Black holes: Timing and spectral properties and evolution

The X-ray spectra of accreting BHs often show a reflection component that is formed when hard X-rays irradiate the disk. This component includes a fluorescent iron emission line, an iron absorption edge, and a broad excess peaking at tens of keV (Lightman and White, 1988; Fabian et al., 1989). Relativistic velocities of the disk material and a redshift from the BH’s gravitational field distort the reflection component (Laor, 1991). Although the the entire reflection component is affected, the distortion is most apparent in the iron K$\alpha$ emission line, and this is clearly seen in both Active Galactic Nuclei (Tanaka et al., 1995) and stellar mass black holes (Miller, 2007).

Although sources do not show reflection components at all times, and some sources do not show them at all, the presence of reflection is common. The majority of sources have reflection components, and they can occur in a variety of spectral states. For example, Cyg X$-$1 exhibits a strong reflection component in its hard state (Nowak et al., 2011; Parker et al., 2015), its soft state (Walton et al., 2016), and its intermediate state (Tomsick et al., 2018). GX 339$-$4 is an active BH transient with a well-studied reflection component in multiple spectral states (e.g., Zdziarski et al., 1998; Parker et al., 2016).

The velocities in the inner portions of the disk can exceed 10% of the speed of light, producing a symmetric distortion of close to 1 keV at the 6.4 keV energy of the iron emission line. The gravitational effect produces an asymmetric distortion, which leads to a red wing of the emission line. A main goal of modeling the iron line and reflection component is to measure the location of the inner edge of the accretion disk ($R_{\rm in}$). If the disk extends to the innermost stable circular orbit (the ISCO), then measuring $R_{\rm in}$ provides a measurement of the BH spin. Even if the disk is slightly truncated and does not reach the ISCO, the $R_{\rm in}$ measurement still provides a lower limit on the BH spin.

Some spectral and timing measurements can be interpreted as the disk becoming highly truncated for sources in the hard state (Tomsick et al., 2009; Plant et al., 2015; De Marco and Ponti, 2016; Xu et al., 2020), while other measurements are not indicative of truncation (Miller et al., 2015). Most observations that suggest a high level of truncation have occurred for sources that are at low luminosities near 0.1% Eddington. At these low luminosities, the iron line profile changes from having a relativistic profile to a relatively narrow line, which provides a low limit on $R_{\rm in}$ (Tomsick et al., 2009; Plant et al., 2015; Xu et al., 2020). However, the relativistic iron line seen at high luminosities in the hard state leaves it unclear precisely when the disk becomes truncated. Determining the location of $R_{\rm in}$ and characterizing its evolution in the hard state is an active area of research (e.g., Wang-Ji et al., 2018).

The iron line and other spectral components are illustrated in Fig. 4. The spectrum shown of MAXI J1820$+$070 in the hard state is based on modeling of a NICER and NuSTAR spectrum. The two direct components are the thermal disk emission at low energies and the coronal emission, which is modeled as being produced by Comptonization by high-energy electrons in the jet with a thermal distribution. Fig. 4 illustrates the source geometry used in work by Buisson et al. (2019). We discuss possible source geometries, which are likely state-dependent, in §4.3. The properties of the reflection component provide motivation for the double-lamppost since many sources have reflection components that combine relativistic reflection from the inner disk and non-relativistic reflection from the outer disk.

NuSTAR has enabled advances in reflection and iron line studies for a number of reasons. NuSTAR’s 3–79 keV bandpass is very well-matched to measuring the reflection component. Its advantages over RXTE are its much better energy resolution as well as improved sensitivity (especially at the high-energy part of the bandpass) due to lower background. NuSTAR also provides advances over CCD instruments like those on XMM-Newton and Suzaku by providing sensitive measurements at higher energies. In addition, photon pile-up can lead to spectral distortion in CCD instruments (Done and Diaz Trigo, 2010; Miller et al., 2010) which is not trivial to account for. Although pile-up can often be mitigated by using certain detector modes or by modeling the pile-up effects, it is a significant source of systematic uncertainty for bright sources. However, NuSTAR has a triggered readout system that is not susceptible to pile-up. The NuSTAR advances in bandpass and avoiding pile-up have occurred along with advances in reflection modeling by making the models more physically realistic (García et al., 2014). These advances have increased the reliability of the modeling technique as a tool to measure the properties in the inner accretion disk, including the inner disk radius and the spins of black holes.

High resolution spectral observations of BHBs revealed the presence of absorption lines (see Fig. 5, Left) blue shifted by up to a few thousand km s^-1, showing that these sources not only produce collimated jets but can also drive winds. The lines are mainly hydrogen- and helium-like iron K${\alpha}$, indicating material that is strongly photoionized by the X-ray illumination from the central source (see reviews by Ponti et al. (2012, 2016); Díaz Trigo and Boirin (2016)).

The wind is an important component in the accretion flow as the mass outflow rate in this component can exceed the mass accretion rate through the disk (Ponti et al., 2012; King et al., 2012). It is also one of the determining factors in the general outburst profile and duration (Dubus et al., 2019; Tetarenko et al., 2020). Winds are observed across the black hole mass scale (King et al., 2013, and references therein), and a comparison of their properties in supermassive BHs and Galactic BHs can provide clues about the driving mechanisms of the winds.

