J. Plasma Fusion Res. SERIES, Vol.4 (2001) 505-508

Theoretical Estimate of Electron Energies Accelerated in an
Oblique Shock Wave
BESSHO Naoki* and OHSAWA yukiharu
Department of Physics, Nagoya University, Nagoya 464_g602, Japan(Received: 5 December 2000 / Accepted: 1g August 2001)

Abstract
Electron acceleration by a magnetosonic shock wave propagating obliquely to a magnetic field isstudied by means of a one-dimensional, relativistic, electromagnetic particle simulation code with full ionand electron dynamics. It is demonstrated by simulations that the shock wave can produce ultrarelativistic
electrons with their Lorentz factors exceeding 100. Detailed physical considerations for this phenomenon
are then given, which clearly indicate that at certain propagation angles the acceleration can become
especially strong.

Keywords:
particle acceleration, ultrarelativistic electron, shock wave, potential formation, particle simulation

1. Introduction

In this paper, we study this phenomenon withtheory and particle simulations. In Sec. 2, we show
simulation results. In Sec. 3, we will discuss the acceleration mechanism in detail. An electron reflected nearHigh-energy electrons, as well as ions, are often
observed in astrophysical plasmas. For instance, in solar
flares, electrons can be accelerated to several tens of
MeV []. From the observations of X-rays and gamma
rays, it is now believed that electrons with energies up
to -l00TeV are produced by supernova remnant
SNl006 [2,3]. Also, in the Crab Nebula high-energy
electrons accelerated up to -100TeV are produced by
the pulsar wind [4].
Motivated by these observations, particle acceleration in nonlinear magnetosonic waves has beenextensively studied. Even though various ion accelera-

the end of the main pulse obtains a large amount ofenergy from both the electric potential and constantelectric field appearing in the wave frame. At certain
shock propagation angles, speeds of the reflected
electrons relative to the shock wave can be quite small;the electrons can then travel a long distance in thedirection parallel to the wave front and perpendicular to
the external magnetic field. This causes particularly
strong electron acceleration.

tion mechanisms have been shown by particle
simulations [5-15], strong electron acceleration has not

2. Simulation

been reported.

To study the shock propagation and electron acceleration, we use a one-dimensional (one space coordinate

Quite recently, however, simulations have demonstrated that a magnetosonic shock wave propagating

and three velocity components), relativistic,

obliquely to a magnetic field can produce ultra-

electromagnetic, particle simulation code with full ion

relativistic electrons [16,171. In this accelerationand electron dynamics. The shock wave propagates inmechanism, large electric potential formed in the shockwave plays an important role,

the x direction in an external magnetic field Be in the (_x," Corresponding author's e-mail: bessho @ plab.phys. nagoya-u.ac.jp)u5@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchBessho N. et al.,I Estimate of Electron Energies Accelerated in an Oblique Shock Wave

awt=2O0100#* son400240100PeyA)6 80

260xl(clrop")
16050

lTLeC

120%n40100Pez
f/leCo" 780

501600150

Te120
J6 so1oo

400500Fig. 2 Phase space plots of electrons, lx, y.l, at various

750

7 10times.

x/( @pr)Fig. 1 Phase space plots ofshows the position x =their maximum values.

trons. The dotted line
at which B, and qtakei401c)sxy) plane; for the definiteness, we assume that B* > 0and B.s > 0. The grid size is 4,Q96; the total numbers ofparticles are N" = Ni = 262,144; the mass ratio is milm" =20ll
II0-20 --f-" +-r-{-Tf"r----t--100; the frequency ratio al""/f)'" is 3 in the upstream
region; and the velocity ratios afe vslc = 0.3, va"lc = 0.1,and vlilc = 0.01. The shock flropagation angle is 0 =j-4-----r----+----ilr

IiIv

45o, where tan9 = BrolB,<t.

n phase space plots (x,Figure I displays ep"), (x, p"), (x, p",), and (x, y" for a shock wave with apropagation speed vr5 = 2.26v ; here, p" is the electron
ntz factor. The dotted
momentum and 7" is the

800
400
Ap"tion where the magnetic
vertical line represents thefield 8., transverse electric eld .E", and the electric
potential g take their maximu values; we denote thissome electrons haveposition by "r-. At this locati> 1OO. Figure 2 shows
very high energies such thatvarious times. Somephase space plots (r, %)end of the main pulse
electrons are reflected nearion; their energies arepositive;diintheand movethe maximum. It thenincreasing. At x = x^, T.

Fig. 3 Time variations of y and y of an accelerated

electron.
decreases rapidly and the electrons move back to thepoint where they were reflected. They repeat thisoscillatory motion. Figure 3 shows time variations of y506

Bessho N. et al., Theotetical Estimate of Electron Energies Accelerated

and yofan accelerated electron. These quantities as wellin an Oblique Shock Wave

can be quite large, where AE3 is defined asr are oscillating after the first reflection. Near point E
where the electron is at x = x., both y and
Ttake theiras

LE3=-eEyo(tt - t)

maximum values.

(5)

Next, we show a theoretical expression for themaximum energy that electrons can obtain and thendiscuss another important process in the acceleration
mechanism. From the energy equation for an electron

3. Physical Considerations
We now discuss the mechanism of the acceleration.
We show in Fig. 4 a schematic diagram of the trajectory

particle,of electron guiding center drawn in the wave frame.
Here, the reflection takes place at point D; the dotted^""'#--eE,vu-eErov*line shows the orbit of a passing electron. As an electron
moves from point A to C, it moves in the negative ydirection because of the E, x Bz drift. It gains kinetic
energy A.E1 from the electric potentialLE,=erplyr)-eeQ) QO).

