and Cosmology

5. Active Galactic Nuclei
180
of relativistic electrons. Electrons in a magnetic field
propagate along a helical, i.e., corkscrew-shaped path,
so that they are continually accelerated by the Lorentz
force. Since accelerated charges emit electromagnetic
radiation, this motion of the electrons leads to the emission
of synchrotron radiation. Because of its importance
for our understanding of the radio emission of AGNs, we
will review some aspects of synchrotron radiation next.
The radiation can be characterized as follows. If
an electron has energy E = γ m e c 2 , the characteristic
frequency of the emission is
ν c = 3γ 2 eB
4πm e c ∼ 4.2 × 106 γ 2 ( B
1G
)
Hz , (5.3)
where B denotes the magnetic field strength, e the
electron charge, and m e = 511 keV/c 2 the mass of the
electron. The Lorentz factor γ , and thus the energy of
an electron, is related to its velocity v via
1
γ := √ . (5.4)
1 − (v/c)
2
For frequencies considerably lower than ν c , the spectrum
of a single electron is ∝ ν 1/3 , whereas it decreases
exponentially for larger frequencies. To a first approximation,
the spectrum of a single electron can
be considered as quasi-monochromatic, i.e., the width
of the spectral distribution is small compared to the
characteristic emission frequency ν c . The synchrotron
radiation of a single electron is linearly polarized, where
the polarization direction depends on the direction of
the magnetic field projected onto the sky. The degree
of polarization of the radiation from an ensemble of
electrons depends on the complexity of the magnetic
field. If the magnetic field is homogeneous in the spatial
region from which the radiation is measured, the
observed polarization may reach values of up to 75%.
However, if the spatial region that lies within the telescope
beam contains a complex magnetic field, with
the direction changing strongly within this region, the
polarizations partially cancel each other out and the
observed degree of linear polarization is significantly
reduced.
To produce radiation at cm wavelengths (ν ∼
10 GHz) in a magnetic field of strength B ∼ 10 −4 G, γ ∼
10 5 is required, i.e., the electrons need to be highly relativistic!
To obtain particles at such high energies, very
efficient processes of particle acceleration must occur
in the inner regions of quasars. It should be mentioned
in this context that comic ray particles of considerably
higher energies are observed (see Sect. 2.3.4). The majority
of cosmic rays are presumably produced in the
shock fronts of supernova remnants. Thus, it is supposed
that the energetic electrons in quasars (and other AGNs)
are also produced by “diffusive shock acceleration”,
where here the shock fronts are not caused by supernova
explosions but rather by other hydrodynamical phenomena.
As we will see later, we find clear indications
in AGNs for outflow velocities that are considerably
higher than the speed of sound in the plasma, so that the
conditions for the formation of shock fronts are satisfied.
Synchrotron radiation will follow a power law if the
energy distribution of relativistic electrons also behaves
like a power law (see Fig. 5.7). If N(E) dE ∝ E −s dE
represents the number density of electrons with energies
between E and E + dE, the power-law index of the resulting
radiation will be α = (s − 1)/2, i.e., the slope in
the power law of the electrons defines the spectral shape
of the resulting synchrotron emission. In particular, an
index of α = 0.7 results for s = 2.4. An electron distribution
with N(E) ∝ E −2.4 is very similar to the energy
distribution of the cosmic rays in our Galaxy, which
may be another indicator for the same or at least a similar
mechanism being responsible for the generation of
this energy spectrum.
Fig. 5.7. Electrons at a given energy emit a synchrotron spectrum
which is indicated by the individual curves; the maximum
of the radiation is at ν c (5.3), which depends on the electron energy.
The superposition of many such spectra, corresponding
to an energy distribution of the electrons, results in a powerlaw
spectrum provided the energy distribution of the electrons
follows a power law