Thermal Stability of Radiation-dominatedStandard Accretion Disks
Shigenobu Hirose (JAMSTEC)
Collaborators:
Julian Krolik (Johns Hopkins University)
Omer Blaes (UC Santa Barbara)

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

Local equilibrium flows in the ↵ model• The ↵ model (Shakura & Sunyaev 1973) considers geometrically thin and opticallythick accretion disks.
• The basic equations are vertically-integrated ones, describing steady (equilibrium) accretion flows in a column:
radiative diffusion

33Ṁ ⌦2 =Wr ⌦ =4⇡
2P/2H⌃⌦2=H2Pa 4 ⌃kB Tc= Tc +2H
32µH
Wr = ↵Pz }| {4acTc43⌃

thermal balancehydrostatic balance
equation of state↵ prescriptionzHTcr
ṀRWr ⌘ wr dz
RP ⌘ pdz

Two possible steady accretion flows• For a given surface density at some radius, there exist two possible steadyaccretion flows, a gas-dominated (low accretion rate) flow and aradiation-dominated (high accretion rate) flow.

/⌃ 11.0000radiation-dominated0.1000Ṁ /ṀEdd

0.0100gas-dominated0.0010/ ⌃5/30.0001↵ = 0.025, ⇠ = 510

310

M/M

4

surface density [gcm102]

510

6= 6.62, r/rG = 30

Radiation-dominated accretion flows in the central regions• The ↵ model predicts that radiation dominates gas when⇣ ↵ ⌘r< 23rG
0.01

221

✓

M/M
1082 ✓
◆ 21

0.1
⌘

◆ 16
21

Ṁ /ṀEdd
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 Ṁ is greater thanabout 1% of ṀEdd .

rM
radiation-dominatedgas-dominatedIssues regarding radiation-dominated accretion flows• Shakura & Sunyaev (1976) raised two issues regarding radiation-dominatedaccretion flows based on their ↵ model:
1. Steady accretion is possible only when the heating rate exactly equals toc⌦2 /T , which is the cooling rate determined by hydrostatic balance.
2. The steady accretion is thermally unstable since Q+ (E) / E andQ (E) / E 1/2 .

• 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 Ṁ /ṀEdd for steady accretion?

?
rMradiation-dominatedgas-dominated“First-principles” calculations of local accretion flows• Local accretion flows are reproduced in the stratified shearing box, by means of3D 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 ⌃.
outflow (no inflow)
vertical: zemergent flux

F =4B Teffg(z) =2Kzz
verticalMHR

radialr

periodic
azimuth: ysurface density

M/M = 6.62, r/rG = 30
X-ray binary case (c.f. Turner 2004)

shearing periodic
radial: x

Time evolution of thermal energy in a series of simulations• 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.

1024

1021

⌃ = 0.25 ⇥ 105 [g cm 2 ]⌃ = 0.54 ⇥ 105 [g cm 2 ]⌃ = 0.75 ⇥ 105 [g cm 2 ]⌃ = 1.07 ⇥ 105 [g cm 2 ]⌃ = 1.24 ⇥ 105 [g cm 2 ]⌃ = 1.52 ⇥ 105 [g cm 2 ]⌃ = 2.15 ⇥ 105 [g cm 2 ]1020⌃ = 0.50 ⇥ 105 [g cm 2 ]thermal energy [erg/cm2]

1023
1022

⌃ = 0.20 ⇥ 105 [g cm 2 ]

1019
1018
0

⌃ = 0.04 ⇥ 105 [g cm 2 ]100

200
orbits300400

“First principles” thermal equilibrium curve
1.0000/⌃ 1
runawaycooling or heating

0.1000runawayheatingṀ /ṀEdd 0.0100(luminous efficiency = 0.1)

/ ⌃5/30.0010

log(Prad /Pgas )0.0001↵ = 0.025, ⇠ = 5

103

104

−2−1

105

01

2

106

surface density [gcm 2 ]• 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.

Radiative advection solves the heating rate issue
1.0
heating0.8c⌦2
T

0.6

0.2

(6.0e+15 erg cm−3 s−1)

0.4
radiativediffusion
radiation
advection

0.0
−0.2
−0.4
−0.6

−0.4

−0.2
0.0
0.2
Height (1.0e+07 cm)

0.4

0.6

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, exactlycompensates for an excess of heating.

