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13.3.1. Self-Absorption

When the apparent source brightness temperature approaches the equivalent kinetic temperature of the relativistic electrons, synchrotron self-absorption becomes important, and part of the radiation is absorbed by the relativistic electrons along the propagation path. Below the self-absorption cutoff frequency, the spectrum is just that of a blackbody with an equivalent temperature Tk = E / k.

From Equation (1.8), E propto nu1/2, so in an opaque synchrotron source the flux density S propto nu2.5, rather than the nu2 law found in thermal sources. In other words, an opaque synchrotron source may be thought of as a body whose equivalent temperature depends on the square root of the frequency. Self-absorption occurs below a frequency nuc where the kinetic temperature

Equation 11a

is equal to the brightness temperature

Equation 11b

Assuming uniform source parameters, the magnetic field can then be estimated from observation of nuc and surface brightness from

Equation 12 (13.12)

where Sm is the maximum flux density in janskys, nuc the cutoff frequency in GHz, and Theta the angular size in milliarcseconds. The quantity delta is a correction for the relativistic Doppler shift if the source is moving with high velocity (see Section 13.3.7). If nu << c, delta ~ 1. The function f (p) only weakly depends on geometry and the value of p, and is about 8 for p ~ 2.5. Variations in opacity throughout the source give an overall spectrum that can be considered as the superposition of many simple regions described by Equation (13.2) and can give rise to the so-called flat or undulating spectra typically observed in compact radio sources (e.g., Condon and Dressel 1973).

The magnetic field in a compact radio source can be determined directly from the observables theta, Sm, and nuc by Equation (13.12). The magnetic energy, EB, can be estimated from Equation (13.8) to be

Equation 13 (13.13)

Similarly, from Equation (13.6), the energy in relativistic electrons is given approximately by

Equation 14 (13.14)

Synchrotron radiation losses lead to a characteristic half-life at a frequency num of

Equation 15 (13.15)

In practice, the use of Equations (13:13) and (13.14) to derive the magnetic and particle energy is difficult due to the strong dependence on angular size and cutoff frequency. However, in almost every case where measurements exist, the particle energy appears to greatly exceed the magnetic energy. But note the dependence on theta and nuc to about the tenth power; so small changes in the geometry may lead to other conclusions. The ratio of Ec / EB may also be reduced considerably with even modest values of delta. See Section 13.3.6.

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