1. ACT III Astrophysical Radiation Magnetohydrodynamics
Being an Opera in Three Acts
Lecture 2: Two Applications
Radiatively Driven Winds/Accretion
Local Magnetic Dynamo
With Words & Music By
Tom Bogdan
Paul Charbonneau
October 10, 2018

2. ACT III Scene 1 Astrophysical Radiation Magnetohydrodynamics
Being an Opera in Three Acts
ACT III Scene 1
We at some times are minions of our theories,
The fault, dear Brutus, is not in ourselves,
But in our stars, that we are underlings.

3. Winds, Accretion Flows & Magnetoconvection
Gravitational Field
Entropy Production
Adiabatic (Reversible)
“An essential difference between the dynamics of radiating and non-radiating fluids is that because photons typically have much longer mean free paths than their material counterparts, they can introduce a fundamental global coupling between widely separated parts of the flow, which must be treated by a full transport theory.”
--Dimitri Mihalas
Internal Energy
Kinetic Energy
Matter
Turbulence
Electro-Magnetic Field
Radiation Field

4. Winds, Accretion Flows Gravitational Field Internal Energy
“I've always enjoyed learning how things work. ... Simply endless puzzles and problems that come to light, some of them trivial, amusing, some of them very important and I take great pleasure in learning them.”
--Eugene Parker
Internal Energy
Kinetic Energy
Matter
Turbulence
“We do not argue with the critic who urges that the stars are not hot enough for this process; we tell him to go and find a hotter place.”
--Arthur Eddington
Radiation Field
It is very hard to do one-dimensional MHD in spherical geometry. Sorry…

5. + + The Radiation Field 𝜕𝐸 𝜕𝑡 +𝛁·𝑭 =𝜅 [4𝜎𝑅𝑇4 −𝑐𝐸]+𝜅 1 𝑐 𝒖·F + ···
Matter Energy + +
Local Thermodynamic Equilibrium
Gray Atmosphere Approximation
1 𝑐 ⋅ 𝜕 𝐼 𝜈 𝜕𝑡 +𝐧·𝛁 𝐼 𝜈 = 𝜂 𝜈 − χ 𝜈 𝐼 𝜈
Thomson Scattering
Radiation Energy
𝜕𝐸 𝜕𝑡 +𝛁·𝑭 =𝜅 [4𝜎𝑅𝑇4 −𝑐𝐸]+𝜅 1 𝑐 𝒖·F + ···
1 𝑐2 ⋅ 𝜕𝑭 𝜕𝑡 +𝛁·ℙ=− 𝜅 𝑐 [ 𝑭 −𝒖{ 4𝜎𝑅 𝑐 𝑇4+ℙ}]+ ···
− 𝜎 𝑐 [ 𝑭 −𝒖{ 2 3 𝐸 +ℙ}]+ ···
Someone still needs to tell us how to determine the radiation pressure tensor!
Higher-order terms in the ratio
of u/c live here.

6. Steady Spherical Flows
Radiation
Mass
𝛁·{ 1 2 𝜌 𝒖 2 𝒖 + 𝜌𝑒+𝑝 𝒖 + 𝜌Φ𝒖 + 𝑭 }=0
𝛁·𝜌𝒖=0
Kinetic
Thermal
Gravitational
Ṁ =4𝜋𝑟2𝜌𝑢
Ė =Ṁ { 1 2 𝑢 2 +𝑒+ 𝑝 𝜌 +Φ}+4𝜋𝑟2𝐹
Nice---we have two constants from our conserved fluxes of energy and mass!!

7. 2 Integrals of the Motion
Ṁ =4𝜋𝑟2𝜌𝑢
Ė =Ṁ { 1 2 𝑢 2 +𝑒+ 𝑝 𝜌 +Φ}+4𝜋𝑟2𝐹

8. One Equation Short… ℒ* Ṁ =4𝜋𝑟2𝜌𝑢 Ė =Ṁ { 1 2 𝑢 2 +𝑒+ 𝑝 𝜌 +Φ}+4𝜋𝑟2𝐹
𝑝= γ−1 𝜌𝑐𝑉𝑇= γ−1 𝜌𝑒 = ℛ 𝜇 𝜌𝑇
Ṁ =4𝜋𝑟2𝜌𝑢
Ė =Ṁ { 1 2 𝑢 2 +𝑒+ 𝑝 𝜌 +Φ}+4𝜋𝑟2𝐹 ℒ*
Φ=− 𝐺𝑀∗ 𝑟
We need one more relation between 𝜌, 𝑢, and 𝑒!

