MAGNETOHYDRODYNAMIC EFFECTS IN PROPAGATING RELATIVISTIC JETS: REVERSE SHOCK AND MAGNETIC ACCELERATION

Relativistic jets are believed to exist in active galactic nuclei (AGNs), black hole binaries, and gamma-ray bursts (GRBs), but their composition is still poorly understood. It has been argued that magnetic fields could play an important dynamic role in these jets (e.g., Lovelace 1976; Blandford 1976; Blandford & Znajek 1977; Blandford & Payne 1982; Usov 1992; Thompson 1994; Mészáros & Rees 1997; Lyutikov & Blandford 2003; Vlahakis & Königl 2003, 2004), but the degree of magnetization, quantified by the magnetization parameter σ (the ratio of electromagnetic to kinetic energy flux), is poorly constrained by observations. GRB afterglow modeling indicates that the ejecta are more magnetized than the ambient medium, suggesting a possibly important dynamic role for magnetic fields in GRB jets (Fan et al. 2002; Zhang et al. 2003; Kumar & Panaitescu 2003; Gomboc et al. 2008).

A useful diagnostic for the degree of jet magnetization can be obtained from the interaction between the decelerating jet and the ambient medium. Added magnetic field pressure in the jet alters the condition for the formation of a reverse shock (RS) as well as its strength (Kennel & Coroniti 1984). Analytical studies of the deceleration of a GRB fireball with arbitrary magnetization (Zhang & Kobayashi 2005, hereafter ZK05; see also Fan et al. 2004 for σ ⩽ 1) suggest novel behavior that does not exist in pure hydrodynamic (HD, σ = 0) models (Sari & Piran 1995; Kobayashi et al. 1999). However, consensus on the conditions required for the existence of the RS or how Poynting flux is transferred to kinetic flux in the interaction region has not yet been achieved (ZK05; Lyutikov 2006; Giannios et al. 2008). We present a one-dimensional study of the interaction between a magnetized relativistic flow and a static, unmagnetized external medium. A Riemann problem is solved both analytically and numerically over a broad range of σ.

We consider a Riemann problem consisting of two uniform initial states (left and right) with discontinuous HD properties specified by the rest-mass density ρ, gas pressure p, specific internal energy u, specific enthalpy h ≡ 1 + u/ρc² + p/ρc², and normal velocity v^N. The right state (the medium external to the jet) is assumed to be a cold fluid with constant density, at rest. Specifically, we select the initial conditions: ρ_R = 1.0ρ₀, p_R = 10⁻²ρ₀c², v^N_R = v^x_R = 0.0, where ρ₀ is an arbitrary normalization constant (our simulations are scale-free) and c is the speed of light. The left state (the propagating relativistic flow) is assumed to have a higher density and pressure than the right state, as well as a relativistic velocity. Specifically, ρ_L = 10²ρ₀, p_L = 1.0ρ₀c², and v^N_L = v^x_L = 0.995c (γ_L ≃ 10). The fluid is described by an adiabatic equation of state p ∝ ρ^Γ with Γ = 4/3.

To investigate the effects of magnetic fields, we consider a perpendicular field component in the jet with B^y = 31.623, 100.0, 316.23, and 447.21 in units of (4πρ₀c²)^1/2 measured in the laboratory frame, corresponding to σ ≡ B²/4πγ²hρc² ≃ B²/4πγ²ρc² being 0.1, 1.0, 10.0, and 20.0, respectively. This field is motivated by the predicted toroidal field domination at the deceleration radius for GRB outflows (e.g., Spruit et al. 2001; ZK05). Increasing σ increases the total (kinetic plus magnetic) energy density of the left (jet) state.

We calculate exact solutions of this problem, using the code of Giacomazzo & Rezzolla (2006), in the region 0.8 ⩽ x ⩽ 1.2 with an initial discontinuity at x = 1.0, where x is in arbitrary units.

3.1. Flow–Medium Interaction

The exact solutions are presented in Figure 1. The four panels display profiles of the gas density, gas (and magnetic) pressure, magnetic field strength B^y, and the Lorentz factor at time t = 0.16.⁷ Different colors represent different σ values: 0.1 (black), 1.0 (red), 2.7 (yellow), 10.0 (green), and 20.0 (blue). The initial Lorentz factor of the left state (jet) is γ_L = 10.

Zoom In Zoom Out Reset image size

For σ = 0.1 (black), the solution shows a right-moving fast shock (FS: forward shock; S_→) and a left-moving fast shock (RS: reverse shock; _←S) relative to the contact discontinuity (C). In the laboratory frame, the contact discontinuity and the two shocks move to the right.

