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

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