Whether we can observe evidence of winds in X-rays depends on the spectral state and disk inclination: wind signatures are preferentially detected in soft states for high inclination sources (see Fig. 5, Done et al., 2007; Ponti et al., 2012). The inclination effect indicates that the plasma has a flat, equatorial geometry above the disk providing large amount of material in the line of sight. Wind signatures are also imprinted in the optical emission lines of He and H as P Cygni profiles and broad emission line wings (Muñoz-Darias et al., 2019, and references therein). These “cold winds” are detected in the hard state. In fact, similar near infrared line features obtained by VLT/X-Shooter indicate that the cold winds are present throughout the outburst in both soft and hard states (Sánchez-Sierras and Muñoz-Darias, 2020). See §4.2 for a discussion of competing models to explain the origin of winds in BHBs.

In addition to the accretion disk, corona and winds observed in the X-ray band, XRBs are also known to launch powerful jets that are most clearly observed through synchrotron emission in the radio to infrared (IR). These jets are capable of carrying away a significant amount of the accretion energy and depositing large amounts of energy into their surrounding environments Gallo et al. (2005); Russell et al. (2007); Tetarenko et al. (2017). Radio emission is only detected in the hard state and during the state transition. Thus, it is clear that the launching mechanism is somehow related to the accretion process, but the causal connection between accretion and ejection and how these relativistic jets are launched, collimated and accelerated remain open questions. Here we describe the properties of the radio and IR emission during a the black hole outburst, and observational efforts to probe the disk-jet connection.

At the beginning of an outburst, when the X-ray emission is spectrally hard, the radio and IR show a steady, compact (AU-sized) jet Dhawan et al. (2000); Corbel et al. (2000); Fender (2001); Stirling et al. (2001); Fender et al. (2004). The radio-to-mm spectral energy distribution (SED), shows a flat or inverted spectrum that extends up to $\sim 10^{13}$ Hz, beyond which the jet becomes optically thin, and the spectrum steepens. This frequency of the spectral break is thought to be set by the location in the jet where particle acceleration begins: higher frequencies correspond to an acceleration zone that is closer to the black hole. In a typical outburst, as the accretion rate increases throughout the hard state, the spectral break is observed to evolve to lower frequencies (down to the radio band), which has been interpreted as being due to the particle-acceleration region moving further from the black hole.

During the transition from hard to soft state, the steady, compact jet turns off, but that is not the end of the story for the jet. In several cases where high-cadence radio monitoring was possible, the end of the state transition was marked by rapid flaring in radio before completely shutting down in the soft state. The flares have been associated with the ejection of discrete knots of plasma moving away from the black hole at relativistic velocities (e.g. Mirabel and Rodríguez (1994); Fender and Belloni (2004); Bright et al. (2020)). No compact radio core is observed in the soft state, but one can observe residual and spatially extended radio emission from ejecta launched during the state transition. In some cases, these knots can reach separations of tens of thousands of times farther than the jet core. In the remarkable case of MAXI J1820$+$070, Bright et al. (2020) discovered that during the state transition, an isolated radio flare from the compact core expanded to a spatially extended, bi-polar, relativistic outflow that was also detected in the X-rays, likely due to shocks from the jet ejecta interacting with the interstellar medium (Corbel et al., 2002; Espinasse et al., 2020).

The dramatic change in both the jet (as observed in radio) and the inner accretion flow (as probed in X-rays) during an outburst motivated studies correlating these two wavelengths in order to better understand their connection. Indeed, the strong correlation between X-ray and radio luminosity suggests that an increase in the mass accretion rate onto the black hole results in an increase in mass loading in the jet Hannikainen et al. (1998); Corbel et al. (2002); Gallo et al. (2003). This same correlation is even seen in supermassive black holes when scaled properly for black hole mass Merloni et al. (2003); Falcke et al. (2004). While early studies suggested X-ray binaries in outburst exist on a single track, such that $L_{\mathrm{radio}}\propto L_{\mathrm{X}}^{\sim 0.7}$ Corbel et al. (2002), in a large population study, Gallo et al. (2012) showed that there were two populations: one that is “radio-loud” and another “radio-quiet”, which may be related to changes in the radiative efficiency of the accretion flow throughout the outburst Casella and Pe’er (2009); Koljonen et al. (2018); Espinasse and Fender (2018) and/or variations in observed luminosity due to inclination affects Zdziarski et al. (2016); Motta et al. (2018); Muñoz-Darias et al. (2013); Heil et al. (2015); Motta et al. (2015).

Beyond correlations in population studies, cross-correlation between the X-ray band and radio/IR/optical bands on timescales as short as milliseconds, have revealed insights into the causal connection between these emitting regions. Several systems show a rapid IR/optical lag of $\sim 0.1$ s behind the X-rays Gandhi et al. (2008, 2017); Vincentelli et al. (2019); Paice et al. (2019), which has been interpreted as being due to the propagation delay between the X-ray corona and the first IR/optical emission zone in the jet, and thus constrains the physical scale over which plasma is accelerated and collimated in the inner jet Jamil et al. (2010); Malzac (2013, 2014). This model can also explain the observed lag on a timescale of $\sim$minutes between radio and sub-mm bands, as the propagation front moves outwards in the jet. In addition to these lags, other tantalizing broadband timing properties are observed in several sources, but their origins remain a topic of debate, including: an anti-correlation of the X-ray and optical/IR at zero-lag (e.g., Veledina et al. (2011); Paice et al. (2021)) and quasi periodic oscillations (QPOs) at same temporal frequency in both X-ray and optical/IR Vincentelli et al. (2021). Simultaneous, fast photometry observations in X-rays and longer wavelengths are challenging to coordinate, and the field is relatively young. Future coordinated campaigns at several points in the outburst have the potential to be a powerful tool for probing disk-jet connection in X-ray binaries.

