Relativistic Radiative Flow in a Luminous Disk

Radiatively driven transfer flow perpendicular to a luminous disk was examined under a fully special relativistic treatment, taking into account radiation transfer. The flow was assumed to be vertical, and the gravity, the gas pressure, and the viscous heating were ignored. In order to construct the boundary condition at the flow top, the magic speed above the flat source was re-examined, and it was found that the magic speed above a moving source can exceed that above a static source ($\sim 0.45~c$). Then, the radiatively driven flow in a luminous disk was numerically solved, from the flow base (disk ``inside''), where the flow speed is zero, to the flow top (disk ``surface''), where the optical depth is zero. For a given optical depth and appropriate initial conditions at the flow base, where the flow starts, a loaded mass in the flow was obtained as an eigenvalue of the boundary condition at the flow top. Furthermore, a loaded mass and the flow final speed at the flow top were obtained as a function of the radiation pressure at the flow base; the flow final speed increases as the loaded mass decreases. Moreover, the flow velocity and radiation fields along the flow were obtained as a function of the optical depth. Within the present treatment, the flow three velocity $v$ is restricted to be within the range of $v < c/\sqrt{3}$, which is the relativistic sound speed, due to the relativistic effect.Comment: 8 pages, 5 figure

Milne-Eddington Solutions for Relativistic Plane-Parallel Flows

Radiative transfer in a relativistic plane-parallel flow, e.g., an accretion disk wind, is examined in the fully special relativistic treatment. Under the assumption of a constant flow speed, for the relativistically moving atmosphere we analytically obtain generalized Milne-Eddington solutions of radiative moment equations; the radiation energy density, the radiative flux, and the radiation pressure. In the static limit these solutions reduce to the traditional Milne-Eddington ones for the plane-parallel static atmosphere, whereas the source function nearly becomes constant as the flow speed increases. Using the analytical solutions, we analytically integrate the relativistic transfer equation to obtain the specific intensity. This specific intensity also reduces to the Milne-Eddinton case in the static limit, while the emergent intensity is strongly enhanced toward the flow direction due to the Doppler and aberration effects as the flow speed increases (relativistic peaking).Comment: 1o pages, 5 figure

Variable Eddington Factor in a Relativistic Plane-Parallel Flow

We examine the Eddington factor in an optically thick, relativistic flow accelerating in the vertical direction. % When the gaseous flow is radiatively accelerated and there is a velocity gradient, there also exists a density gradient. The comoving observer sees radiation coming from a closed surface where the optical depth measured from the observer is unity. Such a surface, called a {\it one-tau photo-oval}, is elongated in the flow direction. In general, the radiation intensity emitted by the photo-oval is non-uniform, and the photo-oval surface has a relative velocity with respect to the position of the comoving observer. Both effects introduce some degree of anisotropy in the radiation field observed in the comoving frame. As a result, the radiation field observed by the comoving observer becomes {\it anisotropic}, and the Eddington factor must deviate from the usual value of 1/3. Thus, the relativistic Eddington factor generally depends on the optical depth $\tau$ and the velocity gradient $du/d\tau$, $u$ being the four velocity. % In the case of a plane-parallel vertical flow, we obtain the shape of the photo-oval and calculate the Eddington factor in the optically thick regime. We found that the Eddington factor $f$ is well approximated by $f(\tau, \frac{du}{d\tau}) = {1/3} \exp (\frac{1}{u} \frac{du}{d\tau})$. % This relativistic variable Eddington factor can be used in various relativistic radiatively-driven flows.Comment: 8 pages, 7 figure

Self-Similar Solutions for ADAF with Toroidal Magnetic Fields

We examined the effect of toroidal magnetic fields on a viscous gaseous disk around a central object under an advection dominated stage. We found self-similar solutions for radial infall velocity, rotation velocity, sound speed, with additional parameter $\beta$ [$=c_{\rm A}^2/(2c_{\rm s}^2)$], where $c_{\rm A}$ is the Alfv\'en speed and $c_{\rm s}$ is the isothermal sound speed. Compared with the non-magnetic case, in general the disk becomes thick due to the magnetic pressure, and the radial infall velocity and rotation velocity become fast. In a particular case, where the magnetic field is dominant, on the other hand, the disk becomes to be magnetically supported, and the nature of the disk is significantly different from that of the weakly magnetized case.Comment: 5pages, 2figures, PASJ 58 (2006) in pres

