Annu. Rev. Astron. Astrophys. 1981. 19: 373-410 Copyright © 1981 by Annual Reviews. All rights reserved

2. SOME BASIC RELATIONS

We assume throughout that the radio emission is due to partially opaque electron synchrotron radiation. For a uniform source of angular size, , a red shift, z, magnetic field, B, and for a power-law distribution of particle energy given by N(E) dE E dE, the frequency, _m, where the flux density reaches a maximum value, S_m, ( ~ 1) is given approximately by ⁽²⁾

(1)

The function f() only weakly depends on , and for = 2, f() ~ 8. At frequencies >> _m, the spectrum is given by the usual power law S , where the spectral index, = (1 - ) / 2. At frequencies << _m, S with = 2.5, a value characteristic of a homogeneous opaque synchrotron source.

Variations in opacity throughout the source lead to an overall spectrum that can be considered as the superposition of many simple regions described by Equation (1) and can give rise to the so-called flat or undulating spectra typically observed over a wide range of frequency.

Synchrotron radiation losses lead to a characteristic electron half-life at _m of

(2)

In very compact sources, in which the radiation energy density is comparable to the magnetic energy density, inverse Compton scattering causes additional electron energy losses, which lead to an upper limit to the observable brightness temperatures of ~ 10¹² K (Kellermann & Pauliny-Toth 1969). In principle, the magnetic field, B, and energy content in the form of relativistic particles, E_p, and magnetic fields, E_m, may be estimated from detailed observation of the brightness distribution at many wavelengths which give the distribution of opacity and brightness temperature. Because the calculated values of B, E_p, and E_m depend on the observables and _m, raised to a power ~ 10, accurate determinations are not feasible with the limited data currently available. Rough estimates indicate B ~ 10^-3±1 gauss, E_p ~ 10^5.2-5.6 ergs, and E_m some 10 orders of magnitude less. Detailed formulae for calculating B, E_p, and E_m from the observables , _m, and S_m including the effect of complex geometries are given by Pacholczyk (1970), Jones et al. (1974), and Moffet (1975).

² Throughout this paper values of flux densities, S, are given in Jy; angular sizes, , in milliarcsec = mas; magnetic fields, B, in gauss, and frequencies, , in GHz. For calculations involving distance, we take H = 50 km s^-1 Mpc^-1 and q₀ = 0. Back.