1. Relativistic Jets in the Universe
Yosuke Mizuno
Institute of Astronomy
National Tsing-Hua University

2. Contents Introduction: Relativistic Jets (observations)
Jet formation / acceleration mechanism
Magnetohydrodynamic (MHD) process
Jet collimation mechanism
Jet is self-collimated?
Dissipation of jets
How magnetic energy converts to jet kinetic energy
Summary

3. Definitions & Units Definitions: Units:
Lorentz factor: G=(1-b2)-1/2 where b=v/c
Gravitational radius of a BH: rg=GMBH/c2 = 2 rS (Schwarzschild radius)
Units:
Magnetic field strength: 1G = 10-4 T
energy: 1erg = 10-7 J
Scale: 1 light year = 1016 m
1 pc = 3 x 1016 m

4. Plasma in the Universe
A plasma is a quasi-neutral gas consisting of positive and negative charged particles
Liquid is heated, atoms vaporize => gas
Gas is heated, atoms collide each other and knock their electrons => decompose into ions & electrons (plasma)
Plasma state: fourth state of matter
Most of (visible) universe is in form of plasma
Plasma form wherever temperatures are high enough or radiation is strong enough to ionize atoms
Magnetohydrodynamics (MHD) is the most simplest description of plasma which approximates to consider as a single fluid and is captured macroscopic behavior

5. Relativistic Regime Kinetic energy >> rest-mass energy
Fluid velocity ~ light speed
Lorentz factor g>> 1
Relativistic jets/ejecta/wind/blast waves (shocks) in AGNs, GRBs, Pulsars
Thermal energy >> rest-mass energy
Plasma temperature >> ion rest mass energy
p/r c2 ~ kBT/mc2 >> 1
GRBs, magnetar flare?, Pulsar wind nebulae
Magnetic energy >> rest-mass energy
Magnetization parameter s >> 1
s = Poyniting to kinetic energy ratio = B2/4pr c2g2
Pulsars magnetosphere, Magnetars
Gravitational energy >> rest-mass energy
GMm/rmc2 = rg/r > 1
Black hole, Neutron star
Radiation energy >> rest-mass energy
E’r /rc2 >>1
Supercritical accretion flow

6. Applicability of Hydrodynamic Approximation
To apply hydrodynamic approximation, we need the condition:
Spatial scale >> mean free path
Time scale >> collision time
These are not necessarily satisfied in many astrophysical plasmas
E.g., solar corona, galactic halo, cluster of galaxies etc.
But in plasmas with magnetic field, the effective mean free path is given by the ion Larmor radius.
Hence if the size of phenomenon is much larger than the ion Larmor radius, hydrodynamic approximation can be used.

7. Astrophysical Jets
Astrophysical jets are a tremendous, elongated and collimated outflows of plasma
Astrophysical jets can be observed in a huge spatial and energetic scale reaching from stellar size to galaxy size
There are many sources for jets
Jets (Outflows) are common feature in the universe
The astrophysical jets seen in AGNs, BHBs, and GRBs have a relativistic speed = relativistic jets
M87 jets (AGNs, optical)

8. M87 = Virgo A Nearby: D ~ 16 Mpc (1 mas = 0.08 pc)
Radio observation
M87 = Virgo A
Nearby: D ~ 16 Mpc (1 mas = 0.08 pc)
AGNs: FR I, Misaligned BL Lac (q ~ 14 deg)
SMBH mass: 6.6 x 109 Msun
VLBA resolution: 20 rs at 43 GHz
Frequency: 43GHz

10. Sources of Jets Stellar Objects Physical Systems Young Stellar Objects
Protostars accreting from disks
HMXBs/LMXBs (microquasars)
NS/BH accreting from disks
Pulsars
Rotating Neutron Star
Gamma-Ray Bursts
Merging NSs or BH forming inside collapsing star
Extragalactic
Active Galactic Nuclei
Accreting supermassive BH in the core of galaxies
jet with relativistic speed

11. Relativistic jets - properties
Highly relativistic jets are lunching from accreting compact objects (BHs/NSs, “central engine”)
Jets transport energy and angular momentum from central engine to remote locations and provide feedback to ISM/IGM
They are fast: Lorentz factors ~ a few to a few tens in AGNs and microquasars (BHBs), and >100 in GRBs
High Lorentz factors mean that special relativistic effects are important (general relativistic effects is also important in the vicinity of compact objects).
Jet phenomenon bridges many orders of magnitude in size: forming in size scale of rg and extending up to rg

