Radiation from accelerated particles in relativistic jets with shocks, shear-flow, and reconnection

We have investigated particle acceleration and shock structure associated with an unmagnetized relativistic jet propagating into an unmagnetized plasma. Strong magnetic fields generated in the trailing jet shock lead to transverse deflection and acceleration of the electrons. We have self-consistently calculated the radiation from the electrons accelerated in the turbulent magnetic fields. We find that the synthetic spectra depend on the bulk Lorentz factor of the jet, the jet temperature, and the strength of the magnetic fields generated in the shock. We have also begun study of electron acceleration in the strong magnetic fields generated by kinetic shear (Kelvin-Helmholtz) instabilities. Our calculated spectra should lead to a better understanding of the complex time evolution and/or spectral structure from gamma-ray bursts, relativistic jets, and supernova remnants.


INTRODUCTION
Recent kinetic simulations have focused on magnetic field generation via electromagnetic plasma instabilities in unmagnetized flows without velocity shears.Three-dimensional (3D) particle-in-cell (PIC) simulations of Weibel turbulence [Nishikawa et al. 2005[Nishikawa et al. , 2009a] ] have demonstrated subequipartition magnetic field generation.We have calculated, selfconsistently, the radiation from electrons accelerated in the turbulent magnetic fields.We found that the synthetic spectra depend on the Lorentz factor of the jet, the jet's thermal temperature, and the strength of the generated magnetic fields [Nishikawa et al. 2011[Nishikawa et al. , 2012]].
Velocity shears also must be considered when studying particle acceleration scenarios, since these trigger the kinetic Kelvin-Helmholtz instability (KKHI).In particular the KKHI has been shown to lead to par-ticle acceleration and magnetic field amplification in relativistic shear flows [Alves et al. 2012;Liang et al. 2012].Furthermore, a shear flow upstream of a shock can lead to density inhomogeneities via the MHD Kelvin-Helmholtz instability (KHI) which may provide important scattering sites for particle acceleration.

THE STANDARD RADIATION MODEL
A synchrotron shock model has been widely adopted as describing the radiation mechanism thought responsible for observed broad-band GRB afterglows [e.g., Zhang & Meszaros 2004;Piran 2005a;Zhang 2007, Nakar 2007].Due to the lack of a first principles theory of collisionless shocks, a purely phenomenological approach to the model of afterglow radiation has been prescribed.Firstly, electrons are as-eConf C121028 Figure 1: Figure 1a shows the spectra for the cases of γ = 10, 20, 50, 100, and 300 with cold (thin lines) and warm (thick lines) electron jets.Figure 1b shows modeled Fermi spectra in νFν units at early (a) to late (e) times [Abdo et al. 2009].The red lines indicate slope in νFν ∼ 1 sumed to be "Fermi" accelerated at relativistic shocks and to have a power-law distribution with index p, where N (E e )dE e ∝ E −p dE e .Secondly, a fraction ξ e (generally taken to be < ∼ 1) of the electrons are assumed to be accelerated, and the total electron energy is assumed to be a fraction ǫ e of the total internal energy in the shocked region.Thirdly, the strength of the magnetic field in the shocked region is unknown, but its energy density (B 2 /8π) is assumed to be a fraction ǫ B of the internal energy.These assumed "microphysics" parameters, p, ξ e , ǫ e and ǫ B , whose values are inferred from the spectral fits to the observations [e.g., Panaitescu & Kumar 2001;Yost et al. 2003], reflect a lack of knowledge of the underlying microphysics [Waxman 2006].It is our intent to place these parameters on a firm physical basis.

SELF-CONSISTENT RADIATION CALCULATION FROM PIC SIMULATIONS
Electrons are accelerated in the electromagnetic fields generated by the Weibel and kinetic Kelvin-Helmholtz instabilities.Radiation can be calculated using the particle trajectories in the self-consistent turbulent magnetic fields.This calculation includes Jitter radiation [Medvedev 2000[Medvedev , 2006] ] which is different from standard synchrotron emission.Radiation from electrons in our simulations is reported in Nishikawa et al. [2011].
We have calculated the radiation spectra directly from our simulations by integrating the expression for the retarded power, derived from Liénard-Wiechert potentials for a large number of representative particles in the PIC representation of the plasma [Jackson 1999;Hededal 2005;Nishikawa et al. 2008aNishikawa et al. ,b, 2009bNishikawa et al. , 2010Nishikawa et al. , 2011;;Martins et al. 2009;Sironi & Spitkovsky 2009;Frederiksen et al. 2010].Initially we verified the technique by calculating radiation from electrons propagating in a uniform parallel magnetic field [Nihikawa et al. 2009b].It should be noted that spectra obtained from colliding jet simulations (fixed contact discontinuity) do not provide spectra in the observer's rest frame, and cannot be compared with observed spectra [Sironi & Spitkovsky 2009b].
The spectra shown in Figure 1a are for emission from jets with Lorentz factors γ = 10, 20, 50, 100, and 300 [Nishikawa et al. 2011[Nishikawa et al. , 2012]].In the figure we show two spectra for each Lorentz factor (represented by the same color line) for initially cold (v jet,th = 0.01c) (thin, lower lines) and initially warm (v jet,th = 0.1c) (thick, upper lines) jet electrons.Here the spectra are calculated for emission along the jet axis (θ = 0 • ).The radiation shows a Bremsstrahlunglike spectrum at low frequencies for the eleven cases (Hededal 2005) because the magnetic fields generated by the Weibel instability are rather weak and jet electron acceleration is modest.A low frequency slope of νF ν = 1 is indicated by the straight red lines.The low frequency slopes in our synthetic spectra are very similar to those of the spectra in Fig. 1b

