We present the basic equations and sample solutions for the steady-state global transonic solutions of the advection-dominated accretion flows (ADAFs) with a causal viscosity prescription. The procedures for the stable numerical calculations and all explicit formula for obtaining the solutions of ADAFs are also presented. We solve the transonic solutions of ADAFs by using the explicit numerical integrations, as in the Runge-Kutta method. In this calculation method, we first solve the physical values at the sonic radius where L'Hopital's rule is used. Then, we numerically solve the coupled differential equations of the radial velocity, the angular momentum, and the sound speed from the sonic radius in the inward and outward directions. By the calculation procedures presented in this paper, we can cover all parameter spaces of the transonic solutions of ADAFs. Sample transonic solutions for ADAF-thick disk and ADAF-thin disk are presented. The explicit formula for the analytical expansion around the singular points, the sonic and the viscous points, are presented. If we set the diffusion timescale to null, the formalism in this study becomes the formalism of the acausal viscosity that is usually used in past studies for calculating of the ADAF structure.
We present self-similar solutions for advection-dominated accretion flows with radial viscous force in the presence of outflows from the accretion flow or infall. The axisymmetric flow is treated in variables integrated over polar sections and the effects of infall and outflows on the accretion flow are parametrised for possible configurations compatible with the self-similar solution. We investigate the resulting accretion flows for three different viscosity laws and derive upper limits on the viscosity parameter alpha. In addition, we find a natural connection to non-rotating and spherical a…
It has recently been demonstrated that the Shakura-Sunyaev prescription for the kinematic viscosity in an advection-dominated accretion disk yields physically reasonable solutions for the structure of the inflow close to the event horizon. In particular, no violations of relativistic causality occur at the horizon. This is somewhat surprising considering the diffusive nature of the angular momentum transport in the Shakura-Sunyaev scenario, and it is therefore natural to ask whether one can also obtain acceptable solutions for the disk structure based on the various alternative models for the …
Thermal instability is examined for advection-dominated one-temperature accretion disks. We consider axisymmetric perturbations with short wavelength in the radial direction. The viscosity is assumed to be sufficiently small for the vertical hydrostatic balance to hold in perturbed states. The type of viscosity is given either by the $α$-viscosity or by a diffusion-type stress tensor. Optically thick disks are found to be in general more unstable than optically thin ones. When the thermal diffusion is present, the optically thin disks become stable, but the optically thick disks are still unst…
We show the results of two-dimensional, azimuthally symmetric simulations of advection dominated fluid flows around black holes. We use the SPH method, and the alpha-prescription for the viscosity, including all terms in the stress tensor. We have performed calculations with 0.001 < alpha < 0.2 and find strong circulation patterns in the flow, on spatial scales that depend on the magnitude of the viscosity.
We present a modified description of advection-dominated accretion flows (ADAF) with a geometrical viscosity. A simplified treatment of the dynamics of the flow in the inner relativistic part is suggested and results are compared with the galactic center source Sgr A*.
We calculate the radiation spectra from advection-dominated accretion flows (ADAFs), taking into account the effects of a global magnetic field. Calculation is based on the analytic model for magnetized ADAFs proposed by Kaburaki, where a large-scale magnetic field controls the accretion process. Adjusting a few parameters, we find that our model can well reproduce the observed spectrum of Sagittarius A$^{*}$. The result is discussed in comparison with those of well-known ADAF models, where the turbulent viscosity controls the accretion process.
