Positron acceleration to ultrarelativistic energies by an oblique magnetosonic shock wave in an electron-positron-ion plasma

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I. INTRODUCTION
[8][9][10][11][12][13][14][15][16][17][18] In a recent paper, 19 it has been shown that an oblique magnetosonic shock wave in an electron-positron-ion ͑e-p-i͒ plasma can accelerate positrons to ultrarelativistic energies.During the acceleration, these positrons stay in the shock transition region and move nearly parallel to the external magnetic field B 0 .The energy increase rate was found to be proportional to E • B.
These features are quite different from those in the surfatron acceleration, [20][21][22][23] where the longitudinal electric field E x accelerates particles in the direction parallel to the wave front and perpendicular to the magnetic field.For the unlimited acceleration, E x / B Ͼ 1 is required. 22his paper describes the positron acceleration found in Ref. 19 in more detail.In Sec.II, we outline some properties of oblique magnetohydrodynamic waves in cold e-p-i plasmas.Linear dispersion relations of the three modes in the frequency regime lower than the electron gyrofrequency are shown.Then, some nonlinear properties of the magnetosonic mode, which is one of the three modes, are mentioned.Although the magnetosonic wave has no electric potential in a pure electron-positron ͑e-p͒ plasma, 8,12,17 it can have large electric potential in an e-p-i plasma, which plays an important role in the particle acceleration.
In Sec.III, we theoretically discuss the positron acceleration.Some positrons are reflected along the magnetic field by E ʈ , the electric field parallel to B. Assuming that vd␥ / dt ӷ ␥͉dv / dt͉ for the reflected positrons, where v is the particle velocity and ␥ is the Lorentz factor, we obtain v and d␥ / dt of relativistic positrons.In this solution, v is nearly parallel to the external magnetic field B 0 , and ␥ increases linearly with time.
In Sec.IV, we study the positron acceleration by using a one-dimensional (one space coordinate and three velocity components), relativistic, electromagnetic particle simulation code with full particle dynamics.It is shown that some positrons are reflected and then accelerated in the shock transition region.The observed maximum energy in the simulations is ␥ ϳ 1000.The energy increase rate d␥ / dt is in good agreement with the theoretical prediction.Also, as the theory predicts, the particle motion is nearly parallel to the external magnetic field.We then examine the dependence of this acceleration on some plasma parameters; shock speed, propagation angle, ion-to-electron mass ratio, and positron density.In Sec.V, we summarize our work.

II. MAGNETOHYDRODYNAMIC WAVES IN AN e-p-i PLASMA
Here, we briefly describe magnetohydrodynamic waves in e-p-i plasmas.From the three-fluid model for a cold e-p-i plasma with full Maxwell equations, we obtain linear dispersion relations of oblique waves as with the subscript j = e (electrons), p (positrons), or i (ions), and ⍀ j is the gyrofrequency including the sign of the charge, The subscript 0 refers to equilibrium quantities.Equation (1) gives six oblique waves.These are shown in Fig. 1, where the ion mass and charge were taken to be m i / m e = 1836 and q i = e.The magnetic field strength is ͉⍀ e ͉ / pe = 3.0.The angle between the wave vector k and the external magnetic field B 0 is = 45°.The positron-to-electron density ratio is n p0 / n e0 = 0.02 for the left panel, while it is n p0 / n e0 = 0.9 for the right panel.These pictures indicate that in the low frequency regime such that Շ ͉⍀ e ͉, we have three modes.In the limit of k → ϱ, these waves have resonance frequencies: for the high-frequency mode, mr Ϸ ͉⍀ e ͉cos , ͑5͒ for the magnetosonic mode, and for the Alfvén mode.At k = 0, the high-frequency mode has a cutoff frequency, hf0 Ϸ Z͑ + ͉͒⍀ e ͉/͑1 + Z 2 ͒, ͑7͒ where = n i0 / ͑n e0 + n p0 ͒, = m e / m i , and Z = q i / e.In the long-wavelength region, the dispersion relation of the magnetosonic wave is given as where v A is the Alfvén speed

͑9͒
The dispersion coefficient m is obtained as

͑10͒
As expected from Eq. (8), nonlinear magnetosonic waves are governed by the Korteweg-de Vries equation. 18The soliton width is ϳc / pi except for the vicinity of = 90°.
The magnetosonic wave has no electric potential in a pure e-p plasma. 8,12,17In an e-p-i plasma, however, it can have a large electric potential.The magnitude of the potential in a solitary wave 18 is given as where M is the Alfvén Mach number.For n i0 / n e0 ϳ 10 −2 , the potential is ϳ10 2 times as large as that in an ordinary electron-ion ͑e-i͒ plasma, ϳ2m i v A 2 ͑M −1͒.Large-amplitude magnetosonic waves will propagate as shock waves.

