connecting radiation to dynamics through simulations with omer blaes, shigenobu hirose, jiming shi...
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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?