12.5.3. Interpretation of Spectral Data
The main features of the radio spectra which must be explained by any theory of the generation of the relativistic particles are:
the relatively sharp concentration of the energy index near ~ 2.6
the extreme values of the index for the Class S spectra of ~ - 0.5 and ~ - 2.0, and the inclusion of nearly all sources in the range -0.5 > > - 1.3
the absence of any sharp high-frequency cut-off in the radio spectra expected from synchrotron radiation losses
the detailed form of the spectra of the transparent sources
the observed index of about -1/4 for the transparent part of the spectrum of the variable sources.
It is tempting to interpret the change in spectral index of -0.5 observed in the Class C- spectra as the effect of synchrotron radiation losses balanced by continuous injection. However, in this case we might expect that the index most commonly observed for the Class S spectra would correspond either to
the low-frequency asymptotic value of the C- spectra if the S spectra refer to young Radio Galaxies sources where b is greater than the maximum observed frequency
the high-frequency asymptotic value of the C- spectra if the S spectra refer to old sources where b is less than the minimum observed frequency.
In fact the median value of for the S spectra of about -0.8 is intermediate between the extremes of the C- spectra, which are about -0.65 and -1.15, so the interpretation is not clear.
It is also possible that the range in spectral index observed in the C- spectra is due to partial self-absorption (see Section 12.5.5) in dense regions of the source which become opaque at relatively short wavelengths. This interpretation is suggested by the fact that a larger fraction of sources with C - spectra contains compact components (as determined, for example, from observations of interplanetary scintillations) than is found for sources with S spectra (Kellermann, Pauliny-Toth, and Williams, 1969).
Finally, it must be remarked that the most recent absolute calibrations indicate that at wavelengths of about one meter and longer most published flux densities are systematically low, so that the true spectral curvature may in fact be much less than has previously been thought.
The value of approximately -1/4 observed for the spectral index of variable sources in the transparent part of the spectrum suggests that this is the initial injection spectrum (so that 0 ~ 1.5). The steeper spectra observed for the more extended sources are then somehow the result of synchrotron radiation losses.
The high probability of finding an observed index steeper than 0 by about 0.5 suggests the continuous injection model. But the limiting value of -1.3 suggests instantaneous injection with = 4/3 [(- 1/4) - 1] = - 1.33. Both of these can be accounted for by assuming that there is a repeated generation of particles with a characteristic period T. Then from Equation (12.15) the spectral index after an elapsed time t >> T have three distinct spectral regions defined by
In case (1) the effect of synchrotron radiation losses is not important, and the index remains equal to its initial value. In case (2) the time scale for radiation loss is longer than the period T between bursts, the injection can be considered quasi-continuous, and the spectrum is in equilibrium since radiation losses are balanced by the injection of new particles. In case (3) synchrotron losses dominate, and the spectrum steepens. The observed curvature in the spectrum of many sources is also consistent with this model.
The problem with the recurring injection model is that the high-frequency part of the spectrum, = (4/3 0 - 1), can be understood only in the case where the electron pitch angles are conserved for times greater than the radiation lifetime. For B = 10-4 and = 1 GHz, this is 106 years. Otherwise, the spectrum must show a sharp cut-off at high frequencies, unless, in fact, the relativistic electrons lose energy not by synchrotron radiation, but by some other mechanism which preserves the power law form of the energy distribution (dE / dt E). The absence of a clear spectral cut-off may also be due to the superposition of a range of cut-offs, such as would occur, for example, if the radio emission came from a number of discrete regions having very different magnetic field strengths.
The observed spectral data are also consistent with a source where multiple injection and radiation losses are important during the initial stages, but followed by a longer period of adiabatic expansion when the spectral shape is preserved. But this interpretation poses serious energy problems, as discussed in Section 12.5.7.
It will be important to determine from future observations at shorter wavelengths at what wavelength the expected spectral cut-off occurs (if anywhere) and so help to specify the mechanism or mechanisms by which the electrons lose energy, and thus place further observational constraints on the radio source lifetimes.