1. Relativistic Variable Eddington Factor Plane-Parallel Case
Jun Fukue
Osaka Kyoiku University

2. Plan of My Talk ０ Astrophysical Jets １ Radiation Hydrodynamics
Moment Formalism
Eddington Approximation and Diffusion Approximation
Variable Eddington Factor and Flux-Limited Diffusion
２　Relativistic Radiation Hydrodynamics
Eddington Approximation in the Comoving Frame
Relativistic Eddington Factor
３　Analytical Approach：One-Tau Photo Oval
One-Tau Region in the Comoving Frame
Linear Analysis
Linear Approximation
Semi-Linear Approximation
４　Results：Comoving Radiation Fields and Eddington Factor
５　Discussion
６　Next Step
2018/12/7
QPO workshop

3. 0 Astrophysical Jets and Other Radiative Phenomena

4. Relativistic Astrophysical Jets
GRS1915
SS433
(YSO)
(CVs, SSXSs)
Crab pulsar
SS 433
microquasar
AGN
quasar
gamma-ray burst
3C273
M87
GRB
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QPO workshop

5. Relativistic Radiative Phenomena
Black Hole Accretion Flow
Relativistic Outflow
Gamma-Ray Burst
Neutrino Torus in Hypernova
Early Universe
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QPO workshop

6. 1 Preparation Moment Fomalism of Radiation Hydrodynamics

7. １． RHD Radiation Hydrodynamics
for
matter
Radiative Transfer
for
radiation
couple
Radiation Hydrodynamics
for
matter+radiation
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QPO workshop

8. １． RHD Fundamental Equation
Boltzman equation
for matter
Transfer equation
for radiation
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9. １． RHD Moment Formalism Moment equations Moment equations for matter
for radiation
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10. １． RHD Closure Relation 1 Closure relation Closure relation for matter
for radiation
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11. in optically thick to thin regimes
１． RHD 　Closure Relation 2
Closure relation
in optically thick to thin regimes
Tamazawa et al. 1975
OK: Physically correct in the limited cases
of tau=0 and infinity.
NG: Quantitatively incorrect in the region
around tau=1.
Levermore and Pomraning 1981
OK: Vector form
convenient for numerical simulations
NG: Diffusion type
cannot apply to an optically thin regime
causality problem
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QPO workshop

12. １． RHD Closure Relation 2 Ohsuga+ 2005 Order of (v/c)1 Time-dependent
Two dimensional
Flux-Limited Diffusion
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QPO workshop

13. 2 Motivation Validity of Eddington Approximation in Moment Fomalism of Relativistic Radiation Hydrodynamics

14. ２． RRHD Moment Formalism
Moment equations
for matter
continuity
momentum
energy
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QPO workshop

15. ２． RRHD Moment Formalism
Moment equations
for radiation
0th moment
1st moment
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16. ２． RRHD Closure Relation 1
Usual closure relation
for radiation
Isotropic assumption may break down
in the relativistic regime
even in the comoving frame.
Eddington Factor
Fukue 2005
Diffusion Approximation
Castor 1972
Ruggles, Bath 1979
Flammang 1982
Tullola+ 1986
Paczynski 1990
Nobili+ 1993, 1994
Numerical Simulations
Eggum+ 1985, 1988
Kley 1989
Okuda+ 1997
Kley, Lin 1999
Okuda 2002
Okuda+ 2005
Ohsuga+ 2005
Ohsuga 2006
In the comoving frame
Diffusion assumption may break down
in the optically thin and/or relativistic regimes
even in the comoving frame.
2018/12/7
QPO workshop

17. ２．RRHD Pathological Behavior
Violation of Eddington Approximation in the Relativistic Moment Formalism
　　　Turolla and Nobili 1988
　　　Turolla et al. 1995
　　　Dullemond 1999
　　　Fukue 2005
2018/12/7
QPO workshop

18. ２．RRHD Singularity at v＝c/√3or
β2=1/3
Deno.=0！
Plane-parallel flow
u=γβ=γv/c： four velocity
F：radiative flux
P：radiation pressure
J：mass flux
2018/12/7
QPO workshop

