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Observations of the BL Lac Jet Acceleration/Collimation Region

Observations of the BL Lac Jet Acceleration/Collimation Region. Review of Marscher et al. (2008) and Related Theoretical Papers David Meier (JPL). Outline. Review of Hydrodynamics (waves, causality, wind principles) Introduction to Magneto-Hydrodynamics (waves, causality, jet principles)

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Observations of the BL Lac Jet Acceleration/Collimation Region

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  1. Observations of the BL Lac Jet Acceleration/Collimation Region Review of Marscher et al. (2008) and Related Theoretical Papers David Meier (JPL)

  2. Outline • Review of Hydrodynamics (waves, causality, wind principles) • Introduction to Magneto-Hydrodynamics (waves, causality, jet principles) • Self-similar jet models (cold, warm; slow, fast) • The Marscher et al. paper (results, interpretation, and significance)

  3. HYDRODYNAMICS

  4. Dispersion Relation for Sound Waves Adiabatic Sound Speed Non-dispersive HD (Sound) Waves

  5. HD Causality • Any k is valid, so • Sound waves are isotropic • But they are Doppler shifted in direction of flow when V  0 • a) Subsonic Flow (V < cs): • Points A and B can both affect each other • The entire region is causally connected • b) Supersonic Flow (V > cs): • Point A can affect point B • But, point B cannot affect point A • Information flows downstream only • Mach Cones and Caustics • Mach cone is similar to light cone: divides the sonic past from the sonic future • Caustic is a vector, tangent to the Mach cone, pointing toward sonic future • Mach cones & caustics appear only when V exceeds cs Bogovalov (1994)

  6. HD Winds moving • The velocity in a single streamline in a smoothly-accelerating wind will eventually pass through the Sonic Point, where V = cs • The full set of such streamlines creates a “Sonic Surface” (SS) • Caustics and Mach cones appear (or disappear) at sonic surfaces • In order for the flow across the sonic surface to be “regular”, an implicit “regularity condition” must be satisfied: the numerator of the wind equation also must = 0 there, or rs = GM/2cs2

  7. IDEAL MAGNETOHYDRODYNAMICS

  8. transverse (Alfvén) longitudinal (magneto-acoustic) Full Dispersion Relation Alfvén Velocity Vector Alfvén Dispersion Relation Non-dispersive MHD Waves: 1. Alfven

  9. Magneto-acoustic Dispersion Relation Fast Magneto-acoustic Speed Slow Magneto-acoustic Speed Phase Speeds along Magnetic Field Non-dispersive MHD Waves: 2. Magneto-acoustic

  10. MHD Causality • Phase velocity remarks • Slow magneto-acoustic velocity is • cs along the magnetic field (sound wave!) • Zero normal to the magnetic field • Fast magneto-acoustic velocity is • VA along magnetic field (but still compressive, not Alfven) • cms = (VA2 + cs2)1/2 normal to the magnetic field • Group velocity remarks • Friedrich’s (polar) diagrams used to determine caustics: • Pick fluid velocity point (magnitude & direction) • Draw tangents from branch curve to point • Sub(magneto)sonic velocities produce no tangents, hence no caustics; flow at that speed is fully causally connected • Special branch of the slow mode (the “cusp” wave) • Transmits information BACKWARD • ITS caustics disappear when V < Vc = cs VA / cms

  11. MHD Winds (Linearly Accelerating) • Assume that V || B • A linearly-accelerating MHD wind consists of 5 regions: • V < Vc (A) • Vc < V < VS (B, C) • VS < V < VA (D) • VA < V < VF (E) • VF < V (F, G, H) • There are three sonic surfaces where caustics appear or disappear • Cusp surface (CS) • Slow Magnetosonic Surface (SMS) • Fast Magnetosonic Surface (FMS) • At the Alfven Surface caustics do not appear/disappear • But, they do change sign • The Alfven Surface, therefore, is a “separatrix surface”

  12. MHD Winds (Collimating Jets) • From Bogovalov (1994) • Adding curvature (collimation) to the MHD wind lifts the degeneracy at the SMS & FMS • Each splits into • A magnetosonic surface and • A separatrix surface, where caustics change direction • Separatrix surfaces • Are physical, not mathematical, surfaces • Are generated by the causal nature of MHD • Act as initial hypersurfaces or internal boundaries • Need to have conditions specified on them that propagate throughout the entire flow • There are, therefore, three important separatrix surfaces that determine the nature of an accelerating, collimating jet • The Alfven Surface (AS) • The Slow Magnetosonic Separatrix Surface (SMSS) • The Fast Magnetosonic Separatrix Surface (FMSS) • And there are still three additional and distinct sonic surfaces (CS, SMS, FMS) • However, note this important point: • The FMS is no longer the “horizon”, where information flow is downstream only • The actual magnetosonic horizon in a collimating jet is the FMSS CS

