Steady Models of Magnetically Supported Black Hole Accretion Disks
and their Application to Bright/Hard State
and Bright/Slow TransitionHiroshi Oda
The Group Meeting @ The ITC Lounge
Table of Contents‣Background Information
✓Basic Picture and General Aspects✓Conventional Theory of BH Accretion Disks
‣Motivation✓Bright/Hard state & Bright/Slow transition✓Recent MHD/Rad.-MHD simulations
➡ Importance of magnetic fields in accretion disks
1.Model & Assumption‣Results
1.Low-β(≡(ppas+prad)/pmag) solutions1. explain the Bright/Hard state & the Bright/Slow transition.
Background Information•Basic Picture of Accretion Disks
•General Aspects of X-ray Spectrul States and State Transitions of Galactic BH Binaries
•Conventional Theory of Accretion DIsks
•Basic equations
•Model for optically thick disks
•Model for optically thin disks
RXTEBasic Picture of Accretion Disks
(around a stellar mass BH)
Companion
Star
BH
Mass Supply
Angular momentum transport due to
viscosity
Heating due to viscosity
Soft X-ray(~1-10 keV)
Hard X-ray(~100keV)
Gravitational E. → Kinetic(Rotation) E. → Thermal E. →Radiation or Accretion onto BH
Suzaku
magnetic stress
dissipation of magnetic energy
→ Magnetic E.
The Origin of the α-viscosity
Turbulent magnetic fieldsTurbulent magnetic fields generated by MRI generated by MRIThe The Maxwell stressMaxwell stress transports the angular momentum. transports the angular momentum.Disk heating via the Disk heating via the dissipation of magnetic energydissipation of magnetic energy..
In quasi-steady stateIn quasi-steady state
~10, ~0.01-0.1,~10, ~0.01-0.1,
Shearing-box
32×64×256
(Hirose+ ’06)
(Machida+ ’00)
200×64×240
Global 3DMHDof opt. thin disks(RIAF)
Local 3D Rad-MHD
of opt. thick disks
(Standard disk)
Density (initial)Density (initial)
Bφ
Turburent Turburent BB
Isosurface of |Isosurface of |BB||
General Aspects of X-ray Spectral States and General Aspects of X-ray Spectral States and State Transitions of Galactic BHBsState Transitions of Galactic BHBs
Low/Hard State:High/Soft State:
Inte
nsi
tyJe
tLo
ren
tz F
act
or
Hardness
HardSoft
Jet
line
Hard-to-Soft Transition:
Relativistic. jet
Energy [ keV ]1 10 100Fl
ux ν
F ν [
keV
/cm
2 s ]
Energy [ keV ]1 10 100FL
ux ν
F ν [
keV
/cm
2 s ]
Schematic Picture ofHardness-Intensity Diagram
Fender+ `04
DBB
Hard tailCutoff Power-Law
SoftExcess
Emission fromoptically thick disks.
Emission fromoptically thin disks.
=Lhard/Lsoft
Basic eqs. of conventional viscous accretion disks
ContinuityEq. of motion
Energy eq.
ϖ-comp.
ϕ-comp.
z-comp.
Viscous force
Surface density
Vertically integrated total pressure
Viscousheating
Assumption: Steady, axisymmetric, Kepler rotation.
, hydrostatic balanceIntegrate in vertical direction.
α-viscosity(Shakura & Sunyaev ’73)
GravityPressuregradient
Cylindrical coordinates
• Standard Disk (High/Soft state)• low M, high Σ• Geometrically thin ( H≪ϖ )• Q+
vis~Q-rad(black body; Teff∝ϖ-3/4)
• pgas dominant, cool (T 〜 107K)
• (prad dominant Std. Disk)• Thermally unstable
• Slim Disk (Slim Disk state)• High M, high Σ• photon trapping
• Geometrically moderately thick (slim)• Q+
vis~Q-adv ( Teff∝ϖ-1/2)
• prad dominant, moderately hot(T 〜107-8K)
Theoretical Models: Optically Thick DIsks
Soft X-ray(Slim Disk)
Advection Heat
moderatelyhot(T 〜 107-
8K)
prad dominant
Soft X-ray(High/Soft)
Heat
Rad
iati
on
cool (T 〜 107K)
pgas dominant
Surface density
Thermal Equilibrium Curves@5rs
Mass
Acc
reti
on
Rate
.
