Direct Particle Acceleration
in Astroplasmas
M. Hoshino
Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo,
Tokyo 113-0033 Japan
Abstract. The high energy particle acceleration mechanisms are discussed by focusing on
the direct acceleration in the astrophysical context. We specifically argue that the relativistic
magnetic reconnection and the shock surfing/surfatron processes can efficiently accelerate
charged particles to a relativistic energy, and that those mechanisms may produce a nonthermal, power-law energy spectrum.
INTRODUCTION
The particle acceleration is known to occur in a diverse variety of astrophysical
sites: the nearest-at-hand examples are the terrestrial aurorae whose typical energies of ~ keV, the terrestrial bow shock and magnetosphere of energies of ~ 1 MeV,
and the radiation belt of energy as high as 100 MeV. In the pulsar magnetosphere
and its surrounding synchrotron nebula, TeV 7—rays are observed. Cosmic-ray
particles with energies in excess of 1020 eV have been detected [1].
Charged particles can be accelerated by utilizing several kinds of free energies.
The most common free energy source is bulk kinetic energy in the form of jets and
winds etc. Shock waves play an significant role on the energy conversion from the
bulk kinetic to the thermal/non-thermal energies, and the diffusive/Fermi shock
acceleration has been discussed as one of the key processes producing the highenergy particles [2, 3, 4]. It predicts a power-law energy spectrum with the powerlaw index of 2 that depends weakly on the shock Mach number, and the index
suggested by the theory is believed to be very close to the observational spectra.
The above supports the diffusive/Fermi shock acceleration model as an origin of
cosmic-ray, but the disadvantage is that the acceleration process is slow, and that
it is hard to keep up with energy losses against radiation processes.
Many other different acceleration processes are also known to exist: magnetic reconnection, electrostatic double layer, unipolar induction, shock surfing/surf at ron,
plasma wakefield, and self-focusing etc. In these mechanisms, charged particles
may be accelerated directly to high energy by continuously traveling toward an
electric field direction, and this kind of acceleration process is called as the "direct" acceleration, contrary to the "stochastic" acceleration such as the diffusive
shock acceleration. The direct acceleration has the advantage of being fast, but it
was usually not obvious how to get an efficient non-thermal particle acceleration
with a power-law energy spectrum. In this paper, we argue recent progress on the
CP634, Science of Superstrong Field Inter actions, edited by K. Nakajima and M. Deguchi
© 2002 American Institute of Physics 0-7354-0089-X/02/$ 19.00
169
direct acceleration that may produce a non-thermal, power-law energy spectrum.
Specifically, we discuss two mechanisms in the family of the direct acceleration,
magnetic reconnection and the shock surfing acceleration.
MAGNETIC RECONNECTION
Let us discuss magnetic reconnection as the first example of the direct acceleration.
Many plasma states in universe have magnetic fields whose structure contains the
neutral sheet where the magnetic field polarity changes its direction. Magnetic
reconnection is believed to play a very important role by converting the magnetic
field energy into plasma kinetic energy. The solar corona and the terrestrial plasma
sheet are the best-studied examples of this process. In addition, for astrophysical
objects, the relativistic reconnection, whose Alfven speed becomes almost the
speed of light, has been recently paid the attention. For example, the relativistic
reconnection in the pulsar wind region of neutron stars has been studied [5, 6]. We
discuss the non-thermal particle acceleration in the relativistic reconnection.
Nonthermal Particles in Relativistic Reconnection
Figure 1 shows a snapshot of the non-linear evolution of the relativistic magnetic
reconnection obtained by a two-dimensional, particle-in-cell code in the ( X , Z )
plane [7]. The plasma is assumed to consist of positrons and electrons, and the
modified Harris equilibrium model is used as the initial condition [7, 8]. The
thickness of the initial plasma sheet is set to be the positron/electron gyro-radius.
The magnetic field lines and the density are shown in Figure 1. An X-type neutral
line is formed at the center of the simulation box, and plasmas are ejected from
the X-type region toward both ±X directions. The behavior of the reconnecting
-20
-10
0
10
x/x
20
Density
aio
FIGURE 1. Snapshot of relativistic magnetic reconnection. The solid lines and the vectors
show magnetic field lines and plasma flow vectors, respectively. The density contour represents
the plasma density, which is normalized by the initial density in the plasma sheet.