9. One Equation Short… ℒ* Ṁ =4𝜋𝑟2𝜌𝑢 Ė =Ṁ { 1 2 𝑢 2 +𝑒+ 𝑝 𝜌 +Φ}+4𝜋𝑟2𝐹
𝑝= γ−1 𝜌𝑐𝑉𝑇= γ−1 𝜌𝑒 = ℛ 𝜇 𝜌𝑇
Ṁ =4𝜋𝑟2𝜌𝑢
Ė =Ṁ { 1 2 𝑢 2 +𝑒+ 𝑝 𝜌 +Φ}+4𝜋𝑟2𝐹 ℒ*
Φ=− 𝐺𝑀∗ 𝑟
We need one more relation between 𝜌, 𝑢, and 𝑒!
𝒖·𝛁𝒖=− 1 𝜌 𝛁𝑝 − 𝛁Φ + 1 𝜌 f

10. The Bondi/Parker Equation-I
𝒖·𝛁𝒖=− 1 𝜌 𝛁𝑝 − 𝛁Φ + 1 𝜌 f
𝑢 ⅆ𝑢 ⅆ𝑟 =− 1 𝜌 ⅆ𝑝 ⅆ𝑟 − ⅆ𝛷 ⅆ𝑟 + 1 𝜌 ⋅𝑓
Radiative Terms Enter Here!
𝑝 𝜌 = γ−1 𝑐𝑉𝑇= γ−1 𝑒 = ℛ 𝜇 𝑇≡𝑎2

11. The Bondi/Parker Equation-II
𝒖·𝛁𝒖=− 1 𝜌 𝛁𝑝 − 𝛁Φ + 1 𝜌 f
𝑢 ⅆ𝑢 ⅆ𝑟 =− 1 𝜌 ⅆ𝑝 ⅆ𝑟 − ⅆ𝛷 ⅆ𝑟 + 1 𝜌 ⋅𝑓
Radiative Terms Enter Here!
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ ⅆ𝑟 𝑝 𝜌 − 𝑝 𝜌2 ⅆ𝜌 ⅆ𝑟 − 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓
𝑝 𝜌 = γ−1 𝑐𝑉𝑇= γ−1 𝑒 = ℛ 𝜇 𝑇≡𝑎2

12. The Bondi/Parker Equation-III
𝒖·𝛁𝒖=− 1 𝜌 𝛁𝑝 − 𝛁Φ + 1 𝜌 f
𝑢 ⅆ𝑢 ⅆ𝑟 =− 1 𝜌 ⅆ𝑝 ⅆ𝑟 − ⅆ𝛷 ⅆ𝑟 + 1 𝜌 ⋅𝑓
Radiative Terms Enter Here!
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ ⅆ𝑟 𝑝 𝜌 − 𝑝 𝜌2 ⅆ𝜌 ⅆ𝑟 − 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ𝑎2 ⅆ𝑟 + 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 − 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓
𝑝 𝜌 = γ−1 𝑐𝑉𝑇= γ−1 𝑒 = ℛ 𝜇 𝑇≡𝑎2

13. The Bondi/Parker Equation-IV
𝒖·𝛁𝒖=− 1 𝜌 𝛁𝑝 − 𝛁Φ + 1 𝜌 f
𝑢 ⅆ𝑢 ⅆ𝑟 =− 1 𝜌 ⅆ𝑝 ⅆ𝑟 − ⅆ𝛷 ⅆ𝑟 + 1 𝜌 ⋅𝑓
Radiative Terms Enter Here!
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ ⅆ𝑟 𝑝 𝜌 − 𝑝 𝜌2 ⅆ𝜌 ⅆ𝑟 − 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ𝑎2 ⅆ𝑟 + 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 − 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓
Someone needs to tell us how to determine the isothermal sound speed!
𝑝 𝜌 = γ−1 𝑐𝑉𝑇= γ−1 𝑒 = ℛ 𝜇 𝑇≡𝑎2