.Energy [ keV ]
1 10 100FLux ν
F ν [
keV
/cm
2 s ] X -ray spectrum
form disks
• (SLE solution)• Q+
vis~Q-rad
• (e.g., brems., Inv.-Comp.)• pgas dominant, hot (Te 〜 10 9 K)• Thermally unstable
• Advection Dominated Accretion Flow(ADAF)/Radiatively Inefficient Accretion FLow(RIAF)
(Low/Hard state)• Low M, low Σ
• The viscously dissipated energy is stored in the gas as entropy and advected inward.
• Geometrically thick.• Q+
vis~Q-adv
• pgas dominant, hot (Te 〜 109K)
Advection Heat
Hard X-ray(Low/Hard)
RIAF
hot(Te 〜 109K)
pgas dominant
Theoretical Models: Optically Thin DIsks
Theoretical Models: Optically Thin DIsks
Surface density
Thermal Equilibrium Curves@5rs
Mass
Acc
reti
on
Rate
.
Energy [ keV ]1 10 100FLux ν
F ν [
keV
/cm
2 s ]
X -ray spectrumform disks
Surface density
Tem
pera
ture
Slim
Std.St
d.
Hard-to-Soft
Thermal Equilibrium Curves@5rs
Thin Opacith Thick
Adv. Heat
Hard X-ray(Low/Hard)
RIAF
hot(Te 〜 109K)
pgas
Soft X-ray(Slim Disk)
Adv. Heat
Slim Disk
moderatelyhot(T 〜 107-8K)
prad
Soft X-ray(High/Soft)
Heat
Rad
.
Std. Diskcool(T 〜 107K)
pgas
Thermal Equilibrium Curves of Accretion Disks
Mass
Acc
reti
on
Rate
Motivation
•Two hard-to-soft transition: Bright/Slow & Dard/Fast
•Bright/Hard state durging the Bright/Slow transition
•Importance of the magnetic fields in such transition: Suggestion form 3D MHD
SoftSoft HardHardRXTE: GX339-4
Belloni+ `06
2004/2005
2002/2003Bright
Dark
Two Distinct Hard-to-Soft Two Distinct Hard-to-Soft TransitionsTransitions
Energy [ keV ]
1 10 100
FLu
x ν
F ν [
keV
/cm
2 s ]
Energy [ keV ]
1 10 100
Flu
x ν
F ν [
keV
/cm
2 s ]
=L9.4-18.5keV/L2.5-6.1keV
Bright/Slow
Transition
occurs at~0.3 LEdd
, takes≲30 days
Dark/Fast
Transition
occurs at≳0.1 LEdd
, takes≳15 daysGierlinski &
Newton `06
Hard-to-Soft Transition of Other Hard-to-Soft Transition of Other ObjectsObjects
=L5-12keV/L3-5keV
Bright/Slow Transition & Bright/Hard StateBright/Slow Transition & Bright/Hard State
[ keV ]1 10 100
Flux ν
F ν [
keV
/cm
2 s ]
High/Soft
Low/Hard
Very High/Steep PL
Slim Disk
Cutoff PL(Ecut 〜 200keV)
Cutoff PL(Ecut 〜 50-200keV)
HR = L5-12keV/L3-5keV
SoftSoft HardHard
L/H
B/H
VH/SPLSlim
H/S
(Gierlinski & Newton`06)X-ray Spectral StatesH-L
Diagram
GX 339-4
Bright/Slow(2002/2003)
Jet
line
Bright/HardBright/Hard
Ecut
Low/Hard Ecut 〜 200keV
Bright/Hard
〜〜 0.2LEdd
Ecut vs. L
Ecut anti-correlates
with L
GX 339-4
(Miyakawa+ `08)
‣Bright/Slow Transition: occurs at ~0.3LEdd
‣Bright/Hard State: Ecut anti-correlates with L (>0.07LEdd). Conventional models (RIAF, Std., Slim) can not reproduce these features.
Dark/Fast(2004/2005)
Lum
inosi
ty
Surface density
Tem
pera
ture
Slim
Std.St
d.
Hard-to-Soft
Thermal Equilibrium Curves@5rs
Thin Opacith Thick
Adv. Heat
Hard X-ray(Low/Hard)
RIAF
hot(Te 〜 109K)
pgas
Soft X-ray(Slim Disk)
Adv. Heat
Slim Disk
moderatelyhot(T 〜 107-8K)
prad
Soft X-ray(High/Soft)
Heat
Rad
.
