**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

At sufficiently high accretion rates
_{BH}, the gas is optically
thick, and the disk radiates as a thermal blackbody:

Here 2 *r*^{2} 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

Parameterizing the above result in terms of the Eddington accretion rate,
_{E}
*L*_{E} /
*c*^{2} = 2.2
( / 0.1)^{-1}
(*M*_{BH}/10^{8}
)
yr^{-1}, and the
Schwarzschild radius,
*R*_{S}
2*GM*_{BH} /
*c*^{2} = 2.95 x 10^{13}
(*M*_{BH}/10^{8}
) cm, gives

In other words, the peak of the blackbody spectrum occurs at a frequency of
_{max} = 2.8 *kT/h*
4 x 10^{16} 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}
10^{8} - 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}.