Hoyle-Lyttleton Accretion onto Accretion Disks

We investigate Hoyle-Lyttleton accretion for the case where the central source is a luminous accretion disk. %In classical Hoyle-Lyttleton accretion onto a ``spherical'' source, accretion takes place in an axially symmetric manner around a so-called accretion axis. The accretion rate of the classical Hoyle-Lyttleton accretion onto a non-luminous object and $\Gamma$ the luminosity of the central object normalized by the Eddington luminosity. %If the central object is a compact star with a luminous accretion disk, the radiation field becomes ``non-spherical''. %Although the gravitional field remains spherical. In such a case the axial symmetry around the accretion axis breaks down; the accretion radius $R_{acc}$ generally depends on an inclination angle $i$ between the accretion axis and the symmetry axis of the disk and the azimuthal angle $\phi$ around the accretion axis. %That is, the cross section of accretion changes its shape. Hence, the accretion rate $\dot{M}$, which is obtained by integrating $R_{acc}$ around $\phi$, depends on $i$. % as well as $M$, $\Gamma$, and $v_\infty$. %In the case of an edge-on accretion ($i=90^{\circ}$), The accretion rate is larger than that of the spherical case and approximately expressed as $\dot{M} \sim \dot{M}_{HL} (1-\Gamma)$ for $\Gamma \leq 0.65$ and $\dot{M} \sim \dot{M}_{HL} (2-\Gamma)^2/5$ for $\Gamma \geq 0.65$. %Once the accretion disk forms and the anisotropic radiation fields are produced around the central object,the accretion plane will be maintained automatically (the direction of jets associated with the disk is also maintained). %Thus, the anisotropic radiation field of accretion disks drastically changes the accretion nature, that gives a clue to the formation of accretion disks around an isolated black hole.Comment: 5 figure

Radiative Transfer and Limb Darkening of Accretion Disks

Transfer equation in a geometrically thin accretion disk is reexamined under the plane-parallel approximation with finite optical depth. Emergent intensity is analytically obtained in the cases with or without internal heating. For large or infinite optical depth, the emergent intensity exhibits a usual limb-darkening effect, where the intensity linearly changes as a function of the direction cosine. For small optical depth, on the other hand, the angle-dependence of the emergent intensity drastically changes. In the case without heating but with uniform incident radiation at the disk equator, the emergent intensity becomes isotropic for small optical depth. In the case with uniform internal heating, the limb brightening takes place for small optical depth. We also emphasize and discuss the limb-darkening effect in an accretion disk for several cases.Comment: 7 pages, 4 figure

Relativistic Radiative Flow in a Luminous Disk II

Radiatively-driven transfer flow perpendicular to a luminous disk is examined in the relativistic regime of $(v/c)^2$, taking into account the gravity of the central object. The flow is assumed to be vertical, and the gas pressure as well as the magnetic field are ignored. Using a velocity-dependent variable Eddington factor, we can solve the rigorous equations of the relativistic radiative flow accelerated up to the {\it relativistic} speed. For sufficiently luminous cases, the flow resembles the case without gravity. For less-luminous or small initial radius cases, however, the flow velocity decreases due to gravity. Application to a supercritical accretion disk with mass loss is briefly discussed.Comment: 7 pages, 5 figure

Interpretation of the expansion law of planetary nebulae

We reproduce the expansion velocity--radius ($V_{\rm{exp}}$--$R_{\rm{n}}$) relation in planetary nebulae by considering a simple dynamical model, in order to investigate the dynamical evolution and formation of planetary nebulae. In our model, the planetary nebula is formed and evolving by interaction of a fast wind from the central star with a slow wind from its progenitor, the AGB star. In particular, taking account of the mass loss history of the AGB star makes us succeed in the reproduction of the observed $V_{\rm{exp}}$-$R_{\rm{n}}$ sequence. As a result, examining the ensemble of the observational and theoretical evolution models of PNe, we find that if the AGB star pulsates and its mass loss rate changes with time (from $\sim 10^{-6.4}M_{\odot}$ yr$^{-1}$ to $\sim 10^{-5}M_{\odot}$ yr$^{-1}$), the model agrees with the observations. In terms of observation, we suggest that there are few planetary nebulae with larger expansion velocity and smaller radius because the evolutionary time-scale of such nebulae is so short and the size of nebulae is so compact that it is difficult for us to observe them.Comment: 16 pages, 18 figure1, PASJ in pres

Relativistic Radiation Hydrodynamical Accretion Disk Winds

Accretion disk winds browing off perpendicular to a luminous disk are examined in the framework of fully special relativistic radiation hydrodynamics. The wind is assumed to be steady, vertical, and isothermal. %and the gravitational fields is approximated by a pseudo-Newtonian potential. Using a velocity-dependent variable Eddington factor, we can solve the rigorous equations of relativistic radiative hydrodynamics, and can obtain radiatively driven winds accelerated up to the {\it relativistic} speed. For less luminous cases, disk winds are transonic types passing through saddle type critical points, and the final speed of winds increases as the disk flux and/or the isothermal sound speed increase. For luminous cases, on the other hand, disk winds are always supersonic, since critical points disappear due to the characteristic nature of the disk gravitational fields. The boundary between the transonic and supersonic types is located at around $\hat{F}_{\rm c} \sim 0.1 (\epsilon+p)/(\rho c^2)/\gamma_{\rm c}$, where $\hat{F}_{\rm c}$ is the radiative flux at the critical point normalized by the local Eddington luminosity, $(\epsilon+p)/(\rho c^2)$ is the enthalpy of the gas divided by the rest mass energy, and $\gamma_{\rm c}$ is the Lorentz factor of the wind velocity at the critical point. In the transonic winds, the final speed becomes 0.4--0.8$c$ for typical parameters, while it can reach $\sim c$ in the supersonic winds.Comment: 6 pages, 5 figures; PASJ 59 (2007) in pres