12. Jet Similarity Similar Morphologies But … Jets from a protostar
Size: Few-light across
Speed: few 100 km/s
Optical: atomic line emission
Jets from a quasars
Size: ~ million light-years across
Speed: ~ c (light speed)
Radio emission: synchrotron emission (non-thermal)
Cygnus A

13. Jet Speeds Sub-relativistic: protostars, v/c ~ 10-3
Mildly-relativistic: SS433, XRBs (v/c = 0.26)
Doppler-shifted emission lines
Highly-relativistic: X-ray binaries (microquasars), ~10% of AGNs (G ~ 2-30)
Doppler beaming (One-sideness)
Illusion of superluminal motion
Gamma-ray flares (to avoid gg-pair production)
Hyper-relativistic: Gamma-Ray Bursts (G ~ )
Gamma-ray variability
Ultra-relativistic: Pulsar jets/winds (G ~ 106)
Modeling of radiation and pulsar nebulae

14. Superluminal Motions in Relativistic Jets
Apparent superluminal motion (Rees 1967)
towards observer q c v

15. Relativistic jets - radiation
Synchrotron radiation – Non-thermal electron + B-field
X-rays/Gamma-rays:
Inverse Compton scattering – non-thermal electrons + photon field
Hadronic emission process(?) – relativistic protons
Very high luminosities ( ergs/s in quasars)
Jets contain highly relativistic charged particles and magnetic field => need efficient particle acceleration mechanisms
Understanding jet emission requires both classical and quantum mechanics as well as special and general relativity

16. Relativistic Jets in AGNs
Jet launched from vicinity of a supper-massive BH (105-9 Msun)
~ 10 % of AGN are radio-loud i.e., have prominent jets
AGN jets can extend from ~ 10-4 pc to several hundred kpc in size
Jet composition is unknown (normal or pair plasma)
Jet powers : erg/s
Jet power integrated over radio source lifetime: erg
Urry & Padovani (1995)

17. Radio galaxies – FR I/FR II dichotomy
Fanaroff & Riley (1974): Connection between radio galaxy luminosity and morphological type
FR I: Prominent two-sided jets, brighter towards center, often distorted with bends => 1.4 GHz luminosities below 1025 W/Hz
FR II: Double-lobed, edge-brightened, prominent hot-spots; relatively weak, straight, one sided jets => 1.4 GHz luminosities over 1025 W/Hz

18. Blazars AGN with relativistic jets seen almost pole-on
Two sub-category: Flat-spectrum radio quasars and BL Lac objects
Jet emission enhanced due to relativistic effects by a factor of thousands
Broad-band SED dominated by non-thermal emission from jets (Synchrotron + Inverse-Compton )
Emission from radio up to TeV gamma-rays
Average SED of blazars (Fossati et al. 98)

19. Relativistic Jets in Microquasars
Observed in radio
Superluminal motion
in microquasar GRS ; Vapp =1.5c
Microquasar is a scaled down (by a factor of 106) version of active galactic nuclei

20. Relativistic Jets in Universe
Mirabel & Rodoriguez 1998

21. Fundamental Problems in Relativistic jets
How are the jets formed near the compact objects (BHs or NSs)? (formation mechanism)
How are the jets accelerated up to relativistic speed? (acceleration mechanism)
How the collimated jet structure make? (collimation mechanism)
How the jets remain stable over large distance? (Jet stability mechanism)
How the jets influence ISM and IGM (feedback)?
How to emit the radiation in the relativistic jets? (Radiation mechanism and particle acceleration mechanism)

22. Jet formation/acceleration mechanism
Jet is formed near the central compact objects (BHs/NSs).
Some accreting matter is getting some force to make jet-like outflows.
Ingredients: rotation, accretion disk, magnetic fields
Jet base: rotating disk or compact objects (BHs/NSs)
The jet formation/acceleration mechanism is still under debate but …
The most promising mechanism is the acceleration/formation by rotating, twisting magnetic fields (magnetohydrodynamic (MHD) process)
Other possibility: gas pressure, radiation pressure, …

