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Relativistic Variable Eddington Factor Plane-Parallel Case

Relativistic Variable Eddington Factor Plane-Parallel Case. Jun Fukue Osaka Kyoiku University. Plan of My Talk. ０　 Astrophysical Jets １　 Radiation Hydrodynamics Moment Formalism Eddington Approximation and Diffusion Approximation Variable Eddington Factor and Flux-Limited Diffusion

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Relativistic Variable Eddington Factor Plane-Parallel Case

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  1. Relativistic Variable Eddington FactorPlane-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 • Moment Formalism • 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 • Linear Analysis • Linear Approximation • Semi-Linear Approximation ５　Discussion ６　Next Step QPO workshop

  3. 0 Astrophysical JetsandOther Radiative Phenomena

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

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

  6. 1 PreparationMoment FomalismofRadiation Hydrodynamics

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

  8. １． RHD Fundamental Equation Transfer equation for radiation Boltzman equation for matter QPO workshop

  9. １． RHD Moment Formalism Moment equations for matter Moment equations for radiation QPO workshop

  10. １． RHD Closure Relation 1 Closure relation for matter Closure relation for radiation QPO workshop

  11. 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 QPO workshop

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

  13. 2 MotivationValidity of Eddington Approximationin Moment FomalismofRelativistic Radiation Hydrodynamics

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

  15. ２． RRHD Moment Formalism Moment equations for radiation • 0th moment • 1st moment QPO workshop

  16. ２． RRHD Closure Relation 1 Usual closure relation for radiation • 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 Isotropic assumption may break down in the relativistic regime even in the comoving frame. In the comoving frame Diffusion assumption may break down in the optically thin and/or relativistic regimes even in the comoving frame. QPO workshop

  17. Violation of Eddington Approximation in the Relativistic Moment Formalism Turolla and Nobili 1988 Turolla et al. 1995 Dullemond 1999 Fukue 2005 ２．RRHDPathological Behavior QPO workshop

  18. ２．RRHDSingularity at v＝c/√3 u2=1/2 or β2=1/3 Deno.=0！ Plane-parallel flow u=γβ=γv/c： four velocity F：radiative flux P：radiation pressure J：mass flux QPO workshop

  19. ２．RRHDValidity 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= fE0： f =1/3 This assumption violates at v～c since the radiation fields become anisotropic even in the comoving frame. 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 QPO workshop

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

  24. 3 Analytical ApproachOne-Tau Photo-Oval

  25. vertical (z) flow velocity (v) up density (ρ) down Shape of region of τ＝１ seen by a comoving observer Plane-Parallel Accelerating Flowin the Vertical Direction v large ρ small surface 光壺 Photo-vessel 光玉 Photo-oval v small ρ large base QPO workshop

  26. ３．Photo OvalLinear Regime Comoving observer at z=z0，β＝β0 QPO workshop

  27. Linear density gradient One-tau range length Optical depth in thes-direction ３．Photo OvalLinear Regime QPO workshop

  28. Shape of photo-oval ３．Photo OvalLinear Regime a= 0.5 0.4 0.3 0.2 0.1 QPO workshop

  29. Breakup condition ３．Photo OvalLinear Regime QPO workshop

  30. Optical depth in thes-direction ３．Photo OvalSemi-Linear Regime QPO workshop

  31. Shape of photo-oval ３．Photo OvalSemi-Linear Regime QPO workshop

  32. Breakup condition ３．Photo OvalSemi-Linear Regime QPO workshop

  33. 4 ResultsComoving Radiation FieldsandVariable Eddington Factor

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

  35. Non-uniformity of the comoving radiative intensity Redshift due to relative velocity ４．Radiation FieldsLinear Regime QPO workshop

  36. Radiataive intensity observed by the comoving observer at τ=τ0 ４．Radiation FieldsComoving Intensity QPO workshop

  37. ４．VEFLinear Regime QPO workshop

  38. 3 × f (β, dβ/dτ) ４．VEFLinear Regime dβ/dτ β QPO workshop

  39. 3 × f (β, dβ/dτ) ４．VEFSemi-Linear Regime dβ/dτ β QPO workshop

  40. 5 Discussion

  41. ５．Discussion他の成分 QPO workshop

  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. QPO workshop

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

  44. +．NextPreliminary Results 3 × f (u, du/dτ) du/dτ u QPO workshop

  45. Sumitomo+ 2007 +．One more periodObservational Appearance of Relativistic Spherical Winds Apparent Photosphere Abramowicz+ 1991 Enchanced Luminosity QPO workshop

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