Thermal Stability of Radiation-dominated Standard Accretion Disks

Shigenobu Hirose (JAMSTEC)

Collaborators: Julian Krolik (Johns Hopkins University)

Omer Blaes (UC Santa Barbara)

ReferencesSH, Krolik, Blaes, ApJ, 691, 1, 2009 SH, Blaes, Krolik, ApJ, 704, 781, 2009 Blaes, Krolik, SH, Shabaltas, ApJ, 733, 110, 2011

r

z

Wr� ⌘Rwr�dz

M

P ⌘RpdzTc

Local equilibrium flows in the ↵ model

H

• The ↵ model (Shakura & Sunyaev 1973) considers geometrically thin and optically

thick accretion disks.

• The basic equations are vertically-integrated ones, describing steady (equilib-

rium) accretion flows in a column:

3

4⇡M⌦2 = �3

2Wr�⌦ =

radiative diffusionz }| {4acT 4

c

3⌃thermal balance

P/2H

H=

⌃⌦2

2hydrostatic balance

P

2H=

a

3T 4

c

+⌃k

B

Tc

2µHequation of state

Wr� = �↵P ↵ prescription

/ ⌃5/3

/ ⌃�1

surface density [gcm�2]

↵ = 0.025, ⇠ = 5

M/MEdd

103 104 105 106

0.0001

0.0010

0.0100

0.1000

1.0000

M/M� = 6.62, r/rG = 30

gas-dominated

radiation-dominated

Two possible steady accretion flows

• For a given surface density at some radius, there exist two possible steady

accretion flows, a gas-dominated (low accretion rate) flow and a

radiation-dominated (high accretion rate) flow.

Radiation-dominated accretion flows in the central regions

r

Mradiation-dominated gas-dominated

• The ↵ model predicts that radiation dominates gas when

r

rG

< 23⇣ ↵

0.01

⌘ 221

✓M/M�108

◆ 221✓0.1

⌘

◆ 1621

M/M

Edd

0.01

! 1621

— This only weakly depends on ↵.

• Radiation-dominated accretion flows are always expected in the central regions

and thus responsible for emissions from accretion disks whose M is greater than

about 1% of MEdd

.

r

Mradiation-dominated gas-dominated

?

Issues regarding radiation-dominated accretion flows

• Shakura & Sunyaev (1976) raised two issues regarding radiation-dominated

accretion flows based on their ↵ model:

1. Steady accretion is possible only when the heating rate exactly equals to

c⌦2/T

, which is the cooling rate determined by hydrostatic balance.

2. The steady accretion is thermally unstable since Q+(E) / E and

Q�(E) / E1/2.

• If this is the case, (steady) standard disks don’t exist at accretion rates

corresponding to the radiation-dominated regime.

• What is the range of M/MEdd

for steady accretion?

radial: x

azimuth: y

vertical: zoutflow (no inflow)

periodic

shearing periodic

g(z) = ��2Kz

radial

vertical

RH r

z

M

emergent flux

surface density�

F = �BT 4e�

M/M� = 6.62, r/rG = 30

“First-principles” calculations of local accretion flows

• Local accretion flows are reproduced in the stratified shearing box, by means of

3D radiation MHD simulations with FLD approximation (ZEUS FLD).

• Unlike the ↵ model, the vertical structure is resolved with “first-principles”

treatment of turbulent heating and vertical heat transport.

• Once ⌦(r) is specified, the only physical parameter is ⌃.

X-ray binary case (c.f. Turner 2004)

⌃ = 0.25⇥ 105 [g cm�2]

⌃ = 1.07⇥ 105 [g cm�2]

⌃ = 2.15⇥ 105 [g cm�2]⌃ = 1.52⇥ 105 [g cm�2]⌃ = 1.24⇥ 105 [g cm�2]

⌃ = 0.75⇥ 105 [g cm�2]⌃ = 0.54⇥ 105 [g cm�2]

⌃ = 0.50⇥ 105 [g cm�2]

⌃ = 0.20⇥ 105 [g cm�2]

⌃ = 0.04⇥ 105 [g cm�2]

0 100 200 300 400orbits

1018

1019

1020

1021

1022

1023

1024

ther

mal

ene

rgy

[erg

/cm

2 ]• With ⌦ fixed, a series of simulations was run, changing ⌃.

• The accretion flows having different ⌃, which were not in thermal equilibrium

initially, found their own thermal equilibria (steady states) with a few exceptions.

Time evolution of thermal energy in a series of simulations

103 104 105 106

0.0001

0.0010

0.0100

0.1000

1.0000

−2 −1 0 1 2

runaway heating

runaway cooling or heating

/ ⌃5/3

/ ⌃�1

log(Prad/Pgas)

surface density [gcm�2]

↵ = 0.025, ⇠ = 5

M/MEdd

• The ↵ model with a single ↵ of 0.025 well describes the obtained steady solutions.

• Runaway heating and/or cooling were observed at some surface densities.

• Steady solutions are also found on the upper branch, where Prad

/Pgas

> 1.

“First principles” thermal equilibrium curve

(luminous efficiency = 0.1)

heating

radiative diffusion

radiation advection

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6Height (1.0e+07 cm)

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

(6.0

e+15

erg

cm

−3 s

−1)

c⌦2

T

Cooling rate by radiative diffusion is nearly constant at c⌦2/T as determined by

the hydrostatic balance.

Heating rate is not equal to the cooling rate by radiative diffusion.