Two-dimensional accretion flows near black holes have been investigated by time-dependent hydrodynamical calculations. We assume that the flow is axisymmetric and that radiative losses of internal energy are negligible, so that the disc is geometrically thick and hot. Accretion occurs due to the overflow of the effective potential barrier near the black hole, similar to the case of the Roche lobe overflowing star in a binary system. We make no pre-assumptions on the properties of the flow, instead our models evolve self-consistently from an initially non-accreting state. The viscosity is due t…
We provide complete and global solutions of transonic flows around black holes, in presence of advection, rotation, heating and cooling. We show that for any degree of advection, there may exist two critical viscosity parameters $α_{c1, c2}$ such that for $α<α_{c1}$ the flow may pass through the inner sonic point only. For $α_{c1} <α<α_{c2}$, the flow may have shock waves, and for $α>α_{c2}$, the flow may again pass through the inner sonic point, depending on flow parameters. No new topologies emerge other than what we found earlier while studying viscous isothermal transonic flows. These find…
We have analyzed pair production in the innermost region of a two-temperature external soft photon comptonized accretion disk. We have shown that, if the viscosity parameter is greater than a temperature and r dependent critical value, the solution to the disk equation is double valued: one, advection dominated and the other, radiation dominated. When the viscosity parameter is below this critical value, the accretion rate has to be confined within a region limited by a minimum and a maximum accretion rates in order to have two steady state solutions. These critical accretion rates are shown t…
We obtain global solutions of radiatively inefficiently accretion flows around black holes. Whether and where convection develops in a flow are self-consistently determined with the mixing-length theory. The solutions can be divided into three types according to the strength of normal viscosity. Type I solution corresponds to large viscosity parameter $α\ga 0.1$, which is purely advection-dominated and with no convection, and has been extensively studied in the literature. Type II solution is for moderate $α\sim 0.01$, which has a three-zone structure. The inner zone is advection-dominated, th…
The $α$ turbulent viscosity formalism for accretion discs must be interpreted as a mean field theory. The extent to which the disc scale exceeds that of the turbulence determines the precision of the predicted luminosity $L_ν$. The assumption of turbulence and use of $\a$ implies: (1) Field line stretching generates a magnetic pressure $\gsim \a^2/6$ of the total pressure generally, and a 1 to 1 relation between $\a$ and the pressure ratio when shearing instabilities dominate the viscosity. (2) Large eddy sizes and speeds in typical advection dominated accretion flows (ADAFs) lead to a lower p…
It has been found that a class of optically-thin two-temperature advection-dominated accretion solutions explains many observations of low-luminosity accreting black holes. Here it is shown that these models give a satisfactory description also of higher luminosity systems, provided the viscosity parameter $α$ is large. The models reproduce the spectra of black hole X-ray binaries in the Low State, and explain the transition from the Low State to the High State at a critical mass accretion rate. The models also show that the X-ray/$γ$-ray spectra of X-ray binaries and active galactic nuclei sh…
A model for advection-dominated accretion flows (ADAFs) in a global magnetic field is proposed. In contrast to the well known ADAF models in which the viscosity of a fluid determines both angular momentum transfer and energy dissipation in the flow, the magnetic field and the electric resistivity, respectively, control them in this model. A manageable set of analytic solutions for the flow and the magnetic field is obtained to vertically non-integrated basic equations. This set describes mathematically a fully advective accretion flow and, in physically plausible situations for most AGNs, it i…
Using mean field MHD, we discuss the energetics of optically thin, two temperature, advection-dominated accretion flows (ADAFs). If the magnetic field is tangled and roughly isotropic, flux freezing is insufficient to maintain the field in equipartition with the gas. In this case, we expect a fraction of the energy generated by shear in the flow to be used to build up the magnetic field strength as the gas flows in; the remaining energy heats the particles. We argue that strictly equipartition magnetic fields are incompatible with a priori reasonable levels of particle heating; instead, the pl…
We solve for the structure of a hot accretion disc with unsaturated thermal Comptonisation of soft photons and with advection, generalizing the classical model of Shapiro et al. The upper limit on the accretion rate due to advection constrains the luminosity to $\simless 0.15 y^{3/5} α^{7/5}$ of the Eddington limit, where $y$ and $α$ are the Compton and viscosity parameters, respectively. The characteristic temperature and Thomson optical depth of the inner flow at accretion rates within an order of magnitude of that upper limit are $\sim 10^9$ K and $\sim 1$, respectively. The resulting spect…
