New Analytical Formulae for Optically-Thin Accretion Flows

In a previous paper, we described new analytic formulae for optically-thick supercritical accretion flows (Watarai 2006, hereafter paper 1). Here we present analytic formulae for optically-thin one-temperature accretion flows including the advection-dominated regime, using the ``semi-iterative'' method described in paper 1. Our analytic formulae have two real solutions. The first solution corresponds to the advection-dominated accretion flow (ADAF), and the second solution corresponds to the radiation-dominated accretion flow described by Shapiro, Lightman, & Eardley (the so-called SLE model). Both solutions are given by a cubic equation for the advection parameter $f$, which is the ratio of the advection cooling rate $Q_{\rm adv}$ to the viscous heating rate $Q_{\rm vis}$, i.e., $f=Q_{\rm adv}/Q_{\rm vis}$. Most previous studies assume that $f$ is constant ($f \sim 1$ for the ADAF). However, it is clear that $f$ should be a function of the physical parameters of the radiative-cooling dominated regime. We found that the ratio $f$ can be written as a function of the radius, mass accretion rate, and viscous parameter $\alpha$. Using this formula, we can estimate the transition radius from the inner optically-thin ADAF to the outer optically-thick standard disk, which can be measured using observations of the quiescent state in black hole X-ray binaries.Comment: 7 pages, 6 figures, accepted for publication in PASJ (December 28, 2006

Optical Light Curves of Luminous Eclipsing Black Hole X-ray Binaries

We examine optical V-band light curves in luminous eclipsing black hole X-ray binaries, using a supercritical accretion/outflow model that is more realistic than the formerly used ones. In order to compute the theoretical light curve in the binary system, we do not only apply the global analytic solution of the disk, but also include the effect of the optically thick outflow. We found that the depth of eclipse of the companion star by the disk changes dramatically when including the effect of the outflow. Due to the effect of outflow, we can reproduce the optical light curve for typical binary parameters in SS433. Our model with outflow velocity v~3000 km/s can fit whole shape of the averaged V-band light curve in SS433, but we found a possible parameter range consistent with observations, such as \dot{M}~5000-10000 L_E/c^2 (with L_E being the Eddington luminosity and $c$ being the speed of light) and T_C~10000K-14000 K for the accretion rate and donor star temperature, respectively. Furthermore, we briefly discuss observational implications for ultraluminous X-ray sources.Comment: 8 pages, 9 figures, accepted for publication in PAS

Where is a Marginally Stable Last Circular Orbit in Super-Critical Accretion Flow?

Impressed by the widespread misunderstanding of the issue, we return to the old question of the location of the inner edge of accretion disk around black hole. We recall the fundamental results obtained in the 1970's and 1980's by Warsaw and Kyoto research groups that proved, in particular, that the inner edge does not coincide with the location of the innermost stable Keplerian circular orbit. We give some novel illustrations of this particular point and of some other fundamental results obtained by Warsaw and Kyoto groups. To investigate the flow dynamics of the inner edge of accretion disk, we carefully solve the structure of the transonic flow and plot the effective potential profile based on the angular-momentum distribution calculated numerically. We show that the flow does not have a potential minimum for accretion rates, {\dot M} > 10 L_E/c^2 (with L_E being the Eddington luminosity and $c$ being the speed of light). This property is realized even in relatively small viscosity parameters (i.e., \alpha ~ 0.01), because of the effect of pressure gradient. In conclusion, the argument based on the last circular orbit of a test particle cannot give a correct inner boundary of the super-critical flow and the inner edge should be determined in connection with radiation efficiency. The same argument can apply to optically thin ADAF. The interpretation of the observed QPO frequencies should be re-considered, since the assumption of Kepler rotation velocity can grossly over- or underestimate the disk rotation velocity, depending on the magnitude of viscosity.Comment: 7 pages, 3 figures, accepted for PAS

Eclipsing light curves for accretion flows around a rotating black hole and atmospheric effects of the companion star

