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

(1)

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

(2)

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

(3)

Parameterizing the above result in terms of the Eddington accretion rate, _E L_E / c² = 2.2 ( / 0.1)^-1 (M_BH/10⁸ ) yr^-1, and the Schwarzschild radius, R_S 2GM_BH / c² = 2.95 x 10¹³ (M_BH/10⁸ ) cm, gives

(4)

In other words, the peak of the blackbody spectrum occurs at a frequency of _max = 2.8 kT/h 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 M_BH and _BH 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 M_BH 10⁸ - 10^9.5 and _BH 0.1 - 1 _E. Seyfert nuclei appear to have lower masses, M_BH 10^7.5 - 10^8.5 , and lower accretion rates, _BH 0.01 - 0.5 _E.