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4.2 Continuum spectroscopy of accreting black holes

There have been at least three different efforts published in the literature that use the luminosities and the continuum spectra of accreting black holes to look for evidence of strong-field phenomena.
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Figure 6: The 2 – 20 keV quiescent luminosities of black-hole candidates (filled circles) and neutron stars (open circles) in units of the Eddington luminosity for different galactic binary systems, as a function of their orbital periods, which are thought to determine the mass transfer rate between the two stars. The systematically lower luminosities of the black-hole systems have been attributed to the presence of the event horizon [106Jump To The Next Citation Point91Jump To The Next Citation Point].

4.2.1 Luminosities of black holes in quiescence and the absence of a hard surface

Low-mass X-ray binaries are stellar systems in which the primary star is a compact object and the secondary star is filling its Roche lobe. Matter is transferred from the companion star to the compact object and releases its gravitational potential energy mostly as high-energy radiation, making these systems the brightest sources in the X-ray sky [12692Jump To The Next Citation Point].

The rate with which mass is transfered from the companion star to the compact object is determined by the ratio of masses of the two stars, the evolutionary state of the companion star, and the orbital separation [173]. On the other hand, the rate with which energy is released in the form of high-energy radiation depends on the rate of mass transfer, on the state of the accretion flow (i.e., whether it is via a geometrically thin disk or a geometrically thick but radiatively inefficient flow), and on whether the compact object has a hard surface or an event horizon. Indeed, for a neutron-star system in steady state, most of the released gravitational potential energy has to be radiated away (only a small fraction heats the stellar core [26]), whereas for a black-hole system, a significant amount of the potential energy may be advected inwards past the event horizon, and hence may be forever lost from the observable universe. For similar systems, in the same accretion state, one would therefore expect black holes to be systematically less luminous than neutron stars [106Jump To The Next Citation Point].

The luminosities of transient black holes and neutron stars in their quiescent states most clearly show this trend. When plotted against the orbital periods of the binary systems, which are used here as observable proxies to the mass transfer rates, sources that are believed to be black holes, based on their large masses, are systematically less luminous (Figure 6View Image and [106Jump To The Next Citation Point6191]). Although the physical mechanism behind the difference in luminosities is still a matter of debate [1061882], the trend shown in Figure 6View Image appears to be a strong, albeit indirect, evidence for the presence of an event horizon in compact objects with masses larger than the highest possible mass of a neutron star.

4.2.2 Hard X-ray spectra of luminous black holes and the presence of an event horizon

Galactic black holes in some of their most luminous states (the so-called very high states) have mostly thermal spectra in the soft X-rays with power-law tails that extend well into the soft γ-rays [68Jump To The Next Citation Point]. It has been hypothesized that these power-law tails are the result of Compton upscattering of soft X-ray photons off the relativistic electrons that flow into the black-hole event horizon with speeds that approach the speed of light and, therefore, constitute an observational signature of the presence of an event horizon (e.g., see [16884Jump To The Next Citation Point]).

A relativistic converging flow has indeed the potential of producing power-law spectral tails (e.g., see [120167123]). However, this mechanism is identical to a second-order Fermi acceleration and hence the power-law tail is a result of multiple scatterings away from the horizon with low energy exchange per scattering rather than the result of very few scatterings of photons with ultrarelativistic electrons near the black-hole horizon [128118]. Moreover, the model spectra always cut off at energies lower than the electron rest mass [84109], whereas the observed spectra extend into the MeV range [68]. Successful theoretical models of the power-law spectra of black holes that are based on Comptonization of soft photons by non-thermal electrons [65], as well as the discovery of similar power-law tails in the spectra of accreting neutron stars that extend to ∼ 100 – 200 keV [4847], have shown conclusively that the observed power-law tails do not constitute evidence of black-hole event horizons.

4.2.3 Measuring the radii of the innermost stable circular orbits of black holes using continuum spectra

The thermal spectrum of a black-hole source in some of its most luminous states is believed to originate in a geometrically thin accretion disk. The temperature profile of such an accretion disk away from the black hole is determined entirely by energy conservation and is independent of the magnitude and properties of the mechanism that transports angular momentum and allows for matter to accrete (as long as this mechanism is local; see [1465]). The situation is very different, however, near the radius of the innermost stable circular orbit (hereafter ISCO).

Inside the ISCO, fluid elements cannot stay in circular orbits but instead quickly loose centrifugal support and rapidly fall into the black hole. The density of the accretion disk inside the ISCO is very small and the viscous heating is believed to be strongly diminished. It is, therefore, expected that only material outside the ISCO contributes to the observed thermal spectrum. The temperature profile of the accretion flow just outside the ISCO depends rather strongly on the mechanism that transports angular momentum outwards and in particular on the magnitude of the torque at the ISCO [80Jump To The Next Citation Point59Jump To The Next Citation Point3Jump To The Next Citation Point]. To lowest order, however, if the entire accretion disk spectrum can be decomposed into a sum of black bodies, each at the local temperature of the radial annulus in which it originates, then the highest temperature will be that of the plasma near the ISCO and the corresponding flux of radiation will be directly proportional to the square of the ISCO radius.

