24P161-B
Eﬀect of Magnetic-Field Gradient on Positron Acceleration
along the Magnetic Field in an Oblique Shock Wave
斜め衝撃波による陽電子加速に対する外部磁場の非一様性の効果
Takashi IWATA, Seiichi TAKAHASHI, and Yukiharu OHSAWA
岩田崇志, 高橋聖一, 大澤幸治
Department of Physics, Nagoya University, Nagoya 464-8602, Japan
名古屋大学大学院理学研究科素粒子宇宙物理学専攻
The eﬀect of nonuniformity of the external magnetic field B0 on ultrarelativistic positron acceleration along themagnetic field due to shock waves in an electron-positron-ion plasma is studied with use of one-dimensional(one space coordinate and three velocities), fully kinetic, fully electromagnetic, particle simulations with a realmass ratio mi /me = 1836. Three cases are examined on shock waves propagating in the positive x direction: 1)
uniform B0 , 2) ∂Bz0 /∂x > 0, and 3) ∂Bz0 /∂x < 0. For the first case, ultrarelativistic acceleration to γ ∼ 104 isdemonstrated with a simulation with the shock speed vsh close to c cos θ, where θ is the angle between the external
magnetic field and the wave normal. Comparison of the second and third cases shows that a greater number ofhigh-energy positrons are created in the ∂Bz0 /∂x < 0 case than in the ∂Bz0 /∂x > 0 case.
1. Introduction
Shock waves propagating obliquely to an external
magnetic field B0 in an electron-positron-ion plasma canaccelerate positrons to ultrarelativistic energies along themagnetic field [1, 2] with the electric field parallel to themagnetic field E∥ = (E · B)/B [3, 4], with the time rate ofchange of the Lorentz factor γ of an accelerated positron
1 dγ c cos θ (E · B)=,Ω p dt
vsh (B · B0 )(1)

where Ω p is the nonrelativistic positron gyrofrequency, c isthe speed of light, vsh is the propagation speed of the shock
wave, θ is the angle between B0 and the wave normal, E isthe electric field, and B is the total magnetic field. In thesimulation of Ref. [2], positron energies reached γ ∼ 2000.
The acceleration is particularly enhanced whenvsh ∼ c cos θ,(2)in which relativistic particles can move with the shock
wave for long periods of time.
If, however, B0 is not uniform, then not all the region can satisfy the condition (2). This paper investigates
the eﬀect of magnetic-field gradient on the positron acceleration, with use of one-dimensional (one space coordinate and three velocities), fully kinetic, fully electromagnetic, particle simulations with the ion-to-electron mass ratio mi /me = 1836.
2. Simulation Results
Figure 1 shows positron phase spaces and the profiles of Bz near shock fronts, where shock waves propagate in the positive x direction in an external magnetic fieldB0 = [Bx0 , 0, Bz0 (x)]. Three diﬀerent cases are compared:1) uniform B0 , 2) ∂Bz0 /∂x > 0, and 3) ∂Bz0 /∂x < 0.
In the uniform B0 case depicted in the top panel, thepropagation angle is taken to be θ = 43◦ , and the shockpropagation speed is observed to be vsh /(c cos θ) = 1.01;
the condition (2) is therefore met . Positron energies reachγ ∼ 104 by the end of the simulation run, ω pe t = 7000.
Since the acceleration has not been saturated, γ will continue to rise if we perform a longer simulation run.
The second panel shows the increasing Bz0 case(∂Bz0 /∂x > 0), with the scale length of the gradient a =1.5 × 104 c/ω pe . The angle θ becomes 43◦ at the positionx0 = 3600c/ω pe . Although the maximum γ is ∼ 8000,
only a small fraction of positrons have γ greater than 4000.
Since the diﬀerence (vsh − c cos θ) keeps increasing as theshock wave propagates away from the position x0 , all thepositrons eventually move to the downstream region after
entering the shock wave.
In the decreasing Bz0 case (bottom panel), the gradientof Bz0 is negative; however, the magnitude of a and otherparameters are taken to be the same as those in the increasing Bz0 case. Here, the acceleration is strongest when theshock front is near the point x0 and has been finished byω pe t = 7000, with the highest energy close to γ ∼ 104 . After the encounter with the shock wave, positrons move tothe downstream region if vsh > c cos θ and go away aheadof the shock front if vsh < c cos θ. Therefore, positrons
that enter the shock wave in the early phase in whichvsh > c cos θ move to the downstream region, while thepositrons encountering the shock wave after vsh has become smaller than c cos θ are reflected forward from theshock front to the upstream region. As shown in the bottom
panel, therefore, there appears a region where the positron
density is quite low.
Although the magnitudes of the field gradient(|∂Bz0 /∂x|) are the same for the second and third panels
in Fig. 1, the decreasing Bz0 case creates higher energy
positrons than the increasing Bz0 case. This arises because

Fig. 2 Orbit in the (x − vsh t, y) plane of a positron in the increasing Bz0 case. The vertical dashed line indicates the peak
position of Bz . This particle stays in the shock transition
region for ω pe t ≃ 2200, after which it goes out to thedownstream region.

Fig. 1 Snapshots of positron phase spaces (x, γ) near shock
fronts. The solid lines show the profiles of Bz . In the top
panel with uniform B0 , which is plotted at ω pe t = 7000,
the highest energy reaches γ ∼ 104 , and the acceleration has not been saturated till the end of the simulation
run. In the second panel with ∂Bz0 /∂x > 0 plotted atω pe t = 6500, the number of positrons with γ > 4000 arequite small. Particles that enter the shock wave eventually move to the downstream region. In the bottom panel
with ∂Bz0 /∂x < 0 at ω pe t = 7000, we find many high energy positrons with γ > 5000 near the shock front, behind
which there is a region of low positron density.

in the former case, positrons that encounter the shock wave
when the diﬀerence (vsh − c cos θ) is about to change frompositive to negative values tend to penetrate deep into theshock wave and thus spend long periods of time in theshock transition region before going out to the upstream
region.
We plot in Fig. 2 the orbit of a positron in the increasing Bz0 case in the (x − vsh t, y) plane. The verticaldashed line represents the position at which Bz takes its
maximum value. After entering the shock transition region, this positron stays there for ω pe t ≃ 2200 and thenmoves to the downstream region.
The orbit of a positron in the decreasing Bz0 case isdepicted in Fig. 3. This particle penetrates deep into theshock transition region and stays in the shock wave for aperiod ω pe t ≃ 4000, after which it is reflected forward tothe upstream region.
3. Summary
With use of one-dimensional, fully kinetic, fully electromagnetic particle simulations, we have studied positronFig. 3 Orbit in the (x − vsh t, y) plane of a positron in the decreasing Bz0 case. After entering the shock transition regionand staying there for rather a long period of time, thisparticle goes back to the upstream region.

acceleration along the magnetic field caused by shockwaves in an electron-positron-ion plasma, with particularattention to the eﬀect of the nonuniformity of the external
magnetic field.
First, we have demonstrated positron acceleration toγ ∼ 104 in a uniform B0 . Since the acceleration has notbeen saturated by the end of the simulation run, positronenergies would continue to rise in a longer simulation. We
have then compared the cases with increasing and decreasing external magnetic fields and found that the positron acceleration is more enhanced in the latter than in the former.
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