As an electron travels along a field line, any change of magnetic field strength causes the pitch angle of the electron to change so as to keep the magnetic flux linked by the orbit constant. If the electron is initially at pitch angle to a field of strength B_0 and travelling at speed v then
B / sin^2 = B_0 / sin^2. (7)
The velocity components perpendicular and parallel to the field are
= v sin (B/B_0); = v (1-B sin^2 /B_0), (8)
The losses are proportional to B^2 sin^2. This needs to be averaged along the orbit, along which B/ sin^2 is a constant, so that the average losses are proportional to
B^3 W(s) ds. (9)
The integral is over all the path s that the electron can reach, with B<B_max where B_max = B_0/sin^2 is the maximum field strength into which the electron can travel (it is mirrored out of stronger fields) and W(s) is proportional to the length of time the electron spends in a region ds of a particular field strength. I will assume that the probability of a field strength B is the same for all electrons, and the time spent in a region of field strength B is inversely proportional to the velocity corresponding to that field strength. The average losses for an electron are then
<B^2 sin^2>_orbits =
_0^B_max B^3 P(B) (1-B/B_max)^-1/2 dB (sin^2 / B_0) -------------------------------------------- _0^B_max P(B) (1-B/B_max)^-1/2 dB
= L(B_0/ sin^2 ). (10)
Fig 3. The mean losses of an electron as a function of the initial value of B_0/sin^2 .
The mean losses are shown as a function of B_0/sin^2 in Fig. 3. Note that average electrons have the largest losses. An electron with a large B_0/sin^2 spends the vast majority of its time in regions of low field strength travelling nearly parallel to the field, and consequently has low average losses. The mean losses in this case are only about 80% of what they were before.
The electron energy spectrum in this case becomes
N(E,,t) = N_0 E^- (1 - Et [^2+L(B/ sin^2)])^(-2). (11)
Fig 4. The synchrotron spectrum from electrons in a random magnetic field with no pitch angle scattering (solid line), with the standard KP model spectrum as a dotted line for comparison.
The total spectrum is again obtained by using equation (11) in equation (2) and averaging over all angles and field strengths. The result is shown in Fig. 4. In this case, spectral aging is systematically slower than in the standard KP model.