We calculate eclipsing light curves for accretion flows around a rotating black hole taking into account the atmospheric effects of the companion star. In the cases of no atmospheric effects, the light curves contain the information of the black hole spin because most of the X-ray photons around 1 keV usually come from the blueshifted part of the accretion flow near the black hole shadow, and the size and the position of the black hole shadow depend on the spin. In these cases, when most of the emission comes from the vicinity of the event horizon, the light curves become asymmetric at ingress and egress. We next investigate the atmospheric absorption and scattering effects of the companion stars. By using the solar-type atmospheric model, we have taken into account the atmospheric effects of the companion star, such as the photoionization by HI and HeI. We found that the eclipsing light curves observed at 1 keV possibly contain the information of the black hole spin. However, in our atmospheric model, the effects of the atmosphere are much larger than the effects of the black hole spin. Therefore, even in the case that the light curves contain the information of the black hole spin, it may be difficult to extract the information of the black hole spin if we do not have the realistic atmospheric profiles, such as the temperature, and the number densities for several elements. Even in such cases, the light-curve asymmetries due to the rotation of the accretion disc exist. Only when we have the reliable atmospheric model, in principle, the information of the strong-gravity regions, such as the black hole spin, can be obtained from the eclipsing light curves.Comment: Takahashi R., Watarai K., 2007, MNRAS, 374, 151

Eclipsing Light-Curve Asymmetry for Black-Hole Accretion Flows

We propose an eclipsing light-curve diagnosis for black-hole accretion flows. When emission from an inner accretion disk around a black hole is occulted by a companion star, the observed light curve becomes asymmetric at ingress and egress on a time scale of 0.1-1 seconds. This light-curve analysis provides a means of verifying the relativistic properties of the accretion flow, based on the special/general relativistic effects of black holes. The ``skewness'' for the eclipsing light curve of a thin disk is $\sim 0.08$, whereas that of a slim disk is $\sim 0$, since the innermost part is self-occulted by the disk's outer rim.Comment: 7 pages, 4 figures, PASJ accepte

Slim Disk: Viscosity Prescriptions and Observational Implications

We examine the effects of the different viscosity prescriptions and the magnitude of the viscosity parameter, $\alpha$, on the structure of the slim disk, and discuss the observational implications on accretion-flow into a stellar-mass black hole. For the range of $\alpha = 10^{-2} \sim 10^{0}$ we calculate the disk spectra and from spectral fitting we derive \Tin, maximum temperature of the disk, \Rin, the size of the region emitting blackbody radiation with \Tin, and $p\equiv -{\rm dln} T_{\rm eff}/{\rm dln}~r$, the slope of the effective temperature distribution. It was founded that the estimated \Tin slightly increases as $\alpha$ increases. This is because the larger the magnitude of viscosity is, the larger becomes the accretion velocity and, hence, the more enhanced becomes advective energy transport, which means less efficient radiative cooling and thus higher temperatures. Furthermore we check different viscosity prescriptions with the form of the viscous stress tensor of $t_{r \varphi} = -\alpha \beta^{\mu}p_{\rm total}$, where $\beta$ is the ratio of gas pressure to total pressure, and $\mu$ is a parameter ($0 \le \mu \le 1$). For $\mu=0$ we have previously found that as luminosity approaches the Eddington, L_{\rm E, \Rin decreases below 3\rg (with $r_{\rm g}$ being Schwarzschild radius) and the effective temperature becomes flatter, $T_{\rm eff} \propto r^{-1/2}$. Such a slim-disk nature does not appear when $\mu$ is large, $\mu \sim 0.5$, even at $L_{\rm E}$. Hence, the temperature of the innermost region of the disk sensitively depends on the $\mu$ value. We can rule out the case with large $\mu~(\sim 0.5)$, since it will not be able to produce a drop in \Rin with an increase in luminosity as was observed in an ultraluminous X-ray source, IC~342, source 1.Comment: 8 pages, 7 figures, accepted for PAS

Accretion Disk Spectra of the Ultra-luminous X-ray Sources in Nearby Spiral Galaxies and Galactic Superluminal Jet Sources