14. The Bondi/Parker Equation-V
𝒖·𝛁𝒖=− 1 𝜌 𝛁𝑝 − 𝛁Φ + 1 𝜌 f
𝑢 ⅆ𝑢 ⅆ𝑟 =− 1 𝜌 ⅆ𝑝 ⅆ𝑟 − ⅆ𝛷 ⅆ𝑟 + 1 𝜌 ⋅𝑓
Radiative Terms Enter Here!
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ ⅆ𝑟 𝑝 𝜌 − 𝑝 𝜌2 ⅆ𝜌 ⅆ𝑟 − 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ𝑎2 ⅆ𝑟 + 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 − 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓
We know how to determine the isothermal sound speed!
𝒖·𝛁𝑒+· 𝑝 𝜌 𝛁·𝒖=𝑇 𝑠
𝑝 𝜌 = γ−1 𝑐𝑉𝑇= γ−1 𝑒 = ℛ 𝜇 𝑇≡𝑎2

15. The Internal Energy Equation-I
𝒖·𝛁𝑒+· 𝑝 𝜌 𝛁·𝒖=𝑇 𝑠
𝑢 ⅆ𝑒 ⅆ𝑟 − 𝑝 𝜌2 ⅆ𝜌 ⅆ𝑟 =𝑇 𝑠
Radiative Terms Enter Here!

16. The Internal Energy Equation-II
𝒖·𝛁𝑒+· 𝑝 𝜌 𝛁·𝒖=𝑇 𝑠
𝑢 ⅆ𝑒 ⅆ𝑟 − 𝑝 𝜌2 ⅆ𝜌 ⅆ𝑟 =𝑇 𝑠
1 γ−1 𝑢 ⅆ𝑎2 ⅆ𝑟 +𝑢 2𝑎2 𝑟 +𝑎2 ⅆ𝑢 ⅆ𝑟 =𝑇 𝑠
Radiative Terms Enter Here!
𝑝 𝜌 = γ−1 𝑐𝑉𝑇= γ−1 𝑒 = ℛ 𝜇 𝑇≡𝑎2

17. The Internal Energy Equation-III
𝒖·𝛁𝑒+· 𝑝 𝜌 𝛁·𝒖=𝑇 𝑠
𝑢 ⅆ𝑒 ⅆ𝑟 − 𝑝 𝜌2 ⅆ𝜌 ⅆ𝑟 =𝑇 𝑠
1 γ−1 𝑢 ⅆ𝑎2 ⅆ𝑟 +𝑢 2𝑎2 𝑟 +𝑎2 ⅆ𝑢 ⅆ𝑟 =𝑇 𝑠
Radiative Terms Enter Here!
− ⅆ𝑎2 ⅆ𝑟 = γ−1 [ 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 ]− 1 𝑢 γ−1 𝑇 𝑠

18. The Bondi/Parker Equation(s)
Kinetic
Thermal
Gravitational
Radiation
Ė =Ṁ { 1 2 𝑢 2 + γ γ−1 𝑎2− 𝐺𝑀∗ 𝑟 }+4𝜋𝑟2𝐹
Ṁ =4𝜋𝑟2𝜌𝑢
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ𝑎2 ⅆ𝑟 +[ 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 ]− 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓 I.
Radiative Terms Enter Here!
II.
− ⅆ𝑎2 ⅆ𝑟 = γ−1 [ 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 ]− 1 𝑢 γ−1 𝑇 𝑠
Now all we need are the radiative terms!

