N(E,,t) = N_0 E^- [1 - Et (^2+B^2 sin^2)]^(-2) (1)
where is the pitch angle, a constant and = 3.18(1+z)^2 is the equivalent magnetic field strength of the microwave background. In the JP model the pitch angles are continually randomized and the above expression is changed by the replacement sin^2 -> <sin^2> = 2/3. The exponent is related to the radio spectral index n by = 2n+1. The depletion of high energy electrons is seen as a fall in the high frequency radio emission below the extrapolation of the low frequency emission.
Electron energy losses are highest in the strong field regions, so the spectrum of the radio emission from those regions steepens most rapidly. Thus at high frequencies the emission is biased towards regions of low field strength. There is a difference between the KP and JP models, as in the former the emission from regions where the magnetic field is perpendicular to the line of sight fades faster.
The total emission is given by
I(,t) = C N(E,t) F(x) dE, (2)
where x = /, with
F(x) = x_x^ K_5/3(z) dz, (3)
_T = c_1 B ^2 sin , (4)
where the energy is defined by
= 1/[ t (^2+B^2 sin ^2)]. (5)
The initial radio spectrum is a power law I~^-n. The total aged spectrum can be characterized by a function A(,x), which simply multiplies the initial spectrum,
I(,t) = A(,x) I(,0). (6)
Here A is a fixed function, so that the shape of the spectrum is the same at all times everywhere, and only the single parameter x is a function of position, while is a constant for any source (assuming no reaccelaration) but can vary between sources.
To get the total spectrum for a source, one must then sum over the entire volume, or equivalently sum over all angles and field strengths. The initial intensity I(,0) (B sin )^n+1 determines the relative contributions to the spectrum from different points.
I assume that the magnetic field can be described by a Gaussian random field. Then the angles are distributed isotropically and the field is drawn from a Maxwellian distribution. In Fig. 1 I show the resulting spectra for both KP and JP models compared with the spectra that would result if the field strength were uniform. Especially in the JP case, the spectra are flatter at high frequencies than in the constant field-strength case. The spectrum initially falls below the constant field-strength case but remains fairly flat. The initial fall is due to increased losses from the regions of highest field strength; the remaining emission then decays more slowly.
Fig 1. Aged spectra as a function of frequency for the KP model (top) and JP model (bottom). The aged spectrum assuming a random field strength is shown as a solid line, and that from a constant field shown as a dotted line for comparison.
The reason for the smoothing out of the sharp spectral decline is quite simple. The resultant spectrum is the superposition of many spectra all with different break frequencies . In the standard KP model, the break frequency depends on pitch angle. In the models in this paper, the break frequency depends in addition on the local field strength. The resultant smearing out of the sharp spectral break depends simply on the presence of a range of field strength and is not specific to a Gaussian field.
In Fig. 2 I show how the spectral index varies with time. The spectral index is that between the two frequencies _1 and _2, plotted against (_1/) which is proportional to time. This figure is then directly comparable to Figs. 2 and 3 of Myers & Spangler (1985), although I have used consistent definitions of in the KP and JP cases (Leahy, Muxlow & Stephens 1989). When plotted in this way, considerable differences between the models are apparent. When the amount of spectral aging is small, the two point spectral index tends to overestimate the age slightly. When the spectral aging is large, the two point spectral index substantially underestimates the age.
Fig 2. Two frequency spectral index as a function of time. Solid lines are KP models, while JP models are dashed lines. The higher two curves are for the unmodified (uniform field strength) models, while the lower curves are for the random field strength case.