1. Particle Acceleration by Relativistic Collisionless Shocks in Electron-Positron Plasmas Graduate school of science, Osaka University Kentaro Nagata

2. Contents Motivation Observation and theory for the Crab nebula Necessity of particle acceleration processes (not Fermi acceleration) Simulation results for relativistic collisionless shocks in electron-positron plasmas Discussion

3. Pulsar nebula The pulsar nebula is brightened by the energy coming from the central neutron star via pulsar wind. One of the best observed and studied pulsar nebula is the Crab nebula because its age is known (951 years) and observations are relatively easy (~2kpc). The Crab nebula in optical three bands (European Southern Observatory)

4. Pulsar magnetosphere Pulsar radius ~ 10km Magnetic field ~ 10 12 G Gamma rays and a strong magnetic field generate a lot of pair plasma The light cylinder is defined by r L =light speed×pulsar period ~ 1500km. In r<r L, the magnetosphere co- rotates and closes. In r>r L, the magnetosphere doesn’t co-rotate and extends far away. Pulsar magnetosphere (P.Goldreich and W.H.Julian, 1969)

5. Underluminous zone and shock The shock is at ~0.1pc from the pulsar. Pair plasmas are carried along the extended magnetic field. ⇒ pulsar wind magnetic field ~3×10 -4 [G] Lorentz factor ~ 3×10 6 at the shock The region inside the shock is underluminous because the pulsar wind is still cold. The magnetic field is almost toridal because the pulsar rotates with high frequency and the wind is highly relativistic. shock X-ray image of the Crab nebula (Chandra)

6. Nebula The thermalized pulsar wind brights by means of synchrotron radiation. Expanding filament structures suggest that the nebula edge expands with a velocity ~2000km/s at ~2pc from the pulsar. The Crab nebula in optical three bands (European Southern Observatory)

7. A theoretical model (KC model) upstream n 0,u 0 (  0 =u 0 ),B 0,P 0 (=0:cold) downstream n 1,u 1 (  1 =1 ),B 1,P 1 ⇒ 7 parameters Rankine-Hugoniot relations ⇒ 4 equations ⇒ 4 parameters in 1 equation One upstream parameter expresses downstream flow velocity and energy by ultra-relativistic Rankine-Hugoniot relations. upstream pulsar wind Rankine-Hugoniot relations adiabatic expansion underluminous zonenebula downstream remnant expansion of nebula B z0 :magnetic field, n 0 :number density of particles,  0 :Lorentz factor of particles (C.F.Kennel, and F.V.Coroniti, 1984) shock(0.1pc) 2pc 2000km/s

8. A theoretical model (KC model) upstream pulsar wind shock(0.1pc) Rankine-Hugoniot relations adiabatic expansion underluminous zonenebula downstream remnant expansion of nebula (C.F.Kennel, and F.V.Coroniti, 1984) Adiabatic expansion is solved by 1-D spherically symmetrical MHD conservation laws. u(z) : flow velocity z=distance/(shock radius) 2pc 2000km/s flow velocity (km/s) 6000 4000 2000

9. A theoretical model (KC model) Boundary condition flow velocity at nebula edge = nebula expansion velocity ⇒ σ~3×10 -3 “kinetic energy dominant” In the same model, upstream flow energy is decided by comparing the theoretical spectrum with the observation,    ~ 3×10 6. upstream pulsar wind Rankine-Hugoniot relations adiabatic expansion underluminous zonenebula downstream remnant expansion of nebula (C.F.Kennel, and F.V.Coroniti, 1984) flow velocity (km/s) 6000 4000 2000 shock(0.1pc) 2pc 2000km/s

10. Spectrum of the Crab nebula The spectrum shows a non-thermal distribution from optical to low gamma range. There must be particle acceleration processes. Multi-wavelength spectrum of the Crab nebula (Aharonian et al 1998) synchrotron radiation IC γ=10 6 γ=10 9

11. Acceleration mechanism When the magnetic field is nearly perpendicular to the flow, downstream particles can not go back to the upstream region (P.D.Hudson, 1965), and the Fermi acceleration doesn’t work efficiently. Possible other acceleration mechanisms shock surfing acceleration, magnetic reconnection parallel shockperpendicular shock magnetic field upstream downstream upstream downstream Fermi acceleration

12. Shock surfing acceleration 1.Magnetized (B z ) charged particles are injected from upstream. Electric field (E y ) satisfies perfect conductivity. 2.If charged particles are trapped at the shock surface, 3.the “E y ” accelerates these particles along the shock surface. ｚ ｘ ｙ BｚBｚ EｙEｙ shock surface ① charged particles ③ +charge ③ -charge ② EｙEｙ shock surface BｚBｚ 4B ｚ ExEx positron

13. Particle motion and electro-magnetic field are consistently solved by the equation of motion and Maxwell equations respectively. Magnetized cold electron-positron plasma is uniformly injected from the left boundary. Initial electro-magnetic fields (E y,B z ) satisfy perfect conductivity. Particles are reflected at the right boundary in order to make back flow. The simulation scheme ｘ ｙ ｚ BｚBｚ EｙEｙ e +,e - injection reflection M.Hoshino, et al, 1992,ApJ,390,454 : relativistic equation of motion : Maxwell equations