We use the standard Runge-Kutta method to solve the set of basic equations describing black hole accretion flows composed of two-temperature plasma. We do not invoke any extra energy transport mechanism such as thermal conduction and do not specify any ad hoc outer boundary condition for the advection-dominated accretion flow (ADAF) solution. We find that in the case of high viscosity and non-zero radiative cooling, the ADAF solution can have an asymptotic approach to the Shakura-Sunyaev disk (SSD) solution, and the SSD-ADAF transition radius is close to the central black hole. Our results fur…
We present global solutions that describe advection-dominated accretion flows around black holes. The solutions are obtained by numerically solving a set of coupled ordinary differential equations corresponding to a steady axisymmetric height-integrated flow. The solutions satisfy consistent boundary conditions at both ends. We obtain well-behaved transonic solutions for a wide range of values of the viscosity parameter $α$, from 0.001 to 0.3. We do not find any need for shocks in our solutions, and disagree with previous claims that viscous accretion flows with low values of $α$ must have sho…
Using a simple thermo-hydrodynamic model that respects relativistic causality, we revisit the analysis of qualitative features of acoustic oscillations in the photon-baryon fluid. The growing photon mean free path introduces transient effects that can be modelled by the causal generalization of relativistic Navier-Stokes-Fourier theory. Causal thermodynamics provides a more satisfactory hydrodynamic approximation to kinetic theory than the quasi-stationary (and non-causal) approximations arising from standard thermodynamics or from expanding the photon distribution to first order in the Thomso…
Cold optically thick accretion disks with hot coronae and radial advection have been investigated. Within the framework of $α$-viscosity models, we assume that all the mass accretion and angular momentum transport take place in the cold disk, but that a fraction of the gravitational energy released is dissipated in the corona. Both the coronal energy dissipation and the advection heat transport have a stabilization effect on the thermal and viscous instabilities of the disk. If that more than $\sim 95$ percent of the total power is dissipated in the corona, then the locally unstable behavior o…
The structure of the boundary layer region between the disc and a comparatively slowly rotating star is studied using a causal prescription for viscosity. The vertically integrated viscous stress relaxes towards its equilibrium value on a relaxation timescale $τ$, which naturally yields a finite speed of propagation for viscous information. For a standard alpha prescription with alpha in the range 0.1-0.01, and ratio of viscous speed to sound speed in the range 0.02-0.5, details in the boundary layer are strongly affected by the causality constraint. We study both steady state polytropic model…
We show that viscous, transonic accretion disks which start with sub-Keplerian angular momentum at the outer boundary reproduce the features of the so-called thick, thin, slim and other accretion disk models depending upon the accretion rates and viscosity parameters. These disks are advection dominated in the pre-shock region and rotation dominated in the post-shock region when viscosity is less than a critical value. For a higher viscosity the shock disappears, and the flow has significant radial velocity only close to the black hole.
Accretion disc theory was first developed as a theory with the local heat balance, where the whole energy produced by a viscous heating was emitted to the sides of the disc. One of the most important new invention of this theory was a phenomenological treatment of the turbulent viscosity, known as ''alpha'' prescription, when the (r$ϕ$) component of the stress tensor was approximated by ($α$ P) with a unknown constant $α$. This prescription played the role in the accretion disc theory as well important as the mixing-length theory of convection for stellar evolution. Sources of turbulence in th…
We write down and solve equations describing steady state, optically thin, advection-dominated accretion onto a Kerr black hole. The mean flow, described by the relativistic fluid equations, is axisymmetric and vertically averaged. The effect of turbulence in the flow is represented by a viscous shear stress. Our treatment differs in several important ways from earlier work: we use a causal prescription for the shear stress, we do not assume the relativistic enthalpy is unity (this is important for rapidly rotating holes), and we use a relativistic equation of state. We present several represe…
We develop a nonlinear generalisation of the causal linear thermodynamics of bulk viscosity, incorporating positivity of the entropy production rate and the effective specific entropy. The theory is applied to viscous fluid inflation (which is necessarily far from equilibrium), and we find thermodynamically consistent inflationary solutions, both exponential and power-law.
