and Cosmology

5. Active Galactic Nuclei
190
Fig. 5.14. Apparent velocity β app = v app /c of a source component
moving with Lorentz factor γ at an angle φ with
respect to the line-of-sight, for four different values of γ .
Overawiderangeinθ, β app > 1, thus apparent superluminal
motion occurs. The maximum values for β app are obtained if
sin θ = 1/γ
Fig. 5.13. Explanation of superluminal motion: a source component
is moving at velocity v and at an angle φ relative to
the line-of-sight. We consider the emission of photons at two
different times t = 0andt = t e . Photons emitted at t = t e will
reach us by Δt = t e (1 − β cos φ) later than those emitted at
t = 0. The apparent separation of the two source components
then is Δr = vt e sin φ, yielding an apparent velocity on the
sky of v app = Δr/Δt = v sin φ/(1 − β cos φ)
the true velocity of the component. For a given value
of v, the maximum velocity v app is obtained if
(sin φ) max = 1 γ , (5.18)
where the Lorentz factor γ = (1 − β 2 ) −1/2 was already
defined in (5.4). The corresponding value for the
maximum apparent velocity is then
( )
vapp = γv. (5.19)
max
Since γ may become arbitrarily large for values of
v → c, the apparent velocity can be much larger than c,
even if the true velocity v is – as required by Special
Relativity – smaller than c. In Fig. 5.14, v app is plotted
as a function of φ for different values of the Lorentz
factor γ .Togetv app > c for an angle φ, we need
1
β>
sin φ + cos φ ≥ √ 1 ≈ 0.707 .
2
Hence, superluminal motion is a consequence of the
finiteness of the speed of light. Its occurrence implies
that source components in the radio jets of AGNs are
accelerated to velocities close to the speed of light.
In various astrophysical situations we find that the
outflow speeds are of the same order as the escape velocities
from the corresponding sources. Examples are the
Solar wind, stellar winds in general, or the jets of neutron
stars, such as in the famous example of SS433 (in
which the jet velocity is 0.26 c). Therefore, if the outflow
velocity of the jets in AGNs is close c, the jets should
originate in a region where the escape velocity has
a comparable value. The only objects compact enough
to be plausible candidates for this are neutron stars and
black holes. And since the central mass in AGNs is considerably
larger than the maximum mass of a neutron
star, a SMBH is the only option left for the central object.
This argument, in addition, yields the conclusion
that jets in AGNs must be formed and accelerated very
close to the Schwarzschild radius of the SMBH.