The pitch angle paradox and radiative life times in a synchrotron source

In synchrotron radiation there is a paradox whether or not the pitch angle of a radiating charge varies. The conventional wisdom is that the pitch angle does not change during the radiation process. The argument is based on Larmor’s radiation formula, where in a synchrotron case the radiation power is along the instantaneous direction of motion of the charge. Then the momentum loss will also be parallel to that direction and therefore the pitch angle of the charge would remain unaffected. The accordingly derived formulas for energy losses of synchrotron electrons in radio galaxies are the standard text-book material for the last 50 years. However, if we use the momentum transformation laws from special relativity, then we find that the pitch angle of a radiating charge varies. While the velocity component parallel to the magnetic field remains unaffected, the perpendicular component does reduce in magnitude due to radiative losses, implying a change in the pitch angle. This apparent paradox is resolved when effects on the charge motion are calculated not from Larmor’s formula but from Lorentz’s radiation reaction formula. We derive the exact formulation by taking into account the change of the pitch angle due to radiative losses. From this we first time derive the characteristic decay time of synchrotron electrons over which they turn from highly relativistic into mildly relativistic ones.