The structure of the inner region of an advection-dominated accretion disk around a nonrotating black hole is explored by applying asymptotic analysis in the region just outside the event horizon. We assume that the viscous transport is described by the standard Shakura-Sunyaev prescription throughout the disk, including the inner region close to the horizon. One of our goals is to explore the self-consistency of this assumption by analyzing the causality of the viscous transport near the black hole. The effects of general relativity are incorporated in an approximate manner by utilizing a pse…
We have analyzed archival ASCA data on the soft X-ray transient source V404 Cyg in quiescence. We find that in the energy range 0.7 to 8.5 keV the spectrum is a hard power-law with a photon spectral index between 1.8 and 2.6 (90% confidence limits). We present a model of V404 Cyg in which the accretion flow has two components: (1) an outer thin disk with a small annular extent, and (2) a large interior region where the flow is advection-dominated. Nearly all the radiation in the infrared, optical, UV and X-ray bands is from the advection-dominated zone; the thin disk radiates primarily in the …
We provide a unified description of thermal equilibria of black hole accretion disks, including the newly-discovered advection-dominated solutions. We classify the solutions on the basis of optical depth and importance of advection cooling. We demonstrate that only four physically distinct topological types of equilibria exist. Two of the types correspond to optically thin and optically thick equilibria, while the other two types are distinguished by whether advection is negligible or dominant. A stable Shakura-Sunyaev disk exists only for accretion rates $\dot M$ below a certain maximum. Howe…
Viscous rotating accretion flows around black holes become advection-dominated when the accretion rate $\dot M$ is sufficiently low. Most of the accretion energy in such flows is stored within the gas and advected radially inward. The temperature is therefore very high, and much of the radiation comes out in hard X-rays and $γ$-rays. We have constructed an advection-dominated accretion flow model for the Galactic Center source Sagittarius A$^*$. The model consists of a $7\times10^5M_\odot$ black hole accreting at $\dot M=1.2\times10^{-5}α\, M_\odot{\rm yr^{-1}}$, where $α$ is the usual viscosi…
We describe new optically thin solutions for rotating accretion flows around black holes and neutron stars. These solutions are advection dominated, so that most of the viscously dissipated energy is advected radially with the flow. We model the accreting gas as a two temperature plasma and include cooling by bremsstrahlung, synchrotron, and Comptonization. We obtain electron temperatures $T_e\sim 10^{8.5}-10^{10}$K. The new solutions are present only for mass accretion rates $\dot M$ less than a critical rate $\dot M_{crit}$ which we calculate as a function of radius $R$ and viscosity paramet…
We aim to illustrate the role of hot protons in enhancing the magnetorotational instability (MRI) via the ``hybrid'' viscosity, which is due to the redirection of protons interacting with static magnetic field perturbations, and to establish that it is the only relevant mechanism in this situation. It has recently been shown by Balbus \cite{PBM1} and Islam & Balbus \cite{PBM11} using a fluid approach that viscous momentum transport is key to the development of the MRI in accretion disks for a wide range of parameters. However, their results do not apply in hot, advection-dominated disks, which…
We carry out the self-similar solutions of viscous-resistive accretion flows around a magnetized compact object. We consider an axisymmetric, rotating, isotheral steady accretion flow which contains a poloidal magnetic field of the central star. The dominant mechanism of energy dissipation is assumed to be the turbulence viscosity and magnetic diffusivity due to magnetic field of the central star. We explore the effect of viscosity on a rotating disk in the presence of constant magnetic diffusivity. We show that the dynamical quantities of ADAFs are sensitive to the advection and viscosity par…
We consider the radial structure of radiatively inefficient hydrodynamic accretion flows around black holes. We show that low-viscosity flows consist of two zones: an outer convection-dominated zone and an inner advection-dominated zone. The transition between these two zones occurs at ~50 Schwarzschild radii.