For σ = 1.0 (red), the solution shows similar profiles (_←SCS_→) as for σ = 0.1. The FS is stronger (due to a higher jump in pressure) and slower (more deceleration relative to the frame of the contact discontinuity), while the RS is weaker but faster. These features are expected from analytical work (ZK05; Giannios et al. 2008), and agree with one-dimensional relativistic magnetohydrodynamic (MHD) simulations (Mimica et al. 2007, 2008).

When the magnetization of the flow exceeds σ = 2.7, the shock profiles change drastically (the significance of this particular value of σ is discussed below). For σ = 10.0 (green) and 20.0 (blue), a prominent left-going rarefaction wave (_←R) is observed, instead of a left-going shock (see also Romero et al. 2005; Mimica et al. 2007). When the rarefaction wave propagates into the jet flow, density and gas pressure decrease, and the flow velocity increases. The terminal Lorentz factor of the left (jet) state and the FS region reaches γ ∼ 14 for σ = 10 and γ>16 for σ = 20. This magnetic acceleration mechanism stems from the magnetic pressure in the flow.⁸

This magnetic acceleration mechanism is solely an MHD effect and requires the magnetic field to generate a rarefaction wave. This is different from the HD/MHD boost mechanism proposed by Aloy & Rezzolla (2006), and further investigated by Mizuno et al. (2008) and Aloy & Mimica (2008). The HD/MHD mechanism is a purely relativistic mechanism, which invokes a relativistic flow perpendicular to the propagation direction of the rarefaction wave. The mechanism discussed here occurs even in the Newtonian case, and acts parallel to the propagation direction of the rarefaction wave. In general, the acceleration efficiency is smaller than that of the HD/MHD boost mechanism (see Section 3.3 for more discussions).

3.2. Conditions for Reverse Shock or Magnetic Acceleration

The magnetic pressure profiles (dotted lines) in Figure 1(b) reveal the physical condition required for the transition from a reverse shock to a rarefaction wave. It is evident that the magnetic pressure increases as σ increases. In the reverse shock cases (σ = 0.1, 1) the upstream magnetic pressure is lower than the gas pressure in the forward shock, while in the rarefaction wave cases (σ = 10, 20) the upstream magnetic pressure exceeds the gas pressure in the FS. Thus, the balance between the upstream magnetic pressure in the unshocked flow region and the FS gas pressure in the shocked medium (ZK05; Romero et al. 2005) provides the condition separating the two regimes. This condition can be derived analytically (see also ZK05). For the interaction between a relativistic flow and an external medium, there exist four physically distinct regions: (1) unshocked medium, (2) shocked medium, (3) shocked flow, and (4) unshocked flow. Hereafter, Q_i denotes the value of a quantity "Q" in region "i". From the relativistic shock jump conditions with Γ = 4/3, one can write u₂/ρ₂c² = (γ₂ − 1) ≃ γ₂ and ρ₂/ρ₁ = 4γ₂ + 3 ≃ 4γ₂. A constant speed across the contact discontinuity requires γ₂ = γ₃, and the relation between the gas pressure and the internal energy gives p₂ = u₂/3. Thus, the thermal pressure generated in the FS region is p₂ = (1/3)(γ₂ − 1)(4γ₂ + 3)ρ₁c² ≃ (4/3)γ²₂ρ₁c². The pressure balance condition is B²₄/8πγ²₄ ∼ p₂, because in pressure balance there is no reverse shock or rarefaction wave, so that regions 4 and 3 are matched as B₄ ≃ B₃ and γ₄ ≃ γ₃ = γ₂. Using the definition σ ≡ σ₄ = B²₄/4πγ²₄ρ₄c², one can derive a critical σ_c value:

$$\begin{equation} \sigma _c = \frac{2}{3}(\gamma _4-1)(4\gamma _4+3) \frac{\rho _1}{\rho _4} \simeq \frac{8}{3}\gamma _4^2 \frac{\rho _1}{\rho _4}= \frac{8}{3}\gamma _L^2 \frac{\rho _R}{\rho _L}. \end{equation} \tag{ 1 }$$

