Classical Electrodynamics · Problem 14.26

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · synchrotron-radiation-crab-nebula

Problem

Consider the synchrotron radiation from the Crab nebula. Electrons with energies up to 101310^{13} eV move in a magnetic field of the order of 10−410^{-4} gauss. (a) For E=1013E = 10^{13} eV, B=3×10−4B = 3\times 10^{-4} gauss, calculate the orbit radius ρ\rho, the fundamental frequency ω0=c/ρ\omega_0 = c/\rho, and the critical frequency ωc\omega_c. What is the energy ℏωc\hbar\omega_c in keV? (b) Show that for a relativistic electron of energy EE in a constant magnetic field the power spectrum of synchrotron radiation can be written P(E, ω)=const(ωE2)1/3f(ωωc)P(E,\,\omega) = \text{const}\left(\frac{\omega}{E^2}\right)^{1/3}f\left(\frac{\omega}{\omega_c}\right) where f(x)f(x) is a cutoff function having the value unity at x=0x=0 and vanishing rapidly for x≫1x\gg 1 [e.g., f≈exp⁡(−ω/ωc)f \approx \exp(-\omega/\omega_c)], and ωc=(3/2)(eB/mc)(E/mc2)2cos⁡θ\omega_c = (3/2)(eB/mc)(E/mc^2)^2\cos\theta, where θ\theta 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∝E−n dEN(E)\, dE \propto E^{-n}\, dE, show that the synchrotron radiation has the power spectrum ⟨P(ω)⟩ dω∝ω−α dω\langle P(\omega)\rangle\, d\omega \propto \omega^{-\alpha}\, d\omega where α=(n−1)/2\alpha = (n-1)/2. (d) Observations on the radiofrequency and optical continuous spectrum from the Crab nebula show that on the frequency interval from ω∼108\omega\sim 10^8 s−1^{-1} to ω∼6×1015\omega\sim 6\times 10^{15} s−1^{-1} the constant α≈0.35\alpha \approx 0.35. At frequencies above 101810^{18} s−1^{-1} the spectrum of radiation falls steeply with α≳1.5\alpha \gtrsim 1.5. Determine the index nn for the electron-energy spectrum, as well as an upper cutoff for that spectrum. Is this cutoff consistent with the numbers of part a? (e) 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 BB is given in milligauss and EE 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?

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