spin measurement by X-ray continuum fitting can also be applied to some SPL state with strong PL component. The stable nature of the measured r_{\mathrm{in}} is proven with the textbook case of LMC X–3, when its X-ray luminosity varied over more than one order of magnitude observed in nearly two decades with many different X-ray instruments, as shown in Fig. 18 [88].

However, when ordered by the observed disk luminosity l_{\mathrm{D}}=L_{\mathrm{disk}}/L_{\mathrm{Edd}}, the measured r_{\mathrm{in}} shows a clear increasing trend when l_{\mathrm{D}}>0.3, as shown in Fig. 19 [88]. The similar trend has also been found in another BHXB GRS 1915+105 [35] and two NSXBs [64, 89]. It was found that r=r_{\mathrm{in}}/r_{\mathrm{g}} indeed increases physically when l_{\mathrm{D}}>0.3, by comparing the evolution of r as a function of l_{\mathrm{D}} over a large range for several BHXBs and NSXBs. Using the blackbody surface emissions of the NSs in these NSXBs is critical in evaluating any possible disk thickening due to high luminosity that would block at least part of the NS surface emission, as well as determining the actual NS mass accretion rate, which turns out to be much less than the disk mass accretion rate; this suggests that the increased radiation pressure is responsible for the increase of r and significant outflow when l_{\mathrm{D}}>0.3 [64]. The same trend is much more pronounced in the super-Eddington accreting ultra-luminous X-ray source NGC1313 X–2, as shown in Fig. 20 together with the data from other BHXBs and NSXBs [90]. However the exact value of r obtained this way should be taken with caution, since the non-negligible energy advection at high accretion rate can modify the disk structure in non-trivial ways, thus making the SSD prescription inaccurate in this case [91].

Figure 20 also shows that as l_{\mathrm{D}} decreases, r again starts to increases. However r increases at higher l_{\mathrm{D}} and with a different slope for a NSXB than for a BHXB, which can be naturally explained as due to the “propeller” effect of the interaction between the NS’s magnetosphere and its accretion disk [92] and the “no-hair” of the BH [90]. Figure 21 shows the radiative efficiencies of various systems; BHXBs may have either higher or lower efficiencies than NSXBs, because a BH has neither solid surface nor magnetic field [5].

It is interesting to compare the BH spin results in our first paper [47] and the most recent literature for the same BHXBs as listed in Table 2: i) For GRS 1915+105, A0620–003 and LMC X–3, both results are fully consistent; ii) For GRO J1655–40, the original result points to an extreme Kerr BH (a_{\ast }\sim 0.93) [47], somewhat different from the most recent result of mildly spinning BH (a_{\ast }=0.65-0.75) [32]. However, the original result was obtained using the BH mass of 7M_{\odot }, about 10% larger than the currently best estimate that was used to obtain an updated BH spin in the most recent literature [32]. From Fig. 15, it can be seen that a_{\ast } would be decreased from 0.93 to  0.87, if the BH mass is decreased by about 10%; actually a_{\ast }=0.85 is also allowed in the new estimate [32]; iii) For Cygnus X–1, our original conclusion that a_{\ast }=0.75 in the high/soft state and a_{\ast }=-0.75 in the low/hard state was based on the assumption that r_{\mathrm{in}} decreased by a factor of two when the source made a transition from its normal low/hard state to the unusual high/soft state [94]. However the more realistic constraint is that r_{\mathrm{in}} changed by more than a factor of 1.8–3.2 during the state transition [94], which implies that a_{\ast } switched from |a_{\ast }|>0.85, fully compatible with the latest result of a_{\ast }>0.95 in the high/soft state [37].

The above would suggest that a very low temperature disk component exists in its hard state spectrum, if Cygnus X–1 indeed harbors a Kerr BH and the accretion disk switches from a retrograde mode to a prograde mode when it makes the hard-to-soft state transition. For a supermassive BH in the center of a galaxy, it is well understood that its mass is mostly gained through accretion in its AGN phase [95]. Therefore random accretion (between prograde and retrograde modes) tends to make its final BH spin close to zero, regardless its initial BH spin. For Cygnus X-1, its current BH mass of around 10–20 M_{\odot } cannot be gained through post-formation accretion, since its observed average mass accretion rate \dot{M}\sim 2\times {10}^{-9}{M}_{\odot } /yr and the age of its companion is much less that 10⁸ yr; actually the age of the companion is estimated to range from 4.6 and 7.8 million years [96]. This means that its post-formation BH mass growth is much less than a fraction of its current mass and thus its current BH spin must be quite close to that at birth. Even if accreting at Eddington rate, to grow its BH spin from 0 and the final mass to be the current observed value, the timescale is around 3.1\times {10}^7 yr and the accreted mass is roughly 7.3 M_{\odot } [37]. If its accretions alternates between prograde and retrograde modes, then its current BH spin should be even closer to its initial spin than that in low mass BHXBs that only stay in one accretion mode due to roche-lobe overflows. Therefore the current high BH spin in Cyg X–1 must be natal; this conclusion is also true for the other highly spinning BHs in other BHXBs [48].

