Addition to 14.9(b): The Crab Nebula, formed in the supernova of 1054, contains electrons with energies up to eV moving in magnetic fields of ~1 mG. Use your result on the radiative lifetime to estimate the time in years that it takes 10 GeV and 1 TeV electrons in the Crab to lose one-half of their energy. Watch your units! The Crab emits ergs/s in radiation. For comparison, the radiative output of the sun is ergs/s. There has been no change in the spectrum of synchrotron radiation emitted by the ultrarelativistic electrons in the nebula over the course of the last 30 years of active observations. What constraints do these observations put on theories of the excitation of the Crab?

 

Comments:

		13.12(a):  It is simplest to calculate using the two-dimensional Fourier representation for the E field in vacuum () as input, and evaluating the Fourier integrals directly. It is not necessary to use the expressions in terms of Bessel functions. You will have to do some algebra after you get the result for E in its initial "natural" form to get the simplified form given by Jackson in the atatement of the problem. A reference you might find useful for general background is "Transition radiation from ultrarelativistic particles", L. Durand, Phys. Rev. D 11, 89 (1975), especially sections A and E.
		14.4 and 14.12: Use vector algebra to simplify the expressions for before introducing explicit forms for , , and . Question: why do the results for P in 14.4(a) and 14.4(b) differ by exactly a factor of 2 for R=a?
		14.12: Treat the particle as relativistic as implied by the formulas given. Expand the result in (b) for << 1 and for << 1 assuming that >> 1. Note you cannot simply set at the beginning and get sensible results, so keep the first term in the expansion of in powers of where necessary. In what ranges of angle are the expansions valid? Explain.

Send comments or questions to: ldurand@theory2.physics.wisc.edu

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© 1997, 1998, 1999 Loyal Durand