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, 173 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]. 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