In summary, BH spin in BHXBs can now be measured reliably with the CF method, when the luminosity of a BHXB is between \sim 0.02 and \sim 0.3 in Eddington unit and their system parameters are well-known a prior. The observed thermal disk spectrum can be modeled directly to obtain the BH spin with the available KERRBB2 model in XSPEC when the X-ray spectrum in dominated by this component, i.e., the source is in the thermal state. When a significant power-law component is present, the inverse Compton scattering process must be taken into account to recover the disk photons scattered into this PL component, with for example the SIMPL/SIMPLR model [37, 97] in XSPEC. Recently another way to measure BH spin using the outburst properties of BHXBs has been proposed [77], which is mentioned briefly in Section 8; however the effectiveness of this method needs to be tested.

4.3 Uncertainties of the continuum fitting method

In spite of the tremendous progress made so far on BH spin measurements in BHXBs, most of these BH spin measurements suffer from considerable uncertainties. Actually the major source of these uncertainties comes primarily from the uncertainties in their BH masses, accretion disk inclination angles, and distances. The accurate BH mass measurement is required because R_{\mathrm{ISCO}} must be in units of r_{\mathrm{g}}=GM/c^2 in Fig. 15. The disk inclination angle and distance are also required because the total luminosity of the disk emission L_{\mathrm{disk}} is needed, in order to estimate the absolute value of a_{\ast }.

So far, all BH masses in X-ray binaries (XRBs) have been estimated using the Kepler’s 3rd law of stellar motion, expressed in the so-called the mass function given in Eq. (1). Since the only direct observables are P_{\mathrm{orb}} and K_{\mathrm{C}}, both M_{\mathrm{C}} and i have to be determined indirectly in order to obtain the BH mass estimate reliably. The companion’s mass M_{\mathrm{C}} can be determined relatively reliably by the observed spectral type of the companion star and i can be estimated by modeling the observed ellipsoidal modulation of the companion’s optical or infrared light curve. The observed ellipsoidal modulation is a consequence of exposing different parts of the pear-shaped companion star to the observer at different orbital phases (see Figs. 1 and 5); the pear-shape is caused by the tidal force of the compact star, which also heats the side of the companion star facing it. For details of BH mass estimates using this method, please refer to Ref. [2].

However, model dependence and other uncertainties (such as accretion disk contamination) cannot be circumvented completely and thus systematic error may exist in determining their system parameters. For example, three optical states, namely “passive”, “loop” and “active” states, have been identified in the normally called “quiescent” state of A0620–003 when its X-ray luminosity is very low; only during the passive state its optical light curve modulation is purely ellipsoidal, i.e., accretion disk contamination is completely negligible, as shown in Fig. 22 [98]. This means that considerable systematic errors in determining its inclination angle may occur unless only the “passive” state data are used. Unfortunately previous observations of BHXBs used to determine their inclinations did not always occur during the passive state, thus systematic errors may be common in previous results [99]. Even for GRO J1655–40, of which all previous observations were made during its passive state [99], its BH mass measured with different observations, or the same data analyzed by different groups are not exactly the same, and even not completely consistent between them, as shown in Fig. 23, which show a scatter of about 20%–30% to the estimated BH mass, much larger than its statistical error of a few percents.

It should be pointed out that the inclination i in Eq. (1) refers to that of the orbital plane of the binary system. Because the accreting material initially carries the angular momentum inherited from the companion star, the formed disk should be co-planar with the orbital plane of the binary system. However, the BH spin in a BHXB cannot be changed significantly by accretion [71] (and see discussion above on Cygnus X–1). Therefore its spin axis may not coincide with the normal direction of the orbital plane of the binary system. In this case the Bardeen-Petterson effect [106], due to frame-dragging by the spinning BH, can rapidly align the normal axis of the inner disk region with the spin axis of the BH, making the inner disk and binary system mis-aligned. Circumstantial evidence already exists for mis-alignment between the two axes, because the orbital inclination of GRO J1655–40 shown in the upper panel of Fig. 23 is significantly different from \sim {80}^{\circ } inferred from its relativistic jets [107], if the jets are powered by extracting the spin energy of the BH via the Blandford–Znajek mechanism [76]. Nevertheless, the orbital inclination is normally used in place of the disk inclination, which is needed in BH spin measurement using the CF method, in order to infer the total disk luminosity and calculate GR correction factors. However, the inner disk inclination is currently not available essentially for any of the known BHXBs. For example, a mis-alignment of {10}^{\circ } from i={70}^{\circ } can cause nearly 50% error to the total disk luminosity, which will translate into nearly 30% error in r_{\mathrm{in}}. A Schwarzschild BH may be estimated to have a_{\ast } falling any where between [-0.5,0.5], if r_{\mathrm{in}} is uncertain within about 30%, according to Fig. 15.