The condition for the existence of a reverse shock is σ < σ_c, which is Equation (31) of ZK05. The condition for a rarefaction wave and magnetic acceleration is σ>σ_c. We adopted ρ₁ = ρ_R = 1.0, ρ₄ = ρ_L = 10², and γ₄ = γ_L = 10.0, so that the critical value is σ_c ≃ 2.7. Our calculations indicate that σ_c marks the transition point where neither a reverse shock nor a rarefaction wave is established (yellow lines in Figure 1). To verify this for a larger parameter space, we investigate the σ dependences of various quantities in detail. Figure 2(a) shows the gas pressure in the region through which the reverse shock/rarefaction wave has propagated. Initial Lorentz factors are γ_L = 5, 10, and 20, respectively. For all cases, we fix the flow density at ρ_L = 10² and increase B (hence σ). The total initial energy density of the flow increases with σ. In all cases, the gas pressure decreases with σ smoothly without a sharp transition from the RS regime (solid lines) to the reverse rarefaction wave regime (dotted lines). The critical magnetization parameters are σ_c ≃ 0.7, 2.7, 10.6 for γ_L = 5, 10, 20, respectively, derived from the analytical solution of Equation (1). We note that in the RS regime, the strength of the shock decreases rapidly with increasing σ. The critical magnetization parameter σ_c increases with γ_L, so that an RS can exist in the high-σ regime if γ_L is sufficiently large (see also ZK05).

Zoom In Zoom Out Reset image size

Another commonly invoked RS condition states that the shock speed in the fluid frame (region 4) is higher than the speed of the Alfvén wave, i.e. γ'_RS,RR > γ'_A, where γ'_RS,RR = (γ₄/γ_RR,RS + γ_RR,RS/γ₄)/2, γ'_A = (1 + σ)^1/2, and γ_RS,RR is the Lorentz factor of the reverse shock or reverse rarefaction wave in the laboratory frame, which can be calculated using the exact solution of Giacomazzo & Rezzolla (2006). Giannios et al. (2008) claim that this condition is different from the pressure balance condition. However, in Figures 2(b) and (c), we present the ratios of gas pressure in the FS region to magnetic pressure in the flow, p_FS/p_B, and γ'_RS,RR/γ'_A, and find that both ratios reach unity at the same critical value σ_c. This suggests that the two reverse shock conditions have the same physical origin, at least for the one-dimensional model studied here.

3.3. Terminal Lorentz Factor and Magnetic Acceleration Efficiency

To better understand the magnetic acceleration mechanism, we plot the Lorentz factor as a function of σ in Figure 2(d). For the magnetic acceleration case, this is the terminal Lorentz factor after acceleration. Because of the dependence of σ_c on γ_L, a higher σ is needed to achieve acceleration for a higher γ_L. The terminal Lorentz factor can be estimated analytically by requiring that the thermal pressure in the FS region balances the magnetic pressure in the region through which the rarefaction wave has propagated. For the terminal Lorentz factor γ_t, this condition can be expressed (roughly) as B²₃/8πγ²_t = (1/3)(γ_t − 1) × (4γ_t + 3)ρ₁c² ≃ (4/3)γ²_tρ₁c². From the definition of σ with B₄ ≃ B₃, this becomes

$$\begin{equation} \gamma _{t} \simeq \left({ 3 \gamma _{4}^{2} \sigma \rho _{4} \over 8 \rho _{1}} \right)^{1/4}. \end{equation} \tag{ 2 }$$

Crosses in Figure 2(d) denote values of terminal Lorentz factors calculated from Equation (2) for model parameters, γ₄ = γ_L = 20, ρ₁ = ρ_R = 1.0, and ρ₄ = ρ_L = 10², in good agreement with the exact solution of the Riemann problem in the reverse rarefaction wave regime.

To investigate the acceleration efficiency, we present in Figure 3 the terminal Lorentz factor γ_t, and its ratio to the initial Lorentz factor (γ_t/γ_L) as a function of the initial flow Lorentz factor γ_L. While a flow with a higher initial Lorentz factor reaches a higher terminal Lorentz factor, a lower initial Lorentz factor implies a higher acceleration efficiency. From Equation (2) it follows that γ_t/γ₄ ≃ (3σρ₄/8ρ₁)^1/4γ^−1/2₄, in good agreement with the exact solution of the Riemann problem in the relativistic regime.

Zoom In Zoom Out Reset image size

We solved the one-dimensional Riemann problem for the deceleration of an arbitrarily magnetized relativistic flow in a static, unmagnetized medium. For the same initial Lorentz factor, the reverse shock becomes progressively weaker with increasing σ, turning into a rarefaction wave when σ ⩾ σ_c, at which point the magnetic pressure in the flow is balanced by the thermal pressure in the forward shock. In the rarefaction wave regime, material in the FS region is accelerated due to the strong magnetic pressure in the flow. This magnetic acceleration mechanism may thus play an important role in the dynamics of strongly magnetized, relativistic flows.

Numerical MHD simulations (e.g., Koide et al. 1999, 2000; Nishikawa et al. 2005; Hardee et al. 2007) are essential to understand magnetized relativistic jets. We performed one-dimensional special relativistic MHD simulations of a relativistic flow propagating in an external medium using the RAISHIN code (Mizuno et al. 2006). The simulation results are in good agreement with the exact solution (Giacomazzo & Rezzolla 2006), serving as a test of the RAISHIN code. As MHD simulations can tackle problems for which an exact solution is not known, we plan to utilize RAISHIN to solve more realistic configurations (e.g., relativistic shells with a finite width and conical geometry, as often envisaged in the GRB problem).