Std. Diskcool(T 〜 107K)
pgas
〜〜 0.40.4αα22LLEddEdd〜〜 0.0010.001LLEddEdd
Bright/Slow Transition [ FAILED ]Bright Hard State [ FAILED ]
TTee ≥ 10 ≥ 1099K in K in RIAFRIAF
Thermal Equilibrium Curves of Accretion Disks
Mass
Acc
reti
on
Rate
Dark/Fast Transition [ OK ]Low/Hard State [ OK ]
Dark/Fast Transition [ OK ]Low/Hard State [ OK ]
Magnetic Pressure Dominated (Low-β) Disks:
Candidates for the Bright/Hard State?““Magnetic Accretion Disks Fall into Two Type” (Shibata+ ’90)Magnetic Accretion Disks Fall into Two Type” (Shibata+ ’90)High β disk: High β disk: BB escapes due to the buoyancy (or Parker). escapes due to the buoyancy (or Parker).Low β disk: Low β disk: BB cannot escapes due to the strong magnetic tension. cannot escapes due to the strong magnetic tension.
Such low-β disks emit hard X-raysSuch low-β disks emit hard X-rays (e.g., Mineshige+ ’95) (e.g., Mineshige+ ’95)
““Formation of Magnetically Supported Disks during Hard-Formation of Magnetically Supported Disks during Hard-to-Soft Transitions in Black Hole Accretion Flows” to-Soft Transitions in Black Hole Accretion Flows” (Machida+ ’06)(Machida+ ’06)
Global 3D MHD of optically thin accretion disks with coolingGlobal 3D MHD of optically thin accretion disks with coolingHot RIAFHot RIAF → → Cool Low-βCool Low-β
Surface density
Low-βLow-β
Low-β
Low-β
Tem
pera
ture
Acc
reti
on
Rate
RIAFRIAF
coolcool
shrinkshrink
TurbulentBφ dominant
β 〜 10
tcool≪tescape B
➡Bφ is almost conserved
β 〜 0.1
✦Early phase
✦Final phase
‣As M increase➡Σ > Σcrit
➡cooling instability➡shrink in z
Time evolution
.
Aims• Construct 1D steady models incorporating magnetic fields.
Aim 2: Transonic solutions (one-temperature; Aim 2: Transonic solutions (one-temperature; TTii==TTee : Oda+ ’07, PASJ) : Oda+ ’07, PASJ)
How does the How does the radial structure of disksradial structure of disks change due to the magnetic change due to the magnetic field?field?
Aim 3: Thermal equilibria & transonic solutions Aim 3: Thermal equilibria & transonic solutions (two-temperature; (two-temperature; TTii≠≠TTee))
No longer validNo longer valid T Tii==TTe e in high temperature, low density region. in high temperature, low density region.Although we consider the bremsstrahlung emission in the 3DMHD and Although we consider the bremsstrahlung emission in the 3DMHD and above-mention models, how about above-mention models, how about the synchrotron emission and the the synchrotron emission and the inverse-Compton effectinverse-Compton effect??Spectrum form the low-β disks?Spectrum form the low-β disks?
Reproduce the spectrum observed in Reproduce the spectrum observed in Bright/Hard stateBright/Hard state??
Aim 1: Local thermal equilibria for opt. thin ~ thick Aim 1: Local thermal equilibria for opt. thin ~ thick disks. (Oda+ ’09, ApJ)disks. (Oda+ ’09, ApJ)
ExistExist thermal equilibrium solutions of low-β disks? thermal equilibrium solutions of low-β disks?Low-β solutions explain Low-β solutions explain Bright/Hard stateBright/Hard state and and Bright/Slow transitionBright/Slow transition??
Basic Eqs.
Grav. Lorentz forceP grad
Steady, azimuthal average
ptot = pgas+prad+pmag
HeatingCooling
ContinuityEq. of motion
Energy eq.
ϖ-comp.
ϕ-comp.
z-comp.
Cylindrical coordinates
Induction eq.
Magnetic Field & Velocity FieldWe assumed that magnetic fields are turbulent
and dominated by the azimuthal component.
[mean field][fluctuating field]
Azimuthal averages of the fluctuating component are zero
‣Decompose into the mean field
and the fluctuating field
Note: is not zero.
Evolution of the turbulent B inside the disk (face on
view)
cv
cvdifferential
rotation
turbulence
turbulence
Kepler rotation instead of the radial component fo the eq. of motion. hydrostatic in z, polytropic, , , are constant in z, integrate in z direction
Maxwell StressAngular momentum swallowed by BH:
Mass accretion rate:
Disk thickness: 2H
Entropy gradient: ξ
Surface density:
Dynamo term Magnetic diffusion term
Basic Eqs.
Steady, azimuthal average
ContinuityEq. of motion
Energy eq.