Linear theory does not explain the stability
α modelQ+ (E) / E
1Q (E) / E 2
unstablesince dQ-/dE < dQ+/dE• 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+ /dEand dQ /dE around an equilibrium.

Toy model with causality reproduces the stable behavior
α modelQ+ (E) / E
1

Q (E) / E 2
unstablesince dQ-/dP < dQ+/dP

dEem (t)
E(t)
=dt
tdiss
tcool (E)demem (t)= R(t)f (E)dttdiss

dE= E(t)dt

1E 2 (t)• Heating variation causes pressure (cooling) variation while it is implicitly
assumed that pressure determines heating in the time-dependent ↵ model.

Validity of the ↵ prescription1022↵ = 0.025

total stress [erg cm 3 ]10211020

1019

1018

1017

log(Prad /Pgas )
−210

−1012

16

1017

1018

1019

1020

1021

1022

1023

thermal pressure [erg cm 3 ]• The ↵ prescription is OK, but only in the time-averaged sense against
different surface densities: i.e. W̄r (P̄ ) / P̄ .

• This is not true as a time-dependent relation (i.e. Wr (P (t)) / P (t)) in an
individual case of certain surface density.

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 numericalalgorithms (Athena and VET closure) for selected three surface densities.
Low ⌃1.0000Medium ⌃/⌃ 1High ⌃runawaycooling or heating

0.1000runawayheatingṀ /ṀEdd 0.0100/ ⌃5/30.0010log(Prad /Pgas )0.0001↵ = 0.025, ⇠ = 5103

104−2−1105

surface density [gcm 2 ]012106

Thermal energy evolution in large ⌃ case1024

thermal energy [erg/cm2]

ZEUS FLD

10Athena VET

23

/⌃ 1

1.0000High ⌃

0.10001022

Ṁ /ṀEdd 0.0100/ ⌃5/30.0010log(Prad /Pgas )0.000110↵ = 0.025, ⇠ = 5

21
10

3

10

4−2−1105

surface density [gcm 2 ]10Large ⌃ (2.15 ⇥ 105 gcm 2 )

20

0100

200orbits

300

400

• Both results agree: the system cannot establish thermal balance and shows
runaway heating.
• Probably, there exists a critical ⌃ beyond which thermal balance is notestablished as the ↵ model predicts.

012106

Thermal energy evolution in small ⌃ case
1024

ZEUS FLD(large initial energy)

energy perturbation

5%

thermal energy [erg/cm2]

0%

10Low ⌃

23

ZEUS FLD/⌃ 11.0000“fragile”

ZEUS FLD(wide box)

1022

10%0.1000

Ṁ /ṀEdd 0.0100

AthenaFLD/ ⌃5/30.0010

log(Prad /Pgas )

1021

1020
00.00012%↵ = 0.025, ⇠ = 510AthenaVET3104−2−11001

5surface density [gcm 2 ]Low ⌃ (0.25 ⇥ 105 gcm 2 )100

200orbits

300400• 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 (Ṁ /ṀEdd ⇡ 1) run also diverges when it is perturbed.

2106

Thermal energy evolution in medium ⌃ casethermal energy [erg/cm2]

1024

1023
Athena VET(large box)

Medium ⌃/⌃ 1

1.0000Athena FLD
0.10001022

ZEUS FLD
Ṁ /ṀEdd 0.0100

1021

ZEUS FLD(wide box)

Athena FLD(wide box)

ZEUS FLD(different initial pert.)/ ⌃5/30.0010log(Prad /Pgas )0.0001↵ = 0.025, ⇠ = 5

10

3

10

4−2−1

10

0

1

5

surface density [gcm 2 ]10Medium ⌃ (1.07 ⇥ 105 gcm 2 )

20

0

100

200orbits

300400• 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 FLDagrees with ZEUS FLD (Ṁ /ṀEdd ⇡ 0.1).

2106

Sumary

Radiation-dominated accretion flows driven by MRI are contrasted with the ↵ modelin 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 atṀ /ṀEdd ⇠ 0.1, but may not at Ṁ /ṀEdd ⇠ 1. (The case of Ṁ /ṀEdd ⇠ 0.1 is still
controversial between ZEUS FLD and Athena VET.)