J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)387–390
000–000

Incessant Shock Acceleration of Fast Ions
Enhanced by Relativistic Effects
USAMI Shunsuke and OHSAWA Yukiharu
Department of Physics, Nagoya University, Nagoya 464-8602, Japan(Received: 9 December 2003 / Accepted: 12 February 2004)

Abstract
Interactions between nonthermal fast ions and an oblique shock wave are studied theoretically and numerically.
Some of the fast ions can be repeatedly accelerated; owing to the relativistic effect that the particle velocity is limitedby the speed of light c while the momentum can grow indefinitely, this acceleration is particularly enhanced when theshock speed vsh and propagation angle θ0 satisfy the relation vsh ~ c cosθ0 . A total of 5000 fast ions with their initial
Lorentz factors γ = 4 are followed with test particle simulations to study the evolution of their momentum and energy
distributions. Some ions obtain energies greater than γ ~ 100.
Keywords:
particle acceleration, shock wave, ultrarelativistic ions, cosmic rays

1. Introduction

We consider an energetic particle that enters the shock
wave from the upstream region. Its speed is assumed to befaster than vsh ; hence, its gyro-radius is greater than the widthof the shock transition region. This particle can return to theupstream region owing to the gyro-motion. Then, it gains
energy from the transverse electric field Ey because it moves
nearly parallel to Ey when it is in the shock region. Taking
the scalar product of the relativistic equation of motion withthe momentum p, we readily obtain

It has been long known that high-energy particles areproduced in various astrophysical plasmas. In association withsolar flares, ions can be accelerated up to 109 ~ 1010 eV andelectrons to 107 ~ 108 eV [1,2]. In supernova remnants,electrons with ~ 100 TeV are produced [3-5]. The observed
highest energy of cosmic rays is ~ 1020 eV [6].
Particle simulations have shown that magnetosonic shockwaves can accelerate particles with various nonstochastic
mechanisms [7-20]. Quite recently, interactions of fast
particles and shock waves have also been studied with theory,
particle simulations, and test particle simulations [21-26]. It
has been shown that fast ions can be incessantly accelerated
to ultrarelativistic energies. Under certain conditions, owing
to the relativistic effect, fast particles can move with the shock
wave for long periods of time, and the acceleration isparticularly enhanced [24-26]. In this paper, we study thisacceleration mechanism. In Sec. 2, we theoretically discuss
this mechanism and derive the energy increase rate of aparticle accelerated by this process. In Sec. 3, with test
particle simulations, we follow trajectories of 5000 fast ionsand show how their momentum and energy distributions
evolve. Section 4 gives a summary of our work.

d ⎛ p2 ⎞= qi p ⋅ E ,dt ⎜⎝ 2 ⎟⎠(1)

where, qi is the ion charge. We suppose that the particle enters
the shock wave at t = tin and goes out to the upstream region
at t = t out . Then, integrating this equation along theunperturbed orbit from t = tin to tout , we find the increase in
the Lorentz factor asδγ =2 qi p1 ⊥ Ey( mi c ) Ω i 12⎛ Ω i1 ( tout − t in ) ⎞⎟,2γ⎠⎝sin ⎜(2)

where p⊥ is the momentum perpendicular to B, mi is the ion
rest mass, Ωi is the ion gyro-frequency, and the subscript 1
denotes the quantities right behind the transition region. Here,
we have neglected the electric field parallel to the magnetic
field E|| , which is weak in magnetohydrodynamic waves.
Time-averaged particle velocity in the x direction is given
by 〈vx 〉 ∼– v0|| cosθ0 , where v0|| is the velocity along B0 . Hence,

2. Theory
We assume that a shock wave propagates in the x
direction with a speed vsh in an external magentic field B0 =B0 (cosθ0 , 0, sinθ 0 ), where the subscript 0 indicates theupstream quantities.