III. POSITRON MOTIONS
In Sec.III A, we describe some properties of positron motions in an oblique shock wave.In Sec.III B, then, we will discuss positron acceleration.The shock wave is supposed to propagate in the x direction with a propagation speed v sh in an external magnetic field B 0 = B 0 ͑cos ,0,sin ͒.

A. Characteristics of positron motions
The width of the transition region ⌬ of an oblique magnetosonic shock wave is of the order of the ion inertial length, ϳc / pi .The gyroradii of nonrelativistic positrons p are thus much smaller than ⌬, while those of ions are comparable to or greater than ⌬.In addition, positrons can make gyromotion many times while they pass through the shock transition region.Hence, the motion of nonrelativistic positrons can be well described with drift approximation.Their guiding center velocity v g may thus be written as where v d is the drift velocity and v ʈ is the velocity parallel to the magnetic field B.
Another important property of positrons is that they can be reflected along the magnetic field by the parallel electric field E ʈ .This occurs because the positron mass is quite small.(Ions can be reflected across the magnetic field. 20,24,25) In fact, from the nonrelativistic equation of motion in the wave frame, we have the following equation:

͑13͒
Here, the subscript w refers to quantities in the wave frame, v wdz is the z component of the drift velocity v wd , and w is the magnetic moment, w = m p v wЌ 2 / ͑2B w ͒ with v wЌ the gyration speed perpendicular to the magnetic field.Also, v rv =−cE wy0 B wz / ͑B x0 B w ͒, and K =−m p v rv 2 / 2. The quantity F w is the integral of E ʈ along B, 26 dx w .͑14͒ Equation (13) indicates that positrons cannot penetrate regions where the values of the right-hand side become negative.Thus, if e͑F w − F w0 ͒ becomes greater than the other terms, positrons will be reflected there.The other terms are of the order of m p v A 2 or m p v Tp 2 (v Tp the thermal speed), whereas eF in a shock wave can exceed m e c 2 (see Ref. 26 or Sec.IV in the present paper).Hence, positron reflection along B can occur in a shock wave.
Reflected positrons would move with the shock wave for long periods of time if the condition v sh ϳ c cos , ͑15͒ is satisfied.Because time averaged particle velocity in the x direction is given as reflected positrons with v ϳ c would not be able to quickly run away ahead of the shock wave with condition (15).

B. Motions of accelerated positrons
We here study the acceleration of reflected positrons and obtain the energy increase rate for such particles.We discuss this in the laboratory frame.
We consider positrons moving with a shock wave; thus, their v x is ͑17͒ We assume that their speeds are very close to c͑␥ ӷ 1͒ and that ͑18͒ Also, we assume that v y is much smaller in magnitude than the other velocity components.Hence, in the relativistic equation of motion, we neglect ␥dv / dt compared with v d␥ / dt, where = x, y, or z.Then, assuming that where ⑀ is a smallness parameter, we obtain the lowest order equations as Here, we have used the relations and which are obtained from Maxwell equations for a stationary wave, where the fields depend only on = x − v sh t.From Eq. (23), we find that these positrons move almost parallel to B 0 , Eliminating v y and v z in Eq. ( 22) with the aid of Eqs.(24)  and (27), we find the time rate of change of ␥ as Here, the particle position and thus E͑͒ and B͑͒ are constant.The particle energy therefore increases linearly with time.The effect of the synchrotron radiation can be neglected in this mechanism if ␥ 2 B Ͻ 10 15 , where B is in Gauss (see the Appendix).
With the aid of Eqs. ( 25) and (26), we can put Eq. (28) into the form We also note that the relation E • B = E • B 0 holds for stationary waves with Eqs. ( 25) and (26).