23. Applicability of MHD Approximation
MHD describe macroscopic behavior of plasmas if
Spatial scale >> ion Larmor radius
Time scale >> ion Larmor period
But MHD can not treat
Particle acceleration
Origin of resistivity
Electromagnetic waves

24. Jet formation/acceleration mechanism (cont.)
Gas or radiation pressure (Blandford & Rees 1974, O’Dell 1984)
push accretion matter to make and accelerate outflows by pressure gradient
Expansion of magnetic tower (Lynden-Bell & Boily 1994)
Mainly toroidal field from start
Acceleration by magnetic pressure
Magnetocentrifugal acceleration (Blandford & Payne 1982)
Mainly poloidal field anchored to disk or rotating objects
Disk or ergosphere of BH acts like crank
Torque transmitted though poloidal field powers jet
Blandford-Znajek process (Blandford & Znajek 1977)
Directly extract the BH rotating energy and convert to outward Poynting flux
Consider force-free limit (MHD Penrose process is similar mechanism, MHD version )

25. Acceleration & Collimation in MHD model
Magnetic field line
Assume: in ideal MHD, plasma is attached with magnetic field
Acceleration
Magneto-centrifugal force
Magnetic pressure
Like expansion of spring
Collimation
Magnetic pinch force
Like shrink lubber band
Centrifugal force
Accretion disk
outflow (jet)
accretion
Jets
Magnetic field lines
Magnetic field lines

26. Jet formation/acceleration mechanism (cont.)
In ideal MHD limit (infinite conductivity), plasma flow (motion) is connected with magnetic field
The rotation of accretion disks or compact objects (BHs / NSs) twisted up the magnetic field into toroidal components
Collapsing, magnetized supernova core (GRBs)
Pulsar magnetosphere
Magnetized accretion disks around neutron stars and black holes
Magnetospheres of rotating black holes
Courtesy to David Meier

27. Relativistic Jets Formation from GRMHD Simulations
Many GRMHD simulations of jet formation (e.g., Hawley & Krolik 2006, McKinney 2006, Hardee et al. 2007) suggest that
a jet spine (Poynting-flux jet) driven by the magnetic fields threading the ergosphere via MHD process or Blandford-Znajek process
may be surrounded by a broad sheath wind driven by the magnetic fields anchored in the accretion disk (mildly-relativistic wind).
High magnetized flow accelerates G >>1, but most of energy remains in B field.
Non-rotating BH Fast-rotating BH
Spine
Sheath
Total velocity
Disk
Density distribution
(McKinney 2006)
Disk Jet/Wind
BH Jet Disk Jet/Wind
(Hardee, Mizuno & Nishikawa 2007)

28. Jet Energetics Gravity, Rotational energy (BH or accretion disk)
Poynting flux (magnetic energy)
Jet kinetic energy
Efficient conversion to EM energy
Easy to get ~ equipartition, hard to get full conversion
Magnetic field is a medium for a transmission not a source

29. Jet Collimation Magnetic hoop stress
Jet is produced by MHD process near the central objects and magnetic field is tightly tied (toroidal field is dominated)
Lorentz force >> plasma pressure & inertia
Huge tension force of wound up magnetic field (hoop stress) compress the flow towards the axis (self-collimation)?
Answer: No!
In the current closure region, the force acts to de-collimation
Need external confinement

30. External Confinement
In BH - accretion disk systems, the relativistic outflows from the black hole and the internal part of the accretion disk could be confined by the mildly-relativistic magnetized wind from the outer parts of the disk.
In GRBs, a relativistic jet from the collapsing core pushes its way through the stellar envelope (confinement).