Accurate determinations of distances of astrophysical objects in the Milky Way are difficult, e.g., for BHXBs that are not standard candles. Normally some absorption features in their spectra, in conjunction with their positions in the galactic coordinates, are used to infer their distances. For example, the distance of GRO J1655–40 is commonly taken as 3.2\pm 0.2 kpc, based primarily on observed absorption lines and somewhat on the dynamics of the observed jets. Critical examinations of all available data related to its distance, however, are in favor of a much closer distance of less than 2 kpc and more likely just 1 kpc [108, 109]. Similar conclusion is also reached to the distance of A0620–003, revising its distance from the commonly accepted 1.2\pm 0.4 kpc to \sim 0.4 kpc, making its possibly the closest BHXB known so far [109]; however, a distance of 1.06\pm 0.12 pc was preferred in a more recent study [30]. Similarly the currently adopted distances of many other BHXBs may also have considerable systematic errors. If true, this would change significantly the current estimates on their masses and spins.

Therefore future improvements of the continuum fitting method depend upon the improved measurements on their BH masses, accretion disk inclination, and distances.

4.4 Future improvements of the continuum fitting method

The mass ratio q can be determined directly according to the law of momentum conservation, i.e.,

\begin{array}{}\text{(1)}& M_{\mathrm{C}}/M_{\mathrm{BH}}=K_{\mathrm{BH}}/K_{\mathrm{C}}\end{array}

if the semi–amplitude of the velocity curve of the BH K_{\mathrm{BH}} can be observed directly, as illustrated in Fig. 1. The orbital inclination i can be then calculated using Eq. (1), avoiding any systematics related to the ellipsoidal light curve modeling.

Since a BH is not directly observable, we can only hope to observe any emission or absorption line feature co-moving with it. The accretion disk certainly moves with the accreting BH. However any line feature of the inner accretion disk suffers from the broadening of disk’s orbital motion and distortions by relativistic effects around the BH, thus making it practically impossible, or difficult at least, for detecting the binary orbital motion of the BH. Orbital motion of double-peaked disk emission lines were observed for NSXB Sco X–1 [110], the BHXBs A0620–003 [111, 112] and GRS 1124–68 [112]. Unfortunately a significant phase offset of velocity modulation was found from that expected based on the observed orbital motion of the companion, though the velocity semi-amplitude is consistent with the expected mass ratio [112]. Soria et al. observed the orbital motion of the double-peaked disk emission line He II λ4686 from GRO J1655–40, and found its velocity modulation phase and semi-amplitude in agreement with the kinematic and dynamical parameters of the system [113]. However one major problem in accurately measuring the orbital motion of the primary from the observed double-peaked emission lines is how to determine reliably the line center, because the lines are typically asymmetric and also variable.

We have recently proposed to observe the Doppler shifts of the absorption lines of the accretion disk winds co-rotating with the BH around its companion star [114], since in many XRBs accretion disk winds are ubiquitous and appear to be rather stable when observed (e.g., in Ref. [115]). We verified this method using Chandra and HST high resolution spectroscopic observations of GRO J1655–40 (shown in Fig. 24) and LMC X–3. Unfortunately the currently available data only covered small portions of their orbital phases and thus do not allow better constraints to their system parameters. Future more observations of these two sources and other sources with detectable absorption lines from their accretion disk winds will allow reliable and precise measurements of the BH masses and orbital inclination angles in accreting BHXBs.

Since the accretion disk has very high density and is ionized near the BH, where the majority of the observed disk emission is produced due to the very deep gravitational potential near the BH, scattering of the primary disk emission is inevitable. The scattered light is polarized and its polarization fraction and position angle depend on the viewing direction (inclination), scattering optical depth and the radius where the scattering occurs 116–119. Ignoring many details, it can be shown that the polarization fraction, P(i), of the observed disk photons (initial disk emission plus the scattered emission) is given by

\begin{array}{}\text{(5)}& {\displaystyle \frac{1}{P(i)}}=1+A{\displaystyle \frac{\cos i}{1-{\cos }^{2}i}}\end{array}

where A is a constant depending upon the scattering optical depth; detailed calculations made by Chandrasekhar [120] gives P({75}^{\circ })=0.04. Note that here the disk photons are from the Raleigh–Jeans part of the multi-color blackbody spectrum with a characteristic shape of f(\nu )\propto {\nu }^{1/3}, i.e., no GR effect is included. We can therefore find the disk inclination angle by measuring the polarization fraction of this part of the disk emission, as shown in Fig. 25. At energies above the Raleigh–Jeans part of the multi-color blackbody spectrum, the polarization is strongly effected by both the inclination and BH spin, as shown in Fig. 26. The continuum spectra are clearly degenerated for the different combinations of inclination and BH spin, but the polarization fraction and angle as functions of energy can clearly distinguish between them [118]. Therefore X-ray spectra-polarimetry observations of BHXBs will certainly make important progresses in measuring BH spin.

