Suppose we have a blob of magnetised plasma emitting synchrotron emission. Let the temperature of the plasma be , so that the Lorentz factor of the electrons is , where is the electron mass and is the speed of light. Let the magnetic field be , and the number of electrons be . The rate at which electrons cool by emitting synchrotron radiation is
where is the classical electron radius. The electrons emit photons with a typical frequency
Electrons can also cool by inverse Compton emission. The cooling rate due to inverse Compton is
This is the same expression as the synchrotron cooling rate, with the magnetic energy density replaced by the photon energy density .
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 . 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 . The magnetic field scales with temperature and frequency as . The energy density of photons in the range is bounded by Black - body energy density . Therefore
For sources observed at a frequency of about 1 GHz, the temperature cannot exceed K. This temperature scales with frequency as , and hence is not very sensitive to it.