COLLISION-INDUCED MAGNETIC RECONNECTION AND A UNIFIED INTERPRETATION OF POLARIZATION PROPERTIES OF GRBs AND BLAZARS

Gamma-ray bursts (GRBs) are the most luminous explosions in the universe, which mark ultra-relativistic jets launched from newborn stellar mass black holes that beam toward Earth (Kumar & Zhang 2015). Blazars are moderately relativistic jets launched from supermassive black holes toward Earth (Ghisellini 2015). The energy composition of these relativistic jets, which is still in debate, is a fundamental property that defines the subsequent energy dissipation process, particle acceleration mechanism, and radiation mechanism, which are directly connected to the observational properties such as light curves, spectra, and polarization characteristics. An important parameter to classify relativistic jets is the magnetization parameter, $\sigma ={E}_{{\rm{em}}}/h$, where $h=\rho {c}^{2}+\hat{\gamma }P/(\hat{\gamma }-1)$ is the specific enthalpy defined in the fluid rest frame, ρ is the rest mass density, P is the gas pressure, $\hat{\gamma }$ is the adiabatic index, and E_em is the EMF energy density calculated by ${E}_{{\rm{em}}}=({{\boldsymbol{B}}}_{{\bf{0}}}^{2}+{{\boldsymbol{E}}}_{{\bf{0}}}^{2})/8\pi$ in the fluid rest frame of each simulation cell. A jet is matter-flux-dominated when σ < 1 (Rees & Meszaros 1994; Kobayashi et al. 1997; Daigne & Mochkovitch 1998; Mészáros & Rees 2000) and is Poynting-flux-dominated (PFD) when σ > 1 (Zhang & Yan 2011; Blandford 1976; Lovelace 1976; Colgate & Li 2004; Giannios et al. 2009).

Recently, evidence suggesting that at least some GRBs that may be produced through magnetic dissipation within moderate PFD jets have been accumulated, including the weak or the lack of bright thermal emission in the spectra of most GRBs (Zhang & Pe'er 2009), linearly polarized emission with high polarization degree (PD) in both GRB prompt emission (Yonetoku et al. 2011) and early optical afterglow (Mundell et al. 2013), as well as the lack of associated high-energy neutrinos with GRBs (Zhang & Kumar 2013; Aartsen et al. 2015). In the blazar field, polarized optical emission has been commonly observed (Abdo et al. 2010; Blinov et al. 2015). A connection between blazar jets and GRB jets has been established through several observational correlations (Nemmen et al. 2012; Zhang et al. 2013; Wu et al. 2016). Theoretically, a possible connection between jets in different scales has been proposed (Mirabel 2004; Zhang 2007), but no detailed studies have been carried out to show that the same physical model can account for the observational data in both GRBs and blazars.

Within the context of GRBs, Zhang & Yan (2011) suggested that collision-induced magnetic reconnection and turbulence (ICMART) can be an efficient mechanism to dissipate energy in a moderate PFD outflow, and the model can overcome some difficulties in the traditional internal shock models and account for many GRB observations. Recently, we (Deng et al. 2015) carried out the first detailed numerical study on collision-induced magnetic dissipation, and demonstrated that such processes can indeed dissipate magnetic energy with a high efficiency, and that local, relativistically boosted regions due to high-σ reconnections can indeed form, which mimic the so-called "mini-jets" invoked in interpreting the rapid variabilities in GRBs (Narayan & Kumar 2009; Zhang & Yan 2011) and blazars (Giannios et al. 2009). It is then encouraging to extend our numerical model to study radiation from the jets and directly compare the model results with observations.

In this Letter, we study radiation properties of jets based on our 3D relativistic MHD numerical simulations of the collision-induced magnetic dissipation model (Deng et al. 2015), paying special attention to the polarization properties of the jets. The observational motivations include the detection of a high-degree of linear polarization as well as a 90° polarization angle (PA) change between two major flares in the prompt γ-ray emission phase of GRB 100826A (Yonetoku et al. 2011) and the detections of a 180° PA swing during strong flaring emission in some blazar sources, especially 3C279 (Abdo et al. 2010; Blinov et al. 2015). Several models (Marscher et al. 2008; Lundman et al. 2014; Marscher 2014; Zhang et al. 2015) have been proposed to interpret these phenomena within the GRB context or blazar context, respectively, but these models introduce very different assumptions, so that there is no unified understanding to the phenomena.

