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 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

0 Astrophysical Jets and Other Radiative Phenomena

Relativistic Astrophysical Jets GRS1915 SS433 (YSO) (CVs, SSXSs) Crab pulsar SS 433 microquasar AGN quasar gamma-ray burst 3C273 M87 GRB 2018/12/7 QPO workshop

Relativistic Radiative Phenomena Black Hole Accretion Flow Relativistic Outflow Gamma-Ray Burst Neutrino Torus in Hypernova Early Universe 2018/12/7 QPO workshop

1 Preparation Moment Fomalism of Radiation Hydrodynamics

１． RHD Radiation Hydrodynamics for matter Radiative Transfer for radiation couple Radiation Hydrodynamics for matter+radiation 2018/12/7 QPO workshop

１． RHD Fundamental Equation Boltzman equation for matter Transfer equation for radiation 2018/12/7 QPO workshop

１． RHD Moment Formalism Moment equations Moment equations for matter for radiation 2018/12/7 QPO workshop

１． RHD Closure Relation 1 Closure relation Closure relation for matter for radiation 2018/12/7 QPO workshop

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 2018/12/7 QPO workshop

１． RHD Closure Relation 2 Ohsuga+ 2005 Order of (v/c)1 Time-dependent Two dimensional Flux-Limited Diffusion 2018/12/7 QPO workshop

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

２． RRHD Moment Formalism Moment equations for matter continuity momentum energy 2018/12/7 QPO workshop

２． RRHD Moment Formalism Moment equations for radiation 0th moment 1st moment 2018/12/7 QPO workshop

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

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

２．RRHD Singularity at v＝c/√3 or 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

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

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 2018/12/7 QPO workshop

２． RRHD Closure Relation 2 What is a closure relation in subrelativistic to relativistic regimes Fukue 2006 Akizuki, Fukue 2007 2018/12/7 QPO workshop

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

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

3 Analytical Approach One-Tau Photo-Oval

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 2018/12/7 QPO workshop

３．Photo Oval Linear Regime Comoving observer at z=z0，β＝β0 2018/12/7 QPO workshop

３．Photo Oval Linear Regime Linear density gradient One-tau range length Optical depth in the s-direction 2018/12/7 QPO workshop

３．Photo Oval Linear Regime Shape of photo-oval a= 0.5 0.4 0.3 0.2 0.1 2018/12/7 QPO workshop

３．Photo Oval Linear Regime Breakup condition 2018/12/7 QPO workshop

３．Photo Oval Semi-Linear Regime Optical depth in the s-direction 2018/12/7 QPO workshop

３．Photo Oval Semi-Linear Regime Shape of photo-oval 2018/12/7 QPO workshop

３．Photo Oval Semi-Linear Regime Breakup condition 2018/12/7 QPO workshop

4 Results Comoving Radiation Fields and Variable Eddington Factor

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 2018/12/7 QPO workshop

４．Radiation Fields Linear Regime Non-uniformity of the comoving radiative intensity Redshift due to relative velocity 2018/12/7 QPO workshop

４．Radiation Fields Comoving Intensity Radiataive intensity observed by the comoving observer at τ=τ0 2018/12/7 QPO workshop

４．VEF　 Linear Regime 2018/12/7 QPO workshop

４．VEF　 Linear Regime 3 × f (β, dβ/dτ) dβ/dτ β 2018/12/7 QPO workshop

４．VEF Semi-Linear Regime 3 × f (β, dβ/dτ) dβ/dτ β 2018/12/7 QPO workshop

5 Discussion

５．Discussion　他の成分 2018/12/7 QPO workshop

　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

+．Next Preliminary Results not β but u=γβ Comoving observer at z=z0，u＝u0 2018/12/7 QPO workshop

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

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