  13. AXISYMETRIC, IDEAL MAGNETOHYDRODYNAMICS

  14. AXISYMMETRIC, STATIONARY, IDEAL MAGNETOHYDRODYNAMICS • Like all conservation laws, MHD is a function of the event point in spacetime (r, θ, ϕ, t) • Full 3-D, time-dependent simulations are the most realistic (Nakamura, Spitkovsky, McKinney, Anninos & Fragile, etc.) • Many have performed 2-D, axisymmetric simulations (∂/∂ϕ = 0), which still afford some realism(r, θ, t) • Time-Independent (stationary; ∂/∂t = 0) MHD studies offer perhaps the best compromise: • Steady-state view of a 2-D, axisymmetric system • Semi-analytic insight into large regions of parameter space • The axisymmetric, stationary equations of ideal MHD are a special and VERY useful set and used for pulsars, jets, black holes, etc. • They have the following properties (not derived here today) …

  15. AXISYMMETRIC, STATIONARY, IDEAL MAGNETOHYDRODYNAMICS (cont.) • If Ω ≠ 0, they produce rotation-driven, outflowing wind (or inflowing accretion) • Along a given magnetic field line, several physical quantities are constant: • Angular velocity of the magnetic field line: Ω = Ωf • The local magnetic flux in a given poloidal area: B dSp • The local mass flux in a given poloidal area: 4π ρ γ V dSp • This leads to an extraordinary result, independent of field strength (Chandrasekhar 1956, Mestel 1961): the poloidal magnetic field and velocity are parallel with the proportionality constant k = 4π ρ γ Vp / Bp • … leading to a closed form for the plasma velocity in terms of the magnetic field V = k B / 4π ρ γ + R Ω eϕ • This is a special case of the “frozen-in field”: in the poloidal plane • Plasma flows along the field and • The field is carried along by the flow • Additional quantities are conserved along B: • Angular momentum per unit mass (including field a.m.) • Total energy (Bernoulli constant) • The adiabatic coefficient KΓ

  16. Table of Important Self-Similar MHD Jet Papers in the Last ¼ Century SELF-SIMILAR, AXISYMMETRIC, STATIONARY, IDEAL MAGNETOHYDRODYNAMICS: The MHD Jet Analogy to the Parker Wind

  17. THE SELF-SIMILAR ASSUMPTION and SELF-SIMILAR MHD JET EQUATIONS • Removes one more degree of freedom, turning the 2-D partial differential equations into 1-D ordinary differential equations • Possible self-similarity assumptions: • Cylindrical Z: presupposes a collimated vertical jet structure • Cylindrical R: useful for accretion disk structure, not jets • Spherical θ: similar to spherical wind (NO collimation) • Spherical r: only choice with equations that allow collimation • Blandford & Payne (1982) chose the latter: • r-self-similarity; θ structure same for every field line • Reduces MHD to only two ordinary differential equations in θ • Standard procedure for deriving any (MHD) wind/jet equation: • Derive the (algebraic) conservation of energy (Bernoulli) equation; then differentiate it to obtain a1 dM / dθ + b1 dψ / dθ = c1 where M is the Alfven Mach numberand ψ is the local magnetic field/velocity angle • Derive another equation that is skew, if not orthogonal, to the differentiated Bernoulli eq. (BP used the Z-component of the momentum equation) to obtain the “cross-field” equation: a2 dM / dθ + b2 dψ / dθ = c2 • Solve for dM / dθ and dψ / dθ to get 2 coupled ordinary differential equations. For example…

  18. THE SELF-SIMILAR ASSUMPTION and SELF-SIMILAR MHD JET EQUATIONS (cont.) dM / dθ = N / D = (c1 b2 - c2 b1 ) / (a1 b2 - a2 b1 ) • Integrate numerically w.r.t θ, applying the regularity condition N = 0 at any θ where D = 0 • Blandford & Payne’s equation was only slightly different: • DNR = 0 at two points: • Alfven “point” (where a single field line crosses the Alfven surface): MNR = ±1 or Vθ= ± Bθ / (4πρ)1/2Vp = ± Bp / (4πρ)1/2 • “Modified Fast Point” (single field line crosses the Fast Magnetosonic Separatrix Surface [FMSS]) Vθ = ± B / (4πρ)1/2 • VERY IMPORTANT: The MFP occurs where the collimation speed toward the axis (Vθ) equals the fast magneto-acoustic speed! • This can occur VERY FAR from the black hole (e.g., 104-5 rg)

