Fusion Res.
Res. SERIES,SERIES, Vol.
Vol. 6 (2004) 000-000
391–394
J. Plasma Fusion

Production of Ultrarelativistic Positrons
by an Oblique Magnetosonic Shock Wavein an Electron-Positron-Ion Plasma
HASEGAWA Hiroki and OHSAWA Yukiharu
Department of Physics, Nagoya University,
Nagoya 464-8602, Japan(Received: 9 December 2003 / Accepted: 23 February 2004)

Abstract
Wave propagation and particle acceleration in a plasma consisting of electrons, positrons, and ions are studied
with theory and particle simulations. With one-dimension, relativistic, electromagnetic, particle simulations, it is foundthat a magnetosonic shock wave propagating obliquely to an external magnetic field can accelerate some positrons toultrarelativistic energies. Also, a theory is presented which explains the energy increase rate and particle motion
observed in the simulations.
Keywords:
positron, oblique shock, particle acceleration, particle simulation, multicomponent plasma

1. Introduction

accelerate positrons to ultrarelativistic energies; in thesimulation, the maximum positron energy was c + 600 ,where c is the Lorentz factor.

Particle simulations show that a magnetosonic shock
wave can accelerate particles with various nonstochastic
mechanisms. It can accelerate some ions by reflection, whichis caused by rapid increase in the electric potential or themagnetic field strength in the shock transition region [1-5]. In
a multi-ion-species plasma with the hydrogen the major
component, a shock wave accelerates all the heavy ions withthe transverse electric field; their maximum speeds are nearly
the same (independent of the particle species) [6,7]. Also, it
can accelerate some electrons to ultrarelativistic energies;
electrons that are reflected near the end of the main pulse aretrapped in the pulse and gain energies from the wave fields[8,9]. If there exist nonthermal energetic ions, a shock wave
can incessantly accelerate them with the transverse electricfield [10-13]. Furthermore, in a plasma containing positrons
as well as electrons and ions, it can accelerate positrons toultrarelativistic energies [14].
Positrons could be found, for instance, in pulsar
magnetospheres, astrophysical jets, or laboratory laserplasmas [15-18]. Electron-positron plasmas have thus beenstudied by many authors [19-23]. In this paper, weinvestigate wave propagation and positron acceleration inelectron-positron-ion (e-p-i) plasmas. In Sec. 2, we describe
the dispersion relations of three low frequency waves
propagating obliquely to an external magnetic field; one ofthem is the magnetosonic wave. In Sec. 3, we theoretically
discuss positron acceleration caused by a magnetosonicshock wave. In Sec. 4, by using relativistic, electromagnetic,
particle simulations, we show that a shock wave can

2. Dispersion relations
From the three-fluid model for a cold e-p-i plasma, wehave three different modes in the frequency regime lower
than the electron gyrofrequency | X e | . Figure 1 shows
dispersion curves for these modes. Here, the electron
gyrofrequency is | X e |/ ~ pe = 3.0 , where ~ pe is the electron
plasma frequency, and the angle between the wave normal
and the external magnetic field B0 is taken to be i = 42 °.
The ion-to-electron mass ratio is m i /m e = 100; we havechosen this value to compare with simulations. In the upper
panel, the positron-to-electron density ratio is n p0 /n e0 = 0.02 ,while in the lower panel, it is n p0 /n e0 = 0.5 . In the frequency
regime ~ L | X e | , we find three modes, which we are notconcerned with in this paper. In the low frequency regime~ K | X e | , we have high-frequency mode (line H),magnetosonic mode (line M), and Alfvén mode (line A).
The high-frequency mode has a cutoff frequency ~ hf0 .
Its value approaches the electron gyrofrequency, ~ hf0 + | X e | ,as the positron density goes to zero, n p0 /n e0 " 0 , while itapproaches the ion gyrofrequency, X i , as n p0 /n e0 " 1. Theresonance frequency of this mode is given by the electron
gyrofrequency | X e | .
The Alfvén mode has a resonance at ~ + X i . Its
dispersion relation in the long-wavelength regime is given by©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: ohsawa@phys.nagoya-u.ac.jp1391

Hasegawa H. and Ohsawa Y., Production of Ultrarelativistic Positrons by an Oblique Magnetosonic Shock Wave in an Electron-Positron-Ion Plasma

where v w is the velocity parallel to the magnetic field, v wdz isthe z component of the drift velocity v wd , and n Bw is themagnetic moment, m p v w2 9 /( 2B w ) with v w9 the gyration
speed. Also, v rv = - cE wy0 B wz /( B x0 B w ) , and K = - m p v r2v /2 .
The quanty F is defined by100

ω/ Ωi

101 θ = 420.1

HMA|Ω e|/ωpe =3Fw = - # Ew ds ,np0/n e0 = 0.02

0.01

where ds is the infinitesimal length along the magnetic field.
This quantity Fw has large positive values in the shock
region. Positrons cannot penetrate regions where Fw has large
positive values, because the left-hand side of eq. (3) must bepositive. Noting that all the terms except Fw in eq. (3) areproportional to m p and that v w 0 and v wd are of the order ofv A , we find that positrons can be reflected by a shock wave.
This reflection can occur in oblique shock waves.
Another important effect of oblique propagation is that it
enables some positrons to move with the shock wave for long
periods of time. In the next subsection, we will discuss thesepositrons.

ω/ Ωi

100
101

H

0.1
0.01

MA

np0/n e0 = 0.5
10 -2

ck/ωpe1Fig. 1 Dispersion relations of low frequency waves inmagnetized e-p-i plasmas with i = 42 ° for two
different positron densities.