QPOs are a tantalizing phenomenology commonly observed during black hole outbursts, where variability power is concentrated in a narrow frequency range (Fig. 6). Their origin is still debated, but much progress has been made in the last decade to understand the physics behind these striking features. QPOs broadly take the form of “low-frequency QPOs” (LFQPO, in the temporal frequency range of $\sim 0.005-40$ Hz) and “high frequency QPOs” (HFQPO, observed at $\sim 40-450$ Hz). There is a rich phenomenology and an admittedly confusing nomenclature, which we attempt to distill below. We refer readers to more thorough descriptions of QPOs in Casella et al. (2005), Belloni (2010) and Ingram and Motta (2019).

While the power spectrum is the amplitude of the Fourier transform of a given light curve, much information is also contained in the phase, and in particular the phase difference between different energy bands (see Fig. 7). For a detailed review on Fourier-resolved time lag techniques and modeling, we encourage readers to see Uttley et al. (2014). Here, we provide a short history of the field, with particular focus on new results from XRBs and how the lags evolve with spectral state.

Fourier-resolved time lags in stellar-mass black holes were first discovered with RXTE in the hard state of the high-mass X-ray binary, Cyg X$-$1 Miyamoto et al. (1988); Nowak et al. (1999); Pottschmidt et al. (2000). In this source, hard photons were observed to lag soft photons on long timescales (low Fourier-frequencies $<1~{}Hz$). Indeed, similar low-frequency hard lags have also been seen in Active Galactic Nuclei (first in Papadakis et al. (2001)), and during the hard state in low-mass X-ray binaries Uttley et al. (2011), but are not typically observed in the soft state of LMXBs, when variability amplitudes are too low. The low-frequency hard lags are typically 0.1 s or longer, and thus are unlikely to arise from a light travel time lag, which would imply that the corona is thousands of gravitational radii ($R_{\rm g}$ = $GM/c^{2}$) Nowak et al. (1999). Thus the prevailing interpretation is that these lags are related to the radial inflow (i.e. viscous) timescale Kotov et al. (2001), as mass accretion rates propagate inwards through the accretion disk Lyubarskii (1997). As such, these lags are often referred to as propagation lags or continuum lags.

More recently, time lag studies have expanded to higher Fourier frequencies, revealing a new phenomenology: the high-frequency soft lag. First discovered with XMM-Newton in the hard state of GX 339$-$4 Uttley et al. (2011), these lags are interpreted as due to the light travel time between the X-ray corona and the accretion disk, though how the observed lags translate to the exact truncation radius of the disk or the height of the corona is still debated (see Section 4.3). De Marco et al. (2015) expanded these studies of GX 339$-$4 in the rise in the hard state, and found that the lag amplitude decreased during the hard state, which was interpreted either as the truncation radius of the disk moving inwards or the corona contracting before the state transition. Pile-up in XMM-Newton timing mode precluded the spectral confirmation of this picture by simultaneous mapping of the iron line profile.

Now NICER has revolutionized spectral-timing studies thanks to its high throughput, good energy resolution with virtually no pile-up that allows us to measure both the time lags and spectra simultaneously. Kara et al. (2019) studied the luminous hard state of the bright XRB MAXI J1820$+$070 and discovered that before the state transition, the reverberation lags evolved to higher frequencies with a smaller amplitude lag. During this same period, the broad iron line profile remained constant, suggesting that the truncation radius did not change, and rather, the vertical extent of the corona decreased. Recently, Wang et al. (2021) and De Marco et al. (2021) discovered that during the state transition, days before the detection of a radio flare and a Type B QPO, the reverberation lag suddenly increased in amplitude (and decreased in Fourier frequency; Fig. 7-left). Both papers suggest this result indicates that the hard-to-soft transition is marked by the corona expanding vertically and launching a jet knot that propagates along the jet stream at relativistic velocities. Most recently, Wang et al. (2022) performed an automated NICER search for time lags in black hole LMXBs, and discovered that the longer reverberation lags during the state transition is not only observed in MAXI J1820+070, but is common behaviour in black hole transients (Fig. 7-right).

Not only are time lags associated with the broadband noise, but also distinctive lags are observed in Type C Altamirano and Méndez (2015); Karpouzas et al. (2021); Méndez et al. (2022); Zhang et al. (2022) and Type B QPOs Stevens et al. (2018); García et al. (2021). Some sources show hard QPO lags, while others show soft lags, which may be determined simply by inclination effects (low orbital inclination sources show hard lags, while high-inclination sources show soft lags; van den Eijnden et al. (2017)) or perhaps is indicative of time-dependent radiative properties of the corona Méndez et al. (2022), where the balance of heating and cooling of the corona can lead to hard or soft QPO lags Karpouzas et al. (2021); Zhang et al. (2022). The latter model was recently applied to $Insight-HXMT$ observations of the low-frequency QPOs of MAXI J1535$-$571 during the transition from HIMS to SIMS, and inferred that the corona extends vertically during the state transition (similar to inferences from the broadband reverberation lags described above; Wang et al. (2021); De Marco et al. (2021); Wang et al. (2022)). Méndez et al. (2022) recently compared the inference from this variable Comptonization model to the radio jet luminosity in GRS 1915$+$105, which also suggests a connection between the powering of the corona and the relativistic jet.