Recent hydrodynamical simulations of radiatively inefficient black hole accretion flows with low viscosity have demonstrated that these flows differ significantly from those described by an advection-dominated model. The black hole flows are advection-dominated only in their inner parts, but convectively dominated at radii R>100R_g. In such flows, the radiative output comes mostly from the convection part, and the radiative efficiency is independent of accretion rate and equals ~0.001. This value gives a limit for how dim an accreting black hole could be. It agrees with recent Chandra observat…
In this note we study the accretion disc that arises in hypercritical accretion of $\dot M\sim 10^8 M_{\rm Edd}$ onto a neutron star while it is in common envelope evolution with a massive companion. In order to raise the temperature high enough that the disc might cool by neutrino emission, Chevalier found a small value of the $α$-parameter, where the kinematic coefficient of shear viscosity is $ν=αc_s H$, with $c_s$ the velocity of sound and $H$ the disc height; namely, $α\sim 10^{-6}$ was necessary for gas pressure to dominate. He also considered results with higher values of $α$, pointing …
We consider effects of the MHD turbulence on the viscosity during the evolution of the thermal-viscous ionization instability in the standard $α$-accretion disks. We consider the possibility that the accretion onto a supermassive black hole proceeds through an outer standard accretion disk and inner, radiatively inefficient and advection dominated flow. In this scenario we follow the time evolution of the accretion disk in which the viscosity parameter $α$ is constant throughout the whole instability cycle, as implied by the strength of MHD turbulence. We conclude that the hydrogen ionization …
We derive from Einstein equation an evolution law for the area of a trapping or dynamical horizon. The solutions to this differential equation show a causal behavior. Moreover, in a viscous fluid analogy, the equation can be interpreted as an energy balance law, yielding to a positive bulk viscosity. These two features contrast with the event horizon case, where the non-causal evolution of the area and the negative bulk viscosity require teleological boundary conditions. This reflects the local character of trapping horizons as opposed to event horizons. Interpreting the area as the entropy, w…
The Misner and Sharp approach to the study of gravitational collapse is extended to the viscous dissipative case in, both, the streaming out and the diffusion approximations. The dynamical equation is then coupled to causal transport equations for the heat flux, the shear and the bulk viscosity, in the context of Israel--Stewart theory, without excluding the thermodynamics viscous/heat coupling coefficients. The result is compared with previous works where these later coefficients were neglected and viscosity variables were not assumed to satisfy causal transport equations. Prospective applica…
We calculate the structure of accretion disc around a rapidly rotating black hole with a super-Eddington accretion rate. The luminosity and height of the disc are reduced by the advection effect. In the case of a large viscosity parameter, alpha > 0.03, the accretion flow strongly deviates from thermodynamic equilibrium and overheats in the central region. With increasing accretion rate, the flow temperature steeply increases, reaches a maximum, and then falls off. The maximum is achieved in the advection dominated regime of accretion. The maximum temperature in the disc around a massive black…
Accretion disk theory was first developed as a theory with the local heat balance, where the whole energy produced by a viscous heating was emitted to the sides of the disk. One of the most important new invention of this theory was the phenomenological treatment of the turbulent viscosity, known the `` alpha'' prescription, where the (r phi) component of the stress tensor was approximated by (alpha p) with a unknown constant alpha. This prescription played the role in the accretion disk theory as well important as the mixing-length theory of convection for stellar evolution. Sources of turbul…
Non-radiating, advection-dominated, accretion flows are convectively unstable. We calculate the two-dimensional (r-theta) structure of such flows assuming that (1) convection transports angular momentum inwards, opposite to normal viscosity and (2) viscous transport by other mechanisms (e.g., magnetic fields) is weak (alpha << 1). Under such conditions convection dominates the dynamics of the accretion flow and leads to a steady state structure that is marginally stable to convection. We show that the marginally stable flow has a constant temperature and rotational velocity on spherical shells…
We consider hot, two-temperature, viscous accretion onto a rotating, unmagnetized neutron star. We assume Coulomb coupling betweenthe protons and electrons, and free-free cooling from the electrons. We show that the accretion flow has an extended settling region which can be described by means of two analytical self-similar solutions: a two-temperature solution which is valid in an inner zone, r<10^{2.5}, where r is the radius in Schwarzchild units; and a one-temperature solution which is valid in an outer zone, r>10^{2.5}. In both zones the density varies as ρ~ r^{-2} and the angular velocity…
We show that, unlike the results presented previously in the literature, the transition from an outer Shakura-Sunyaev disk (SSD) to an advection-dominated accretion flow (ADAF) is possible for large values of the viscosity parameter $α>0.5$. The transition is triggered by thermal instability of a radiation-pressure-supported SSD. The transition radius is close to the central black hole. We confirm our qualitative prediction by actually constructing global bimodal SSD-ADAF solutions.