  19. Supersonic Solar Wind Sonic Point Sonic Radius Hydrostatic Solar Wind Modified Fast Point (Vθ= -Vfast) Alfven Point (Vjet = VAlfven) Modified Slow Point (Vθ = Vslow) What happens beyond the MFP? Poynting Flux Dominated Vjet = Vfast = (VA2+cS2)1/2(not a singular point) Collimation Shock !! Kinetic Energy Flux Dominated RECAP SO FAR: SELF-SIMILAR JET ACCELERATION AND COLLIMATION THEORY • After launching, jet continues to be accelerated and collimated by the rotating magnetic field • The process is similar to the Parker solar wind • MHD jets have 3 singular points (Blandford & Payne 1982; Vlahakis & Konigl 2004): MSP, AP, MFP SMSS AS FMSS • Side Notes: • FR II jets appear to be Kinetic Energy Flux Dominated  Strong collimation shock disrupted jet in the nucleus at ~MFP (i.e., 104-5 rg) • Some FR I jets appear to be Poynting Flux (magnetically) Dominated  Weak or absent MFP • But, FR I sources are likely to be a very heterogeneous lot

  20. SELF-SIMILAR MHD JET MODELS • Blandford & Payne (1982) summary: • Assumed cold plasma (p = 0), so did not have a Modified Slow Point (SMSS) • Assumed Keplerian rotation at the base of the outflow, so had a specific Ω(R) and Bϕ(R) at base • Assumed final jet was cylindrical, so there never was a true MFP either (i.e., θMFP = 0)! • I.e., the solution had only an Alfven point • Typical model results: • Jet Total Luminosity: LT ≈ 2.4 Ψ2out Ωout / R0,max = 2.4 B2out R30,max Ωout • Jet Mass Loss Rate: ΔM ≈ 0.02 Ψ2out / R30,max Ωout = 0.02 B2out R0,max / Ωout • Jet Torque (A.M. loss):G ≈ 0.51 Ψ2out / R0,max = 0.51 B2out R30,max • Li, Chiueh, & Begelman (1992) summary: • Added relativistic flow; NOTE: self-similarity assumption was NOT compatible with gravity (not even Newtonian gravity); So, gravity is not included in relativistic self-similar MHD • Used a true cross-field equation, instead of Z-component momentum equation • Similar denominator to BP, but with relativistic expressions for Alfven and Fast speeds; also much more complex numerator; ALSO sought cylindrical solutions and ignored the MFP • Obtained Lorentz factors up to  ~ 50 • Typical results similar to BP, but with R0,max replaced withRL,out; that is, scale radius is now the LIGHT CYLINDER radius of the outermost magnetic field line • Jet Total Luminosity:LT ≈ 1.6 Ψ2out Ωout / RL, out = 1.6 B2out R4L, out Ω2out / c (BZ expression) • Jet Mass Loss Rate:ΔM ≈ 0.08 Ψ2out / R3L, out Ωout = 0.08 B2out c / Ω2out

  21. SELF-SIMILAR MHD JET MODELS (cont.) • Vlahakis, Tsinganos, Sauty, & Trussoni (2000) summary: • Assumed warm plasma (0 < p < B2/8π), so did have a Modified Slow Point (SMSS): D  Vθ4 - Vθ2 cms2 + cs2 V2A,θ • Also achieved a true Modified Fast Point (FMSS) • So, the solution had all three singular points • Actually used the polar angle θ as the dependent variable, rather than Z or R • MFP Specific Energy vs. Z Diagram Bogovalov-type Causality Diagram for MHD Jet Model

  22. SELF-SIMILAR MHD JET MODELS (cont.) • Vlahakis & Konigl(2003,2004) summary: • Can be considered to be a combination of Li et al. (1992) and Vlahakis et al. (2000): relativistic AND warm flow, with MSP and MFP • These are the models to use for AGN, microquasars, etc. γ ≈ 40 Model for 3C 345: Specific Energy and Velocity Components vs. R Diagrams

  23. Modified Fast Point (Vθ= -Vfast) Alfven Point (Vjet = VAlfven) What happens beyond the MFP? Poynting Flux Dominated Poynting Flux Dominated Vjet = Vfast = (VA2+cS2)1/2(not a singular point) Collimation Shock !! Kinetic Energy Flux Dominated SELF-SIMILAR MHD JET MODELS: SUMMARY • Basic Model: • Three separatrix surfaces (manifested as modified singular surfaces in the equations) • Three sonic surfaces (cusp, classical slow, classical fast) SMSS (SMP) AS (AP) FMSS (MFP) • What happens after the MFP is a mystery; at least 2 possibilities: • Convergence creates a strong collimation shock, converting magnetic energy to particle energy and jet to kinetically dominated • Flow remains magnetically dominated (bounce instead of shock)