14. Simulation results (  =10 -1 ) γ/γ 0 relativistic Maxwellian log(N) 1 0 1 3 0.6 3 injection wall upstream downstream magnetic field B z /B z0 electron energy   electrostatic field E x /E y0 x/(upstream gyro radius) 0 50 A shock surface is generated. The averaged of fields satisfy the R-H relations. The spectrum is nearly relativistic Maxwellian. shock front

15. Simulation results (  =10 -4 ) 1 0 0 20 30 injection wall upstream downstream magnetic field B z /B z0 electron energy   electrostatic field E x /E y0 x/(upstream gyro radius) 0 shock front 80 20 γ/γ 0 relativistic Maxwellian log(N) nonthermal particles The amplitude of the fields is very large. The averaged fields still satisfies R-H relations. Nonthermal high energy particles appear at the shock front.

16. Particle trajectory and time variation of energy BA space ⇒ time ⇒ energy (γ/γ 0 )→ shock front ~ c/2 (electrostatic field E x ) B A upstream downstream particle A ： not trapped by the shock front and not accelerated particle B ： trapped by the shock front and accelerated The particles are accelerated at the shock front The average of accelerating electric field 〈 E y /E y0 〉 ~1.7 > 0 114

17. Past studies for the shock surfing acceleration in electron-positron plasma M.Hoshino (2001) showed that this phenomenon is characterized by , when  <<1 particle energy spectrum is nonthermal, accelerated particles are trapped at the shock surface, where a magnetic neutral sheet (MNS) exists. M.Hoshino, 2001, Prog. Phys. Sup. 143, 149

18. Trapping by a magnetic neutral sheet BzBz x x y electron magnetic neutral sheet (MNS) Particles are trapped because their gyro motions change its direction through the MNS. All that time, motional electric field E y accelerates the particles along the MNS. In the shock transition region, E y has a finite value. On the other hand, in the down stream E y is almost zero. This is the reason why the particle acceleration occurs only at the shock front. BzBz BzBz

19. Relation between  and MNS x x y BzBz current electron ～ B z0 generated magnetic field (B) MNS The reflected particles begin to gyrate and induce an additional magnetic field. By definition of, when  is small, the initial magnetic field is small and the particle kinetic energy is large. Then initial magnetic field ＜ generated magnetic field, and a MNS is generated. We focus on the shock transition region in case  = 10 -4 << 1.

20. Enlargement of the shock transition region injection wall upstreamdownstream shock transition region velocity u x /u x0 shock front

21. Shock transition region The injected particles go through the shock front and some of them return to the front. Trapping and acceleration occur at the shock front, not the whole shock transition region. u x /u x0 E y /E y0 B z /B z0 accelerated particles MNS thermalize injection shock front

22. ー E y ー B z +The position of accelerated particle space ⇒ time ⇒ upstream downstream Electromagnetic fields around the trapped particle

23. Enlargement of the shock front E y /E y0 B z /B z0 particle energy γ/γ 0 inertia length (c/ω p )× ２

24. The structure of the shock front (in downstream frame) The acceleration by E y is unclear, because E y has a large positive and negative amplitude around the MNS. The MNS propagate upstream with velocity ~ c/2, so we should discuss in the MNS frame (= shock frame). x/ （ gyro radius ） 9.869.96 E y /E y0 B z /B z0 0 60 particle energy γ/γ 0 0 20 steady state Maxwell eqs ⇒ evaluate E y accelerated particles inertia length (c/ω p )× ２

25. Averaged E y ’ is constantly positive in the acceleration region, 〈 E y ’/E y0 〉～ 0 － 7. The averaged E y ’ which accelerate particles in this frame 〈 E y ’/E y0 〉～ 2 ⇒ consistent with the above estimate. the Lorentz transformation of the shock frame x/ （ gyro radius ） 9.869.96 0 20 E y ’/E y0 B z ’/B z0 0 50 particle energy γ/γ 0 accelerated particles inertia length (c/ω p )× ２ The structure of shock front (in shock frame)

26. Summary The acceleration phenomenon is characterized by  which is defined as the ratio of the magnetic field energy density and the particle kinetic energy density. The additional magnetic field induced by gyrating particles overcome the initial magnetic field when  is small, and then a magnetic neutral sheet is generated. Therefore this acceleration process works effectively in the Crab nebula,  ~ 10 -3. Some particles are accelerated by motional electric field E y while the magnetic neutral sheet traps them. The magnetic neutral sheet is kept stationary at the shock front. We confirmed that the averaged E y is positive in the shock frame and that the particle acceleration occurs at the shock frame.

27. To go further The condition of trapping and detrapping for accelerated particles ⇒ Spectrum form (maximum energy, power-low or thermal) 2-D simulation ⇒ Weibel instability Magnetic field consisting of opposing two regions (sector structure) ⇒ coupling with reconnection, etc.

28. Measuring the magnetic field strength The spectrum has a break frequency, ~10 13. The lifetime of the Crab is  ~ 930 yr. The spectrum of the Crab nebula P.L.Marsden, et al Radio IR Optical