I re-examine the brightness temperature problem in PKS 0405-385 which is an extreme intra-day variable radio quasar with an inferred brightness temperature of \sim 5 \times 10 ^ { 14 } K at 5 GHz , well above the Compton catastrophe limit of \sim 10 ^ { 11 } K reached when the synchrotron photon energy density exceeds the energy density of the magnetic field .
If one takes into account the uncertainty in the distance to the ionized clouds responsible for interstellar scintillation causing rapid intra-day variability in PKS 0405-385 it is possible that the brightness temperature could be as low as \sim 10 ^ { 13 } K at 5 GHz , or even lower .
The radio spectrum can be fitted by optically thin emission from mono-energetic electrons , or an electron spectrum with a low-energy cut-off such that the critical frequency of the lowest energy electrons is above the radio frequencies of interest .
If one observes optically thin emission along a long narrow emission region , the average energy density in the emission region can be many orders of magnitude lower than calculated from the observed intensity if one assumed a spherical emission region .
I discuss the physical conditions in the emission region and find that the Compton catastrophe can then be avoided using a reasonable Doppler factor .
I also show that MeV to 100 GeV gamma-ray emission at observable flux levels should be expected from extreme intra-day variable sources such as PKS 0405-385 .