31. Collimation vs Acceleration
For jet collimation, external confinement is necessary
Without external confinement, the flow is near radial and acceleration stops at an early stage (Tomimatsu 1994; Beskin et al. 1998)
The gas pressure profile of external confinement medium is the important parameter
The spatial distribution of confining gas pressure determines the shape of the jet flow boundary, magnetic field configuration and acceleration rate (Tchekovskoy et al. 2009, 2010; Komissarov et al. 2009; Lyubarsky 2009,2010).
Optimal collimation  pressure decrease slowly along jets
Optimal acceleration  pressure decrease rapidly along jets
=> Collimation and acceleration of jet are related (poloidal) magnetic field configuration

32. Magnetic Acceleration
Acceleration of jet by magnetic field depends on how rapidly magnetic flux surface separate from one another
Faster than radial
K.E/P.E increases (more acceleration, rapid energy conversion)
But not collimated
Slower than radial
K.E/P.E decreases (less acceleration)
Maintain collimated structure

33. Effects of external confinement
2D RMHD simulations (Komissarov et al. 2009)
Parabolic (z ∝ r2)
Acceleration: slow, collimation: OK
Conical (z ∝ r):
Acceleration: fast, collimation: X
Lorentz factor
Some part of jets can convert Poynting flux to Kinetic Energy but most can’t.
Energy conversion is too slow to become kinetic energy dominated, it is unreasonably long distance = inconsistent of observations.
We need to consider some sort of dissipation (rapid energy conversion)

34. Observed Jet structure
Global structure of M87 jet (Asada & Nakamura 2012)
The parabolic structure (z ∝ r1.7) maintains over 105 rs, external confinement is worked.
The transition of streamlines presumably occurs beyond the gravitational influence of the SMBH (= Bondi radius)
Stationary feature HST-1 is a consequence of the jet recollimation due to the pressure imbalance at the transition
In far region, jet stream line is conical (z ∝ r)
Parabolic streamline
(confined by ISM?)
Conical streamline
(unconfined, free expansion)
Over-collimation at HST-1 stationary knot (recollimation shock?)
HST-1 region

35. Observed Jet structure (cont.)
In M87 jet, the asymptotic acceleration from non-relativistic (0.01c) to relativistic speed (0.99c) occurs over rs
This is very slow acceleration = consistent with theoretical results?
The absence of bulk-Comptonization spectral signatures in blazars implies that Lorentz factors >10 must be attained at least ~1000 rg (Sikora et al. 05).
But according to spectral fitting, jets are already matter-dominated at ~1000 rg (Ghisellini et al 10).
Transition of Sub- to super-luminal motion in M87 jet
Asada & Nakamura (2014)

36. Dissipation in the Jet Time-dependent energy injection to jet
=> Internal shocks in jets
Sudden change of confined external medium spatial profile
=> Recollimation shock/ rarefaction acceleration
Magnetic field reversal or deformation of ordered magnetic field
=> Magnetic reconnection
MHD Instabilities in jets
Kelvin-Helmholtz instability at jet boundary
Current-Driven Kink instability at jet interior
=> Turbulence in the jets and/or magnetic reconnection?

37. Dissipation in the Jet: Energetics
Tapping kinetic energy
Internal shock
Recollimation shock
Kelvin-Helmholtz instability
Tapping magnetic energy
Rarefaction acceleration
CD kink instability
Magnetic reconnection
Prefer dissipation mechanism for Poynting-dominated jet (conversion from Poynting flux to Kinetic energy)

38. Rarefaction Acceleration
If the external confined media is suddenly disappeared and jet becomes free expansion (parabolic => conical), the rarefaction wave is formed by overpressure of jet and propagates in the jets during the transition from jet boundary.
Rarefaction wave converts jet thermal & magnetic energies to jet kinetic energy efficiently (e.g., Aloy & Rezzolla 2006; Mizuno et al. 2008; Komissarov et al. 2010), so-called rarefaction acceleration
The mechanism is favor for GRBs and may be possible for AGNs.
stationary feature by recollimation shock?
Komissarov et al. 2010
Rarefaction wave propagates from the jet boundary
Change channel shape from parabolic to conical
Lorentz factor

39. CD Kink Instability
Well-known instability in laboratory plasma (TOKAMAK), astrophysical plasma (Sun, jet, pulsar etc).
In configurations with strong toroidal magnetic fields, current-driven (CD) kink mode (m=1) is unstable.
This instability excites large-scale helical motions that can be strongly distort or even disrupt the system
For static cylindrical force-free equilibria, well known Kruskal-Shafranov (KS) criterion
Unstable wavelengths:
l > |Bp/Bf |2pR
However, rotation and shear motion could significant affect the instability criterion
Distorted magnetic field structure may trigger of magnetic reconnection.
Schematic picture of CD kink instability
3D RMHD simulation of CD kink instability in PWNe (Mizuno et al. 2011)

