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

5. HD Causality Any k is valid, so a) Subsonic Flow (V < cs):
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. Non-dispersive MHD Waves: 1. Alfven
Full Dispersion Relation
transverse (Alfvén)
longitudinal (magneto-acoustic)
Alfvén Velocity Vector
Alfvén Dispersion Relation

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

10. MHD Causality Phase velocity remarks Group 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. Thermal / Relativistic Properties
SELF-SIMILAR,
AXISYMMETRIC, STATIONARY, IDEAL MAGNETOHYDRODYNAMICS:
The MHD Jet Analogy to the Parker Wind
Table of Important Self-Similar MHD Jet Papers in the Last ¼ Century
Thermal / Relativistic Properties
Non-Relativistiic
Relativistic
Cold (p = 0)
Blandford & Payne (1982)
(2 singular points: AS, FMSS)
Li, Chiueh, Begelman (1992)
(2 singular points)
Warm (0 < p < B2/8π)
Vlahakis et al. (2000)
(3 singular points: add SMSS)
Vlahakis & Konigl (2003)
(3 singular points)

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 number and 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/  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., rg)

19. RECAP SO FAR: SELF-SIMILAR JET ACCELERATION AND COLLIMATION THEORY
Sonic Radius
Hydrostatic Solar Wind
Supersonic Solar Wind
Sonic Point
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
Modified Fast Point (Vθ = -Vfast)
Poynting Flux Dominated
Alfven Point (Vjet = VAlfven)
What happens beyond the MFP?
Kinetic Energy Flux Dominated
Collimation Shock !!
Vjet = Vfast = (VA2+cS2)1/ (not a singular point)
Modified Slow Point (Vθ = Vslow)
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., 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 ≈ Ψ2out Ωout / R0,max = 2.4 B2out R30,max Ωout
Jet Mass Loss Rate: ΔM ≈ Ψ2out / R30,max Ωout = B2out R0,max / Ωout
Jet Torque (A.M. loss): G ≈ Ψ2out / R0,max = 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 with RL,out; that is, scale radius is now the LIGHT CYLINDER radius of the outermost magnetic field line
Jet Total Luminosity: LT ≈ Ψ2out Ωout / RL, out = 1.6 B2out R4L, out Ω2out / c (BZ expression)
Jet Mass Loss Rate: ΔM ≈ Ψ2out / R3L, out Ωout = 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. 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)
Modified Fast Point (Vθ = -Vfast)
Poynting Flux Dominated
Alfven Point (Vjet = VAlfven)
What happens beyond the MFP?
Kinetic Energy Flux Dominated
Collimation Shock !!
Vjet = Vfast = (VA2+cS2)1/ (not a singular point)
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)
Poynting Flux Dominated

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 rg in length or more
M87  – 140 mas in size
Cen A  – 60 mas in size
BL Lac  – 0.6 mas in size (foreshortened)
FR II jets are expected to collimate quickly  collimation regions rg in length, AND BE FARTHER AWAY
Cyg A  1 – 10 μas in size
3C 35  – 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 (~ , ) 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

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. They don’t know this to be the case!!
MARSCHER et al. (cont.)
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
They don’t know this to be the case!!

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