The magnetic acceleration mechanism discussed here also applies in the Newtonian MHD limit. The transition point from a reverse shock to a rarefaction wave is then also given by the pressure balance condition, and the terminal velocity of the flow can be estimated from the Newtonian shock jump condition as $v_{t}= c_{s1} (p_2/p_1 - 1) \sqrt{2/\Gamma /[(\Gamma +1)(p_2/p_1) +(\Gamma -1)]}$, where c_s1 = (Γp₁/ρ₁)^1/2 is the sound speed in the upstream medium (see also Hawley et al. 1984). Defining σ = (B²₄/8π)/(ρ₄v²₄/2) in the Newtonian limit, one can derive the terminal velocity of the flow, v_t, determined by balance between the magnetic pressure and the pressure in the forward shock region, p₂ = B²₃/8π ≃ B²₄/8π = σρ₄v²₄/2. For a strong shock (p₂ ≫ p₁), the terminal velocity can be approximated as $v_{t} \simeq \sqrt{ \sigma (\rho _{4}/\rho _1)/ (\Gamma +1)} v_4$. For v_t = v₄, one derives σ_c ≃ (Γ + 1)ρ₁/ρ₄. For Γ = 5/3, typical for nonrelativistic shocks, this expression is consistent with Equation (1) for γ_L = 1. Although the general physics is the same, the dependence of the terminal velocity (v_t/c)γ_t is different in the Newtonian and relativistic case.

Our results have implications for understanding the deceleration of strongly magnetized outflows, possibly present in GRBs and AGNs. Exact solutions indicate that the condition for the existence of a reverse shock is σ < σ_c, as suggested by ZK05 (see Giannios et al. 2008). The paucity of bright optical flashes in GRBs (e.g., Roming et al. 2006) may, among other interpretations, be attributed to highly magnetized flows. Furthermore, the magnetic acceleration mechanism discussed here suggests that σ and γ are not independent parameters at the deceleration radius. For high-σ flows, the ejecta would experience magnetic acceleration at small radii, before reaching the coasting regime, so that the coasting Lorentz factor (i.e., the "initial" Lorentz factor for the afterglow) is at least the "terminal" Lorentz factor defined by Equation (2). As a result, the high-σ and low-γ part of parameter space is suppressed. This implies that some region in the ξ − σ parameter space⁹ of GRB models (ZK05; Giannios et al. 2008) is suppressed as well. Here we only focus on one-dimensional models with Cartesian geometry. Implications for GRB models will be discussed in more detail when this Riemann problem is solved in the conical jet geometry.

Variable emission (down to minute timescales) observed in some TeV blazars suggests very high Lorentz factors in AGN jets (Aharonian et al. 2007). Models for the production of TeV emission appeal to high Lorentz factor jet cores surrounded by lower Lorentz factor sheaths (e.g., Ghisellini et al. 2005) or rapid jet deceleration (Georganopoulos et al. 2005) in order to reconcile the required jet Lorentz factors with lower Lorentz factors suggested by proper motion studies. Our results suggest the possibility of magnetic acceleration occurring where highly magnetized jet material overtakes more weakly magnetized jet material. In this case the magnetically accelerated Lorentz factor behind the forward shock can significantly exceed the Lorentz factor of the overtaken jet material.

In this study we considered a static, unmagnetized medium but for a weakly magnetized medium our conclusions hold as well. However, the conclusions will not apply in the very large σ regime when the underlying MHD approximation breaks down, e.g., σ∼ several 100s in the problem of GRBs (Spruit et al. 2001; Zhang & Mészáros 2002).

We thank S. Kobayashi, H. Sol, and L. Rezzolla for helpful comments. Y.M. and B.Z. acknowledge NASA NNG05GB67G, NNG05GB68G, and NNX08AE57A for partial support during Y.M.'s stay at UNLV. Y.M., P.H., and K.I.N. acknowledge partial support by NSF AST-0506719, AST-0506666, NASA NNG05GK73G, NNX07AJ88G, and NNX08AG83G. B.G. thanks CSPAR-UAH/NSSTC for hospitality during the preparation of part of this work and acknowledges the DFG grant SFB/Transregio 7 for partial support. S.N. is partially supported by Grants-in Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science and Technology 19047004, 19104006, and 19740139. The simulations were performed on Columbia Supercomputer at NAS Division at NASA Ames Research Center and Altix3700 BX2 at YITP in Kyoto University.