High-throughput instruments with moderate energy resolution are revolutionizing our understanding of LMXBs, allowing for modeling of both the timing and spectral products. Time lags are an example of one such technique where the energy and time-dependence of the photons are accounted for in the models. Frequency-resolved spectroscopy is another technique that allows obtaining the energy spectrum of the X-ray flux variations on different time scales. It has been applied to very bright sources to probe the geometry of the black hole vicinity (Revnivtsev et al., 1999; Baby and Ramadevi, 2022), as well as to study the origin of QPOs (Axelsson and Done, 2016). Other examples of important spectral-timing techniques include: phase-resolved spectroscopy (e.g. Stevens et al. (2018); Ingram et al. (2016)), cross-spectral analysis (e.g. Mastroserio et al. (2019)), and the bispectrum (e.g. (Arur and Maccarone, 2022)).

The observations with the OSSE instrument on the Compton Gamma-ray Observatory provided the first evidence of the distinct hard X-ray (from $\sim$10 keV to a few hundred keVs) - soft $\gamma$-ray (from a few hundred keVs to a few MeVs) behaviour in the hard and the soft states of BHBs. In Cyg X$-$1, the hard state shows a break at around 100 keV while the soft state emission has no break up to the MeV range Grove et al. (1998). This behavior has been seen in many sources, and can also be observed for MAXI J1820$+$070 in Fig. 3. The spectra are consistent with thermal Comptonization in the hard state, thereby providing a measure of the electron temperature and the optical thickness in the corona, and non-thermal Comptonization in the soft state from electrons with an initial power-law energy distribution Coppi (1999).

The game changer in understanding the high energy emission behavior of BHBs has been the $INTEGRAL$ Observatory with a combination of sensitive instruments operating in the hard X-ray and soft $\gamma$-ray bands (as well as a softer X-ray instrument) and its ability to perform uninterrupted observations with a large field of view. A recent review by Motta et al. (2021) discusses the contribution of the $INTEGRAL$ observatory in BHB research in detail, and only a few highlights are presented here.

One of the most important contributions of $INTEGRAL$ has been the confirmation and detailed investigations of a soft $\gamma$-ray tail in the spectra of several BHBs in the hard and the hard intermediate states extending up to the MeV energies. Equally important discovery made by $INTEGRAL$ is the polarization measurement of Cyg X$-$1 in the soft $\gamma$-ray band showing that the polarisation fraction increases with energy (from less than 20% below $\sim$230 keV, to greater than 75% above $\sim$400 keV, Laurent et al. (2011); Jourdain et al. (2012)). Soft X-ray (at 2.6 keV and 5.2 keV) polarization measurements of Cyg X$-$1 performed with $OSO-8$ indicate a relatively low (compared to soft $\gamma$-ray measurements) polarization level of 2-5% with low significance Long et al. (1980), consistent with the decreased polarization fraction with decreasing energy.

Historical soft $\gamma$-ray band observations also claimed narrow and broad features associated with positron annihilation; a broad feature in Cyg X$-$1 measured with $HEAO1$ Nolan and Matteson (1983), a narrow line for Nova Muscae Sunyaev et al. (1992) and a broad and variable feature in 1E1740.7–2942 Bouchet et al. (1991) measured with SIGMA on $GRANAT$. Recently, using the SPI instrument on $INTEGRAL$, a detection of a broadened 511 keV annihilation line during a strong flaring activity in the 2015 outburst of V404 Cyg has been claimed Siegert et al. (2016), yet other groups failed to detect the same feature in the same dataset Roques and Jourdain (2019); Jourdain et al. (2017). Comprehensive searches for annihilation line features with IBIS and SPI on $INTEGRAL$ did not find any point source, and the emission in the associated band is consistent with diffuse positron-annihilation De Cesare (2011); Tsygankov and Churazov (2010). MAXI J1820$+$070 is no exception as a detailed analysis of the soft $\gamma$-ray region extending up to 2 MeV with PICSiT and SPI on $INTEGRAL$ only provided a strong upper limit to an annihilation line feature. The non-detection, together with the Comptonization model parameters allowed a calculation of the electron-positron pair density which is turned out to be low Zdziarski et al. (2021b).

Finally, observational campaigns have been performed during the state transitions with $RXTE$ as well as with $INTEGRAL$ in the hard X-ray band to characterize the evolution of the break frequency (or cut-off frequency depending on the spectral model used) and other spectral parameters to understand the possible physical changes in the accretion environment during transitions. GX 339$-$4 in the 2004 outburst showed a rapid transition ($<$10 hrs) from the HIMS to SIMS with significant changes happening in the break frequency and the timing properties Belloni et al. (2006). The evolution of the break energy during the transitions are quite complex, decreasing during the hard state to the HIMS, and then increasing, disappearing, or moving beyond the detection range towards the transition to SIMS Motta et al. (2021).