19. The Bondi/Parker Equation(s)
Kinetic
Thermal
Gravitational
Ė =Ṁ { 1 2 𝑢 2 + γ γ−1 𝑎2− 𝐺𝑀∗ 𝑟 }
Ṁ =4𝜋𝑟2𝜌𝑢
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ𝑎2 ⅆ𝑟 +[ 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 ]− 𝐺𝑀∗ 𝑟2 I.
II.
− ⅆ𝑎2 ⅆ𝑟 = γ−1 [ 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 ]
…but first let’s do it without radiation…

20. The Bondi/Parker Equation-No Radiation
Woops, what happens if γ=1?!
The Bondi/Parker Equation-No Radiation
Kinetic
Thermal
Gravitational
Now evaluate the energy and the mass flux at the critical point where r=rc and u=uc. If you can also determine a(rc) and ρ(rc) by some means---voilà!!
Ė =Ṁ { 1 2 𝑢 2 + γ γ−1 𝑎2− 𝐺𝑀∗ 𝑟 }
Ṁ =4𝜋𝑟2𝜌𝑢
Pressure Wins I.
𝑢 ⅆ𝑢 ⅆ𝑟 −γ 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 =γ 2𝑎2 𝑟 − 𝐺𝑀∗ 𝑟2
Parker
Supersonic
ⅆ𝑢 ⅆ𝑟
𝑢−γ 𝑎2 𝑢
Bondi
Subsonic
Gravity Wins

22. Isothermal Wind & Accretion
Ė =Ṁ { 1 2 𝑢 2 + γ γ−1 𝑎2− 𝐺𝑀∗ 𝑟 }
Now just some constant!
Ṁ =4𝜋𝑟2𝜌𝑢
Pressure Wins
𝑢 ⅆ𝑢 ⅆ𝑟 − 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 = 2𝑎2 𝑟 − 𝐺𝑀∗ 𝑟2 U
Parker
Supersonic
ⅆ ⅆ𝑟 [ 𝑢2 2 −𝑎2 log 𝑢] = ⅆ ⅆ𝑟 [2𝑎2 log 𝑟+ 𝐺𝑀∗ 𝑟 ]
Bondi
1 𝑈 exp 1 2 𝑈2=𝑅2 exp [ 2 𝑅 − ]
Subsonic
Gravity Wins R

23. The Bondi/Parker Equation(s)
Kinetic
Thermal
Gravitational
Radiation
Ė =Ṁ { 1 2 𝑢 2 + γ γ−1 𝑎2− 𝐺𝑀∗ 𝑟 }+4𝜋𝑟2𝐹
Ṁ =4𝜋𝑟2𝜌𝑢
𝑢 ⅆ𝑢 ⅆ𝑟 =− ⅆ𝑎2 ⅆ𝑟 +[ 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 ]− 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓 I.
Radiative Terms Enter Here!
II.
− ⅆ𝑎2 ⅆ𝑟 = γ−1 [ 2𝑎2 𝑟 + 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 ]− 1 𝑢 γ−1 𝑇 𝑠

24. Radiatively Driven Winds-Phenomenology
Kinetic
Thermal
Gravitational
Radiation
Ė =Ṁ { 1 2 𝑢 2 + γ γ−1 𝑎2− 𝐺𝑀∗ 𝑟 }+4𝜋𝑟2𝐹
Ṁ =4𝜋𝑟2𝜌𝑢
𝑢 ⅆ𝑢 ⅆ𝑟 −γ 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 =γ 2𝑎2 𝑟 − 𝐺𝑀∗ 𝑟2 + 1 𝜌 ⋅𝑓− 1 𝑢 γ−1 𝑇 𝑠 I. 𝛼
𝑢 ⅆ𝑢 ⅆ𝑟
CAK Theory of Stellar Winds
𝑢 ⅆ𝑢 ⅆ𝑟 −γ 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 =γ 2𝑎2 𝑟 − 𝐺𝑀∗ 𝑟2 + 𝐶 𝑟2 +𝐿𝑢 ⅆ𝑢 ⅆ𝑟
Continuum Scattering
Resonance Line Absorption