ϖ-comp.
ϕ-comp.
z-comp.
Induction eq.
ptot = pgas+prad+pmag
Cylindrical coordinates
Magnetic fluxadvection rate
proportional to total pressure (gas+radiation+magnetic)
∵Following the results of 3DMHD(Machida et al. 2006)When the disk shrink due to the cooling instability, tcool≪tescape B
→conserving the azimuthal component of magnetic fields→The gas pressure decreases, while the magnetic pressure increases
Then, the ϖφ-component of the Maxwell stress is proportional to the total pressure
Note: In the conventional theory, this term is proportional to (gas+radiation) pressure Decrease in T→Decrease in pgas, prad→Decrease in the Maxwell stress
Maxwell Stress
BHKey point
If we rewrite this relationin term of the kinematic viscosity
Note: If we fixDecrease in T and pgas+prad➡Decrease in pmag
Inconsistent with the results of 3DMHDAt least, when the disk shrinks in the vertical directiondue to the cooling instability, the magnetic pressure decreases.
Magnetic Flux Advection Rate
Following the results of the 3DMHD (Machida+ ’06)
Dynamo term Magnetic diffusion termIf we ignore the dynamo term and the magnetic diffusion term, Φ is
constant.However, Φ is not always conserved in the radial direction due to the
presence of the dynamo term and the magnetic diffusion term.
( )
Note: ς~1 in the 3DMHD
We parametrize the radial dependence Φ introducing a parameter ς
.
..
Heating rate
If the magnetic pressure is high, the heating rate can also be large even when the gas pressure and the radiation pressure are low.
Cooling rate
In the opt. thick limit: Black body
In the opt. thin limit: Brems.
Advection termFollowing the result of 3DMHD(Machida+ ’04), entropy gradient ξ=1
ξ> 0: heat advection act as cooling at certain radius.ξ< 0: heat advection act as heating at certain radisu.
Energy Eq.
Key point
( e.g., Hubeny 1990; Narayan & Yi 1995;
Abramowicz et al. 1996)
Result: New Thermal Equilibrium Solution Result: New Thermal Equilibrium Solution Connecting an Opt. Thin Region and an Opt. Connecting an Opt. Thin Region and an Opt.
Thick RegionThick Region
Hard X-ray(Bright/Hard)
Opt. thin low-β disk
ソフト X 線(High/Soft?)
Opt. thick low-β disk
hotter than Std. disk
(T 〜 107-8K)
BH
cooler than RIAF(T 〜 107-
11K)
Surface density
Tem
pera
ture
Thermal Equilibrium Curves@5rs
Thin Opacith Thick
Adv. Heat
Hard X-ray(Low/Hard)
RIAF
hot(Te 〜 109K)
pgas
Soft X-ray(Slim Disk)
Adv. Heat
Slim Disk
moderatelyhot(T 〜 107-8K)
prad
Soft X-ray(High/Soft)
Heat
Rad
.
Std. Diskcool(T 〜 107K)
pgas
Mass
Acc
reti
on
Rate
Heat
Rad
.
pmag
BHHeat
Rad
.
pmag
RIAF
RIAF
Std.
Std.
Slim
SlimLo
w-β
Low-β
ζ= 0(dotted) 0.3(dashed) 0.6(thick solid) Conventional(thin solid)
‣Conventional model:
➡No Q+ balances Qrad
‣This model:
➡Q+ balances Qrad when the magnetic pressure is high.
Why Can We Obtain the Low-β Why Can We Obtain the Low-β Solutions?Solutions?
Energy eq.
@high density, low temperature
0 ~0 ~
Surface density
Tem
pera
ture
Thermal Equilibrium Curves@5rs
Thin Opacith Thick
Mass
Acc
reti
on
Rate
BHHeat
Rad
.
pmag
RIAF
RIAF
Std.
Std.
