Steady Models of Magnetically Supported Black Hole Accretion Disks and their Application to...

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


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


Thin Opacith Thick

Adv. Heat


RIAF

hot(Te 〜 109K)

pgas


Adv. Heat

Slim Disk

moderatelyhot(T 〜 107-8K)

prad


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


Thin Opacith Thick

Adv. Heat


RIAF

hot(Te 〜 109K)

pgas


Adv. Heat

Slim Disk


prad


Heat

Rad

.


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


Energy eq.

ϖ-comp.

ϕ-comp.

z-comp.


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


Energy eq.

ϖ-comp.

ϕ-comp.

z-comp.

Induction eq.

ptot = pgas+prad+pmag


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


Thin Opacith Thick

Adv. Heat


RIAF

hot(Te 〜 109K)

pgas


Adv. Heat

Slim Disk


prad


Heat

Rad

.


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


Thin Opacith Thick

Mass

Acc

reti

on

Rate

BHHeat

Rad

.

pmag

RIAF

RIAF

Std.

Std.

Slim

Slim

Low

-β

Low-β


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


Mass

Acc

reti

on

Rate

Lum

inosi

ty


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


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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