Suppose we have a blob of magnetised plasma emitting synchrotron emission. Let the temperature of the plasma be k T, so that the Lorentz factor of the electrons is  \gamma_e = k T / m_e c^2 , where m_e is the electron mass and c is the speed of light. Let the magnetic field be B, and the number of electrons be N_e. The rate at which electrons cool by emitting synchrotron radiation is

 L_s \approx N_e r_0^2 c B^2 \gamma_e^2

where  r_0 is the classical electron radius. The electrons emit photons with a typical frequency

 \nu_s \approx \frac{c}{r_0} \sqrt{\frac{B^2 r_0^3}{m_e c^2}} \gamma_e^2

Electrons can also cool by inverse Compton emission. The cooling rate due to inverse Compton is

 L_c \approx N_e r_0^2 c u \gamma_e^2

This is the same expression as the synchrotron cooling rate, with the magnetic energy density  B^2 replaced by the photon energy density  u .

When inverse Compton cooling rate exceeds the synchrotron cooling rate, synchrotron emission is suppressed. This happens when the radiation energy density is comparable to the magnetic energy density  u \approx B^2 . Observationally, this condition manifests itself as an upper bound on the brightness temperature of radio sources.

Suppose an astrophysical source is observed at a frequency \nu_s. The magnetic field scales with temperature and frequency as  B \propto \nu_s/ T^2 . The energy density of photons in the  nu_s range is bounded by Black - body energy density  u \approx T \nu_s^3 . Therefore

 \frac{L_c}{L_s} \approx \frac{u}{B^2} \approx \left(\frac{k T}{m_e c^2} \right)^5 \left(\frac{\nu_s r_0}{c} \right)<1 .

For sources observed at a frequency of about 1 GHz, the temperature cannot exceed 10^{12} K. This temperature scales with frequency as  \nu_s^{-1/5} , and hence is not very sensitive to it.