Slim
Slim
Low
-β
Low-β
ζ= 0(dotted) 0.3(dashed) 0.6(thick solid) Conventional(thin solid)
The Opt. Thin Low-β Disk:The Opt. Thin Low-β Disk: Bright/Hard State & Bright/Slow Bright/Hard State & Bright/Slow TransitionTransition
=L5-12keV/L3-5keV
Lum
inosi
ty
Hardness-Luminosity Diagram
RIA
F
Std
.Slim
SLE
Low-β
〜〜 0.10.1LLEddEdd
Low/Hard
Bright/Hard
High/Soft
Slim VH/SPL
GX 339-4
〜〜 0.10.1LLEddEdd
50 100 200 [ keV ]
Low/Hard
XTE J1550-564
Jet
line
Bright/Slow
Hardness Ratio
Gierlinski & Newton (‘06)
20
10
5
2
1 0.5
[10
37 e
rg s
-1] Ecut vs. L
Large ς: Bright/Slow transition‣RIAF→Opt. thin low-β→Opt. thick disk‣T anti-correlates with M in opt. thin low-β➡explain Bright/Hard state
Small ς (≒conventional model): Dark/Fast ‣RIAF→Opt. thick low-β
If the magnetic flux escape with the jet around the jet line,the disk undergoes a transition to equilibrium state
withsmaller ς. Bright/Hard
Electron Temperature
Thermal Equilibrium Curves@5rs
Mass
Acc
reti
on
Rate
Lum
inosi
ty
ζ= 0(dotted) 0.3(dashed) 0.6(thick solid) Conventional(thin solid)
Miyakawa+ (‘08)
Temperature
Method: Radial Structure of the Method: Radial Structure of the DisksDisks
5rs 10rs 100rs
5rs 10rs 100rs
solutions for fixed M andvarious ϖ
Local thermal equilibrium curve: solutions for fixed ϖ and various M
Radial dependenceWhen 3 solutions are found for the same M: Since we focus on the Hard-to-Soft transition,➡we choose the RIAF solution.
Surface densityMass
Acc
reti
on
Rate
Surface density Surface density
Surf
ace
den
sity
Radius
.
.
Φ increase inward (ζ=0.6)
RIAF
Low-β
RIAF
RIAF
RIAF
RIAF
Low-β
Low-β
Low-β
Low-β
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
Result: Radial Structure of the Result: Radial Structure of the DisksDisks
Φ remain constant (ζ=0)
RIAF
Low-β
Low M: The disk is in the RIAF at every radius.
High M: The disk undergoes a transition from RIAF to Low-β from the outer radii. [Φ increase inward(ζ=0.6)]Opt. thin low-β[Φ remain const.(ζ=0)]Opt. thick low-β (Teff ∝ ϖ-3/4)
Higher M: The disk undergoes a transition to Slim from the inner radii.
Slim
Tem
pera
ture
Opti
cal d
ep
th
Radius Radius
.
.
.
.
.
Bright/Slowtransition
Dark/Fast transition
Mass
Acc
reti
on
Rate
The Opt. Thick Low-β Disk: Thermal DiskThe Opt. Thick Low-β Disk: Thermal Disk
Radiation mechanism: Black body radiationRadial distribution of the effective temperature: Teff∝ϖ-3/4 Same as that for the Standard
disk!X-ray spectrum in Low/Hard state of Cyg X-1(Makishima et al.
(2008)
?RIAF
Std.
Slim
SLE
Low-β
Limit cycle observed in GRS 1915+105:
Slim ⇔ Opt. thick low-β?(Smaller variation in luminosity
compared to that expected from Slim⇔Std.)
Surface density
Thermal Equilibrium Curves@5rs
Mass
Acc
reti
on
Rate ζ= 0(dotted)
0.3(dashed) 0.6(thick solid) Conventional(thin solid)
SummarySummaryWe construct 1D steady models incorporating We construct 1D steady models incorporating magnetic fields on the basis of the results of 3D magnetic fields on the basis of the results of 3D MD simulations.MD simulations.‣ We assume that the Maxwell stress (therefore the heating rate) is We assume that the Maxwell stress (therefore the heating rate) is
proportional the total pressure.proportional the total pressure.‣ We prescribe the magnetic flux advection rate (instead of We prescribe the magnetic flux advection rate (instead of β) β) to determine to determine
the azimuthal magnetic flux.the azimuthal magnetic flux.
RIAFSt
d.
Slim
SLE
Low-β
Opt. thin low-β disksOpt. thin low-β disks✓ exist at high mass accretion rate. exist at high mass accretion rate.
Temperature anti-correlates with mass Temperature anti-correlates with mass accretion rate.accretion rate.
explain the Bright/Slow transition explain the Bright/Slow transition and the and the Bright/Hard stateBright/Hard state
‣ Opt. thick low-β disksOpt. thick low-β disks✓ The radial distribution of the effective The radial distribution of the effective
temperature is same as that for the temperature is same as that for the standard disk.standard disk.➡ origin of the origin of the DBB componentDBB component??➡ Limit-cycle between Limit-cycle between Slim diskSlim disk⇔⇔Opt. Opt.
thick diskthick disk??
Thin Opacith Thick
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