3. THEORETICAL BACKGROUND RELATED TO CLUSTER MAGNETIC FIELDS

3.1. Synchrotron radiation

The synchrotron emission is produced by the spiraling motion of relativistic electrons in a magnetic field. It is therefore the easiest and more direct way to detect magnetic fields in astrophysical sources. The total synchrotron emission from a source provides an estimate of the strength of magnetic fields while the degree of polarization is an important indicator of the field uniformity and structure.

An electron of energy = m_e c² (where is the Lorentz factor) in a magnetic field experiences a × force that causes it to follow a helical path along the field lines, emitting radiation into a cone of half-angle ^-1 about its instantaneous velocity. To the observer, the radiation is essentially a continuum with a fairly peaked spectrum concentrated near the critical frequency

(1)

The synchrotron power emitted by a relativistic electron is

(2)

where is the pitch angle between the electron velocity and the magnetic field direction while c₁ and c₂ depend only on fundamental physical constants

(3)

In practical units:

(4) (5)

From Eq. 1, it is derived that electrons of 10⁴ in magnetic fields of B 1 G produce synchrotron radiation in the optical domain, whereas electrons of 10⁵ in magnetic fields of B 10 G radiate in the X-rays (see Fig. 1). Therefore at a given frequency, the energy (or Lorentz factor) of the emitting electrons depends directly on the magnetic field strengths. The higher is the magnetic field strength, the lower is the electron energy needed to produce emission at a given frequency. In a magnetic field of about B 1 µG, a synchrotron radiation detected for example at 100 MHz, is produced by relativistic electrons with 5000.

Figure 1. Electron Lorentz factor = / me c2 (left-hand axis) and energy (right-hand axis) versus synchrotron critical frequency for magnetic field strengths in the range 10-6 - 100 G ( = 90°).

For an homogeneous and isotropic population of electrons with a power-law energy distribution, i.e. with the particle density between and + d given by

(6)

the total intensity spectrum, in regions which are optically thin to their own radiation, varies as:

(7)

where the spectral index = ( - 1) / 2. Below the frequency where the synchrotron emitting region becomes optically thick, the total intensity spectrum can be described by:

(8)

The synchrotron emission radiating from a population of relativistic electrons in a uniform magnetic field is linearly polarized. In the optically thin case, the degree of intrinsic linear polarization, for a homogeneous and isotropic distribution of relativistic electrons with a power-law spectrum as in Eq. 6, is:

(9)

with the electric (polarization) vector perpendicular to the projection of the magnetic field onto the plane of the sky. For typical values of the particle spectral index, the intrinsic polarization degree is ~ 75 - 80%. In the optically thick case:

(10)

and the electric vector is parallel to the projected magnetic field.

In practice, the polarization degree detected in radio sources is much lower than expected by the above equations. A reduction in polarization could be due to a complex magnetic field structure whose orientation varies either with depth in the source or over the angular size of the beam. For instance, if one describes the magnetic field inside an optically thin source as the superposition of two components, one uniform B_u, the other isotropic and random B_r, the observed degree of polarization can be approximated by [7]:

(11)

A rigorous treatment of how the degree of polarization is affected by the magnetic field configuration is presented by Sokoloff et al. [8, 9].