Initial Conditions and Previous Results
Recently the kinetic KH instability (KKHI) has been investigated using a relativistic counterstreaming velocity shear setup with γ 0 = 3 [Alves et al. 2012].In this RPIC simulation the velocity shear occurs at the edges of a velocity field with v 0 pointing in the positive x (x 1 ) direction in the middle of the simulation box, with upper and lower quarters of the simulation box containing a velocity field with v 0 pointing in the negative x 1 direction as indicated by the arrows in Fig. 2c.Initially, the system was charge and current neutral.The simulation box dimensions were 250 × 80 × 80(c/ω p ) 3 , where ω p = (4πn 2 e /m e ) 1/2 is the plasma frequency, with a resolution of 4 cells per c/ω p .Periodic boundary conditions were applied in all directions.Figure 2c shows our simulation model where the sheath plasma can be stationary or moving in the same direction as the jet core.
Figure 2a shows the magnetic field line topology for this relativistic case.Alves et al. found that KKHI modulations are less noticeable in the relativistic regime because they are masked by a strong DC magnetic field component (negligible in the subrelativistic regime) with a magnitude greater than the AC component,.As the amplitude of the KKHI modulations grows the electrons from one flow cross the shear-surfaces and enter the counter-streaming flow.In their simulations the protons being heavier (m p /m e = 1836) are unperturbed.DC current sheets which point in the direction of the proton velocity form around the shear-surfaces.These DC current sheets induce a DC component in the magnetic field shown in Fig. 2b.The DC magnetic field is dominant in the relativistic scenario because a higher DC current is set up by the crossing of electrons with a larger initial flow velocity and also because the growth rate of the AC dynamics is lower by γ 3/2 0 compared with a subrelativistic case.It is very important to note that this DC magnetic field is not captured in MHD [e.g., Zhang et al. 2009) or fluid theories because it results from intrinsically kinetic phenomena.Furthermore, since the DC field is stronger than the AC field, a kinetic treatment is clearly required in order to fully capture the field structure generated in unmagnetized or weakly magnetized relativistic flows with velocity shear.This characteristic field structure will also lead to a distinct radiation signature [Sironi & Spitkovsky 2009b;Martins et al. 2009;Frederiksen et al. 2010;Nishikawa et al. 2011Nishikawa et al. , 2012]].

Our New Core-Sheath Jet KKHI Results
The simulation setup for our study of velocity shears (not counter-streaming shear flows as used by Alves et al. [2012] and Liang et al. [2012]) is shown in Fig. 2c.In our simulation a relativistic jet plasma is surrounded by a sheath plasma.This setup is similar to the setup of our RMHD simulations [Mizuno et al. 2007].In our initial simulation the jet core has v core = 0.9978c (γ = 15) pointing in the positive x direction in the middle of the simulation box as in Alves et al. 2012.Unlike Alves et al. the upper and lower quarter of the simulation box contain a stationary, v sheath = 0, sheath plasma.Our setup allows for motion of the sheath plasma in the positive x direction.
Overall, this structure is similar in spirit, although not in scale, to that proposed for active galactic nuclei (AGN) relativistic jet cores surrounded by a slower moving sheath, and is also relevant to gamma-ray burst (GRB) jets.In particular, we note that this structure is also relevant to the jet-in-a-jet or needles in a jet scenarios [Giannios et al. 2009, and papers therein], which have been invoked to provide smaller scale high speed structures within a much larger more slowly moving AGN jet.Similar smaller scale structures within GRB jets are also conceivable.
This more realistic setup is different from the initial conditions used by the previous simulations with counter-steaming flows Alves et al. [2012], and hence allows us to compute synthetic spectra in the observer frame.As mentioned by Alves et al. [2012], in a non-counterstreaming or unequal density counterstreaming setup the growing kinetic KHI will propagate with the flow.For GRB jets, the relativistic jet core will have much higher density relative to the ex-eConf C121028   3b shows By (red), Bx (black), and Bz (blue) at x = 500∆ and y = 100∆.Figure 3c shows the x component of the electric current (jet flow is out of the page).The current is positive on the core side and negative on the sheath side of the velocity shear.The positive current is stronger than the negative current, leading to By as shown in Figure 3b.The arrows show the magnetic field in the plane.
ternal medium.On the other hand, for an AGN jet the relativistic core is less dense than the surrounding sheath.
We have performed a simulation using a system with (L x , L y , L z ) = (1005∆, 205∆, 205∆) and with an ion to electron mass ratio of m i /m e = 20.Figure 3 shows the magnetic field structures generated by the relativistic electron-ion core with γ = 15 and with a stationary sheath plasma at time t = 70 ω −1 pe .Figure 3a shows the magnitude of B y plotted in the y − z plane (jet flow is out of the page) at the center of the simulation box, x = 500∆.Figure 3d shows the magntude of B y in the x − z plane (jet flow in the +xdirection indicated by the arrow) at the center of the simulation box, y = 100∆.Figure 3b shows B y (red), B x (black), and B z (blue) magnetic field components at x = 500∆ and y = 100∆.Figure 3d shows the x component of the current.Relativistic jet flow is out of the page and positive (red) current flows along the jet side, whereas negative (blue) current flows along the sheath side.Positive currents are stronger than the negative currents, leading to B y as shown in Figs.3a and 3d.In forthcoming work we will obtain synthetic spectra from particles accelerated by KKHI as we have done for shock simulations [Nishikawa et al. 2011[Nishikawa et al. , 2012]].