There are a significant number of exceptions to the standard path taken by BH transients in the HID. For example, there are many cases of “failed outbursts” that do not reach the SIMS and soft state, and there are also “hard-only” outbursts that do not even make it into the HIMS (Motta et al., 2021). These differences in the HIDs occur on an outburst-by-outburst basis, and Fig. 8 shows different outburst types for the same source. While some sources may more often have one type or another type of outburst, a single BH transient can exhibit regular, failed, and hard-only outbursts. Thus, the difference may be related to different mass accretion rates between outbursts or perhaps different properties of the disc (e.g., magnetic field) or jet.

There are also some interesting accreting BH sources that have unusual aspects of their HID behavior. Cyg X$-$1 is a persistent source and makes transitions between the hard and soft states, but its luminosity remains unchanged to within about 15% (Zhang et al., 1997). Thus, it occupies a relatively narrow range of intensity in the HID. GRS 1915$+$105 can show extremely rapid flux oscillations with distinctive and repeating patterns. At times, these oscillations occur while GRS 1915+105 is restricted to the bright and soft part of the HID. Combining X-ray, radio, and near-IR measurements provides evidence for a connection between the X-ray oscillations, which are presumably related to disk instabilities, and jet production (Fender and Belloni, 2004).

Although less common, there are also cases where the hardness-intensity path differs from regular outbursts. Low-soft states have been seen in the persistent BH systems 1E 1740.7–2942 and GRS 1758–258. Both of these sources spend the vast majority of their time in the hard state, but they will occasionally decrease in flux and transition to the soft state (Smith et al., 2001). This has also been seen in the BH transient Swift J1753.5–0127, which showed a very similar behavior at one point during an outburst that lasted over a decade (Shaw et al., 2016). It has been suggested that this phenomenon may be due to the presence of two accretion flows, a thin disk and a hot halo (Smith et al., 2002). The low-soft states can be naturally produced if the time scale for changes in mass accretion rate to propagate through the disk are different in the two flows.

Another BH transient that shows low-soft states is 4U 1630–47 (Tomsick et al., 2014). Approximately 20 outbursts have been seen from this source going back to 1969. In 1998, it showed a regular outburst rising in the hard state and moving in a counter-clockwise direction through the HID (Fig. 8). However, most of the 4U 1630–47 outbursts show the opposite pattern with the source taking a clockwise path through the HID. Although the reason for this difference is not known, it has some relationship to the the strength of the jet because 4U 1630–47 typically shows weak or undetectable radio emission during outbursts, but the strongest radio emission seen from 4U 1630–47 occurred during the 1998 outburst (Hjellming et al., 1999).

Accreting black holes often have a strong thermal component in their spectra (see Fig. 3), especially (but not exclusively) when they are emitting at high luminosities. This component is very well-described by a multi-temperature disk, which is approximated by adding the contributions of blackbody emission from many annuli in the disk. A theoretical temperature vs. radius profile (Shakura and Sunyaev, 1973) can be used to determine the temperature for each annulus. The simplest and often used version of this model is the “disk-blackbody” model (Mitsuda et al., 1984), and it is specified by the inner disk temperature and a normalization. The normalization depends on the apparent inner radius, the distance to the source, and the inclination of the disk. With corrections described in Kubota et al. (1998), a physical value for the inner radius ($R_{\rm in}$) can be estimated. Measuring $R_{\rm in}$ is of great interest since it is a key disk geometry parameter, having implications for what happens during state transitions as well as providing an opportunity to measure the spin of the BH, $a_{*}$ (Middleton, 2016).

Using the measurement of $R_{\rm in}$ to determine $a_{*}$ is only justified if the inner edge of the optically thick disk is at the innermost stable circular orbit (ISCO). In addition to the properties of the thermal component, there are other observational lines of evidence that the disc reaches near to or to the ISCO. The red wing of the iron emission line (see §3.1) shows that the emission reflected from the disc comes from the direct vicinity of the BH. Similarly, the HFQPOs must originate from very close to the ISCO. Measurements of $R_{\rm in}$ obtained by modeling the thermal component (“continuum fitting”) often show constant $R_{\rm in}$ values over large ranges of luminosity, strongly suggesting that the inner edge reaches a limiting value (Steiner et al., 2010), which is naturally interpreted as the ISCO.

Using continuum fitting to determine $a_{*}$ requires a significant amount of additional knowledge of the binary BH system. To determine $R_{\rm in}$ in physical units requires knowledge of the distance to the system as well as the disk inclination, which is usually assumed to be the same as the binary inclination. Once $R_{\rm in}$ is determined in physical units, then the mass of the BH is needed to calculate $R_{\rm g}$, and determine $R_{\rm in}$/$R_{\rm g}$, which determines $a_{*}$. Measurements of $a_{*}$ using the continuum fitting technique was accomplished for ten sources (McClintock et al., 2014) with a range from $a_{*}=0.12\pm 0.19$ (A 0620–00) to $>$0.95 (Cyg X$-$1). Recently, an update of the Cyg X$-$1 distance (Miller-Jones et al., 2021) has led to a revision of the spin estimate to $a_{*}>0.9985$ (Zhao et al., 2021).