The effect of a possible hydrodynamical wind on the nature of hot accretion disk solutions is studied. It is found that the advection dominated branch, in the presence of a wind, maintains the self-similar solution for the disk structure with the temperature, $θ\propto 1/r$, optical depth $τ\propto r^{P-1/2}$ and accretion rate $\dot m \propto r^P$. Based on global solutions, cooling due to wind energy loss and advection are found to be equally important. For a wide range of viscosity and wind parameters, the temperature is about one-tenth of the virial value and $P \approx 0.9$, independent o…
We investigate the effects of a large-scale magnetic field with open field lines on the steady-state structure of a radiation-dominated accretion disk, using self-similarity technique. The disk is supposed to be turbulent and possesses an effective viscosity and an effective magnetic diffusivity. We consider the extreme case in which the generated energy due to viscous and magnetic dissipation is balanced by the advection cooling. While the magnetic field outside of the disk is treated in a phenomenological way, the internal field is determined self-consistently. Magnetized and nonmagnetized s…
We describe hot, optically-thin solutions for one-temperature accretion disks around black holes. We include cooling by synchrotron, bremsstrahlung, and Comptonization. Our solutions are thermally and viscously stable, with gas temperatures on the order of $T \sim 10^9-10^{10.7}$K. The thermal stability is a direct result of the inclusion of synchrotron cooling. The new solution branch is related to the advection-dominated solution for a two-temperature gas described by Narayan \& Yi (1995b). It is present only for mass accretion rates less than some critical $\dot{M}_{crit}$ which depends on …
Nonlinear time-dependent calculations have been carried out in order to study the evolution of the thermal instability for optically thick, transonic, slim accretion discs around black holes. In the present calculations we have investigated only the original version of the slim disc model with low viscosity. This version does not yet contain several important non-local effects but our aim is to use it as a standard reference against which to compare the results from forthcoming studies in which additional effects will be added one by one thus giving a systematic way of understanding the contri…
We show that recently published assertions that advection dominated accretion flows (ADAFs) require the presence of strong winds are unfounded because they assume that low radiative efficiency in flows accreting at low rates onto black holes implies vanishing radial energy and angular momentum fluxes through the flow (which is also formulated in terms of the `Bernoulli function' being positive). This, however, is a property only of self-similar solutions which are an inadequate representation of global accretion flows. We recall general properties of accretion flows onto black holes and show t…
The mass of the central black hole in the active galaxy NGC 4258 (M106) has been measured to be $M=3.6\times10^7\Msun$ (Miyoshi et al. 1995). The Eddington luminosity corresponding to this mass is $L_E=4.5\times10^{45}$ erg s$^{-1}$. By contrast the X-ray luminosity of the nucleus of NGC 4258 between $2-10$ keV is $(4\pm 1)\times10^{40}~{\rm erg\,s^{-1}}$ while the optical/UV luminosity is less than $1.5\times10^{42} ~{\rm erg\,s^{-1}}$. The luminosity of NGC 4258 is therefore extremely sub-Eddington, $L\sim10^{-5}L_E$ in X-rays and $L\sim3\times10^{-4} L_E$ even if we take the maximum optical…
A toy model of a disk undergoing steady state accretion onto a black hole is presented. The disk is in a hydrostatic equilibrium for all radii r > r_{in}, with the inner disk radius located between the marginally stable and marginally bound orbits: r_{ms} > r_{in} > r_{mb}. Matter flows from the disk through a narrow cusp at r_{ms} and falls freely into the black hole, carrying with it no thermal energy. At radii larger than r_{out} the disk is assumed to radiate away all locally generated heat, and therefore the disk is geometrically thin for r > r_{out}. We assume that no heat generated in t…