  24. OBSERVING THE JET ACCELERATION & COLLIMATION REGION

  25. QUESTIONS TO TRY TO ANSWER WITH OBSERVATIONS • Is the MHD model at all viable? • Can we detect a helical, well-ordered magnetic field in this region? • Is there any evidence of rapid rotation of the jet plasma or features? • Where does the gamma-ray emission come from? Shocks? Emission deep in the collimation or launching region? • What happens at the MFP and beyond? • Does a strong, field-destroying shock develop? • Or is there a gentler transition, leaving the jet still in a Poynting flux/magnetically-dominated state? • Does this answer depend on the type of source (e.g., FR I, FR II) • NOTES: • FR I jets are expected to collimate slowly  collimation regions 103-5 rg in length or more • M87  1.4 – 140 mas in size • Cen A  0.6 – 60 mas in size • BL Lac  0.006 – 0.6 mas in size (foreshortened) • FR II jets are expected to collimate quickly  collimation regions 10 - 100 rg in length, AND BE FARTHER AWAY • Cyg A  1 – 10 μas in size • 3C 35  0.4 – 4 mas in size

  26. MARSCHER et al. (2008, Nature, 452, 966) • Observed BL Lac with VLBA, UMRAO/Metsahovi, Steward/Crimea, XTE, & ?? (TeV) • 2 γ-ray flares (~2005.82, 2005.88) detected • Outbursts also seen in radio, optical (R-band), & X-ray during γ-ray flaring period • Particularly important results: • γ-ray flaring period occurs during the birth of a new VLBI component • First γ-ray flare occurs simultaneously with a 240 degree polarization rotation in the optical • Optical polarization reaches 15%  VERY strong magnetic field • Unfortunately, 2nd γ-ray flare has no optical polarization data 2005.88 2005.82

  27. MARSCHER et al. (cont.) • Marscher et al.’s interpretation: • Very similar to self-similar jet model picture, but with time-dependent features • The birth of a VLBI component generates a ‘pulse’ that travels along the jet that produces a “Moving emission feature” • Rotating R-band polarization feature is this pulse rotating around in its helical path through the rotating helical magnetic field • First γ-ray flare is a beaming event of the “moving emission feature” • Second γ-ray flare occurs when pulse passes through a “standing shock” • Acceleration & collimation region is between 104 and 105 rg • Gamma-rays are produced by shocks in the jet, not near the central engine

  28. MARSCHER et al. (cont.) They don’t know this to be the case!! • My additions: • The “moving emission feature” is actually an MHD slow-mode shock, which is constrained to travel ONLY along the helical magnetic field (recall properties of slow-mode waves) • The coherent 240 degree polarization swing is strong evidence for a well-ordered helical magnetic field; knowing the “beaming angle” we could calculate the field pitch angle • The “standing shock” is probably a collimation shock produced near the MFP, and represents the place where the jet nozzle ends and free jet flow begins • However, the lack of polarization data during and after 2nd flare is a severe loss • We cannot tell whether plasma is strongly magnetized or becomes turbulent and, therefore, cannot tell for certain whether a true MFP exists or not • BL Lac’s parent is probably an FR I; FR II objects are likely to be quite different

  29. MARSCHER et al. (cont.) • Importance and significance of the Marscher et al. (and future such) observations • First time we have peered into the “central jet engine” and seen a little bit of how it works • This is the ONLY method we have (probably for decades) to probe the acceleration & collimation region of ANY ASTROPHYSICAL JET (AGN, protostellar, microquasar, symbiotic, GRB, SN, nor PN) • The high-resolution imaging of the VLBA is essential for determining the location of optical/X-/γ-ray features • Multi-wavelength observations, esp. optical & hard X-/γ-ray are essential for probing the internal structure of the jet • Space VLBI will provide even higher resolution, allowing perhaps a probe of an FR II-class object • One of the most important mysteries to solve is why are FR II sources so kinetically dominated, if all jets are magnetically accelerated and collimated? • Where is the magnetic energy converted into non-thermal internal energy? (In the MHD model it’s converted into kinetic energy in the acceleration/collimation region. • Our old, bright VLBI source friends are probably the best candidates for this type of work: • They are bright, giving very good S/N • We are peering down jet engine nozzle • Helical field easily identified with multi-frequency polarization observations • They evolve on relatively short time scales • All we need is the highest angular VLBI resolution and best (u,v)-coverage possible

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