Corresponding author’s e-mail: usami@plab.phys.nagoya-u.ac.jp1387©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchUSAMI S. et al., Incessant Shock Acceleration of Fast Ions Enhanced by Relativistic Effects

a fast particle with4v0|| cosθ0 –∼ vsh ,(3)Bz 2

can move with the shock wave for some period of time.
However, as the particle crosses the shock front, its p ||increases owing to the magnetic structure [22-26]. The
particle therefore escapes from the shock wave to theupstream region. However, for a shock wave withvsh ~ c cosθ0 ,01

Ey 0.5(4)

fast ions cannot quickly go away ahead of the shock wave,
because the particle speed is limited by the speed of light c.
Hence, the acceleration processes can be repeated many times.
For a fast particle barely entering the shock wave, the time
period in which it is in the shock wave is quite short, tout – tin<< 2πγ /Ωi0 . Accordingly, its gyro-period is approximately
given by 2πγ /Ωi0 . The fast ion suffers an energy jump (2) ineach gyro-period. The energy increase rate average overrelativistic gyro-period is therefore given as1dγΩ i 0 dt=g vshπc,

0200

600

1000x/(c/ωpe)
Fig. 1 Profiles of Bz and Ey in an oblique shock wave.

(5)

100

where g is a numerical factor smaller than unity. Here, wehave eliminated Ey using the relation Ey = (vsh /c)(Bz1 – Bz0),
which results from Faraday’s law. Equation (5) indicates thatγ linearly grows with time, if the time dependence of g isweak.

γ50

3. Simulation
To study long-time behavior of fast ions incessantly
accelerated by an oblique shock wave, we have carried out
test particle simulations [25,26]. In this method, we obtain
the electric and magnetic fields of a shock wave from onedimensional, relativistic, electromagnetic particle simulations.
Then, assuming that the shock wave propagates steadily, we
follow long-time trajectories of energetic particles in thesefields.
The parameters of the particle simulation were asfollows. The total system length was Lx = 8192∆g , where ∆gis the grid spacing. The total number of electrons was Ne =576000. As space plasmas, the code contained helium ions aswell as hydrogen ions, and the helium number density was
10 % of the hydrogen density. The ion-to-electron mass ratios
were mH /me = 50 and mHe /me = 200. The ratio of the electron
gyro-frequency to the plasma frequency was ⏐Ωe0 ⏐/ωpe = 1.5
in the upstream region. The Alfvén speed is thus vA/c = 0.20,
where vA is defined using the mass density in the H-Heplasma. The electron skin depth was c/(ωpe∆g) = 4. The
external magnetic field was in the (x, z) plane with θ0 = 61°.
The time step was ωpe∆t = 0.05. We then excited a shock
wave with vsh = 2.4vA , which is close to c cosθ0 . The profilesof Bz and Ey in this wave are shown in Fig. 1.
Using the wave fields propagating with the speed vsh ,we followed the trajectories of test energetic hydrogen ionswhich are initially in the upstream region. The number ofthese particles was N = 5000, and their initial energy was00Fig. 23000

6000ΩH0t9000

Stepwise increase in γ of an incessantly accelerated fastion.

taken to be γ0 = 4; their initial energy distribution was f (γ ) =Nδ (γ – γ0) with an isotropic momentum distribution. The
plasma parameters were the same as those in the particle
simulation.
Figure 2 shows time variation of γ of an incessantlyaccelerated ion. The energy increased stepwise from γ0 = 4 toγ0 = 120; when the calculation was terminated at ΩH0t = 9000,
the acceleration was not finished.
Figure 3 displays energy distribution functions at ΩH0t =0 and at ΩH0t = 9000. By ΩH0t = 9000, 3.4 % of the fast ionshave been accelerated to energies higher than γ = 50. Themaximum energy is γ ∼– 130.
Figure 4 shows time evolution in the momentum space( p|| , p⊥). Since the initial particle energies are the same, γ0 =4, all the particles are on the half circle in the ( p|| , p⊥) planeat ΩH0t = 0. We see from the lower panel (ΩH0t = 9000) that2388

Particles

Particles

USAMI S. et al., Incessant Shock Acceleration of Fast Ions Enhanced by Relativistic Effects

10 4
10 3
10 2
10140

ΩH0t=0p

⊥

200

10 4
10 3
10 2
1010Fig. 3ΩH0t=040

9000

9000p

⊥

200040γ8050120p

100150//Fig. 4

Evolution of particles in the momentum space. Particles
spread along the line v0|| = vsh /cosθ0 .

