 Solution

ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2022/4/c/solution

The magnetic Lorentz force is perpendicular to velocity, so it does no work and

γ

remains constant over an orbit when radiative losses are neglected. For the transverse momentum,

p_{⊥} = γ m_{e} v_{⊥}

and

∣ \dot{p}_{⊥} ∣ = ω_{g} p_{⊥} = e B v_{⊥}

. Hence

ν_{g} = \frac{ω _{g}}{2 π} = \frac{e B}{2 πγ m _{e}} .

Since

10 μ G = 1 0^{- 9} T

ν_{g} = 2.80 \times 1 0^{- 5} Hz, ω_{g} = 1.76 \times 1 0^{- 4} s^{- 1}, T_{g} = 3.57 \times 1 0^{4} s ≃ 9.9 h .

This is the gyromotion frequency, not the much higher characteristic synchrotron radiation frequency.

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