Among the several spectral states of accreting black holes, the thermal state (§4.1) is of particular interest. A feature of this state is that the X-ray spectrum is dominated by a soft blackbody-like component that is emitted by (relatively) cool optically-thick gas, presumed to be located in the accretion disk. A minor nonthermal (power-law) component in the spectrum, possibly from a hot optically thin corona, is energetically unimportant. The thermal state is believed to match very closely the classic thin accretion disk model of Shakura & Sunyaev (1973) and Novikov & Thorne (1973). This theoretical model has been widely studied for many decades and its physics is well understood.

The idealized thin disk model describes an axisymmetric radiatively efficient accretion flow in which, for a given black hole mass M, mass accretion rate M and black hole spin parameter a* (§1), we can calculate precisely the total luminosity of the disk: ¿disk = nMc2, where n is a function only of a*. We can also calculate precisely the local radiative flux Fdisk(R) emitted at radius R by each surface of the disk. Moreover, the accreting gas is optically thick, and the emission is thermal and blackbody-like, making it straightforward to compute the spectrum of the emission. Most importantly, the inner edge of the disk is located at the innermost stable circular orbit (ISCO) of the black hole space-time, whose radius Risco (in gravitational units) is a function only of the spin of the black hole: RISCO/(GM/c2) = £(a*), where £ is a monotonically decreasing function of a* (see Fig. 4). Thus, if we measure the radius of the disk inner edge, and if we know the mass M of the black hole, we can immediately obtain a*. This is the principle behind the continuum-fitting method of estimating black hole spin, which was first described by Zhang et al. (1997; see also Gierlinski et al. 2001).

Before discussing how to measure RISCO of a disk, we remind the reader how one measures the radius R* of a star. Given the distance D to the star, the radiation flux Fobs received from the star, and the temperature T of the continuum radiation, the luminosity of the star is given by

where a is the Stefan-Boltzmann constant. Thus, from Fobs and T we immediately obtain the ratio R2/D2, the solid angle subtended by the star. Then, if we know the distance to the star, we obtain the stellar radius R*. For accurate results we must allow for limb-darkening and other non-blackbody effects in the stellar emission by computing a stellar atmosphere model, a minor detail.

The same principle applies to an accretion disk, but with some differences. First, since Fdisk(R) varies with radius, the radiation temperature T also varies with R. But the precise variation is known, so it is easily incorporated into the model. Second, since the bulk of the emission is from the inner regions of the disk, the effective area of the radiating surface is directly proportional to the square of the disk inner radius, Aeff = CRfSCO, where the constant C is known. Third, the observed flux Fobs depends not only on the luminosity and the distance, but also on the inclination i of the disk to the line-of-sight. Allowing for these differences, one can write a relation for the disk problem similar in spirit to eq. (1). Therefore, in analogy with the stellar case, given Fobs and a characteristic T (from X-ray observations), one obtains the solid angle subtended by the ISCO: cos i R2SCO/D2. If we know i and D, we obtain RisCO, and if we also know M, we obtain a* [via £(a*), Fig. 1]. This is the basic idea of the method.

Note that RISCO varies by a factor of 6 between a* =0 and a* = 1. This means that the solid angle subtended by the ISCO varies by a whopping factor of 36 (or even 81 if we include negative values of a* to allow for counter-rotating disks, see Fig. 1). The method is thus potentially very sensitive. Also, all one asks the X-ray spectral data to provide are the characteristic temperature T of the radiation and the X-ray flux Fobs (or equivalently the normalization of the spectrum). These quantities can be obtained robustly from appropriately selected data. In contrast, other proposed approaches to estimate black hole spin require very intricate modeling of X-ray spectra (e.g., Brenneman & Reynolds 2006).

Zhang et al. (1997) first argued that the relativistic jets and extraordinary X-ray behavior of GRS 1915 are due to the high spin of its black hole primary. In their approximate analysis, they found that both GRS 1915 and GRO J1655-40 had high spins, a* > 0.9. Subsequently, Gierlinski et al. (2001) estimated the spin of GRO J1655-40 and LMC X-3. Recently, our group has firmly established the methodology pioneered by Zhang et al. and Gierlinski et al. by constructing relativistic accretion disk models (Li et al. 2005; Davis et al. 2005) and by modeling in detail the effects of spectral hardening (Davis et al. 2005, 2006). We have made these analysis tools publicly available via XSPEC (kerrbb and bhspec; Arnaud 1996). Using this modern methodology, spins have been estimated for three stellar-mass black holes: GRO J1655-40 (a* = 0.65 - 0.75) and 4U 1543-47 (a* = 0.75 - 0.85; Shafee et al. 2006, hereafter S06), and LMC X-3 (a* < 0.26; Davis et al. 2006).

Most recently, our group estimated the spin of the extraordinary microquasar GRS 1915 +105 and conclude that its compact primary is a rapidly-rotating black hole. We find a lower limit on the dimensionless spin parameter of a* > 0.98. Our result is robust in the sense that it is independent of the details of the data analysis and insensitive to the uncertainties in the mass and distance of the black hole.

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