2.1. MHD Code Introduction

The 3D special relativistic MHD (SRMHD) code, which we are using here, is named "LA-COMPASS" (Li & Li 2003), which was first developed at Los Alamos National Laboratory. It uses the higher-order Godunov-type finite-volume methods to solve the ideal MHD equations as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\partial ({\rm{\Gamma }}\rho )}{\partial t}+{\rm{\nabla }}\cdot ({\rm{\Gamma }}\rho {\boldsymbol{V}})=0,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\displaystyle \frac{{{\rm{\Gamma }}}^{2}h}{{c}^{2}}{\boldsymbol{V}}+\displaystyle \frac{{\boldsymbol{E}}\times {\boldsymbol{B}}}{4\pi c}\right)+{\rm{\nabla }}\cdot \left[\displaystyle \frac{{{\rm{\Gamma }}}^{2}h}{{c}^{2}}{\boldsymbol{V}}\otimes {\boldsymbol{V}}\right.\\ &&\quad +\left.\left(p+\displaystyle \frac{{B}^{2}+{E}^{2}}{8\pi }\right){\boldsymbol{I}}-\displaystyle \frac{{\boldsymbol{E}}\otimes {\boldsymbol{E}}+{\boldsymbol{B}}\otimes {\boldsymbol{B}}}{4\pi }\right]=0,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left({{\rm{\Gamma }}}^{2}h-p-{\rm{\Gamma }}\rho {c}^{2}+\displaystyle \frac{{B}^{2}+{E}^{2}}{8\pi }\right)\\ &&\quad +{\rm{\nabla }}\cdot \left[({{\rm{\Gamma }}}^{2}h-{\rm{\Gamma }}\rho {c}^{2}){\boldsymbol{V}}+\displaystyle \frac{c}{4\pi }{\boldsymbol{E}}\times {\boldsymbol{B}}\right]=0,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}+c{\rm{\nabla }}\times {\boldsymbol{E}}=0,\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\boldsymbol{E}}=-\displaystyle \frac{{\boldsymbol{V}}}{c}\times {\boldsymbol{B}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&p=(\hat{\gamma }-1)u,\end{eqnarray} \tag{ 6 }$$

where ${\boldsymbol{B}}$, ${\boldsymbol{E}}$, and ${\boldsymbol{V}}$ are the vectors of magnetic field, electric field, and fluid velocity, respectively, and Γ and u are the Lorentz factor and internal energy density, respectively (see more in Deng et al. 2015).

2.2. Problem Setup

We set up the two-blob collision problem similar to Deng et al. (2015). We consider that the black hole central engine of a GRB or a blazar launches a PFD jet with both toroidal and poloidal magnetic field components. The jets are episodic, probably due to variations of the accretion rate, intrinsic episodic magnetic activities from the accretion disk (Yuan & Zhang 2012), or current-driven kink instabilities during jet propagation (Guan et al. 2014; Mizuno et al. 2014), forming discrete high-σ blobs with both toroidal and poloidal field components (Li et al. 2006). The collisions between the two blobs lead to discharge of magnetic energy through reconnection (Zhang & Yan 2011; Deng et al. 2015), which lead to efficient particle acceleration (Guo et al. 2014) and synchrotron radiation. We simulate the collisions in the jet comoving frame, which is the center of mass frame of the two blobs. The magnetic field configuration of each magnetic blob is initialized using the model from Li et al. (2006). We introduce the equations in the cylindrical coordinates $(r,\phi ,z)$ first and then transfer them to the Cartesian coordinates in our simulations. From the center (r = 0) of each blob, the field is assumed to be axisymmetric with the r- and z-components of the poloidal field:

$$\begin{eqnarray}&&{B}_{r}=-\displaystyle \frac{1}{r}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial z}=2{B}_{0}\displaystyle \frac{{zr}}{{r}_{0}^{2}}\mathrm{exp}\left(-\displaystyle \frac{{r}^{2}+{z}^{2}}{{r}_{0}^{2}}\right),\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&{B}_{z}=\displaystyle \frac{1}{r}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial r}=2{B}_{0}\left(1-\displaystyle \frac{{r}^{2}}{{r}_{0}^{2}}\right)\mathrm{exp}\left(-\displaystyle \frac{{r}^{2}+{z}^{2}}{{r}_{0}^{2}}\right),\end{eqnarray} \tag{ 8 }$$

