Synchrotron Emission Mechanism

Synchrotron radiation is emitted whenever an electron has a relativistic velocity in a region with a magnetic field component perpendicular to it's velocity. The acceleration will be perpendicular to the directions of both the velocity of the electron and the magnetic field component. As the acceleration is perpendicular to the instantaneous velocity, the magnitude of the velocity does not change and in general the electron moves in a helical path with axis parallel to the direction of the magnetic field. (See figure 5.1).

Figure 5.1: Emission of Synchrotron Radiation.

Due to the the relativistic velocity, the gyration frequency of the election is reduced due to its increased effective mass. Mathematically, if the electron were non-relativistic, its cyclotron frequency would be given as,

(5..1)

where $B$ is the magnetic field, $m$ is the mass of the electron, $e$ is the charge of the electron and $\theta$ is the angle between the electron velocity vector and the magnetic field vector.

Due to relativistic effects, the effective mass of the electron will increase and the reduced gyration frequency will be given as,

(5..2)

If the electron is highly relativistic, the forward lobe of its dipole radiation compresses into a cone with axis along the velocity vector of the electron as it moves in the magnetic field. The half-angle of this cone will be given by

$$\rm {\xi = \sqrt{(1 - v^2/c^2)} = mc^2/E}$$ (5..3)

The observer detects synchrotron radiation pulses whenever this cone of radiation sweeps across him.

The synchrotron emission from most of the radio sources is observed to vary with frequency according to a simple power law. This can be explained theoretically by the electron energy distribution also follows a power law with an appropriate index. If the number of electrons with energy between between an energy and $E + dE$ be given by,

(5..4)

Then it can be shown that the the synchrotron flux density $S$ at a particular frequency is given by,

$$\rm {S_\nu \propto \nu^{-(\gamma-1)/2} = \nu^{-\alpha}}$$ (5..5)

Where $\alpha$ is called the radio spectral index.

Typical radio sources follow a power law in most of their spectrum. They generally show departures in the power law at either of the ends the observable frequency range.

The ways in which this power law spectrum is modified involve processes that affect the electron energy distribution as well as those that directly affect the radio spectrum. Some such process are briefly discussed below. (Ref : Alan T. Moffet, "Strong non-thermal radio emission from galaxies").