Compton Limit

Suppose we have a blob of magnetised plasma emitting synchrotron emission. Let the temperature of the plasma be $kT$, 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.