170
(a) 107
106
t / TA = 80.8
t / TA = 92.4
t/TA=11.5
t / TA = 80.8
| 105
swl
103
102
10
15
20
25
30
10
E/mc2
e/mc2
FIGURE 2. Energy spectra in the reconnection region, (a) Dashed and solid lines indicate
energy spectra over the whole simulation box at t/r^ = 11.5 and 80.6, respectively. Dotted line
shows the partial energy of particles around the X-type region, (b) Energy spectra in the double
logarithmic scales. Note that the spectra are approximated by the power-law spectrum with its
index of 1.
magnetic field is basically same as those studied in non-relativistic MHD and
particle simulations. Note that the outflow speed in this relativistic reconnection
case reaches up to 0.91c.
Figure 2 shows the energy spectra integrated over pitch angle in the reconnection
region. The dashed line in Figure 2a shows the spectrum at I/TA = 11.5 in the early
stage of reconnection, which is well described by a Maxwellian distribution, while
the solid line represents the spectrum at I/TA — 80.8 in the non-linear stage. TA is
the Alfven transit time of the plasma sheet. One can find not only strong plasma
heating but also non-thermal high energy tail. The dotted line represents a partial
energy spectrum obtained from the particles around the X-type region, and we
find that most of the high energy particles are produced around the X-type region.
Figure 2b shows the partial energy spectra around the X-type region in the double
log scales at I/TA = 80.8 and 92.4. Both spectra are well approximated by a powerlaw spectrum N(e) oc e~s with s~I. We also observed that the maximum energy
grows in time, and the energy is described by £max ~ eEoc(t — to), where e, EQ, c
and to are the electric charge, the electric field around the X-type region, the speed
of light, and the onset time of reconnection, respectively.
Acceleration in X-type Region
Let us discuss how the non-thermal power-law energy spectrum can be produced
near the X-type region. First we assume that the particle can be accelerated
by the reconnection electric field Ey, and gain quickly a relativistic energy. The
acceleration rate of the relativistic particle with the energy s can be expressed by,
(i)
171
Secondly we evaluate the life time of particle around the X-type region. Since the
particle motion around the X-type region can be described by a Speiser orbit, we
may assume that the life time of particle can be expressed by the gyro-period of the
relativistic particle. Then the loss rate with which the accelerated particles escape
from the X-type acceleration region can be given by
1 dN(e)
N(e)
dt
eBz me2
'
v
~ ~c "' v~ ~e ~"
m
where TV is the particle number and Bz is the reconnecting magnetic field component.
From the above two equations, we can obtain the energy spectrum as
N(e)(*£~s,
s~Bz/Ey~i.
(3)
Since we discuss the particle acceleration around the X-type region where Bz ~ Ey,
we get the power-law index s ~ 1. The particle trajectory near the X-type region is
very sensitive to the reconnecting magnetic field structure, and it would be required
for further discussion to determine the energy spectrum in details.
SHOCK SURFING ACCELERATION
The next example of the direct acceleration is the shock surfing acceleration. It
is believed that the wave-particle interaction inside the shock front layer where
strong plasma wave/turbulence is expected can provide the non-thermal particle
acceleration as well as the plasma heating/thermalization. As to the shock front
acceleration processes, the shock drift acceleration and the shock surfing acceleration are often discussed. Both mechanisms utilize a motional electric field for
accelerating a charged particle. In the case of the shock surfing, the particle can
be effectively trapped in the shock front region, and its energy would be expected
to reach up to a relativistic energy.
Nonthermal Electrons in High Mach Number Shock
Figure 3 shows a snapshot of the nonlinear stage of a magnetosonic shock front
region, obtained by the one-dimensional, particle-in-cell simulation where both ions
and electrons are treated as particle [9, 10]. In the simulation, a low-entropy, highspeed plasma consisting of electrons and ions is injected from the left boundary
region which travels towards positive x. The downstream right boundary condition
is a wall where particles and waves are reflected. Under the interaction between
the plasma traveling towards positive x and the reflected particles and waves from
the right-hand boundary, the shock wave is produced, and it propagates backward
in the — x direction. The left-hand panel in Figure 3 shows the shock front region
from X/(c/upe) = 150 to 300, while the right-hand panel is its enlarged picture.