where, B₀ and r₀ are the normalization factor for the magnetic strength and the characteristic radius of the magnetic blob, respectively. The poloidal field is closed, which keeps the net global poloidal flux zero. The toroidal field configuration is motivated by the consideration of a rapidly rotating central engine, whose spin shears the poloidal flux to form the toroidal flux. It has the form

$$\begin{eqnarray}&&{B}_{\phi }=\displaystyle \frac{\alpha {\rm{\Phi }}}{{r}_{0}r}={B}_{0}\;\alpha \displaystyle \frac{r}{{r}_{0}}\mathrm{exp}\left(-\displaystyle \frac{{r}^{2}+{z}^{2}}{{r}_{0}^{2}}\right)\;,\end{eqnarray} \tag{ 9 }$$

where the parameter α controls the toroidal-to-poloidal flux ratio. In our simulation, we set α = 3, which suggests that the two flux components are roughly equal to each other (Li et al. 2006).

The initial conditions for the GRB and blazar simulations are presented in Table 1. The simulation box size is 10L₀ × $10{L}_{0}$ × 10L₀ for the GRB case and $10{L}_{0}$ × $10{L}_{0}$ × 20L₀ for the blazar case. The total simulation time is 160t₀ for both cases. We also assume a comoving observer in the x-axis direction and a bulk motion along the z-axis. The bulk Lorentz factor is set to 300 for the GRB case and 20 for the blazar case, respectively. From the MHD simulations, we can roughly estimate the observed luminosity assuming that the radiation is isotropic in the jet comoving frame. For the GRB case, the average observed luminosity from 0 to 100 s is ${\bar{L}}_{{\rm{\Omega }}}$ = ${{\rm{\Gamma }}}^{4}\cdot {dP}^{\prime} /d{\rm{\Omega }}^{\prime}$ ≈ ${{\rm{\Gamma }}}^{4}\cdot 5\times {10}^{40}$ erg s⁻¹ ≈ 4 × 10⁵⁰ erg s⁻¹; for the blazar case, the average observed luminosity from 0 to 20 days is ${\bar{L}}_{{\rm{\Omega }}}$ = ${{\rm{\Gamma }}}^{4}\cdot {dP}^{\prime} /d{\rm{\Omega }}^{\prime}$ ≈ ${{\rm{\Gamma }}}^{4}\cdot 6.5\times {10}^{40}$ erg s⁻¹ ≈ 10⁴⁶ erg s⁻¹. Both values are typical for the respective systems.

Table 1.  The Parameters of Our MHD Simulations in the Simulation (Jet Comoving) Frame

Case	GRB 100826A		Blazar 3C279
Parameter	Code	Physical	Code	Physical
σc	6	6	2	2
$\bar{B}$	0.25	750 (G)	0.35	0.35 (G)
$| {V}_{z}|$	0.3	0.3c	0.8	0.8c
t0	1	375 (s)	1	9.5 × 105 (s)
L0	1	1.13 × 1013 (cm)	1	2.86 × 1016 (cm)
xs	0.01	1.13 × 1011 (cm)	1	2.86 × 1016 (cm)
r0	2	2.25 × 1013 (cm)	2.5	7.15 × 1016 (cm)
α	3	3	3	3
P	10−2	9 × 104 (erg cm−3)	10−2	10−2 (erg cm−3)
ρbkg	10−1	10−15 (g cm−3)	10−1	1.1 × 10−22 (g cm−3)

Note. For each parameter, we give the normalization relationship between the dimensionless code units and the physical units. Figure 1 shows the model geometry. Here, σ_c is the blob σ value before the collision; $\bar{B}$ is the typical average magnetic field strength in the reconnection region; $| {V}_{z}|$ is the magnitude of the initial velocity of each blob; α is the toroidal-to-poloidal flux ratio factor introduced in Equation (9); t₀ and L₀ are the time and length normalization factors; x_s is the initial misalignment between the center of the two blobs in the x direction; r₀ is the initial radius of each blob before collision; P is the uniform initial thermal pressure in the entire simulation box, and ρ_bkg is the uniform initial mass density outside the blobs. The density inside the blob is determined by σ.

