and Cosmology

5.3 The Central Engine: A Black Hole
than the gravitational force. Hence the disk will locally
rotate with approximately the Kepler velocity. Since
a Kepler disk rotates differentially, in the sense as the
angular velocity depends on radius, the gas in the disk
will be heated by internal friction. In addition, the same
friction causes a slight deceleration of the rotational velocity,
whereby the gas will slowly move inwards. The
energy source for heating the gas in the disk is provided
by this inward motion – namely the conversion of potential
energy into kinetic energy, which is then converted
into internal energy (heat) by friction.
According to the virial theorem, half of the potential
energy released is converted into kinetic energy; in the
situation considered here, this is the rotational energy
of the disk. The other half of the potential energy can
be converted into internal energy. We now present an
approximately quantitative description of this process,
specifically for accretion onto a black hole.
Temperature Profile of a Geometrically Thin, Optically
Thick Accretion Disk. When a mass m falls from
radius r + Δr to r, the energy
ΔE = GM •m
− GM •m
r r + Δr ≈ GM •m Δr
r r
is released. Here M • denotes the mass of the SMBH,
assumed to dominate the gravitational potential, so that
self-gravity of the disk can be neglected. Half of this energy
is converted into heat, E heat = ΔE/2. If we assume
that this energy is emitted locally, the corresponding
luminosity is
ΔL = GM •ṁ
Δr , (5.11)
2r 2
where ṁ denotes the accretion rate, which is the mass
that falls into the black hole per unit time. In the
stationary case, ṁ is independent of radius since otherwise
matter would accumulate at some radii. Hence the
same amount of matter per unit time flows through any
cylindrical radius.
If the disk is optically thick, the local emission corresponds
to that of a black body. The ring between r and
r + Δr then emits a luminosity
ΔL = 2 × 2πr Δr σ SB T 4 (r), (5.12)
where the factor 2 originates from the fact that the disk
has two sides. Combining (5.11) and (5.12) yields the
radial dependence of the disk temperature,
( ) GM• ṁ 1/4
T(r) =
8πσ SB r 3 .
A more accurate derivation explicitly considers the dissipation
by friction and accounts for the fact that part
of the generated energy is used for heating the gas,
where the corresponding thermal energy is also partially
advected inwards. Except for a numerical correction
factor, the same result is obtained,
( ) 3GM• ṁ 1/4
T(r) =
8πσ SB r 3 , (5.13)
which is valid in the range r ≫ r S . Scaling r with the
Schwarzschild radius r S , we obtain
( ) 3GM• ṁ 1/4 ( ) r −3/4
T(r) =
8πσ SB rS
3 .
r S
By replacing r S with (3.31) in the first factor, this can
be written as
(
T(r) =
3c 6 ) 1/4
64πσ SB G 2 ṁ 1/4 M•
−1/2
( r
r S
) −3/4
.
(5.14)
From this analysis, we can immediately draw a number
of conclusions. The most surprising one may be
the independence of the temperature profile of the disk
from the detailed mechanism of the dissipation because
the equations do not explicitly contain the viscosity.
This fact allows us to obtain quantitative predictions
based on the model of a geometrically thin, optically
thick accretion disk. 4 The temperature in the disk increases
inwards ∝ r −3/4 , as expected. Therefore, the
total emission of the disk is, to a first approximation,
a superposition of black bodies consisting of rings with
4 The physical mechanism that is responsible for the viscosity is unknown.
The molecular viscosity is far too small to be considered as the
primary process. Rather, the viscosity is probably produced by turbulent
flows in the disk or by magnetic fields, which become spun up by
differential rotation and thus amplified, so that these fields may act as
an effective friction. In addition, hydrodynamic instabilities may act
as a source of viscosity. Although the properties of the accretion disk
presented here – luminosity and temperature profile – are independent
of the specific mechanism of the viscosity, other disk properties definitely
depend on it. For example, the temporal behavior of a disk in
the presence of a perturbation, which is responsible for the variability
in some binary systems, depends on the magnitude of the viscosity,
which therefore can be estimated from observations of such systems.
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