Investigations of Faraday Rotation Maps of Extended Radio Sources ...

8 CHAPTER 1. INTRODUCTION
This synchrotron emission spectrum can then be related to the energy density of
relativistic electrons ɛ re = ∫ dE n e (E)E and the magnetic field B ⊥ (Burbidge 1956;
Pacholczyk 1970; Leahy 1991). Suppose that the total flux density S ν emitted by the
source in a frequency range ν 1 < ν < ν 2 and the angular size of the source Θ are
measured, one can obtain the relativistic energy density ɛ re
ɛ re =
∫ ν2
ν 1
dE n e (E)E = aB −3/2 Θ 2 S ν (ν 0 ), (1.21)
where ν 0 is a characteristic frequency between ν 1 and ν 2 and the proportionality factor
a is a known function which depends on ν 1 , ν 2 and α (see e.g. chap. 7, Pacholczyk
1970).
If the relativistic particle energy can be determined by an additional method, the
estimation of the magnetic field strength follows from Eq. (1.21). Often this necessary
information is not directly available. Therefore, an additional assumption is often used
by considering the total energy budget
ɛ tot = ɛ re + ɛ rp + ɛ B , (1.22)
where ɛ B = B 2 /8π is the magnetic energy density, ɛ re and ɛ rp are the relativistic
energy density of the electrons and of the heavy particles, respectively. Assuming
that the energy density of heavy particles is proportional to the energy density of the
relativistic electrons ɛ rp = kɛ re , where k ∼ 100 for relativistic particles in our Galaxy
(Beck et al. 1996), Eq. (1.22) reduces to
ɛ tot = (1 + k)ɛ re + ɛ B . (1.23)
In order to estimate the magnetic field strength, one can either assume equipartition
between the energies (1 + k)ɛ re = ɛ B or minimise ɛ tot with respect to B. The latter
results in
∂ɛ tot
∂B
= 0 = −(1 + k)
3c
2 Θ2 S ν (ν 0 ) B −5/2 + 1
4π
B, (1.24)
where Eq. (1.21) was used, leading to a magnetic field estimate of the form B 7/2 ∝
Θ 2 S ν (ν 0 ). However, the equipartition assumption leads to a similar result which deviates
only by a factor of 8/6. As an example using equipartition arguments, Niklas
(1995) determined a mean magnetic field strength of 9 ± 3 µG from a sample of 74
spiral galaxies, however larger field strength have been locally detected in Galaxies
(Beck et al. 1996).
The assumption of equipartition or minimum energy is continously under debate.
It is not clear if this assumption is valid over all length and time scales. On large length
scales or long time scales equilibrium might have developed whereas on small length
or short time scales this may not be true. However, in our Galaxy the assumption of
equipartition can be tested, since independent measurement of the relativistic electron
energy density and of the relativistic proton energy density exist. A combination of
these information yields an excellent agreement with the estimates using the equipartition
assumption (Beck 2001).
Another possibility to use synchrotron emission in order to estimate the magnetic
field strength for extragalactic sources is the comparison between inverse Compton
(IC) X-ray emission and radio synchrotron emission (Harris & Grindlay 1979;