Relativistic MHD Jets

Transcription

1 Formation and Kinematic Properties of Relativistic MHD Jets Nektarios Vlahakis Outline ideal MHD in general semianalytical modeling r-self similarity AGN outflows GRB outflows z-self similarity Crab-like pulsar winds summary meet the observations

2 Ideal Magneto-Hydro-Dynamics How the jet is collimated and accelerated? Need to examine outflows taking into account matter: velocity V, rest density ρ 0, pressure P, specific enthalpy ξc 2 electromagnetic field: E, B ideal MHD equations: Maxwell: B = 0 = E + B c t, B = E c t + 4π c J, E = 4π c J 0 Ohm: E + V c B = 0 mass conservation: (γρ 0) t specific entropy conservation: ( ) momentum: γρ 0 t + V + (γρ 0 V) = 0 ( ) ( P t + V ρ Γ 0 ) = 0 (ξγv) = P + J 0 E + J B c

3 Integration assume axisymmetry ( / φ = 0, E φ = 0) steady state ( / t = 0) introduce the magnetic flux function A (A = const is a poloidal field-streamline) the full set of ideal MHD equations can be partially integrated to yield five fieldline constants: 1 the mass-to-magnetic flux ratio (continuity equation) 2 the field angular velocity (Faraday + Ohm) 3 the specific angular momentum ( ˆφ component of momentum equation) 4 the total energy-to-mass flux ratio (momentum equation along V) 5 the adiabat (entropy equation) two integrals remain to be performed, involving the Bernoulli and transfield force-balance boundary conditions?

4 r self-similarity If the boundary conditions on the conical disk surface θ = θ i are power laws: B r = C 1 r F 2, B φ = C 2 r F 2, V r /c = C 3, V θ /c = C 4, V φ /c = C 5, ρ 0 = C 6 r 2(F 2), P = C 7 r 2(F 2), then the variables r, θ are separable and the system reduces to ODEs. The solution should cross the Alfvén and the modified fast singular points. [Blandford & Payne (nonrelativistic) Li, Chiueh, & Begelman (1992) and Contopoulos (1994) (cold) Vlahakis & Königl (2003, astro-ph/ , ) (including thermal/radiation effects)] F (the only parameter of the model) controls the current distribution: I ϖb φ r F 1 F > 1: current-carrying jet (near the rotation axis) F < 1: return-current (possibly at large ϖ)

5 ( ' ?:@? A <:=< >.0/21 axis of rotation BDC!#"$ 8:98 ; % & )*+,- disk

6 AGN outflows (Vlahakis & Königl in preparation) (modeling the sub-pc-scale jet in NGC 6251) VLBI measurements show sub-pc scale acceleration of the radio-jet in NGC 6251, from V (r = 0.53pc) = 0.13c to V (r = 1pc) = 0.42c [Sudou, H., et al. 2000, PASJ, 52, 989] Adopting the best fit model of Melia et al. 2002, ApJ, 567, 811 (consistent with the limits set by Jones et al. 1986, ApJ, 305, 684) and assuming n r 2 we find temperature: T = [ r(pc) sound speed: C s c = specific enthalpy: ξ = /3 o 0.026] K [ ] 2/3 r(pc) [ ] 4/3 r(pc) Thus, for 0.53pc< r < 1pc the flow is supersonic and the quantity ξγ 1 changes from at r = 0.53pc to at r = 1pc. As for hydrodynamic flows ξγ 1 = const., the conclusion is that the flow is not hydrodynamically accelerated. We propose the magnetic acceleration as a plausible explanation of the observations.

7 ϖωβ φ /Ψ Α c 2 =(Poynting flux)/(mass flux)c 2 z(pc) inner poloidal fieldline outer poloidal fieldline ξγ=(enthalpy+kinetic)/(mass flux)c ϖ(pc) 10 0 γ=lorentz factor ϖ/ϖ A 10 1 Β φ (G) 10 1 Β z (G) magnetic Β ϖ (G) (dyn cm 3 ) centrifugal 10 7 pressure ϖ/ϖ A ϖ/ϖ A

8 GRB outflows (including time dependence, e ±, radiation) 10 4 EB φ /(4πγρ 0 cv p )=(Poynting flux)/(mass flux)c γ 10 4 ρο /100 g cm 3 kt/m e c 2 B p /10 14 G 10 1 ξ=(enthalpy)/(rest mass)c ϖω/c z(cm) ϖ=ϖ i slow Alfven classical fast τ ± =1 γ=ξ i ϖ=ϖ τ= τ τ τ M/10 7 M s ϖ 1 ϖ ϖ(cm) ϖ 2 3 ϖ 4 ϖ5 ϖ 6 ϖ 7 ϖ sp r out /r in ϖ 1 < ϖ < ϖ 6 : Thermal acceleration - force free magnetic field (γ ϖ, ρ 0 ϖ 3, T ϖ 1, ϖb φ = const, parabolic shape of fieldlines: z ϖ 2 ) ϖ 6 < ϖ < ϖ 8 : Magnetic acceleration (γ ϖ, ρ 0 ϖ 3 ) ϖ = ϖ 8 : cylindrical regime - equipartition γ ( EB φ /4πγρ 0 V p )

9 Collimation Acceleration The flow is centrifugally accelerated for V φ V p V p c 2. Thermal acceleration is important for γ ξ i. For γ ξ i, ξ 1 and the magnetic acceleration takes over. How efficient is the magnetic acceleration? (σ =?) For F > 1 the flow reaches asymptotically a rough equipartition between kinetic and Poynting fluxes (σ 1). The Lorentz force is capable of collimating the flow reaching cylindrical asymptotics (the collimation is possible for γ a few 10, following γ 2 ϖ R). For F < 1, the acceleration is more efficient. The collimation is not so strong and the flow eventually approaches conical asymptotics. Is the 100% acceleration efficiency possible (σ = 0)? Super-Alfvénic asymptotic solutions show that it is!

10 Crab-like pulsar winds Integrate transfield force-balance equation under the z self-similar ansatz z = F 1 (A)F 2 (ϖ). 1.4e+18 G z(cm) e+15 σ 1.2e+18 (a) 8e+14 (b) 1e+18 z(cm) 8e+17 6e+17 ( ϖb φ ) 4 Φ=4 6e+14 4e+14 4e+17 2e+17 ( ϖb φ ) 3 2e+14 ( ϖb φ ) ( ϖb φ ) 1 2 Φ=1 0 Φ= e e e e e+14 ϖ(cm) ϖ(cm) J < 0 J > 0

11 Summary of the previous results The shape is determined close to the source (J < 0) Collimation is possible The acceleration continues at larger distances (J > 0) The magnetic acceleration is eficient r self-similar: does not cover both (J 0) cases (F > 1 is preferable) Alternatives: z self similar (captuers both cases) θ self-similar: applies to thermally driven flows near the axis (inside the light cylinder) Fully numerical studies

12 Meet the observations boundary conditions on the disk line shape z = z(ϖ), B, T, ξ(e ± or e p +?), V C s, V, B, ρ 0, P V φ = ϖω, size (M BH ), Ṁ as function of distance along each fieldline bulk flow synchrotron emission (knowing B in space) positions of the shocks γ 2 c t (knowing γ) (Source variability t?) final value of σ (asymptotic B B in shocks) polarizarion asymptotic width opening angle

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