Ultra-luminous Compact X-ray Sources (ULXs) in nearby spiral galaxies and Galactic superluminal jet sources share the common spectral characteristic that they have unusually high disk temperatures which cannot be explained in the framework of the standard optically thick accretion disk in the Schwarzschild metric. On the other hand, the standard accretion disk around the Kerr black hole might explain the observed high disk temperature, as the inner radius of the Kerr disk gets smaller and the disk temperature can be consequently higher. However, we point out that the observable Kerr disk spectra becomes significantly harder than Schwarzschild disk spectra only when the disk is highly inclined. This is because the emission from the innermost part of the accretion disk is Doppler-boosted for an edge-on Kerr disk, while hardly seen for a face-on disk. The Galactic superluminal jet sources are known to be highly inclined systems, thus their energy spectra may be explained with the standard Kerr disk with known black hole masses. For ULXs, on the other hand, the standard Kerr disk model seems implausible, since it is highly unlikely that their accretion disks are preferentially inclined, and, if edge-on Kerr disk model is applied, the black hole mass becomes unreasonably large (> 300 M_solar). Instead, the slim disk (advection dominated optically thick disk) model is likely to explain the observed super-Eddington luminosities, hard energy spectra, and spectral variations of ULXs. We suggest that ULXs are accreting black holes with a few tens of solar mass, which is not unexpected from the standard stellar evolution scenario, and that their X-ray emission is from the slim disk shining at super-Eddington luminosities.Comment: ApJ, accepte

New Analytical Formula for Supercritical Accretion Flows

We examine a new family of global analytic solutions for optically thick accretion disks, which includes the supercritical accretion regime. We found that the ratio of the advection cooling rate, $Q_{\rm adv}$, to the viscous heating rate, $Q_{\rm vis}$, i.e., $f=Q_{\rm adv}/Q_{\rm vis}$, can be represented by an analytical form dependent on the radius and the mass accretion rate. The new analytic solutions can be characterized by the photon-trapping radius, \rtrap, inside which the accretion time is less than the photon diffusion time in the vertical direction; the nature of the solutions changes significantly as this radius is crossed. Inside the trapping radius, $f$ approaches $f \propto r^0$, which corresponds to the advection-dominated limit ($f \sim 1$), whereas outside the trapping radius, the radial dependence of $f$ changes to $f \propto r^{-2}$, which corresponds to the radiative-cooling-dominated limit. The analytical formula for $f$ derived here smoothly connects these two regimes. The set of new analytic solutions reproduces well the global disk structure obtained by numerical integration over a wide range of mass accretion rates, including the supercritical accretion regime. In particular, the effective temperature profiles for our new solutions are in good agreement with those obtained from numerical solutions. Therefore, the new solutions will provide a useful tool not only for evaluating the observational properties of accretion flows, but also for investigating the mass evolution of black holes in the presence of supercritical accretion flows.Comment: 14 pages, 7 figures, accepted for publication in the Astrophysical Journa

Model for Relaxation Oscillations of Luminous Accretion Disk in GRS1915+105: Variable Inner Edge

To understand the bursting behavior of the microquasar GRS 1915+105, we calculate time evolution of a luminous, optically thick accretion disk around a stellar mass black hole undergoing limit-cycle oscillations between the high- and low- luminosity states. We, especially, carefully solve the behavior of the innermost part of the disk, since it produces significant number of photons during the burst, and fit the theoretical spectra with the multi-color disk model. The fitting parameters are \Tin (the maximum disk temperature) and \Rin (the innermost radius of the disk). We find an abrupt, transient increase in \Tin and a temporary decrease in \Rin during a burst, which are actually observed in GRS 1915+105. The precise behavior is subject to the viscosity prescription. We prescribe the radial-azimuthal component of viscosity stress tensor to be $T_{r \phi}=-\alpha \Pi (p_{\rm gas}/p)^{\mu}$, with $\Pi$ being the height integrated pressure, $\alpha$ and $\mu$ being the parameter, and $p$ and $p_{\rm gas}$ being the total pressure and gas pressure on the equatorial plane, respectively. Model with $\mu=0.1$ can produce the overall time changes of \Tin and \Rin, but cannot give an excellent fit to the observed amplitudes. Model with $\mu=0.2$, on the other hand, gives the right amplitudes, but the changes of \Tin and \Rin are smaller. Although precise matching is left as future work, we may conclude that the basic properties of the bursts of GRS 1915+105 can be explained by our ``limit-cycle oscillation'' model. It is then required that the spectral hardening factor at high luminosities should be about 3 at around the Eddington luminosity instead of less than 2 as is usually assumed.Comment: 11 pages, 5 figures, accepted for publication in Ap