Besides using polarization measurements to obtain inner disk inclination (and BH spin), the broad iron K-alpha measurements can also be used to do so [121–123], because iron K-alpha emission is believed to come from the fluorescent emission of the disk as a result of illumination by a hard X-ray source above the disk. Naturally this emission is sensitive to the disk inclination (and BH spin). However, compared to the polarization measurement method, this method is less straight forward and may suffer from systematic uncertainties in deriving the disk inclination (and BH spin), because complicated modeling of the hard X-ray component and line emissivity from the disk is required. Therefore it is essential to use all methods discussed above to measure both inclination angles (orbital plane and inner disk) and BH spin. Studying the relationship between the results obtained with different methods is also important in its own rights, in order to understand accretion disk physics, its interaction with the BH in its center, production of relativistic jets, the origin of the BH spin, and ultimately the formation mechanisms of BHs and BHXB systems.

The recent dispute on the distances of GRO J1655-40 and A0620–003 exemplifies the difficulty of determining the distances of BHXBs using mostly absorption lines [108, 109]. We have recently suggested a method of using the delay time between the X-ray fluxes of an XRB and its X-ray scattering halo by interstellar dust to infer its distance [124, 125]. However this method may suffer from our incomplete knowledge of the distribution of interstellar medium. Ideally precise astrometry can determine their distances model-independently, by measuring their trigonometric parallaxes. Recently the distance to Cygnus X–1 was determined reliably and accurately this way ({1.86}_{-0.11}^{+0.12} kpc; [126]); which is key to the consequent measurement of its BH mass and spin [36, 37]. Currently it remains challenging to measure the trigonometric parallaxes of objects at distances beyond several kpc where most BHXBs are located. Future high precision astrometry missions are expected to improve the distance estimates to these BHXBs significantly.

Therefore we expect that the improved measurements discussed above on their BH masses, inner disk inclination, and distances will allow future improvements of BH spin measurement with the CF method.

4.5 Possible application of the continuum fitting method to AGNs

More recently, the CF method is also applied to constrain the BH spin in an active galactic nucleus (AGN), which is powered by matter accretion onto the central supermassive BH in its center [127]. However the BH spin inferred this way for an supermassive BH is quite uncertain, because: i) The peak temperature of the accretion disk inversely increases with the mass of a BH. Therefore, for a supermassive, its temperature would be in the ultraviolet energy range which will be strongly absorbed and hard to observe; ii) The system parameters of a supermassive BH (e.g., the mass of the BH and the inclination angle of the accretion disk) have larger uncertainties; iii) The uncertain mechanism for some components (e.g., the soft X-ray excess) in AGN spectra also increases the difficulty; iv) Some emission and absorption lines may distort the continuum spectrum substantially; and v) In some cases the contribution of its host galaxy to the observed total continuum spectrum cannot be removed satisfactorily.

The polarized continuum of an AGN should be a pure accretion disk continuum, at least in the optical to near infrared band [128]. One possible way to measure the BH spin in an AGN with the CF method is to combine the observed polarized optical to near infrared continuum spectrum with the observed total UV continuum spectrum to get a broad band continuum spectrum of an AGN [129]. In principle the broadened Balmer edge features and the total UV spectrum can be used to constrain the disk inclination angle and fraction of host galaxy contamination, respectively [129]. However the quality of the currently available data is still insufficient to allow accurate determination of BH spin in AGNs with this method.

Nevertheless the principal method of measuring BH spin in AGNs should be using the reflected broad iron line and continuum components [121–123]. The main reason is that i can also be determined simultaneously and r_{\mathrm{in}} obtained this way is already in units of r_{\mathrm{g}}, thus avoiding naturally the uncertainties caused by the BH mass and inclination in the CF method. However cross calibration can be done between the CF and reflection fitting methods if both can be applied to the same supermassive BH.