A. Simulation method and parameters
We now study the positron acceleration in a shock wave by means of a one-dimensional (one space coordinate and three velocity components), relativistic, electromagnetic particle simulation code with full particle dynamics. 27The method to initiate shock waves was described in detail in the preceding papers. 26,28,29That is, the particles near the left boundary of a simulation box initially have rightward momenta.They push the neighboring particles and excite a shock wave, which then evolves in a self-consistent manner.The simulation parameters are as follows.The total system length is L x = 8192⌬ g , where ⌬ g is the grid spacing.The ion-to-electron mass ratio is m i / m e = 100, with m p = m e .(These are rest masses.)The number of electrons is N e = 614, 400, with N e = N i + N p .The initial thermal velocities are v Te / c = v Tp / c = 0.08 and v Ti / c = 0.008.The magnetic field strength is ͉⍀ e ͉ / pe = 3.0 in the upstream region.The electron skin depth is c / pe =4⌬ g .The time step is sufficiently small, pe ⌬t = 0.01, so that ⌬t is much smaller than the electron gyroperiod even in the shock region.As in the theoretical model, the external magnetic field is in the ͑x , z͒ plane, and waves propagate in the x direction.

B. Simulation result
Figure 2 shows profiles of B z in a shock wave at various times.Here, the shock propagation angle is = 42°.The positron density is taken to be n p0 / n e0 = 0.02; thus, v A / c = 0.301.The shock speed is observed to be v sh = 2.42v A ; hence, v sh ϳ c cos is satisfied.The main pulse propagates nearly steadily.Figure 3 displays field profiles at pe t = 800.The fields E y , B z , , and F have similar profiles.The maximum value of F is much greater than m p v A 2 , suggesting that positrons can be reflected along B.
We show in Fig. 4 phase space plots ͑x , ␥͒ of positrons and profiles of B z .Here, some positrons are accelerated up to ␥ ϳ 700.In the shock transition region, positrons are reflected by the large positive F; in the top panel, we find no positrons behind the shock transition region.As these reflected positrons move with the shock wave, they gain ener- gies from the wave.(In the upstream region, positrons have a Maxwellian velocity distribution with a thermal velocity much lower than the speed of light.) Figure 5 shows time variations of ␥ and v of a positron accelerated to ␥ ϳ 700.After the encounter with the shock wave at pe t Ӎ 250, this particle was reflected.Its ␥ linearly increased with time from pe t Ӎ 250 to pe t Ӎ 1600.The energy increase rate is d␥ / d͑ pe t͒ = 0.41, which is close to the theoretical value d␥ / d͑ pe t͒ = 0.38 estimated from Eq. (28).
Here, in the theoretical estimate, we have used simulation data for the field values; ͗E x / B 0 ͘ = 1.14, ͗B y / B 0 ͘ = 1.18, and ͗B z / B 0 ͘ = 2.45, where the brackets denote time averaging.
The thin horizontal lines in the panels of v x and v z represent the velocities v x = v sh and v z = v sh tan , respectively.These pictures indicate that v x Ϸ v sh on average and that the particle moves nearly parallel to the external magnetic field.From the third panel, we find that v y is negative and is small in magnitude for pe t Շ 1800.These are consistent with the theory.
Figure 6 shows the trajectory of this positron in the ͑x − v sh t , y͒ plane.After the reflection, this particle moves in the negative y direction along the shock front with hypotrochoidlike orbit.In the final stage, however, the trajectory changes to epitrochoidlike orbit; at this stage, ␥ shows a stepwise increase as in the incessant acceleration; [30][31][32][33] in Fig. 5, ␥ jumped at pe t Ӎ 1900.
Ions and electrons are also accelerated by this shock wave with other different mechanisms; for the ion and electron acceleration mechanisms, see Refs. 24, 28, and 34-37 and Refs.26, 38, and 39, respectively.Figure 7 shows phase space plots of ions (the upper panel) and of electrons (the lower panel).Some ions are reflected across the magnetic field and are accelerated to ␥ ϳ 5. On the other hand, some electrons gained ultrarelativistic energies, ␥ ϳ 500.
Next, we investigate parameter dependence of the positron acceleration.Figure 8 shows the maximum value of ␥ as a function of the shock speed v sh .The other parameters are the same as those in the above simulation.Here, the dashed vertical line represents the shock speed v sh = c cos .For a fairly wide range of v sh , ␥ exceeds 100.In particular, positron energies become very high, ␥ ϳ 1000, when v sh ϳ c cos .
Figure 9 shows the maximum value of ␥ as a function of the propagation angle .The propagation speeds for these shocks were fixed to be v sh / c Ӎ 0.73; the other parameters were unchanged.For these parameters, ␥ exceeds 100 for 30°Շ Շ 50°.Similarly to Fig. 8, ␥ has a peak near the angle satisfying the relation v sh = c cos ; this angle is indicated by the dashed vertical line.
Figure 10 shows the maximum value of ␥ as a function of the mass ratio m i / m e .Here, v A , v sh , and were fixed to be the same as those in Fig. 2   When the positron density is high, n p0 ϳ n i0 , this acceleration mechanism is less effective.Figure 11 displays positron phase space plots ͑x , ␥͒ at pe t = 2100 for n p0 / n e0 = 0.02, 0.1, and 0.5.Again, v A , v sh , and are the same as those in Fig. 2. For n p0 / n e0 = 0.5, the fraction of accelerated positrons is small, and their maximum energy is rather low, ␥ Շ 300.(The acceleration to ␥ տ 100 in this case is caused by the interaction between E y and gyromotion perpendicular to B. [30][31][32][33] ) That the acceleration along B is weak is due to the fact that, as n p0 rises, the quantity F becomes small and non-stationary; Fig. 12 compares profiles of F for three different values of n p0 / n e0 .