Radiation advection, which is not considered in the (static) ↵ model, exactly

compensates for an excess of heating.

Radiative advection solves the heating rate issue

α model

unstable since dQ-/dE < dQ+/dE

Q+(E) / E

Q�(E) / E12

• Heating and cooling variations are highly stochastic and the observed stability

may not be explained simply by means of a linear theory that compares dQ+/dE

and dQ�/dE around an equilibrium.

Linear theory does not explain the stability

α model

unstable since dQ-/dP < dQ+/dP

dE

dt= E(t)� E

12 (t)

dE

dt=

em

(t)

tdiss

� E(t)

tcool

(E)

dem

dt= R(t)f(E)� e

m

(t)

tdiss

Toy model with causality reproduces the stable behavior

Q+(E) / E

Q�(E) / E12

• Heating variation causes pressure (cooling) variation while it is implicitly

assumed that pressure determines heating in the time-dependent ↵ model.

1017 1018 1019 1020 1021 1022 10231016

1017

1018

1019

1020

1021

1022

−2 −1 0 1 2

↵ = 0.025

log(Prad/Pgas)

to

ta

lstre

ss

[e

rg

cm

�3]

thermal pressure [erg cm�3]

Validity of the ↵ prescription

• The ↵ prescription is OK, but only in the time-averaged sense against

different surface densities: i.e. Wr�(P ) / P .

• This is not true as a time-dependent relation (i.e. Wr�(P (t)) / P (t)) in an

individual case of certain surface density.

103 104 105 106

0.0001

0.0010

0.0100

0.1000

1.0000

−2 −1 0 1 2

runaway heating

runaway cooling or heating

/ ⌃5/3

/ ⌃�1

log(Prad/Pgas)

surface density [gcm�2]

↵ = 0.025, ⇠ = 5

M/MEdd

Medium ⌃

Low ⌃

High ⌃

Athena VET (Jiang et al. 2013) vs. ZEUS FLD (Hirose et al. 2009)variable Eddington tensor flux-limited diffusion

• Jiang et al. (2013) did similar simulations with more sophisticated numerical

algorithms (Athena and VET closure) for selected three surface densities.

0 100 200 300 400orbits

1020

1021

1022

1023

1024th

erm

al e

nerg

y [e

rg/c

m2 ]

Large ⌃ (2.15⇥ 105 gcm�2)

ZEUS FLD

Athena VET

Thermal energy evolution in large ⌃ case

• Both results agree: the system cannot establish thermal balance and shows

runaway heating.

• Probably, there exists a critical ⌃ beyond which thermal balance is not

established as the ↵ model predicts.

103 104 105 106

0.0001

0.0010

0.0100

0.1000

1.0000

−2 −1 0 1 2

/ ⌃5/3

/ ⌃�1

log(Prad/Pgas)

surface density [gcm�2]

↵ = 0.025, ⇠ = 5

M/MEdd

High ⌃

Thermal energy evolution in small ⌃ case

Low ⌃ (0.25⇥ 105 gcm

�2)

ZEUS FLD

ZEUS FLD (wide box)

ZEUS FLD (large initial energy)

Athena VET

Athena FLD

“fragile”

0 100 200 300 400orbits

1020

1021

1022

1023

1024th

erm

al e

nerg

y [e

rg/c

m2 ]

energy perturbation

0%

5%

10%

2%103 104 105 106

0.0001

0.0010

0.0100

0.1000

1.0000

−2 −1 0 1 2

/ ⌃5/3

/ ⌃�1

log(Prad/Pgas)

surface density [gcm�2]

↵ = 0.025, ⇠ = 5

M/MEdd

Low ⌃

• Athena runs (VET and FLD) show runaway cooling.

• ZEUS runs show diverging results, depending on the initial condition or box size.

• The equilibrium ZEUS FLD (M/MEdd

⇡ 1) run also diverges when it is perturbed.

0 100 200 300 400orbits

1020

1021

1022

1023

1024th

erm

al e

nerg

y [e

rg/c

m2 ]

ZEUS FLD (wide box)

ZEUS FLD

Athena FLD

Athena VET (large box)

Athena FLD (wide box)

ZEUS FLD (different initial pert.)

Medium ⌃ (1.07⇥ 105 gcm�2)

Thermal energy evolution in medium ⌃ case

103 104 105 106

0.0001

0.0010

0.0100

0.1000

1.0000

−2 −1 0 1 2

/ ⌃5/3

/ ⌃�1

log(Prad/Pgas)

surface density [gcm�2]

↵ = 0.025, ⇠ = 5

M/MEdd

Medium ⌃

• ZEUS runs converge, showing thermally stability, in spite of changing the initial

perturbation or box size.

• Athena runs diverge, depending on the closure or box size, although Athena FLD

agrees with ZEUS FLD (M/MEdd

⇡ 0.1).

Sumary

Radiation-dominated accretion flows driven by MRI are contrasted with the ↵ model

in the following properties:

• Radiation advection: essential to achieve thermal equilibrium

• Stochasticity: a simple linear theory may not be applied.

• Causality: stress variation causes pressure variation, not vice versa as assumed

in the ↵ model.

Consequently, radiation-dominated accretion flows can be thermally stable at

M/MEdd

⇠ 0.1, but may not at M/MEdd

⇠ 1. (The case of M/MEdd

⇠ 0.1 is still

controversial between ZEUS FLD and Athena VET.)