There is a robust limit on the maximal frequency of synchrotron radiation from electrons that can be achieved whenever the mechanism for accelerating electrons is through shock heating (as is the most common case in astrophysics). Interestingly, this frequency is independent on the strength of the magnetic field in which the electrons are accelerating and radiating.

Up to same constant, which depends on the details of acceleration (such as the strength of the magnetic field in the downstream compared to the upstream), the time it takes for an electron to double its energy is roughly the shock crossing time. Since in both the upstream and the downstream, the electron's motion is basically gyration around the magnetic field, the shock crossing time is of the order of the gyration time: $ t_h\approx\frac{2\gamma m_e c}{eB} $. This time becomes longer as the electron accelerates and $ \gamma $ increases.

As the electron accelerates in the magnetic field it also radiates by synchrotron emission, causing it to lose energy. The energy loss rate by synchrotron is approximately the electron's energy over the synchrotron power: $ t_c=\frac{\gamma m_e c^2}{\sigma_T c \gamma^2 \frac{B^2}{8\pi}}=\frac{8 \pi m_e c}{\sigma_T \gamma B^2} $. This time decreases with the electron's energy.

Since the electron takes both longer and longer times to heat and shorter and shorter times to cool as its energy increases, it is obvious that there is a maximal possible equilibrium energy where these rates meet. Equating $ t_h, t_c $ yields:

$ E_{max}=m_e c^2 \gamma_{max}\approx m_e c^2 \bigg(\frac{4\pi e}{\sigma_T B}\bigg)^{1/2} \approx 5\times 10^{13} \mbox{eV} \left(\frac{B}{\mbox{Gauss}} \right)^{-1/2} $.

Plugging $ \gamma_{max} $ into the formula for the typical synchrotron frequency, we obtain the maximal synchrotron frequency:

$ \nu_{syn,max}=\frac{eB\gamma_{max}^2}{2\pi m_e c}=\frac{2 e^2}{\sigma_T m_e c}\approx 100\mbox{MeV} $.

One astrophysical scenario where this limit is thought to be reached is the crab nebula.

We conclude with two notes.

First, in case the shocked material within which the electrons are gyrating has some bulk relativistic motion towards us, then the maximal frequency above should be multiplied by the Lorentz factor of the bulk. This is relevant to GRB afterglows and possibly to their prompt emission phase as well. Recent observations of extremely high energy photons (up to $ \approx 100 $GeV) likely rule out synchrotron emission as the radiation mechanism for the very high energy photons.

Second, since $ \sigma_T \propto m_e^{-2} $, the maximal frequency increases linearly with mass and will be over three orders of magnitude higher for protons. However, one should keep in mind that proton synchrotron is extremely inefficient compared to electron synchrotron.

### Hillas Criterion Edit

Fermi acceleration can also be truncated by the escape of a charged particle from the magnetic region. As a particle gains more energy, its Larmor radius increases. When the Larmor radius is comparable to the size of the system, it can escape. Hence the maximum energy a particle can attain is

$ \varepsilon \le L q B $

where $ \varepsilon $ is the energy of the particle, $ q $ is its charge, $ B $ is the magnetic field and $ L $ is the size of the system.