Stepwise increase in γ of an incessantly accelerated fastion.

2K / K0

the particle distribution spreads along the line v0|| = vsh /cosθ0 ,which is hyperbola in the ( p|| , p⊥) plane. Some particles havequite large p⊥ at ΩH0t = 9000. They are in the shock wave at
this moment.
Figure 5 shows the time variation of the total energy ofthese particles. Even around ΩH0t = 9000, it is increasingsteadily.

1.5

4. Summary1

We have studied the interactions between fast ions andoblique shock waves. When the shock propagation speed isvsh ~ c cosθ0 , some fast ions are incessantly accelerated toultrarelativistic energies. The energy increase rate for theseparticles was theoretically obtained. The evolution of 5000
fast particles was also investigated with test particle
simulations. The acceleration mechanism discussed in thispaper could be important around pulsars where the magnetic
fields are strong.

03000

6000

9000

ΩH0t
Fig. 5References

Time variation of total energy K of a set of 5000 fastions. Here, K0 is the initial energy.

[7] D. Biskamp and H. Welter, Nucl. Fusion 12, 663 (1972).
[8] D.W. Forslund, K.B. Quest, J.M. Brackbill and K. Lee,J. Geophys. Res. 89, 2142 (1984).
[9] R.L. Tokar, S.P. Gary and K.B. Quest, Phys. Fluids 30,2569 (1987).
[10] Y. Ohsawa, J. Phys. Soc. Jpn. 55, 1047 (1986).
[11] Y. Ohsawa, Phys. Fluids 28, 2130 (1985).
[12] Y. Ohsawa, J. Phys. Soc. Jpn. 59, 2782 (1990).
[13] T. Kawashima, S. Miyahara and Y. Ohsawa, J. Phys. Soc.
Jpn. 72, 1664 (2003).
[14] M. Toida and Y. Ohsawa, J. Phys. Soc. Jpn. 64, 2036
(1995).
[15] M. Toida and Y. Ohsawa, Sol. Phys. 171, 161 (1997).

[1] D.J. Forrest, W. Vestrand, E.L. Chupp, J. Cooper and G.
Share, Adv. Space Res. 6, 115 (1986).
[2] E.L. Chupp, H. Debrunner, E. Fluckiger et al.,Astrophys. J. 318, 913 (1987).
[3] T. Tanimori, Y. Hayami, S. Kamei et al., Astrophys. J.
Lett. 497, L25 (1998).
[4] T. Tanimori, K. Sakurazawa, S.A. Dazeley et al.,Astrophys. J. Lett. 492, L33 (1998).
[5] H. Muraishi, T. Tanimori, S. Yanagita et al., Astron.
Astrophys. 354, L57 (2000).
[6] M. Takeda, N. Hayashida, K. Honda et al., Phys. Rev.
Lett. 81, 1163 (1998).
3389

USAMI S. et al., Incessant Shock Acceleration of Fast Ions Enhanced by Relativistic Effects[22] T. Masaki, H. Hasegawa and Y. Ohsawa, Phys. Plasmas
7, 529 (2000).
[23] R.B. Decker, Space Sci. Rev. 48, 195 (1988).
[24] S. Usami, H. Hasegawa and Y. Ohsawa, Phys. Plasmas
8, 2666 (2001).
[25] S. Usami and Y. Ohsawa, Phys. Plasmas 9, 1069 (2002).
[26] S. Usami and Y. Ohsawa, Phys. Plasmas 11, 918 (2004).

[16] N. Bessho, K. Maruyama and Y. Ohsawa, J. Phys. Soc.
Jpn. 68, 1 (1999).
[17] N. Bessho and Y. Ohsawa, Phys. Plasmas 6, 3076 (1999).
[18] N. Bessho and Y. Ohsawa, Phys. Plasmas 7, 4004 (2000).
[19] N. Bessho and Y. Ohsawa, Phys. Plasmas 9, 979 (2002).
[20] H. Hasegawa, S. Usami and Y. Ohsawa, Phys. Plasmas
10, 3455 (2003).
[21] K. Maruyama, N. Bessho and Y. Ohsawa, Phys. Plasmas
5, 3257 (1998).

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