While improved distances and masses for more systems will lead to more opportunities for $a_{*}$ measurements from continuum fitting, the systematic uncertainties need to be considered carefully. For example, spectral hardening of the disc continuum by the disc atmosphere leads to a higher observed temperature than the effective temperature of the disc. A correction using the spectral hardening factor, $f_{\rm col}$, is typically made when using the continuum fitting technique, but it has been argued that the uncertainty in $f_{\rm col}$ is large enough to lead to significant uncertainty in $a_{*}$ (Salvesen and Miller, 2021). In addition, there have been challenges to the assumption of spin-orbit alignment (Walton et al., 2016; Salvesen and Pokawanvit, 2020). One avenue to further improvements is the joint measurements of $a_{*}$ with the continuum fitting and reflection (iron line modeling) techniques (Parker et al., 2016). Spectro-polarimetry is another type of measurement that is expected to provide advances in this area (see Chapter 5).

The two main unresolved discussion points regarding the winds in BHBs are the driving mechanism of the wind (thermal, magnetic or radiation pressure) and the disappearance of X-ray absorption lines in the hard state.

The bright soft state provides ample, possibly unobscured ionizing radiation to lift matter from the disk surface through Compton heating (Begelman et al., 1983). For this thermal driving mechanism the wind is launched at large distances (10⁴ - 10⁵ $R_{g}$) from the central black hole. The launch radius will be a fraction of the Compton radius (the distance such that escape speed equals the isothermal sound speed at the inverse Compton temperature, Woods et al., 1996), and the exact value of the fraction depends on properties of the illuminating X-rays Done et al. (2018).

The other possible mechanisms to drive winds are radiation pressure and magnetic processes. For Galactic BHs, line driving is not efficient due to low UV flux, and high ionization (Done et al., 2007); however, under certain conditions radiation pressure can transfer momentum to free electrons (Reynolds, 2012; Done et al., 2018). On the other hand, magnetic wind launching mechanisms are viable possibilities. In fact, for GRO J1655$-$40, a single observation revealed a rich series of absorption lines (see Fig. 5) from a dense, highly ionized wind, initially interpreted as magnetically driven (Miller et al., 2006) as the calculated wind launch radius was much smaller than the Compton radius inferred for the given source luminosity prohibiting thermal winds. Magnetically driven winds close to the black hole are implied in other sources such as IGR J17091$-$3624 (King et al., 2012; Miller et al., 2012) and 4U 1630$-$47 (King et al., 2014). However, the magnetic origin in the single GRO J1655$-$40 observation has been challenged by multiple authors. If the wind is optically thick, the luminosity will be underestimated (it could be close to Eddington) leading to complications in the interpretation of the data due to radiative mechanisms (see (Done et al., 2018) for additional discussion regarding arguments against magnetic origin). Perhaps both origins are possible within a magneto-thermal hybrid wind, with magnetic launching from the inner parts of the disk and thermal launching from the outer parts of the disk in the same source (Neilsen and Homan, 2012; Trueba et al., 2019).

The disappearance of the X-ray absorption features in the hard state has been linked to the over-ionization of the material, photoionization instabilities, geometrical obscuration of the outer disk by a large scale-height corona, and the details of the driving mechanism Ponti et al. (2012); Chakravorty et al. (2013, 2016). Over-ionization in a static absorber is ruled out by detailed photoionization computations (Neilsen and Homan, 2012), and observationally the hard state luminosity during the rise of the outburst is similar to the soft state luminosities with wind features (see Fig. 5).

Perhaps the lack of winds in the hard state and the origin of the driving mechanism of the wind are interlinked. The hard states show ubiquitous jets, and an analysis of accretion ejection behavior in GRS 1915$+$105 led to the to claim that both jets and winds are part of the same magnetic system and the degree of collimation determines whether we see a jet or wind (e.g. “jet-wind dichotomy” Neilsen and Lee, 2009). However, winds and optically thin jets are known to coexist (Kalemci et al., 2016; Homan et al., 2016), and cold winds may be present in all states. Some magnetohydrodynamic simulations indicate that that the inclusion of magnetic fields into a thermally driven wind has the effect of suppressing the wind in low-plasma beta regions Waters and Proga (2018). There are rare claims of hard state wind features in the X-ray spectra such as for Swift J1658.2-–4242 with NuSTAR (Xu et al., 2018) and for V404 Cyg and GRS 1915$+$105 with Chandra (Homan et al., 2016). Perhaps more sensitive instruments (such as Xrism and later Athena) will provide more insight to the properties and the origin of the hard state winds.

