  The occurrence of the inverse Compton catastrophe when the synchrotron
brightness temperature exceeds a threshold value, usually estimated to be
10^{12} K, appears to be in contradiction with observation because: (i) the
threshold is substantially exceeded by several intra-day variable radio
sources, but the inverse Compton emission is not observed, (ii) powerful,
extra-galactic radio sources of known angular size do not appear to congregate
close to the predicted maximum brightness temperature. We re-examine the
parameter space available to synchrotron sources using a population of
monoenergetic electrons, in order to see whether the revised threshold
temperature is consistent with the data. The electron distribution and the
population of each generation of scattered photons are computed using spatially
averaged equations. We confirm our previous finding that intrinsic brightness
temperatures T_{\rm B}~10^{14} K can occur without catastrophic cooling. We
show that substantially higher temperatures cannot be achieved either in
transitory solutions or in solutions that balance losses with a powerful
acceleration mechanism. Depending on the observing frequency, we find strong
cooling can set in at a range of threshold temperatures and the imposition of
the additional constraint of equipartition between particle and magnetic field
energy is not warranted by the data. Postulating a monoenergetic electron
distribution, which approximates one that is truncated below a certain Lorentz
factor, \gamma_{min}, alleviates several theoretical difficulties associated
with the inverse Compton catastrophe, including those mentioned above.
