Cygnus X-1 is a binary star system that is comprised of a black hole and a massive giant companion star in a tight orbit. Building on our accurate distance measurement reported in the preceding paper, we first determine the radius of the companion star, thereby constraining the scale of the binary system. To obtain a full dynamical model of the binary, we use an extensive collection of optical photometric and spectroscopic data taken from the literature. By using all of the

A typical galaxy is thought to contain tens of millions of stellar-mass black holes, the collapsed remnants of once massive stars, and a single nuclear supermassive black hole. Both classes of black holes accrete gas from their environments. The accreting gas forms a flattened orbiting structure known as an accretion disk. During the past several years, it has become possible to obtain measurements of the spins of the two classes of black holes by modeling the X-ray emission from their accretion disks. Two methods are employed, both of which depend upon identifying the inner radius of the accretion disk with the innermost stable circular orbit (ISCO), whose radius depends only on the mass and spin of the black hole. In the Fe Kα method, which applies to both classes of black holes, one models the profile of the relativistically-broadened iron line with a special focus on the gravitationally redshifted red wing of the line. In the continuumfitting method, which has so far only been applied to stellar-mass black holes, one models the thermal X-ray continuum spectrum of the accretion disk. We discuss both methods, with a strong emphasis on the continuum-fitting method and its application to stellar-mass black holes. Spin results for eight stellar-mass black holes are summarized. These data are used to argue that the high spins of at least some of these black holes are natal, and that the presence or absence of relativistic jets in accreting black holes is not entirely determined by the spin of the black hole.

The compact primary in the X-ray binary Cygnus X-1 was the first black hole to be established via dynamical observations. We have recently determined accurate values for its mass and distance, and for the orbital inclination angle of the binary. Building on these results, which are based on our favored (asynchronous) dynamical model, we have measured the radius of the inner edge of the black hole's accretion disk by fitting its thermal continuum spectrum to a fully relativistic model of a thin accretion disk. Assuming that the spin axis of the black hole is aligned with the orbital angular momentum vector, we have determined that Cygnus X-1 contains a near-extreme Kerr black hole with a spin parameter a * > 0.95 (3σ). For a less probable (synchronous) dynamical model, we find a * > 0.92 (3σ). In our analysis, we include the uncertainties in black hole mass, orbital inclination angle and distance, and we also include the uncertainty in the calibration of the absolute flux via the Crab. These four sources of uncertainty totally dominate the error budget. The uncertainties introduced by the thin-disk model we employ are particularly small in this case given the extreme spin of the black hole and the disk's low luminosity.

The black-hole binary system LMC X-3 has been observed by virtually every X-ray mission since the inception of X-ray astronomy. Among the persistent sources, LMC X-3 is uniquely both habitually soft and highly variable. Using a fully relativistic accretion-disk model, we analyze hundreds of spectra collected during eight X-ray missions that span 26 years. For a selected sample of 391 RXTE spectra we find that to within ≈ 2 percent the inner radius of the accretion disk is constant over time and unaffected by source variability. Even considering an ensemble of eight X-ray missions, we find consistent values of the radius to within ≈ 4 − 6 percent. Our results provide strong evidence for the existence of a fixed inner-disk radius. The only reasonable inference is that this radius is closely associated with the general relativistic innermost stable circular orbit (ISCO). Our findings establish a firm foundation for the measurement of black hole spin.

We report a direct and accurate measurement of the distance to the X-ray binary Cygnus X-1, which contains the first black hole to be discovered. The distance of 1.86 +0.12 −0.11 kpc was obtained from a trigonometric parallax measurement using the Very Long Baseline Array. The position measurements are also sensitive to the 5.6 d binary orbit and we determine the orbit to be clockwise on the sky. We also measured the proper motion of Cygnus X-1 which, when coupled to the distance and Doppler shift, gives the three-dimensional space motion of the system. When corrected for differential Galactic rotation, the non-circular (peculiar) motion of the binary is only about 21 km s −1 , indicating that the binary did not experience a large "kick" at formation.

We analyze photometry of the Soft X-ray Transient A0620-00 spanning nearly 30 years, including previously published and previously unpublished data. Previous attempts to determine the inclination of A0620 using subsets of these data have yielded a wide range of measured values of i. Differences in the measured value of i have been due to changes in the shape of the light curve and uncertainty regarding the contamination from the disk. We give a new technique for estimating the disk fraction and find that disk light is significant in all light curves, even in the infrared. We also find that all changes in the shape and normalization of the light curve originate in a variable disk component. After accounting for this disk component, we find that all the data, including light curves of significantly different shapes, point to a consistent value of i. Combining results from many separate data sets, we find i = 51. • 0 ± 0.9, implying M = 6.6 ±0.25M ⊙ . Using our dynamical model and zero-disk stellar VIH magnitudes, we find d = 1.06±0.12kpc. Understanding the disk origin of non-ellipsoidal variability may assist with making reliable determinations of i in other systems, and the fluctuations in disk light may provide a new observational tool for understanding the three-dimensional structure of the accretion disk.

We present a dynamical model of the high mass X-ray binary LMC X-1 based on high-resolution optical spectroscopy and extensive optical and nearinfrared photometry. From our new optical data we find an orbital period of P = 3.90917±0.00005 days. We present a refined analysis of the All Sky Monitor data from RXTE and find an X-ray period of P = 3.9094 ± 0.0008 days, which is consistent with the optical period. A simple model of Thomson scattering in the stellar wind can account for the modulation seen in the X-ray light curves. The V − K color of the star (1.17 ± 0.05) implies A V = 2.28 ± 0.06, which is much larger than previously assumed. For the secondary star, we measure a radius of R 2 = 17.0±0.8 R ⊙ and a projected rotational velocity of V rot sin i = 129.9±2.2 km s −1 . Using these measured properties to constrain the dynamical model, we find an inclination of i = 36.38±1.92 • , a secondary star mass of M 2 = 31.79±3.48 M ⊙ , and a black hole mass of 10.91 ± 1.41 M ⊙ . The present location of the secondary star in a temperature-luminosity diagram is consistent with that of a star with an initial mass of 35 M ⊙ that is 5 Myr past the zero-age main sequence. The star nearly fills its Roche lobe (≈ 90% or more), and owing to the rapid change in radius with time in its present evolutionary state, it will encounter its Roche lobe and begin rapid and possibly unstable mass transfer on a timescale of a few hundred thousand years.

The evolution of massive stars is influenced by the mass lost to stellar winds over their lifetimes. These winds limit the masses of the stellar remnants (such as black holes) that the stars ultimately produce. We use radio astrometry to refine the distance to the black hole X-ray binary Cygnus X-1, which we find to be 2.22−0.17+0.18 kiloparsecs. When combined with archival optical data, this implies a black hole mass of 21.2 ± 2.2 solar masses, higher than previous measurements. The formation of such a high-mass black hole in a high-metallicity system (within the Milky Way) constrains wind mass loss from massive stars.