5 Further developments on hot accretion flows

In this section I briefly summarize some further developments on hot accretion flows, which are believed to be responsible for the PL component of the spectra in BHXBs. The multi-waveband spectra of the hard state of the BHXB XTE J1118+480 was modeled with the TID with hot accretion flow (ADAF) geometry [panel (a) in Fig. 27] and TID (with ADAF) plus jet model [panel (b) in Fig. 27], with {\dot{m}}_{\mathrm{D}}=0.05 and {\dot{m}}_{\mathrm{jet}}=5\times {10}^{-3}{\dot{m}}_{\mathrm{D}} [93]. The steep UV spectrum provides clear evidence for a large truncation radius for the SSD (r=600), and the radio to infrared spectrum dominates the jet emission, which also contribute to the hard PL component [93]. Such low accretion rate gives very low SSD luminosity l_{\mathrm{D}}\sim 5\times {10}^{-4} (since the disk radiation efficiency \eta \sim 1/r; see Fig. 17), which is far below the turn-over luminosity of l_{\mathrm{D}}\sim {10}^{-2} shown in Fig. 20. Therefore the inferred truncation radius of XTE J1118+480 agrees with the extrapolation of data points of XTE J1817–330 down to very low disk luminosity. Large truncation radii are also reported from several other sources in the hard state (e.g., in Ref. [130]). However, the exact values of these truncation radii may have large uncertainties, since no direct detection of the inner disk peak emission was available, unlike the strong case of XTE J1118+480 [93]. For this reason I did not include these reported values in Fig. 20.

Panel (b) in Fig. 27 also shows that the radiation from both the hot accretion flow and the jet contribute to the X-ray emission. However the former is roughly proportional to {\dot{m}}^2, whereas the latter to \dot{m}. Thus with the decrease of \dot{m}, the contribution from the jet becomes more and more important, thus the X-ray radiation will be dominated by the jet (when l\lesssim {10}^{-5}–{10}^{-6}), as shown in Fig. 28 [93]. The observational data of very low-luminosity AGNs clearly show a correlation between radio and X-ray with a correlation index of \sim 1.2 [131], in excellent agreement with the prediction shown in Fig. 28 [93].

It is well known that the highest luminosity an ADAF can produce is only about 3\%L_{\mathrm{Edd}}. However the observed highest luminosity a hard state can reach can be 10\%L_{\mathrm{Edd}} or even higher, which can be described by LHAF shown in Fig. 29, where several previously known solutions of accretion flows are all unified in a single scheme [133]. The hard state spectrum of XTE J1550–564 with L\sim 6\%L_{\mathrm{Edd}} is explained by LHAF very well, including the X-ray spectral slope and the value of the cutoff energy [134].

6 Further developments on corona formation

While the formation of the SSD in a BHXB is reasonably well understood, the formation of the corona, which is the hot accretion flow discussed in Section 5, remains less understood. Over the last more than 10 years, Liu and her collaborators have developed a model to explain the formation and evolution of the corona in a BHXB or AGN, in which the mass accretion rate \dot{m} in units of the Eddington ratio drives the variations of the complex accretion flows by the interaction between the cold SSD and the hot corona [60, 135–139]. Specifically, the coupling between the hot corona and the cold disk leads to mass exchange between them. The gas in the thin disk is heated up and evaporates into the corona as a consequence of thermal conduction from the hot corona, or the corona gas condenses into the disk as a result of overcooling by, for example, external inverse Compton scattering. If \dot{m} is low, evaporation occurs and can completely remove the thin disk, leaving only the hot corona in the inner region and a truncated thin disk in outer region; this provides a mechanism for ADAF at low \dot{m}. If \dot{m} is high, the gas in the corona partially condenses to the disk due to strong Compton cooling, resulting in disk dominant accretion. The model naturally explains the different structures of accretion flow in different spectral states as shown in Fig. 30 [60, 135, 137–144]. The hysteresis observed in spectral state transitions can also be explained by different irradiations from different evolution history under the same scenario [145–147].

Figure 30 is significantly different from the illustration of accretion flow structures in different spectral state in Fig. 12 on three aspects: i) At intermediate \dot{m}, the SSD is broken by ADAF into two parts, an outer disk and an inner disk; ii) The inner disk boundary here is always located very close to the BH, except in the very low \dot{m} hard or quiescent state, which is very different from the TID scenario depicted in Fig. 12; iii) The corona here covers essentially the whole accretion disk, especially the inner disk region, whereas in Fig. 12 the corona is mostly located inward from the inner disk boundary. The observed soft X-ray component in the low/hard state can be explained by the existence of the inner disk [139, 140, 142, 144]. As I have discussed above, the inner disk boundary radius inferred with the CF method in the presence of a strong PL component is actually consistent with that in the soft state when the PL component is weak, after taking into account the Compton scattering in the corona [65, 87]. Actually the essential assumption behind the broad iron line/reflection fitting method of determining BH spin is that the inner disk boundary is at the ISCO when the PL component is strong. All these tend to support the existence of the inner disk at intermediate \dot{m}. However it remains to be demonstrated that if the whole SSD is indeed broken into the two parts at intermediate \dot{m}, as illustrated in Fig. 30.

