Scattering of Ultrastrong Electromagnetic Waves by Magnetized Particles

Observations of powerful radio waves from neutron star magnetospheres raise the question of how strong waves interact with particles in a strong background magnetic field $B_{\mathrm{bg}}$. This problem is examined by solving the particle motion in the wave. Remarkably, waves with amplitudes $E_0>B_{\mathrm{bg}}$ pump particle energy via repeating resonance events, quickly reaching the radiation reaction limit. As a result, the wave is scattered with a huge cross section. This fact has implications for models of fast radio bursts and magnetars. Particles accelerated in the wave emit $\gamma$ rays, which can trigger an $e^{\pm }$ avalanche and, instead of silent escape, the wave will produce x-ray fireworks.

Figure 1

Two types of particle motion in strong waves ($E_0>B_{\mathrm{bg}}$), illustrated using a wave with $a_0=30$ and ${\xi }_{\mathrm{rise}}=150(2\pi /\omega )$. The leading edge of the wave packet is at $\xi =0$; the particle’s coordinate $\xi =t-z/c$ grows with time to the left. Left: ${\omega }_B=10\omega$. One can see large $|\overline{\mathit{u}}|\gg |\mathrm{\Delta }\mathit{u}|$ pumped by resonances that happen every Larmor time ${\xi }_L\approx t_L=2\pi /{\omega }_L$. One of the resonances is indicated by the vertical dotted line. The behavior of $u^{\mu }$ near the resonance is also shown as a function of proper time $\tau$. Right: ${\omega }_B=0.2\omega$. Then, the motion has $|\overline{\mathit{u}}|\ll |\mathrm{\Delta }\mathit{u}|$ and remains regular (doubly periodic in $\xi$, with fast $\omega$ and slow ${\omega }_L={\omega }_B/\overline{\gamma }$). The large number of $\omega$ oscillations shown in the figure ($N=600$) merge, forming the black stripes, whose thickness demonstrates the oscillation amplitude $|\mathrm{\Delta }\mathit{u}|\sim a_0$. The $\omega$ oscillations of $u_{\xi }$ are small (the red stripe is thin).

Figure 2

Same model as in Fig. 1 (${\omega }_B=10\omega$), but now with radiation reaction. We chose $r_e\omega /c={10}^{-4}{\omega }_B^2/a_0^5{\omega }^2$, which gives ${\gamma }_{\mathrm{RRL}}\approx 1370$ (horizontal dashed line). The evolution of $\overline{\gamma }$ consists of stochastic jumps $\delta \gamma$ (resonances) followed by gradual losses. Bottom: $\delta \gamma$ vs wave phase $\varphi$ at the resonance. The result confirms Eq. (11) with $H\approx 2.6$ (dotted curve).

Figure 3

Development of radiation reaction with increasing $q$ in waves with $\omega >{\omega }_B$. Averaging $⟨\dots ⟩$ is performed over a time longer than the particle gyration in $B_{\mathrm{bg}}$; $t_{\mathrm{em}}\equiv ⟨\gamma ⟩/⟨{\dot{\gamma }}_{\mathrm{em}}⟩$. The plot was constructed by solving a sequence of models with varying $r_e\omega /c$ at fixed $a_0=30$ and ${\omega }_B=0.1\omega$; however, the same result holds for other choices of $a_0\gg 1$ and ${\omega }_B<\omega$.

Figure 4

Wave pumps $⟨\gamma {⟩}_{\mathrm{ens}}$ of a particle ensemble with initial $T\ne 0$ ($a_0=30$, $\omega {\xi }_{\mathrm{rise}}/2\pi =10$, ${\omega }_B=10\omega$). Three models are shown: with no radiative losses for $\tilde{T}\equiv kT/m{c}^2=0.01$ (blue) and 1 (black), and with losses for $\tilde{T}=1$, ${\gamma }_{\mathrm{RRL}}=1370$ (red). Black dotted line shows the acceleration law $⟨\gamma {⟩}_{\mathrm{ens}}\propto {\xi }^{3/7}$.