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We use ${P}_{m}=-{\boldsymbol{J}}\cdot {\boldsymbol{E}}$ as a normalization reference to inject relativistic particles⁴ , where ${\boldsymbol{J}}$ is the current density and ${\boldsymbol{E}}$ is the electric field strength. The negative value of P_m implies magnetic energy dissipation. Figure 1 shows two cuts of the P_m evolution and the corresponding zoom-in magnetic field line configuration for the GRB and blazar cases, respectively. The collision-induced magnetic reconnection happens in the contact area of the two blobs around the middle plane (z = 0), where the magnetic energy dissipation mainly concentrates. Based on the distribution of the negative P_m value, we define a reconnection region (inner black rectangle box) that is the major magnetic energy release region to produce strong flaring emission for GRB prompt emission and blazar flares. We also define a background region (outer black rectangle boxes) that contributes to the background emission. The magnetic field topology in the strong dissipation region evolves significantly between the two snapshots for both cases.

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In each time step and each simulation cell with a negative P_m value, we inject non-thermal electrons with certain power-law distributions as follows:

$$\begin{eqnarray}n({\gamma }_{e})=\left\{\begin{array}{ll}{n}_{0}{({\gamma }_{e}/{\gamma }_{m})}^{-1}, & {10}^{2}\lt {\gamma }_{e}\lt {\gamma }_{m},\\ {n}_{0}{({\gamma }_{e}/{\gamma }_{m})}^{-3.2}, & {\gamma }_{m}\lt {\gamma }_{e}\lt {10}^{6},\end{array}\right.\end{eqnarray} \tag{ 10 }$$

for the GRB jet (Golenetskii et al. 2010; Uhm & Zhang 2014), and

$$\begin{eqnarray}&&n({\gamma }_{e})={n}_{0}{\gamma }_{e}^{-2},\quad \quad {10}^{3}\lt {\gamma }_{e}\lt 5\times {10}^{4},\end{eqnarray} \tag{ 11 }$$

for the blazar jet (Böttcher et al. 2013). In the reconnection region, the normalization factor n₀ is calculated with the assumption that the injected particle energy rate is about one-half of the P_m values. In the background region, on the other hand, n₀ is set to be uniform and normalized to the background emission level from the observations. The minimum Lorentz factor ${\gamma }_{m}$ is set to 1.4 × 10⁴ for the GRB case⁵ to roughly match the observed peak energy (E_p) of GRB 100826A (Golenetskii et al. 2010).

Next, we use a 3D multi-zone polarization-dependent ray-tracing radiation code (Zhang et al. 2014) named "3DPol" to calculate the detailed radiation and polarization evolution properties based on the MHD simulations. The "3DPol" code takes magnetic field and particle population as inputs and calculates Stokes parameters at each spatial point. By adding up the polarized signals that arrive at the observer at the same time, the time-dependent PD and PA can be calculated. This code performs full 3D calculations of the polarization properties by taking into account detailed radiation and polarization transfer, light travel time effects, and time, space, and frequency dependencies.

Our results and the referenced observations are shown in Figures 2 and 3 for the GRB and blazar cases, respectively. For the GRB case, the simulated light curve shows multiple global pulses (corresponding to the "slow" component of GRB emission; Gao et al. 2012), which is similar to the observations. The simulated PA on average rotates by 90° between the two intervals of the light curves, which matches the observation quite well. In addition, the calculated PD also shows an increasing trend between the two intervals, consistent with the observations. For the blazar case, our results also match the observations well. In particular, the following three key features are reproduced. 1) The PA shows a 180° angle swing. 2) The PD nearly drops to zero around the flare's peak time (around t ∼ 10 days). 3) In the light curve, the background level after the flare is much lower than it was before the flare.

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Our results can be understood in terms of the magnetic field configuration evolution during the collision processes. Since we assume that the observer views the system from the x-axis in the jet comoving frame (the 1/Γ cone direction in the observer frame), only the radiation power from the B_z and B_y components contribute to the observed emission. The PA is roughly controlled by the dominant component of these two components, and the PD is related to the relative ratio between them. Since we inject the same energy distribution of non-thermal electrons in each cell, the synchrotron radiation powers of the two components in a certain region are related to ${\sum }_{i=1}^{n}{n}_{0}(i)\ast {B}_{y}{(i)}^{2}$ and ${\sum }_{i=1}^{n}{n}_{0}(i)\ast {B}_{z}{(i)}^{2}$, respectively, where n is the total number of cells in this region, with n₀ introduced in Equations (10) and (11).