Recently it has been suggested that the ADAF and corona shown in Fig. 30 may have clumpy structures, as shown in Fig. 31 [148]. The “clumpy” model (Fig. 31) has mainly two different consequences from the “uniform” model (Fig. 30): i) The inner disk is transient in the “clumpy” model; ii) The “clumpy” model can explain the variabilities observed in X-ray binaries (such as the state transitions discussed in Section 7 and radio-loud AGNs (such as BL Lac objects).

Determining the structure of the corona in an BHXB observationally remains difficult; the fundamental issue on whether the corona covers mostly the accretion disk or the central compact object still remains unclear so far. Since the observed similar spectral and state transitions between some NS and BH XRBs are quite similar [149], it is reasonable to assume that they have similar coronae. Recently we have used type I X-ray bursts from low-mass NSXBs to show that X-ray bursts experience negligible Comptonization and that the corona cools rapidly during the rising phase of X-ray bursts and is then heated up rapidly during the rising phase of X-ray bursts, as shown in the upper panel of Fig. 32 for IGR J1747–721 [150]. These results suggest that the corona cannot cover the central compact object completely (lower panel of Fig. 32) and that the destruction and formation time scales of the corona are as short as seconds; such short time scales are quite difficult to understand in the above discussed evaporation model, in which the time scales are related to the viscous time scales of the accretion disk. However, this short time scale is consistent with a corona produced by magnetic reconnections in the accretion disk, in a similar way to the solar corona heating [151]; this conclusion was based on the inferred accretion flow structure of a BHXB shown in panel (a) of Fig. 33, in comparison with the atmospheric structure of the Sun shown in panel (b) of Fig. 33.

It is worth noticing that the purported coronae in both XRBs and AGNs have strikingly similar properties, e.g., they all have electron temperatures of the order of hundreds keV, in spite that the temperatures, inner disk radii and variability time scales of their cold accretiondisks all scale with their BH masses (and accretion rates) as predicted in the SSD model. This means that their coronae are scale independent. It is perhaps not coincidental that the electrons’ velocities in a corona are approximately the same as the Keplerian orbital velocities of the inner disk, which are also roughly the same as the launching velocities of jets. Of course these velocities are also the varialized velocities of the central BHs. It is plausible that turbulent small scale magnetic fields lifts the plasma in the “warm layer” shown in Fig. 33 to form the corona, which is then launched into the jets by the rotating large scale magnetic fields through either the BP or BZ mechanism; during these processes the magnetic fields are mostly responsible for changing the directions of motion of plasmas by the Lorentz force. If the corona is clumped as discussed in Section 6 [148], then the plasma channeled into the jets should be clumpy and thus the ejected jets should be episodic and have knotted structures, in agreement with observations; this scenario is illustrated in Fig. 34. Alternatively episodic ejections may also occur in the disk, similar to the coronal mass ejection on the Sun [152].

Interestingly, the soft X-ray excess (SXE) frequently observed in AGNs can be interpreted as the warm layer found in XRBs, since the inferred plasmas parameters, with electron temperature of 0.1 to 0.3 keV and Compton scattering optical depth of around 10, are similar to that of XRBs and universal among AGNs with different BH masses, but seem to be related only with \dot{m} [153–155]. Therefore their warm layers are also scale independent. Indeed, magnetic flares, perhaps caused by magnetic reconnections in the cold disk, can produce the warm layers in both XRBs and AGNs [156, 157].

7 Further developments on state transitions

To further understand the mechanism of the spectral state transitions, it is important to study those spectral state transitions in individual BHXBs during different outbursts. The advantages are obvious – uncertainties in our estimates of black hole mass, the binary properties such as the orbital period, or the source distance will not play any role in producing the observed diverse transition properties in individual BHXBs, which can only be driven by the accretion process under different initial conditions. The sources which firstly allowed such a study were the BH transients GX 339–4 and XTE J1550–564, the NS transient Aquila X–1, and the flaring NS low-mass XRB (LMXB) 4U 1705–44, in which a remarkable correlation between the luminosity of the hard-to-soft transition and the peak luminosity of the following soft state was found [158–160]. More recently, a comprehensive study of the hard-to-soft spectral state transitions detected in all the bright XRBs in a period of about five years with simultaneous X-ray monitoring observations with the RXTE/ASM and the Swift/BAT confirmed the correlation between the hard-to-soft transition and the peak luminosity of the following soft state [161, 162], as shown in the upper panel of Fig. 35 [161]. More important was the discovery of the correlation between the transition luminosity and the rate-of-change of the luminosity during the rising phase of an outburst of transients or a flare of persistent sources, as shown in the lower panel of Fig. 35 [161].

