CONNECTING RADIATION TO DYNAMICS THROUGH

SIMULATIONS

with Omer Blaes, Shigenobu Hirose, Jiming Shi and Jim Stone

Dynamics vs. Thermodynamics

Angular momentum transport is the primary driver of accretion—Does thermodynamics matter?

Yes—

It determines the equation of state

Radiation forces can determine the vertical profile of disk material

Both issues can affect the magnetic saturation level

How to Include Thermodynamics

Requirements:

Good energy conservation

Radiation transfer solution

Include radiation in momentum, energy conservation

Specifics of Implementation(Hirose, K. & Stone 2006)

Shearing box approximation

Finite difference MHD (Zeus)

Total and internal energy equations

Internal energy acquires gridscale losses at every operator-split update

Thermally-averaged opacity

Time-implicit flux-limited diffusion approximation (multi-grid solver)

Program Overview:Disk Vertical Structure as Function of pr/pg

What are the vertical profiles of density, stress, dissipation rate, pressure?

What determines the magnetic saturation level? Is it numerically well-defined?

What are the characteristics of the fluctuations, both in time and in position?

Is the structure stable: dynamically, thermally, or in terms of inflow?

Simplest Example: A Gas-Dominated Disk

Time-Averaged Density and Dissipation Profiles

Q=½/ § ¡ 1=2

Time-Averaged Pressure Profiles

Saturated Parker Modes in the Upper Layers

Does Pressure Control Magnetic Stress?Or Does Magnetic Stress Control Pressure?

Magnetic leads Pressure leads

Is the Stress Numerically Well-Defined?

standard resolution

2 x

2 x r

2 x z

2 x (r,,z)

independent of resolution

See also Davis, Stone & Pessah (2009) for the same conclusion

Box = (2Hx8Hx8H) Standard = (32x64x256)

Buoyancy Creates Vertical Coherence

black curve = standard resolution

red curve = 2 x (r,,z)

Thermodynamic Convergence Requires Better Resolution

When Radiation Forces Matter

Radiation-Dominance Is Generic(Shakura & Sunyaev 1973)

r=rg < 170(L=LE )16=21(M=M ¯ )2=21

Radiation pressure exceeds gas pressure for

That is, for the most interesting parts of all bright accretion disks around black holes

– Model Predicts Thermal Instability When pr > pg

$\int dz Q \propto p_r h$$

Shakura & Sunyaev 1976

In the model,

ZdzQ » ­

ZdzTrÁ » ®prh

When radiation pressure dominates, h / F =

ZdzQ

And pr » Qtcool » Q(h=c)¿ » (¿=c)ZdzQ

Thermal Instability

But Even When pr ~ 10pg , No Runaway!

Hirose, K. & Blaes (2009)

tcool = 15 orbits

Trends with pr/pg: Density Profile

pr=pg » 10pr=pg » 0:015

Trends with pr/pg: Pressure Profiles

pr=pg » 0:015 pr=pg » 10

Trends with pr/pg: Dissipation Profile

pr=pg » 0:015 pr=pg » 10

Trends with pr/pg: Dissipation Profile

»´ F(2=3)c(aT 4

0 )=¿

Trends with pr/pg: Stress vs. Pressure

Magnetic leads Pressure leads

A New Dimensional Analysis

Two quantities with dimensions stress:

p (Shakura & Sunyaev 1973)

c (Shakura & Sunyaev 1976) ;

Q = c2/required for radiation-supported hydrostatic balance

Fluctuation analysis reveals (weak) role of p; c does not fluctuate

Both enter through hydrostatic balance.

Averaging over T >> tcool Restores

Is the Radiation-Dominated Branch Inflow Unstable? (Lightman & Eardley 1974)

Mass conservation + angular momentum conservation give diffusion equation for surface density:

@§@t

+1R@@R

½1¼­

@@R

hr2­ _M (§ )

i¾

=0

Linearizing gives a negative effective diffusion coefficient when the accretion rate varies inversely with the surface density.

Linear Analysis of Inflow in Shearing Boxes Incomplete

Growth rate ~ k2; what about horizontal heat flow?

Shearing box can show bunching, but no net inflow

There is a maximum permitting thermal balance;

what happens if max ?

Summary

Thermodynamics dynamics• Hydrostatic balance/magnetic buoyancy determines

vertical structure, influences magnetic saturation (two characteristic stresses)

• MHD dynamics/hydrostatic balance drives thermodynamics, stabilizing radiation-dominated disks

• MHD dynamics control surface density=optical depth, influence thermal state, which influences magnetic saturation: is radiation-dominated inflow stable?