Summary and Discussion
Fig. 4 shows KKHI growth, current filament merger and strong magnetic field generation in our simulation.The structure of KKHI seems similar to that found in counter-streaming simulations.In order to examine growth rates and wave propagation, we have rederived the linear theory to describe our core-sheath jets.
We have extended the analysis presented in Gruzinov (2008) to core-sheath electron-proton plasma flows allowing for different core and sheath densities eConf C121028 n jt and n am , respectively, and core and sheath velocities v jt and v am , respectively.In this analysis the protons are considered to be free-streaming whereas the electron fluid quantities and fields are linearly perturbed.The dispersion relation becomes: where ω p,jt and ω p,am are the plasma frequencies (ω 2 p ≡ 4πne 2 /γ 3 m) of jet and ambient electrons, respectively, k is the wave number parallel to the jet flow, and γ jt and γ am are Lorentz factors of jet and ambient electrons, respectively.Analytic solutions are not available except in the low (ω ≪ ω p and kc ≪ ω p ) and high (ω ≫ ω p and kc ≫ ω p ) frequency and wavenumber limits.Equation (2) provides an analytic solution to the dispersion relation in the small wavenumber limit.Here the real part gives the phase velocity and the imaginary part gives the temporal growth rate and directly shows the dependence of the growth rate on the velocity difference across the shear surface.
in Abdo et al. [2009], and show change with the Lorentz factor like the temporal evolution observed by Fermi as shown in Fig. 1b.Here we have not included radiation losses [Jaroschek et al. 2009; Medvedev & Spitkovsky 2009].

Figure 2 :
Figure2: Figure2ashows magnetic field lines generated in the relativistic shear fromAlves et al. [2012].Figure2bshows the electron density in orange (blue) of the plasma that flows in the positive (negative) x1 direction.In Fig.2bdarker regions in the color map indicate high electron density, whereas lighter regions indicate low electron density.Figure2cshows our simulation model where the sheath plasma can be stationary or moving in the same direction as the jet core.

Figure 3 :
Figure 3: Magnetic field structure generated by a relativistic electron-ion jet core with γ = 15 and stationary sheath plasma at simulation time t = 70 ω −1 pe .The magnitude of By is plotted (a) in the y − z plane (jet flow out of the page) at the center of the simulation box, x = 500∆ (d) in the x − z plane (jet flow in the +x-direction indicated by the arrow) at the center of the simulation box, y = 100∆.Figure3bshows By (red), Bx (black), and Bz (blue) at x = 500∆ and y = 100∆.Figure3cshows the x component of the electric current (jet flow is out of the page).The current is positive on the core side and negative on the sheath side of the velocity shear.The positive current is stronger than the negative current, leading to By as shown in Figure3b.The arrows show the magnetic field in the plane.
Figure 3: Magnetic field structure generated by a relativistic electron-ion jet core with γ = 15 and stationary sheath plasma at simulation time t = 70 ω −1 pe .The magnitude of By is plotted (a) in the y − z plane (jet flow out of the page) at the center of the simulation box, x = 500∆ (d) in the x − z plane (jet flow in the +x-direction indicated by the arrow) at the center of the simulation box, y = 100∆.Figure3bshows By (red), Bx (black), and Bz (blue) at x = 500∆ and y = 100∆.Figure3cshows the x component of the electric current (jet flow is out of the page).The current is positive on the core side and negative on the sheath side of the velocity shear.The positive current is stronger than the negative current, leading to By as shown in Figure3b.The arrows show the magnetic field in the plane.

Figure 4 :
Figure 4: Figure 4 shows the time evolution of current filaments (Jx) in the area denoted by small box in Fig 3c.KKHI starts to grow at t = 30ω −1 pe (a) and current filaments have merged by t = 70ω −1 pe (d).The maximum current density (simulation units) is (a) ±0.105 at t = 30ω −1 pe , (b) ±0.167 at t = 40ω −1 pe , (c) ±0.296 at t = 50ω −1 pe , and (d) ±0.625 at t = 70ω −1 pe .The arrows show the magnetic field (By, Bz) (the length of the arrows are not scaled to the strength of the magnetic fields).