The above correlation implies that in most cases it is the rate-of-increase of the mass accretion rate, rather than the mass accretion rate itself, determines the hard-to-soft spectral transition; this is depicted in Fig. 36 [161]. In addition, the discovery of the relation between the hard X-ray peak flux and the waiting time of transient outburst in the BH transient GX 339–4 (as shown in Fig. 37) supports that the total mass in the disk determines the peak soft state luminosity of the following outburst [159]. Therefore spectral states should be understood in the non-stationary accretion regime, which would be described by both \dot{m} and \ddot{m}, as well as the total mass in the disk before an outburst.

8 Further developments on thermal stability of SSD

The SSD model predicts that when the accretion rate is over a small fraction of the Eddington rate, which corresponds to L\gtrsim 0.06L_{\mathrm{Edd}}, the inner region of the disk is radiation-pressure-dominated and then both secularly [163] thermally unstable [164, 165]. However, observations of the high/soft state of black hole X-ray binaries with luminosity well within this regime (0.01L_{\mathrm{Edd}}\lesssim L\lesssim 0.5L_{\mathrm{Edd}}) indicate that the disk has very little variability, i.e., quite stable [166]. It has been well established that the accretion flow in this state is described by the SSD model [4, 167]. Radiation magnetohydrodynamic simulations of a vertically stratified shearing box have confirmed the absence of the thermal instability [168]. Recently, the thermal stability is revisited by linear analysis, by taking into account the role of magnetic field in the accretion flow [169]. By assuming that the field responds negatively to a positive temperature perturbation, it was found that the threshold of accretion rate above which the disk becomes thermally unstable increases significantly, compared with the case of not considering the role of magnetic field. This accounts for the stability of the observed sources with high luminosities, as shown in Fig. 38. If the magnetic pressure is less than about 24% of the total pressure, then this model can explain the “heart-beat” limit-cycle instability observed in GRS 1915+105 at its highest luminosity; this peculiar source holds the highest accretion rate (or luminosity) among BHXBs.

Observations of GRS 1915+105 showed that t_{\mathrm{high}} (the duration of the outburst phase) is comparable to t_{\mathrm{low}} (duration of the quiescent phase) and L_{\mathrm{high}} is 3 to 20 times larger than L_{\mathrm{low}} [170]. However, numerical calculations showed that t_{\mathrm{high}} is less than 5 percent of t_{\mathrm{low}}, and L_{\mathrm{high}} is around two orders of magnitude larger than L_{\mathrm{low}}. Some efforts have been made to improve the theory in order to explain observations either by some artificial viscosity prescription [171] or by additional assumption of the energy exchange between the disk and corona [172]. Taking into account the stress evolution process, it was found that the growth rate of thermally unstable modes can decrease significantly owing to the stress delay, which may help to understand the “heart-beat” limit-cycle variability of GRS 1915+105 [173]. The limit-cycle properties are found to be dominated by the mass-supply rate (accretion rate at the outer boundary) and the value of the α-viscosity parameter in the SSD model that assumes that the viscous torque is proportional to the total pressure [77]. It was also found that only the maximal outburst luminosity (in Eddington units) is positively correlated with the spin of a BH, providing another way to probe BH spin [77]; this is mainly due to the smaller inner disk radius and thus higher radiative efficiency for larger a_{\ast } as shown in Figs. 15, 16 and 17.

9 Unification and outlook

Despite of the many progresses made over the last decades on the study of BHXBs and microquasars, there are still many outstanding and unresolved issues, which can be pierced by two big pictures. One big picture is related to the astrophysics of BHXBs and microquasars, which is centered on understanding the accretion-outflow (wind or jet) connections at different accretion rates for different types of astrophysical systems involving BHs. The other big picture is related to the fundamental physics of BHXBs and microquasars, which is centered on identifying astrophysical BHs and testing theories of strong gravity. These two pictures are actually also entangled together, because a strong gravity theory, such as GR, is needed to describe the astrophysical aspects of these systems, and we need to understand the astrophysics of BHXBs and microquasars before we can start to test any strong gravity theory [174].

Some necessary elaborations and points:

•\dot{m}\gg 10: the two-phase clumpy thick disk is in the form of neutrino dominated accretion flow (NDAF), which can extend to ISCO due to the extremely low opacity of neutrinos. The large scale magnetic field lines are wound-up by the spinning BH and rotating disk. The discontinuous jet is produced by the BZ mechanism. Most likely no such ultra-Eddington AGN exists.