40. CD Kink Instability in Jets
Mizuno et al. (2009)
Helical structure is developed by CD kink instability.
Growth rate of CD kink instability is small
Magnetic energy in the jets converts thermal and kinetic energies by development of instability (via turbulent structure)
Jet structure is strongly deformed but may be not disrupted entirely (depends on magnetic pitch, density, and flow profiles) (Mizuno et al. 09, 11, 12, 14).
Density
+ B-field
Mizuno et al. (2014)
Regular helical magnetic field is strongly deformed via CD kink instability => may be triggered magnetic reconnection
jet
Density
+ B-field

41. Magnetic reconnection
Ideal MHD gives frozen in magnetic fields.
Resistive MHD (non-ideal MHD) allows diffusion of fields.
Magnetic reconnection occurs through diffusion in narrow current sheets.
Magnetic reconnection is the process of a rapid rearrangement of magnetic field topology.
Magnetic energy could be converted to thermal and kinetic energies
Problem: how differently oriented magnetic field lines could come close each other?
Global MHD instability
Alternating B-field formed at jet begging.

42. Magnetic Reconnection (cont.)
Dissipation of alternating magnetic fields in the jet (e.g., Giannios & Spruit 05,07; Giannios 06,08; McKinney & Uzdensky 2012) j B g
Alternating polarity by misaligned dipole field
multipolar field (local closed field)
Ordered field broken by instability
Mckinney & Uzdensky (2012)

43. Ultra-Fast TeV Flare in Blazars
Ultra-Fast TeV flares are observed in some Blazars (AGN jets).
Vary on timescale as sort as
tv~3min << Rs/c ~ 3M9 hour
For the TeV emission to escape pair creation Γem>50 is required (Begelman, Fabian & Rees 2008)
But PKS , Mrk 501 show “moderately” superluminal ejections (vapp ~several c)
Emitter must be compact and extremely fast
Model for the Fast TeV flaring
Magnetic Reconnection inside jet (Giannios et al. 2009)
PKS (Aharonian et al. 2007)
See also Mrk501, PKS
Giannios et al.(2009)

44. Advantage of Magnetic Reconnections
Magnetic reconnection easily provides large radiative efficiencies and strong variability inferred in AGNs and GRBs.
Reconnection at high s produces relativistic “jets in a jet”; this could account for the fast TeV variability of blazars (Giannios et al 2010)
Questions: why fast reconnection occur in jets?
If we consider anomalous resistivity in small dissipation region, we get fast reconnection. But we do not know what is anomalous resistivity.
This is still unsolved problem in physics.

45. Relativistic Magnetic Reconnection using RRMHD Code
Mizuno 2013, ApJS
To handle relativistic magnetic reconnection numerically, we need to perform resistive relativistic MHD simulations.
Initial condition
Consider Pestchek-type magnetic reconnection
anti-parallel magnetic field
Anomalous resistivity for triggering magnetic reconnection
Results
B-filed：typical X-type topology
Density：Plasmoid
Reconnection outflow: ~0.8c

46. The Five Regions in AGN Jets
Hot Spot/Lobe: ~109 rS (~100 kpc; or 20’)
 Outer jet is not Poynting-Flux Dominated
Kinetic-Flux-Dominated (KFD) Jet: ~103 – 109 rS
(0.1 – 105 pc; 1 mas – 20’)
Transition Region: ~102.50.5 rS (< 0.1 pc; < 1 mas)
Poynting-Flux Dominated (PFD)  KFD
MHD Acceleration/Collimation Region: ~10 – 102.50.5 rS
(1 – < 100 mpc; 10 as – < 1 mas)
The Jet “Nozzle”
Jet Launching Region: The Accretion Flow; ~5 – 50 rS
(0.5 – 5 mpc; 5 – 50 as)
Probably unresolved or slightly resolved

47. Five Regions in AGN Jets
Modified from Graphic
courtesy David Meier
Jet Launching Region
Jet Collimation Region (10 –100  Launching Region)
Alfven Point
Sheath
Modified Fast Point
Kinetic Energy Flux Dominated
with Tangled (?) Field
Fast MS Point
Collimation Shock
Slow
MS Point
High speed spine
Poynting Flux Dominated
CD Unstable Magnetic Helicity Driven Region
Combined CD/KH
Unstable Region
KH Unstable Velocity Shear Driven Region 47

