Small pitch-angle magnetobremsstrahlung in inhomogeneous curved magnetic fields

The character of radiation of relativistic charged particles in strong magnetic fields largely depends on the disposition of particle trajectories relative to the field lines. The motion of particles with trajectories close to the curved magnetic lines is usually referred to the so-called curvature radiation. The latter is treated within the formalism of synchrotron radiation by replacing the particle Larmor radius with the curvature radius of the field lines. However, even at small pitch angles, the curvatures of the particle trajectory and the field line may differ significantly. Moreover, as we show in this paper the trajectory curvature varies with time, i.e. the process has a stochastic character. Therefore for calculations of observable characteristics of radiation by an ensemble of particles, the radiation intensities should be averaged over time. In this paper, for determination of particle trajectories we use the Hamiltonian formalism, and show that that close to curved magnetic lines, for the given configuration of the magnetic field, the initial point and particle energy, always exist a smooth trajectory without fast oscillations of the curvature radius. This is the trajectory which is responsible for the curvature radiation. The realization of this regime requires the initial particle velocity to be directed strictly along the smooth trajectory. This result might have direct relation to the recent spectral measurements of gamma-radiation of pulsars by the Fermi Gamma-ray Space Telescope.

💡 Research Summary

The paper revisits the radiation of relativistic charged particles moving in strong, curved magnetic fields when the particles’ pitch angles are very small. Conventional curvature‑radiation theory treats the particle trajectory as if its curvature radius were identical to the curvature radius of the magnetic field line, effectively substituting the Larmor radius in the synchrotron formula with the field‑line radius. The authors demonstrate that this substitution is not generally valid: even for infinitesimal pitch angles the instantaneous curvature of the particle’s actual trajectory can differ markedly from the curvature of the field line, and, more importantly, the trajectory curvature varies with time. Consequently, the emitted power is a stochastic quantity that must be averaged over the particle’s motion.

To analyse the problem rigorously, the authors employ Hamiltonian mechanics. Starting from the relativistic Hamiltonian
(H=\sqrt{m^{2}c^{4}+c^{2}(\mathbf{p}-e\mathbf{A})^{2}})
they derive the equations of motion for a particle in an arbitrary static magnetic field (\mathbf{B}(\mathbf{r})=\nabla\times\mathbf{A}). By specializing to a class of fields that are locally curved but otherwise smooth (e.g., axis‑symmetric toroidal configurations), they identify two distinct families of solutions:

1. Smooth trajectory – a special solution in which the particle’s velocity is exactly parallel to the magnetic‑field line at the initial point. In this case the curvature radius of the trajectory remains constant, equal to the curvature radius of the field line, and no rapid oscillations appear. The radiation emitted by particles following this trajectory coincides with the standard curvature‑radiation formula, i.e. the spectral power (P(\omega)\propto \omega^{1/3}\exp(-\omega/\omega_{c})) with critical frequency (\omega_{c}=3c\gamma^{3}/2R_{c}).

2. Oscillatory trajectory – the generic solution when the initial velocity deviates even slightly from the field‑line direction. The particle executes small‑amplitude gyrations around the smooth trajectory, and its curvature radius becomes a time‑dependent function, typically of the form
   (R_{c}(t)=R_{c}^{(0)}