V. SUMMARY
We have studied positron acceleration in an oblique magnetosonic shock wave in an e-p-i plasma.Nonlinear oblique magnetosonic pulses propagate with pulse width ⌬ ϳ c / pi and electric potential e տ 2m i v A 2 ͑M −1͒.From the relativistic equation of motion, then, we have found that positrons can be accelerated in the shock transition region, mov-ing nearly parallel to the external magnetic field.The energy increase rate is proportional to E • B. Then, we have demonstrated this acceleration with a one-dimensional, relativistic, electromagnetic particle simulation code with full particle dynamics.It has been shown that an oblique shock wave reflect some positrons and then accelerate them to ultrarelativistic energies with this mechanism; positrons with ␥ ϳ 1000 have been observed.Further, we have examined the parameter dependence of this acceleration.For fairly wide ranges of v sh and , positrons are accelerated to energies ␥ տ 100; the acceleration is particularly enhanced when v sh ϳ c cos .Also, the acceleration is strong when the positron density is rather low, n p0 / n e0 Շ 0.1.

ACKNOWLEDGMENTS
This work was carried out by the joint research program of the National Institute for Fusion Science and was supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.

APPENDIX: COMPARISON BETWEEN THE ENERGY INCREASE RATE AND RADIATION LOSS RATE
The energy loss rate by the synchrotron radiation from a particle with velocity perpendicular to the magnetic field, v = v Ќ , is given by 40 where B ͑G͒ is the magnetic field strength in Gauss.The synchrotron radiation from a positron accelerated to energy ␥ by the present mechanism will be much smaller than Eq.(A1), because the particle velocity is nearly parallel to B 0 .On the other hand, from Eqs.Assuming that v Ќ ϳ c, v y ϳ −0.1c, v sh ϳ c cos , and tan ϳ 1, we have − d␥/d͑⍀ p t͒ syn d␥/d͑⍀ p t͒ acc ϳ 10 −15 ϫ ␥ 2 B ͑G͒ .͑A4͒ We can neglect the effect of the synchrotron radiation if ␥ 2 B ͑G͒ Ͻ 10 15 .

FIG. 1 .
FIG.1.Dispersion relations for magnetohydrodynamic waves in e-p-i plasmas.The propagation angle is = 45°.The positron densities are n p0 / n e0 = 0.02 and n p0 / n e0 = 0.9 in the left and right panels, respectively.In the low frequency regime, we have three modes: the high-frequency mode (line H), magnetosonic mode (line M), and Alfvén mode (line A).

FIG. 2 .
FIG. 2. Profiles of B z of an oblique shock wave at various times.

FIG. 8 .
FIG.8.The maximum ␥ vs shock speed v sh .The other parameters are the same as those in Fig.2.

3 ͓e 2 /2 v sh tan c 2 ͑
FIG. 10.The maximum ␥ vs mass ratio m i / m e .The propagation angle and Alfvén speed v A are fixed to be = 42°and v A / c = 0.301.