Electron acceleration to energies beyond GeV by a relativistic ion beam instability

Synchrotron emission suggests the presence of TeV electrons at various astrophysical objects. We propose a mechanism for the acceleration of electrons to ultrahigh energies (UHE) by intense electrostatic waves (ESWs). The latter are driven by dense proton beams that move at relativistic speeds relative to a background plasma and the electrons are accelerated by their nonlinear interaction with the ESWs. We follow the evolution of the wave instability by means of particle-in-cell (PIC) simulations. After the instability has saturated, we obtain spatially confined electron voids in which secondary instabilities develop due to resonant interactions between the beams and the background protons, generating intense ESWs which accelerate electrons to ultrarelativistic speeds within times of a few hundred inverse plasma frequencies.

Figure 1

The power spectrum $A(n,t)$ for the simulation with the beam density $n_b=n_{\mathit{el}}$ (left plot) and the simulation with $n_b=7.1n_{\mathit{el}}$ (right plot). In both simulations, an ESW initially grows at the wave number $k_u$. This ESW is sufficiently strong to produce a harmonic at $k=2k_u$. In the left plot, the ESW stabilizes after its initial saturation. In the right plot we find growing low-$k$ waves at $t\approx 600\times 2\pi ∕{\omega }_p$. At $t\approx 800\times 2\pi ∕{\omega }_p$, the spectrum is increasingly bursty and the waves spread over a large $k$ interval. After $t\approx 1050\times 2\pi ∕{\omega }_p$, the spectrum exhibits a continuous power law.

Figure 2

The electric field $E$ at time $t=350\times 2\pi ∕{\omega }_p$ (upper left panel), $t=600\times 2\pi ∕{\omega }_p$ (upper right panel), $t=890\times 2\pi ∕{\omega }_p$ (lower left panel), and $t=1300\times 2\pi ∕{\omega }_p$ (lower right panel). Two spikes which move with $v\approx 0.9c$ are visible at $t=600\times 2\pi ∕{\omega }_p$. These spikes evolve initially into the low-$k$ oscillations at $t=890\times 2\pi ∕{\omega }_p$ and thereafter into the extremely strong fluctuations at all spatial scales at $t=1300\times 2\pi ∕{\omega }_p$.

Figure 3

The base-10 logarithm of the evolving maximum electric field of the largest spike in our simulation. Between the times $t=550\times 2\pi ∕{\omega }_p$ and $t=700\times 2\pi ∕{\omega }_p$ the wave amplitude grows exponentially. The increased growth after $t=700\times 2\pi ∕{\omega }_p$ is linked to the triggering of oscillations with a large spatial extent.

Figure 4

The charge density fluctuations of the proton beam with $v_b=0.99c$ normalized to the electron density (upper plot), and the charge density distribution of the electrons with $v_x>0$ normalized to the total electron charge density (lower plot), at $t=600\times 2\pi ∕{\omega }_p$. Both plots show the same spatial interval of the simulation box. We observe that the depletions of the charge density in the proton beam at $x∕{\lambda }_u\approx 8$ and $x∕{\lambda }_u\approx 50$ are associated with intervals in which no electrons with $v_x>0$ are present.

Figure 5

The electron momentum space density (upper plot) and the momentum space density summed over all proton species (lower plot), at time $t=1300\times 2\pi ∕{\omega }_p$. Both distributions are given in terms of physical particles, i.e., we have multiplied the number of computational particles by a factor that expresses the ratio between simulation particle numbers and true particle numbers. The electrons have a flat distribution up to $p_x∕m_{e}c\approx {10}^3$. Above this momentum, the electron distribution has a spectrum that is well approximated by a power law, viz. $P\left(p_x\right)=C_1{(p_x∕m_{e}c)}^{-3.4}$, which is the line we have overplotted. The lower plot shows that the protons have been fully thermalized.