Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reinterpretation of the Fermi acceleration of cosmic rays in terms of the ballistic surfing acceleration in supernova shocks

Krzysztof Stasiewicz

TL;DR

This work challenges the primacy of first-order Fermi acceleration for cosmic rays by reframing energization in quasi-perpendicular shocks as ballistic surfing acceleration (BSA), a mechanism that operates outside the shock ramp and is consistent with electrodynamics. By constructing a test-particle model in the shock frame, the authors derive a per-cycle energy gain $\Delta K \approx 2 q E_y (r_{cu}-r_{cd})$ and show that the resulting spectrum follows $dN/dK \propto K^{s}$ with $s=-(2 c_B-1)/(c_B-1)$, where $c_B$ is the magnetic compression. The knee of the spectrum arises when the gyroradius matches the shock size, with $K_L \sim \frac{q c}{2} \langle L B_d \rangle$, allowing $s \approx -2.5$ below the knee and $s \approx -3$ above; the acceleration to knee times are of order $\sim 300$ years, and the model can accommodate knee energies around $5\times10^{15}$ eV for plausible shock parameters. Overall, the paper argues that Fermi/DSA is physically invalid in these contexts and should be replaced by the BSA framework for modeling cosmic-ray acceleration in quasi-perpendicular shocks, with implications for interpreting CR spectra and knee formation.

Abstract

The applicability of first-order Fermi acceleration in explaining the cosmic ray spectrum has been reexamined using recent results on shock acceleration mechanisms from the Multiscale Magnetospheric mission in Earth's bow shock. It is demonstrated that the Fermi mechanism is a crude approximation of the ballistic surfing acceleration (BSA) mechanism. While both mechanisms yield similar expressions for the energy gain of a particle after encountering a shock once, leading to similar power-law distributions of the cosmic ray energy spectrum, the Fermi mechanism is found to be inconsistent with fundamental equations of electrodynamics. It is shown that the spectral index of cosmic rays is determined by the average magnetic field compression rather than the density compression, as in the Fermi model. It is shown that the knee observed in the spectrum at an energy of 5x10^{15} eV could correspond to ions with a gyroradius comparable to the size of shocks in supernova remnants. The BSA mechanism can accurately reproduce the observed spectral index s = -2.5 below the knee energy, as well as a steeper spectrum, s = -3, above the knee. The acceleration time up to the knee, as implied by BSA, is on the order of 300 years. First-order Fermi acceleration does not represent a physically valid mechanism and should be replaced by ballistic surfing acceleration in applications or models related to quasi-perpendicular shocks in space. It is noted that BSA, which operates outside of shocks, was previously misattributed to shock drift acceleration (SDA), which operates within shocks.

Reinterpretation of the Fermi acceleration of cosmic rays in terms of the ballistic surfing acceleration in supernova shocks

TL;DR

This work challenges the primacy of first-order Fermi acceleration for cosmic rays by reframing energization in quasi-perpendicular shocks as ballistic surfing acceleration (BSA), a mechanism that operates outside the shock ramp and is consistent with electrodynamics. By constructing a test-particle model in the shock frame, the authors derive a per-cycle energy gain and show that the resulting spectrum follows with , where is the magnetic compression. The knee of the spectrum arises when the gyroradius matches the shock size, with , allowing below the knee and above; the acceleration to knee times are of order years, and the model can accommodate knee energies around eV for plausible shock parameters. Overall, the paper argues that Fermi/DSA is physically invalid in these contexts and should be replaced by the BSA framework for modeling cosmic-ray acceleration in quasi-perpendicular shocks, with implications for interpreting CR spectra and knee formation.

Abstract

The applicability of first-order Fermi acceleration in explaining the cosmic ray spectrum has been reexamined using recent results on shock acceleration mechanisms from the Multiscale Magnetospheric mission in Earth's bow shock. It is demonstrated that the Fermi mechanism is a crude approximation of the ballistic surfing acceleration (BSA) mechanism. While both mechanisms yield similar expressions for the energy gain of a particle after encountering a shock once, leading to similar power-law distributions of the cosmic ray energy spectrum, the Fermi mechanism is found to be inconsistent with fundamental equations of electrodynamics. It is shown that the spectral index of cosmic rays is determined by the average magnetic field compression rather than the density compression, as in the Fermi model. It is shown that the knee observed in the spectrum at an energy of 5x10^{15} eV could correspond to ions with a gyroradius comparable to the size of shocks in supernova remnants. The BSA mechanism can accurately reproduce the observed spectral index s = -2.5 below the knee energy, as well as a steeper spectrum, s = -3, above the knee. The acceleration time up to the knee, as implied by BSA, is on the order of 300 years. First-order Fermi acceleration does not represent a physically valid mechanism and should be replaced by ballistic surfing acceleration in applications or models related to quasi-perpendicular shocks in space. It is noted that BSA, which operates outside of shocks, was previously misattributed to shock drift acceleration (SDA), which operates within shocks.
Paper Structure (5 sections, 10 equations, 3 figures)

This paper contains 5 sections, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Ballistic surfing acceleration
  3. BSA and formation of the cosmic ray spectrum
  4. Discussion
  5. Conclusions

Figures (3)

  • Figure 1: Ion trajectories and heating by TTT and BSA in an oblique shock with a shock angle of $\eta=85^\circ,\; \chi_{B}=8$, the cross-shock electric field $\chi_S=1.5$, compression $c_B=4$, and thickness ratio $r_{ci}/D=1$, without waves. (a): Trajectories of two ions injected at $x=-40$ with a sonic Mach number $M=8$ and additional velocities: $u_x=+20$ (blue) and $u_y=-20$ (red) in units of the upstream ion thermal velocity $v_{Ti}$. (b): Total kinetic energies of ions along the respective trajectories. (c): Thermal (gyration) energies of ions along trajectories. The black curve shows theoretical adiabatic heating for the shock profile given by $b(x)\equiv B(x)/B_u$. (d): Parallel energies of ions along trajectories. Typical values upstream of the bow shock are: $T_i\approx 20$ eV, $v_{Ti}\approx 60$km s$^{-1}$ , $r_{ci}\approx 100$ km, $B_u=5$ nT.
  • Figure 2: BSA and TTT of ions in the same format as in Fig. \ref{['Ff1']} but for a shock angle $\eta=70^\circ$. The red ion is reflected upstream. Ion gyration energy is $u_\perp^2=400$ and the drift energy $M^2=64$ at the starting position $x=-40$.
  • Figure 3: The explanation of ballistic surfing acceleration: Three ions with a perpendicular velocity of $v_\perp=50v_{Ti}$ are injected at incident angles of $0^\circ$ (black), $+10^\circ$ (blue), and $-10^\circ$ (red) into the perpendicular shock located at position $x=0$. The parameters are set as follows: $c_B=4$, Mach number $M=3,\; r_{ci}/D=1$, and the cross-shock electric field $\chi_S=1.5$. The energy gain after one gyration across the shock is $\Delta K\approx 2qE_y(r_{cu}-r_{cd})$, which leads directly to Eq. (\ref{['DK']}). SDA occurs within the ramp denoted by vertical lines, where $|x|/D < 1$, while BSA operates in the gradient-free zone for $|x|/D \geq 1$.