172
-5
-10
10
o 8
s
'^\J^^^
-4t
10
-10
-20
——^1^^
^
w
0
-10
-20
^J
225
230
X/(c/«,J
220
240
X/(c/o,pe)
FIGURE 3. High Mach number, magnetosonic shock structure for MA = 32: an overall shock
front region (left), and an enlarged picture in the shock transition region (right).
The leftward and the rightward regions are respectively the shock upstream and
downstream. From the top, the ion phase space diagram in (X, Uix), the electron
phase space diagrams in (X, Uex), and in (X, \Ue\) where \Ue\ is the magnitude
of electron velocity. The bottom three panels are the transverse electric field Ey,
the magnetic field Bz, and the electric field Ex. The plasma four velocity U is
normalized by the upstream flow velocity UQ = ^o/y 1 — (^o/ c ) 2 - The magnetic field,
and the electric field are normalized by the upstream magnetic field BQ and the
upstream motional electric field EQ = V$BQ/C, respectively. The spatial scale is
normalized by the electron inertia length c/ujpe in upstream.
In the shock front region in Figure 3, we can find two ion components in the ion
phase space diagram; one is the cold ion flowing into the shock downstream, the
other is the reflected ion going away from the shock front. A part of the incoming
ions is reflected due to both the polarization electrostatic field and the compressed
magnetic field in shock downstream [11, 12]. The magnetic overshoot structure
can be found around X/(c/ujpe) ~ 248. In addition to such ion dynamics around
this shock transition region, we find the electron hole in the electron phase space
diagram in (X,Uex) around X/(c/upe) ~ 223, and its corresponding electrostatic
solitary wave (ESW) in the electric field Ex. It is understood that the ESW can
be formed by a nonlinear Buneman instability between the reflected ions and the
incoming electrons for a super-critical shock of MA > 3 [9].
Looking at the electron hole structure in the shock transition region in details,
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4
103
I
10°
10 -1
10
10
1
2
3
7/7o
4
5
FIGURE 4. Shock downstream energy spectrum for electrons. Dotted lines indicate
Maxwellian distribution as reference.
we find that some electrons are accelerated and gain a large amount of energy.
After the strong acceleration in the shock transition region, the electrons are
transported downstream. Figure 4 shows the downstream electron energy spectrum.
The energy 7 = Jl + U2. is normalized by the incident electron bulk energy 70 =
1/y 1 — (vo/c)2. The dot-dashed line is the Maxwellian fit for the spectrum as
reference, and the bottom dotted line is the so-called one-count level. We can
clearly find the enhancement of the nonthermal population above the Maxwellian
level over 7/70 > 1.8.
Electron "Shock Surfing Acceleration"
The above electron acceleration can be understood by the shock surfing/surfatron acceleration. Let us quickly review the idea of the shock surfing
[13, 14]. The shock surfing mechanism has been extensively studied for the ion
acceleration [15, 16, 17, 18, 19, 20]. Due to the inertia difference between ions
and electrons flowing into the shock, the polarization electric field normal to the
shock front is formed. An ion having a small velocity normal to the shock front,
i.e., l/2Mv% < e<ps, can be reflected from the shock front to the shock upstream,
where (/>s is the electrostatic shock front potential induced by the inertia difference
between ions and electrons. During the gyro-motion in upstream, ions gain their
energy from the motional electric field parallel to the shock front. If the width of
the shock front potential <j)s is thin, the electric force can overcome the Lorentz
force, and then the multiple reflection can occurs.
What we discuss here is the "electron" shock surfing acceleration [10, 21, 22].
The above "standard" shock surfing acceleration cannot apply for the electron
acceleration, because the electron cannot be reflected from the shock front by
174
EX upstream
solitary wave
FIGURE 5. An illustration of electron shock surfing mechanism under an action of electrostatic solitary wave.
the shock front potential 0S. We propose a new scheme of "electron" shock surfing
acceleration under the action of ESW. Since the electron hole is a positively charged
structure, and an electron can be trapped if raet^/2 < e</>esw, where 0esw is the scalar
potential for the electrostatic solitary wave (ESW). Furthermore, the propagation
velocity of ESW differs from the plasma bulk velocity, and ESW together with the
trapped electrons can stay longer time in the shock transition region. Namely, in
the frame moving with ESW, the convection electric field is not zero. Therefore,
we think that the "shock surfing" mechanism is occurring for electrons.