~ - v A (1 + v A2 /c 2 ) - 1/2 k cos i,

3.2 Positron acceleration
Here, we consider positrons staying in the shocktransition region and assume that their velocities satisfy therelations v . c and v x . v sh & v y . We will derive theenergy increase rate for these particles.
Becuase c is quite large, we neglect cdv v /dt comparedwith v v dc/dt , where v = x, y , or z . Then, using the relations
E y = (v sh /c)( B z - B z0 ) and E z = - (v sh /c) B y , we may write
the zeroth-order equation of motion as(1)

where v A is the Alfvén speed in the e-p-i plasma defined asv A = B0 /( 4r !j n j m j )1/2 , c is the speed of light, and k is
the wave number.
The dispersion relation of the magenetosonic wave isapproximated by~ - v A (1 + v A2 /c 2 ) - 1/2 k,

(4)

vydcv
= eE x (p) + e c B z (p) - e cz B y (p), (5)
dtv
v
(6)
0 = e cz B x0 - e csh B z0 ,vydc
(7)=- e c B x0 ,m p vzdt(2)

m p v shin the long-wavelength regime, ck /~ pe % 1. Also, it has aresonance at ~ + X e cos i. The shock wave studied in thispaper is the magnetosonic mode.

3. Acceleration theory

where p is the particle position, p = x - v sh t . Equation (6)gives

3.1 Positron reflection
In magnetohydrodynamic waves, the electric fieldparallel to the magnetic field is quite weak. The ions are thus
rarely reflected along the magnetic field in shock waves.
They are reflected across the magnetic field. On the other
hand, positrons can be easily reflected by the shock wave
along the magnetic field because their mass is small.
Suppose that a shock is propagating in the x direction
with a propagation speed v sh in an external magnetic fieldB0 = B0 ( cos i, 0, sin i ), where the subscript 0 denotes
equilibrium quantities. Then, in the wave frame, the electricfield is given as E w = (E wx , - (v sh /c) B wz0 , 0), where thesubscript w refers to quantities in the wave frame. From theequation of motion for the positrons, one obtains thefollowing equation [8,14],v z /v sh = B z0 /B x0 = tan i,(8)which indicates that the velocity of an accelerated positron isnearly parallel to the external magnetic field. Also,substituting eqs. (7) and (8) in (5) yields the time rate ofchange of ceB 2 J E x (p) - v sh B z0 B y (p)/( cB x0 ) NO
dc= m p vx0sh KK(9)O.
dt
B z0 B z (p) + B x20
PLThis shows that c would grow linearly with time if theparticle position p is constant.

4. Simulation
To further study the positron acceleration, we use a one
dimension (three velocities), relativistic, electromagnetic,
particle simulation code with full particle dynamics. Thesystem length is L x = 8192D g with D g the grid spacing. The
total number of electrons is N e = 614, 000 . The ion-to-1 m (v - v ) 2 =- e (F - F ) - K - n B
rvw
w0Bww
2 p w
(3)
Ew
0y2+ 1 m p v w2 0 - 1 m p v wd- m p c B (v wdz - v wz0 ),22x02392

Hasegawa H. and Ohsawa Y., Production of Ultrarelativistic Positrons by an Oblique Magnetosonic Shock Wave in an Electron-Positron-Ion Plasma

600 ωpe t =800γ 300

γ 300Bz
3 B000101600

γ 3000-20v/c6Bz
3 B0-1001020(x-xm)/(c/ ωpe)

01030vx /c6006006vy /cFig. 2 Phase space plots (x, c) of positrons. The solid lines
indicate profiles of B z .

electron mass ratio is m i /m e = 100 , with charges
q i = q p = - q e . The electron skin depth is c/~ pe = 4D g . The
magnetic field strength is X e /~ pe = 3.0 in the upstream
region with i = 42 °. The Alfvén speed for these parametersis v A /c = 0.30 . The time step is ~ pe Dt = 0.01 , which ismuch smaller than the periods of electron plasma oscillationand gyromotion.
Figure 2 shows phase space plots of positrons at two
different times for a shock wave with v sh = 2.4v A . Here, x m
is the x position at which the magnetic field has its maximum
value. The solid lines indicate profiles of B z . Many positrons
are reflected in the shock transition region and thenaccelerated to ultrarelativistic energies; the maximum energy
in this example is c + 600 .
Figure 3 displays time variations of c and velocities of apositron that has been accelerated to c + 600 . This particle
encountered the shock wave at ~ pe t + 200 and wasreflected. Its speed quickly reached v + c . The panels of v xand v z indicate that the particle moves with the shock speed
in the x direction, v x - v sh , and that its velocity is nearlyparallel to the external magnetic field, v z + v sh tan i, which
agrees with eq. (8). We also note that the magnitude of v y isquite small.
We obtain time-averaged velocities G v y /c H = - 0.12
and G v z /c H = 0.66 from the simulation data from~ pe t = 250 to ~ pe t = 1600 . Substituting these values in eq.
(7), we find the energy increase rate as dc/d (~ pe t) = 0.41.
This is also in good agreement with the observed value
dc/d (~ pe t) = 0.41 for the same time period.

vsh

00vz /c-11vsh tan θ00800

ωpe t

1600

Fig. 3 Time variations of c , v , v x , vy , and vz of anaccelerated positron.

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5. Summary
We have studied the wave propagation and positron
acceleration in an electron-positron-ion plasma. It is foundthat an oblique magnetosonic shock wave can acceleratesome positrons to ultrarelativistic energies; in oursimulations, positron energy reached c + 600 . Particle
motion and energy increase rate predicted by our theory arein good agreement with the simulation results.

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