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3.1 Fitting the Spectra of Accretion Disks

As material falls toward a black hole, it is believed to settle into an accretion disk in which angular momentum is dissipated by viscosity. From the virial theorem, half of the gravitational potential energy U is radiated. Therefore the luminosity is

Equation 1 (1)

At sufficiently high accretion rates MdotBH, the gas is optically thick, and the disk radiates as a thermal blackbody:

Equation 2 (2)

Here 2 pi r2 is the surface area of the disk and sigma is the Stefan-Boltzmann constant. The effective temperature of the disk as a function of radius r is therefore

Equation 3 (3)

Parameterizing the above result in terms of the Eddington accretion rate, MdotE ident LE / epsilon c2 = 2.2 (epsilon / 0.1)-1 (MBH/108 sun) sun yr-1, and the Schwarzschild radius, RS ident 2GMBH / c2 = 2.95 x 1013 (MBH/108 sun) cm, gives

Equation 4 (4)

In other words, the peak of the blackbody spectrum occurs at a frequency of numax = 2.8 kT/h appeq 4 x 1016 Hz, where k is Boltzmann's constant and h is Planck's constant. This peak is near 100 Å or 0.1 keV. In fact, the spectra of many AGNs show a broad emission excess at extreme ultraviolet or soft X-ray wavelengths. This ``big blue bump'' has often been identified with the thermal emission from the accretion disk. A fit to the luminosity and the central frequency of the big blue bump gives MBH and MdotBH but not each separately. Corrections for disk inclination and relativistic effects further complicate the analysis. This method is therefore model-dependent and provides only approximate masses. Typical values for quasars are MBH appeq 108 - 109.5 sun and MdotBH appeq 0.1 - 1 MdotE. Seyfert nuclei appear to have lower masses, MBH appeq 107.5 - 108.5 sun, and lower accretion rates, MdotBH appeq 0.01 - 0.5 MdotE.

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