48. Summary
Relativistic jet is formed and accelerated by the MHD process near the compact objects.
Flow is dominated by the Poynting flux (EM energy)
No self-collimation: need external confinement
External confinement is crucial for efficient jet collimation and acceleration of Poynting dominated outflows.
Relativistic jet accelerates gradually => need dissipation to convert EM energy of jets
Magnetic reconnection is a key for the dissipation

49. Frontier of research Numerical simulations:
GRMHD simulations are possible
Additional physics: Resistivity, radiation, microphysics, …
Effects of time-dependence, non-asymmetry (3D)
GR radiation transfer calculation is a key tool to connect MHD simulations and observations (Need correct radiation process including particle acceleration)
Observations: mm & submm-VLBI trying to observe BH shadow & jet launching region (EHT, GLT, Black Hole Cam projects)

50. BH shadow image (no jet)
Takahashi et al. (2004)
Pu et al. (2014)
a=0
a=0.5
a=0.999
Optically thick
Geometrically thin
i=10 deg
i=85 deg
Optically thin
Geometrically thick
i=85 deg

51. High resolution VLBI observations
Source
SgrA*
M87
Cen A
BH Mass (Msun)
0.045 x 108
66 x 108
0.45 x 108
Distance
0.008 pc
16.7 Mpc
3.4 Mpc rS
11 mas
8 mas
0.3 mas
Size of shadow
52 mas
40 mas
1.5 mas
Submm-VLBI
GLT
In radio VLBI observation, angular resolution ∝ l/D, l: wavelength, D: baseline
Resolution ~ 50 mas at l ~1mm (300GHz) & D=4000 km
In GLT project (with other sub-mm array), D > 9000 km
=> 20 mas at 345 GHz
SMA
ALMA

53. M87: Jet Launching & Collimation Region
Walker et al. (2008)
Beam: ~ 0.4 x 0.2 mas mas ~ pc ~ 42Rs
Jet launching region:
< (0.4 mas/sin 150) ~ 200 Rs
Jet collimation region:
~ (7.5 mas/sin 150) ~ 4000 Rs
Bulk Lorentz factor: ~ 2 – 3
@ 15o jet viewing angle
5 mas ~ 1.55 pc ~ 2700 Rs
5 mas  0.4 pc ~700Rs
TeV < 200 Rs
Blazar Model: Jorstad et al.(2007)
Bulk Lorentz factor: Γ ~
Particle Lorentz factor: γ > 105
4000 Rs 53

54. Jet Wobbling
15 GHz VLBI image of 4C+12.50
High resolution VLBI observations of some AGN jets show regular or irregular swings of innermost jet structural position angle (jet wobbling).
Parsec scale AGN jet curvatures and helical-like structures (inner parsec or large scale) are believed to be triggered by changes in direction at the jet nozzle.
Physical origin of jet wobbling (Agudo 2009)
Accretion disk precession
Orbital motion of accretion system (binary BH?)
Jet instabilities
Position angle oscillation of 3C273
P.A.
year

55. Instability of Relativistic Jets
When jets propagate from magnetosphere of compact object (BH, NS), there are possibility to grow of two major instabilities
Kelvin-Helmholtz (KH) instability
Important at the shearing boundary flowing jet and external medium
Current-Driven (CD) instability
Important in existence of twisted magnetic field
Twisted magnetic field is expected jet formation simulation & MHD theory
Kink mode (m=1) is most dangerous in such system
 Instability of relativistic jet is important for understanding observed many jet phenomena & structure
quasi-periodic wiggles, knots, filaments, limb brightening, jet disruption etc
Limb brightening of M87 jets (observation)

56. CD Kink Instability in Jets (Newtonian)
Appl et al. (2000)
Consider force-free field with different radial pitch profile in the rest frame of jet
maximum growth rate: Gmax=0.133 vA/P0,
unstable wave length: lmax=8.43P0
Wave number
Growth rate for m=-1~-4 in constant pitch case.
(P0=a in our notation:
Magnetic pitch =RBz/Bf )
Growth rate for m=-1 mode as a function of wavenumber with different pitch profile
Maximum growth rate and unstable wave number for m=-1 kink as a function of magnetic Pitch