While the general picture for the soft state is a non-truncated optically thick disc extending to the ISCO, there is no consensus on the accretion geometry in the hard state. In the context of the advection-dominated accretion flow (ADAF) model, a paradigm emerged where the disc begins to truncate in the intermediate state, $R_{\rm in}$ increases further in the hard state, and then reaches $10^{2}$ to $10^{4}$ Schwarzschild radii below Eddington ratios of $\sim$0.01 (Esin et al., 1997). Some measurements, such as the drop in the temperature of the thermal disc component and the drop in the QPO frequencies as the luminosity drops support this picture (Dinçer et al., 2014). In addition, other measurements can be explained by the truncated disc picture, including the size and evolution of the X-ray lags (De Marco and Ponti, 2016) as well as changes in the iron line, which has been found to change from a relativistic profile to a narrow line at Eddington ratios of 0.001-0.003 (Tomsick et al., 2009; Xu et al., 2020). An analysis of the frequency-resolved spectra of MAXI J1820$+$070 also indicated a reflector moving closer to the inner edge of the accretion disk in the hard-to-soft transition (Axelsson and Veledina, 2021). However, the relativistic iron lines that are seen in the bright ($\sim$0.01-0.1 $L_{\rm Edd}$) hard states (Miller et al., 2015; García et al., 2015) strongly contradict the idea that disc truncation is necessary for sources to show hard state properties.

Another critical addition to our understanding of BHB spectral states in the last two decades is that a compact jet is present in the hard state. The jet dominates the spectrum at radio wavelengths (Fender, 2001) and can dominate up to at least the near-IR (Corbel and Fender, 2002; Gandhi et al., 2011). Although it very likely extends to X-rays and perhaps gamma-rays, it is unclear what fraction of the X-ray and gamma-ray emission we see comes from the jet (see also §4.5). The question of how much X-ray emission comes from the jet compared to a more extended corona is important since the reflection modeling that is used to derive the inner radius of the disc depends on the source geometry (Dauser et al., 2013).

As described in §3.1, narrow and broad iron line components are often seen in hard state spectra, and one explanation that has been advanced for MAXI J1820$+$070 is that there is X-ray emission coming from different heights in the jet that illuminate both the inner and outer parts of a non- or mildly-truncated disc (Kara et al., 2019; Buisson et al., 2019). However, it has been shown that the MAXI J1820$+$070 can also be described by other geometries. The narrow and broad iron line components have also been modeled with a moderately truncated disk and illumination by multiple X-ray components (Zdziarski et al., 2021a). Within the reflection interpretation, the broad iron line does not allow for a highly truncated disc, but these two analyses show that complex models can lead to different numerical answers for $R_{\rm in}$.

In addition to the questions of disc truncation, high-energy emission from the jet, and source geometry in general, another question is regarding the disc inclination and whether the inner disc is aligned with the binary orbit, if it is set by the BH spin, or if the inner disc precesses, which has been suggested as a QPO mechanism (Ingram et al., 2017) (see also §4.6). Overall, there are still many open questions related to geometry of accretion in hard state, and answering the open questions related to the hard state accretion and source geometry is likely to require a combination of observations, theory, and numerical simulations.

In Section 3, we described several key observations that demonstrate a connection between the hot, optically thin corona at $R<100R_{\rm g}$ and the relativistic jet observed at much larger distances ($10^{7-8}R_{\rm g}$). These include:

• •

  The X-ray/radio flux correlation in Active Galactic Nuclei and XRBs, known as the ‘fundamental plane’ (Merloni et al. (2003); §3.3)

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  The low reflection fraction observed in the hard state, which has been interpreted as due to the corona/jet base accelerating away from the accretion disk. This beams much of the X-ray emission away from the disk (Markoff et al. (2005); §3.1).

• •

  The close temporal coincidence of the Type B QPOs during the state transition, and radio flares from the jet (Homan et al. (2020); §3.4).

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  The correlation between the size of the X-ray corona (probed by X-ray reverberation), and the radio luminosity (Wang et al. (2021); De Marco et al. (2021); §3.5).

Because of these and other observables, several models have been proposed that relate the corona and the jet. For instance, Markoff et al. (2005) suggest that the X-ray corona is the base of the relativistic jet. Thermal seed photons (and possibly also synchrotron photons from electrons in the jet) will be inverse-Compton upscattered in the jet base to produce the observed hard X-ray photons. This model maintains a spatially compact corona/jet-base that is required to explain the broad reflection features. Kylafis et al. (2008) and Reig et al. (2018) suggest a similar model where disk photons are upscattered in a jet, though require that the low-frequency hard lags (Section 3.1.6) are due to light travel time delays in the jet, and therefore require a much larger, extended jet. Petrucci et al. (2008) also suggest a connection between the hard X-ray emitting region and the relativistic jet, but propose that the hard X-rays originate in the plane of the accretion disk, in the so-called jet-emitting disk (JED), a highly magnetized, low-density inner accretion flow. The large large-scale jets then tap the mechanical energy released in the JED, via the Blandford-Payne mechanism Blandford and Payne (1982). The observational implication of the JED model is a truncated thin disk in the hard state, which is distinct from the jet-base model where the thin disk can extend to the ISCO. Because of these different observational predictions, future mutli-wavelength spectral-timing observations, paired with improved numerical simulations will reveal how the corona and jet are coupled.

The most popular models that have been put forward to explain the origin of the high-energy tail in the hard and hard intermediate states of BHBs are Comptonization from a non-thermal electron population and synchrotron emission from a compact jet. The energy dependent polarization measurement for Cyg X$-$1 above 400 keV nearly reaches the theoretical maximum for synchrotron radiation with ordered magnetic fields and strongly favors a jet origin beyond this energy as neither thermal nor non-thermal Compton scattering would provide polarization fractions close to the observed 75%. A jet origin for Cyg X$-$1’s MeV emission has interesting physical implications as it implies magnetic field strengths above the equipartition and a very efficient particle acceleration Zdziarski et al. (2014). There are no other reports of polarization measurement in hard X-rays/soft $\gamma$-rays for any other BH sources, and whether the high polarization is a peculiarity of Cyg X$-$1, or a common phenomena in all BH sources remains to be seen. A future MeV polarimeter such as the future Compton Spectrometer and Imager (COSI) (Tomsick et al., 2021) will allow for sensitive polarization measurements of BH transients.