•\dot{m}\sim 1: the thin disk is truncated by radiation pressure, which drives near-spherical winds out. BL3Q is predicted as the early phase of a quasar, when the radiation pressure drives all gas in the radiation cone out so that no broad-line region (BLR) can be formed [179]. Actually all gas mixed with dust in the radiation cone can be blown out at substantially below Eddington rate, since the dust has a very high opacity to emissions from visible to UV. There BL3Q should be generic in the early phase of a quasar’s active cycle, if the quasar activity is triggered by enhanced mass inflow consisting of a mixture of gas and dust.

•\dot{m}\sim 0.3: the thin disk extends to around ISCO and the two-phase corona is clumpy, so the jet is produced mostly via the BZ mechanism and is discontinuous. The dusty torus and BLR for in a FSRQ are not shown.

•\dot{m}\sim 0.1: the thin disk extends to around ISCO and near-equatorial wind is thermally driven out; small scale and turbulent magnetic fields may be responsible for launching the plasma out of the disk via magnetic reconnections. The dusty torus and BLR in a RQQ are not shown.

•\dot{m}\sim 0.03: the thin disk is made of an inner and outer part; the inner disk extends to around ISCO. The two-phase ADAF is clumpy, so the jet produced via a mixture of the BP and BZ mechanisms is discontinuous. The dusty torus and BLR in a BL Lac are not shown.

•: “Q” refers to the very low luminosity quiescent state that normally displays a hard power-law spectrum. The large scale magnetic fields rotating with the truncated thin disk and thick ADAF channel the continuous plasma to form collimated and continuous jets via the BP mechanism. There is neither dusty torus nor BLR in LLAGNs [179].

Figure 39 shows the relation between the collimation-corrected gamma-ray luminosity and the kinetic power for AGNs and gamma-ray bursts (GRBs) [175]. For at least some of GRBs, a super-Eddington accreting compact stellar object, probably a BH with around 10M_{\odot }, is believed to be responsible for powering the highly relativistic jets, which produces intense gamma-rays through violent collisions of blobs in the jets [176]. The two types of AGNs shown here, flat-spectrum radio quasars (FSRQs) and BL Lacs, have all been observed to have mildly relativistic jets. Their central engines are supermassive BHs with around {10}^{7-9}{M}_{\odot } with accretion rates just below (for FSRQs) or far below (for BL Lacs) Eddington rate [177]. The good correlation over about 10 orders of magnitudes between these very different systems with very different accretion rates suggests that there must be some common mechanisms responsible for the accretion-outflow (wind/jet) connections for all accreting BHs [178].

Putting together all related phenomenologies and some theoretical modeling of BHXBs and microquasars discussed in this article, an unification scheme is illustrated in Fig. 40 with the major elements in the accretion-outflow connections in different types of astrophysical systems harboring both stellar mass BHs and supermassive BHs with accretion rates over several orders of magnitudes. The types of accreting stellar mass BHs include BHXBs and microquasars in different spectral states, as well as ultra-luminous X-ray sources (ULXs) (some of which are most likely ultra-Eddington accreting BHs [180]), and GRBs. The types of accreting supermassive mass BHs include low-luminosity AGNs (LLAGNs), BL Lac objects, normal radio quiet quasars (RQQs), FSRQs and broad-line-less luminous quasars (BL3Qs) [179] (I create the acronym “BL3Q” here just for fun).

This scheme is not a theoretical model, or even toy model yet. It, however, can be used as one possible chain to pierce many observed phenomenologies together, providing a possible frame work for further theoretical developments on accretion-outflow connections. Future observations will scrutinize this unification scheme and revise it inevitably. Of particular importance is the reliable determinations of the mass and spin of accreting BHs. The BH mass allows us to determine the accretion rate in units of Eddington rate, i.e., \dot{m}, which is a key parameter to the unification scheme of accretion-outflow connections. The BH spin is of course another key parameter, because it can determine the radiative efficiency of the disk and jet power in the BZ mechanism.

Once we have well determined BH parameters and a consistent and predictable theory describing the observed accretion-outflow connections, we are then ready to study the fundamental physics of BHs with BHXBs and microquasars, such as the properties of event horizon, space-time around Schwarzschild and Kerr BHs, BH spin energy (and mass) extraction. For example, both the broad iron line and CF fitting methods can be used to measure a BH’s spin. However the two methods are equivalent only if the metric is accurately described with Kerr metric. Different spin measurements for a BH with the two different methods would invalidate Kerr metric, provided that we understand the accretion disk physics thoroughly. On the other hand, a correct metric is of course required to describe accurately the accretion flow around BHs. This is one example of the entanglement or interplay between astrophysics and fundamental physics of accreting BHs. The stake is high, but the job is difficult.

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