25. Radiatively Driven Winds-Phenomenology
As this approaches unity we reach the “Eddington Limit” for accretion.
Pressure Wins I.
𝑢 ⅆ𝑢 ⅆ𝑟 −γ 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 =γ 2𝑎2 𝑟 − 𝐺𝑀∗ 𝑟2 + 𝐶 𝑟2 +𝐿𝑢 ⅆ𝑢 ⅆ𝑟
Parker
Supersonic
ⅆ𝑢 ⅆ𝑟
=γ 2𝑎2 𝑟 −(1−ε) 𝐺𝑀∗ 𝑟2
𝑢(1−𝐿)− γ𝑎2 𝑢
Bondi
Line Absorption increases the critical speed.
Continuum Scattering reduces the critical radius.
Subsonic
Gravity Wins

26. Flows with Thomson Scattering
…but what about the energy density in the radiation field?
Flows with Thomson Scattering
𝜎= 𝜎𝑇𝜌 𝑚
𝐹= ℒ∗ 4𝜋 𝑟 2
𝜀= ℒ∗ ℒ 𝐸
𝐶 𝑟2 = 𝛾𝜎𝐹 𝜌𝑐
Pressure Wins I.
𝑢 ⅆ𝑢 ⅆ𝑟 −γ 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 =γ 2𝑎2 𝑟 − 𝐺𝑀∗ 𝑟2 + 𝐶 𝑟2
Parker
Supersonic
ⅆ𝑢 ⅆ𝑟
=γ 2𝑎2 𝑟 −(1−ε) 𝐺𝑀∗ 𝑟2
𝑢− γ𝑎2 𝑢
Bondi
Continuum Scattering reduces the critical radius.
Subsonic
Gravity Wins

27. Thomson Scattering-The Rest of the Story
𝛁·ℙ=− 𝜎 𝑐 𝑭 + ···
1 𝑈 exp 1 2 𝑈2=𝑅2 exp [ 2 𝑅 − ]
𝜎= 𝜎𝑇𝜌 𝑚
ⅆ 𝑃 𝑟𝑟 ⅆ𝑟 + 3 𝑃 𝑟𝑟 −𝐸 𝑟 =− 𝜎𝐹 𝑐
𝐹= ℒ∗ 4𝜋 𝑟 2
The Eddington Approximation
𝐸=𝑃 𝑟𝑟 + 𝑃 𝜃𝜃 + 𝑃 𝜙𝜙 = 3 𝑃 𝑟𝑟 U
Parker U2≈4 log R E≈ R-3 /(log R)1/2
ⅆ𝐸 ⅆ𝑟 =− 3𝜌𝜎𝑇ℒ∗ 4𝜋 𝑟 2 𝑚𝑐
For the Bondi Accretion Flow F/E = cH/J tends to zero as r goes to infinity.
For the Parker Wind F/E = cH/J tends to infinity as r goes to infinity…but cH/J cannot exceed c !!!! Now what???
Bondi U≈R-2 E≈ R-1
ⅆ𝐸 ⅆ𝑟 =− 3Ṁ𝜎𝑇ℒ∗ 16𝜋2 𝑟 4 𝑢𝑚𝑐 R

28. Radiatively Driven Winds-CAK Theory
P Cygni
𝑢 ⅆ𝑢 ⅆ𝑟 −γ 𝑎2 𝑢 ⅆ𝑢 ⅆ𝑟 −𝐿[𝑢 ⅆ𝑢 ⅆ𝑟 ]𝛼=γ 2𝑎2 𝑟 − 𝐺𝑀∗ 𝑟2
𝛼≈0.4
CAK Critical Point
Line Center
Parker Isothermal Wind Critical Point
𝜈′ = Γu 𝜈 (1− 1 𝑐 n·𝒖 )

29. ACT III Scene 2 Astrophysical Radiation Magnetohydrodynamics
Being an Opera in Three Acts
I will not lend thee a quadrature!
Why then the universe is my oyster---
Which I with computer shall open,
I will retort the knowledge in equipage.