Figure 5 summarizes our idea of the electron shock surfing mechanism. Top panel
shows an electron's trajectory in the x — y plane. The magnetic field is polarized
perpendicular to the x — y plane, and the plasmas are convecting towards positive
x. The shock upstream is the left-hand side, while the downstream is the righthand side. Bottom panel shows the electric field Ex along the x axis, and ESW in
association with its electron hole in phase space is depicted in the center. Due to the
nature of the electron hole, the electron charge density is slightly lower than the ion
one, and ESW has a bipolar signature with diverging electric field. If an electron
convecting towards the ESW structure is reflected by both the Lorentz force and
the electric field Ex and is trapped inside the ESW structure, it is successively
accelerated towards the negative Ey direction. As increasing the electron's velocity
vy by the shock surfing/surfatron acceleration, it can be de-trapped from ESW
when the Lorentz force evyBz/c becomes larger than the electric force eEx, and then
it is convecting towards downstream and becomes quickly an isotropic, gyrotropic
distribution.
To illustrate the efficiency of the above electron shock surfing mechanism, let us
discuss the x component of the equation of motion,
dPx/dt = e(Ex + VyBz/c},
175
(4)
where Px — meUex is the electron momentum. In our interest, Ex on the righthand side can be replaced by the electrostatic solitary wave (ESW) produced in
the shock transition region. If the electric force of eEesw is larger than the Lorentz
force of (e/c)vyBu, the electrons are trapped and gain their energy. During the
non-adiabatic acceleration phase, the velocity vy increases. If the electron satisfies
the condition of e£esw < (e/c)vyBQ, it can escape from ESW and will be convected
downstream.
The amplitude of ESW may be estimated by equating the wave energy density
to the drift energy density of the incoming electron [23], and we obtain, E^SW/STT ~
m e nV^ 2 a/2, where Vd ~ 2v$ is the relative velocity between the reflected ions and
the incoming electrons. The energy conversion factor a is of order of O(10~1), and
we assume a = 1/4 ~ (rag/raj)1/3 [23,24]. The electric force FE = eEx can be given
by,
*"
m
e
ln\
(5)
and the Lorentz force FB in the shock transition region is,
FB ~ eVyBo/c.
(6)
By equating FE to FB, we obtain the maximum velocity for fy,max as,
% for 2MA^a(me/mi) < 1
for 2MAJa(mJmi} > 1
If MA > (l/2)ymi/(m e a), then FE > FB is always satisfied, and the maximum
speed becomes the speed of light. In this case, the electrons cannot escape from
the ESW region, and the electrons continue to accelerate until they reach the edge
of a global shock structure. The condition of the "unlimited electron acceleration"
for the real mass ratio of mi/me = 1830 is given by MA > 43 ^ 75.
DISCUSSIONS
The relativistic particle acceleration mechanism discussed in this paper may be
applied for more general situations. For simplicity, we discussed the relativistic
reconnection for the electron-positron plasmas, but the similar acceleration would
be possible for the ion-electron plasmas. If VA ~ c, then the ion gyro-radius on
the reconnecting magnetic field becomes comparable to the size of the acceleration
region with E > 5, and we can expect a strong non-thermal ion acceleration as
well.
In the shock surfing acceleration, we discussed the electron acceleration through
the trapping of electron by the electrostatic solitary wave. This process can occur
only for the ion-electron shocks. For a shock wave in the electron-positron plasmas
that can be expected in the high energy wind region such as the Crab nebula, the
electrostatic solitary wave cannot be excited in the shock transition layer due to the
176
symmetry of the electric charge [25, 26]. However, it is discussed that the Alfvenic
structure with a current sheet can be formed in the shock front region, and that the
current sheet plays a key role as a counter part of the electrostatic solitary wave
in the ion-electron plasmas [21]. The formation of a power-law energy spectrum in
the shock surfing is still an open question, but it has been pointed out that shock
surfing can produce a power-law energy spectrum for the ion shock surfing case
[18]. It would be occurring the similar process for the other shock surfing cases.
The non-thermal particle enegization in the direct acceleration process is still
controversial, and we have only begun to investigate the potentially rich consequences of the astroplasma particle acceleration mechanism. It obviously will be
important to understand the relationship between the stochastic acceleration and
the direct acceleration.
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