Radio spectrum of the Crab nebula as an evidence for fast initial spin of its pulsar

The origin of relativistic electrons in the Crab nebula which are producing the broad-band flat radio spectrum of this prototype plerion has proved difficult to understand. Here I show that these electrons can be naturally explained as a relic population of the pulsar wind electrons that have lost most of their energy in the expanding nebula because of intensive radiative and adiabatic cooling in the past. The observed radio spectrum suggests that the initial slowing-down time of the pulsar was about 30 yrs (or less), which implies that it has been born with a spin period P_0 ~ 3-5 ms, several times shorter than presently believed. Consistency of these results with the current data and the historical records is discussed.