Figure 4 shows the evolution of the above two quantities for both cases. The final results should be controlled by the superposition (solid lines) from both the reconnection (dashed–dotted lines) and background (dotted lines) regions. For the GRB case (left panel), the observation (Abdo et al. 2010) shows a low level of the background emission, so we inject a small amount of non-thermal particles in the background region. Therefore, the radiation is dominated by the reconnection region. In this panel, there are two major power release intervals. The first one forms during the initial collision-induced strong magnetic reconnection process. Due to the small misalignment (x_s) and high-σ environment, the blobs bounce back significantly. This process temporally suppresses energy release due to reconnection. After that, the imbalance due to the small misalignment is gradually built up, which triggers the later strong reconnection again to produce the second major flare (see a detailed analysis of this process in Section 4.2 and Figure 12 in Deng et al. 2015). This episodic power release process gives the multiple global pulses. On the other hand, the dominant component in the reconnection region switches from the B_y-dominated radiation to the B_z-dominated radiation between the two intervals, which is mainly due to the magnetic reconnection that reconnects the B_y component to form the B_z component gradually. This is the main reason to cause the 90° PA change as observed.

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For the blazar case (right panel), the observation (Abdo et al. 2010) shows a relatively high level of background emission, so we inject more non-thermal particles in the background region to mimic the observation. Therefore, the superposition of the statistical quantities from the two regions is different from the GRB case. From the light curve point of view, due to the larger misalignment and moderate-σ adopted in the simulation, the bouncing effect is relatively weak and more significant rotation is triggered, so that power release is relatively continuous in the reconnection region that gives rise to a single major flare in the light curve. The systematic decay of the magnetic field strength during the reconnection dissipation process causes the continuous decay of the background power release, which can explain the observed significantly lower level of the background emission after the flare than before. From the PA point of view, during the flare, the reconnection changes the dominant radiation component from B_y to B_z gradually. However, at the late stage of the flare, B_z domination becomes weaker and is even surpassed by the B_y component again. This is mainly due to the relatively faster rotation caused by the larger misalignment, which rotates the system more than 45° at the late stage. So B_z becomes less dominating in the rotation comoving frame, or even turns into B_y domination for the fixed observer in the simulation frame. This double switching of the dominant component in the reconnection region causes the 180° PA swing. From the PD point of view, the background contribution which is mainly B_y dominant for this case is relatively high, so that the B_z dominated emission from the reconnection region around the peak stage of the flare is nearly balanced by the B_y dominated background emission. Therefore, the PD nearly drops to zero around the flare's peak time (around t ∼ 10 days). We note that besides this representative case, many other blazars show significant PA swing along with relatively strong flare(s) (Blinov et al. 2015), which can be also potentially interpreted with our model.

In reality, different GRB and blazar events may have different parameters (e.g., σ, α, x_s, etc.), or even different global magnetic field structures. Based on our study, we find that as long as the change of magnetic field topology is mainly due to the magnetic reconnection process within a single global reconnection location, the dominant magnetic field component would switch and give a significant (e.g., 90°) PA change. For more complicated cases involving multiple blobs colliding and triggering many global reconnection events consecutively, the polarization evolution would be more complicated.

In summary, using MHD simulations, we are able to evolve the magnetic field topology of the GRB and blazer systems self-consistently by fully taking into account the collision-induced reconnection and rotation effect due to misalignment. The unified interpretation of the polarization properties of both GRBs and blazars suggest that similar underlying physics is in operation in black hole relativistic jet systems. In particular, collision-induced magnetic reconnection may hold the key to understanding a variety of phenomenology.

We thank the referee Pawan Kumar for helpful suggestions. This work is supported by NASA through grants NNX15AK85G and NNX14AF85G and by the LANL/LDRD program and Institutional Computing Programs at LANL and by DOE/Office of Fusion Energy Science through CMSO. We are grateful to Dr. Yonetoku for sharing the data with us. We thank important technical support from Shengtai Li. We also thank helpful discussion and suggestions from Z. Lucas Uhm, Bin-Bin Zhang, Fan Guo, and Xiaocan Li.