An electron of Lorentz factor moves with pitch angle in a jet threaded by a magnetic…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres

Stars from the Inside, Ch Hot and fast electrons: bremsstrahlung, Compton scattering and synchrotron emission

Problem

An electron of Lorentz factor γ=104\gamma=10^{4} moves with pitch angle α=90∘\alpha=90^\circ in a jet threaded by a magnetic field of magnitude B=5×10−8 TB=5\times10^{-8}\,\mathrm{T}, and also sits in an ambient radiation field of energy density Urad=1×10−4 J m−3U_{\rm rad}=1\times10^{-4}\,\mathrm{J\,m^{-3}} in the frame used here. Let νg0=eB/2πme\nu_{g0}=eB/2\pi m_e be the non-relativistic gyrofrequency and ωB,rel=eB/(γme)=2πνg0/γ\omega_{B,\rm rel}=eB/(\gamma m_e)=2\pi\nu_{g0}/\gamma the electron's actual (relativistic) angular gyration frequency. Treat β=v/c≈1\beta=v/c\approx1. ((a)) The electron's radiation is beamed into a cone of half-angle 1/γ1/\gamma about its instantaneous velocity, and it sweeps through this angle at angular rate ωB,rel\omega_{B,\rm rel}. Combine the time to sweep this angle with the retardation (light-travel-time) compression factor (1−β)≈1/2γ2(1-\beta)\approx1/2\gamma^2 for a nearly head-on ray to show that the observed pulse duration is Δt≈1/(2γ3ωB,rel)\Delta t\approx1/(2\gamma^3\omega_{B,\rm rel}), and hence that the characteristic angular frequency of the radiation is ωc∼1/Δt∝γ2×2πνg0\omega_c\sim1/\Delta t\propto\gamma^2\times2\pi\nu_{g0}. ((b)) The pitch-angle-averaged synchrotron power of one electron has the same functional form as the pitch-angle-averaged inverse-Compton power, with the magnetic energy density UB=B2/2μ0U_B=B^2/2\mu_0 in place of the radiation energy density UradU_{\rm rad}: Psyn=43σTcγ2β2UB,PIC=43σTcγ2β2Urad.P_{\rm syn}=\tfrac43\sigma_Tc\gamma^2\beta^2U_B,\qquad P_{\rm IC}=\tfrac43\sigma_Tc\gamma^2\beta^2U_{\rm rad}. Show that the ratio Psyn/PICP_{\rm syn}/P_{\rm IC} depends only on the two energy densities, not on γ\gamma. ((c)) Using ωc≈32γ2(2πνg0)sin⁡α\omega_c\approx\tfrac32\gamma^2(2\pi\nu_{g0})\sin\alpha, compute the characteristic frequency νc=ωc/2π\nu_c=\omega_c/2\pi of this electron's synchrotron emission, and compute Psyn/PICP_{\rm syn}/P_{\rm IC} for the stated BB and UradU_{\rm rad}.

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