The case for Compton scattering from a non-thermal population of electrons have already been put forward to explain the tail in the soft state, and also have been invoked in the hard state as well. For lower quality data (e.g. during the outburst decays with lower luminosity) the presence of non-thermal or hybrid Comptonization is implied by poor fits with thermal only Comptonization models, and a significant improvement in reduced $\chi^{2}$ with addition of a non-Maxwellian electron energy distribution (using the $eqpair$ model Coppi (1999), for example). With the much higher quality data, such as the case for MAXI J1820$+$070, the picture becomes more complicated. The analysis of SPI data at different energy bands indicated a two component emission, modeled as a thermal Comptonization plus a cut-off power-law with a claim that the power-law has possibly a jet origin as it is only prevalent at high energies (100 - 300 keV) in the hard state. On the other hand, an analysis including lower energies using $NuSTAR$ and higher energies using SPI and PicSIT on $INTEGRAL$ led to an alternative picture with two Comptonizing regions, one closer to the BH which is producing both thermal and non-thermal Comptonization, and the other possibly over the disk at a larger radii with thermal Comptonization, without the need of a separate power-law component Zdziarski et al. (2021b).

Non-thermal tails in the electron energy distribution invoked to explain the hard tails would also invoke enhanced synchrotron radiation by the same electrons, possibly not only altering the seed photon energy distribution (much less than the disk photon energies in the hard state), but may also be producing near-infrared emission at the same energies where the jets could also contribute Veledina et al. (2013).

Many models have been forward to explain the origin of QPOs (for both HFQPOs and LFQPOs, even extending same models to explain the QPOs in neutron star systems), but in general, the models can be classifies as “geometrical”, or “intrinsic”. In the geometrical models, one of the accretion/ejection component’s shape varies, and in the intrinsic models, the geometry is stable within the QPO oscillation timescale, however, the disk or the corona emission changes due to changes in the local properties. An extensive list of both geometrical and intrinsic models and their principles have been given in a recent review, Ingram and Motta (2019)), and here only models relevant to general evolution of X-ray spectral and timing will be discussed.

The Type C QPOs are observed almost every state, and currently the leading model to explain the origin of this QPO is the Lense-Thirring Precession (LTP) model arising from the frame dragging in the vicinity of a spinning BH Ingram et al. (2009). This model assumes a thick accretion flow (or a corona) inside a truncated disk misaligned with the equatorial plane of the BH causing a nodal precession of the entire inner flow due to the inequality between the orbital and vertical epicyclic frequencies. The epicyclic frequencies are the natural frequencies of a test mass disturbed slightly from the black hole equatorial plane, and in Kerr metric the orbital, radial and vertical frequencies are slightly different (Ingram and Motta, 2019). There are strong observational evidences for geometrical origin through the inclination dependence on the QPO RMS, and the iron line centroid frequency dependence on the QPO phase, both consistent with the predictions of LTP (though the latter is observed only in one source). There are also supporting arguments in favor of LTP, and against some of the intrinsic models. First of all, there is no sign of direct QPO modulation in the emission from the disk through frequency-resolved spectroscopy Sobolewska and Życki (2006), in contrast, the Type C QPOs can be observed beyond 200 keV Ma et al. (2021), therefore the modulated emission is coming from the corona. The LTP predicts a highest QPO frequency based on the spin, therefore this highest frequency should vary from source to source which is the case so far. It is important to stress that if LTP is indeed the correct model for Type C QPOs, most BH disk axis should be misaligned with the spin axis, and should stay misaligned (with at least 5^∘) for a long time as the accretion brings them back to alignment. There are observational and theoretical arguments in favor of misalignment which can be found in Ingram & Motta 2019 Ingram and Motta (2019) and references therein.

Type B QPO properties also depend on inclination, pointing to a geometrical origin, yet their RMS is higher for the lower inclinations systems whereas the opposite is observed for Type C QPOs Motta et al. (2015). One model associates Type B QPO to the precession of the jet Motta et al. (2015), as opposed to the precession of the thick flow in Type C QPOs.

For the HFQPOs, the 2:3 ratio in GRO J1655$-$40 (and weaker detection claims in other sources) suggested models invoking resonances related to test particle orbits, and for the case of GRO J1655$-$40, with the accompanying low frequency QPOs, a simple relativistic precession model has been suggested in which the LFQPO fundamental frequency, the lower, and the higher HFQPOs are the Lense-Thirring precession, periastron precession, and the orbital frequency at a characteristic radius, respectively Motta et al. (2014). Knowing all three frequencies allowed to determine the spin, characteristic radius and mass of the black hole, and the inferred mass is in agreement with the mass determined through optical observations. The HFQPO pair can also be explained within the framework of a precessing torus with frequencies corresponding to the breathing mode and vertical epicyclic frequency of the inner flow Fragile et al. (2016).