30. Electro-Magnetic Field
Magnetoconvection
Gravitational Field
Entropy Production
Adiabatic (Reversible)
“It is nice to know that the computer understands the problem, but I would like to understand it too.”
--Eugene Wigner
Internal Energy
Kinetic Energy
Matter
Turbulence
“The first principle is that you must not fool yourself---and you are the easiest person to fool.”
--Richard Feynman
Electro-Magnetic Field
Radiation Field
“Das ist nicht nur nicht richtig; es ist nicht einmal falsch!”
--Wolfgang Pauli

31. Meet MURaM Specs: 4 Mm x 6 Mm x 6 Mm x 30 min 15.7 km / pixel
Fǁ = 6.3 x 1010 erg/cm2/sec
gǁ = 2.74 x 104 cm/sec2
<Bǁ> = 0
Chemical Composition:

32. Horizontal and Temporal Average
Constant Temperature Gradient
Optically Thin
Optically Thin
Horizontal and Temporal Average
Optically Thick
Constant Temperature
How the Radiation “sees” MuRAM
How the MHD “sees” MuRAM

33. 𝜕𝐸 𝜕𝑡 +𝛁·𝑭 =𝜅 [4𝜎𝑅𝑇4 −𝑐𝐸]+𝜅 1 𝑐 𝒖·F + ···
The Radiation Field ↗Ꝏ ↗Ꝏ
𝜕𝐸 𝜕𝑡 +𝛁·𝑭 =𝜅 [4𝜎𝑅𝑇4 −𝑐𝐸]+𝜅 1 𝑐 𝒖·F + ···
1 𝑐2 ⋅ 𝜕𝑭 𝜕𝑡 +𝛁·ℙ =− 𝜅 𝑐 [ 𝑭 −𝒖{ 4𝜎𝑅 𝑐 𝑇4+ℙ}]+ ··· ↗Ꝏ
ℙ≈ 1 3 𝐸 𝕀

34. The Radiation Field: Radiation Diffusion
Gray Atmosphere Approximation
1 𝑐 ⋅ 𝜕 𝐼 𝜈 𝜕𝑡 +𝐧·𝛁 𝐼 𝜈 = 𝜂 𝜈 − χ 𝜈 𝐼 𝜈
ℙ≈ 1 3 𝐸 𝕀
constant
𝛁·ℙ≈− 𝜅 𝑐 𝑭
𝑭 = − 4𝜎𝑅𝑇3 3𝜅 𝛁𝑇 ≈
constant
4𝜎𝑅𝑇4≈𝑐𝐸
<𝑭> = <− 4𝜎𝑅𝑇3 3𝜅 𝛁𝑇>

35. Horizontal and Temporal Average
<𝑭> = <− 4𝜎𝑅𝑇3 3𝜅 𝛁𝑇>
Optically Thin
Horizontal and Temporal Average
[<T2>]½
<T>
Optically Thick

36. Horizontal and Temporal Average
<𝑭> = < − 4𝜎𝑅𝑇3 3𝜅 >𝛁<𝑇>
Optically Thin
Horizontal and Temporal Average
[<T2>]½
<T>
Optically Thick

37. Horizontal and Temporal Average
<𝑭> = <− 4𝜎𝑅𝑇3 3𝜅 𝛁𝑇>
Optically Thin
Horizontal and Temporal Average
[<T2>]½
<T>
Optically Thick

38. The Radiation Field: “A Chromosphere”
Optically Thin
Optically Thick
How plane-parallel hydrostatic acolytes “describe” the Chromosphere…
How MURaM describes the Chromosphere!

39. The Material: Mass Flux
𝜕 𝜕𝑡 𝜌+𝛁·𝜌𝒖=0
𝜕 𝜕𝑡 <𝜌>+𝛁·<𝜌𝒖>=0
𝜕 𝜕𝑥ǁ <𝜌𝑢ǁ>=0
<𝜌𝑢ǁ>=𝐹ǁ

40. ρ(x,t) = <ρ>(xǁ) + ρ’(x,t) u(x,t) = <u>(xǁ) + u’(x,t)
𝜕 𝜕𝑡 𝜌+𝛁·𝜌𝒖=0
Optically Thick
<ρ><uǁ> = -< ρ’u’ǁ> + <ρuǁ>
Optically Thin
<ρuǁ>