19. ２．RRHD Validity of Closure Relation
The cause of the singularity
is the Eddington approximation
in the comoving frame.
P0：radiation pressure in the comoving frame
E0：radiation energy density
　　P0= f E0： f =1/3
This assumption violates at v～c
since the radiation fields become anisotropic
even in the comoving frame.
2018/12/7
QPO workshop

20. ２． RRHD Closure Relation 2
What is a closure relation
in subrelativistic to relativistic regimes
Tamazawa+ 1975
f=(1+τ)/(1+3τ)
τ→τ/[γ(1+β)]
Fukue 2006
Akizuki, Fukue 2007
Abramowicz+ 1991
Koizumi, Umemura 2007
Fukue 2007; this study
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QPO workshop

21. ２． RRHD Closure Relation 2
What is a closure relation
in subrelativistic to relativistic regimes
Fukue 2006
Akizuki, Fukue 2007
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QPO workshop

22. ２． RRHD Closure Relation 2
What is a closure relation
in subrelativistic to relativistic regimes
mean free path l=
Koizumi, Umemura 2007
2018/12/7
QPO workshop

23. ２． RRHD Closure Relation 2
What is a closure relation
in subrelativistic to relativistic regimes
dβ/dτ β
Fukue 2007; this study
2018/12/7
QPO workshop

24. 3 Analytical Approach One-Tau Photo-Oval

25. Plane-Parallel Accelerating Flow in the Vertical Direction
v large
ρ small
vertical (z) flow
velocity (v) up
density (ρ) down
Shape of region of τ＝１ seen by a comoving observer
surface 光壺
Photo-vessel 光玉
Photo-oval
v small
ρ large
base
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26. ３．Photo Oval Linear Regime
Comoving observer
at z=z0，β＝β0
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27. ３．Photo Oval Linear Regime
Linear density gradient
One-tau range length
Optical depth in the s-direction
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28. ３．Photo Oval Linear Regime
Shape of photo-oval a=
0.5
0.4
0.3
0.2
0.1
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29. ３．Photo Oval Linear Regime
Breakup condition
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30. ３．Photo Oval Semi-Linear Regime
Optical depth in the s-direction
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31. ３．Photo Oval Semi-Linear Regime
Shape of photo-oval
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32. ３．Photo Oval Semi-Linear Regime
Breakup condition
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33. 4 Results Comoving Radiation Fields and Variable Eddington Factor

34. ４．Radiation Fields Comoving Radiation Fields
Radiative intensity in the comoving frame
Redshift due to relative velocity between the comoving observer and the inner wall of photo-oval
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QPO workshop

35. ４．Radiation Fields Linear Regime
Non-uniformity of the comoving radiative intensity
Redshift due to relative velocity
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36. ４．Radiation Fields Comoving Intensity
Radiataive intensity observed by the comoving observer at τ=τ0
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37. ４．VEF　 Linear Regime
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38. ４．VEF　 Linear Regime
3 × f (β, dβ/dτ)
dβ/dτ β
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39. ４．VEF Semi-Linear Regime
3 × f (β, dβ/dτ)
dβ/dτ β
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40. 5 Discussion

41. ５．Discussion　他の成分
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42. 　Concluding Remarks
We have semi-analytically examined the relativistic Eddington factor of the plane-parallel flow under the linear approximation.
We proved that it decreases in proportion to the velocity gradient in the subrelativistic regime.
We will study in the future an extremely relativistic regime, an optically thin case, a spherical flow, and so on.
2018/12/7
QPO workshop

43. +．Next Preliminary Results
not β
but u=γβ
Comoving observer at z=z0，u＝u0
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QPO workshop

44. +．Next Preliminary Results
3 × f (u, du/dτ)
du/dτ u
2018/12/7
QPO workshop

45. +．One more period　Observational Appearance of Relativistic Spherical Winds
Sumitomo+ 2007
Apparent Photosphere
Abramowicz+ 1991
Enchanced Luminosity
2018/12/7
QPO workshop