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Figure 7: The spectra emerging from geometrically thin accretion disks around black holes with different spins, but with the same accretion luminosity [41Jump To The Next Citation Point].From left to right, the curves correspond to spins (a ∕M) of 0, 0.2, 0.4, 0.6, 0.78, 0.881, 0.936, 0.966, and 0.99. The spin values were chosen to give roughly equal variation in the position of the spectral peak for spins > 0.8. The other parameters that determine the model are the viscosity parameter, α = 0.01, the inclination of the observer, cosi = 0.5, the mass of the black hole, M = 10M ⊙, and the accretion luminosity, L = 0.1LEdd. The peak energies of the spectra increase with increasing spin, as a consequence of the fact that the ISCO radius decreases with spin.

Phenomenological fits of multi-temperature black-body models to the observed spectra of black holes provide strong support to the above interpretation. When model spectra are fit to observations of any given black hole in luminosity states that differ by more than one order of magnitude, the inferred ISCO radius remains approximately constant [86]. For systems with a dynamically measured mass and with a known distance, such an observation can lead to a measurement of the physical size of the ISCO and hence of the spin of the black hole (see Figure 7View Image[187Jump To The Next Citation Point64].

There are a number of complications associated with producing the model spectra of multi-temperature black-body disks that are required in measuring spectroscopically the ISCO radius around a black hole. First, as discussed above, the temperature profile of an accretion disk at the region around the ISCO depends very strongly on the details of the mechanism of angular momentum transport, which are poorly understood [80593]. Second, the vertical structure of the disk at each annulus, which determines the emerging spectrum, may or may not be in hydrostatic equilibrium near the ISCO, as it is often assumed, and its structure depends strongly on the external irradiation of the disk plasma by photons that originate in other parts of the disk. Finally, material in the inner accretion disk is highly ionized and often far from local thermodynamic equilibrium, generating spectra that can be significantly different from black bodies [72].

There have been a number of approximate models of multi-temperature accretion disks that take into account some of these effects, in a phenomenological or ab initio way. The models of Li et al. [87], based on the alpha model for angular momentum transport, assume that the local emission from each annulus is a black body at the local temperature, and take into account the strong lensing of the emitted photons by the central black hole. On the other hand, the models of Davis et al. [41Jump To The Next Citation Point] are the result of ionization-equilibrium and radiative-transfer calculations at each annulus; they are based on the alpha model for angular momentum but allow for non-zero torques at the ISCO, and take into account the strong lensing of photons by the black hole.

Given the flux F of the accretion disk measured by an observer on Earth, the color temperature Tcol that corresponds to the innermost region in the disk that is emitting (which presumably is near the ISCO), the distance D to the source, and the mass M of the black hole, the spin α of the black hole can be inferred [187] by equating the radius of the ISCO, i.e.,

∘ ------ 2GM { } rISCO = ---2-- 3 + A2 ± [(3 − A1 )(3 + A1 + 2A2)]1∕2 , (20 ) c
to the one inferred spectroscopically (since F ∼ T4R2) by
[ ]1∕2[ ]2 ----F---- fcolfGR(θ,α-)- rspec = D 2σg (θ,α) Tcol . (21 )
Here A1 = 1 + (1 − a2)1∕3[(1 + a)1∕3 + (1 − a)1∕3], A2 = (3a2 + A21)1∕2, a is the specific angular momentum per unit mass for the black hole, and the positive (negative) sign is taken for prograde (retrograde) disks. In these equations, σ is the Stefan–Boltzmann constant and θ is the inclination of the observer with respect to the symmetry axis of the accretion disk. The functions g(θ,α) and fGR (θ,α) are correction factors for the flux and the temperature, respectively, that need to be calculated when going from an accretion disk annulus to a distant observer and incorporate the combined effects of gravitational lensing, gravitational redshift, and Doppler boosting of the disk photons. Given a thickness of the accretion disk, both these transfer functions can be computed to any desired degree of accuracy. Finally, the factor fcol measures the ratio of the color temperature of the spectrum (as measured by fitting a black body to the observed spectrum) to the effective temperature in that annulus in the accretion disk (which is a measure of the total radiation flux emerging from that annulus). Computing the value of the factor fcol is the goal of the recent calculations of the ionization equilibrium and radiative transfer in accretion disks [41].

Fitting these spectral models to a number of observations of black-hole candidates with dynamically measured masses has resulted in approximate measurements of their spins: a > 0.7 for GRS 1915+105 [9693]; a = 0.75 – 0.85 for 4U 1543–47 [145Jump To The Next Citation Point]; a = 0.65 – 0.75 for GRO J1655–40 [145]. It is remarkable that all inferred values of the black-hole spins are high, comparable to the maximum allowed by the Kerr solution.

Equations (20View Equation) and (21View Equation) demonstrate the strong dependence of the inferred values of black-hole spins on various observable quantities (the mass of, distance to, and inclination of the black hole, as well as the flux, and temperature of its disk spectrum) and on a model parameter (the color correction factor fcol). Numerical simulations of magnetohydrodynamic flows onto black holes are finely tuned to resolve the length and timescales of phenomena that occur in the vicinity of the horizon of a black hole (see, e.g., [6042]). When such models incorporate accurate multi-dimensional radiative transfer, they will provide the best theoretical spectra to be compared directly to observations (see, e.g., [20]). Moreover, monitoring of the same sources at long wavelengths will improve the measurements of their masses and distances. Finally, combination of this with other methods based on line spectra and the rapid variability properties of accreting black holes will enable us to tighten the uncertainties in the various model parameters and observed quantities that enter Equation (20View Equation) and measure with high precision the spins of galactic black holes.

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