(> 0).

(6)and from the calculations of wave structure, we find themaximum electron energy as,-^ _eF^* mrTac2 - m rc(v,n Bro / cB,o)y*v*o
mc'1l - (v Bro I cB,o) (8,^/ B^)) ' Q)"(1)"nwhere the subscript 0 refers to the values in the far

At the same time, it loses energy A-82 because of theupstream region. The quantity F- is the maximum value

electric field E eof F, which is defined as

LEz=-eEro(tc - t)(< 0).

Q)

F=-The net change in the energy is therefore

LE- LEt+ LE2.

(8)

with E1 being the electric field parallel to the magnetic
field. The quantity F takes its maximum value F. at x =xr. An electron reflected near the end of the main pulse
can have this amount of energy at the location where B.
takes its maximum value.
Equation (7) indicates that yen can be extremely(3)

Even though the magnitudes of AE1 and A,E2 are quite
large, they almost cancel when an electron moves withdrift approximation; in particular, in a perpendicularpulse they have exactly the same magnitude and M= 0.
However, if an electron is reflected and moves from D
to E, then it would gain energies from both E, and En6.
As a result, the increase in energy,

6g= L,E, + LE.. ,[ q,hlarge when the denominator

h=m"czfl-(v"nB,o/ cB,o)(8.-/f.;1,(9)is close to zero. We discuss the physical meaning of ft -(4)0.

The guiding center velocity u, of an electron may
be written as

vs=vd+v,,8/8,(10)where v11 is the velocity along the magnetic field and rois the drift velocity, which can be approximated as
vo= cExB l82,

Erol(1 1)

in the present case. The r component of rn is therefore
given byIv^---acE,n B,"BtB<_,_trtrux

V"-8,"(r2)

The first term on the right-hand side is always negative,

while the second term is positive if u1 > 0. Usually, v11 isnegative in the wave frame [16]. However, reflected
electrons have positive v11 and have positive v*. In thiscase, the particle can move from point D to point E. Ifr_za

Fig. 4 Schematic diagram of electron guidingorbit.

507

Estimate of Electron Energies Accelerated in an Oblique Shock Wave

Bessho N. €t d/..

References

the conditioncEro B,*,B,oB2B

tll S.R. Kane, E.L. Chupp, D.J. Forrest, G.H' Share
t2l K.Koyama, R. Petre, E.V' Gotthelf et al.,Nattteholds (here, vn is approximatedc), then v* becomesquite small. Substitution of thein Eq. (13) yields

tionE6=-vsrBzolc

Y.r' 4and E. Rieger, Astrophys. J. Lett. 300' L95 (1986)'(13)-(London) 378, 255 (1995).

t3l T. Tanimori, Y. Hayami, S. Kamei et aI.,
Astrophys. J. Lett. 497,L25 (1998).

[4] T. Tanimori, K. Sakurazawa,S.A.Dazeley et al',,(14)'- , B B^
This gives h - O; note that , - B^ for a largeBro

Astrophys. J. Lett. 492,L33, (1998).
t5l D. Biskamp and H. Welter, Nucl. Fusion 12,663(r972).

t6l D.W. Forslund, K.B' Quest' J.U. Brackbill and K'

amplitude shock wave. That is, becomes close to zero
when vr, - 0. The condition ft - can be rewritten as

Lee, J. Geophys. Res. 89,2142 (1984).
l7l B. Lembege and J.M. Dawson, Phys. Fluids B I'(15)ccos0-ysh.

1001 (1989).

t8l Y. Ohsawa, Phys. Fluids 2E,2130 (1985).

If v- is small, then it takes very long time for thefrom point D to Point
particle to move from xp to.x6E). Because the y comPonent of 6 is given by,,Ydv-- cE, B'o "B'1

(in the wave frame, theE,= 0), the particle moves in,

t9l Y.Ohsawa, J. Phys. Soc. Jpn. 55' 1047 (1986)'
t10l Y. Ohsawa, J. Phys. Soc. Jpn' 59,2782 (1990).
tl1l R.Z. Sagdeev and V.D. Shapiro, Zh. Eksp' Teor.
Fiz. Pis'ma Red. 17, 387 (1973) UETP Lett. 17'(16)279 (1973)1.

t12l M.A. Lee, V.D. Shapiro and R.Z. Sagdeev, J.

term vanishes because

Geophys. Res. 101, 4777 (1996).

y direction during thist13l M. Toida and Y. Ohsawa, Sol. Phys. 171, 16lperiod by

Ly --,l,',uo"t, ",

dt.

(1997).

(r7)

tl4l K. Maruyama, N. Bessho and Y. Ohsawa, Phys.
Plasmas 5,3257 (1998).

t15l T. Masaki, H. Hasegawa and Y. Ohsawa, Phys.
Plasmas 7, 529 (20OO).

by -eEsLy. It can bequite large if the time (tE - ti i long. Hence, when v,- 0 (or h - 0), reflected el rons can have large

The particle gains energy fromt16l N. Bessho and Y. Ohsawa, Phys. Plasmas 6,3076(1999).

energies.

l17l N. Bessho and Y. Ohsawa, Phys. Plasmas 7,4004(2000).

508