57. 4D General Relativistic MHD Equation
General relativistic equation of conservation laws and Maxwell equations:　　　
　　　　　　　∇n ( r U n ) = 0　　　 (conservation law of particle-number)
　　　　　　　∇n T mn = (conservation law of energy-momentum)
　　　　　　　∂mFnl + ∂nFlm + ∂lF mn = 0
　　　　　　　∇mF mn = - J n
Ideal MHD condition: FnmUn = 0
metric：　ds2=-a2 dt2+gij (dxi+b i dt)(dx j+b j dt)
Equation of state : p=(G-1) u
(Maxwell equations)
r : rest-mass density. p : proper gas pressure. u: internal energy. c: speed of light.
h : specific enthalpy, h =1 + u + p / r.
G: specific heat ratio.
Umu : velocity four vector. Jmu : current density four vector.
∇mn : covariant derivative. gmn : 4-metric. a : lapse function, bi : shift vector, gij : 3-metric
Tmn : energy momentum tensor, Tmn = pgmn + r h Um Un+FmsFns -gmnFlkFlk/4.
Fmn : field-strength tensor,

58. Conservative Form of GRMHD Equations (3+1 Form)
(Particle number conservation)
(Momentum conservation)
(Energy conservation)
(Induction equation)
U (conserved variables)
Fi (numerical flux)
S (source term)
√-g : determinant of 4-metric
√g : determinant of 3-metric

59. Detailed Features of the Numerical Schemes
Mizuno et al. 2006a, 2011c and progress
RAISHIN utilizes conservative, high-resolution shock capturing schemes (Godunov-type scheme) to solve the 3D GRMHD equations (metric is static)
* Reconstruction: PLM (Minmod & MC slope-limiter function), convex ENO, PPM, WENO5, MP5, MPWENO5
* Riemann solver: HLL, HLLC, HLLD approximate Riemann solver
* Constrained Transport: Flux CT, Fixed Flux-CT, Upwind Flux-CT
* Time evolution: Multi-step Runge-Kutta method (2nd & 3rd-order)
* Recovery step: Koide 2 variable method, Noble 2 variable method, Mignore-McKinney 1 variable method
* Equation of states: constant G-law EoS, variable EoS for ideal gas

60. RAISHIN Code (3DGRMHD)
Mizuno et al. 2006a, 2011c, & progress
RAISHIN utilizes conservative, high-resolution shock capturing schemes (Godunov-type scheme) to solve the 3D ideal GRMHD equations (metric is static)
Ability of RAISHIN code
Multi-dimension (1D, 2D, 3D)
Special & General relativity (static metric)
Different coordinates (RMHD: Cartesian, Cylindrical, Spherical and GRMHD: Boyer-Lindquist of non-rotating or rotating BH)
Different schemes of numerical accuracy for numerical model (spatial reconstruction, approximate Riemann solver, constrained transport schemes, time advance, & inversion)
Using constant G-law and approximate Equation of State (Synge-type)
Parallel computing (based on OpenMP, MPI)

61. M87 Jet & BH Basic facts: D ~ 16.7 Mpc , 1” ~ 80 pc
Marshall et al. (x-ray)
Perlman et al. (optical)
Nuclear region: Mbh ~ 6 x 109 Msol , Rs ~ 0.6 milli-pc
Gebhardt & Thomas (2009)
Biretta, Junor & Livio (2002)
Junor, Biretta & Livio (1999)
Ly, Walker & Wrobel (2004)
Zhou et al. (radio)
1 pc
0.03 pc

62. M87: BH & Jet Collimation Region
Radio Galaxy
FR Type
MBH (M)
D (pc)
RS/as
Collimation Region Size
M87/Vir A I
6.0  109
1.67  107
7.5
0.10 – 10 mas
Cen A
2.0  108
3.4  106
1.2
0.06 – 6 mas
Cyg A II
2.5  109
2.2  108
0.2
2.2 – 44 as
Sgr A GC
2.5  106
8,500
5.9
0.07 – 7 mas
M87 Black Hole & Jet Launching Region Largest Angular Size! 62