and Cosmology

5. Active Galactic Nuclei
194
For accretion to occur at all, we need L < L edd .
Remembering that the Eddington luminosity is proportional
to M • we can turn the above argument around: if
a luminosity L is observed, we conclude L edd > L,or
M • > M edd :=
σ T
L
4πGcm
( p
)
≈ 8 × 10 7 L
10 46 M ⊙ .
erg/s
(5.24)
Therefore, a lower limit for the mass of the SMBH can
be derived from the luminosity. For luminous AGNs,
like QSOs, typical masses are M • 10 8 M ⊙ , while
Seyfert galaxies have lower limits of M • 10 6 M ⊙ .
Hence, the SMBH in our Galaxy could in principle
provide a Seyfert galaxy with the necessary energy.
In the above definition of the Eddington luminosity
we have implicitly assumed that the emission of
radiation is isotropic. In principle, the above argument
of a maximum luminosity can be avoided, and
thus luminosities exceeding the Eddington luminosity
can be obtained, if the emission is highly anisotropic.
A geometrical concept for this would be, for example,
accretion through a disk in the equatorial plane and the
emission of a major part of the radiation along the polar
axes (see Fig. 5.18). Models of this kind have indeed
been constructed. It was shown that the Eddington limit
may be exceeded by this, but not by a large factor. How-
Fig. 5.18. A sketch of the innermost region of an accretion
disk. Because of high temperatures in this region, radiation
pressure can dominate the gas pressure inside the disk; this
leads to an inflation into a thick disk. Radiation from the thick
part of the disk can then hit the thin parts and be partially
reflected. This reflection is a plausible explanation of the X-ray
spectra of AGNs
ever, the possibility of anisotropic emission has another
very important consequence. To derive a value for the
luminosity from the observed flux of a source, the relation
L = 4πDL 2 S is applied, which is explicitly based on
the assumption of isotropic emission. But if this emission
is anisotropic and thus depends on the direction to
the observer, the true luminosity may differ considerably
from that which is derived under the assumption of
isotropic emission. Later we will discuss the evidence
for anisotropic emission in more detail.
Eddington Accretion Rate. If the conversion of infalling
mass into energy takes place with an efficiency ɛ,
the accretion rate can be determined,
ṁ =
L
ɛ c 2 ≈ 0.18 1 ɛ
(
L
10 46 erg/s
)(
M⊙
)
. (5.25)
1yr
Since the maximum efficiency is of order ɛ ∼ 0.1, this
implies accretion rates of typically several Solar masses
per year for very luminous QSOs. If L is measured in
units of the Eddington luminosity, we obtain with (5.23)
ṁ =
L ( 1.3 × 10 38 erg/s
L edd ɛc 2
) ( )
M •
≡
M ⊙
L ṁ edd ,
L edd
(5.26)
where in the last step the Eddington accretion rate has
been defined,
ṁ edd = L edd
ɛc 2 ≈ 1 ɛ 2 × 10−9 M • yr −1 . (5.27)
Growth Rate of the SMBH Mass. The Eddington accretion
rate is the maximum accretion rate if isotropic
emission is assumed, and it depends on the assumed efficiency
ɛ. We can now estimate a characteristic time in
which the mass of the SMBH will significantly increase,
t evo := M ( )
• L −1
ṁ ≈ ɛ 5 × 10 8 yr , (5.28)
L edd
i.e., even with efficient energy production (ɛ ∼ 0.1), the
mass of a SMBH can increase greatly on cosmologically
short time-scales by accretion. However, this is
not the only mechanism which can produce SMBHs of
large mass. They can also be formed through the merger
of two black holes, each of smaller mass, as would be
expected after the merger of two galaxies if both partners
hosted a SMBH in its center. This aspect will be
discussed more extensively later.