41. The Material: Mean Stratification
<ρ>

42. The Material: Mean Flows
<uǁ>
|<u⊥>|

43. The Material: Turbulence

44. The Material: Anisotropic Turbulenceǁ ꓕ
Isotropic ꓕ ǁ

45. The Magnetic Field c𝛁 ⤫𝑬=− 𝜕𝑩 𝜕𝑡 𝛁·𝑩=0 c𝛁 ⤫𝑩=4𝜋𝑱 𝛁·<𝑩>=0
𝜕 𝜕𝑥ǁ <𝐵ǁ>=0
Area·<𝐵ǁ>=Φǁ

46. 0=Φǁ <|B|2/8πp> <|B|2>½ <Bǁ> ≈ 0 |<B⊥>|
High Beta
Low Beta
Optically Thin
0=Φǁ
Optically Thick
<|B|2>½
<Bǁ> ≈ 0
|<B⊥>|

47. The Magnetic Field: Local Dynamoǁ ǁ ꓕ ꓕ
Isotropic

48. CURTAIN CALL Astrophysical Radiation Magnetohydrodynamics
Being an Opera in Three Acts
CURTAIN CALL

49. Momentum Conservation
Microphysics is required to relate these quantities back to things we know…
𝜕 𝜕𝑡 𝜌𝒖+𝛁·(𝑝+𝜌𝒖𝒖)=−𝜌 𝛁Φ + f
𝛁·𝔾= 𝜌𝛁Φ
Newtonian gravity can not store any momentum…
1 𝑐2 ⋅ 𝜕𝑺 𝜕𝑡 +𝛁·𝕄= −𝜌 𝑒 𝑬− 1 𝑐 𝑱⤫𝑩
1 𝑐2 ⋅ 𝜕𝑭 𝜕𝑡 +𝛁·ℙ= 1 𝑐 0 ∞ 𝑑𝜈 𝑑𝐧 𝐧[ 𝜂 𝜈 − χ 𝜈 𝐼 𝜈 ]
𝜕𝕻 𝜕𝑡 +𝛁⋅ℿ=0
Macrophysics is about conserving what needs to be conserved…

50. Energy Conservation 𝜕 𝜕𝑡 1 2 𝜌 𝒖 2 +𝛁· 1 2 𝜌 𝒖 2 𝒖=−𝒖·𝛁𝑝−𝜌𝒖·𝛁Φ+𝒖·𝒇
Microphysics is required to relate these quantities back to things we know…
𝜕 𝜕𝑡 1 2 𝜌 𝒖 2 +𝛁· 1 2 𝜌 𝒖 2 𝒖=−𝒖·𝛁𝑝−𝜌𝒖·𝛁Φ+𝒖·𝒇
𝜕 𝜕𝑡 𝜌𝑒+𝛁· 𝜌𝑒+𝑝 𝒖=+𝒖·𝛁𝑝+𝜌𝑇 𝑠
Newtonian gravity needs mass to store energy…
𝜕 𝜕𝑡 1 2 𝜌Φ+𝛁·(𝜌Φ𝒖+𝑮)=𝜌𝒖·𝛁Φ
𝜕 𝜕𝑡 1 8𝜋 ( 𝑬 𝑩 2 )+𝛁· 𝑐 4𝜋 𝑬⤫𝑩=−𝑱·𝑬
𝜕𝐸 𝜕𝑡 +𝛁·𝑭 = 0 ∞ 𝑑𝜈 𝑑𝐧 [ 𝜂 𝜈 − χ 𝜈 𝐼 𝜈 ]
Macrophysics is about conserving what needs to be conserved…
𝜕𝔈 𝜕𝑡 +𝛁⋅𝕱=0

51. The “Golden Rule of RMHD”
“Always evaluate interactions between the matter and the classical fields in the comoving, e.g., rest- frame, of the material!!!” but… “Solve your equations in whatever is the most convenient frame of reference for your objectives.”

52. Corollary to the “Golden Rule of RMHD”
“You better know how to transform coordinates, physical quantities, fields, differential and integral operators (and anything else you can think of) between any two frames of reference, under all conditions.”

53. A Semi-Final Thought…! Ouch!
Zut!…now did I use a Galilean Boost instead of a Lorentz Boost?!…I wonder what this will do to the MHD drive…

54. A Final Thought…!
c Δt
c Δt

55. The End!