Consider the synchrotron radiation from the Crab nebula. Electrons with energies up to 10¹³eV move in a magnetic field of the order of 10^?4 gauss.

(a) For E = 10¹³eV, ? = 3 X 10^?4 gauss, calculate the orbit radius ?, the fundamental frequency w₀ = c/p, and the critical frequency w_c. What is the energy hw_c in keV?

(b) Show that for a relativistic electron of energy E in a constant magnetic field the power spectrum of synchrotron radiation can be written

where f(x) is a cutoff function having the value unity at x = 0 and vanishing rapidly for x >> 1 [e.g., f ? exp(-w/w_c)], and w_c = (3/2)(eB/mc)(E/mc²)²cos ?, where ? is the pitch angle of the helical path. Cf. Problem 14.17a.

(c) If electrons are distributed in energy according to the spectrum N(E) dE ? ?^?n dE, show that the synchrotron radiation has the power spectrum

> dw ? w^?? dw

where ? = (n - 1)/2.

(d) The half-life of a particle emitting synchrotron radiation is defined as the time taken for it to lose one half of its initial energy. From the result of Problem 14.9b, find a formula for the half-life of an electron in years when ? is given in milligauss and E in GeV. What is the half-life using the numbers from part a? How does this compare with the known lifetime of the Crab nebula? Must the energetic electrons be continually replenished? From what source?

1/3 P(E, w) = const